JOHANNESKEPLER ¨ LINZ UNIVERSITAT

JKU

Technisch-Naturwissenschaftliche Fakult¨ at

Electromagnetic-Acoustic Resonators for Remote, Multi-Mode Solid and Liquid Phase Sensing DISSERTATION zur Erlangung des akademischen Grades

Doktor im Doktoratsstudium der

Technischen Wissenschaften

Eingereicht von:

Dipl.–Ing. Frieder Lucklum Angefertigt am:

Institut f¨ur Mikroelektronik und Mikrosensorik Beurteilung:

Univ.–Prof. Dipl.–Ing. Dr. techn. Bernhard Jakoby (Betreuung) Prof. Dr.–Ing. Bernd Henning

Linz, Januar, 2010

Preface The following work describes my research efforts in the last few years as a PhD candidate of the Institute for Microelectronics and Microsensors at the Johannes Kepler University Linz, Austria. The work builds upon a diploma thesis carried out at the Otto-von-Guericke University Magdeburg, Germany. No scientific research can be successfully undertaken without guidance, support, collaboration, and discussion. As such, I want to sincerely thank Prof. Bernhard Jakoby, my thesis adviser, for accepting and guiding this research topic, and my father Dr. Ralf Lucklum for introducing this research topic to me. The initial results were achieved under the supervision of Prof. Peter Hauptmann of the University of Magdeburg and Prof. Nico de Rooij of the University of Neuchˆatel. During this time I had the pleasure to collaborate with several colleagues in Germany, Switzerland and Austria, and I am grateful for all the help and fruitful discussions. I want to acknowledge Dr. Ulrike Hempel, Dr. S¨oren Hirsch and Dr. Alexandra Homsy for their great support during the initial stages. I thoroughly enjoyed the freedom and flexibility in the group of Prof. Jakoby and I am very thankful for experiencing the different and challenging aspects of research, teaching and organization. All my colleagues in Linz contributed to this productive and pleasant atmosphere. I want to especially thank Dr. Erwin Reichel for the close collaboration in our similar research projects, as well as my other office mates Dr. Ju ¨rgen Kasberger and Johann Mayrw¨oger. Furthermore, I’d like to acknowledge Hans Katzenmayer, Bernhard Mayrhofer and Dr. Wolfgang Hilber for their support with the fabrication of different devices and elements for this research. Last but not least, I want to thank my family for guiding me to a scientific career and for their love and support.

To my family.

Abstract Electromagnetic excitation of acoustic resonator sensors is a unique method alternative to piezoelectric and electrostatic excitation. Based on electromagnetic-acoustic transducers, the continuous, resonant operating mode of these sensors offers distinctive advantages and possibilities, which have only recently been demonstrated. Sensor element and excitation as well as detection circuitry can be spatially separated. Only conductive materials or layers are required, and the mechanical forces can be impressed in a variety of directions. This results in multiple degrees of freedom, which allows the excitation and detection of fundamentally different modes of vibration in the resonator element. This multi-mode operation leads to the possibility to measure multiple physical properties in terms of their acoustic load with no or minimal changes to the sensor setup. In this work, the theory behind this method has been investigated. Suitable models and simulations have been derived in order to explain and predict measurement results. Three different resonator devices have been fabricated and analyzed for modes of flexural and shear vibration. Here, the focus has been put on simple micromachined silicon membranes and inexpensive aluminum and brass plates. Suitable vibration modes were utilized for mass detection, liquid property sensing, and liquid level measurements as a proof of concept for the multi-mode sensor principle. Additionally, prototype metal resonator arrays were used to demonstrate simultaneous measurement of different physical properties.

Kurzfassung Die elektromagnetische Anregung von akustisch resonanten Sensoren ist eine neue Alternative zur klassischen piezoelektrischen oder elektrostatischen Wandlung. Sie basiert auf den klassischen elektromagnetisch-akustischen, oder elektrodynamischen, Wandlern, wobei die kontinuierliche, resonante Anregung dieser Sensoren besondere Vorteile und M¨oglichkeiten er¨offnet. Hierbei sind Sensorelement und Anrege- und Auswerteschaltung r¨aumlich voneinander getrennt. Fu ¨r das Sensorelement wird nur ein leitf¨ahiges Material oder eine leitf¨ahige Schicht ben¨otigt, und die mechanischen Kr¨afte k¨onnen in die verschiedensten Richtungen eingepr¨agt werden. Dadurch ergeben sich mehrere Freiheitsgrade, was die Anregung fundamental verschiedener Schwingungsmodi in ein und demselben Sensorelement erlaubt. Dieser multi-mode Betrieb erm¨oglicht, mehrere verschiedene physikalischen Eigenschaften als ¨ Anderungen der akustischen Last zu messen, ohne oder nur mit geringfu ¨gigen ¨ Anderungen am Messaufbau. In dieser Arbeit wurde die Theorie hinter diesem Messprinzip beleuchtet, und es wurden geeignete Modelle und Simulationen erstellt, um die erzielten Messergebnisse zu erkl¨aren und vorherzusagen. Es wurden drei verschiedene Typen von Resonatorelementen hergestellt und bezu ¨glich ihrer Schwingungsmoden untersucht. Das Hauptaugenmerk lag dabei auf einfachen, mikromechanischen Siliziummembranen und kostengu ¨nstigen Aluminium- und Messingpl¨attchen. Geeignete detektierte Schwingungsmodi konnten fu ¨r die Mikrow¨agung und die Messung von Flu ¨ssigkeitseigenschaften und Fu ¨llst¨anden genutzt werden. Diese Experimente zeigen die Eignung des resonanten Messprinzips elektromagnetischer Anregung. Mittels Prototypen eines Resonatorarrays aus Metallfolien wurde weiterhin die gleichzeitige Anregung und Messung verschiedener Schwingungsmoden und Eigenschaften demonstriert.

Contents List of Tables

X

List of Figures

XII

List of Abbreviations

XVIII

List of Symbols 1 Introduction 1.1 State of Art Resonator Sensors . . . 1.1.1 Piezoelectric Transducers . . . 1.1.2 Capacitive Transducers . . . . 1.1.3 Electromagnetic Transducers . 1.2 Motivation . . . . . . . . . . . . . . . 1.3 Thesis Outline . . . . . . . . . . . . .

XIX

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2 Theory and Modeling 2.1 Fundamental Physical Background . . . . 2.1.1 Mechanical Resonant Vibration . . 2.1.2 Resonator Excitation and Detection 2.2 Multi-mode Excitation Methods . . . . . . 2.3 Finite Element Simulation . . . . . . . . . 2.4 Equivalent Circuit Modeling . . . . . . . . 2.5 Summary . . . . . . . . . . . . . . . . . . 3 Basic Resonator Operation 3.1 Design and Fabrication . . . . . . . . 3.1.1 Silicon Membrane Resonators 3.1.2 Metal Plate Elements . . . . . 3.1.3 Resonator Array Sensors . . .

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VIII

CONTENTS

3.2

Eigenmode Analysis . . . . . . . . . 3.2.1 Silicon Membrane Resonators 3.2.2 Metal Plate Elements . . . . . 3.2.3 Resonator Array Sensors . . . 3.3 Experimental Excitation Setup . . . 3.3.1 Array Multi-Mode Setup . . . 3.4 Eigenmode Excitation Results . . . . 3.4.1 Thickness Shear Modes . . . . 3.4.2 Face Shear Modes . . . . . . . 3.4.3 Flexural Plate Modes . . . . . 3.4.4 Array Multi-Mode Excitation 3.5 Summary . . . . . . . . . . . . . . . 4 Mass Microbalance Application 4.1 Analytical Description . . . . . 4.2 Suitable Modes of Vibration . . 4.3 Equivalent Circuit Model . . . . 4.4 Results and Discussion . . . . . 4.5 Summary . . . . . . . . . . . .

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5 Liquid Phase Density-Viscosity Sensing 5.1 Analytical Description . . . . . . . . . . 5.2 Suitable Modes of Vibration . . . . . . . 5.3 Equivalent Circuit Model . . . . . . . . . 5.4 Results and Discussion . . . . . . . . . . 5.4.1 Metal Plate Elements . . . . . . . 5.4.2 Silicon Membrane Resonators . . 5.5 Summary . . . . . . . . . . . . . . . . . 6 Liquid Level Measurement 6.1 Suitable Modes of Vibration 6.2 Equivalent Circuit Model . . 6.3 Results and Discussion . . . 6.3.1 Preliminary Results .

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65 65 68 69 70 72

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73 73 75 77 78 78 81 83

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85 86 87 89 89

CONTENTS

IX

6.3.2 Measurement Setup . . . . . . . . . . . . . . . . . . . . 89 6.3.3 Measurement and Simulation Comparison . . . . . . . 91 6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 7 Conclusions 97 7.1 Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 Bibliography

101

A Appendix 115 A.1 Additional tables . . . . . . . . . . . . . . . . . . . . . . . . . 115 A.2 MATLAB routines . . . . . . . . . . . . . . . . . . . . . . . . 119 Curriculum Vitae

123

List of Tables 2.1

Electro-mechanical equivalences with the Mobility Analog. . . 29

3.1

Changes to the radial thickness shear mode of (100) Si membrane at 15 MHz due to increasing copper layer thickness. . . 40

3.2

Comparison of operating frequency, signal response and Qfactor for selected resonator elements excited at radial TSM in air; with different selected geometries and operating TSM harmonics for the silicon membranes. . . . . . . . . . . . . . . 57

4.1

Copper layer thickness calculated from frequency shift of radial TSM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.1

Frequency shift and damping behavior of circular FSM at 142 kHz in aluminum due to liquid loading. . . . . . . . . . . 79

5.2

Frequency shift and damping behavior of diagonal FSM at 217 kHz in aluminum due to liquid loading. . . . . . . . . . . 80

5.3

Influence of density and viscosity on resonant response of diagonal FSM of (111) Si membrane. . . . . . . . . . . . . . . . 82

A.1 Resonator material properties. . . . . . . . . . . . . . . . . . . 115 A.2 Planar excitation coil parameters (Ct → 0). . . . . . . . . . . . 116

A.3 Successfully excited and detected flexural and face shear eigenmodes of simple 15x15 mm aluminum plate, with radial, diagonal and circular excitation setup. . . . . . . . . . . . . . . . . 116 A.4 Successfully excited and detected TSM eigenmodes of simple 15x15 mm aluminum plate, with radial, diagonal and circular excitation setup. . . . . . . . . . . . . . . . . . . . . . . . . . 117 A.5 Successfully excited and detected flexural and face shear eigenmodes of round mesa aluminum plate, with radial, diagonal and circular excitation setup. . . . . . . . . . . . . . . . . . . 117

LIST OF TABLES

A.6 Successfully excited and detected TSM eigenmodes of round mesa aluminum plate, with radial, diagonal and circular excitation setup. . . . . . . . . . . . . . . . . . . . . . . . . . . . A.7 Successfully excited and detected flexural and face shear eigenmodes of round mesa brass (CuZn) plate, with radial, diagonal and circular excitation setup. . . . . . . . . . . . . . . . . . A.8 Successfully excited and detected TSM eigenmodes of round mesa brass (CuZn) plate, with radial, diagonal and circular excitation setup. . . . . . . . . . . . . . . . . . . . . . . . . A.9 Electroplating settings at 22◦ C and results for additional copper layer thickness and smoothness on Si membranes. . . . .

XI

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. 118 . 118

List of Figures 1.1 1.2 1.3 1.4 1.5

Acoustic sensor principle of input quantity reflected in acoustic wave, which is converted to an electrical signal. . . . . . . . .

1

Classification scheme used for different resonant acoustic wave sensors in this work. . . . . . . . . . . . . . . . . . . . . . . .

3

Standard quartz crystal resonator sensor for thickness shear mode operation. . . . . . . . . . . . . . . . . . . . . . . . . . .

5

Modified electrode geometry for lateral field excitation of a quartz resonator with bare sensing surface. . . . . . . . . . . .

6

Capacitive micromachined ultrasonic transducer (CMUT), single cell design (left) and portion of single array element with hundreds of individual cells (right). . . . . . . . . . . . . . . .

9

2.1

Fundamental eigenmodes for thin plates and membranes. . . . 18

2.2

Impedance of the primary coil with no conductive material present (left), with a conductive material present resulting in eddy current induction (center), and with a conductive resonator present resulting in eddy current induction (large peak), Lorentz force generation, resonant vibration, and movement induction of a secondary eddy current (small peaks) (right). 21

2.3

Effect of different phase responses in the normalized impedance of the primary coil around face shear (left), flexural (center) and thickness shear resonance (right) of a silicon sensor element. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.4

Principle of resonance excitation with a planar spiral coil and perpendicular magnetic field resulting in radial Lorenz forces. . 24

2.5

Modified principle for generation of linear, diagonal Lorentz forces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.6

Complete FEM excitation model with planar spiral coil placed at the center of the resonator element. . . . . . . . . . . . . . 26

LIST OF FIGURES

XIII

2.7

Exemplary excitation simulation of a radial TSM at 15 MHz in a circular silicon plate, with eddy current induction (left), Lorentz force generation (middle) and resulting thickness shear mode vibration (right). . . . . . . . . . . . . . . . . . . . . . . 27

2.8

Fundamental face shear mode shapes of a round silicon membrane with displacement of diagonal/linear (left), circular (center) and radial (right) symmetry for a circular plate. . . . . . . 28

2.9

From mechanical description and schemata to equivalent circuit with the Mobility Analog (left to right). . . . . . . . . . . 31

2.10 Electromagnetic-acoustic transducer with electrical equivalent circuit (left) and mechanical equivalent circuit (right). . . . . . 32 2.11 Electromagnetic-acoustic equivalent circuit, with eddy current induction described by electromagnetic transformer T , coupling between current i, voltage u and force F , velocity v in the resonator modeled by electro-mechanical transformer XT .

33

2.12 Comparison of impedance amplitude (left) and phase (right) for equivalent circuit simulation and measurements of face shear mode (top) and thickness shear mode (bottom). . . . . . 34 3.1

Simplified 7 step silicon fabrication process; substrate (grey), oxide mask (red), photoresist (yellow), metal (purple). . . . . . 38

3.2

SEM image of sidewall of round silicon membrane with characteristic DRIE structure. . . . . . . . . . . . . . . . . . . . . 39

3.3

Changes to a radial face shear mode of a (100) silicon membrane with increasing copper layer thickness. . . . . . . . . . . 41

3.4

Brass and aluminum metal plates of 1 mm thickness with round mesa structure of 15 mm diameter milled in center. . . 42

3.5

Design of 2x2 resonator array setup with individual coils and magnets for each element. . . . . . . . . . . . . . . . . . . . . 43

3.6

Metal prototype arrays with different coupling springs and element size of 10x10 mm (top) and different corresponding excitation coil arrays on polymer foils with 100 µm metalization thickness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

XIV

3.7

3.8 3.9 3.10

3.11

3.12 3.13 3.14

3.15

3.16

3.17

3.18

LIST OF FIGURES

Results of eigenmode analysis for round (left) and quadratic (right) 15x15 mm (100) silicon membrane showing diagonal (top), radial (middle) and circular (bottom) face shear modes. Mode shapes for round (100) silicon membrane with diagonal TSM at 15 MHz. . . . . . . . . . . . . . . . . . . . . . . . . . First and second harmonic of diagonal FSM in round aluminum mesa resonator at 81 kHz and 171 kHz. . . . . . . . . Eigenmode shape of circular FSM at 142 kHz (left) and radial flexural plate mode at 295 kHz (right) of round aluminum mesa resonator. . . . . . . . . . . . . . . . . . . . . . . . . . . Simulated mode shape for radial face shear mode (left) and circular thickness shear mode (right) of a single metal array element. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Geometric arrangement for electromagnetic-acoustic resonator setup with remote, non-contact excitation and detection. . . . Design and realization of excitation setup for radial and circular shear modes. . . . . . . . . . . . . . . . . . . . . . . . . Lorentz force distribution (arrows) with different coil and magnet layouts for radial (left), diagonal/linear (center), and circular (right) vibration mode shapes. . . . . . . . . . . . . . . . Simultaneous excitation of different modes of vibration in each array element at different excitation frequencies (left), and of the same vibration mode at the same resonant frequency (right). Normalized impedance of quadratic coil with standard magnetic field resulting in radial TSM at 15.42 MHz (left) and with alternated magnetic field resulting in diagonal TSM at 15.36 MHz (right) in a quadratic high-Q silicon resonator. . . Normalized impedance of a standard spiral coil at the center resulting in a radial TSM at 15.65 MHz (left) and placed out of center resulting in a circular TSM at 15.61 MHz (right) in a round (100) silicon membrane. . . . . . . . . . . . . . . . . . Impedance spectrum of circular and diagonal TSM in aluminum mesa element and quadratic plate, respectively. . . . .

46 47 48

49

50 50 51

53

55

58

58 59

LIST OF FIGURES

3.19 Normalized impedance of diagonal FSM at 222 kHz (left) and circular FSM at 459 kHz (right) in a quadratic high-Q silicon resonator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.20 Normalized impedance of diagonal FSM at 143 kHz (left) and 2nd circular FSM at 369 kHz (right) in a mesa aluminum plate. 3.21 Normalized impedance of radial flexural mode at 248 kHz in a quadratic high-Q silicon resonator (left) and at 288 kHz for an aluminum mesa resonator (right). . . . . . . . . . . . . . . 3.22 Experimental results for individual excitation of a circular FSM in all elements of a metal array anchored at the corners at ≈234 kHz, with noticeable cross-coupling between individual elements as evidenced by the minor peaks. . . . . . . . . . . . 3.23 Simultaneous excitation of a circular face shear mode in (100) silicon and a radial flexural mode in aluminum around ≈250 kHz, and different frequency shifts with DI water loading due to different sensitivities. . . . . . . . . . . . . . . . . . . . . .

XV

60 60

61

62

63

4.1

Diagonal, circular and radial TSM (left to right) for round (100) silicon membrane at ≈ 15 MHz. . . . . . . . . . . . . . . 69 4.2 Electromagnetic-acoustic equivalent circuit with added ideal mass element. . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.3 Shift of unloaded fundamental TSM frequency after coating with a 300 nm polystyrene monolayer. . . . . . . . . . . . . . 71 5.1

Displacement profiles in a viscous liquid for a diagonal thickness shear mode (left) and a diagonal face shear mode (right). 5.2 Optimal modes of vibration in viscous liquids: Circular FSM of aluminum resonator (left) and diagonal FSM of Si membrane. 5.3 Electromagnetic-acoustic equivalent circuit with added ideal mass element. . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Normalized impedance spectrum of primary coil for liquid loading of circular FSM at 142 kHz in aluminum resonator yielding frequency shifts and damping as summarized in Table 5.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

74 76 78

80

5.5

Measurement results for diagonal FSM of (111) silicon membrane for different alcohols in terms of density-viscosity product vs. frequency shift and normalized impedance. . . . . . . . 81

5.6

Measurement results for diagonal FSM of (111) silicon membrane for different alcohols in terms of density-viscosity product vs. frequency shift and impedance damping. . . . . . . . . 83

6.1

Experimental setup for liquid level measurements with resonator element integrated into liquid reservoir and excited by planar coil and external magnet. . . . . . . . . . . . . . . . . . 87

6.2

Electromagnetic-acoustic equivalent circuit, with eddy current induction described by electromagnetic transformer T , coupling between current i, voltage u and force F , velocity v in the resonator modeled by electro-mechanical transformer XT , and the standing longitudinal wave in a liquid above the resonator represented by a lossy transmission line with a wave impedance Za . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

6.3

Preliminary results during density-viscosity experiments showing a strong dependence on volume in liquid measurement cell. 89

6.4

Experimental setup for liquid level measurements with resonator element integrated into liquid reservoir and excited by planar coil and external magnet. . . . . . . . . . . . . . . . . . 90

6.5

Impedance measurement (left) and simulation (right) for flexural resonance at 275 kHz with liquid volume at maximum around 900 µl, corresponding to a height at maximum of 2.75 mm = λ/2. Each curve corresponds to a change of 25 µl water or 76 µm height, respectively. . . . . . . . . . . . . . . . 91

6.6

Impedance measurement (left) and simulation (right) for flexural resonance at 275 kHz with liquid volume at maximum around 2700 µl, corresponding to a height at maximum of 8.25 mm = 3/2λ. Here, each curve corresponds to a change of 50 µl water or 154 µm height. . . . . . . . . . . . . . . . . . . 92

LIST OF FIGURES

6.7

XVII

Comparison of phase measurement (left) and simulation (right) corresponding to figure 6.6. . . . . . . . . . . . . . . . . . . . 93

List of Abbreviations Symbol

Description

AC BAW BHF CMOS CMUT DI DRIE EMAT FSM FEM IDT KOH MARS MEMS MST NdFeB NDT PCB PMMA PZT QCM QCR RF RIE SAW SEM TSM

alternating current bulk acoustic wave buffered hydrofluoric acid complementary metal oxide semiconductor capacitive micromachined ultrasonic transducer deionized (water) deep reactive ion etching electromagnetic acoustic transducer face shear mode finite element method interdigital transducer potassium hydroxide magnetic acoustic resonator sensors microelectromechanical systems microsystems technology neodymium iron boron (permanent magnet) non-destructive testing printed circuit board poly(methyl methacrylate) lead zirconium titanate quartz crystal microbalance quartz crystal resonator radio frequency reactive ion etching surface acoustic wave scanning electron microscope thickness shear mode

List of Symbols Symbol

Description

Unit

A B B0 c c C d D E E f ∆f F fL fR FL G h H hR i j J k k L m

surface area magnetic flux density (magnetic field) static magnetic flux density wave velocity elastic stiffness constant capacitance piezoelectric charge constant electric displacement field electric field elastic modulus frequency frequency shift force Lorentz force density resonance frequency Lorentz force conductance friction coefficient magnetic field intensity resonator thickness time variant current imaginary unit current density wave number spring constant inductance mass

m2 T T m/s Pa F m/V C/m2 V/m Pa Hz Hz N N/m3 Hz N S m/N s A/m m A — A/m2 1/m N/m H kg

XX

LIST OF SYMBOLS

Symbol

Description

Unit

n n q Q r r R S t T u u v wk Wk wp Wp x, y, z X Y Y Z ∆Z Zi Za Zm

harmonic number elastic compliance electric charge quality factor radius or distance mechanical resistance electrical resistance mechanical strain time variable mechanical stress mechanical displacement time variant voltage velocity kinetic energy density kinetic energy potential energy density potential energy Cartesian coordinates Lorentz force coupling coefficient movement induction coupling coefficient admittance impedance impedance damping input impedance acoustic impedance mechanical impedance

— 1/Pa C — m N s/m Ω — s Pa m V m/s J/m3 J J/m3 J m N/A V s/m S Ω Ω Ω Pa s/m N s/m

γ δ ε η

propagation constant penetration depth permittivity shear viscosity

— m F/m Pa s

XXI

LIST OF SYMBOLS

Symbol

Description

Unit

κ λ µ µR ν φ ρ ρq σ Θ ω ωR

compressibility wavelength magnetic permeability shear stiffness Poisson ratio phase angle mass density charge density specific conductivity impedance phase angle angular frequency angular resonance frequency

1/Pa m H/m Pa — rad kg/m3 C/m3 S/m rad 1/s 1/s

∇· ∇× ℑ ℜ

divergence operator curl operator imaginary part of complex value real part of complex value

m−1 m−1 — —

1 Introduction The field of acoustics encompasses the complete spectrum of mechanical vibrations and elastic waves in gases, liquids and solids, ranging from infrasound and audible frequencies, over ultrasonic devices, up to radio frequency (RF), or hypersonic, applications at several Gigahertz. The application of acoustics permeates throughout our daily lives. Beyond audio applications, ultrasonic devices are used for frequency standards, imaging systems, material testing, sensors, and RF filters, multiplexers, switches. With the advent of microelectromechanical systems (MEMS) and microsystems technology (MST) [1], research in acoustic devices has focused on new materials, electromechanical transduction methods, and miniaturization of acoustic transducers [2, 3]. Today acoustic devices have replaced or introduced various functional components in computers, cars, and cell phones, remote controls, and other hand-held systems. Acoustic sensors are devices where the measured properties are reflected as changes to the characteristics of an elastic wave [4, 5]. A wide range of possible measurands affect an acoustic wave, including acoustic properties such as wave velocity and polarization, but also non-acoustic parameters like bio-chemical, thermal, and mechanical properties, e.g. mass, position, level, acceleration, velocity, flow, concentration, pressure, density, and viscosity, as well as electric and magnetic properties, for example conductivity, permittivity, electric or magnetic fields [6]. The acoustic wave can propagate in the sensor element itself, radiate into the surrounding environment, or travel

Input quantity (mass, density, flow, elasticity, viscosity, distance, level, ...)

Influence on acoustic wave / impedance (velocity, frequency, amplitude, phase, ...)

Transduction to electrical property (frequency, amplitude, phase)

Electrical signal (voltage, current)

Figure 1.1: Acoustic sensor principle of input quantity reflected in acoustic wave, which is converted to an electrical signal.

2

1. Introduction

as a combination of both. The different measurands can influence either the acoustic properties of the sensor element itself or of the surrounding medium. The acoustic transducer then converts this effect, and thus the detected quantity, into an electrical signal (Fig. 1.1) [7, 8, 9]. In general, the acoustic wave carries the information of interest. We can distinguish between pulsed and continuous generation of acoustic waves, where characteristics like delay time, phase shift, polarization, and insertion loss, or resonance frequency, quality factor, and damping are evaluated, respectively. Furthermore, we can differentiate excitation principles, sensor materials, fabrications methods, active or passive operation, operating frequency, and more. To characterize different sensors, several classification schemes exist, e.g. [10], but an encompassing scheme is difficult to achieve for acoustic sensors. Ultrasonic sensors generally refer to emitters and receivers of acoustic waves radiated into a surrounding medium, but can also simply describe all acoustic sensors operating at ultrasonic frequencies. Acoustic wave sensors on the other hand generally refer to devices where the dominant effect is given by a modification of the acoustic wave propagating in the transducer element. Acoustic microsensors are then a special form of miniaturized acoustic wave sensors utilizing microfabrication technology, but can also include devices radiating waves into a medium. The term resonant sensors is also often used as a global description of acoustic sensors, referring to devices where the resonant properties of the sensor element are exploited. Resonant sensors generally encompass classic macroscopic resonators, sensors in oscillatory circuits, acoustic microsensors, some, but not all ultrasonic sensors, as well as micromechanical resonators. The latter are again partially interchangable with acoustic microsensors, but also include, e.g., accelerometers or pressure sensors, which are usually not classified as acoustic wave sensors. A selection of different acoustic sensors and applications is given in the following review papers [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23] and references therein. The devices presented in this work can generally be described as resonant acoustic wave sensors. Depending on geometry, excitation method and transducer design, bulk acoustic waves (BAW), surface acoustic waves (SAW), or flexural waves are generated in the sensor element. The resulting vibration

3

1.1. State of Art Resonator Sensors

Acoustic Sensors

Transduction Principle

Piezoel.

El.static

El.magn.

Mode of Vibration

Other

Flexural

Shear

Extensional

Figure 1.2: Classification scheme used for different resonant acoustic wave sensors in this work.

of the resonator surface interacts with the surrounding medium and is influenced by the overall acoustic properties acting at the device surface. This acoustic loading of the elastic wave in the resonator is the actual measured property and can be described in terms of a material property, the characteristic acoustic impedance Za defined as the product of wave velocity and density, or as force acting on a surface area divided by wave velocity. This characteristic impedance has to be replaced with an effective impedance in case of a finite surrounding medium, to account for effects of, e.g., reflection and interference of the acoustic wave [24, 25, 26]. Depending on the mode of vibration, acoustic wave sensors can be employed as, e.g., masssensitive deposition rate monitors, liquid phase sensors for viscosity and density, for measuring viscoelastic properties, and analyzing biochemical species and processes, see e.g. [27, 28, 29, 30, 31, 32, 33, 34]. Due to the difficulty of nomenclature and classification, and following the focus of this work, we will primarily distinguish between different acoustic transduction principles and the mode of vibration of the sensor element as depicted in figure 1.2.

1.1

State of Art Resonator Sensors

A variety of transduction methods exist today for generating and detecting elastic waves. The most widely employed method is piezoelectric excitation. The most popular and extensively investigated acoustic transducer is the piezoelectric quartz crystal resonator (QCR). Like the quartz crystal, many

4

1. Introduction

acoustic sensors have been derived from electronic components, where they are used as highly accurate frequency standards, delay lines, or other RF components. These components have usually been slightly modified to permit sensing of acoustic loads in contact with the resonator [35, 36, 37]. With the development of new excitation methods, novel designs and applications became possible. One such example is capacitive excitation, which became feasible only in recent years with the advent of microfabrication of mechanical elements. The third major method for generating ultrasonic waves is electromagnetic excitation, which we are focusing on in this thesis. Other methods include mechanical and thermal ultrasound generation, magnetostrictive transducers, Laser generation and detection of ultrasound, or spark excitation [38, 39].

1.1.1

Piezoelectric Transducers

Piezoelectricity, discovered by the brothers Curie in 1880, is a material property that combines the electrical and mechanical domain. The electrical charges within a piezoelectric material are spatially separated to form dipoles, but are evenly distributed resulting in no net polarization. However, piezoelectric materials lack a center of inversion symmetry. Applying a mechanical strain thus changes the distribution of the charges and results in a net polarization. This mechanism has been termed the direct piezoelectric effect. Conversely, applying an electrical field to the material results in a mechanical displacement of the charges. These two effects are described by the following constitutive relations: Di = dij Tj + εTij Ej t Si = n E ij Tj + dij Ej .

(1.1) (1.2)

The electrical displacement D is coupled to mechanical stress T by the direct piezoelectric coefficient d and to the electric field E by the permittivity ε. Conversely, the mechanical strain S is coupled to mechanical stress via the elastic compliance n and to the electric field via the transposed piezoelectric coefficient.

1.1.1. Piezoelectric Transducers

5

Figure 1.3: Standard quartz crystal resonator sensor for thickness shear mode operation.

Piezoelectricity can be found in naturally occurring crystalline materials like quartz, but also silk, wood and others. For industrial use artificially grown materials are primarily used, e.g. synthesized quartz, zinc oxide and Langasite crystals, ceramic materials like lead zirconium titanate (PZT), as well as certain polymers. For ultrasonic transducers, quartz and PZT are most commonly employed. Depending on the crystal cutting angle, the transducer substrate displays different properties, e.g. AT-cut quartz is widely used because of its temperature stability due to compensating thermal coefficients of the piezoelectric parameters. The cut angle of a piezoelectric crystal and the corresponding electrode geometry also defines the mode of vibration of the resonator. Acoustic waves in a piezoelectric material are usually excited by applying a voltage to electrodes attached to the faces. As an example shown in figure 1.3, an often used variant is an AT-cut quartz disc with electrodes applied to both main faces, exciting a thickness-shear bulk acoustic wave in the disc. To excite a surface acoustic wave, e.g., an ST-cut quartz with interdigital electrodes (IDT) patterned on one surface can be used. A variation of the classic QCR design is the lateral field excitation of the thickness shear mode (TSM) in a quartz disc in figure 1.4 [40, 41, 42]. Here, the electrodes are confined to a single face of the disc, the surface stays bare and is thus better suited for sensing applications. Other combinations of crystal cut and electrode geometry include flexural vibration modes in, e.g., discs, cantilevers or tuning forks, and shear-horizontal

6

1. Introduction

Figure 1.4: Modified electrode geometry for lateral field excitation of a quartz resonator with bare sensing surface.

Love-wave propagation in thin, piezoelectric waveguides [43]. In all these resonators designs, the mode of vibration is predefined by substrate, especially the crystal cut, and electrode geometries, and there is very little or no option to excite a piezoelectric resonator in a fundamentally different mode. This is an advantage for the suppression of undesired, spurious vibration modes, but restricts the versatility of a single sensor element. A recently developed alternative excitation method applies a planar coil to bare, electrodeless AT-cut quartz discs [44, 45]. There are two separate mechanisms involved here. In air the fundamental excitation of a quartz is achieved by the planar coil acting as an antenna and the generated electromagnetic wave coupling into the piezoelectric material. The electric field component E is in plane with the resonator. But a vertical electric displacement component D appears in the disc due to the tensor nature of the piezoelectric permittivity. This vertical electrical component is then the cause of the converse piezoelectric effect creating a shear mode vibration. The second mechanism is emphasized during liquid phase measurements. Here, the electromagnetically excited resonance decreases strongly, but a second resonance peak is enhanced. The electric potential distribution of the coil and the virtual ground of the upper face of the quartz create an electrode setup similar to QCR and an equivalent acoustic excitation, with the only difference being the potential gradient of the coil compared to a constant potential of an electrode. Given the tensor nature of the electromechanical

7

1.1.2. Capacitive Transducers

properties in a piezoelectric material, a planar spiral coil remotely excites two different resonances in a piezoelectric resonator, one due to the electromagnetic and another due to the electric field of the coil. With this setup, the mode of vibration can be modified by adjusting the alignment of disc and excitation coil. Furthermore, disc and coil can be spatially separated, allowing for remote excitation. Another advantage is the possibility to excite very high harmonics of the fundamental vibration mode up to GHz frequencies. The coil can also be deposited on one side of the quartz disc itself, increasing the electro-mechanical coupling but losing the non-contact nature and variability of exciting different modes [46]. The measurement mechanism for these sensors is generally the evaluation of the complex, frequency dependent electrical impedance. Changes to the acoustic loading of the sensor element, and thus changes to the acoustic impedance, are reflected in the electrical impedance of the device. Piezoelectric ultrasonic sensors can be considered as the most thoroughly investigated and best documented ultrasonic transducers today. With new developments in materials science and fabrication technologies, these devices continue to be the most commonly used ultrasonic components in industry and research.

1.1.2

Capacitive Transducers

Capacitive excitation is governed by the Coulomb force, the attraction and repulsion of opposite and similar charges, respectively, and subsequently by ~ While this electrostatic the force acting on a charge q in an electric field E. force is usually negligible in the macroscopic domain, it scales very well into the microscopic domain. The attracting force F due to a voltage U between two charged plates of cross-section A and distance r is given by 2

~ = εA U . (1.3) F~ = q · E 2r2 While macroscopic capacitive transducers were unsuitable for sensing applications due to the high electric field strength (on the order of 106 V/cm) and thus very high voltages required, with microfabrication technology it became possible to reduce the characteristic dimensions and thus decrease the distance of the electrodes into the micron and submicron range [47]. A much

8

1. Introduction

lower operating voltage is required (tens to a few hundred Volts) to achieve strong enough forces for ultrasonic actuation. Electro-mechanical coupling factors comparable to piezoelectric excitation have been achieved. The basic design is relatively simple. Using surface micromachining, a top electrode is patterned onto a thin membrane, with the bottom electrode located on the substrate forming a capacitor cell. To create the (vacuum) gap between top and bottom electrode, a sacrificial layer technique is usually utilized [48]. This gap has to be evacuated to reduce the damping of the membrane vibration. Applying a voltage between upper and lower electrode pulls the membrane towards the substrate. To generate ultrasonic waves the membrane is usually pre-stressed by a bias voltage and excited by an alternating voltage. The capacitive transducer is a wide-band transducer that allows the generation and detection of a large variety of frequencies. The major advantage of capacitive ultrasonic transducers is the ability to fabricate them using standard cleanroom processes, while miniaturized piezoelectric transducers require special materials and processes. Capacitive micromachined ultrasonic transducers (CMUT) today are fabricated in CMOS compatible processes, thus enabling easy integration with microelectronic components [49, 50]. Furthermore, unlike small piezoelectric substrates, CMUTs can be realized on large silicon wafers, thus allowing for large transducer arrays instead of single sensor elements (Fig. 1.5). The development of CMUT arrays has been especially useful for improving ultrasonic imaging techniques and high intensity focused ultrasound in medicine, and have thus become the industry standard for these applications today, e.g. [51]. The primary mode of vibration of capacitive resonators is the flexural plate mode for radiating compressional waves into a material. With changes to the transducer design, it is also possible to excite further Lamb waves in a membrane structure [52]. But in general these transducers are limited to flexural vibration, which limits the application as liquid phase sensors due to high damping of the generated ultrasonic waves. Due to their high bandwidth they are however not limited to vibrating at their resonance frequencies. The main use of capacitive transducers is the generation and detection of

9

1.1.3. Electromagnetic Transducers

Figure 1.5: Capacitive micromachined ultrasonic transducer (CMUT), single cell design (left) and portion of single array element with hundreds of individual cells (right).

ultrasound in different media for imaging applications, material testing, and focused ultrasound.

1.1.3

Electromagnetic Transducers

The third major mechanism for exciting acoustic waves is based on generating Lorentz forces, which are complementary to Coulomb forces in an electromagnetic field, and the term magnetic direct generation has been given to this coupling mechanism between electromagnetic and acoustic waves by Quinn [53]. The method of magnetic direct generation of acoustic waves in conductive materials was developed from studies in low temperature physics. Also termed electrodynamic, inductive, or simply electromagnetic transducers, these devices have been employed especially in non-destructive testing (NDT) and evaluation for several decades, for example in the automotive industry [38]. The Lorentz force F~L on a charge q moving with velocity ~v ~ originally in an electromagnetic field with the magnetic field component B, formulated by Maxwell and later Lorentz [54, 55], is given by ~ ~ ~ (1.4) FL = q E + ~v × B .

If the net charge of the material is zero, only a magnetic component results in a force in the resonator. Thus, the Lorentz force density can be written as ~ f~L = J~ × B,

(1.5)

10

1. Introduction

where the current density J~ = ρq~v is given as product of the moving charge density ρq and its velocity. As can be seen in the equations, electromagnetic generation of acoustic waves necessitates a conductive part of the resonator where the required current can be applied. This primary current can be directly applied to the sample, where the conductive paths act similar to electrodes for piezoelectric and capacitive excitation. Depending on the current path layout and the applied magnetic field, a variety of Lorentz force directions can be realized, and thus a variety of vibrations modes can be excited. After the design of the current paths, an element is however again limited to a specific family of vibration modes. The common application of an electromagnetic acoustic transducer (EMAT) is the non-contact mode of operation [56]. Prior to generating the Lorentz forces, the necessary current is remotely induced in the material. This again requires a conductive substrate material or a conductive layer in a composite element. These eddy currents are induced by means of an alternating current flowing through a primary coil in the vicinity of the sample. The eddy current flow and thus the Lorentz force direction is directly dependent on the layout of the primary coil and the direction of the applied magnetic field [57, 58, 59]. Therefore, besides the remote nature of excitation, all kinds of vibration modes can be excited without changes to the acoustic sample. The primary alternating current, be it eddy current or directly applied current, also generates a time-variant magnetic field. This field can be utilized for Lorentz force excitation. Since both current density and magnetic flux density are now time-variant, the Lorentz force and thus the acoustic wave will oscillate at double the frequency of the primary current. However, in order to achieve suitably large magnetic flux densities, very high currents in the range of kiloamps are necessary. These currents can generally only be achieved in pulsed applications and therefore this method is not suitable for continuous, resonant excitation [60]. The other option is to supply the magnetic field by means of an external magnet. With modern permanent rare-earth magnets high static remanent flux densities are possible, e.g. NdFeB magnets with up to 1.5 T. The placement of the external magnet introduces another degree of freedom for determining the direction of the

1.2. Motivation

11

Lorentz forces. The detection mechanism works as follows. The sample moving in an external magnetic field gives rise to a secondary eddy current, which in turn induces a secondary voltage in the primary coil geometry. The combined eddy current density in a static magnetic field can be written as ~ ~ ~ (1.6) J(t) = σ E(t) + ~v (t) × B .

For EMAT excitation the induced secondary voltage is dependent on the electro-mechanical coupling efficiency as well as the electromagnetic coupling efficiency of the eddy current induction (see also the equivalent circuit in fig. 2.11). In general, the transduction efficiency is considerably smaller than with piezoelectric or capacitive transducers. For sensing applications, we are therefore looking at continuous, resonant excitation to overcome the weaker coupling with higher quality factors. The main application of EMAT sensors continues to be in NDT applications, for detecting material defects, evaluating layer adhesion and delamination, thickness measurements, and material characterization, especially where direct contact to the sample is difficult, e.g. due to corrosion, coatings, dirt, or high temperatures.

1.2

Motivation

Both piezoelectric and capacitive transducers have been extensively investigated in recent years, yielding many new, exciting developments. Especially for acoustic, resonant sensing, both methods however show certain limitations that cannot be easily overcome. These are primarily the restriction of a resonator design to specific modes of vibration, the necessity for direct electric contacts to the resonator and thus problems with insulation, crosscouping, parasitic effects, especially in liquid environments, as well as the high material cost due to multi-layer cleanroom processing or the application of non-standard, piezoelectric materials. Electromagnetic resonators on the other hand require only a modicum of conductivity in the substrate, or in a layer on the substrate material. Therefore, the resonator material can be chosen only in terms of its mechanical, elastic properties.

12

1. Introduction

By combining magnetic direct generation with a suitable acoustic resonator material, the reduced transduction efficiency can be overcome. By driving the primary coil with an alternating continuous current instead of a pulsed current, it is possible to excite a mechanical resonance in the element [61, 62, 63, 64]. It is advantageous to tune the electrical resonance to coincide with mechanical resonance, for example by bridging the primary coil with a parallel capacitance, because the detectable signal response is then improved by the Q-factors of both resonances. This combination of utilizing a single planar spiral coil was applied by Stevenson et al. and termed magnetic acoustic resonator sensors (MARS) as a new sensing methodology [65]. The advantages of such an acoustic sensor over traditional piezoelectric transducers such as the QCR are obviously the absence of direct electric contacts to the resonator, no need for piezoelectric materials, a bare sensing surface, the ability to utilize a large variety of different materials and material combinations which have been exempted before, and the possibility of exciting a variety of different modes of vibration. These advantages can qualify the use of such devices as detectors in the traditional gravimetric regime to measure mass deposition or selective adsorption of molecules and particles, as well as sensors in the so-called non-gravimetric regime to measure material or interfacial changes due to chemical and biochemical interactions. Electromagnetic transducers can be used to excite all kinds of acoustic waves in materials, often requiring only changes to the layout and alignment of primary coil and magnetic field. Similarly, we will show that electromagneticacoustic resonator sensors can be employed for different applications requiring different modes of vibrations without any changes to the resonator element or major changes to the excitation setup. For example, the difference between exciting shear vibration modes of radial, linear or circular symmetry is primarily the positioning of the excitation coil and corresponding magnets. A multi-mode excitation setup that can be used for a variety of measurands and applications is therefore possible. With the introduction of array devices, multiple different modes can also be simultaneously excited in each element, and thus different measurands can be evaluated in one measurement.

1.3. Thesis Outline

1.3

13

Thesis Outline

In the following chapters we will detail the research efforts in modeling and simulating the excitation and detection mechanisms involved, investigate suitable electromagnetic-acoustic resonator and transducer designs, present experimental results, and finish with an outlook. Chapter 2 will cover the fundamental physical background as well as introduce possible multi-mode excitation methods, the finite element simulation employed for mode shape analysis, and the electrical equivalent circuit used to model and predict the experimental results. In chapter 3 we will thoroughly explain the resonator sensor operation. Starting with design and fabrication methods for our different resonators, silicon membrane elements, metal plates and brass sheet resonator arrays, we will analyze the different eigenmodes for these devices, describe the experimental setups used and how we excited the different thickness shear, face shear and flexural plate modes of vibration. In the second part of this thesis we take a detailed look at three applications of our resonator sensors. In chapter 4 the devices are investigated as mass microbalance. Our primary focus has been liquid phase measurement though, in chapter 5 we analyze the sensor response to changes in density and viscosity of a liquid in contact with a resonator. We have also utilized the same sensor elements for precise liquid level measurements, and corresponding acoustic wave velocity, which is described in chapter 6. The thesis is then concluded with an outlook over possible applications and future research.

2 Theory and Modeling Electromagnetic-acoustic transduction is a complex process that cannot be analytically solved as a whole. Therefore we employ different simulation methods to properly model this behavior. The mechanical, elastic properties of our different resonator elements are numerically simulated with the finite elements method (FEM), where the focus is placed on eigenmode analysis to evaluate possible modes of vibration and the necessary Lorentz force distribution to excite them. Around these eigenmode resonances both excitation and detection mechanism can be modeled with a lumped elements equivalent circuit. The basic setup consists of a primary excitation coil, an external permanent magnet and the resonator element. In the following we will discuss the mechanisms involved in exciting and detecting the acoustic resonances, and describe our models and simulation.

2.1 2.1.1

Fundamental Physical Background Mechanical Resonant Vibration

Mechanical resonators and acoustic loads are described by the equations of vibrations and waves. A simple mass-spring system can be expressed by a combination of Hooke’s law and Newton’s equation of motion. The resulting equation describes the unperturbed mechanical oscillation of the system with mass m and spring constant k. k d2 x + x=0 dt2 m

(2.1)

A solution to this differential equation can be written as: x = A sin (ω0 t + φo ) ,

(2.2)

where ω0 = k/m is the angular frequency of the harmonic motion and the two independent parameters are the oscillation amplitude A and the initial phase angle φ0 . The energy stored in a mechanical system is the sum of the

16

2. Theory and Modeling

potential energy Wp and the kinetic energy Wk . For the simple mass-spring system, the potential energy is stored in the spring: Z x 1 (2.3) Wp = kx dx = kA2 sin2 (ω0 t + φo ) , 2 0 and the kinetic energy given by the mass moving with velocity v: 1 1 2 Wk = mv 2 = mvm cos2 (ω0 t + φo ) . 2 2

(2.4)

Expanding the simple model to a more general approach requires the addition of a damping element, e.g. a mechanical dashpot. This mechanical resistance r, or its inverse, the friction coefficient h, leads to the equation of motion for a damped system. Introducing complex notation for periodic functions we can replace the trigonometric functions, e.g. x = A cos(ω, t) = Aℜ {exp(jωt)}, √ with the imaginary unit j = −1. This leads to an equation in the form of: d2 x 1 dx + + ω0 x = F ejωt , 2 dt hm dt

(2.5)

where energy would be dissipated and thus the vibration amplitude reduced with every cycle without external input. The necessary energy to keep the system oscillating is introduced by periodic external excitation forces F ejωt . For the solution of this forced vibration, we can derive a mechanical impedance Zm : k F jωt , (2.6) Zm = e = r + j ωm − v ω

Now we can define a resonance frequency ωR = 2πfR which occurs when kinetic and potential energy stored in mass and spring, respectively, are equal and thus only the real part of the impedance remains. This resonance occurs when the driving frequency coincides with the natural frequency of the oscillator, i.e. ωR = ω0 . The quality of the resonance can be described by the Q-factor, which is a measure for the ratio of stored energy and dissipated energy in the system. It can be defined as the ratio of resonance frequency and bandwidth BW or in terms of the mechanical constants as: Q=

ωR ωR m = . BW r

(2.7)

17

2.1.1. Mechanical Resonant Vibration

We can use Q as measure of the strength of a resonance and the efficiency of excitation. In a suitable medium, the basic oscillatory motion can propagate as an elastic, or acoustic, wave. The constitutive equation for a simple one-dimensional wave in the x-direction can thus be written as [8, 7]: 2 ∂ 2u 2∂ u =c , ∂t2 ∂x2

(2.8)

with displacement u and wave propagation velocity c. For an elastic solid we introduce the stress tensor Tkl = Fk /Al and the strain tensor Skl = 1/2 (∂uk /∂xl + ∂ul /∂xk ), where the indices k and l take on the x, y or z direction. The diagonal terms, e.g. Txx , represent compressional stress and strain, while the non-diagonal terms represent shear stress or strain. Both stress and strain tensor are symmetric, therefore only six of the nine elements are unique and the number of indices can be reduced to one. The constitutive equation of elasticity, expanded into three dimensions and utilizing this reduced tensor notation (due to symmetry), is given by Ti = cij Sj ,

(2.9)

where cij are the elastic stiffness constants of the material. As the stress is defined as a force per area, we can use Newton’s law to rewrite the wave equation in an elastic medium as ∂ 2u ∂ 2u ρ 2 = cij 2 . ∂t ∂x

(2.10)

In an elastic medium, the wave velocity in a specific direction is thus given p by c = cij /ρ. As with a simple vibration, an elastic wave propagating through a medium results in kinetic and potential energy being stored in the medium. The kinetic energy density wk (energy per unit volume) is defined similar to equation 2.4 1 ∂ 2 ui 1 (2.11) wk = ρv 2 = ρ 2 . 2 2 ∂t The potential energy is stored in a strained volume, a change of potential energy density is given by dwp = Ti dSi . Following the formula for potential

18

2. Theory and Modeling

Figure 2.1: Fundamental eigenmodes for thin plates and membranes.

energy stored in as simple mass-spring system, we find that 1 wp = cij Si Sj . 2

(2.12)

By balancing these energy densities we can again calculate the resonance frequency, as discussed later in chapter 4. Depending on the propagation direction and polarization (direction of the particle displacement) of the acoustic wave, a resonator element can experience a variety of fundamental vibration modes. For thin plates and membranes, these modes are flexural, extensional, face shear and thickness shear motion, which are schematically depicted in figure 2.1. These fundamental modes can be found in different mode shapes, i.e. the geometric shape of the displacement, and of course with many higher harmonics and superposed combinations of different modes. Analytical solutions for the displacement, velocity or energy distribution are only possible for a select few with specific boundary conditions [66, 67]. In general, the complexity of geometry, boundary conditions, mode shapes and distribution of the external forces requires non-analytical approaches to model the mechanical resonators and acoustic loads.

2.1.2. Resonator Excitation and Detection

2.1.2

19

Resonator Excitation and Detection

A coupling mechanism between electromagnetic and acoustic waves has been first observed in the late 1930s and confirmed by the late 1960s. This effect was then termed magnetic direct generation of acoustic waves by Quinn [53]. The fundamental mechanisms involved can be explained by Maxwell’s equations. According to Ampere’s law, a magnetic field is created by the movement of electric charges. ~ ~ = J~ + ∂ D ∇×H ∂t

(2.13)

~ are current densities J~ and time variant The curls of a magnetic field H ~ This magnetic field, generated here by electric curelectric displacements D. rents in the primary coil, is time variant in a harmonic manner characterized by the frequency of the driving current. It therefore is the source of an in~ according to Faraday’s law of induction with µ as the duced electric field E permeability of the material penetrated by this electromagnetic field. ~ ~ = −µ ∂ H ∇×E ∂t

(2.14)

In a conductive material this induction gives rise to the electromotive force (emf) which affects the free charges of the conductor. This emf, or induced voltage, causes electrical currents in the conductor. According to Lenz’s law, these eddy currents are circulating in such a direction to produce a magnetic field that directly opposes the change in the primary magnetic flux. In the case of a planar spiral coil for example, the eddy currents in a conductive layer placed in parallel above the coil will flow in the opposite clockwise direction as the primary current in the coil. The penetration depth, or skin depth, of these eddy currents can be derived from the electromagnetic wave equation resulting from Maxwell’s equations 2.13 and 2.14 [68], in complex notation for the electric field written as σ~ E = 0, ∇ E + ω µε 1 − j ωε 2~

2

(2.15)

20

2. Theory and Modeling

with angular frequency ω, permittivity ε, conductivity σ, and j as the imaginary unit. We can define a complex propagation constant γ as r σ √ (2.16) γ = α + jβ = jω µε 1 − j . ωε A basic plane wave solution of equation 2.15 in the z-direction with only an x component is given by Ex (z) = E + e−γz + E − eγz +. For a good conductor, where the conductive current is much larger than the displacement current, we can use the approximation σ ≫ ωε to reduce equation 2.16 to r r σ ωµσ √ = (1 + j) . (2.17) γ = α + jβ ≈ jω µε jωε 2 The penetration depth δ of the electromagnetic wave is then defined as the depth in the penetrated material where the amplitude of the fields has decayed by 1/e or 36.8%. For a conducting material it thus defines the required conductive thickness for optimal electromagnetic coupling at a specific operating frequency, and is given by r 2 1 . (2.18) δ= = α ωµσ After eddy current generation, the second mechanism involved requires a superposed magnetic field. Corresponding to the Lorentz force law a charged ~ or moving through a magnetic field B ~ with particle q in an electric field E the velocity ~vq is subjected to a force F~L [54, 55]. This force acts as the external driving force in equation 2.5 for the resonator and is given by: ~ ~ ~ (2.19) FL = q E + ~vq × B .

~ of the electromagnetic The Coulomb force due to the electric component E field created by the primary current vanishes if the net charge of the material is zero and only a magnetic component results in a force in the resonator. The primary Lorentz force density f~L for electromagnetic excitation can then be written as ~ f~L = J~e × B, (2.20)

where the eddy current density J~e = ρ~vq is given as product of movable charge density ρ and its velocity ~vq . These Lorentz forces transfer a periodic

21

2.1.2. Resonator Excitation and Detection

643

25 20 15 10 5 7.5

10.0

12.5

15.0

Frequency f / MHz

17.5

20.0

Measured Impedance Zi /Ω

Measured Impedance Zi /Ω

Measured Impedance Zi /kΩ

650 30 600 550 500 450 400 350

7.5

10.0

12.5

15.0

Frequency f / MHz

17.5

20.0

642

641

640

639

15.0

15.2

15.4

15.6

Frequency f / MHz

15.8

Figure 2.2: Impedance of the primary coil with no conductive material present (left), with a conductive material present resulting in eddy current induction (center), and with a conductive resonator present resulting in eddy current induction (large peak), Lorentz force generation, resonant vibration, and movement induction of a secondary eddy current (small peaks) (right).

momentum to the moving free charges, which in turn interact with the crystal lattice of the conductive material and cause it to vibrate with the frequency of the primary current in the coil. Thus, an acoustic wave is generated in the sensor element. At driving frequencies corresponding to the natural frequencies of the resonator mechanical resonance is achieved. For the case of a thickness shear mode vibration, the resonant wave propagates between the upper and lower faces of the sample. A detailed theoretical investigation of this combination of the electrical, mechanical and acoustic domains, including magnetization and magnetostriction effects, can be found in [69]. To explain the detection mechanism, we have to understand the movement induction in the vibrating plate in the static magnetic field. Figure 2.2 shows the changes to the impedance of a planar coil. When operated in air without a conductive material present, the impedance is only determined by the electrical inductance, resistance and capacitance of the coil setup (left). Once a conductive material is placed in the vicinity, eddy currents will be induced as described above, resulting in a transformer-like setup where the magnetic field generated by the eddy currents opposes the magnetic field of the primary coil, leading to a mutual inductance which changes the impedance measured at the primary coil (center). When a conductive resonator is placed in the vicinity and superposed with a static magnetic field, Lorentz forces will be generated that can excite resonant vibrations at the eigenmodes of the resonator. These vibrations in

22

2. Theory and Modeling

a magnetic field naturally represent a movement of charges in a magnetic field, which in turn induces a secondary voltage according to the general law of induction. The induced voltage results in secondary eddy currents in the vibrating element. This effect is generally negligible, due to the mechanical vibration amplitude being several orders of magnitudes smaller than the eddy current vibration amplitude. In resonance however, the mechanical vibration amplitude is strongly enhanced according to the Q-factor of the mechanical resonator. In principle, the secondary current is 180◦ out-of-phase to the primary current, however the phase of mechanical vibration and electrical eddy current must be taken into account. This effect appears as additional peaks in figure 2.2 right. These peaks are employed as the major sensor signal, reflecting important characteristics of the resonator in its specific vibration mode, including the acoustic load acting on the resonator. Note that the secondary eddy currents also give rise to Lorentz forces, which, in general, weaken the excitation force. The total eddy current density J~t in a magnetic field can be written as the sum of primary and secondary ~ e, eddy current, due to the electrical field induced by the primary current, E and due to movement induction with a local resonator velocity, ~v , respectively. This equation also represents Ohm’s law in a moving frame of reference: ~ ~ ~ Jt = σ Ee + ~v (t) × B . (2.21)

The total current density in turn results in a secondary time-variant magnetic field that induces a voltage in the primary coil, which changes the impedance accordingly. Outside mechanical resonance, only the primary eddy current determines the impedance change, while in resonance additional currents are superposed to the primary eddy current (Fig. 2.2). Amplitude and phase of the secondary currents depend on primary coil layout, Lorentz force distribution, the specific vibration mode shape, and the resonator material properties, such as conductivity, elasticity, electron mobility, as well as acoustic boundary conditions induced by the surrounding medium. An analytical solution is only possible for very simple, special cases, e.g. a single linear conducting wire, therefore we will focus on numerical models to describe the complex

23

2.2. Multi-mode Excitation Methods

0.03 0.02 0.01 0 −0.01 −0.02

220

221

222

223

Frequency f / kHz

224

4

Normalized Impedance Zi /Ω

0.25

Normalized Impedance Zi /Ω

Normalized Impedance Zi /Ω

0.04

0.2 0.15 0.1 0.05 0 −0.05 −0.1

246

247

248

249

Frequency f / kHz

250

3 2 1 0 −1 −2

15.410

15.415

15.420

Frequency f / MHz

15.425

Figure 2.3: Effect of different phase responses in the normalized impedance of the primary coil around face shear (left), flexural (center) and thickness shear resonance (right) of a silicon sensor element.

measurement setup. The effect of different phase responses of the mechanical vibration can be seen in figure 2.3, showing both reduction (impedance amplitude increases) and improvement (impedance amplitude decreases) of the primary eddy current. The mechanical resonance can thus be detected by an RF analyzer circuit monitoring the coil impedance. Additionally, the signal response is also amplified by driving the setup at electrical resonance. This is achieved by bridging the planar spiral coil with a (tuning) capacitor and setting the electrical resonance to the same frequency as the mechanical resonance, and has been termed magnetic acoustic resonator sensors (MARS) as a new sensing methodology [65].

2.2

Multi-mode Excitation Methods

Linear, or diagonal, radial, and circular symmetry are the three fundamental mode shapes for in-plane, shear vibration (see, e.g., Fig. 2.7). Flexural motion can be more complex. To realize a true multi-mode resonator, we need to realize a corresponding Lorentz force distribution for all these fundamental mode shapes, i.e. also forces of linear, radial and circular direction. With the advantage of spatial separation of resonator and excitation, we have multiple degrees of freedom, which are the geometry of the primary coil(s), the direction of the external magnetic field(s), and the alignment of resonator, coils and magnets. We also have to consider the electromagnetic coupling

24

2. Theory and Modeling

Figure 2.4: Principle of resonance excitation with a planar spiral coil and perpendicular magnetic field resulting in radial Lorenz forces.

efficiency between primary current and induced eddy currents though, also described by primary inductance and mutual inductance, respectively. The most common coil geometry employed for this sensor principle has been a planar spiral coil, due to high electromagnetic coupling, with a vertically perpendicular magnetic field. However, applying the Lorentz force law to the induced eddy currents in the resonator, this layout yields in-plane radial forces when centered under the resonator element (Fig. 2.4). These radial forces will preferably excite radially symmetric vibration mode shapes with in-plane, shear components. According to finite element method (FEM) simulation, this radial vibration, however, is coupled to an out-of-plane movement at the center of the element due to the conservation-of-mass principle. This vibration component generates unwanted compressional waves, which radiate energy into the surrounding medium, leading to additional damping in liquids. Therefore, a number of alternative vibration modes are of particular interest for liquid phase sensing, especially the diagonal and circular shear modes. The distribution of the Lorentz forces is affected by the distribution of the eddy currents and the direction of the magnetic field. Therefore, in order to generate non-radial forces and excite different modes of vibration, the coil geometry has to be adapted and the magnetic fields have to be changed accordingly. Eddy currents basically mirror the layout of the primary coil. Therefore, quadratic or rectangular spiral coils coupled with an alternated permanent magnetic field are suitable for excitation and detection of linear, or diagonal, lateral shear modes. These coils induce components of linear eddy current

2.2. Multi-mode Excitation Methods

25

flow, which result in a linear Lorentz force distribution. The alternated magnetic field accounts for the necessity of a coil structure for high inductance and the thus alternating flow directions. Multiple parallel, linear conducting lines would also result in linear Lorentz forces, however the non-contact, electromagnetic eddy current coupling efficiency would be minimal due to very low inductances. The linear, or diagonal, vibration is similar to the thickness shear mode shape in QCM sensors and will lead to improved liquid phase sensing capabilities. Fig. 2.5 shows the modified principle for generating diagonal forces and therefore a diagonal shear mode. There are also other coil geometries possible for generating these forces, such as linear solenoids or meander line structures, which suffer from a decreased electromagnetic coupling. At the edges of the vibration area a linear movement will however also result in out-of-plane components due to conservation of mass. The excitation of pure circular shear modes, or torsional modes, is most suitable for pure in-plane, shear vibration, due to the lack of singularities, especially if the whole resonator element twists itself. To generate the necessary circumferential forces, radial eddy currents are required. Placing planar, spiral coil(s) out of center of a resonator yields alternated, radial current components. Superposed by vertically perpendicular, alternated magnetic fields results in a force distribution with a dominating circumferential component [70]. At proper frequencies, these forces will result in a circular shear motion, or also called torsional vibration mode.

Figure 2.5: Modified principle for generation of linear, diagonal Lorentz forces.

26

2. Theory and Modeling

Figure 2.6: Complete FEM excitation model with planar spiral coil placed at the center of the resonator element.

2.3

Finite Element Simulation

The experimental setup has been modeled by means of the finite elements method (FEM) using the software suites ANSYS and COMSOL Multiphysics [71, 72]. The primary goal was to develop a model to describe and visualize the excitation mechanism and predict the possible modes of vibration. The complete model is shown in figure 2.6. The FEM simulation of the excitation mechanism is divided into three separate steps. These are the eddy current induction, Lorentz force generation due to a superposed magnetic field, and the mechanical vibration of the element [73]. The effect of the secondary eddy currents on the mechanical vibration is neglected. The induction is simulated in a harmonic electromagnetic analysis, with the Lorentz forces calculated by superposition with the static magnetic field [74]. In the final step, these forces are applied to the resonator element in a transient mechanical analysis to visualize the possible resonant modes of vibration that can be excited by a certain Lorentz force distribution [75]. The primary focus of FEM simulation was the investigation of the eigenfrequencies of the different resonator elements in a mechanical modal analysis. This simulation revealed the possible vibration mode shapes for each element depending on geometry

2.3. Finite Element Simulation

27

Figure 2.7: Exemplary excitation simulation of a radial TSM at 15 MHz in a circular silicon plate, with eddy current induction (left), Lorentz force generation (middle) and resulting thickness shear mode vibration (right).

and boundary conditions, and thus showed the necessary force distribution. Selected mode excitation was also verified with the complete excitation simulation as described above (e.g. Fig. 2.7), but in general successful excitation was verified primarily via experimental means. Special simulation challenges appeared with the elastic anisotropy of some materials, e.g. (100) silicon. While isotropic materials can also be roughly modeled in two dimension, anisotropic materials require an appropriate three-dimensional model which leads to greatly increased computational requirements. The simulation has shown three distinct fundamental vibration symmetries at different fundamental frequencies for the shear modes of a clamped circular or rectangular membrane. The face and thickness shear modes can result in diagonal, circular and radial displacements. For the face shear modes the fundamental frequencies of these vibrations are very different from each other (Fig. 2.8), while the three thickness shear modes are excited at approximately the same frequency. The excitation simulation shows that radial Lorentz forces only excite radial shear modes efficiently, as it is expected. The diagonal and circular shear modes are superposed by higher harmonics of other modes and no pronounced resonant mode shape can be excited. Furthermore, the analysis shows a small out-of-plane displacement at the center of the radial vibration, resulting from mass conservation effects. Due to compressional vibration of the element towards a singularity at the center, but limited compressibility of the resonator material, radial motion

28

2. Theory and Modeling

Figure 2.8: Fundamental face shear mode shapes of a round silicon membrane with displacement of diagonal/linear (left), circular (center) and radial (right) symmetry for a circular plate.

experiences this out-of-plan buckling. This undesirable vertical displacement is of similar amplitude as the primary shear movement and therefore not negligible. For liquid phase sensing this additional component will radiate compressional waves into the liquid and create a higher attenuation. Therefore, a setup consisting of a planar spiral coil is not the ideal solution for liquid phase measurements, as we show in later chapters. FEM simulation has primarily been used as a tool to predict the possible modes of vibration for a given resonator geometry and to design our excitation setups accordingly, as well as to limit the amount of measurement data necessary to find a specific mode. A more detailed discussion of the different possible mode shapes for each resonator element and application will be given in the following chapters.

2.4

Equivalent Circuit Modeling

The second modeling aspect is the investigation of the feedback mechanism into a measurable electrical quantity. Building an FEM model incorporating this mechanism is theoretically possible, but not feasible in terms of required computation steps. The movement induction of a secondary current and the resulting induced secondary voltage in a primary coil could have been simulated, but the combination with the actually excited mode shape and the superposition of eddy current and secondary current in the moving resonator would have resulted in memory and computational requirements beyond our capacities. Coming from electrical engineering, a far more ac-

29

2.4. Equivalent Circuit Modeling Table 2.1: Electro-mechanical equivalences with the Mobility Analog. voltage

u

v

velocity

current

i

F

force

inductance

L

n

elastic compliance

capacitance

C

m

inertial mass

resistance

R

h

friction coefficient

G

r = 1/h

conductance impedance

Z = R + j ωL −

L

1 ωC

u = jωLi

Z m = h + j ωn −

mech. impedance

n

m u=

1 i jωC

v=

1 F jωm

h

R

Voltage Law

v = jωnF

C

Current Law

mech. resistance 1 ωm

u = Ri P ⋆i = 0

P

◦u = 0

v = hF P ⋆F = 0 P

◦

v=0

sum of nodal forces loop velocities

cessible approach is to depict this electro-mechanical system as an electric equivalent circuit, where excitation coil, eddy currents, mechanical resonator, and acoustic load are modeled as lumped elements. A similar approach has been successfully utilized to model piezoelectric acoustic microsensors as well, e.g. [76, 77, 78, 79]. The electrical, mechanical and acoustic domains are governed by differential equations of similar form and they can be described by the same mathematical network model [80]. When choosing the network coordinates force F and velocity v, the mechanical equation of motion can be written as a parallel mechanical circuit (Fig. 2.9): Z 1 dv 1 v dt = F (t) , (2.22) m + v+ dt h n with m as inertial mass, h as mechanical damping or friction admittance, and elastic compliance n, the inverse spring constant. In this form, termed Mo-

30

2. Theory and Modeling

bility Analog, the similarity to the governing equation of a parallel electrical circuit is evident. Z 1 1 du u dt = i (t) (2.23) + u+ C dt R L Here, the relationship between current i and voltage u is influenced by the electrical circuit elements of inductance L, resistance R, and capacitance C. As we can see, with this method the flux of the system is given by force and current, while the potential at a network node is given by velocity and voltage. The equivalent circuit description of the mechanical system follows the same laws as an electrical circuit. Just as Kirchhoff’s law states that the sum of all currents flowing into an electrical network node is zero, similarly the sum of all forces acting upon a mechanical network node is zero. And as the sum of voltages in a closed loop within an electrical circuit is zero, the sum of velocities in a closed loop in a mechanical circuit is zero as well. A complete analogy is given in table 2.1. Both systems exhibit an isomorphic, i.e. structurally identical, behavior. It is also possible to utilize the converse, so-called Impedance Analog, where the network flux and potential are given by velocity and force, respectively. Both analogies will yield equivalent results and can also be translated using the dual (dot) method [81]. The Mobility Analog however is more intuitive, e.g. mechanically connected objects move with the same velocity, i.e. are thus ’wired’ in parallel (Fig. 2.9). The mechanical circuit can thus be analyzed using the same network tools and programs used for electrical circuits without additional verification or modification. The same analytic method can also be applied to electro-acoustic analogies, where the flux component is given by the pressure. Using electrical network theory and methods, it is advantageous to also switch from the time-domain description to a complex, frequency-domain description, as used in table 2.1. While this limits the model to sinusoidal functions, any periodic function can be decomposed into a sum of sinousoidal functions with the help of a Fourier series, thus the conversion is possible without loss of generality. Complex functions, or phasor calculus, simplify many calculations of harmonic signals, reducing trigonometric functions in the time-domain to the time-invariant parameters amplitude, phase and fre-

31

2.4. Equivalent Circuit Modeling

F F v

m

F m

n

r

m

n

r

r

n

v

v

Figure 2.9: From mechanical description and schemata to equivalent circuit with the Mobility Analog (left to right).

quency (ω). Kirchhoff’s circuit laws, Ohm’s law, power law are equally applicable to networks with time-harmonic flux and potential. Complex variables are here written with an underscore, with j as the imaginary unit in order to not confuse it with current i. With the isomorphism of electrical, mechanical and acoustic network it is now possible to combine all domains into a single model, which can then be analyzed as a whole. To establish a quantitative relationship between flux and potential of the different domains, we have to define proportionality constants between the different network coordinates. (2.24) u = Yv 1 F i = (2.25) X The potential equation relates voltage u to velocity v with the coupling coefficient Y , describing the movement induction as in eq. 2.21. The flux equation relates force F to current i and is a different description of the Lorentz force law (eq. 2.20) for our application with a second coupling coefficient X. In general, this proportionality can also be factored directly into the element properties, i.e. a mass element in kg can be converted into a capacitance in F. With this coupling we can successfully model the complete system of electrical excitation, mechanical resonator and acoustic load. For a so-called electrodynamic transducer, i.e. an electromagnetic transducer utilizing Lorentz forces, the electrical side is made up of inductance and resistance of the current carrying path, driven by a primary AC voltage

32

2. Theory and Modeling

i

u

L

R

FL

uv

F

m

h

n

v

Figure 2.10: Electromagnetic-acoustic transducer with electrical equivalent circuit (left) and mechanical equivalent circuit (right).

u. The mechanical side is driven by a Lorentz force source F L , which is a function of the primary current, the external magnetic field B0 and the current path geometry, and is modeled as a parallel circuit of mass, compliance and friction coefficient, resulting in a mechanical vibration with velocity v. Conversely, this vibration in the magnetic field leads to the induction of a secondary voltage uv resulting in a secondary current superposed to the primary current i. The mechanical side is governed by the differential equation 1 v − jωmv − rv. F = F L (i, B0 ) − (2.26) jωn The electrical side can be written as u = Ri + jωLi + uv (v, B0 ) .

(2.27)

As we can see, the Lorentz force generation is modeled as current-controlled current source, while the secondary movement induction is modeled as a voltage-controlled voltage source. These two elements can consequently be combined in an electro-mechanical transformer element, with the parameters being the two coupling coefficients X and Y , based on magnetic field and geometry. However, a complete EMAT resonator also utilizes eddy current induction, i.e. the primary current is not impressed into the mechanical element but remotely induced. This mechanism can be modeled by a lossy electromagnetic transformer. Note that this lumped elements model is only valid under the assumption of frequency independent elements. Furthermore, for specific coupling coefficients X and Y , the model only serves as a reasonable approximation in the

33

2.4. Equivalent Circuit Modeling

Ri AC

Zi

Rc

T1

T

Ct Lc

Re Le

k

T3

FL

i uv

{X, Y } T2

v

h

n

m

T4

XT

Excitation circuit

Eddy currents

Acoustic resonator

Figure 2.11: Electromagnetic-acoustic equivalent circuit, with eddy current induction described by electromagnetic transformer T , coupling between current i, voltage u and force F , velocity v in the resonator modeled by electro-mechanical transformer XT .

vicinity of a particular resonance frequency. For different modes of vibration, different coupling coefficients have to be fitted to the measurement data. The resulting complete electromagnetic-acoustic equivalent circuit is shown in figure 2.11. We measure the input impedance Zi with an impedance analyzer represented as the AC source and input resistance Ri . The electrical circuit consists of the primary coil inductance Lc and resistance Rc , with an external tuning capacitor Ct in parallel to achieve electrical resonance according to the MARS principle [65]. The induced eddy currents experience a secondary inductance Le and resistance Re in the conductive part of the resonator. A current-controlled current source couples the eddy currents i and Lorentz forces FL , while a voltage-controlled voltage source is responsible for the induction of a secondary voltage uv due to the resonator velocity v, which are contained in the electro-mechanical transformer XT . In the frequency domain this electro-mechanical coupling can be described by the following complex equations. F L = X (B0 , . . .) i

(2.28)

uv = Y (B0 , . . .) v

(2.29)

The coupling coefficients X and Y are in our case dependent on the magnetic flux density B0 and the geometry of Lorentz force generation and the vibration mode shape. On the mechanical side the resonator is modeled

34

2. Theory and Modeling

14.4

33.5 Measurement Simulation

33.0

14.2 32.5 14

Phase / °

Impedance Amplitude / Ω

Measurement Simulation

13.8

32.0 31.5 31.0

13.6 30.5 13.4

287.4

287.6

287.8

Frequency / kHz

288.0

30.0

288.2

708

287.8

Frequency / kHz

288.0

288.2

Measurement Simulation

19.2

706

19.1

Phase / °

Impedance Amplitude / Ω

287.6

19.3 Measurement Simulation

707

705 704

19.0 18.9

703

18.8

702

18.7

701

287.4

15.41215.41415.41615.41815.42015.42215.42415.426

Frequency / MHz

18.6

15.41215.41415.41615.41815.42015.42215.42415.426

Frequency / MHz

Figure 2.12: Comparison of impedance amplitude (left) and phase (right) for equivalent circuit simulation and measurements of face shear mode (top) and thickness shear mode (bottom).

as a parallel resonant circuit, with vibrating mass m, elastic compliance n, and damping h calculated from geometry and the material properties, e.g. density and elastic stiffness. An acoustic load on the resonator gives rise to additional elements, which depend on vibration mode and load properties, e.g. mass or viscous loading for shear vibrations result in parallel mass and damping elements. Figure 2.12 shows very good agreement between equivalent circuit simulation and measurements for exemplary radial FSM and TSM resonances without an acoustic load, which are described in detail in chapter 3. The model is suitable for all excited modes of vibration, including flexural, face shear, and thickness shear modes. The appropriate load elements will be introduced in the respective chapters, i.e. additional elements for mass loading, viscous loading and compressional wave propagation.

2.5. Summary

2.5

35

Summary

The principle of electromagnetic excitation of acoustic waves is a well known mechanism. The application to resonant sensors and exploitation of the resonant vibration in a sensor element is the new approach used in our research. As resonant sensors, we can utilize our devices in a similar fashion as established piezoelectric or capacitive transducers with the added benefit of unique advantages such as spatial separation, bare sensor surfaces, and excitation of fundamentally different vibrations in the same resonator. The physical background is given by eddy current induction and Lorentz force generation, whereas the excited resonant vibration influences the eddy currents, and thus the induced voltage in a primary coil. In the equivalent circuit model we established, the coupling between electrical and mechanical domain is given by two transformers. These are an electromagnetic transformer describing the eddy current coupling between primary coil and conductive element, and an electro-mechanical transformer describing the Lorentz force excitation and movement induction between eddy currents and resonant vibration. Together with an FEM analysis of the mechanical eigenmodes of the resonator, this simple equivalent circuit model is suitable to describe the physical mechanisms and predict and analyze sensor measurements.

3 Basic Resonator Operation Electromagnetic-acoustic excitation can be applied to any kind of resonator material, the only requirement being a conductive substrate or layer for eddy current induction. Therefore, we can focus on different aspects for different applications, e.g. inexpensive mass-produced sensor elements, substrate materials with high elastic stiffness and minimal temperature coefficients, or materials suitable for integration in specific measurement setups. In this chapter we introduce the three main sensor elements employed in our experiments and describe the design and fabrication process employed, the eigenmode simulation results for each device and the appropriate experimental setups to excite the different possible modes of vibration. As a resonator substrate with excellent elastic properties and very low temperature coefficients similar to quartz crystal resonators we utilized silicon. In a prototype process silicon membranes were etched and a conductive copper layer applied. These devices have shown acoustic resonances with very high quality factors (Q) comparable to standard AT-cut quartz sensors. For lowcost applications we chose simple, polished aluminum and brass plates. To improve the suitability as resonators, we additionally milled a mesa-structure into these plates, to trap the vibrational energy in the sensing region. The third focus has been placed on multi-mode excitation. For this application we devised prototype resonator arrays out of thin metal nickel silver sheets, where we successfully realized simultaneous excitation of different modes of vibration, which allows for the simultaneous measurement of multiple physical properties of an analyte.

3.1 3.1.1

Design and Fabrication Silicon Membrane Resonators

In order to achieve optimal experimental results, a microfabrication process was set up to create high-Q silicon membrane elements [82, 83]. This

38

3. Basic Resonator Operation

Figure 3.1: Simplified 7 step silicon fabrication process; substrate (grey), oxide mask (red), photoresist (yellow), metal (purple).

prototype process also aimed for a low-cost fabrication. Single mask photolithography for structuring the silicon substrate has been used together with a minimum number of standard process steps to validate the feasibility of that goal. The anisotropy of the material silicon has been one reason why it has not yet been investigated as a resonator excited by magnetic direct generation. Therefore both standard (1 0 0) silicon wafers with anisotropic mechanical properties and (111) wafers with isotropic properties have been acquired. Ten 4′′ (100 mm) double polished wafers of the (1 0 0) crystalline orientation were sourced from a wafer batch of Siltronic AG. These phosphorus n-doped wafers displayed a spread resistivity of 20.3 Ω cm and had a specified substrate thickness of 383.4 µm. The surface flatness of these Czochralski grown wafers is given with a maximum deviation of ±0.5% and a total thickness variation of < 5 µm. For the (1 1 1) orientation only a single double polished wafer could be acquired from the Technical University of Chemnitz, Germany. Figure 3.1 shows the simplified fabrication process. Starting with the cleaned silicon substrates (gray), a 1 µm thick, thermal oxide layer (red) was grown on the wafers and structured in the photolithography step. First, a 8 µm thick photoresist (AZ4562 ; yellow/light grey) was spun onto the wafers. In a S¨ uss MA/BA6 mask aligner the 5′′ transparency mask with the structure for 12 round and 12 quadratic 15 × 15 mm membrane elements was used to

3.1.1. Silicon Membrane Resonators

39

Figure 3.2: SEM image of sidewall of round silicon membrane with characteristic DRIE structure.

expose the resist. After development the opened oxide was then structured in a BHF etch bath. Both remaining oxide and resist layer created the etch mask for the reactive ion etching (RIE) process, which was chosen due its intrinsic advantages of steep sidewalls, sharp edges and smooth membrane surfaces. RIE was also necessary to anisotropically etch (111) substrates. The last clean room step was to deposit a 1 µm thick layer of copper metalization on the opposite side (purple), after removing all remaining resist and oxide. The wafers were then ready to be diced. The membranes were characterized with scanning electron microscopy (SEM) and an optical profilometer for surface smoothness, sidewall steepness and transition and possible defects. These investigations showed well-defined membranes with very sharp transitions from sidewall to membrane surface (figure 3.2), which promise only minimal acoustic dissipation and damping effects. The metal layer thickness is suitable for an efficient excitation of thickness shear modes at several tens of MHz, while the acoustic load represented by this copper layer does not significantly influence the resonant behavior except a frequency shift to slightly lower frequencies due to the added surface mass, see chapter 4. However, for face shear modes in the range of several

40

3. Basic Resonator Operation

hundreds of kHz, this metal thickness is much smaller than the penetration depth of the eddy currents (see eq. 2.18), and therefore the excitation is less efficient. Further fabrication steps were needed to realize a new optimization of conductive layer thickness and damping characteristics to efficiently excite and detect these face shear mode vibrations. To increase the copper layer thickness, we have utilized a galvanic setup (PPG 10/3-B 3 L, Heimerle+Meule, Germany) for miniature samples to electroplate several additional microns of copper onto the silicon elements [84]. With the 1 µm copper as a seed layer and using a copper-phosphorus electrode and the acidic copper bath CU 500, we achieved a deposition rate of 1 − 3 µm per minute, which was adjusted to achieve a high uniformity and surface smoothness. While the face shear modes of the unplated membranes are barely detectable, the thickness shear modes (TSM) show strong resonance peaks and can be used for mass sensing applications in gaseous environments. Besides the frequency shift due to the added mass, the efficiency of mode excitation and corresponding quality factors are also affected by an increase of copper layer thickness and changes to the copper surface. Here, an optimum of additional deposition for the excitation of TSM vibrations was obtained at a few µm (Tab. 3.1). The coupling efficiency increases up to a point where the additional inhomogeneity and surface roughness introduces a surpassing amount of damping to the resonant behavior. Figure 3.3 shows the changes to a radial face shear mode at ≈ 250 kHz with increasing copper layer thickness in terms of peak impedance change ∆Z and peak phase shift ∆Θ detected in the primary impedance of the excitation coil. Transduction efficiency, which is characterized in terms of ∆Z and ∆Θ Table 3.1: Changes to the radial thickness shear mode of (100) Si membrane at 15 MHz due to increasing copper layer thickness. Cu layer

fR / MHz

Zi / mΩ

Θ / m◦

1µm

15.89

691.00

52.07

≈ 2µm

15.57

965.79

105.90

≈ 4µm

15.15

228.81

33.06

41

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 −0.1 −0.2

~15 µm

~4 µm ~2 µm

increasing Cu thickness

240

242

244

246

Frequency (kHz)

248

250

1.2 1.0 0.8 0.6 0.4 0.2 0 252

Phase shift (°)

Normalized Impedance (Ω)

3.1.1. Silicon Membrane Resonators

Figure 3.3: Changes to a radial face shear mode of a (100) silicon membrane with increasing copper layer thickness.

is increasing in a fairly linear fashion with increasing thickness, while the influence on the resonance quality factor is still negligible up to several ten µm. The measured Q is also dependent on the electromagnetic eddy current coupling and the Q of the primary coil, thus a reduction of mechanical Q may not be readily apparent without further analysis (see MARS principle). With these changes to our sensor elements it was possible to transfer the results of pure metallic resonators to the composite silicon membranes and take advantage of the stiffer elastic properties of silicon resulting in larger quality factors and better defined resonance peaks. Although electroplating techniques do not allow a precise prediction of deposition rate and layer thickness, we were able to use the fitted simulation models to evaluate the deposition process. Additionally we used the Sauerbrey equation for mass microbalance to correlate the frequency shift of the thickness shear mode resonances to a corresponding additional mass (see chapter 4). The comparison between FEM simulation, approximation from deposition parameters, e.g. applied current density and total coating area, and acoustic microbalance are the basis of our thickness estimate.

42

3. Basic Resonator Operation

Figure 3.4: Brass and aluminum metal plates of 1 mm thickness with round mesa structure of 15 mm diameter milled in center.

3.1.2

Metal Plate Elements

Metal plates have been used as a proof of concept in the earliest stage of this research. Due to the high conductivity of complete sensor substrate, the electro-mechanical transduction coefficients are much higher than for composite resonators. However, the elastic properties of metals and especially the temperature coefficients are worse then, e.g., the silicon membranes. As very simple, inexpensive resonators we have continued to employ these elements in our experiments. To improve the acoustic resonant behavior we processed aluminum and brass plate elements in an in house milling process to create 1 mm thick, polished, coplanar mesa-shaped structures, see figure 3.4. A central, circular mesa structure of 15 mm diameter was defined by milling a 500 µm deep, 2 mm wide trench into the substrate. This design effectively traps the acoustic energy to the mesa structure [70], similar to the energy trapping in quartz crystals due to the electrode geometry. This method reduces undesired vibrational components at the outer clamping area and confines the vibration to the actual sensing region.

3.1.3

Resonator Array Sensors

To utilize multiple different vibration modes and their different sensitivities we devised a sensor array setup consisting of multiple resonator elements

3.1.3. Resonator Array Sensors

43

Figure 3.5: Design of 2x2 resonator array setup with individual coils and magnets for each element.

anchored to a common frame, as shown in figure 3.5. With a variety of excitation options, each element can vibrate in the same or different resonant mode, depending on the Lorentz force distribution and excitation frequency applied to each element. We have fabricated and developed different coil designs and magnet layouts corresponding to suitable Lorentz force distributions. These include coil arrays of planar spiral and quadratic design to generate radial and diagonal forces across each element, as well as larger single planar spiral and quadratic coils that can simultaneously excite circular and diagonal forces in each element. The magnet placement has to be adjusted accordingly. While the setup shown in figure 3.5 generates radial forces, a different alignment of the permanent magnets changes these to a linear horizontal direction. For flexural plate modes an external magnetic field in-plane with the sensor array is most suitable, requiring the placement of the magnets on the sides of the array in a vertical position. Prototype metal resonator 2x2 arrays were fabricated out of nickel silver sheets of 100 µm thickness with different designs for resonator element and spring couplers connecting to the common frame [85]. For one design

44

3. Basic Resonator Operation

Figure 3.6: Metal prototype arrays with different coupling springs and element size of 10x10 mm (top) and different corresponding excitation coil arrays on polymer foils with 100 µm metalization thickness.

the spring couplers were placed at the edges of the elements creating a short, stiff connection to the frame. Another type utilized a double U-shaped spring on each side, which reduced coupling effects due to higher damping of waves radiated into the springs. However, for better decoupling, different designs will be necessary. The metal sheets were patterned and etched in a standard photolithographic process. Additionally, a similar process was used to fabricate multiple primary coils and coil arrays for simultaneous excitation of the resonator elements. These designs were realized on copper-coated polymer foils and standard PCBs. A selection of these arrays and elements is shown in figure 3.6. As discussed before, another aspect to consider is the penetration depth of eddy currents (see 2.18). While this is usually not a problem with conductive resonators, for very thin substrates or composite resonators the thickness of the conductive layer greatly influences the potential vibration modes. The penetration depth of eddy currents δ is inversely proportional to the square

3.2. Eigenmode Analysis

45

root of frequency f , as well as conductivity σ and permeability µ. In order to efficiently induce eddy currents and couple the resonator to the primary current, a larger metal thickness is therefore necessary at lower resonant frequencies. As previously shown, this effect has to be accounted for when the excitation of face shear modes or flexural modes in the range of a few hundred kHz is desired for a composite resonator, such as the copper coated silicon membrane. Additional fabrication steps such as electroplating or substrate doping are required to achieve the necessary metal layer thickness. On the other hand, these additional processes may lead to a decreased efficiency in generating higher frequency vibration modes, such as thickness shear modes at a few MHz, due to an increase in surface roughness, inhomogeneities and an acoustic impedance mismatch between substrate and conductive layer. For a composite sensor array as proposed above, a prior decision on the desired modes of vibration is necessary, so that an appropriate metal layer can be applied.

3.2 3.2.1

Eigenmode Analysis Silicon Membrane Resonators

Silicon membranes have been fabricated with a variety of thicknesses and in round and quadratic shape in both (1 0 0) and (1 1 1) orientation. The metal copper layer thickness also varies between 1 µm and up to 20 µm. The eigenmodes of all these combinations have been analyzed as preparation for our experiments. As explained in chapter 2, the fundamental face shear modes and thickness shear modes of diagonal, circular and radial symmetry have been of foremost interest. Figure 3.7 shows the results of the undamped eigenmode analysis for round and quadratic (1 0 0) silicon membranes with 3 µm copper layer thickness. The eigenfrequencies for these face shear modes are found at 192 kHz and 206 kHz for the diagonal mode shape, at 239 kHz and 245 kHz for the radial symmetry, and at 428 kHz and 438 kHz for the first of two fundamental circular mode shapes, respectively. As we can see, the difference between round and quadratic membrane shape is rather minimal.

46

3. Basic Resonator Operation

Figure 3.7: Results of eigenmode analysis for round (left) and quadratic (right) 15x15 mm (100) silicon membrane showing diagonal (top), radial (middle) and circular (bottom) face shear modes.

The anisotropy of elasticity for crystalline (1 0 0) silicon has been fully taken into account. However, comparison between the two substrate orientations shows only minimal differences between elastically anisotropic (1 0 0)

3.2.2. Metal Plate Elements

47

Figure 3.8: Mode shapes for round (100) silicon membrane with diagonal TSM at 15 MHz.

and elastically isotropic (1 1 1) silicon (for shear vibrations in the x-y-plane [86]). The main difficulty for the simulation has been the very high aspect ratio of these elements, with thicknesses in the order of a few to a hundred micron and lateral dimensions of a few millimeter. Establishing an appropriate finite element mesh with enough detail but reasonable numbers of elements becomes difficult with such aspect ratios. While we can specifically define the mesh grid in ANSYS and thus take care of this challenge, COMSOL works best with a free, automatic mesh. However, COMSOL allows the abstract inclusion of dimensional scale factors, whereas we can scale the vertical dimension by a certain factor to reduce the aspect ratio to a size that the automatic mesh algorithm can handle properly. This is especially important for thickness shear modes, where the vertical dimension needs a finer step size to accurately describe the shear deformation. Figures 2.8 and 3.8 shows simulated TSM shapes for a round silicon membrane.

3.2.2

Metal Plate Elements

We focus on the FEM results for a round aluminum resonator with an active circular mesa structure of 15 mm diameter [87]. This element was cast to

48

3. Basic Resonator Operation

Figure 3.9: First and second harmonic of diagonal FSM in round aluminum mesa resonator at 81 kHz and 171 kHz.

form the energy trapping mesa structure in the center and dimensioned to fit the existing planar coils. The first diagonal face shear mode appears at 81 kHz for this element. However, the primary shear mode is superposed by a slight out-of-plane movement, induced by the element geometry. The 2nd harmonic of this diagonal face shear mode can be found at 171 kHz. A slight vertical component in the active area is also evident here (Fig. 3.9). For our liquid phase experiments, we have primarily utilized two different modes of vibration, the circular face shear mode and a radial flexural plate mode (Fig. 3.10). Both modes are very different in the behavior in a fluid environment. The circular FSM shows no out-of-plane vibration component both in the eigenmode and in the excitation analysis. The acoustic energy is well trapped in the central mesa structure. This behavior is ideal for sensing the properties of a liquid, such as viscosity and density, with evanescent shear waves. There is primarily only viscous damping of this resonance, additional damping due to destructive interference of compressional waves is minimal. The total volume and mass of the liquid on the resonator has no significant influence on the sensor response, and a directly proportional relationship between the square root of density and viscosity and the frequency shift of the resonance can be established. These results have also been achieved by other groups, and similar conclusions could be drawn. Circular face shear mode or torsional plate resonators are suitable sensor devices for measuring

3.2.3. Resonator Array Sensors

49

Figure 3.10: Eigenmode shape of circular FSM at 142 kHz (left) and radial flexural plate mode at 295 kHz (right) of round aluminum mesa resonator.

liquid properties and distinguishing between different fluids [88]. The radial flexural plate mode on the other hand is dominated by out-ofplane components of the vibration. Due to the coupling with a radial shear movement, the in-plane forces generated by a circular spiral coil are highly efficient at exciting this mode of vibration. The vertical movement radiates compressional waves into the liquid and is thus highly sensitive to the surface level height of the liquid. This device is suitable for measuring small samples of liquid volume with high precision. The influence of liquid property changes is negligible compared to the effect of changes in liquid height.

3.2.3

Resonator Array Sensors

For both the metal array prototype and a silicon array we have simulated the respective eigenmodes of the array elements. Similar to single plate resonators, there exists a variety of flexural plate modes, as well as face shear and thickness shear modes of radial, diagonal, and circular symmetry. Depending on the spring coupling of the elements to the array frame, these modes can be slightly distorted by and not fully decoupled from other elements. For quadratic elements coupled at the corners (see Fig. 3.6) the radial face shear mode and the circular face and thickness shear modes are of interest (Fig. 3.11). The fundamental shear eigenmodes of interest are similar to single ele-

50

3. Basic Resonator Operation

Figure 3.11: Simulated mode shape for radial face shear mode (left) and circular thickness shear mode (right) of a single metal array element.

ments, the radial symmetry mode with strong flexural components for liquid volume sensing, the diagonal symmetry mode with larger vibration amplitudes, and the circular symmetry for viscosity-density sensing. Combinations of different modes and higher harmonics have also been investigated theoretically and experimentally, since in some cases the necessary force distribution is easier to generate than for the fundamental modes. Cross-coupling between individual elements is reduced if our spring coupler design is used instead of fixed connections at the corners to the main frame, but further analysis is necessary.

3.3

Experimental Excitation Setup

Our experimental setup consists of primary planar induction coils, external permanent magnets and the resonator element, as shown in figure 3.12. The Liquid Reservoir

Conductive Resonator Impedance Analyzer

Primary Excitation Coil Permanent Magnet

Figure 3.12: Geometric arrangement for electromagnetic-acoustic resonator setup with remote, non-contact excitation and detection.

3.3. Experimental Excitation Setup

51

Figure 3.13: Design and realization of excitation setup for radial and circular shear modes.

excitation coils were either wound from enameled copper wire (0.12–0.19 mm diameter, Roadrunner Electronics Ltd., UK) or, alternatively, lithographically etched on 1 mm thick PCB substrates or 50 µm thin polymer foils. The coils created for this setup vary in number of turns, wire thickness and also geometric realization. The corresponding magnet layout is realized by using several permanent magnets placed together and aligned with the primary coil. The permanent magnets used were rare-earth NdFeB magnets of different sizes corresponding with the size of the resonator element (e.g. 20x20x10 mm) with a measured static magnetic flux density at the sensor element of 0.25 − 0.5 T.

The primary coil is bridged by a parallel trimming capacitor to be able to adjust the electrical resonance to fit the mechanical resonance according the MARS principle. With this principle it is possible to effectively multiply the Q-factors and achieve a stronger signal response. The complete electrical setup was connected to an Agilent 4294A Precision Impedance Analyzer via an Agilent 42942A Terminal Adapter to excite and detect the mechanical resonances. In order to minimize interferences from parasitic wire and connector capacitances, the connection to the impedance analyzer has been kept very short, using two basic wires from contact pads on the PCB to a clamping fixture on the terminal adapter. Suitable, non-magnetic holders for the permanent magnets, coils, and transducers, as well as fluid measurement cells, were fabricated in PMMA and other plastic materials (Fig. 3.13). The advantages of this setup are quickly interchangeable spiral coils for

52

3. Basic Resonator Operation

different frequency ranges and simple connectors to use with different test fixtures or more advanced custom electronics. This setup can also be used for exciting piezoelectric resonators. The permanent magnet does not interfere with magneto-piezoelectric excitation [44, 83]. We measured the impedance spectrum of this setup around the mechanical resonance used for the respective experiment. The secondary induction due to the resonating plate yields an additional contribution, which is superposed to the impedance characteristics of the primary coil. In order to better visualize the resonance curve of interest in the measured impedance spectrum, we normalize the measurement by subtracting the electrical impedance of the coil: Z i (f2 ) − Z i (f1 ) · (f − f1 ) . (3.1) Z i,norm (f ) = Z i (f ) − Z i (f1 ) + f2 − f1

Here, Z i is the complex input impedance and f1 and f2 are the start and stop frequencies of the measured frequency spectrum, resulting in the normalized input impedance Z i,norm shown in all following measurements. Any offset due to slight changes in alignment, proximity, or location of the experimental setup is hence eliminated and only the changes due to the mechanical resonance are reflected properly. This normalization is applied to both impedance and phase measurements. The absolute impedance varies greatly depending on type of primary coil, operating frequency, excitation setup and alignment, mode shape, and electrical resonance tuning. For flexural and face shear modes it is usually in the range of a few 10 Ω, for thickness shear modes a few hundred up to 1000 Ω. The most common coil geometry employed for this sensor principle has been a planar spiral coil. However, applying the Lorentz force law to the induced eddy currents in the resonator, this layout yields radial forces when centered under the resonator element. These radial forces will preferably excite radial vibration mode shapes. According to finite element method (FEM) simulation, a radial thickness shear eigenmode exists for silicon resonator elements, however, this radial vibration is coupled to an out-of-plane movement at the center of the element due to the conservation-of-mass principle. This vibration component generates unwanted compressional waves in

3.3. Experimental Excitation Setup

53

Figure 3.14: Lorentz force distribution (arrows) with different coil and magnet layouts for radial (left), diagonal/linear (center), and circular (right) vibration mode shapes.

liquids, which lead to additional damping. Therefore, a number of alternative vibration modes are of particular interest for liquid phase sensing. Our FEM simulation results show that diagonal, circular and radial symmetry are the three primary mode shapes for face shear and thickness shear mode vibrations (see above). Therefore, we have concentrated our efforts into realizing experimental setups to accommodate these mode shapes. The distribution of the Lorentz forces is affected by the distribution of the eddy currents and the direction of the magnetic field. Therefore, in order to generate non-radial forces, the coil geometry has to be adapted and the magnetic fields changed accordingly, as explained in chapter 2. We developed a setup of quadratic or rectangular spiral coils coupled with multiple alternated permanent magnets for excitation and detection of diagonal, lateral shear modes. This vibration is similar to the thickness shear mode in QCM sensors and will lead to improved liquid phase sensing capabilities. In figure 3.14 three different designs for the microfabrication of the primary excitation coil geometry are given [89]. These coils have been realized in a photolithographic process. The corresponding magnet layout is visualized by the colored areas and realized by using several permanent magnets placed together and aligned with the primary coil. The complete setup for these designs is similar to that comprised of a single spiral coil. Besides the coil geometries presented here, other possibilities have also been investigated. For the diagonal shear mode these include linear solenoids and meander line structures with single and alternated magnetic fields, respectively. For the circular or torsional mode, other geometries include toroidal layouts for excitation and a secondary spiral detection coil and also

54

3. Basic Resonator Operation

dual ellipsoidal designs based on the results with a spiral coil but optimized towards a symmetrical excitation. Earlier work on EMAT transducers suggests even more possible design alternatives. The primary challenge lies with the electromagnetic coupling efficiency. For an efficient excitation, an appropriate force distribution is necessary. The resulting vibration must however also lead to a secondary induction in parallel to the primary current flow. And as the coupling between primary coil and resonator element is achieved via the dynamic magnetic fields of excitation current and eddy currents, the energy transfer needs to be maximized. In electrical terms, this is achieved by a high inductance and mutual inductance. This excludes several possible designs for excitation and/or detection coil. For precise and reproducible measurements a temperature stable setup is required, primarily due to the temperature coefficients of density, viscosity and ultrasonic velocity. Most experiments shown in the following have been carried out at a temperature of 25 ± 0.5◦ C. To further minimize the influence of the temperature dependence, we have also developed a thermostat based, temperature stabilized experimental setup operated at ±0.05 K for simultaneous measurements with different acoustic resonator sensors [90]. The measurement cells for liquid phase sensing required non-magnetic material such as PMMA, other plastics, or aluminum. One further requirement was to keep the resonator element as close as possible to the primary coil, i.e. for setups with the resonators placed inside a cell, the bottom wall had to be very thin and was usually realized with a 100 µm thin plastic foil. The resonator element was simply placed on the bottom in an arbitrary, free-moving fashion and the coil and magnet were then aligned appropriately. The other option to realize a measurement cell was to use the resonator itself as the bottom of the cell. Then hollow plastic cylinders were attached to the surface forming a closed cavity. Here, the distance to the primary coil was already minimal. Difficulties arose from the proper attachment of the cylinder without strongly perturbing the resonant vibration. Therefore, the cylinder was attached to the regions of minimal displacement according the eigenmode analysis of the specific vibration mode described before.

3.3.1. Array Multi-Mode Setup

55

Figure 3.15: Simultaneous excitation of different modes of vibration in each array element at different excitation frequencies (left), and of the same vibration mode at the same resonant frequency (right).

3.3.1

Array Multi-Mode Setup

The setup for experimental measurements with array resonators is similar to our experiments with single resonator elements and thus allows for easy comparison of the results in terms of achievable excitation efficiency, resonance quality factor, and resulting impedance changes. Here, the impedance of a single large primary coil or coil array is also monitored with the impedance analyzer. Alternatively, each individual coil and element can also be driven by individual AC sources and analyzed by separate detection electronics. An advantage of the sensor array setup is the simultaneous single and multi-mode excitation, i.e. of similar and different vibration modes, respectively. Figure 3.15 shows how to achieve multi-mode excitation with a coil array design, where each coil can be driven at a different frequency, and explains the single mode excitation with a series connection of each individual coil driven at the same frequency. Alternatively, a single large coil can also be used to excite the individual elements simultaneously. Most suitable are rectangular planar coils that induce linear, diagonal vibrations, or a round spiral with a radial current direction underneath each element resulting in circumferential forces.

3.4

Eigenmode Excitation Results

In the following section we will present selected examples of successfully excited thickness shear, face shear and flexural plate modes. All measurements

56

3. Basic Resonator Operation

have been carried out in air. The undamped FEM eigenmode simulation described above was used to predict the eigenfrequencies for the different mode shapes of interest and verified experimentally. In general, we achieved good agreement between FEM prediction, excitation theory and experiments. And overview of detected vibration modes can be found in the appendix.

3.4.1

Thickness Shear Modes

We have successfully excited diagonal, circular and radial thickness shear modes in silicon membranes and aluminum or brass plates. The detected fundamental resonance frequencies vary between 5 and 30 MHz for the microfabricated membranes both in elastically isotropic (1 1 1) and anisotropic (1 0 0) silicon substrates, with higher harmonics detected up to and above 100 MHz as well. Since the electro-mechanical transduction is of inductive nature, the resonance frequency of the electrical circuit is the limiting factor for excitation of higher harmonics, i.e. above the electrical resonance of the primary coil we have dominant capacitive coupling and the resonance peaks vanish. This also is the main difference to piezoelectric excitation, where the coupling mechanism is of capacitive nature, and thus higher harmonics can be detected easily. All these TSM resonances display very high Q-factors up to about 200,000 due to the high intrinsic elastic stiffness and low loss of silicon. Silicon of (1 1 1) crystalline orientation shows only mediocre results. Both Q factor and amplitude of the signal response are several times less than for anisotropic (1 0 0) silicon. Table 3.2 lists different resonator materials and geometries and compares operating frequency, signal response and quality factor. While a QCR still gives a larger signal due to the different excitation mechanism, it is surpassed by the membranes in Q-factor of these resonances. Additionally the silicon resonators can be excited in even and uneven harmonics of the fundamental TSM [83]. For exciting the diagonal mode shape, the quadratic coil geometry results in a higher signal amplitude but also in additional spurious modes, whereas the rectangular layout excites a single resonance but at reduced signal strength due to lower mutual inductance between sensor element and

57

3.4.1. Thickness Shear Modes

Table 3.2: Comparison of operating frequency, signal response and Q-factor for selected resonator elements excited at radial TSM in air; with different selected geometries and operating TSM harmonics for the silicon membranes. Material

dR / µm

Geometry

fR / MHz

harmonic

Zi / mΩ

Q / 104

Quartz (AT-cut)

165

round

10.108

1

105.47

2.38

Aluminum

700

quad

2.380

1

18.81

0.78

Silicon (100)

390

quad

14.907

2

2.83

2.98

Silicon (100)

290

quad

20.360

2

6.71

6.03

Silicon (100)

190

round

15.648

1

8.82

6.26

Silicon (100)

190

round

31.561

2

1.41

7.78

Silicon (100)

190

quad

15.417

1

10.44

7.01

Silicon (100)

190

quad

30.761

2

2.13

7.19

Silicon (100)

90

round

35.087

1

1.20

5.44

Silicon (100)

90

quad

34.318

1

0.80

6.02

Silicon (111)

300

round

24.613

3

0.36

1.89

larger coil area. In figure 3.16 the radial TSM resonance for a quadratic (100) silicon membrane at 15.42 MHz is compared to the diagonal mode shape of this element at 15.36 MHz. Both distinct resonance peaks only appear for their respective coil geometry and magnet placement (see Fig. 3.14) and disappear completely when excited by an unsuitable setup, as, e.g., a quadratic spiral coil with a single magnet aligned in center of the resonator excites the radial modes, when exchanging with two alternated magnets, the radial modes disappear but the diagonal resonance peaks appear. The resonance strength of the diagonal mode is smaller than for the radial mode. The Q-factor of both resonances is similar. Using out-of-center spiral coil excitation it is possible to excite a circular TSM resonance, or torsional mode, in a round (100) silicon membrane. Figure 3.17 shows the comparison of the radial shear mode and the circular mode shape at 15.65 MHz and 15.61 MHz, respectively. The results show that the excitation with a single spiral coil out of center does not result in an ideal coupling between coil and resonator element. The increased impedance

58

3. Basic Resonator Operation

0.5

Normalized Impedance Zi /Ω

Normalized Impedance Zi /Ω

4 3 2 1 0 −1 −2

15.410

15.415

15.420

Frequency f / MHz

0.4 0.3 0.2 0.1 0 −0.1 −0.2

15.425

15.350

15.355

15.360

Frequency f / MHz

15.365

Figure 3.16: Normalized impedance of quadratic coil with standard magnetic field resulting in radial TSM at 15.42 MHz (left) and with alternated magnetic field resulting in diagonal TSM at 15.36 MHz (right) in a quadratic high-Q silicon resonator.

of the signal level is the result of a reduced mutual inductance due to the loss of coupling efficiency. The primary reason for this reduction in performance in our measurement setup is the size of resonator element and planar coil, which have been optimized for the radial mode and are therefore not completely overlapping for the circular mode setup. Spurious components in the resonance signal can be explained by the asymmetric force distribution leading to the excitation of unwanted modes. With alternated magnetic fields and dual-circular coil layouts, this asymmetry can be overcome and pure circular shear motion is possible. 1

Normalized Impedance Zi /Ω

Normalized Impedance Zi /Ω

3 2 1 0 −1 −2 −3 15.640

15.645

15.650

Frequency f / MHz

15.655

0.8 0.6 0.4 0.2 0 −0.2 −0.4

15.600

15.605

15.610

Frequency f / MHz

15.615

Figure 3.17: Normalized impedance of a standard spiral coil at the center resulting in a radial TSM at 15.65 MHz (left) and placed out of center resulting in a circular TSM at 15.61 MHz (right) in a round (100) silicon membrane.

59

3.4.1. Thickness Shear Modes

0.08

Normalized Impedance Zi /Ω

Normalized Impedance Zi /Ω

0.3

0.2

0.1

0

−0.1

1.5250

1.5255

1.5260

Frequency f / MHz

1.5265

0.06 0.04 0.02 0 −0.02 −0.04

2.322

2.324

2.326

Frequency f / MHz

2.328

Figure 3.18: Impedance spectrum of circular and diagonal TSM in aluminum mesa element and quadratic plate, respectively.

Aluminum and brass plate resonators have also been successfully excited in the diagonal and torsional mode of vibration (Fig. 3.18). The experimental results and conclusions are similar to those achieved with silicon resonators. The issue of elastic anisotropy does not seem to have any negative influence on the generation of any of the investigated mode shapes. The lower elastic stiffness of aluminum results in lower quality factors compared to silicon resonators. However, eddy currents permeate deeper into the conductive substrate compared to a thin conductive layer, resulting in a higher transduction efficiency with metal resonators. A major drawback of pure metal resonators or composites with larger amounts of metal is the high temperature dependency of these elements. The temperature coefficients of aluminum for example are several orders of magnitude larger than silicon or quartz glass. Therefore, a temperature stabilized setup is necessary for reproducible experiments with metal resonators. Of note is that the Q-factor reported here is the minimal estimate taking the lower 3 dB frequency of the maximum peak resonance and the upper 3 dB frequency limit of the minimum peak. Due to the phase shift not occurring at the exact center frequency of the actual mechanical resonance, the bandwidth and thus the Q-factors can only be estimated. Equivalent circuit simulations resulting in a similar impedance spectrum show that the actual resonance quality of the ’virtual current’ in the mechanical part of the

60

3. Basic Resonator Operation

0.08

Normalized Impedance Zi /Ω

Normalized Impedance Zi /Ω

0.04 0.03 0.02 0.01 0 −0.01 −0.02

220

221

222

223

Frequency f / kHz

224

0.06 0.04 0.02 0 −0.02

456

458

460

462

Frequency f / kHz

Figure 3.19: Normalized impedance of diagonal FSM at 222 kHz (left) and circular FSM at 459 kHz (right) in a quadratic high-Q silicon resonator.

equivalent circuit is twice or more of the reported values.

3.4.2

Face Shear Modes

The excitation of face shear modes does not differ much from thickness shear mode excitation. The only difference is the required larger metal thickness for efficient eddy current coupling, due to eigenfrequencies in the the kHz region, and thus much larger eddy current penetration depths. Setup geometry and alignment to carter for the different mode shapes is similar, i.e. spiral coils with singular magnets placed at center excite radial symmetric modes, alternated magnets are required for diagonal, linear vibrations, and out-of0.5

Normalized Impedance Zi /Ω

Normalized Impedance Zi /Ω

0.4 0.3 0.2 0.1 0 −0.1

142.0 142.5 143.0 143.5 144.0 144.5 145.0

Frequency f / kHz

0.4 0.3 0.2 0.1 0 −0.1 −0.2

367

368

369

Frequency f / kHz

370

371

Figure 3.20: Normalized impedance of diagonal FSM at 143 kHz (left) and 2nd circular FSM at 369 kHz (right) in a mesa aluminum plate.

61

3.4.3. Flexural Plate Modes

0.8

Normalized Impedance Zi /Ω

Normalized Impedance Zi /Ω

0.25 0.2 0.15 0.1 0.05 0 −0.05 −0.1

246

247

248

249

Frequency f / kHz

250

0.6 0.4 0.2 0 −0.2

287.4

287.6

287.8

288.0

Frequency f / kHz

288.2

Figure 3.21: Normalized impedance of radial flexural mode at 248 kHz in a quadratic high-Q silicon resonator (left) and at 288 kHz for an aluminum mesa resonator (right).

center placement of the coil is necessary to generate circular forces. One difference to TSM is the superposition with flexural plate modes. Since both FSM and flexural modes are propagating waves along the lateral dimensions of the elements, the operating frequencies are similar. Thus, higher harmonics of a flexural mode can easily interfere with and superpose a pure in-plane shear motion, such as evidenced for radial face shear modes. Figures 3.19 and 3.20 show the impedance curves of diagonal and circular face shear modes for silicon and aluminum resonators, respectively.

3.4.3

Flexural Plate Modes

Flexural plate modes are excited with the standard circular planar spiral coil generating radial in-plane forces. For radial symmetric flexural plate modes there always exist radial in-plane displacement components into which the Lorentz forces can couple. Alternatively, we can place the magnets on the sides of the resonator to create a lateral magnetic field and use linear eddy currents to generate out-of-plane forces. These setups however suffer from a reduced magnetic flux density due to the distance to the magnets (size of the elements) and more complex setups with magnetic circuits are required for an efficient excitation. Figure 3.21 shows two examples of flexural motion utilizing a radial shear mode superposed with a flexural mode resulting in a strong vertical displacement at the center. Compared to diagonal or cir-

62

3. Basic Resonator Operation

0.6

1.4

Array element 1 Array element 2 Array element 3 Array element 4

0.4 0.3 0.2 0.1

1

0.8

0.6

0.4

0

0.2

−0.1

0

−0.2 230

231

232

Frequency / kHz

233

234

Array element 1 Array element 2 Array element 3 Array element 4

1.2

Phase shift / °

Impedance change / Ω

0.5

−0.2

230

231

232

233

234

235

236

237

238

Frequency / kHz

Figure 3.22: Experimental results for individual excitation of a circular FSM in all elements of a metal array anchored at the corners at ≈234 kHz, with noticeable cross-coupling between individual elements as evidenced by the minor peaks.

cular shear modes, the excitation efficiency and electro-mechanical coupling is much better due to coil and resonator element aligned centered to each other.

3.4.4

Array Multi-Mode Excitation

Shown in figure 3.22 is the impedance spectrum for a radial face shear mode in a metal array with stiff coupling at the corners, corresponding to the simulated mode in figure 3.11. Here, every element is excited individually [85]. As evidenced by the minor peaks of idle elements for each individual excitation, there is considerable cross-coupling to neighboring elements with this anchor design and small variations in the resonance frequency of each element. The latter is induced by fabrication tolerances and material inhomogeneities. For this vibration mode, the evaluation of amplitude and phase signal is equally possible, however the impedance changes depend on the frequency characteristics of the primary coil(s). As described above, simultaneous single-mode excitation can be achieved by connecting multiple coils in series or parallel, or by a single large coil driving all elements. We have investigated several combinations of resonator elements and excitation layout in air and with a liquid load of a droplet of DI water. Shown in Fig. 3.23 is the simultaneous measurement of similar

63

3.4.4. Array Multi-Mode Excitation

0.20

in air DI water

0.06

in air DI water

Al − flexural mode

0.04 0.02

∆f ∆f

0

−0.02

Impedance Phase

Impedance Phase

0.15 0.10

∆f

0.05 0

Si − FSM diagonal

−0.05

(100) Si

(111) Si −0.10

480

500

520

540

Frequency / kHz

560

210

220

230

240

250

Frequency / kHz

260

270

Figure 3.23: Simultaneous excitation of a circular face shear mode in (100) silicon and a radial flexural mode in aluminum around ≈250 kHz, and different frequency shifts with DI water loading due to different sensitivities.

circular face shear vibration modes in different (100) and (111) Si resonator elements, and the similar behavior of these resonances can be seen, resulting in simultaneous, multiple measurements for increased redundancy. Of greater interest is the excitation of different modes of vibration exhibiting different sensitivities. For example, the circular face shear mode in silicon elements is sensitive to the density-viscosity product of a liquid load. On the other hand, a radial flexural mode in an aluminum plate radiates compressional waves into the liquid, which are reflected at the surface boundary, and thus it is sensitive to the liquid height. This volume sensitivity results in much larger frequency shifts than those due to other liquid properties. A combined setup of a circular shear mode resonator and a flexural mode resonator can thus simultaneously measure density, viscosity, and liquid volume (Fig. 3.23). Other properties which can influence acoustic resonators are for example temperature, compressibility, conductivity, and permittivity. With an electromagnetically excited sensor array it is possibly to remotely measure these physical parameters in a single experiment at the same time by simply evaluating the complete impedance spectrum of the excitation circuit.

64

3.5

3. Basic Resonator Operation

Summary

Resonator design and experimental setup are tied very closely to the FEM and equivalent circuit models. By analyzing the eigenmodes of a resonator geometry, we have qualitatively determined the necessary force distribution for a successful excitation and designed our experimental setup accordingly. We have focused on three primary resonator devices, inexpensive aluminum and brass plates, simple micromachined silicon membranes, and thin, suspended metal resonator arrays. For composite resonators with conductive layers, such as our silicon membranes, we have shown how to modify the elements to be suitable in different frequency ranges. A large variety of different vibration modes, harmonics and superposition of modes have been successfully excited and detected. The suitability of specific modes for different sensor applications will be shown in the following chapters. We have demonstrated the feasibility of electromagnetic excitation of acoustic sensor array devices. As shown, it is possible to excite similar and different modes of vibration in these array elements simultaneously leading to increased redundancy and measurement of multiple physical properties with a single sensor. A miniaturized (silicon) resonator array can be a new alternative to established acoustic sensor devices taking advantage of the remote nature of electromagnetic excitation and the simultaneous measurement of multiple physical properties such as added mass, density, viscosity, temperature, and liquid volume. The proposed silicon sensor arrays can be fabricated in a standard micromachining process on an SOI wafer.

4 Mass Microbalance Application The primary sensor application of bulk wave resonators has traditionally been mass sensing. Added mass on the resonator surface results in changes in the elastic and geometric properties, leading to a shift of the resonant frequency. With a high frequency resolution a very high mass resolution down to a few pico- and femtograms can be achieved. Due to this sensitivity bulk resonators, especially quartz discs, are in use as thickness monitors for various deposition techniques, e.g. evaporation, sputtering, etc. [27]. More advanced applications include the detection of particles and (bio-)molecules adsorbed on the resonator surface to measure concentration, surface coverage, selective absorption, process speed, or many other parameters. In order to validate the suitability of electromagnetic excitation compared to piezoelectric or capacitive excitation, we have also investigated our resonator elements in terms of a simple mass microbalance. While liquid phase applications have been the focus of our study, we can show that our technology can also be easily applied to this kind of measurements. Additionally, the mass changes can be exploited with thickness shear modes as one independent parameter of a multi-mode sensor array.

4.1

Analytical Description

The electromagnetic-acoustic transducers can be used as resonant sensors similar to quartz resonators. The acoustic part of the transduction scheme is similar to QCR sensors, only the transformation of the mechanical resonance into an electrical signal is different. Acoustic loads put into contact with the sensor surface change the properties of wave propagation and result in a sensor response in form of a frequency shift. In the most simple case, the acoustic sensor acts as a mass balance, which is well described for thickness shear motion. A TSM resonator can be described by the general wave equation, which

66

4. Mass Microbalance Application

can be simplified for high-Q materials, where damping is negligible. The wave propagation is limited to the thickness dimension z and displacement, e.g. for a diagonal mode, is limited to a lateral dimension x. From the equations of motion 2.10 one obtains a wave equation for displacement ux with mass density and shear stiffness of the resonator, ρR and µR respectively. ∂ 2 ux (z, t) ∂ 2 ux (z, t) = ρ (4.1) R ∂z 2 ∂t2 Solving this equation yields a wave function with the wave number k and the boundary condition of stress-free surfaces ∂ux /∂z = 0, i.e. the resonator is operated without any restoring forces. µR

ux (z, t) = ux0 cos (kz) exp (jωt)

(4.2)

The TSM resonance frequency fR of harmonic n can be calculated as a function of resonator thickness hR r µR n fR = , (4.3) 2hR ρR and the shear wave phase velocity vs is given by r µR . vs = ρR

(4.4)

In resonance, thickness shear mode resonators have their displacement maxima at the surfaces, making them very sensitive to changes to the surface properties. Different models exist to describe these TSM resonators, e.g., very basic physical models and equivalent circuit models, delivering similar results [24, 25, 26, 31, 35, 77, 78]. Following the energy balance approach as described in chapter 2, we have a simple analytical description of the resonance behavior. According to the Rayleigh hypothesis, mechanical resonance is achieved when the kinetic energy is balanced by the potential energy of the system [7]. Energy is periodically exchanged between both forms at resonance, similar to other types of oscillators. Added mass in form of a thin, rigid layer at the surface causes an increase in the kinetic energy (displacement maximum), with no changes to the potential energy. Therefore, the resonance frequency changes to rebalance kinetic and potential energy.

67

4.1. Analytical Description

The kinetic energy density is defined by the particle velocity in all directions. Following from equation 2.11, we can sum the kinetic energies from infinitesimal slices across the resonator thickness. The peak kinetic energy density wk is therefore given by: ! Z hR ω 2 u2x0 ρR hR ω2 2 ux (z) dz = ρl ux0 + ρR ρl + , (4.5) wk = 2 2 2 0 where ρl is the areal mass density (mass per area) of the acoustic load on the surface of the TSM resonator. The peak potential energy density on the other hand occurs at the oscillation point of maximum displacement and zero particle velocity. From equation 2.12 and integrating over the thickness it can be therefore written as Z hR µR k 2 u2x0 hR 1 2 2 sin2 (kz) dz = . (4.6) wp = µR k ux0 2 4 0 Balancing peak kinetic and peak potential energy density results in a relationship between resonant frequency and acoustic mass load. We obtain ω 2 2ρl R , (4.7) =1+ ω hR ρR

where the unperturbed resonant frequency ωR , i.e. for no acoustic load, is given as r nπ µR . (4.8) ωR = hR ρR Equation 4.7 can be seen as a universal description of TSM resonator mass sensitivity. For weak mass loading, i.e. ρl ≪ hR ρR , the universal solution can be approximated with a Taylor series expansion as a linear relationship between frequency shift ∆f and added mass. ρl ∆f =− fR hR ρR

(4.9)

This equation recover’s the Sauerbrey formula [27], who first defined the resulting frequency shift as follows: ∆f = −nfR2 √

2 ∆m . ρ R µR A

(4.10)

68

4. Mass Microbalance Application

Here, the frequency shift ∆f is a function of the fundamental thickness shear mode frequency fR divided by the mass density and shear stiffness of the resonator, ρR and µR respectively, and the surface mass density ∆m/A acting as the acoustic load, operated at the n-th harmonic. The Sauerbrey equation is only valid if the additional mass can be treated as an extended thickness of the resonator, i.e. the layer must be sufficiently thin, rigid and homogeneous. This equation reflects a direct proportional relation between frequency shift and mass loading of a sensor element. According to Sauerbrey’s investigation any material capable of transmitting acoustic waves and resonating in a thickness shear mode experiences these frequency shifts, with the strength of the shift mainly influenced by the elastic properties and the intrinsic loss of the material. Silicon was also chosen for its higher elasticity and lower temperature coefficients of the elastic stiffness than for example quartz glass. The thin metallic layer also acts as an acoustic load for the silicon resonator, changing its fundamental frequency. The mechanical resonance of the metallic layer itself has no influence on the sensor resonance though, due to its minimal thickness. As the behavior and properties of mass loading are well known for piezoelectric TSM sensors, this vibration mode has been the primary focus of initial studies for the concept of electromagnetic-acoustic resonators.

4.2

Suitable Modes of Vibration

Added mass on the surface of the resonator influences every mode of vibration. As a linear relationship for thin mass layers has been derived for TSM vibrations, as outlined above, we have also utilized these modes for mass sensing. For electromagnetic coupling the planar spiral coil exciting radial modes has been most efficient and these modes are therefore used in this application. Face shear motion and flexural motion follow different relations as the kinetic and potential energies are differently distributed, the Sauerbrey equation is only valid for thickness shear motion. The (100) silicon membranes operate in the fundamental thickness shear modes around 15 MHz. At that frequency the penetration depth of the eddy currents is only a few microns. As discussed in the previous chapter, the

4.3. Equivalent Circuit Model

69

Figure 4.1: Diagonal, circular and radial TSM (left to right) for round (100) silicon membrane at ≈ 15 MHz.

optimal copper thickness for our elements is between 1 and 2 microns. Because a rigid mass layer only contributes a frequency shift of the resonance and no damping, we can utilize the radial TSM of our high-Q silicon membranes with no additional copper coating for our most precise measurements. Circular and diagonal modes are equally suited but result in a lower sensor signal due to the reduced coupling efficiency (Fig. 4.1). The radial TSM displays an out-of-plane component at the center of the element due to conservation of mass, which however does not contribute a considerable error to the frequency shift for solid, homogeneous mass loads in vacuum and gaseous environments.

4.3

Equivalent Circuit Model

To account for additional surface mass the basic equivalent circuit has to be extended by an additional lumped mass element. To use a lumped element and not assume a variable mass distribution, the added mass has to be considered as a homogeneous surface layer across the active resonator region, similar to the assumptions for the analytical Sauerbrey equation. The modified equivalent circuit is shown in figure 4.2. The lumped elements equivalent circuit cannot account for randomly distributed mass loads, since the position of the added mass in regards to the resonance mode shape strongly influences the resulting frequency shift. A more detailed finite element approach is required in such situations.

70

4. Mass Microbalance Application

T Ri AC

Zi

T1

Rc

Re

Ct Lc

Le

T3

i uv

k

FL {X, Y} T2

v

h

n

m

m

T4

XT

Excitation circuit

Eddy currents

Acoustic resonator

Mass load

Figure 4.2: Electromagnetic-acoustic equivalent circuit with added ideal mass element.

4.4

Results and Discussion

We have utilized our resonators as mass sensors in the process of electroplating the silicon membranes for application as face shear mode elements, as explained in chapter 3. Besides the frequency shift due to the added mass, the efficiency of mode excitation and corresponding quality factors are also affected by an increase of copper layer thickness and changes to the copper surface. Figure 3.3 shows the changes to a radial face shear mode at ≈ 250 kHz with increasing copper layer thickness. From these measurements we can calculate the added mass, or of more interest, the average added copper layer thickness (Tab. 4.1). To verify the use of silicon resonators as acoustic sensors, several elements have been coated with a monolayer of 300 nm large, spherical polystyrene particles [91, 92]. Polystyrene particles have been used due to rigid, very densely packed deposition as a monolayer, with thus a pre-defined thickness. Table 4.1: Copper layer thickness calculated from frequency shift of radial TSM. Sample (111) Si 296µm quad

(100) Si 185µm quad,

f0 / MHz

f1 / MHz

∆f / kHz

dCu /µm

8.511

8.191

320.528

3.8

8.511

8.044

467.468

5.0

15.890

15.570

319.585

2.0

15.890

15.152

737.617

3.3

71

4.4. Results and Discussion

Normalized Impedance Zi /Ω

5 4

unloaded 300 nm PS

∆f

3 2 1 0 −1 −2

15.410

15.415

15.420

Frequency f / MHz

15.425

Figure 4.3: Shift of unloaded fundamental TSM frequency after coating with a 300 nm polystyrene monolayer.

With appropriate deposition techniques this layer also represents an acoustic load for the sensor corresponding with the Sauerbrey equation 4.5. The frequency shift attained by a QCM sensor with the same coating was -4.65 kHz. Sauerbrey predicts a shift of -4.83 kHz for such a quartz crystal resonator. Deviations from theory are primarily down to not ideal polystyrene layer homogeneity, which cannot be avoided with the current coating methodology. The confidence level of polystyrene layer thickness and deposited mass is 90% for this process, i.e. at least 90% of the surface area is covered with a monolayer of tightly packed polysterene particles. Deviations result from holes, packing mismatch, and particle stacking. For a quadratic (1 0 0) silicon membrane with a thickness of 190 µm the measured frequency shift resulted in -6.6 kHz, whereas theory predicts a change of -6.95 kHz (Fig. 4.3). These measured values lie within 95% of the theoretical values and verify the Sauerbrey like characteristics also for electromagnetic-acoustic resonances in silicon membranes. Damping is minimal, which also is an indicator for a smooth coating of the elements. Silicon resonators are equally well suited for film thickness monitoring as QCM sensors.

72

4.5

4. Mass Microbalance Application

Summary

Sensors based on the resonance principle have found a large variety of applications as mass microbalance. We have demonstrated that electromagnetically excited resonators can also serve as microbalance in a gaseous environment. Here, the most effective mode is the radial TSM. The well described mass sensitivity relations can be applied in a similar way. The measurement yields information for example on the thickness of a deposited layer, a metal or a chemically sensitive film, the latter pointing at the mass increase due to selective adsorption of molecules when the resonator is working as a chemical sensor. As the primary focus of this research has been liquid phase applications, we did not pursue this idea much further. However, since the microbalance application is very attractive in liquid environment as well, vibration modes without out-of-plane vibration components are favorable. We will utilize face shear modes for the determination of liquid material properties which exist at lower frequencies. In terms of a multi-mode device, it is simply a matter of increasing the operating frequency with no changes to the measurement setup to excite the thickness shear mode and to achieve a sufficient mass sensitivity.

5 Liquid Phase Density-Viscosity Sensing The remote, non-contact nature of excitation and detection truly begins to shine with measurements in liquids. Other transducer methods always introduce design challenges, e.g. electric contacts and insulation, capacitive coupling, and leakage. With electromagnetic-acoustic transducers, the sensor element can be simply placed inside the measurement chamber, while the electric circuitry is spatially separated on the outside. Therefore, the main focus of this thesis has been the application of these resonator devices for liquid phase sensing. Since acoustic sensors such as quartz crystal resonators have proven to be sensitive to density and viscosity, our primary interest has been to validate the suitability of our devices for this application as well. In-plane shear motion of a surface loaded with a liquid layer results in a frequency shift and damping of the mechanical resonance proportional to the square root of density and viscosity. Out-of-plane vibration must be avoided. Both thickness shear modes (TSM) and face shear modes (FSM) of our aluminum and silicon resonator elements can be suitable for these measurements, however the strong viscous damping of acoustic shear waves in liquids has to be taken into account. Therefore, the low transduction efficiency of electromagnetic resonators needs to be carefully addressed in terms of resonator design, vibration mode, and experimental setup.

5.1

Analytical Description

In a Newtonian liquid for 1D shear excitation in the x-y plane, the shear velocity gradient ∂vx /∂z and the shear stress Txz are related by a constant parameter, the shear viscosity η. Txz = −η

∂vx ∂z

(5.1)

74

5. Liquid Phase Density-Viscosity Sensing

Viscous Liquid

ρ, η

z

Penetration Depth

Viscous Liquid

δ

ρ, η

x

Resonator ρ R, µR

z

Penetration Depth

δ

x

Resonator ρR , cR

Displacement Profile ux(z)

Displacement Profile ux (x,y)

Figure 5.1: Displacement profiles in a viscous liquid for a diagonal thickness shear mode (left) and a diagonal face shear mode (right).

Therefore, to measure viscosity a shearing surface an evanescent shear wave in the liquid can be used. In an incompressible liquid, the resulting velocity field vx can be calculated by solving the Navier-Stokes equation for onedimensional flow: ∂vx ∂ 2 vx . (5.2) η 2 =ρ ∂z ∂t With a periodic driving force at the solid-liquid interface we can solve this wave equation as: z z (5.3) vx (z, t) = vx0 cos − ωt exp − , δ δ with the penetration depth as:

δ=

r

2η . ωρ

(5.4)

The complex shear stress acting on the surface to generate the shear wave is thus given by: ηvx0 ∂vx = Txz = −η (1 + j) . (5.5) ∂z z=0 δ

The resulting displacement profiles of diagonal shear motion is shown in figure 5.1. Face shear and thickness shear modes differ in terms of the energy stored in the resonator, but evanescent shear waves in the liquid follow the same equations for a similar surface motion.

5.2. Suitable Modes of Vibration

75

Stored and dissipated energy can be described by a complex acoustic impedance Za as the ratio of shear stress and velocity at the boundary of a semi-infinite viscous liquid. r ωρη Txz (1 + j) (5.6) Za = − = vx z=0 2

Since the impedance is related to the additional kinetic and potential energy, √ these are also related to the square root of density and viscosity, ρη. As shown in the previous chapter for a simple mass loading, the energy balance is directly related to the resulting frequency changes. Consequently, the frequency shift and damping due to a viscous load will be a function of this density-viscosity product as well. √ ∆f ∝ ρη (5.7) √ (5.8) ∆Z ∝ ρη

From energy balance or using an equivalent circuit method, an approximation of the frequency shift of thickness shear modes is possible, yielding the equation of Kanazawa and Gordon [28, 93]: s fR3 ρη 1 ∼ . (5.9) ∆f = − n πρR µR Here, the frequency shift ∆f is a function of the thickness shear mode frequency fR divided by the mass density and shear stiffness of the resonator, ρR and µR respectively, and the density-viscosity product ρη acting as the acoustic load, operated at the n-th harmonic. For face shear modes no simple analytical relationship can be derived, especially due to the dependence on the specific two-dimensional shape of vibration. The general relation to the density-viscosity product stays applicable, but the proportionality factors have to be evaluated for each specific resonator element and mode shape.

5.2

Suitable Modes of Vibration

For pure liquid property sensing as described above, we require a vibration mode with ideally only shear vibration at the surface. Any out-of-plane

76

5. Liquid Phase Density-Viscosity Sensing

component radiates compressional waves and results in secondary frequency shifts and damping due to liquid level, compressibility and other parameters. In case of a semi-infinite liquid these compressional waves just decrease the acoustic energy in the resonator, thereby increasing the insertion loss. In case of finite dimensions of the liquid the situation becomes more involved. Since damping of compressional waves is quite small in many pure liquids and hence the penetration depth much larger than geometric dimensions, the liquid must be understood as liquid cavity with acoustic properties dependent on frequency, liquid level and material properties like compressibility. This cross sensitivity may result in secondary frequency shifts or additional damping. Here we focus on suitable face shear eigenmodes for the circular aluminum resonator with a central mesa structure and the modified silicon membranes. The optimal modes of vibration for these elements are shown in figure 5.2. The circular FSM of the aluminum plate shows no out-of-plane vibration both in the eigenmode and in the excitation analysis. The acoustic energy is well trapped in the central mesa structure and this is ideal for sensing the liquid properties with shear waves. There is primarily only viscous damping of this resonance, additional damping due to dissipated energy is minimal. The total volume and mass of the liquid on the resonator has no significant influence on the sensor response, and a direct relationship between the density-viscosity product and the frequency shift of the resonance can be es-

Figure 5.2: Optimal modes of vibration in viscous liquids: Circular FSM of aluminum resonator (left) and diagonal FSM of Si membrane.

5.3. Equivalent Circuit Model

77

tablished. Circular FSM or torsional plate resonators are thus suitable sensor devices for measuring liquid properties and distinguishing between different fluids. Radial and diagonal modes of the plate resonator exhibit considerable out-of-plane motion. In our experiments with the modified silicon transducers we have focused on circular and round membranes in (100) and (111) silicon. Here, both the circular and diagonal modes are primarily shear motion with negligible out-ofplane components, while the significant out-of-plane vibration of the radial modes results in additional damping due to compressional waves radiated into the liquid. Due to lower electro-mechanical transduction efficiency of thickness shear modes, the diagonal FSM has been primarily used in our experiments. Compared to the excitation of a circular mode, the eddy current coupling is considerably stronger for a diagonal mode, because the primary coil is placed directly centered to the resonator and not out-of-center, see chapter 3. Theoretically, thickness shear modes are equally suited for liquid property sensing. For our method and devices however, the coupling efficiency is too low to overcome the strong damping of high frequency evanescent shear waves. With our current experimental setups, we are not able to measure frequency shift and damping of TSM resonances in a truly reproducible fashion. The measured resonance peak strengths and the signal to noise ratio are simply too small for exact calculation of frequency shift and impedance amplitude. Therefore our experiments focus on the face shear modes described above.

5.3

Equivalent Circuit Model

Acoustic loading by a viscous liquid can be modeled by adding additional mass ml and damping hl elements to the equivalent mechanical circuit (Fig. 5.3). For more complex liquids, this can be replaced by a general complex impedance, e.g. to also account for elastic energy storage in the liquid. The additional mass and damping elements are both functions of the densityviscosity product ρη, as well as dependent on the geometry of the mode

78

5. Liquid Phase Density-Viscosity Sensing

T Ri AC

Zi

T1

Rc

Re

Ct Lc

Le

T3

i uv

k

{X, Y} T2

FL v

h

n

m

hl

ml

T4

XT

Excitation circuit

Eddy currents

Acoustic resonator

Viscous Load

Figure 5.3: Electromagnetic-acoustic equivalent circuit with added ideal mass element.

shape. These parameters have to be fitted for different resonator devices, and the density and viscosity can then be extracted from the fitting function. Furthermore, this is again a one-dimensional model. Therefore, the different mode shapes of circular or diagonal shear vibrations also has to be taken into account. Out-of-plane components of the vibration must be negligible to be able to use this simple model for comparison with measurement data. The next chapter 6 explains the additional elements necessary to account for compressional waves radiated into the liquid. That extension of the model can also be used to include the effect of small flexural motion on the measurement of liquid properties. Ideally, multiple sensor elements are used in different vibration modes to independently measure, e.g., density, viscosity and liquid level, and with further data processing we can then eliminate the parasitic influences on the individual measurement parameters. Alternatively, for a similar densities and equal measurement volumina, the flexural components result in a constant offset of the frequency shift and damping, which can thus be easily subtracted.

5.4 5.4.1

Results and Discussion Metal Plate Elements

The metal mesa plate elements have been our first devices for experimental validation of sensing in the liquid phase, due to their higher electro-

79

5.4.1. Metal Plate Elements Table 5.1: Frequency shift and damping behavior of circular FSM at 142 kHz in aluminum due to liquid loading. ρ / kg/m3

η / mPas

Zi / mΩ

∆f / Hz

Ethanol

775

1.12

122.07

−31.3

Propanol

785

1.51

112.21

−62.5

Glycerine 1:10

815

3.65

88.55

−77.5

Glycerine 1:5

930

10.92

52.37

−136.3

Liquid

mechanical coupling factors. The stronger coupling is required to overcome the high damping in viscous liquids. For these measurements, the mesa plates were combined with plastic tubular cylinders to form a liquid measurement cell with the resonators forming the bottom. Depending on the mode of vibration, the plastic cylinders were attached on the outer edge of the mesa groove or the outer edge of the metal plate. E.g., for the circular FSM the resonator motion is minimal at the outer edge of the mesa groove, but larger at the outer edge of the plate, therefore the plastic cylinders were attached to the region of minimal vibration to minimize the interference and damping of the desired circular mode shape. The height of the measurement liquids has been kept constant to minimize interference due to spurious out-of-plane vibration components. For the metal plate elements, the circular and diagonal face shear mode shape have given the best results [94, 95]. In both cases, the vibration is effectively trapped in the mesa structure of the resonator element. Table 5.1 and figure 5.4 show the measurement results of the circular face shear mode for different liquids, in terms of frequency shift ∆f and damping. The normalized impedance Zi (eq. 3.1) is given as measure of damping and changes to the Q-factor of the resonance. Similar results have been obtained for the diagonal FSM, where the excitation and detection efficiency is not as high (Table 5.2). The results show a nearly linear relationship between √ frequency shift ∆f and square root of density-viscosity ρη, however the measurement error remains too large to serve as proof of this dependency. Both modes are nevertheless well suited to distinguish between different fluids

80

Normalized Impedance Zi /Ω

5. Liquid Phase Density-Viscosity Sensing

0.14 0.12 0.1

in air Ethanol Propanol Glycerine 1:10 Glycerine 1:5

increasing viscosity

0.08 0.06 0.04 0.02 0 −0.02

141.65

141.70

141.75

141.80

141.85

Frequency f / kHz

141.90

141.95

Figure 5.4: Normalized impedance spectrum of primary coil for liquid loading of circular FSM at 142 kHz in aluminum resonator yielding frequency shifts and damping as summarized in Table 5.1.

and to determine the density-viscosity product. The influence of sample mass and volume is negligible and parasitic effects influencing the sensor response have not been observed. The metal plates work very well, are easy to integrate in a measurement cell and easy to clean and replace. They are best suited for larger liquid volumina and lower frequency vibration modes. The exact relationship between density-viscosity product and frequency shift and damping has to be experimentally derived for each specific setup, as the vibration mode shape strongly influences the proportionality factor for frequency shift and damping. The boundary conditions of the specific setup (e.g. clamped, simply supported or Table 5.2: Frequency shift and damping behavior of diagonal FSM at 217 kHz in aluminum due to liquid loading. ρ / kg/m3

η / mPas

Zi / mΩ

∆f / Hz

Ethanol

775

1.12

13.47

−47.5

Propanol

785

1.51

12.14

−57.0

Glycerine 1:10

815

3.65

13.18

−100.0

Glycerine 1:5

930

10.92

9.32

−152.5

Liquid

81

60 40 20 0 −20 −40

in air

1-Propanol

0.6 0.5 0.4 0.3 0.2 0.1 0

Phase shift Θ/◦

Normalized Impedance Zi /mΩ

5.4.2. Silicon Membrane Resonators

221.0 221.2 221.4 221.6 221.8 222.0 222.2 222.4 222.6

Frequency f / kHz

Figure 5.5: Measurement results for diagonal FSM of (111) silicon membrane for different alcohols in terms of density-viscosity product vs. frequency shift and normalized impedance.

free edges) also influence the mode shape considerably, therefore a specific calibration is necessary for each specific resonator elements and measurement cell.

5.4.2

Silicon Membrane Resonators

After modifying multiple silicon elements with increased copper layer thickness as described in chapter 3, they became suitable for face shear mode excitation and thus liquid phase sensing. Excitation was achieved with a quadratic planar spiral coil (15x15 mm) and two alternated NdFeB permanent magnets (measured flux density of 0.4 T), resulting in linear in-plane forces (see Fig. 3.14). The resonance experiences a distinct and reproducible frequency shift and damping characteristics when put in contact with a fluid. We have achieved reproducible results for the investigation of small volumes of different alcohols. The measurement cell was a quadratic, 20x20 mm, plastic cylinder (PMMA) with a thin plastic foil at the bottom to minimize distance to the excitation coil, into which the silicon resonator element was placed. Excitation and detection was performed remotely on the outside of the liquid reservoir, i.e. there was no direct contact between the electric

82

5. Liquid Phase Density-Viscosity Sensing

Table 5.3: Influence of density and viscosity on resonant response of diagonal FSM of (111) Si membrane. ρ / kg/m3

η / mPas

Zi / mΩ

Θ / m◦

∆f / Hz

1-Propanol

803.5

1.898

13.2

94.20

−262.5

1-Butanol

809.8

2.524

12.2

87.93

−337.5

1-Pentanol

814.4

3.441

10.4

77.54

−350.0

1-Heptanol

821.9

5.942

8.5

61.56

−425.0

1-Octanol

827.0

7.368

7.8

56.56

−450.0

1-Decanol

829.7

10.974

6.9

47.36

−562.5

Liquid

parts and the analyte. This again demonstrates a strong advantage of electromagnetic excitation, where difficulties of electrical insulation, crosstalk, parasitic capacitances, etc. are circumvented. Flexural components of the vibration result in an offset corresponding to the liquid volume, which can be considered constant for the same volume of the different test liquids, though. For our experimental results shown here, we have utilized the first diagonal face shear mode at 222 kHz of a 300 µm thick (111) Si membrane, where the electromechanical coupling efficiency was strongest and only a minimal amount of flexural movement could perturb the measurements [84]. In figure 5.5 we have plotted the normalized impedance and phase curves for a single measurement. We have investigated six different primary alcohols from 1Propanol to 1-Decanol. Table 5.3 lists the density and viscosity of these alcohols at a temperature of 25◦ C, as well as the resulting frequency shift compared to the resonance in air and the normalized impedance magnitude and phase peaks. The liquid volume was kept constant at 100 µl put inside the custom designed liquid reservoir. The silicon membrane and the reservoir were emptied, cleaned with isopropanol, and dried after each measurement to remove all traces of the previous liquid. The measurement time was kept the same for all liquids to minimize effects due to evaporation. Figure 5.6 visualizes the resulting frequency shift and additional viscous damping for increasing dynamic viscosity and density. As the characteristic acoustic impedance of a shear wave in a viscous liquid depends on the density-

83

5.5. Summary

700

Impedance damping ∆Z/mΩ

650 600

12

550 500

10

450 400

8

350

Frequency shift ∆f / Hz

Impedance Damping Frequency Shift

14

300 6

250 1

1.5

Density-viscosity

2

√

2.5

ρη / Pa s

3/2

3

m

200

−1

Figure 5.6: Measurement results for diagonal FSM of (111) silicon membrane for different alcohols in terms of density-viscosity product vs. frequency shift and impedance damping.

visosity product ρη, it is a reasonable assumption that the shown parameters are also functions of ρη. A linear correlation of the frequency shift can be fitted to the square root of density and viscosity, as suggested by theory. The Q-factor, or the damping, is characterized in terms of the normalized impedance peak ∆Z in the frequency response of Zi , which also shows a nearly linear relationship.

5.5

Summary

By utilizing a different set of vibration modes compared to mass sensing, namely pure face shear vibrations, we have demonstrated the use of these sensor devices for liquid phase measurements. Different technologies for the resonator element have been considered (metallic disks, polymer foils, micromachined membranes) and found suitable for this application at different operating frequencies and mode shapes. Similar to other acoustic resonators,

84

5. Liquid Phase Density-Viscosity Sensing

the electromagnetic-acoustic transducer is in principle a viable sensor for measuring liquid properties as density and viscosity. These transducers can take full advantage of the remote nature of excitation and detection, as well as the inexpensive fabrication. The resonator element can be fully immersed into the liquid analyte, while the more expensive excitation and detection setup is placed on the outside. At the same time, the detection efficiency is considerably lower than that of conventional wired piezoelectric transducers. Therefore, dedicated sensor electronics are required. Using single crystalline silicon elements as electromagnetic-acoustic resonators yields the advantage of higher Q-factors than other suitable materials. However, for efficient excitation the conductive layer of the substrate needs to be adjusted to the specific frequency range of operation. We have demonstrated that additional electroplating on our silicon membranes improves the coupling efficiency for face shear modes considerably, while resulting in a negligible reduction of Q. These modified elements are therefore also suitable for liquid phase sensing at higher sensitivity and accuracy than polymer foils or metal plates. As proof of concept we validated the theoretical relationship of resonance frequency and impedance amplitude to the density-viscosity product of different alcohols.

6 Liquid Level Measurement Liquid level measurement is a key aspect for many applications in the processing industry, in analytical chemistry, and the pharmaceutical industry to determine, e.g., the exact liquid volume in a reaction chamber. A large variety of measurement principles exist, ranging from mechanical, electrical, and optical methods to ultrasonic and radar measurements [96, 97, 98]. However, many of these principles are not suitable for sensing very small amounts or changes of liquid volume and liquid level. Ultrasonic transducers allow for very precise liquid level measurements, where the sensitivity can be improved by increasing the operating frequency. If the liquid level is a known property, we can in turn use the measurement results to evaluate acoustic wave velocity and acoustic impedance in the liquid [18]. Ultrasonic liquid level sensor devices are primarily based on piezoelectric or capacitive excitation methods. In general, the geometric electrode configuration limits each transducer to a single mode of vibration and therefore limits the sensitivity of the sensor element. In this chapter we introduce the application of our electromagnetic-acoustic resonators as liquid level sensors. Here, out-of-plane flexural vibrations are generated to radiate compressional, longitudinal waves into the liquid. In contrast to acoustic shear waves, these display a very large penetration depth and are highly sensitive to the liquid level. We measure the frequency shifts and changed damping characteristics due to interference effects instead of the time-of-flight measurements usually employed by ultrasonic liquid level sensors. Our experiments show, that by using continuous, resonant excitation the achievable resolution is not limited by the wavelength of the ultrasonic waves radiated into the liquid. We again employ an experimental setup consisting of a simple resonating element placed at the bottom of a fluidic chamber, which is remotely excited by a planar coil. We have found that while out-of-plane motion is an unwanted component for mass and viscosity sensors, with specific flexural modes we can utilize the

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same experimental setup as liquid level sensors as well. This presents us with a unique and novel advantage, the possibility to measure multiple properties with the same, single sensor element. With sensor arrays instead of single elements, we can also measure these properties simultaneously by exciting different vibration modes in each element.

6.1

Suitable Modes of Vibration

As described in chapter 2, depending on setup of primary coil, magnets, and resonator, a multitude of Lorentz force distributions can be realized. Flexural, out-of-plane motion generates longitudinal, compressional waves which are radiated into a liquid. Compared to ultrasonic shear waves, the much larger penetration depth in viscous liquids causes the compressional wave to be reflected at boundaries of the liquid to return and interfere with the resonator motion. We employ continuous excitation generating standing acoustic waves and taking advantage of this interference behavior. As the generated mode of vibration is depending on external parameters such as the alignment of coil and magnets, and the driving frequency, the actual sensor element does not have to be exchanged for sensing different properties. The frequency f of the eigenmodes of the resonator is determined by the material properties, elasticity coefficients and density, and the geometry of the element. The ultrasonic velocity c of compressional waves in the liquid analyte is determined by the square root of compressibility κ and density p ρ: c = 1/(κρ) and we can calculate the wavelength λ of the longitudinal wave with λ = c/f . Due to interference effects of longitudinal waves reflected from the liquid surface, it can be expected that the resonance behavior cycles through a repetitive pattern whenever the level increases by λ/2. Within this cycle constructive and destructive interference are alternating. The minimum and maximum interference correspond to a level difference of λ/4. Due to this repetitive pattern, a single measurement is ambiguous in terms of absolute height, however, depending on mode shape, frequency shift and amplitude damping can be evaluated as two distinct parameters. The most efficiently excited flexural vibration eigenmode with our ex-

6.2. Equivalent Circuit Model

87

Figure 6.1: Experimental setup for liquid level measurements with resonator element integrated into liquid reservoir and excited by planar coil and external magnet.

perimental setup is a radial flexural face shear mode. This mode features radial in-plane vibration with a strong flexural component at the center of the resonator, due to conservation of mass. Therefore, a planar spiral coil and perpendicular magnetic field is the ideal excitation setup and the experiments also show that the radial flexure mode is one of the best coupled modes of vibration. For the aluminum mesa plates the radial flexure mode has an eigenfrequency of 289 kHz (Fig. 6.1).

6.2

Equivalent Circuit Model

To model the behavior of standing compressional waves radiated into a liquid and reflected back to the resonator surface we have amended our equivalent circuit from chapter 2. The liquid level sensitivity and interference behavior is given by a lossy transmission line, where the characteristic wave impedance Za corresponds to the acoustic impedance of the liquid and line length l corresponds to the liquid level [99]. The surface boundary condition for the acoustic wave at the liquid/air boundary is included in the load resistance Rr . Due to the complexity of the vibration mode shape, we also take into account the viscous shear load as shown in chapter 5, for clarity these

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6. Liquid Level Measurement

T Ri AC

Zi

T1

Rc

Re

Ct Lc

Le k

T3

i uv

Za

FL {X, Y} T2

v

h

n

T4

l

XT

Excitation circuit

Eddy currents

Rr(l)

m

Acoustic resonator

Acoustic wave

Figure 6.2: Electromagnetic-acoustic equivalent circuit, with eddy current induction described by electromagnetic transformer T , coupling between current i, voltage u and force F , velocity v in the resonator modeled by electro-mechanical transformer XT , and the standing longitudinal wave in a liquid above the resonator represented by a lossy transmission line with a wave impedance Za .

elements have been omitted from figure 6.2, but included in the simulation. Strictly speaking, the mechanical shear and normal velocities are different variables depending on the 3D mode shape, however, for a particular dominant mode we can assume that they are linked by an approximate constant factor and thus both can be represented by one lumped variable in the same 1D equivalent circuit. The transmission line represents a one-dimensional approximation for the interference of compressional waves and resonator movement. Effects not included in this simple demonstration of the transmission line modeling are multiple transmission paths depending on the three-dimensional mode shape of the resonator vibration. Other effects include mode conversion at the liquid surface boundary or resonator surface. These effects are partially approximated by introducing a dependence of the reflection coefficients on the length l, i.e. Rr (l). The simulation procedure consists of calculating resonator mass and compliance from density, elasticity and geometry and fitting the damping parameter to measurement data in air. Then we calculate the acoustic impedance and wave velocity of the liquid from density, compressibility and geometry and enter these as the transmission line parameters, including the liquid level as its length. Finally we fit the reflection and coupling coefficients to measurement data at this liquid level.

89

6.3. Results and Discussion

Normalized Impedance Zi /Ω

0.05 0µl

0.04 increasing fluid volume

0.03

1500µl 1400µl

0.02 0.01

900µl

1100µl 1200µl

800µl

1000µl

1300µl

0 265

270

275

280

Frequency f / kHz

285

Figure 6.3: Preliminary results during density-viscosity experiments showing a strong dependence on volume in liquid measurement cell.

6.3 6.3.1

Results and Discussion Preliminary Results

While analyzing the efficiency of different modes of vibration for densityviscosity measurements, we experienced much larger frequency shifts than anticipated for viscous loads for specific vibration modes. Further investigation revealed these frequency shifts to be dependent on the liquid volume inserted into our measurement cells. Figure 6.3 shows such combined measurement results. Upon detailed analysis of the employed mode shape, we concluded the dominance of radiated compressional waves as described above for this vibration [100].

6.3.2

Measurement Setup

To utilize an electromagnetic-acoustic resonating element as a liquid sensor, we can simply place the element inside a fluidic chamber and remotely excite it from the outside as outlined above. The resonator can also be integrated into or already be part of the chamber. The only requirement is that the resonating element must be conductive and that the eddy current induction is

90

6. Liquid Level Measurement

Figure 6.4: Experimental setup for liquid level measurements with resonator element integrated into liquid reservoir and excited by planar coil and external magnet.

not shielded by the chamber walls. Figure 6.4 shows the measurement principle with a resonator placed inside the liquid chamber. The photo depicts a measurement setup, where the liquid chamber is part of the plate resonator and placed above primary induction coil (0.12–0.19 mm diameter, Roadrunner Electronics Ltd., UK) and permanent NdFeB magnet with a measured static magnetic flux density at the sensor element of 0.375 T. The liquid volume has been inserted with microliter pipettes (Eppendorf epResearch3 Pack ) in steps of 25±0.25 µl or 50±0.35 µl. The fluidic chamber for the aluminum resonator was a round plastic cylinder with a diameter of 20.5 mm. All geometrical parameters were measured with a micrometer gauge. This results in a liquid level increase of about 76 µm per 25 µl added water volume. Uncertainties in liquid level are derived from uncertainties in added liquid volume and geometrical parameters of the fluid chamber, which has to be taken into account for calibration of the device. According to the calculations from above, the resonance pattern should cycle with every 900 µl added volume, corresponding to a liquid level change of λ/2. For precise measurements a temperature stable setup is required, primarily due to the temperature coefficients of density and ultrasonic velocity. The experiments shown here have been carried out at a temperature of 25±0.5◦ C.

91

6.3.3. Measurement and Simulation Comparison

0.06

Normalized Impedance Zi /Ω

Normalized Impedance Zi /Ω

0.06 0.05 0.04 0.03 0.02 0.01 0 −0.01

271

272

273

274

275

Frequency f / kHz

276

277

0.05 0.04 0.03 0.02 0.01 0 −0.01

271

272

273

274

275

Frequency f / kHz

276

277

Figure 6.5: Impedance measurement (left) and simulation (right) for flexural resonance at 275 kHz with liquid volume at maximum around 900 µl, corresponding to a height at maximum of 2.75 mm = λ/2. Each curve corresponds to a change of 25 µl water or 76 µm height, respectively.

In standard literature, a volumetric temperature expansion coefficient of DIwater of 0.256 ml · l−1 K−1 can be found, leading to a height deviation of ±0.384 µm/ml, or 0.013 % for this setup. On the other hand, from literature values we can estimate a temperature coefficient for the ultrasonic velocity of 2.67 ms−1 K−1 around 25◦ C, leading to an error of ±1.34 ms−1 , or 0.089 %, in the velocity. These primary effects partially compensate each other, leading to a systematic frequency error of ±209 Hz, or 0.076 % at our compressional mode operating frequency of 275 kHz. To further minimize the influence of the temperature dependence, we have also been developing a thermostat based, temperature stabilized experimental setup operated at ±0.05 K for simultaneous measurements with different acoustic resonator sensors.

6.3.3

Measurement and Simulation Comparison

As our sensor elements we utilize aluminum plate resonators and silicon membranes with copper coating as the conductive layer. For liquid level measurement we excited the resonators at a radially symmetric flexural mode of vibration at 275 kHz. This mode of vibration is efficiently excited with a planar spiral coil and a perpendicular magnetic field in the normal direction, resulting in radial Lorentz forces. Figure 6.5 shows the comparison between

92

6. Liquid Level Measurement

0.12

Normalized Impedance Zi /Ω

Normalized Impedance Zi /Ω

0.12 0.1 0.08 0.06 0.04 0.02 0 −0.02

270

272

274

276

Frequency f / kHz

278

0.1 0.08 0.06 0.04 0.02 0 −0.02

270

272

274

276

Frequency f / kHz

278

Figure 6.6: Impedance measurement (left) and simulation (right) for flexural resonance at 275 kHz with liquid volume at maximum around 2700 µl, corresponding to a height at maximum of 8.25 mm = 3/2λ. Here, each curve corresponds to a change of 50 µl water or 154 µm height.

measured and simulated impedance curves for an aluminum resonator and a liquid loading around 900 µl, with 25 µl difference between the different spectra. For both, measurement and equivalent circuit simulation a maximum impedance is achieved for a liquid level of λ/2, or 2.75 mm. The frequency shifts for each 25 µl step are around 150 Hz. At maximum impedance the resonance curves achieve Q-factors of about 300-400. At these measurement parameters and with a temperature deviation of 209 Hz, equipment accuracy of 5 Hz, and measurement points every 12.5 Hz (depending on frequency span), we can calculate a liquid level resolution of 6.3 µm at an accuracy of about 100 µm. With increasing liquid level, distortion due to secondary propagation paths is reduced. Only the normal components of the compressional wave pattern are reflected to the resonator surface, other components are increasingly dampened by multiple reflections and longer propagation paths. The resulting frequency spectrum is shown in figure 6.6 for the impedance amplitude and figure 6.7 for the impedance phase. Here, the maximum of the envelope is around 2700 µl, corresponding to a height of 3λ/2 = 8.25 mm. The correlation between measurement and simulation is improved compared to lower liquid levels, most likely due to the decrease of the influence of multiple,

93

6.3.3. Measurement and Simulation Comparison

Normalized Phase Θ/◦

Normalized Phase Θ/◦

0.2 0.1 0 −0.1 −0.2

268

270

272

274

276

Frequency f / kHz

278

280

0.1 0 −0.1 −0.2

268

270

272

274

276

Frequency f / kHz

278

280

Figure 6.7: Comparison of phase measurement (left) and simulation (right) corresponding to figure 6.6.

reflected propagation paths. The frequency shifts increase to 750 Hz for a 50 µl step at envelope maximum, and the resonance curve shows a Q-factor up to 1000. With a frequency step size of 15 Hz, the liquid level resolution therefore improves to 3 µm, at an accuracy of about 40 µm, the latter dominated by temperature effects. This corresponds to a liquid volume resolution of 1000 nl for our experimental setup. The measurements in figures 6.5 and 6.6 show a cyclic interference behavior as predicted. There are however some noticeable differences in the envelope around each interference cycle, which are primarily the result of the 1D-approximation of the three-dimensional radiation pattern of the compressional wave. Mainly the width of the envelope and the frequency shifting between different liquid levels cannot be exactly reproduced. However, the general behavior is successfully modeled. The equivalent circuit simulation can be used to predict the cyclic interference pattern accurately and to model the primary acoustic wave behavior. Silicon elements have been successfully excited to a similar mode of vibration and have shown the same cyclic interference behavior. However, the coupling coefficients for flexural and face shear modes are weaker than those obtained with aluminum resonators. A modified design and experimental setup is required to fully utilize the better elastic properties and thus the larger intrinsic mechanical Q-factor of these devices. Other modes of flexural

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6. Liquid Level Measurement

vibration have also been investigated for both silicon and aluminum resonators. The resonant vibration of face shear modes in our plate resonators is much stronger though, purely flexural modes induce only a weak secondary voltage. Finally, we would like to note another possible application of the sensor. As described by Hauptmann et al. for different piezoelectric transducers [18], if the liquid level is a known property, we can in turn use the measurement results to evaluate acoustic wave velocity and acoustic impedance in the liquid. Acoustic velocity and impedance primarily depend on the product and fraction of density and compressibility, respectively. From our liquid level resolution we can estimate an acoustic velocity resolution down to 0.4 m/s, or 270 ppm. From measurements of the in-plane circular face shear mode we can extract the density-viscosity product. Therefore, by analyzing frequency shift and damping of two vibration modes in the same resonator element, we can successfully establish an accurate description of the liquid analyte in terms of three fundamental physical properties without requiring multiple sensor elements, literature values, or complex algorithms.

6.4

Summary

In this chapter we have presented a novel approach to ultrasonic liquid level measurements utilizing electromagnetic-acoustic resonator sensors. Instead of pulsed ultrasonic waves, where the resolution is limited by the wavelength, we continuously excite a standing compressional wave in the liquid and measure frequency shifts and damping due to interference effects of the reflected wave at the resonator surface. With this method we have achieved a liquid level resolution of 3 µm at a height of 10 mm. Our experiments show the suitability of this method to measure and control very small amounts of liquids. To further improve level resolution and accuracy, a better temperature stability of the measurement setup and specialized sensor electronics for smaller frequency steps around resonance are required. Our equivalent circuit model from chapter 2 has been refined with a onedimensional approximation of the compressional wave radiated into the liquid

6.4. Summary

95

to predict the cyclic interference pattern seen in the measurements. A lossy transmission line with a length corresponding to the liquid level sufficiently models this behavior. The elements of the complete equivalent circuit are based on the material properties of excitation coil, resonator element, and liquid analyte, while the coupling coefficients and acoustic damping elements were used as fit parameters. We have successfully demonstrated a suitable correlation of experimental and simulated results.

7 Conclusions Electromagnetic-acoustic resonators have been established as the third major family of resonant acoustic sensors. In the work presented in this thesis, we have shown how to take advantage of the unique features offered by this method of excitation. Of major interest, especially for liquid phase sensing, is the ability to spatially separate resonator element from excitation and readout electronics. We have demonstrated how this remote excitation can be exploited and shown that it greatly reduces the packaging complexity and problematics, e.g. by removing problems with electric insulation, parasitic elements, or other undesired effects. Resonator elements can be simply placed inside measurement cells or even be part of them, no further is packaging required. Since electromagnetic-acoustic excitation is also possible for many kinds of resonator materials, including low cost metal plates or plastic foils, this opens the door to treating the vibrating element as an easily replaceable part of the measurement setup. Experimenting with complex, aggressive, and polluted analytes becomes more feasible, as rigorous cleaning and preparation methods are no longer necessary. Furthermore, for measurements with higher precision, which is basically limited by the elastic quality and intrinsic loss of the resonator, we have demonstrated the use of single crystalline silicon membranes, which exhibit better elastic properties than even quartz, and are easier and cheaper to fabricate and modify. Another advantage that ties into the spatial separation is the ability to work with a bare resonator surface, which is not perturbed by metalization or passivation layers. This adds a great deal of freedom for the possible surface chemistry, especially on inert materials like silicon. However, from our point of view, the major and most unique advantage lies with the ability to excite a wide variety of fundamentally different vibration modes in the same resonator element. Due to the different physical mechanisms at work and the spatial separation of the excitation sources, i.e. resonator, coils and magnets, we have several degrees of freedom that

98

7. Conclusions

influence the distribution of currents, forces and displacement. The geometry of the coil and its alignment to the conductive surface define the eddy current distribution, which, together with the direction of the external magnetic fields, defines the distribution of Lorentz forces. The forces then again have to match the desired vibration mode shape both in terms of direction and driving frequency. Changing these degrees of freedom determines which vibration mode is suppressed, and which is successfully excited. With this flexibility, the same vibrating element and the same excitation setup can be used for fundamentally different vibrations, e.g. pure shear movement or dominant flexural motion. Since the surface motion is dependent on different physical properties, e.g. density, viscosity, level, and compressibility in this case, one and the same sensor is suitable to measure multiple physical properties. That is very much unlike established ultrasonic devices, where for multiple measurands, multiple sensors with different designs are generally required. As proof of concept, we have demonstrated the suitability of our resonator designs to act as mass-microbalance, density-viscosity sensors, and liquid level measurement devices. The designs and experimental setups were based on theoretical and finite element analysis of suitable eigenmodes and the requisite Lorentz force distribution. For prediction and verification of experimental results, we have developed an equivalent circuit model based on the physical properties of the electrical and mechanical components and the acoustic loads. The circuit is based on electro-mechanical analogies and can thus be easily analyzed and expanded with the powerful network theory tools of electrical engineering. Our colleagues and other groups have also demonstrated different approaches to electromagnetic-acoustic resonators, e.g. by impressing a current into cantilevers, suspended bridges and membranes [23, 63, 101, 102, 103]. While losing the non-contact advantage and some degrees of freedom, these methods can display improved coupling efficiency and also be operated in non-resonant modes.

7.1. Perspectives

7.1

99

Perspectives

Future work on this topic should focus on establishing calibration curves for the different resonators and modes of operation. Establishing analytical relationships between surface mass or density-viscosity and resonance frequency shift similar to Sauerbrey’s and Kanazawa’s equation for quartz crystal resonators requires the analytical description of the complete sensing mechanism. Since we recovered the known linear relation of frequency shift to added mass and the square root of the liquid density-viscosity product, a constant factor of proportionality, or transduction factor can be expected. The same applies to liquid level sensing. It also relies on changes in the acoustic load, here due to constructive and destructive interference of compressional acoustic waves in the liquid cavity. Consequently, the acoustic load is frequency dependent, just in contrast to density-viscosity measurements where the frequency dependence of the acoustic load is rather weak. Although frequency shift measurement is the prevailing measurement technique, the impedance maximum carries independent information of the system under investigation, first of all information about acoustic energy dissipation. Besides material related absorption this parameter also has the capability to be used for estimation of other attenuation mechanisms, e.g., amount of acoustic wave radiation or acoustic wave scattering. Again, an analytical relation between acoustic properties and its electrical representation is required. This part is more challenging, since frequency measurement does not rely on the height of the resonance peak. The analytical deviation of the transduction factor or transduction function requires respective relations dealing with mutual inductance of the planar coil and its counterpart formed in the resonator as well as analytical approximations of the mechanical displacement (pattern) of the resonator. On the other hand, the liquid density-viscosity product also shows the same square root dependence with regard to change in the coil impedance maximum which coincides with the behavior of the acoustic load. With knowledge of the transduction parameters, analytically derived from calibration curves, absolute values of mass, density, viscosity, or compressibil-

100

7. Conclusions

ity (changes) can be determined. With a clever design of the measurement cell the above products can be separated, e.g. [104]. Once a complete set of calibration curves, or transfer functions, has been created, we can directly analyze multiple physical measurands of an unknown liquid. This furthermore allows for a new class of applications. As described before, if one measurand is a known property, we can in turn use the measurement results to evaluate other paramaters, e.g., acoustic wave velocity and acoustic impedance instead of liquid level. Therefore, by analyzing frequency shift and damping of multiple vibration modes in the same resonator element, we can successfully establish an accurate description of the liquid analyte in terms of multiple fundamental physical properties.

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A Appendix A.1

Additional tables Table A.1: Resonator material properties. Variable c11 c12 c44 EAl ECu EN iAg ESi νAl νCu νN iAg νSi ρAl ρCu ρN iAg ρSi σAl σCu

Value 166 GPa 64 GPa 80 GPa 72 GPa 110 GPa 125 GPa 170 GPa 0.34 0.343 0.30 0.26 2710 kg/m3 8960 kg/m3 8670 kg/m3 2329 kg/m3 3.550 · 107 S/m 5.998 · 107 S/m

Description elastic stiffness coefficient of silicon elastic stiffness coefficient of silicon elastic stiffness coefficient of silicon elastic modulus of aluminum elastic modulus of copper elastic modulus of nickel-silver elastic modulus of silicon (isotropic) Poisson ratio of aluminum Poisson ratio of copper Poisson ratio of nickel-silver Poisson ratio of silicon (isotropic) density of aluminum density of copper density of nickel-silver density of silicon conductivity of aluminum conductivity of copper

116

A. Appendix

Table A.2: Planar excitation coil parameters (Ct → 0). Geometry Lc / µH Cp / pF Rc / mΩ org spiral N63 23.33 7.3 7990 org spiral N63 1.0 5.1 124.8 new spiral round 3.3 7.1 5200 new spiral rect 3.2 7.6 5000 new linear N44 0.1185 4.5 15.7 new linear N44 0.1109 4.3 139 new radial 1 0.1060 6.3 37.5 new radial 2 0.1192 5.3 14.9 new radial 3 0.1113 6.1 13.6

Table A.3: Successfully excited and detected flexural and face shear eigenmodes of simple 15x15 mm aluminum plate, with radial, diagonal and circular excitation setup.

radial 143 204 284 416 517 530 545

Flex, FSM / kHz diagonal circular 1 circular 2 22.9 9.2 202 40.4 50.9 339 64.8 92.5 369 138 132 388 143 210 487 280 245 518 339 288 571 369 386 388 462 435 571 487 710 497 575 643 697

117

A.1. Additional tables

Table A.4: Successfully excited and detected TSM eigenmodes of simple 15x15 mm aluminum plate, with radial, diagonal and circular excitation setup.

radial 2.35 2.375 2.38 2.385

TSM / MHz diagonal circular 1 circular 2 2.32 2.34 2.32 2.42 2.36 2.415 2.43 2.48

Table A.5: Successfully excited and detected flexural and face shear eigenmodes of round mesa aluminum plate, with radial, diagonal and circular excitation setup.

radial 15.5 48 101 168 217 264 287 457 621

Flex, FSM / kHz diagonal circular 1 circular 2 14.7 64.3 46.0 30.9 142 144 48.0 536 218 81.9 699 265 144 286 169 359 217 398 223 435 275 476 370 494 398 513 465 609 493 513 699

Table A.6: Successfully excited and detected TSM eigenmodes of round mesa aluminum plate, with radial, diagonal and circular excitation setup. TSM / MHz radial diagonal circular 1 circular 2 1.57 1.52 1.525 1.55 1.60 1.63 1.71

118

A. Appendix

Table A.7: Successfully excited and detected flexural and face shear eigenmodes of round mesa brass (CuZn) plate, with radial, diagonal and circular excitation setup.

radial 68.5 172 192 305 417

Flex, FSM / kHz diagonal circular 1 circular 2 30.9 30.9 68.5 55.4 41.8 88.4 112.5 55.5 112.4 145 95.6 145 266 112.4 193 344 318 266 403 291 468 318

Table A.8: Successfully excited and detected TSM eigenmodes of round mesa brass (CuZn) plate, with radial, diagonal and circular excitation setup. TSM / MHz radial diagonal circular 1 circular 2 1.105 1.099 1.130 1.146

Table A.9: Electroplating settings at 22◦ C and results for additional copper layer thickness and smoothness on Si membranes. Si sample (100) 190 µm (111) 300 µm (100) 390 µm (100) 190 µm (100) 190 µm (111) 300 µm (100) 190 µm (111) 300 µm (100) 190 µm (100) 190 µm

Q Q Q Q Q Q Q Q Q R

electroplating settings Cu thickness TSM freq. / MHz 1 V, 1.6 A, 10 min ≈ 20 − 25 µm 16.05 →n/a 0.5 V, 0.5 A, 1 min ≈ 4 − 6 µm 8.51 → 8.19 2 V, 2.6 A, 1 min ≈ 5 µm 7.45 →n/a 0.5 V, 0.6 A, 2 min ≈ 6 µm 15.89 → 15.57 1 V, 1.4 A, 2 min ≈ 13 µm 16.06 →n/a 1 V, 1.6 A, 1 min ≈ 5 µm 8.19 → 8.05 1 V, 1.5 A, 1 min ≈ 4 µm 15.57 → 15.15 0.4 V, 0.35 A, 10 min 8.05 → 7.99 0.4 V, 0.38 A, 10 min 0.4 V, 0.42 A, 10 min

quality bad good bad good bad good good good medium medium

A.2. MATLAB routines

A.2

MATLAB routines

Save impedance analyzer measurement - ZA save.m % function [ZA] = ZA_save (title, shorttitle) title = input(’Title: ’, ’s’); shorttitle = input(’Short Title: ’, ’s’); % Create a tcpip object. ZA = instrfind(’Type’, ’tcpip’, ’RemoteHost’, ’140.78.120.12’, ’RemotePort’, 5025, ’Tag’, ’’); if isempty(ZA) ZA = tcpip(’140.78.120.12’, 5025); else fclose(ZA); ZA = ZA(1) end % Connect to ZA. fopen(ZA); % Communicating with ZA. % data1 = query(ZA, ’*IDN?’); % fprintf(ZA, ’BWFACT 3’); fprintf(ZA, ’BEEPDONE on’); fprintf(ZA, fprintf(ZA, fprintf(ZA, fprintf(ZA, fprintf(ZA, fprintf(ZA, fprintf(ZA, fprintf(ZA,

’SING’); ’*WAI’); ’TRAC A’); [’TITL "’, title, ’"’]); ’AUTO’); ’TRAC B’); ’AUTO’); ’*WAI’);

fprintf(ZA, [’SAVDASC "’, shorttitle, ’"’]); fprintf(ZA, [’SAVDTIF "’, shorttitle, ’"’]); fprintf(ZA, ’*WAI’); input(’Save in progress’); % Disconnect from ZA. fclose(ZA);

Save impedance analyzer sweep - ZA sweep.m % function [ZA] = ZA_sweep (title, shorttitle, start, stop, span) title = input(’Title: ’, ’s’); shorttitle = input(’Short Title: ’, ’s’); start = input(’Start Freq: ’); stop = input(’Stop Freq: ’); span = input(’Span Freq: ’); % Create a tcpip object. ZA = instrfind(’Type’, ’tcpip’, ’RemoteHost’, ’140.78.120.12’, ’RemotePort’, 5025, ’Tag’, ’’); if isempty(ZA) ZA = tcpip(’140.78.120.12’, 5025); else fclose(ZA); ZA = ZA(1) end step = (stop-start)/span; % Connect to ZA.

119

120

A. Appendix

fopen(ZA); % Communicating with ZA. % data1 = query(ZA, ’*IDN?’); % fprintf(ZA, ’BWFACT 3’); fprintf(ZA, ’BEEPDONE off’); i=0; while (i 0) freq(k)=d{1}; data(k)=d{2} + i*d{3}; else break; end; k=k+1; end; fclose(f);

Read measuremed phase data - readdata p.m function [freq,data] = readdata_p (fnam) f=fopen(fnam,’rt’); s=fgets(f);

A.2. MATLAB routines

while (length(findstr(’TRACE: B’,s))==0) s=fgets(f); end; while (length(findstr(’Frequency’,s))==0) s=fgets(f); end; k=1; while (~feof(f) ) d=textscan(f,’%n %n %n\n\r’); if (length(d{1}) > 0) freq(k)=d{1}; data(k)=d{2} + i*d{3}; else break; end; k=k+1; end; fclose(f);

Normalize data - norm ZA.m function [norm_data] = norm_ZA (data) s = data(1); e = data(801); rise = (e-s)/801; i=1; while(i

JKU

Technisch-Naturwissenschaftliche Fakult¨ at

Electromagnetic-Acoustic Resonators for Remote, Multi-Mode Solid and Liquid Phase Sensing DISSERTATION zur Erlangung des akademischen Grades

Doktor im Doktoratsstudium der

Technischen Wissenschaften

Eingereicht von:

Dipl.–Ing. Frieder Lucklum Angefertigt am:

Institut f¨ur Mikroelektronik und Mikrosensorik Beurteilung:

Univ.–Prof. Dipl.–Ing. Dr. techn. Bernhard Jakoby (Betreuung) Prof. Dr.–Ing. Bernd Henning

Linz, Januar, 2010

Preface The following work describes my research efforts in the last few years as a PhD candidate of the Institute for Microelectronics and Microsensors at the Johannes Kepler University Linz, Austria. The work builds upon a diploma thesis carried out at the Otto-von-Guericke University Magdeburg, Germany. No scientific research can be successfully undertaken without guidance, support, collaboration, and discussion. As such, I want to sincerely thank Prof. Bernhard Jakoby, my thesis adviser, for accepting and guiding this research topic, and my father Dr. Ralf Lucklum for introducing this research topic to me. The initial results were achieved under the supervision of Prof. Peter Hauptmann of the University of Magdeburg and Prof. Nico de Rooij of the University of Neuchˆatel. During this time I had the pleasure to collaborate with several colleagues in Germany, Switzerland and Austria, and I am grateful for all the help and fruitful discussions. I want to acknowledge Dr. Ulrike Hempel, Dr. S¨oren Hirsch and Dr. Alexandra Homsy for their great support during the initial stages. I thoroughly enjoyed the freedom and flexibility in the group of Prof. Jakoby and I am very thankful for experiencing the different and challenging aspects of research, teaching and organization. All my colleagues in Linz contributed to this productive and pleasant atmosphere. I want to especially thank Dr. Erwin Reichel for the close collaboration in our similar research projects, as well as my other office mates Dr. Ju ¨rgen Kasberger and Johann Mayrw¨oger. Furthermore, I’d like to acknowledge Hans Katzenmayer, Bernhard Mayrhofer and Dr. Wolfgang Hilber for their support with the fabrication of different devices and elements for this research. Last but not least, I want to thank my family for guiding me to a scientific career and for their love and support.

To my family.

Abstract Electromagnetic excitation of acoustic resonator sensors is a unique method alternative to piezoelectric and electrostatic excitation. Based on electromagnetic-acoustic transducers, the continuous, resonant operating mode of these sensors offers distinctive advantages and possibilities, which have only recently been demonstrated. Sensor element and excitation as well as detection circuitry can be spatially separated. Only conductive materials or layers are required, and the mechanical forces can be impressed in a variety of directions. This results in multiple degrees of freedom, which allows the excitation and detection of fundamentally different modes of vibration in the resonator element. This multi-mode operation leads to the possibility to measure multiple physical properties in terms of their acoustic load with no or minimal changes to the sensor setup. In this work, the theory behind this method has been investigated. Suitable models and simulations have been derived in order to explain and predict measurement results. Three different resonator devices have been fabricated and analyzed for modes of flexural and shear vibration. Here, the focus has been put on simple micromachined silicon membranes and inexpensive aluminum and brass plates. Suitable vibration modes were utilized for mass detection, liquid property sensing, and liquid level measurements as a proof of concept for the multi-mode sensor principle. Additionally, prototype metal resonator arrays were used to demonstrate simultaneous measurement of different physical properties.

Kurzfassung Die elektromagnetische Anregung von akustisch resonanten Sensoren ist eine neue Alternative zur klassischen piezoelektrischen oder elektrostatischen Wandlung. Sie basiert auf den klassischen elektromagnetisch-akustischen, oder elektrodynamischen, Wandlern, wobei die kontinuierliche, resonante Anregung dieser Sensoren besondere Vorteile und M¨oglichkeiten er¨offnet. Hierbei sind Sensorelement und Anrege- und Auswerteschaltung r¨aumlich voneinander getrennt. Fu ¨r das Sensorelement wird nur ein leitf¨ahiges Material oder eine leitf¨ahige Schicht ben¨otigt, und die mechanischen Kr¨afte k¨onnen in die verschiedensten Richtungen eingepr¨agt werden. Dadurch ergeben sich mehrere Freiheitsgrade, was die Anregung fundamental verschiedener Schwingungsmodi in ein und demselben Sensorelement erlaubt. Dieser multi-mode Betrieb erm¨oglicht, mehrere verschiedene physikalischen Eigenschaften als ¨ Anderungen der akustischen Last zu messen, ohne oder nur mit geringfu ¨gigen ¨ Anderungen am Messaufbau. In dieser Arbeit wurde die Theorie hinter diesem Messprinzip beleuchtet, und es wurden geeignete Modelle und Simulationen erstellt, um die erzielten Messergebnisse zu erkl¨aren und vorherzusagen. Es wurden drei verschiedene Typen von Resonatorelementen hergestellt und bezu ¨glich ihrer Schwingungsmoden untersucht. Das Hauptaugenmerk lag dabei auf einfachen, mikromechanischen Siliziummembranen und kostengu ¨nstigen Aluminium- und Messingpl¨attchen. Geeignete detektierte Schwingungsmodi konnten fu ¨r die Mikrow¨agung und die Messung von Flu ¨ssigkeitseigenschaften und Fu ¨llst¨anden genutzt werden. Diese Experimente zeigen die Eignung des resonanten Messprinzips elektromagnetischer Anregung. Mittels Prototypen eines Resonatorarrays aus Metallfolien wurde weiterhin die gleichzeitige Anregung und Messung verschiedener Schwingungsmoden und Eigenschaften demonstriert.

Contents List of Tables

X

List of Figures

XII

List of Abbreviations

XVIII

List of Symbols 1 Introduction 1.1 State of Art Resonator Sensors . . . 1.1.1 Piezoelectric Transducers . . . 1.1.2 Capacitive Transducers . . . . 1.1.3 Electromagnetic Transducers . 1.2 Motivation . . . . . . . . . . . . . . . 1.3 Thesis Outline . . . . . . . . . . . . .

XIX

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2 Theory and Modeling 2.1 Fundamental Physical Background . . . . 2.1.1 Mechanical Resonant Vibration . . 2.1.2 Resonator Excitation and Detection 2.2 Multi-mode Excitation Methods . . . . . . 2.3 Finite Element Simulation . . . . . . . . . 2.4 Equivalent Circuit Modeling . . . . . . . . 2.5 Summary . . . . . . . . . . . . . . . . . . 3 Basic Resonator Operation 3.1 Design and Fabrication . . . . . . . . 3.1.1 Silicon Membrane Resonators 3.1.2 Metal Plate Elements . . . . . 3.1.3 Resonator Array Sensors . . .

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VIII

CONTENTS

3.2

Eigenmode Analysis . . . . . . . . . 3.2.1 Silicon Membrane Resonators 3.2.2 Metal Plate Elements . . . . . 3.2.3 Resonator Array Sensors . . . 3.3 Experimental Excitation Setup . . . 3.3.1 Array Multi-Mode Setup . . . 3.4 Eigenmode Excitation Results . . . . 3.4.1 Thickness Shear Modes . . . . 3.4.2 Face Shear Modes . . . . . . . 3.4.3 Flexural Plate Modes . . . . . 3.4.4 Array Multi-Mode Excitation 3.5 Summary . . . . . . . . . . . . . . . 4 Mass Microbalance Application 4.1 Analytical Description . . . . . 4.2 Suitable Modes of Vibration . . 4.3 Equivalent Circuit Model . . . . 4.4 Results and Discussion . . . . . 4.5 Summary . . . . . . . . . . . .

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5 Liquid Phase Density-Viscosity Sensing 5.1 Analytical Description . . . . . . . . . . 5.2 Suitable Modes of Vibration . . . . . . . 5.3 Equivalent Circuit Model . . . . . . . . . 5.4 Results and Discussion . . . . . . . . . . 5.4.1 Metal Plate Elements . . . . . . . 5.4.2 Silicon Membrane Resonators . . 5.5 Summary . . . . . . . . . . . . . . . . . 6 Liquid Level Measurement 6.1 Suitable Modes of Vibration 6.2 Equivalent Circuit Model . . 6.3 Results and Discussion . . . 6.3.1 Preliminary Results .

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65 65 68 69 70 72

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73 73 75 77 78 78 81 83

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85 86 87 89 89

CONTENTS

IX

6.3.2 Measurement Setup . . . . . . . . . . . . . . . . . . . . 89 6.3.3 Measurement and Simulation Comparison . . . . . . . 91 6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 7 Conclusions 97 7.1 Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 Bibliography

101

A Appendix 115 A.1 Additional tables . . . . . . . . . . . . . . . . . . . . . . . . . 115 A.2 MATLAB routines . . . . . . . . . . . . . . . . . . . . . . . . 119 Curriculum Vitae

123

List of Tables 2.1

Electro-mechanical equivalences with the Mobility Analog. . . 29

3.1

Changes to the radial thickness shear mode of (100) Si membrane at 15 MHz due to increasing copper layer thickness. . . 40

3.2

Comparison of operating frequency, signal response and Qfactor for selected resonator elements excited at radial TSM in air; with different selected geometries and operating TSM harmonics for the silicon membranes. . . . . . . . . . . . . . . 57

4.1

Copper layer thickness calculated from frequency shift of radial TSM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.1

Frequency shift and damping behavior of circular FSM at 142 kHz in aluminum due to liquid loading. . . . . . . . . . . 79

5.2

Frequency shift and damping behavior of diagonal FSM at 217 kHz in aluminum due to liquid loading. . . . . . . . . . . 80

5.3

Influence of density and viscosity on resonant response of diagonal FSM of (111) Si membrane. . . . . . . . . . . . . . . . 82

A.1 Resonator material properties. . . . . . . . . . . . . . . . . . . 115 A.2 Planar excitation coil parameters (Ct → 0). . . . . . . . . . . . 116

A.3 Successfully excited and detected flexural and face shear eigenmodes of simple 15x15 mm aluminum plate, with radial, diagonal and circular excitation setup. . . . . . . . . . . . . . . . . 116 A.4 Successfully excited and detected TSM eigenmodes of simple 15x15 mm aluminum plate, with radial, diagonal and circular excitation setup. . . . . . . . . . . . . . . . . . . . . . . . . . 117 A.5 Successfully excited and detected flexural and face shear eigenmodes of round mesa aluminum plate, with radial, diagonal and circular excitation setup. . . . . . . . . . . . . . . . . . . 117

LIST OF TABLES

A.6 Successfully excited and detected TSM eigenmodes of round mesa aluminum plate, with radial, diagonal and circular excitation setup. . . . . . . . . . . . . . . . . . . . . . . . . . . . A.7 Successfully excited and detected flexural and face shear eigenmodes of round mesa brass (CuZn) plate, with radial, diagonal and circular excitation setup. . . . . . . . . . . . . . . . . . A.8 Successfully excited and detected TSM eigenmodes of round mesa brass (CuZn) plate, with radial, diagonal and circular excitation setup. . . . . . . . . . . . . . . . . . . . . . . . . A.9 Electroplating settings at 22◦ C and results for additional copper layer thickness and smoothness on Si membranes. . . . .

XI

. 117

. 118

. 118 . 118

List of Figures 1.1 1.2 1.3 1.4 1.5

Acoustic sensor principle of input quantity reflected in acoustic wave, which is converted to an electrical signal. . . . . . . . .

1

Classification scheme used for different resonant acoustic wave sensors in this work. . . . . . . . . . . . . . . . . . . . . . . .

3

Standard quartz crystal resonator sensor for thickness shear mode operation. . . . . . . . . . . . . . . . . . . . . . . . . . .

5

Modified electrode geometry for lateral field excitation of a quartz resonator with bare sensing surface. . . . . . . . . . . .

6

Capacitive micromachined ultrasonic transducer (CMUT), single cell design (left) and portion of single array element with hundreds of individual cells (right). . . . . . . . . . . . . . . .

9

2.1

Fundamental eigenmodes for thin plates and membranes. . . . 18

2.2

Impedance of the primary coil with no conductive material present (left), with a conductive material present resulting in eddy current induction (center), and with a conductive resonator present resulting in eddy current induction (large peak), Lorentz force generation, resonant vibration, and movement induction of a secondary eddy current (small peaks) (right). 21

2.3

Effect of different phase responses in the normalized impedance of the primary coil around face shear (left), flexural (center) and thickness shear resonance (right) of a silicon sensor element. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.4

Principle of resonance excitation with a planar spiral coil and perpendicular magnetic field resulting in radial Lorenz forces. . 24

2.5

Modified principle for generation of linear, diagonal Lorentz forces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.6

Complete FEM excitation model with planar spiral coil placed at the center of the resonator element. . . . . . . . . . . . . . 26

LIST OF FIGURES

XIII

2.7

Exemplary excitation simulation of a radial TSM at 15 MHz in a circular silicon plate, with eddy current induction (left), Lorentz force generation (middle) and resulting thickness shear mode vibration (right). . . . . . . . . . . . . . . . . . . . . . . 27

2.8

Fundamental face shear mode shapes of a round silicon membrane with displacement of diagonal/linear (left), circular (center) and radial (right) symmetry for a circular plate. . . . . . . 28

2.9

From mechanical description and schemata to equivalent circuit with the Mobility Analog (left to right). . . . . . . . . . . 31

2.10 Electromagnetic-acoustic transducer with electrical equivalent circuit (left) and mechanical equivalent circuit (right). . . . . . 32 2.11 Electromagnetic-acoustic equivalent circuit, with eddy current induction described by electromagnetic transformer T , coupling between current i, voltage u and force F , velocity v in the resonator modeled by electro-mechanical transformer XT .

33

2.12 Comparison of impedance amplitude (left) and phase (right) for equivalent circuit simulation and measurements of face shear mode (top) and thickness shear mode (bottom). . . . . . 34 3.1

Simplified 7 step silicon fabrication process; substrate (grey), oxide mask (red), photoresist (yellow), metal (purple). . . . . . 38

3.2

SEM image of sidewall of round silicon membrane with characteristic DRIE structure. . . . . . . . . . . . . . . . . . . . . 39

3.3

Changes to a radial face shear mode of a (100) silicon membrane with increasing copper layer thickness. . . . . . . . . . . 41

3.4

Brass and aluminum metal plates of 1 mm thickness with round mesa structure of 15 mm diameter milled in center. . . 42

3.5

Design of 2x2 resonator array setup with individual coils and magnets for each element. . . . . . . . . . . . . . . . . . . . . 43

3.6

Metal prototype arrays with different coupling springs and element size of 10x10 mm (top) and different corresponding excitation coil arrays on polymer foils with 100 µm metalization thickness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

XIV

3.7

3.8 3.9 3.10

3.11

3.12 3.13 3.14

3.15

3.16

3.17

3.18

LIST OF FIGURES

Results of eigenmode analysis for round (left) and quadratic (right) 15x15 mm (100) silicon membrane showing diagonal (top), radial (middle) and circular (bottom) face shear modes. Mode shapes for round (100) silicon membrane with diagonal TSM at 15 MHz. . . . . . . . . . . . . . . . . . . . . . . . . . First and second harmonic of diagonal FSM in round aluminum mesa resonator at 81 kHz and 171 kHz. . . . . . . . . Eigenmode shape of circular FSM at 142 kHz (left) and radial flexural plate mode at 295 kHz (right) of round aluminum mesa resonator. . . . . . . . . . . . . . . . . . . . . . . . . . . Simulated mode shape for radial face shear mode (left) and circular thickness shear mode (right) of a single metal array element. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Geometric arrangement for electromagnetic-acoustic resonator setup with remote, non-contact excitation and detection. . . . Design and realization of excitation setup for radial and circular shear modes. . . . . . . . . . . . . . . . . . . . . . . . . Lorentz force distribution (arrows) with different coil and magnet layouts for radial (left), diagonal/linear (center), and circular (right) vibration mode shapes. . . . . . . . . . . . . . . . Simultaneous excitation of different modes of vibration in each array element at different excitation frequencies (left), and of the same vibration mode at the same resonant frequency (right). Normalized impedance of quadratic coil with standard magnetic field resulting in radial TSM at 15.42 MHz (left) and with alternated magnetic field resulting in diagonal TSM at 15.36 MHz (right) in a quadratic high-Q silicon resonator. . . Normalized impedance of a standard spiral coil at the center resulting in a radial TSM at 15.65 MHz (left) and placed out of center resulting in a circular TSM at 15.61 MHz (right) in a round (100) silicon membrane. . . . . . . . . . . . . . . . . . Impedance spectrum of circular and diagonal TSM in aluminum mesa element and quadratic plate, respectively. . . . .

46 47 48

49

50 50 51

53

55

58

58 59

LIST OF FIGURES

3.19 Normalized impedance of diagonal FSM at 222 kHz (left) and circular FSM at 459 kHz (right) in a quadratic high-Q silicon resonator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.20 Normalized impedance of diagonal FSM at 143 kHz (left) and 2nd circular FSM at 369 kHz (right) in a mesa aluminum plate. 3.21 Normalized impedance of radial flexural mode at 248 kHz in a quadratic high-Q silicon resonator (left) and at 288 kHz for an aluminum mesa resonator (right). . . . . . . . . . . . . . . 3.22 Experimental results for individual excitation of a circular FSM in all elements of a metal array anchored at the corners at ≈234 kHz, with noticeable cross-coupling between individual elements as evidenced by the minor peaks. . . . . . . . . . . . 3.23 Simultaneous excitation of a circular face shear mode in (100) silicon and a radial flexural mode in aluminum around ≈250 kHz, and different frequency shifts with DI water loading due to different sensitivities. . . . . . . . . . . . . . . . . . . . . .

XV

60 60

61

62

63

4.1

Diagonal, circular and radial TSM (left to right) for round (100) silicon membrane at ≈ 15 MHz. . . . . . . . . . . . . . . 69 4.2 Electromagnetic-acoustic equivalent circuit with added ideal mass element. . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.3 Shift of unloaded fundamental TSM frequency after coating with a 300 nm polystyrene monolayer. . . . . . . . . . . . . . 71 5.1

Displacement profiles in a viscous liquid for a diagonal thickness shear mode (left) and a diagonal face shear mode (right). 5.2 Optimal modes of vibration in viscous liquids: Circular FSM of aluminum resonator (left) and diagonal FSM of Si membrane. 5.3 Electromagnetic-acoustic equivalent circuit with added ideal mass element. . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Normalized impedance spectrum of primary coil for liquid loading of circular FSM at 142 kHz in aluminum resonator yielding frequency shifts and damping as summarized in Table 5.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

74 76 78

80

5.5

Measurement results for diagonal FSM of (111) silicon membrane for different alcohols in terms of density-viscosity product vs. frequency shift and normalized impedance. . . . . . . . 81

5.6

Measurement results for diagonal FSM of (111) silicon membrane for different alcohols in terms of density-viscosity product vs. frequency shift and impedance damping. . . . . . . . . 83

6.1

Experimental setup for liquid level measurements with resonator element integrated into liquid reservoir and excited by planar coil and external magnet. . . . . . . . . . . . . . . . . . 87

6.2

Electromagnetic-acoustic equivalent circuit, with eddy current induction described by electromagnetic transformer T , coupling between current i, voltage u and force F , velocity v in the resonator modeled by electro-mechanical transformer XT , and the standing longitudinal wave in a liquid above the resonator represented by a lossy transmission line with a wave impedance Za . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

6.3

Preliminary results during density-viscosity experiments showing a strong dependence on volume in liquid measurement cell. 89

6.4

Experimental setup for liquid level measurements with resonator element integrated into liquid reservoir and excited by planar coil and external magnet. . . . . . . . . . . . . . . . . . 90

6.5

Impedance measurement (left) and simulation (right) for flexural resonance at 275 kHz with liquid volume at maximum around 900 µl, corresponding to a height at maximum of 2.75 mm = λ/2. Each curve corresponds to a change of 25 µl water or 76 µm height, respectively. . . . . . . . . . . . . . . . 91

6.6

Impedance measurement (left) and simulation (right) for flexural resonance at 275 kHz with liquid volume at maximum around 2700 µl, corresponding to a height at maximum of 8.25 mm = 3/2λ. Here, each curve corresponds to a change of 50 µl water or 154 µm height. . . . . . . . . . . . . . . . . . . 92

LIST OF FIGURES

6.7

XVII

Comparison of phase measurement (left) and simulation (right) corresponding to figure 6.6. . . . . . . . . . . . . . . . . . . . 93

List of Abbreviations Symbol

Description

AC BAW BHF CMOS CMUT DI DRIE EMAT FSM FEM IDT KOH MARS MEMS MST NdFeB NDT PCB PMMA PZT QCM QCR RF RIE SAW SEM TSM

alternating current bulk acoustic wave buffered hydrofluoric acid complementary metal oxide semiconductor capacitive micromachined ultrasonic transducer deionized (water) deep reactive ion etching electromagnetic acoustic transducer face shear mode finite element method interdigital transducer potassium hydroxide magnetic acoustic resonator sensors microelectromechanical systems microsystems technology neodymium iron boron (permanent magnet) non-destructive testing printed circuit board poly(methyl methacrylate) lead zirconium titanate quartz crystal microbalance quartz crystal resonator radio frequency reactive ion etching surface acoustic wave scanning electron microscope thickness shear mode

List of Symbols Symbol

Description

Unit

A B B0 c c C d D E E f ∆f F fL fR FL G h H hR i j J k k L m

surface area magnetic flux density (magnetic field) static magnetic flux density wave velocity elastic stiffness constant capacitance piezoelectric charge constant electric displacement field electric field elastic modulus frequency frequency shift force Lorentz force density resonance frequency Lorentz force conductance friction coefficient magnetic field intensity resonator thickness time variant current imaginary unit current density wave number spring constant inductance mass

m2 T T m/s Pa F m/V C/m2 V/m Pa Hz Hz N N/m3 Hz N S m/N s A/m m A — A/m2 1/m N/m H kg

XX

LIST OF SYMBOLS

Symbol

Description

Unit

n n q Q r r R S t T u u v wk Wk wp Wp x, y, z X Y Y Z ∆Z Zi Za Zm

harmonic number elastic compliance electric charge quality factor radius or distance mechanical resistance electrical resistance mechanical strain time variable mechanical stress mechanical displacement time variant voltage velocity kinetic energy density kinetic energy potential energy density potential energy Cartesian coordinates Lorentz force coupling coefficient movement induction coupling coefficient admittance impedance impedance damping input impedance acoustic impedance mechanical impedance

— 1/Pa C — m N s/m Ω — s Pa m V m/s J/m3 J J/m3 J m N/A V s/m S Ω Ω Ω Pa s/m N s/m

γ δ ε η

propagation constant penetration depth permittivity shear viscosity

— m F/m Pa s

XXI

LIST OF SYMBOLS

Symbol

Description

Unit

κ λ µ µR ν φ ρ ρq σ Θ ω ωR

compressibility wavelength magnetic permeability shear stiffness Poisson ratio phase angle mass density charge density specific conductivity impedance phase angle angular frequency angular resonance frequency

1/Pa m H/m Pa — rad kg/m3 C/m3 S/m rad 1/s 1/s

∇· ∇× ℑ ℜ

divergence operator curl operator imaginary part of complex value real part of complex value

m−1 m−1 — —

1 Introduction The field of acoustics encompasses the complete spectrum of mechanical vibrations and elastic waves in gases, liquids and solids, ranging from infrasound and audible frequencies, over ultrasonic devices, up to radio frequency (RF), or hypersonic, applications at several Gigahertz. The application of acoustics permeates throughout our daily lives. Beyond audio applications, ultrasonic devices are used for frequency standards, imaging systems, material testing, sensors, and RF filters, multiplexers, switches. With the advent of microelectromechanical systems (MEMS) and microsystems technology (MST) [1], research in acoustic devices has focused on new materials, electromechanical transduction methods, and miniaturization of acoustic transducers [2, 3]. Today acoustic devices have replaced or introduced various functional components in computers, cars, and cell phones, remote controls, and other hand-held systems. Acoustic sensors are devices where the measured properties are reflected as changes to the characteristics of an elastic wave [4, 5]. A wide range of possible measurands affect an acoustic wave, including acoustic properties such as wave velocity and polarization, but also non-acoustic parameters like bio-chemical, thermal, and mechanical properties, e.g. mass, position, level, acceleration, velocity, flow, concentration, pressure, density, and viscosity, as well as electric and magnetic properties, for example conductivity, permittivity, electric or magnetic fields [6]. The acoustic wave can propagate in the sensor element itself, radiate into the surrounding environment, or travel

Input quantity (mass, density, flow, elasticity, viscosity, distance, level, ...)

Influence on acoustic wave / impedance (velocity, frequency, amplitude, phase, ...)

Transduction to electrical property (frequency, amplitude, phase)

Electrical signal (voltage, current)

Figure 1.1: Acoustic sensor principle of input quantity reflected in acoustic wave, which is converted to an electrical signal.

2

1. Introduction

as a combination of both. The different measurands can influence either the acoustic properties of the sensor element itself or of the surrounding medium. The acoustic transducer then converts this effect, and thus the detected quantity, into an electrical signal (Fig. 1.1) [7, 8, 9]. In general, the acoustic wave carries the information of interest. We can distinguish between pulsed and continuous generation of acoustic waves, where characteristics like delay time, phase shift, polarization, and insertion loss, or resonance frequency, quality factor, and damping are evaluated, respectively. Furthermore, we can differentiate excitation principles, sensor materials, fabrications methods, active or passive operation, operating frequency, and more. To characterize different sensors, several classification schemes exist, e.g. [10], but an encompassing scheme is difficult to achieve for acoustic sensors. Ultrasonic sensors generally refer to emitters and receivers of acoustic waves radiated into a surrounding medium, but can also simply describe all acoustic sensors operating at ultrasonic frequencies. Acoustic wave sensors on the other hand generally refer to devices where the dominant effect is given by a modification of the acoustic wave propagating in the transducer element. Acoustic microsensors are then a special form of miniaturized acoustic wave sensors utilizing microfabrication technology, but can also include devices radiating waves into a medium. The term resonant sensors is also often used as a global description of acoustic sensors, referring to devices where the resonant properties of the sensor element are exploited. Resonant sensors generally encompass classic macroscopic resonators, sensors in oscillatory circuits, acoustic microsensors, some, but not all ultrasonic sensors, as well as micromechanical resonators. The latter are again partially interchangable with acoustic microsensors, but also include, e.g., accelerometers or pressure sensors, which are usually not classified as acoustic wave sensors. A selection of different acoustic sensors and applications is given in the following review papers [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23] and references therein. The devices presented in this work can generally be described as resonant acoustic wave sensors. Depending on geometry, excitation method and transducer design, bulk acoustic waves (BAW), surface acoustic waves (SAW), or flexural waves are generated in the sensor element. The resulting vibration

3

1.1. State of Art Resonator Sensors

Acoustic Sensors

Transduction Principle

Piezoel.

El.static

El.magn.

Mode of Vibration

Other

Flexural

Shear

Extensional

Figure 1.2: Classification scheme used for different resonant acoustic wave sensors in this work.

of the resonator surface interacts with the surrounding medium and is influenced by the overall acoustic properties acting at the device surface. This acoustic loading of the elastic wave in the resonator is the actual measured property and can be described in terms of a material property, the characteristic acoustic impedance Za defined as the product of wave velocity and density, or as force acting on a surface area divided by wave velocity. This characteristic impedance has to be replaced with an effective impedance in case of a finite surrounding medium, to account for effects of, e.g., reflection and interference of the acoustic wave [24, 25, 26]. Depending on the mode of vibration, acoustic wave sensors can be employed as, e.g., masssensitive deposition rate monitors, liquid phase sensors for viscosity and density, for measuring viscoelastic properties, and analyzing biochemical species and processes, see e.g. [27, 28, 29, 30, 31, 32, 33, 34]. Due to the difficulty of nomenclature and classification, and following the focus of this work, we will primarily distinguish between different acoustic transduction principles and the mode of vibration of the sensor element as depicted in figure 1.2.

1.1

State of Art Resonator Sensors

A variety of transduction methods exist today for generating and detecting elastic waves. The most widely employed method is piezoelectric excitation. The most popular and extensively investigated acoustic transducer is the piezoelectric quartz crystal resonator (QCR). Like the quartz crystal, many

4

1. Introduction

acoustic sensors have been derived from electronic components, where they are used as highly accurate frequency standards, delay lines, or other RF components. These components have usually been slightly modified to permit sensing of acoustic loads in contact with the resonator [35, 36, 37]. With the development of new excitation methods, novel designs and applications became possible. One such example is capacitive excitation, which became feasible only in recent years with the advent of microfabrication of mechanical elements. The third major method for generating ultrasonic waves is electromagnetic excitation, which we are focusing on in this thesis. Other methods include mechanical and thermal ultrasound generation, magnetostrictive transducers, Laser generation and detection of ultrasound, or spark excitation [38, 39].

1.1.1

Piezoelectric Transducers

Piezoelectricity, discovered by the brothers Curie in 1880, is a material property that combines the electrical and mechanical domain. The electrical charges within a piezoelectric material are spatially separated to form dipoles, but are evenly distributed resulting in no net polarization. However, piezoelectric materials lack a center of inversion symmetry. Applying a mechanical strain thus changes the distribution of the charges and results in a net polarization. This mechanism has been termed the direct piezoelectric effect. Conversely, applying an electrical field to the material results in a mechanical displacement of the charges. These two effects are described by the following constitutive relations: Di = dij Tj + εTij Ej t Si = n E ij Tj + dij Ej .

(1.1) (1.2)

The electrical displacement D is coupled to mechanical stress T by the direct piezoelectric coefficient d and to the electric field E by the permittivity ε. Conversely, the mechanical strain S is coupled to mechanical stress via the elastic compliance n and to the electric field via the transposed piezoelectric coefficient.

1.1.1. Piezoelectric Transducers

5

Figure 1.3: Standard quartz crystal resonator sensor for thickness shear mode operation.

Piezoelectricity can be found in naturally occurring crystalline materials like quartz, but also silk, wood and others. For industrial use artificially grown materials are primarily used, e.g. synthesized quartz, zinc oxide and Langasite crystals, ceramic materials like lead zirconium titanate (PZT), as well as certain polymers. For ultrasonic transducers, quartz and PZT are most commonly employed. Depending on the crystal cutting angle, the transducer substrate displays different properties, e.g. AT-cut quartz is widely used because of its temperature stability due to compensating thermal coefficients of the piezoelectric parameters. The cut angle of a piezoelectric crystal and the corresponding electrode geometry also defines the mode of vibration of the resonator. Acoustic waves in a piezoelectric material are usually excited by applying a voltage to electrodes attached to the faces. As an example shown in figure 1.3, an often used variant is an AT-cut quartz disc with electrodes applied to both main faces, exciting a thickness-shear bulk acoustic wave in the disc. To excite a surface acoustic wave, e.g., an ST-cut quartz with interdigital electrodes (IDT) patterned on one surface can be used. A variation of the classic QCR design is the lateral field excitation of the thickness shear mode (TSM) in a quartz disc in figure 1.4 [40, 41, 42]. Here, the electrodes are confined to a single face of the disc, the surface stays bare and is thus better suited for sensing applications. Other combinations of crystal cut and electrode geometry include flexural vibration modes in, e.g., discs, cantilevers or tuning forks, and shear-horizontal

6

1. Introduction

Figure 1.4: Modified electrode geometry for lateral field excitation of a quartz resonator with bare sensing surface.

Love-wave propagation in thin, piezoelectric waveguides [43]. In all these resonators designs, the mode of vibration is predefined by substrate, especially the crystal cut, and electrode geometries, and there is very little or no option to excite a piezoelectric resonator in a fundamentally different mode. This is an advantage for the suppression of undesired, spurious vibration modes, but restricts the versatility of a single sensor element. A recently developed alternative excitation method applies a planar coil to bare, electrodeless AT-cut quartz discs [44, 45]. There are two separate mechanisms involved here. In air the fundamental excitation of a quartz is achieved by the planar coil acting as an antenna and the generated electromagnetic wave coupling into the piezoelectric material. The electric field component E is in plane with the resonator. But a vertical electric displacement component D appears in the disc due to the tensor nature of the piezoelectric permittivity. This vertical electrical component is then the cause of the converse piezoelectric effect creating a shear mode vibration. The second mechanism is emphasized during liquid phase measurements. Here, the electromagnetically excited resonance decreases strongly, but a second resonance peak is enhanced. The electric potential distribution of the coil and the virtual ground of the upper face of the quartz create an electrode setup similar to QCR and an equivalent acoustic excitation, with the only difference being the potential gradient of the coil compared to a constant potential of an electrode. Given the tensor nature of the electromechanical

7

1.1.2. Capacitive Transducers

properties in a piezoelectric material, a planar spiral coil remotely excites two different resonances in a piezoelectric resonator, one due to the electromagnetic and another due to the electric field of the coil. With this setup, the mode of vibration can be modified by adjusting the alignment of disc and excitation coil. Furthermore, disc and coil can be spatially separated, allowing for remote excitation. Another advantage is the possibility to excite very high harmonics of the fundamental vibration mode up to GHz frequencies. The coil can also be deposited on one side of the quartz disc itself, increasing the electro-mechanical coupling but losing the non-contact nature and variability of exciting different modes [46]. The measurement mechanism for these sensors is generally the evaluation of the complex, frequency dependent electrical impedance. Changes to the acoustic loading of the sensor element, and thus changes to the acoustic impedance, are reflected in the electrical impedance of the device. Piezoelectric ultrasonic sensors can be considered as the most thoroughly investigated and best documented ultrasonic transducers today. With new developments in materials science and fabrication technologies, these devices continue to be the most commonly used ultrasonic components in industry and research.

1.1.2

Capacitive Transducers

Capacitive excitation is governed by the Coulomb force, the attraction and repulsion of opposite and similar charges, respectively, and subsequently by ~ While this electrostatic the force acting on a charge q in an electric field E. force is usually negligible in the macroscopic domain, it scales very well into the microscopic domain. The attracting force F due to a voltage U between two charged plates of cross-section A and distance r is given by 2

~ = εA U . (1.3) F~ = q · E 2r2 While macroscopic capacitive transducers were unsuitable for sensing applications due to the high electric field strength (on the order of 106 V/cm) and thus very high voltages required, with microfabrication technology it became possible to reduce the characteristic dimensions and thus decrease the distance of the electrodes into the micron and submicron range [47]. A much

8

1. Introduction

lower operating voltage is required (tens to a few hundred Volts) to achieve strong enough forces for ultrasonic actuation. Electro-mechanical coupling factors comparable to piezoelectric excitation have been achieved. The basic design is relatively simple. Using surface micromachining, a top electrode is patterned onto a thin membrane, with the bottom electrode located on the substrate forming a capacitor cell. To create the (vacuum) gap between top and bottom electrode, a sacrificial layer technique is usually utilized [48]. This gap has to be evacuated to reduce the damping of the membrane vibration. Applying a voltage between upper and lower electrode pulls the membrane towards the substrate. To generate ultrasonic waves the membrane is usually pre-stressed by a bias voltage and excited by an alternating voltage. The capacitive transducer is a wide-band transducer that allows the generation and detection of a large variety of frequencies. The major advantage of capacitive ultrasonic transducers is the ability to fabricate them using standard cleanroom processes, while miniaturized piezoelectric transducers require special materials and processes. Capacitive micromachined ultrasonic transducers (CMUT) today are fabricated in CMOS compatible processes, thus enabling easy integration with microelectronic components [49, 50]. Furthermore, unlike small piezoelectric substrates, CMUTs can be realized on large silicon wafers, thus allowing for large transducer arrays instead of single sensor elements (Fig. 1.5). The development of CMUT arrays has been especially useful for improving ultrasonic imaging techniques and high intensity focused ultrasound in medicine, and have thus become the industry standard for these applications today, e.g. [51]. The primary mode of vibration of capacitive resonators is the flexural plate mode for radiating compressional waves into a material. With changes to the transducer design, it is also possible to excite further Lamb waves in a membrane structure [52]. But in general these transducers are limited to flexural vibration, which limits the application as liquid phase sensors due to high damping of the generated ultrasonic waves. Due to their high bandwidth they are however not limited to vibrating at their resonance frequencies. The main use of capacitive transducers is the generation and detection of

9

1.1.3. Electromagnetic Transducers

Figure 1.5: Capacitive micromachined ultrasonic transducer (CMUT), single cell design (left) and portion of single array element with hundreds of individual cells (right).

ultrasound in different media for imaging applications, material testing, and focused ultrasound.

1.1.3

Electromagnetic Transducers

The third major mechanism for exciting acoustic waves is based on generating Lorentz forces, which are complementary to Coulomb forces in an electromagnetic field, and the term magnetic direct generation has been given to this coupling mechanism between electromagnetic and acoustic waves by Quinn [53]. The method of magnetic direct generation of acoustic waves in conductive materials was developed from studies in low temperature physics. Also termed electrodynamic, inductive, or simply electromagnetic transducers, these devices have been employed especially in non-destructive testing (NDT) and evaluation for several decades, for example in the automotive industry [38]. The Lorentz force F~L on a charge q moving with velocity ~v ~ originally in an electromagnetic field with the magnetic field component B, formulated by Maxwell and later Lorentz [54, 55], is given by ~ ~ ~ (1.4) FL = q E + ~v × B .

If the net charge of the material is zero, only a magnetic component results in a force in the resonator. Thus, the Lorentz force density can be written as ~ f~L = J~ × B,

(1.5)

10

1. Introduction

where the current density J~ = ρq~v is given as product of the moving charge density ρq and its velocity. As can be seen in the equations, electromagnetic generation of acoustic waves necessitates a conductive part of the resonator where the required current can be applied. This primary current can be directly applied to the sample, where the conductive paths act similar to electrodes for piezoelectric and capacitive excitation. Depending on the current path layout and the applied magnetic field, a variety of Lorentz force directions can be realized, and thus a variety of vibrations modes can be excited. After the design of the current paths, an element is however again limited to a specific family of vibration modes. The common application of an electromagnetic acoustic transducer (EMAT) is the non-contact mode of operation [56]. Prior to generating the Lorentz forces, the necessary current is remotely induced in the material. This again requires a conductive substrate material or a conductive layer in a composite element. These eddy currents are induced by means of an alternating current flowing through a primary coil in the vicinity of the sample. The eddy current flow and thus the Lorentz force direction is directly dependent on the layout of the primary coil and the direction of the applied magnetic field [57, 58, 59]. Therefore, besides the remote nature of excitation, all kinds of vibration modes can be excited without changes to the acoustic sample. The primary alternating current, be it eddy current or directly applied current, also generates a time-variant magnetic field. This field can be utilized for Lorentz force excitation. Since both current density and magnetic flux density are now time-variant, the Lorentz force and thus the acoustic wave will oscillate at double the frequency of the primary current. However, in order to achieve suitably large magnetic flux densities, very high currents in the range of kiloamps are necessary. These currents can generally only be achieved in pulsed applications and therefore this method is not suitable for continuous, resonant excitation [60]. The other option is to supply the magnetic field by means of an external magnet. With modern permanent rare-earth magnets high static remanent flux densities are possible, e.g. NdFeB magnets with up to 1.5 T. The placement of the external magnet introduces another degree of freedom for determining the direction of the

1.2. Motivation

11

Lorentz forces. The detection mechanism works as follows. The sample moving in an external magnetic field gives rise to a secondary eddy current, which in turn induces a secondary voltage in the primary coil geometry. The combined eddy current density in a static magnetic field can be written as ~ ~ ~ (1.6) J(t) = σ E(t) + ~v (t) × B .

For EMAT excitation the induced secondary voltage is dependent on the electro-mechanical coupling efficiency as well as the electromagnetic coupling efficiency of the eddy current induction (see also the equivalent circuit in fig. 2.11). In general, the transduction efficiency is considerably smaller than with piezoelectric or capacitive transducers. For sensing applications, we are therefore looking at continuous, resonant excitation to overcome the weaker coupling with higher quality factors. The main application of EMAT sensors continues to be in NDT applications, for detecting material defects, evaluating layer adhesion and delamination, thickness measurements, and material characterization, especially where direct contact to the sample is difficult, e.g. due to corrosion, coatings, dirt, or high temperatures.

1.2

Motivation

Both piezoelectric and capacitive transducers have been extensively investigated in recent years, yielding many new, exciting developments. Especially for acoustic, resonant sensing, both methods however show certain limitations that cannot be easily overcome. These are primarily the restriction of a resonator design to specific modes of vibration, the necessity for direct electric contacts to the resonator and thus problems with insulation, crosscouping, parasitic effects, especially in liquid environments, as well as the high material cost due to multi-layer cleanroom processing or the application of non-standard, piezoelectric materials. Electromagnetic resonators on the other hand require only a modicum of conductivity in the substrate, or in a layer on the substrate material. Therefore, the resonator material can be chosen only in terms of its mechanical, elastic properties.

12

1. Introduction

By combining magnetic direct generation with a suitable acoustic resonator material, the reduced transduction efficiency can be overcome. By driving the primary coil with an alternating continuous current instead of a pulsed current, it is possible to excite a mechanical resonance in the element [61, 62, 63, 64]. It is advantageous to tune the electrical resonance to coincide with mechanical resonance, for example by bridging the primary coil with a parallel capacitance, because the detectable signal response is then improved by the Q-factors of both resonances. This combination of utilizing a single planar spiral coil was applied by Stevenson et al. and termed magnetic acoustic resonator sensors (MARS) as a new sensing methodology [65]. The advantages of such an acoustic sensor over traditional piezoelectric transducers such as the QCR are obviously the absence of direct electric contacts to the resonator, no need for piezoelectric materials, a bare sensing surface, the ability to utilize a large variety of different materials and material combinations which have been exempted before, and the possibility of exciting a variety of different modes of vibration. These advantages can qualify the use of such devices as detectors in the traditional gravimetric regime to measure mass deposition or selective adsorption of molecules and particles, as well as sensors in the so-called non-gravimetric regime to measure material or interfacial changes due to chemical and biochemical interactions. Electromagnetic transducers can be used to excite all kinds of acoustic waves in materials, often requiring only changes to the layout and alignment of primary coil and magnetic field. Similarly, we will show that electromagneticacoustic resonator sensors can be employed for different applications requiring different modes of vibrations without any changes to the resonator element or major changes to the excitation setup. For example, the difference between exciting shear vibration modes of radial, linear or circular symmetry is primarily the positioning of the excitation coil and corresponding magnets. A multi-mode excitation setup that can be used for a variety of measurands and applications is therefore possible. With the introduction of array devices, multiple different modes can also be simultaneously excited in each element, and thus different measurands can be evaluated in one measurement.

1.3. Thesis Outline

1.3

13

Thesis Outline

In the following chapters we will detail the research efforts in modeling and simulating the excitation and detection mechanisms involved, investigate suitable electromagnetic-acoustic resonator and transducer designs, present experimental results, and finish with an outlook. Chapter 2 will cover the fundamental physical background as well as introduce possible multi-mode excitation methods, the finite element simulation employed for mode shape analysis, and the electrical equivalent circuit used to model and predict the experimental results. In chapter 3 we will thoroughly explain the resonator sensor operation. Starting with design and fabrication methods for our different resonators, silicon membrane elements, metal plates and brass sheet resonator arrays, we will analyze the different eigenmodes for these devices, describe the experimental setups used and how we excited the different thickness shear, face shear and flexural plate modes of vibration. In the second part of this thesis we take a detailed look at three applications of our resonator sensors. In chapter 4 the devices are investigated as mass microbalance. Our primary focus has been liquid phase measurement though, in chapter 5 we analyze the sensor response to changes in density and viscosity of a liquid in contact with a resonator. We have also utilized the same sensor elements for precise liquid level measurements, and corresponding acoustic wave velocity, which is described in chapter 6. The thesis is then concluded with an outlook over possible applications and future research.

2 Theory and Modeling Electromagnetic-acoustic transduction is a complex process that cannot be analytically solved as a whole. Therefore we employ different simulation methods to properly model this behavior. The mechanical, elastic properties of our different resonator elements are numerically simulated with the finite elements method (FEM), where the focus is placed on eigenmode analysis to evaluate possible modes of vibration and the necessary Lorentz force distribution to excite them. Around these eigenmode resonances both excitation and detection mechanism can be modeled with a lumped elements equivalent circuit. The basic setup consists of a primary excitation coil, an external permanent magnet and the resonator element. In the following we will discuss the mechanisms involved in exciting and detecting the acoustic resonances, and describe our models and simulation.

2.1 2.1.1

Fundamental Physical Background Mechanical Resonant Vibration

Mechanical resonators and acoustic loads are described by the equations of vibrations and waves. A simple mass-spring system can be expressed by a combination of Hooke’s law and Newton’s equation of motion. The resulting equation describes the unperturbed mechanical oscillation of the system with mass m and spring constant k. k d2 x + x=0 dt2 m

(2.1)

A solution to this differential equation can be written as: x = A sin (ω0 t + φo ) ,

(2.2)

where ω0 = k/m is the angular frequency of the harmonic motion and the two independent parameters are the oscillation amplitude A and the initial phase angle φ0 . The energy stored in a mechanical system is the sum of the

16

2. Theory and Modeling

potential energy Wp and the kinetic energy Wk . For the simple mass-spring system, the potential energy is stored in the spring: Z x 1 (2.3) Wp = kx dx = kA2 sin2 (ω0 t + φo ) , 2 0 and the kinetic energy given by the mass moving with velocity v: 1 1 2 Wk = mv 2 = mvm cos2 (ω0 t + φo ) . 2 2

(2.4)

Expanding the simple model to a more general approach requires the addition of a damping element, e.g. a mechanical dashpot. This mechanical resistance r, or its inverse, the friction coefficient h, leads to the equation of motion for a damped system. Introducing complex notation for periodic functions we can replace the trigonometric functions, e.g. x = A cos(ω, t) = Aℜ {exp(jωt)}, √ with the imaginary unit j = −1. This leads to an equation in the form of: d2 x 1 dx + + ω0 x = F ejωt , 2 dt hm dt

(2.5)

where energy would be dissipated and thus the vibration amplitude reduced with every cycle without external input. The necessary energy to keep the system oscillating is introduced by periodic external excitation forces F ejωt . For the solution of this forced vibration, we can derive a mechanical impedance Zm : k F jωt , (2.6) Zm = e = r + j ωm − v ω

Now we can define a resonance frequency ωR = 2πfR which occurs when kinetic and potential energy stored in mass and spring, respectively, are equal and thus only the real part of the impedance remains. This resonance occurs when the driving frequency coincides with the natural frequency of the oscillator, i.e. ωR = ω0 . The quality of the resonance can be described by the Q-factor, which is a measure for the ratio of stored energy and dissipated energy in the system. It can be defined as the ratio of resonance frequency and bandwidth BW or in terms of the mechanical constants as: Q=

ωR ωR m = . BW r

(2.7)

17

2.1.1. Mechanical Resonant Vibration

We can use Q as measure of the strength of a resonance and the efficiency of excitation. In a suitable medium, the basic oscillatory motion can propagate as an elastic, or acoustic, wave. The constitutive equation for a simple one-dimensional wave in the x-direction can thus be written as [8, 7]: 2 ∂ 2u 2∂ u =c , ∂t2 ∂x2

(2.8)

with displacement u and wave propagation velocity c. For an elastic solid we introduce the stress tensor Tkl = Fk /Al and the strain tensor Skl = 1/2 (∂uk /∂xl + ∂ul /∂xk ), where the indices k and l take on the x, y or z direction. The diagonal terms, e.g. Txx , represent compressional stress and strain, while the non-diagonal terms represent shear stress or strain. Both stress and strain tensor are symmetric, therefore only six of the nine elements are unique and the number of indices can be reduced to one. The constitutive equation of elasticity, expanded into three dimensions and utilizing this reduced tensor notation (due to symmetry), is given by Ti = cij Sj ,

(2.9)

where cij are the elastic stiffness constants of the material. As the stress is defined as a force per area, we can use Newton’s law to rewrite the wave equation in an elastic medium as ∂ 2u ∂ 2u ρ 2 = cij 2 . ∂t ∂x

(2.10)

In an elastic medium, the wave velocity in a specific direction is thus given p by c = cij /ρ. As with a simple vibration, an elastic wave propagating through a medium results in kinetic and potential energy being stored in the medium. The kinetic energy density wk (energy per unit volume) is defined similar to equation 2.4 1 ∂ 2 ui 1 (2.11) wk = ρv 2 = ρ 2 . 2 2 ∂t The potential energy is stored in a strained volume, a change of potential energy density is given by dwp = Ti dSi . Following the formula for potential

18

2. Theory and Modeling

Figure 2.1: Fundamental eigenmodes for thin plates and membranes.

energy stored in as simple mass-spring system, we find that 1 wp = cij Si Sj . 2

(2.12)

By balancing these energy densities we can again calculate the resonance frequency, as discussed later in chapter 4. Depending on the propagation direction and polarization (direction of the particle displacement) of the acoustic wave, a resonator element can experience a variety of fundamental vibration modes. For thin plates and membranes, these modes are flexural, extensional, face shear and thickness shear motion, which are schematically depicted in figure 2.1. These fundamental modes can be found in different mode shapes, i.e. the geometric shape of the displacement, and of course with many higher harmonics and superposed combinations of different modes. Analytical solutions for the displacement, velocity or energy distribution are only possible for a select few with specific boundary conditions [66, 67]. In general, the complexity of geometry, boundary conditions, mode shapes and distribution of the external forces requires non-analytical approaches to model the mechanical resonators and acoustic loads.

2.1.2. Resonator Excitation and Detection

2.1.2

19

Resonator Excitation and Detection

A coupling mechanism between electromagnetic and acoustic waves has been first observed in the late 1930s and confirmed by the late 1960s. This effect was then termed magnetic direct generation of acoustic waves by Quinn [53]. The fundamental mechanisms involved can be explained by Maxwell’s equations. According to Ampere’s law, a magnetic field is created by the movement of electric charges. ~ ~ = J~ + ∂ D ∇×H ∂t

(2.13)

~ are current densities J~ and time variant The curls of a magnetic field H ~ This magnetic field, generated here by electric curelectric displacements D. rents in the primary coil, is time variant in a harmonic manner characterized by the frequency of the driving current. It therefore is the source of an in~ according to Faraday’s law of induction with µ as the duced electric field E permeability of the material penetrated by this electromagnetic field. ~ ~ = −µ ∂ H ∇×E ∂t

(2.14)

In a conductive material this induction gives rise to the electromotive force (emf) which affects the free charges of the conductor. This emf, or induced voltage, causes electrical currents in the conductor. According to Lenz’s law, these eddy currents are circulating in such a direction to produce a magnetic field that directly opposes the change in the primary magnetic flux. In the case of a planar spiral coil for example, the eddy currents in a conductive layer placed in parallel above the coil will flow in the opposite clockwise direction as the primary current in the coil. The penetration depth, or skin depth, of these eddy currents can be derived from the electromagnetic wave equation resulting from Maxwell’s equations 2.13 and 2.14 [68], in complex notation for the electric field written as σ~ E = 0, ∇ E + ω µε 1 − j ωε 2~

2

(2.15)

20

2. Theory and Modeling

with angular frequency ω, permittivity ε, conductivity σ, and j as the imaginary unit. We can define a complex propagation constant γ as r σ √ (2.16) γ = α + jβ = jω µε 1 − j . ωε A basic plane wave solution of equation 2.15 in the z-direction with only an x component is given by Ex (z) = E + e−γz + E − eγz +. For a good conductor, where the conductive current is much larger than the displacement current, we can use the approximation σ ≫ ωε to reduce equation 2.16 to r r σ ωµσ √ = (1 + j) . (2.17) γ = α + jβ ≈ jω µε jωε 2 The penetration depth δ of the electromagnetic wave is then defined as the depth in the penetrated material where the amplitude of the fields has decayed by 1/e or 36.8%. For a conducting material it thus defines the required conductive thickness for optimal electromagnetic coupling at a specific operating frequency, and is given by r 2 1 . (2.18) δ= = α ωµσ After eddy current generation, the second mechanism involved requires a superposed magnetic field. Corresponding to the Lorentz force law a charged ~ or moving through a magnetic field B ~ with particle q in an electric field E the velocity ~vq is subjected to a force F~L [54, 55]. This force acts as the external driving force in equation 2.5 for the resonator and is given by: ~ ~ ~ (2.19) FL = q E + ~vq × B .

~ of the electromagnetic The Coulomb force due to the electric component E field created by the primary current vanishes if the net charge of the material is zero and only a magnetic component results in a force in the resonator. The primary Lorentz force density f~L for electromagnetic excitation can then be written as ~ f~L = J~e × B, (2.20)

where the eddy current density J~e = ρ~vq is given as product of movable charge density ρ and its velocity ~vq . These Lorentz forces transfer a periodic

21

2.1.2. Resonator Excitation and Detection

643

25 20 15 10 5 7.5

10.0

12.5

15.0

Frequency f / MHz

17.5

20.0

Measured Impedance Zi /Ω

Measured Impedance Zi /Ω

Measured Impedance Zi /kΩ

650 30 600 550 500 450 400 350

7.5

10.0

12.5

15.0

Frequency f / MHz

17.5

20.0

642

641

640

639

15.0

15.2

15.4

15.6

Frequency f / MHz

15.8

Figure 2.2: Impedance of the primary coil with no conductive material present (left), with a conductive material present resulting in eddy current induction (center), and with a conductive resonator present resulting in eddy current induction (large peak), Lorentz force generation, resonant vibration, and movement induction of a secondary eddy current (small peaks) (right).

momentum to the moving free charges, which in turn interact with the crystal lattice of the conductive material and cause it to vibrate with the frequency of the primary current in the coil. Thus, an acoustic wave is generated in the sensor element. At driving frequencies corresponding to the natural frequencies of the resonator mechanical resonance is achieved. For the case of a thickness shear mode vibration, the resonant wave propagates between the upper and lower faces of the sample. A detailed theoretical investigation of this combination of the electrical, mechanical and acoustic domains, including magnetization and magnetostriction effects, can be found in [69]. To explain the detection mechanism, we have to understand the movement induction in the vibrating plate in the static magnetic field. Figure 2.2 shows the changes to the impedance of a planar coil. When operated in air without a conductive material present, the impedance is only determined by the electrical inductance, resistance and capacitance of the coil setup (left). Once a conductive material is placed in the vicinity, eddy currents will be induced as described above, resulting in a transformer-like setup where the magnetic field generated by the eddy currents opposes the magnetic field of the primary coil, leading to a mutual inductance which changes the impedance measured at the primary coil (center). When a conductive resonator is placed in the vicinity and superposed with a static magnetic field, Lorentz forces will be generated that can excite resonant vibrations at the eigenmodes of the resonator. These vibrations in

22

2. Theory and Modeling

a magnetic field naturally represent a movement of charges in a magnetic field, which in turn induces a secondary voltage according to the general law of induction. The induced voltage results in secondary eddy currents in the vibrating element. This effect is generally negligible, due to the mechanical vibration amplitude being several orders of magnitudes smaller than the eddy current vibration amplitude. In resonance however, the mechanical vibration amplitude is strongly enhanced according to the Q-factor of the mechanical resonator. In principle, the secondary current is 180◦ out-of-phase to the primary current, however the phase of mechanical vibration and electrical eddy current must be taken into account. This effect appears as additional peaks in figure 2.2 right. These peaks are employed as the major sensor signal, reflecting important characteristics of the resonator in its specific vibration mode, including the acoustic load acting on the resonator. Note that the secondary eddy currents also give rise to Lorentz forces, which, in general, weaken the excitation force. The total eddy current density J~t in a magnetic field can be written as the sum of primary and secondary ~ e, eddy current, due to the electrical field induced by the primary current, E and due to movement induction with a local resonator velocity, ~v , respectively. This equation also represents Ohm’s law in a moving frame of reference: ~ ~ ~ Jt = σ Ee + ~v (t) × B . (2.21)

The total current density in turn results in a secondary time-variant magnetic field that induces a voltage in the primary coil, which changes the impedance accordingly. Outside mechanical resonance, only the primary eddy current determines the impedance change, while in resonance additional currents are superposed to the primary eddy current (Fig. 2.2). Amplitude and phase of the secondary currents depend on primary coil layout, Lorentz force distribution, the specific vibration mode shape, and the resonator material properties, such as conductivity, elasticity, electron mobility, as well as acoustic boundary conditions induced by the surrounding medium. An analytical solution is only possible for very simple, special cases, e.g. a single linear conducting wire, therefore we will focus on numerical models to describe the complex

23

2.2. Multi-mode Excitation Methods

0.03 0.02 0.01 0 −0.01 −0.02

220

221

222

223

Frequency f / kHz

224

4

Normalized Impedance Zi /Ω

0.25

Normalized Impedance Zi /Ω

Normalized Impedance Zi /Ω

0.04

0.2 0.15 0.1 0.05 0 −0.05 −0.1

246

247

248

249

Frequency f / kHz

250

3 2 1 0 −1 −2

15.410

15.415

15.420

Frequency f / MHz

15.425

Figure 2.3: Effect of different phase responses in the normalized impedance of the primary coil around face shear (left), flexural (center) and thickness shear resonance (right) of a silicon sensor element.

measurement setup. The effect of different phase responses of the mechanical vibration can be seen in figure 2.3, showing both reduction (impedance amplitude increases) and improvement (impedance amplitude decreases) of the primary eddy current. The mechanical resonance can thus be detected by an RF analyzer circuit monitoring the coil impedance. Additionally, the signal response is also amplified by driving the setup at electrical resonance. This is achieved by bridging the planar spiral coil with a (tuning) capacitor and setting the electrical resonance to the same frequency as the mechanical resonance, and has been termed magnetic acoustic resonator sensors (MARS) as a new sensing methodology [65].

2.2

Multi-mode Excitation Methods

Linear, or diagonal, radial, and circular symmetry are the three fundamental mode shapes for in-plane, shear vibration (see, e.g., Fig. 2.7). Flexural motion can be more complex. To realize a true multi-mode resonator, we need to realize a corresponding Lorentz force distribution for all these fundamental mode shapes, i.e. also forces of linear, radial and circular direction. With the advantage of spatial separation of resonator and excitation, we have multiple degrees of freedom, which are the geometry of the primary coil(s), the direction of the external magnetic field(s), and the alignment of resonator, coils and magnets. We also have to consider the electromagnetic coupling

24

2. Theory and Modeling

Figure 2.4: Principle of resonance excitation with a planar spiral coil and perpendicular magnetic field resulting in radial Lorenz forces.

efficiency between primary current and induced eddy currents though, also described by primary inductance and mutual inductance, respectively. The most common coil geometry employed for this sensor principle has been a planar spiral coil, due to high electromagnetic coupling, with a vertically perpendicular magnetic field. However, applying the Lorentz force law to the induced eddy currents in the resonator, this layout yields in-plane radial forces when centered under the resonator element (Fig. 2.4). These radial forces will preferably excite radially symmetric vibration mode shapes with in-plane, shear components. According to finite element method (FEM) simulation, this radial vibration, however, is coupled to an out-of-plane movement at the center of the element due to the conservation-of-mass principle. This vibration component generates unwanted compressional waves, which radiate energy into the surrounding medium, leading to additional damping in liquids. Therefore, a number of alternative vibration modes are of particular interest for liquid phase sensing, especially the diagonal and circular shear modes. The distribution of the Lorentz forces is affected by the distribution of the eddy currents and the direction of the magnetic field. Therefore, in order to generate non-radial forces and excite different modes of vibration, the coil geometry has to be adapted and the magnetic fields have to be changed accordingly. Eddy currents basically mirror the layout of the primary coil. Therefore, quadratic or rectangular spiral coils coupled with an alternated permanent magnetic field are suitable for excitation and detection of linear, or diagonal, lateral shear modes. These coils induce components of linear eddy current

2.2. Multi-mode Excitation Methods

25

flow, which result in a linear Lorentz force distribution. The alternated magnetic field accounts for the necessity of a coil structure for high inductance and the thus alternating flow directions. Multiple parallel, linear conducting lines would also result in linear Lorentz forces, however the non-contact, electromagnetic eddy current coupling efficiency would be minimal due to very low inductances. The linear, or diagonal, vibration is similar to the thickness shear mode shape in QCM sensors and will lead to improved liquid phase sensing capabilities. Fig. 2.5 shows the modified principle for generating diagonal forces and therefore a diagonal shear mode. There are also other coil geometries possible for generating these forces, such as linear solenoids or meander line structures, which suffer from a decreased electromagnetic coupling. At the edges of the vibration area a linear movement will however also result in out-of-plane components due to conservation of mass. The excitation of pure circular shear modes, or torsional modes, is most suitable for pure in-plane, shear vibration, due to the lack of singularities, especially if the whole resonator element twists itself. To generate the necessary circumferential forces, radial eddy currents are required. Placing planar, spiral coil(s) out of center of a resonator yields alternated, radial current components. Superposed by vertically perpendicular, alternated magnetic fields results in a force distribution with a dominating circumferential component [70]. At proper frequencies, these forces will result in a circular shear motion, or also called torsional vibration mode.

Figure 2.5: Modified principle for generation of linear, diagonal Lorentz forces.

26

2. Theory and Modeling

Figure 2.6: Complete FEM excitation model with planar spiral coil placed at the center of the resonator element.

2.3

Finite Element Simulation

The experimental setup has been modeled by means of the finite elements method (FEM) using the software suites ANSYS and COMSOL Multiphysics [71, 72]. The primary goal was to develop a model to describe and visualize the excitation mechanism and predict the possible modes of vibration. The complete model is shown in figure 2.6. The FEM simulation of the excitation mechanism is divided into three separate steps. These are the eddy current induction, Lorentz force generation due to a superposed magnetic field, and the mechanical vibration of the element [73]. The effect of the secondary eddy currents on the mechanical vibration is neglected. The induction is simulated in a harmonic electromagnetic analysis, with the Lorentz forces calculated by superposition with the static magnetic field [74]. In the final step, these forces are applied to the resonator element in a transient mechanical analysis to visualize the possible resonant modes of vibration that can be excited by a certain Lorentz force distribution [75]. The primary focus of FEM simulation was the investigation of the eigenfrequencies of the different resonator elements in a mechanical modal analysis. This simulation revealed the possible vibration mode shapes for each element depending on geometry

2.3. Finite Element Simulation

27

Figure 2.7: Exemplary excitation simulation of a radial TSM at 15 MHz in a circular silicon plate, with eddy current induction (left), Lorentz force generation (middle) and resulting thickness shear mode vibration (right).

and boundary conditions, and thus showed the necessary force distribution. Selected mode excitation was also verified with the complete excitation simulation as described above (e.g. Fig. 2.7), but in general successful excitation was verified primarily via experimental means. Special simulation challenges appeared with the elastic anisotropy of some materials, e.g. (100) silicon. While isotropic materials can also be roughly modeled in two dimension, anisotropic materials require an appropriate three-dimensional model which leads to greatly increased computational requirements. The simulation has shown three distinct fundamental vibration symmetries at different fundamental frequencies for the shear modes of a clamped circular or rectangular membrane. The face and thickness shear modes can result in diagonal, circular and radial displacements. For the face shear modes the fundamental frequencies of these vibrations are very different from each other (Fig. 2.8), while the three thickness shear modes are excited at approximately the same frequency. The excitation simulation shows that radial Lorentz forces only excite radial shear modes efficiently, as it is expected. The diagonal and circular shear modes are superposed by higher harmonics of other modes and no pronounced resonant mode shape can be excited. Furthermore, the analysis shows a small out-of-plane displacement at the center of the radial vibration, resulting from mass conservation effects. Due to compressional vibration of the element towards a singularity at the center, but limited compressibility of the resonator material, radial motion

28

2. Theory and Modeling

Figure 2.8: Fundamental face shear mode shapes of a round silicon membrane with displacement of diagonal/linear (left), circular (center) and radial (right) symmetry for a circular plate.

experiences this out-of-plan buckling. This undesirable vertical displacement is of similar amplitude as the primary shear movement and therefore not negligible. For liquid phase sensing this additional component will radiate compressional waves into the liquid and create a higher attenuation. Therefore, a setup consisting of a planar spiral coil is not the ideal solution for liquid phase measurements, as we show in later chapters. FEM simulation has primarily been used as a tool to predict the possible modes of vibration for a given resonator geometry and to design our excitation setups accordingly, as well as to limit the amount of measurement data necessary to find a specific mode. A more detailed discussion of the different possible mode shapes for each resonator element and application will be given in the following chapters.

2.4

Equivalent Circuit Modeling

The second modeling aspect is the investigation of the feedback mechanism into a measurable electrical quantity. Building an FEM model incorporating this mechanism is theoretically possible, but not feasible in terms of required computation steps. The movement induction of a secondary current and the resulting induced secondary voltage in a primary coil could have been simulated, but the combination with the actually excited mode shape and the superposition of eddy current and secondary current in the moving resonator would have resulted in memory and computational requirements beyond our capacities. Coming from electrical engineering, a far more ac-

29

2.4. Equivalent Circuit Modeling Table 2.1: Electro-mechanical equivalences with the Mobility Analog. voltage

u

v

velocity

current

i

F

force

inductance

L

n

elastic compliance

capacitance

C

m

inertial mass

resistance

R

h

friction coefficient

G

r = 1/h

conductance impedance

Z = R + j ωL −

L

1 ωC

u = jωLi

Z m = h + j ωn −

mech. impedance

n

m u=

1 i jωC

v=

1 F jωm

h

R

Voltage Law

v = jωnF

C

Current Law

mech. resistance 1 ωm

u = Ri P ⋆i = 0

P

◦u = 0

v = hF P ⋆F = 0 P

◦

v=0

sum of nodal forces loop velocities

cessible approach is to depict this electro-mechanical system as an electric equivalent circuit, where excitation coil, eddy currents, mechanical resonator, and acoustic load are modeled as lumped elements. A similar approach has been successfully utilized to model piezoelectric acoustic microsensors as well, e.g. [76, 77, 78, 79]. The electrical, mechanical and acoustic domains are governed by differential equations of similar form and they can be described by the same mathematical network model [80]. When choosing the network coordinates force F and velocity v, the mechanical equation of motion can be written as a parallel mechanical circuit (Fig. 2.9): Z 1 dv 1 v dt = F (t) , (2.22) m + v+ dt h n with m as inertial mass, h as mechanical damping or friction admittance, and elastic compliance n, the inverse spring constant. In this form, termed Mo-

30

2. Theory and Modeling

bility Analog, the similarity to the governing equation of a parallel electrical circuit is evident. Z 1 1 du u dt = i (t) (2.23) + u+ C dt R L Here, the relationship between current i and voltage u is influenced by the electrical circuit elements of inductance L, resistance R, and capacitance C. As we can see, with this method the flux of the system is given by force and current, while the potential at a network node is given by velocity and voltage. The equivalent circuit description of the mechanical system follows the same laws as an electrical circuit. Just as Kirchhoff’s law states that the sum of all currents flowing into an electrical network node is zero, similarly the sum of all forces acting upon a mechanical network node is zero. And as the sum of voltages in a closed loop within an electrical circuit is zero, the sum of velocities in a closed loop in a mechanical circuit is zero as well. A complete analogy is given in table 2.1. Both systems exhibit an isomorphic, i.e. structurally identical, behavior. It is also possible to utilize the converse, so-called Impedance Analog, where the network flux and potential are given by velocity and force, respectively. Both analogies will yield equivalent results and can also be translated using the dual (dot) method [81]. The Mobility Analog however is more intuitive, e.g. mechanically connected objects move with the same velocity, i.e. are thus ’wired’ in parallel (Fig. 2.9). The mechanical circuit can thus be analyzed using the same network tools and programs used for electrical circuits without additional verification or modification. The same analytic method can also be applied to electro-acoustic analogies, where the flux component is given by the pressure. Using electrical network theory and methods, it is advantageous to also switch from the time-domain description to a complex, frequency-domain description, as used in table 2.1. While this limits the model to sinusoidal functions, any periodic function can be decomposed into a sum of sinousoidal functions with the help of a Fourier series, thus the conversion is possible without loss of generality. Complex functions, or phasor calculus, simplify many calculations of harmonic signals, reducing trigonometric functions in the time-domain to the time-invariant parameters amplitude, phase and fre-

31

2.4. Equivalent Circuit Modeling

F F v

m

F m

n

r

m

n

r

r

n

v

v

Figure 2.9: From mechanical description and schemata to equivalent circuit with the Mobility Analog (left to right).

quency (ω). Kirchhoff’s circuit laws, Ohm’s law, power law are equally applicable to networks with time-harmonic flux and potential. Complex variables are here written with an underscore, with j as the imaginary unit in order to not confuse it with current i. With the isomorphism of electrical, mechanical and acoustic network it is now possible to combine all domains into a single model, which can then be analyzed as a whole. To establish a quantitative relationship between flux and potential of the different domains, we have to define proportionality constants between the different network coordinates. (2.24) u = Yv 1 F i = (2.25) X The potential equation relates voltage u to velocity v with the coupling coefficient Y , describing the movement induction as in eq. 2.21. The flux equation relates force F to current i and is a different description of the Lorentz force law (eq. 2.20) for our application with a second coupling coefficient X. In general, this proportionality can also be factored directly into the element properties, i.e. a mass element in kg can be converted into a capacitance in F. With this coupling we can successfully model the complete system of electrical excitation, mechanical resonator and acoustic load. For a so-called electrodynamic transducer, i.e. an electromagnetic transducer utilizing Lorentz forces, the electrical side is made up of inductance and resistance of the current carrying path, driven by a primary AC voltage

32

2. Theory and Modeling

i

u

L

R

FL

uv

F

m

h

n

v

Figure 2.10: Electromagnetic-acoustic transducer with electrical equivalent circuit (left) and mechanical equivalent circuit (right).

u. The mechanical side is driven by a Lorentz force source F L , which is a function of the primary current, the external magnetic field B0 and the current path geometry, and is modeled as a parallel circuit of mass, compliance and friction coefficient, resulting in a mechanical vibration with velocity v. Conversely, this vibration in the magnetic field leads to the induction of a secondary voltage uv resulting in a secondary current superposed to the primary current i. The mechanical side is governed by the differential equation 1 v − jωmv − rv. F = F L (i, B0 ) − (2.26) jωn The electrical side can be written as u = Ri + jωLi + uv (v, B0 ) .

(2.27)

As we can see, the Lorentz force generation is modeled as current-controlled current source, while the secondary movement induction is modeled as a voltage-controlled voltage source. These two elements can consequently be combined in an electro-mechanical transformer element, with the parameters being the two coupling coefficients X and Y , based on magnetic field and geometry. However, a complete EMAT resonator also utilizes eddy current induction, i.e. the primary current is not impressed into the mechanical element but remotely induced. This mechanism can be modeled by a lossy electromagnetic transformer. Note that this lumped elements model is only valid under the assumption of frequency independent elements. Furthermore, for specific coupling coefficients X and Y , the model only serves as a reasonable approximation in the

33

2.4. Equivalent Circuit Modeling

Ri AC

Zi

Rc

T1

T

Ct Lc

Re Le

k

T3

FL

i uv

{X, Y } T2

v

h

n

m

T4

XT

Excitation circuit

Eddy currents

Acoustic resonator

Figure 2.11: Electromagnetic-acoustic equivalent circuit, with eddy current induction described by electromagnetic transformer T , coupling between current i, voltage u and force F , velocity v in the resonator modeled by electro-mechanical transformer XT .

vicinity of a particular resonance frequency. For different modes of vibration, different coupling coefficients have to be fitted to the measurement data. The resulting complete electromagnetic-acoustic equivalent circuit is shown in figure 2.11. We measure the input impedance Zi with an impedance analyzer represented as the AC source and input resistance Ri . The electrical circuit consists of the primary coil inductance Lc and resistance Rc , with an external tuning capacitor Ct in parallel to achieve electrical resonance according to the MARS principle [65]. The induced eddy currents experience a secondary inductance Le and resistance Re in the conductive part of the resonator. A current-controlled current source couples the eddy currents i and Lorentz forces FL , while a voltage-controlled voltage source is responsible for the induction of a secondary voltage uv due to the resonator velocity v, which are contained in the electro-mechanical transformer XT . In the frequency domain this electro-mechanical coupling can be described by the following complex equations. F L = X (B0 , . . .) i

(2.28)

uv = Y (B0 , . . .) v

(2.29)

The coupling coefficients X and Y are in our case dependent on the magnetic flux density B0 and the geometry of Lorentz force generation and the vibration mode shape. On the mechanical side the resonator is modeled

34

2. Theory and Modeling

14.4

33.5 Measurement Simulation

33.0

14.2 32.5 14

Phase / °

Impedance Amplitude / Ω

Measurement Simulation

13.8

32.0 31.5 31.0

13.6 30.5 13.4

287.4

287.6

287.8

Frequency / kHz

288.0

30.0

288.2

708

287.8

Frequency / kHz

288.0

288.2

Measurement Simulation

19.2

706

19.1

Phase / °

Impedance Amplitude / Ω

287.6

19.3 Measurement Simulation

707

705 704

19.0 18.9

703

18.8

702

18.7

701

287.4

15.41215.41415.41615.41815.42015.42215.42415.426

Frequency / MHz

18.6

15.41215.41415.41615.41815.42015.42215.42415.426

Frequency / MHz

Figure 2.12: Comparison of impedance amplitude (left) and phase (right) for equivalent circuit simulation and measurements of face shear mode (top) and thickness shear mode (bottom).

as a parallel resonant circuit, with vibrating mass m, elastic compliance n, and damping h calculated from geometry and the material properties, e.g. density and elastic stiffness. An acoustic load on the resonator gives rise to additional elements, which depend on vibration mode and load properties, e.g. mass or viscous loading for shear vibrations result in parallel mass and damping elements. Figure 2.12 shows very good agreement between equivalent circuit simulation and measurements for exemplary radial FSM and TSM resonances without an acoustic load, which are described in detail in chapter 3. The model is suitable for all excited modes of vibration, including flexural, face shear, and thickness shear modes. The appropriate load elements will be introduced in the respective chapters, i.e. additional elements for mass loading, viscous loading and compressional wave propagation.

2.5. Summary

2.5

35

Summary

The principle of electromagnetic excitation of acoustic waves is a well known mechanism. The application to resonant sensors and exploitation of the resonant vibration in a sensor element is the new approach used in our research. As resonant sensors, we can utilize our devices in a similar fashion as established piezoelectric or capacitive transducers with the added benefit of unique advantages such as spatial separation, bare sensor surfaces, and excitation of fundamentally different vibrations in the same resonator. The physical background is given by eddy current induction and Lorentz force generation, whereas the excited resonant vibration influences the eddy currents, and thus the induced voltage in a primary coil. In the equivalent circuit model we established, the coupling between electrical and mechanical domain is given by two transformers. These are an electromagnetic transformer describing the eddy current coupling between primary coil and conductive element, and an electro-mechanical transformer describing the Lorentz force excitation and movement induction between eddy currents and resonant vibration. Together with an FEM analysis of the mechanical eigenmodes of the resonator, this simple equivalent circuit model is suitable to describe the physical mechanisms and predict and analyze sensor measurements.

3 Basic Resonator Operation Electromagnetic-acoustic excitation can be applied to any kind of resonator material, the only requirement being a conductive substrate or layer for eddy current induction. Therefore, we can focus on different aspects for different applications, e.g. inexpensive mass-produced sensor elements, substrate materials with high elastic stiffness and minimal temperature coefficients, or materials suitable for integration in specific measurement setups. In this chapter we introduce the three main sensor elements employed in our experiments and describe the design and fabrication process employed, the eigenmode simulation results for each device and the appropriate experimental setups to excite the different possible modes of vibration. As a resonator substrate with excellent elastic properties and very low temperature coefficients similar to quartz crystal resonators we utilized silicon. In a prototype process silicon membranes were etched and a conductive copper layer applied. These devices have shown acoustic resonances with very high quality factors (Q) comparable to standard AT-cut quartz sensors. For lowcost applications we chose simple, polished aluminum and brass plates. To improve the suitability as resonators, we additionally milled a mesa-structure into these plates, to trap the vibrational energy in the sensing region. The third focus has been placed on multi-mode excitation. For this application we devised prototype resonator arrays out of thin metal nickel silver sheets, where we successfully realized simultaneous excitation of different modes of vibration, which allows for the simultaneous measurement of multiple physical properties of an analyte.

3.1 3.1.1

Design and Fabrication Silicon Membrane Resonators

In order to achieve optimal experimental results, a microfabrication process was set up to create high-Q silicon membrane elements [82, 83]. This

38

3. Basic Resonator Operation

Figure 3.1: Simplified 7 step silicon fabrication process; substrate (grey), oxide mask (red), photoresist (yellow), metal (purple).

prototype process also aimed for a low-cost fabrication. Single mask photolithography for structuring the silicon substrate has been used together with a minimum number of standard process steps to validate the feasibility of that goal. The anisotropy of the material silicon has been one reason why it has not yet been investigated as a resonator excited by magnetic direct generation. Therefore both standard (1 0 0) silicon wafers with anisotropic mechanical properties and (111) wafers with isotropic properties have been acquired. Ten 4′′ (100 mm) double polished wafers of the (1 0 0) crystalline orientation were sourced from a wafer batch of Siltronic AG. These phosphorus n-doped wafers displayed a spread resistivity of 20.3 Ω cm and had a specified substrate thickness of 383.4 µm. The surface flatness of these Czochralski grown wafers is given with a maximum deviation of ±0.5% and a total thickness variation of < 5 µm. For the (1 1 1) orientation only a single double polished wafer could be acquired from the Technical University of Chemnitz, Germany. Figure 3.1 shows the simplified fabrication process. Starting with the cleaned silicon substrates (gray), a 1 µm thick, thermal oxide layer (red) was grown on the wafers and structured in the photolithography step. First, a 8 µm thick photoresist (AZ4562 ; yellow/light grey) was spun onto the wafers. In a S¨ uss MA/BA6 mask aligner the 5′′ transparency mask with the structure for 12 round and 12 quadratic 15 × 15 mm membrane elements was used to

3.1.1. Silicon Membrane Resonators

39

Figure 3.2: SEM image of sidewall of round silicon membrane with characteristic DRIE structure.

expose the resist. After development the opened oxide was then structured in a BHF etch bath. Both remaining oxide and resist layer created the etch mask for the reactive ion etching (RIE) process, which was chosen due its intrinsic advantages of steep sidewalls, sharp edges and smooth membrane surfaces. RIE was also necessary to anisotropically etch (111) substrates. The last clean room step was to deposit a 1 µm thick layer of copper metalization on the opposite side (purple), after removing all remaining resist and oxide. The wafers were then ready to be diced. The membranes were characterized with scanning electron microscopy (SEM) and an optical profilometer for surface smoothness, sidewall steepness and transition and possible defects. These investigations showed well-defined membranes with very sharp transitions from sidewall to membrane surface (figure 3.2), which promise only minimal acoustic dissipation and damping effects. The metal layer thickness is suitable for an efficient excitation of thickness shear modes at several tens of MHz, while the acoustic load represented by this copper layer does not significantly influence the resonant behavior except a frequency shift to slightly lower frequencies due to the added surface mass, see chapter 4. However, for face shear modes in the range of several

40

3. Basic Resonator Operation

hundreds of kHz, this metal thickness is much smaller than the penetration depth of the eddy currents (see eq. 2.18), and therefore the excitation is less efficient. Further fabrication steps were needed to realize a new optimization of conductive layer thickness and damping characteristics to efficiently excite and detect these face shear mode vibrations. To increase the copper layer thickness, we have utilized a galvanic setup (PPG 10/3-B 3 L, Heimerle+Meule, Germany) for miniature samples to electroplate several additional microns of copper onto the silicon elements [84]. With the 1 µm copper as a seed layer and using a copper-phosphorus electrode and the acidic copper bath CU 500, we achieved a deposition rate of 1 − 3 µm per minute, which was adjusted to achieve a high uniformity and surface smoothness. While the face shear modes of the unplated membranes are barely detectable, the thickness shear modes (TSM) show strong resonance peaks and can be used for mass sensing applications in gaseous environments. Besides the frequency shift due to the added mass, the efficiency of mode excitation and corresponding quality factors are also affected by an increase of copper layer thickness and changes to the copper surface. Here, an optimum of additional deposition for the excitation of TSM vibrations was obtained at a few µm (Tab. 3.1). The coupling efficiency increases up to a point where the additional inhomogeneity and surface roughness introduces a surpassing amount of damping to the resonant behavior. Figure 3.3 shows the changes to a radial face shear mode at ≈ 250 kHz with increasing copper layer thickness in terms of peak impedance change ∆Z and peak phase shift ∆Θ detected in the primary impedance of the excitation coil. Transduction efficiency, which is characterized in terms of ∆Z and ∆Θ Table 3.1: Changes to the radial thickness shear mode of (100) Si membrane at 15 MHz due to increasing copper layer thickness. Cu layer

fR / MHz

Zi / mΩ

Θ / m◦

1µm

15.89

691.00

52.07

≈ 2µm

15.57

965.79

105.90

≈ 4µm

15.15

228.81

33.06

41

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 −0.1 −0.2

~15 µm

~4 µm ~2 µm

increasing Cu thickness

240

242

244

246

Frequency (kHz)

248

250

1.2 1.0 0.8 0.6 0.4 0.2 0 252

Phase shift (°)

Normalized Impedance (Ω)

3.1.1. Silicon Membrane Resonators

Figure 3.3: Changes to a radial face shear mode of a (100) silicon membrane with increasing copper layer thickness.

is increasing in a fairly linear fashion with increasing thickness, while the influence on the resonance quality factor is still negligible up to several ten µm. The measured Q is also dependent on the electromagnetic eddy current coupling and the Q of the primary coil, thus a reduction of mechanical Q may not be readily apparent without further analysis (see MARS principle). With these changes to our sensor elements it was possible to transfer the results of pure metallic resonators to the composite silicon membranes and take advantage of the stiffer elastic properties of silicon resulting in larger quality factors and better defined resonance peaks. Although electroplating techniques do not allow a precise prediction of deposition rate and layer thickness, we were able to use the fitted simulation models to evaluate the deposition process. Additionally we used the Sauerbrey equation for mass microbalance to correlate the frequency shift of the thickness shear mode resonances to a corresponding additional mass (see chapter 4). The comparison between FEM simulation, approximation from deposition parameters, e.g. applied current density and total coating area, and acoustic microbalance are the basis of our thickness estimate.

42

3. Basic Resonator Operation

Figure 3.4: Brass and aluminum metal plates of 1 mm thickness with round mesa structure of 15 mm diameter milled in center.

3.1.2

Metal Plate Elements

Metal plates have been used as a proof of concept in the earliest stage of this research. Due to the high conductivity of complete sensor substrate, the electro-mechanical transduction coefficients are much higher than for composite resonators. However, the elastic properties of metals and especially the temperature coefficients are worse then, e.g., the silicon membranes. As very simple, inexpensive resonators we have continued to employ these elements in our experiments. To improve the acoustic resonant behavior we processed aluminum and brass plate elements in an in house milling process to create 1 mm thick, polished, coplanar mesa-shaped structures, see figure 3.4. A central, circular mesa structure of 15 mm diameter was defined by milling a 500 µm deep, 2 mm wide trench into the substrate. This design effectively traps the acoustic energy to the mesa structure [70], similar to the energy trapping in quartz crystals due to the electrode geometry. This method reduces undesired vibrational components at the outer clamping area and confines the vibration to the actual sensing region.

3.1.3

Resonator Array Sensors

To utilize multiple different vibration modes and their different sensitivities we devised a sensor array setup consisting of multiple resonator elements

3.1.3. Resonator Array Sensors

43

Figure 3.5: Design of 2x2 resonator array setup with individual coils and magnets for each element.

anchored to a common frame, as shown in figure 3.5. With a variety of excitation options, each element can vibrate in the same or different resonant mode, depending on the Lorentz force distribution and excitation frequency applied to each element. We have fabricated and developed different coil designs and magnet layouts corresponding to suitable Lorentz force distributions. These include coil arrays of planar spiral and quadratic design to generate radial and diagonal forces across each element, as well as larger single planar spiral and quadratic coils that can simultaneously excite circular and diagonal forces in each element. The magnet placement has to be adjusted accordingly. While the setup shown in figure 3.5 generates radial forces, a different alignment of the permanent magnets changes these to a linear horizontal direction. For flexural plate modes an external magnetic field in-plane with the sensor array is most suitable, requiring the placement of the magnets on the sides of the array in a vertical position. Prototype metal resonator 2x2 arrays were fabricated out of nickel silver sheets of 100 µm thickness with different designs for resonator element and spring couplers connecting to the common frame [85]. For one design

44

3. Basic Resonator Operation

Figure 3.6: Metal prototype arrays with different coupling springs and element size of 10x10 mm (top) and different corresponding excitation coil arrays on polymer foils with 100 µm metalization thickness.

the spring couplers were placed at the edges of the elements creating a short, stiff connection to the frame. Another type utilized a double U-shaped spring on each side, which reduced coupling effects due to higher damping of waves radiated into the springs. However, for better decoupling, different designs will be necessary. The metal sheets were patterned and etched in a standard photolithographic process. Additionally, a similar process was used to fabricate multiple primary coils and coil arrays for simultaneous excitation of the resonator elements. These designs were realized on copper-coated polymer foils and standard PCBs. A selection of these arrays and elements is shown in figure 3.6. As discussed before, another aspect to consider is the penetration depth of eddy currents (see 2.18). While this is usually not a problem with conductive resonators, for very thin substrates or composite resonators the thickness of the conductive layer greatly influences the potential vibration modes. The penetration depth of eddy currents δ is inversely proportional to the square

3.2. Eigenmode Analysis

45

root of frequency f , as well as conductivity σ and permeability µ. In order to efficiently induce eddy currents and couple the resonator to the primary current, a larger metal thickness is therefore necessary at lower resonant frequencies. As previously shown, this effect has to be accounted for when the excitation of face shear modes or flexural modes in the range of a few hundred kHz is desired for a composite resonator, such as the copper coated silicon membrane. Additional fabrication steps such as electroplating or substrate doping are required to achieve the necessary metal layer thickness. On the other hand, these additional processes may lead to a decreased efficiency in generating higher frequency vibration modes, such as thickness shear modes at a few MHz, due to an increase in surface roughness, inhomogeneities and an acoustic impedance mismatch between substrate and conductive layer. For a composite sensor array as proposed above, a prior decision on the desired modes of vibration is necessary, so that an appropriate metal layer can be applied.

3.2 3.2.1

Eigenmode Analysis Silicon Membrane Resonators

Silicon membranes have been fabricated with a variety of thicknesses and in round and quadratic shape in both (1 0 0) and (1 1 1) orientation. The metal copper layer thickness also varies between 1 µm and up to 20 µm. The eigenmodes of all these combinations have been analyzed as preparation for our experiments. As explained in chapter 2, the fundamental face shear modes and thickness shear modes of diagonal, circular and radial symmetry have been of foremost interest. Figure 3.7 shows the results of the undamped eigenmode analysis for round and quadratic (1 0 0) silicon membranes with 3 µm copper layer thickness. The eigenfrequencies for these face shear modes are found at 192 kHz and 206 kHz for the diagonal mode shape, at 239 kHz and 245 kHz for the radial symmetry, and at 428 kHz and 438 kHz for the first of two fundamental circular mode shapes, respectively. As we can see, the difference between round and quadratic membrane shape is rather minimal.

46

3. Basic Resonator Operation

Figure 3.7: Results of eigenmode analysis for round (left) and quadratic (right) 15x15 mm (100) silicon membrane showing diagonal (top), radial (middle) and circular (bottom) face shear modes.

The anisotropy of elasticity for crystalline (1 0 0) silicon has been fully taken into account. However, comparison between the two substrate orientations shows only minimal differences between elastically anisotropic (1 0 0)

3.2.2. Metal Plate Elements

47

Figure 3.8: Mode shapes for round (100) silicon membrane with diagonal TSM at 15 MHz.

and elastically isotropic (1 1 1) silicon (for shear vibrations in the x-y-plane [86]). The main difficulty for the simulation has been the very high aspect ratio of these elements, with thicknesses in the order of a few to a hundred micron and lateral dimensions of a few millimeter. Establishing an appropriate finite element mesh with enough detail but reasonable numbers of elements becomes difficult with such aspect ratios. While we can specifically define the mesh grid in ANSYS and thus take care of this challenge, COMSOL works best with a free, automatic mesh. However, COMSOL allows the abstract inclusion of dimensional scale factors, whereas we can scale the vertical dimension by a certain factor to reduce the aspect ratio to a size that the automatic mesh algorithm can handle properly. This is especially important for thickness shear modes, where the vertical dimension needs a finer step size to accurately describe the shear deformation. Figures 2.8 and 3.8 shows simulated TSM shapes for a round silicon membrane.

3.2.2

Metal Plate Elements

We focus on the FEM results for a round aluminum resonator with an active circular mesa structure of 15 mm diameter [87]. This element was cast to

48

3. Basic Resonator Operation

Figure 3.9: First and second harmonic of diagonal FSM in round aluminum mesa resonator at 81 kHz and 171 kHz.

form the energy trapping mesa structure in the center and dimensioned to fit the existing planar coils. The first diagonal face shear mode appears at 81 kHz for this element. However, the primary shear mode is superposed by a slight out-of-plane movement, induced by the element geometry. The 2nd harmonic of this diagonal face shear mode can be found at 171 kHz. A slight vertical component in the active area is also evident here (Fig. 3.9). For our liquid phase experiments, we have primarily utilized two different modes of vibration, the circular face shear mode and a radial flexural plate mode (Fig. 3.10). Both modes are very different in the behavior in a fluid environment. The circular FSM shows no out-of-plane vibration component both in the eigenmode and in the excitation analysis. The acoustic energy is well trapped in the central mesa structure. This behavior is ideal for sensing the properties of a liquid, such as viscosity and density, with evanescent shear waves. There is primarily only viscous damping of this resonance, additional damping due to destructive interference of compressional waves is minimal. The total volume and mass of the liquid on the resonator has no significant influence on the sensor response, and a directly proportional relationship between the square root of density and viscosity and the frequency shift of the resonance can be established. These results have also been achieved by other groups, and similar conclusions could be drawn. Circular face shear mode or torsional plate resonators are suitable sensor devices for measuring

3.2.3. Resonator Array Sensors

49

Figure 3.10: Eigenmode shape of circular FSM at 142 kHz (left) and radial flexural plate mode at 295 kHz (right) of round aluminum mesa resonator.

liquid properties and distinguishing between different fluids [88]. The radial flexural plate mode on the other hand is dominated by out-ofplane components of the vibration. Due to the coupling with a radial shear movement, the in-plane forces generated by a circular spiral coil are highly efficient at exciting this mode of vibration. The vertical movement radiates compressional waves into the liquid and is thus highly sensitive to the surface level height of the liquid. This device is suitable for measuring small samples of liquid volume with high precision. The influence of liquid property changes is negligible compared to the effect of changes in liquid height.

3.2.3

Resonator Array Sensors

For both the metal array prototype and a silicon array we have simulated the respective eigenmodes of the array elements. Similar to single plate resonators, there exists a variety of flexural plate modes, as well as face shear and thickness shear modes of radial, diagonal, and circular symmetry. Depending on the spring coupling of the elements to the array frame, these modes can be slightly distorted by and not fully decoupled from other elements. For quadratic elements coupled at the corners (see Fig. 3.6) the radial face shear mode and the circular face and thickness shear modes are of interest (Fig. 3.11). The fundamental shear eigenmodes of interest are similar to single ele-

50

3. Basic Resonator Operation

Figure 3.11: Simulated mode shape for radial face shear mode (left) and circular thickness shear mode (right) of a single metal array element.

ments, the radial symmetry mode with strong flexural components for liquid volume sensing, the diagonal symmetry mode with larger vibration amplitudes, and the circular symmetry for viscosity-density sensing. Combinations of different modes and higher harmonics have also been investigated theoretically and experimentally, since in some cases the necessary force distribution is easier to generate than for the fundamental modes. Cross-coupling between individual elements is reduced if our spring coupler design is used instead of fixed connections at the corners to the main frame, but further analysis is necessary.

3.3

Experimental Excitation Setup

Our experimental setup consists of primary planar induction coils, external permanent magnets and the resonator element, as shown in figure 3.12. The Liquid Reservoir

Conductive Resonator Impedance Analyzer

Primary Excitation Coil Permanent Magnet

Figure 3.12: Geometric arrangement for electromagnetic-acoustic resonator setup with remote, non-contact excitation and detection.

3.3. Experimental Excitation Setup

51

Figure 3.13: Design and realization of excitation setup for radial and circular shear modes.

excitation coils were either wound from enameled copper wire (0.12–0.19 mm diameter, Roadrunner Electronics Ltd., UK) or, alternatively, lithographically etched on 1 mm thick PCB substrates or 50 µm thin polymer foils. The coils created for this setup vary in number of turns, wire thickness and also geometric realization. The corresponding magnet layout is realized by using several permanent magnets placed together and aligned with the primary coil. The permanent magnets used were rare-earth NdFeB magnets of different sizes corresponding with the size of the resonator element (e.g. 20x20x10 mm) with a measured static magnetic flux density at the sensor element of 0.25 − 0.5 T.

The primary coil is bridged by a parallel trimming capacitor to be able to adjust the electrical resonance to fit the mechanical resonance according the MARS principle. With this principle it is possible to effectively multiply the Q-factors and achieve a stronger signal response. The complete electrical setup was connected to an Agilent 4294A Precision Impedance Analyzer via an Agilent 42942A Terminal Adapter to excite and detect the mechanical resonances. In order to minimize interferences from parasitic wire and connector capacitances, the connection to the impedance analyzer has been kept very short, using two basic wires from contact pads on the PCB to a clamping fixture on the terminal adapter. Suitable, non-magnetic holders for the permanent magnets, coils, and transducers, as well as fluid measurement cells, were fabricated in PMMA and other plastic materials (Fig. 3.13). The advantages of this setup are quickly interchangeable spiral coils for

52

3. Basic Resonator Operation

different frequency ranges and simple connectors to use with different test fixtures or more advanced custom electronics. This setup can also be used for exciting piezoelectric resonators. The permanent magnet does not interfere with magneto-piezoelectric excitation [44, 83]. We measured the impedance spectrum of this setup around the mechanical resonance used for the respective experiment. The secondary induction due to the resonating plate yields an additional contribution, which is superposed to the impedance characteristics of the primary coil. In order to better visualize the resonance curve of interest in the measured impedance spectrum, we normalize the measurement by subtracting the electrical impedance of the coil: Z i (f2 ) − Z i (f1 ) · (f − f1 ) . (3.1) Z i,norm (f ) = Z i (f ) − Z i (f1 ) + f2 − f1

Here, Z i is the complex input impedance and f1 and f2 are the start and stop frequencies of the measured frequency spectrum, resulting in the normalized input impedance Z i,norm shown in all following measurements. Any offset due to slight changes in alignment, proximity, or location of the experimental setup is hence eliminated and only the changes due to the mechanical resonance are reflected properly. This normalization is applied to both impedance and phase measurements. The absolute impedance varies greatly depending on type of primary coil, operating frequency, excitation setup and alignment, mode shape, and electrical resonance tuning. For flexural and face shear modes it is usually in the range of a few 10 Ω, for thickness shear modes a few hundred up to 1000 Ω. The most common coil geometry employed for this sensor principle has been a planar spiral coil. However, applying the Lorentz force law to the induced eddy currents in the resonator, this layout yields radial forces when centered under the resonator element. These radial forces will preferably excite radial vibration mode shapes. According to finite element method (FEM) simulation, a radial thickness shear eigenmode exists for silicon resonator elements, however, this radial vibration is coupled to an out-of-plane movement at the center of the element due to the conservation-of-mass principle. This vibration component generates unwanted compressional waves in

3.3. Experimental Excitation Setup

53

Figure 3.14: Lorentz force distribution (arrows) with different coil and magnet layouts for radial (left), diagonal/linear (center), and circular (right) vibration mode shapes.

liquids, which lead to additional damping. Therefore, a number of alternative vibration modes are of particular interest for liquid phase sensing. Our FEM simulation results show that diagonal, circular and radial symmetry are the three primary mode shapes for face shear and thickness shear mode vibrations (see above). Therefore, we have concentrated our efforts into realizing experimental setups to accommodate these mode shapes. The distribution of the Lorentz forces is affected by the distribution of the eddy currents and the direction of the magnetic field. Therefore, in order to generate non-radial forces, the coil geometry has to be adapted and the magnetic fields changed accordingly, as explained in chapter 2. We developed a setup of quadratic or rectangular spiral coils coupled with multiple alternated permanent magnets for excitation and detection of diagonal, lateral shear modes. This vibration is similar to the thickness shear mode in QCM sensors and will lead to improved liquid phase sensing capabilities. In figure 3.14 three different designs for the microfabrication of the primary excitation coil geometry are given [89]. These coils have been realized in a photolithographic process. The corresponding magnet layout is visualized by the colored areas and realized by using several permanent magnets placed together and aligned with the primary coil. The complete setup for these designs is similar to that comprised of a single spiral coil. Besides the coil geometries presented here, other possibilities have also been investigated. For the diagonal shear mode these include linear solenoids and meander line structures with single and alternated magnetic fields, respectively. For the circular or torsional mode, other geometries include toroidal layouts for excitation and a secondary spiral detection coil and also

54

3. Basic Resonator Operation

dual ellipsoidal designs based on the results with a spiral coil but optimized towards a symmetrical excitation. Earlier work on EMAT transducers suggests even more possible design alternatives. The primary challenge lies with the electromagnetic coupling efficiency. For an efficient excitation, an appropriate force distribution is necessary. The resulting vibration must however also lead to a secondary induction in parallel to the primary current flow. And as the coupling between primary coil and resonator element is achieved via the dynamic magnetic fields of excitation current and eddy currents, the energy transfer needs to be maximized. In electrical terms, this is achieved by a high inductance and mutual inductance. This excludes several possible designs for excitation and/or detection coil. For precise and reproducible measurements a temperature stable setup is required, primarily due to the temperature coefficients of density, viscosity and ultrasonic velocity. Most experiments shown in the following have been carried out at a temperature of 25 ± 0.5◦ C. To further minimize the influence of the temperature dependence, we have also developed a thermostat based, temperature stabilized experimental setup operated at ±0.05 K for simultaneous measurements with different acoustic resonator sensors [90]. The measurement cells for liquid phase sensing required non-magnetic material such as PMMA, other plastics, or aluminum. One further requirement was to keep the resonator element as close as possible to the primary coil, i.e. for setups with the resonators placed inside a cell, the bottom wall had to be very thin and was usually realized with a 100 µm thin plastic foil. The resonator element was simply placed on the bottom in an arbitrary, free-moving fashion and the coil and magnet were then aligned appropriately. The other option to realize a measurement cell was to use the resonator itself as the bottom of the cell. Then hollow plastic cylinders were attached to the surface forming a closed cavity. Here, the distance to the primary coil was already minimal. Difficulties arose from the proper attachment of the cylinder without strongly perturbing the resonant vibration. Therefore, the cylinder was attached to the regions of minimal displacement according the eigenmode analysis of the specific vibration mode described before.

3.3.1. Array Multi-Mode Setup

55

Figure 3.15: Simultaneous excitation of different modes of vibration in each array element at different excitation frequencies (left), and of the same vibration mode at the same resonant frequency (right).

3.3.1

Array Multi-Mode Setup

The setup for experimental measurements with array resonators is similar to our experiments with single resonator elements and thus allows for easy comparison of the results in terms of achievable excitation efficiency, resonance quality factor, and resulting impedance changes. Here, the impedance of a single large primary coil or coil array is also monitored with the impedance analyzer. Alternatively, each individual coil and element can also be driven by individual AC sources and analyzed by separate detection electronics. An advantage of the sensor array setup is the simultaneous single and multi-mode excitation, i.e. of similar and different vibration modes, respectively. Figure 3.15 shows how to achieve multi-mode excitation with a coil array design, where each coil can be driven at a different frequency, and explains the single mode excitation with a series connection of each individual coil driven at the same frequency. Alternatively, a single large coil can also be used to excite the individual elements simultaneously. Most suitable are rectangular planar coils that induce linear, diagonal vibrations, or a round spiral with a radial current direction underneath each element resulting in circumferential forces.

3.4

Eigenmode Excitation Results

In the following section we will present selected examples of successfully excited thickness shear, face shear and flexural plate modes. All measurements

56

3. Basic Resonator Operation

have been carried out in air. The undamped FEM eigenmode simulation described above was used to predict the eigenfrequencies for the different mode shapes of interest and verified experimentally. In general, we achieved good agreement between FEM prediction, excitation theory and experiments. And overview of detected vibration modes can be found in the appendix.

3.4.1

Thickness Shear Modes

We have successfully excited diagonal, circular and radial thickness shear modes in silicon membranes and aluminum or brass plates. The detected fundamental resonance frequencies vary between 5 and 30 MHz for the microfabricated membranes both in elastically isotropic (1 1 1) and anisotropic (1 0 0) silicon substrates, with higher harmonics detected up to and above 100 MHz as well. Since the electro-mechanical transduction is of inductive nature, the resonance frequency of the electrical circuit is the limiting factor for excitation of higher harmonics, i.e. above the electrical resonance of the primary coil we have dominant capacitive coupling and the resonance peaks vanish. This also is the main difference to piezoelectric excitation, where the coupling mechanism is of capacitive nature, and thus higher harmonics can be detected easily. All these TSM resonances display very high Q-factors up to about 200,000 due to the high intrinsic elastic stiffness and low loss of silicon. Silicon of (1 1 1) crystalline orientation shows only mediocre results. Both Q factor and amplitude of the signal response are several times less than for anisotropic (1 0 0) silicon. Table 3.2 lists different resonator materials and geometries and compares operating frequency, signal response and quality factor. While a QCR still gives a larger signal due to the different excitation mechanism, it is surpassed by the membranes in Q-factor of these resonances. Additionally the silicon resonators can be excited in even and uneven harmonics of the fundamental TSM [83]. For exciting the diagonal mode shape, the quadratic coil geometry results in a higher signal amplitude but also in additional spurious modes, whereas the rectangular layout excites a single resonance but at reduced signal strength due to lower mutual inductance between sensor element and

57

3.4.1. Thickness Shear Modes

Table 3.2: Comparison of operating frequency, signal response and Q-factor for selected resonator elements excited at radial TSM in air; with different selected geometries and operating TSM harmonics for the silicon membranes. Material

dR / µm

Geometry

fR / MHz

harmonic

Zi / mΩ

Q / 104

Quartz (AT-cut)

165

round

10.108

1

105.47

2.38

Aluminum

700

quad

2.380

1

18.81

0.78

Silicon (100)

390

quad

14.907

2

2.83

2.98

Silicon (100)

290

quad

20.360

2

6.71

6.03

Silicon (100)

190

round

15.648

1

8.82

6.26

Silicon (100)

190

round

31.561

2

1.41

7.78

Silicon (100)

190

quad

15.417

1

10.44

7.01

Silicon (100)

190

quad

30.761

2

2.13

7.19

Silicon (100)

90

round

35.087

1

1.20

5.44

Silicon (100)

90

quad

34.318

1

0.80

6.02

Silicon (111)

300

round

24.613

3

0.36

1.89

larger coil area. In figure 3.16 the radial TSM resonance for a quadratic (100) silicon membrane at 15.42 MHz is compared to the diagonal mode shape of this element at 15.36 MHz. Both distinct resonance peaks only appear for their respective coil geometry and magnet placement (see Fig. 3.14) and disappear completely when excited by an unsuitable setup, as, e.g., a quadratic spiral coil with a single magnet aligned in center of the resonator excites the radial modes, when exchanging with two alternated magnets, the radial modes disappear but the diagonal resonance peaks appear. The resonance strength of the diagonal mode is smaller than for the radial mode. The Q-factor of both resonances is similar. Using out-of-center spiral coil excitation it is possible to excite a circular TSM resonance, or torsional mode, in a round (100) silicon membrane. Figure 3.17 shows the comparison of the radial shear mode and the circular mode shape at 15.65 MHz and 15.61 MHz, respectively. The results show that the excitation with a single spiral coil out of center does not result in an ideal coupling between coil and resonator element. The increased impedance

58

3. Basic Resonator Operation

0.5

Normalized Impedance Zi /Ω

Normalized Impedance Zi /Ω

4 3 2 1 0 −1 −2

15.410

15.415

15.420

Frequency f / MHz

0.4 0.3 0.2 0.1 0 −0.1 −0.2

15.425

15.350

15.355

15.360

Frequency f / MHz

15.365

Figure 3.16: Normalized impedance of quadratic coil with standard magnetic field resulting in radial TSM at 15.42 MHz (left) and with alternated magnetic field resulting in diagonal TSM at 15.36 MHz (right) in a quadratic high-Q silicon resonator.

of the signal level is the result of a reduced mutual inductance due to the loss of coupling efficiency. The primary reason for this reduction in performance in our measurement setup is the size of resonator element and planar coil, which have been optimized for the radial mode and are therefore not completely overlapping for the circular mode setup. Spurious components in the resonance signal can be explained by the asymmetric force distribution leading to the excitation of unwanted modes. With alternated magnetic fields and dual-circular coil layouts, this asymmetry can be overcome and pure circular shear motion is possible. 1

Normalized Impedance Zi /Ω

Normalized Impedance Zi /Ω

3 2 1 0 −1 −2 −3 15.640

15.645

15.650

Frequency f / MHz

15.655

0.8 0.6 0.4 0.2 0 −0.2 −0.4

15.600

15.605

15.610

Frequency f / MHz

15.615

Figure 3.17: Normalized impedance of a standard spiral coil at the center resulting in a radial TSM at 15.65 MHz (left) and placed out of center resulting in a circular TSM at 15.61 MHz (right) in a round (100) silicon membrane.

59

3.4.1. Thickness Shear Modes

0.08

Normalized Impedance Zi /Ω

Normalized Impedance Zi /Ω

0.3

0.2

0.1

0

−0.1

1.5250

1.5255

1.5260

Frequency f / MHz

1.5265

0.06 0.04 0.02 0 −0.02 −0.04

2.322

2.324

2.326

Frequency f / MHz

2.328

Figure 3.18: Impedance spectrum of circular and diagonal TSM in aluminum mesa element and quadratic plate, respectively.

Aluminum and brass plate resonators have also been successfully excited in the diagonal and torsional mode of vibration (Fig. 3.18). The experimental results and conclusions are similar to those achieved with silicon resonators. The issue of elastic anisotropy does not seem to have any negative influence on the generation of any of the investigated mode shapes. The lower elastic stiffness of aluminum results in lower quality factors compared to silicon resonators. However, eddy currents permeate deeper into the conductive substrate compared to a thin conductive layer, resulting in a higher transduction efficiency with metal resonators. A major drawback of pure metal resonators or composites with larger amounts of metal is the high temperature dependency of these elements. The temperature coefficients of aluminum for example are several orders of magnitude larger than silicon or quartz glass. Therefore, a temperature stabilized setup is necessary for reproducible experiments with metal resonators. Of note is that the Q-factor reported here is the minimal estimate taking the lower 3 dB frequency of the maximum peak resonance and the upper 3 dB frequency limit of the minimum peak. Due to the phase shift not occurring at the exact center frequency of the actual mechanical resonance, the bandwidth and thus the Q-factors can only be estimated. Equivalent circuit simulations resulting in a similar impedance spectrum show that the actual resonance quality of the ’virtual current’ in the mechanical part of the

60

3. Basic Resonator Operation

0.08

Normalized Impedance Zi /Ω

Normalized Impedance Zi /Ω

0.04 0.03 0.02 0.01 0 −0.01 −0.02

220

221

222

223

Frequency f / kHz

224

0.06 0.04 0.02 0 −0.02

456

458

460

462

Frequency f / kHz

Figure 3.19: Normalized impedance of diagonal FSM at 222 kHz (left) and circular FSM at 459 kHz (right) in a quadratic high-Q silicon resonator.

equivalent circuit is twice or more of the reported values.

3.4.2

Face Shear Modes

The excitation of face shear modes does not differ much from thickness shear mode excitation. The only difference is the required larger metal thickness for efficient eddy current coupling, due to eigenfrequencies in the the kHz region, and thus much larger eddy current penetration depths. Setup geometry and alignment to carter for the different mode shapes is similar, i.e. spiral coils with singular magnets placed at center excite radial symmetric modes, alternated magnets are required for diagonal, linear vibrations, and out-of0.5

Normalized Impedance Zi /Ω

Normalized Impedance Zi /Ω

0.4 0.3 0.2 0.1 0 −0.1

142.0 142.5 143.0 143.5 144.0 144.5 145.0

Frequency f / kHz

0.4 0.3 0.2 0.1 0 −0.1 −0.2

367

368

369

Frequency f / kHz

370

371

Figure 3.20: Normalized impedance of diagonal FSM at 143 kHz (left) and 2nd circular FSM at 369 kHz (right) in a mesa aluminum plate.

61

3.4.3. Flexural Plate Modes

0.8

Normalized Impedance Zi /Ω

Normalized Impedance Zi /Ω

0.25 0.2 0.15 0.1 0.05 0 −0.05 −0.1

246

247

248

249

Frequency f / kHz

250

0.6 0.4 0.2 0 −0.2

287.4

287.6

287.8

288.0

Frequency f / kHz

288.2

Figure 3.21: Normalized impedance of radial flexural mode at 248 kHz in a quadratic high-Q silicon resonator (left) and at 288 kHz for an aluminum mesa resonator (right).

center placement of the coil is necessary to generate circular forces. One difference to TSM is the superposition with flexural plate modes. Since both FSM and flexural modes are propagating waves along the lateral dimensions of the elements, the operating frequencies are similar. Thus, higher harmonics of a flexural mode can easily interfere with and superpose a pure in-plane shear motion, such as evidenced for radial face shear modes. Figures 3.19 and 3.20 show the impedance curves of diagonal and circular face shear modes for silicon and aluminum resonators, respectively.

3.4.3

Flexural Plate Modes

Flexural plate modes are excited with the standard circular planar spiral coil generating radial in-plane forces. For radial symmetric flexural plate modes there always exist radial in-plane displacement components into which the Lorentz forces can couple. Alternatively, we can place the magnets on the sides of the resonator to create a lateral magnetic field and use linear eddy currents to generate out-of-plane forces. These setups however suffer from a reduced magnetic flux density due to the distance to the magnets (size of the elements) and more complex setups with magnetic circuits are required for an efficient excitation. Figure 3.21 shows two examples of flexural motion utilizing a radial shear mode superposed with a flexural mode resulting in a strong vertical displacement at the center. Compared to diagonal or cir-

62

3. Basic Resonator Operation

0.6

1.4

Array element 1 Array element 2 Array element 3 Array element 4

0.4 0.3 0.2 0.1

1

0.8

0.6

0.4

0

0.2

−0.1

0

−0.2 230

231

232

Frequency / kHz

233

234

Array element 1 Array element 2 Array element 3 Array element 4

1.2

Phase shift / °

Impedance change / Ω

0.5

−0.2

230

231

232

233

234

235

236

237

238

Frequency / kHz

Figure 3.22: Experimental results for individual excitation of a circular FSM in all elements of a metal array anchored at the corners at ≈234 kHz, with noticeable cross-coupling between individual elements as evidenced by the minor peaks.

cular shear modes, the excitation efficiency and electro-mechanical coupling is much better due to coil and resonator element aligned centered to each other.

3.4.4

Array Multi-Mode Excitation

Shown in figure 3.22 is the impedance spectrum for a radial face shear mode in a metal array with stiff coupling at the corners, corresponding to the simulated mode in figure 3.11. Here, every element is excited individually [85]. As evidenced by the minor peaks of idle elements for each individual excitation, there is considerable cross-coupling to neighboring elements with this anchor design and small variations in the resonance frequency of each element. The latter is induced by fabrication tolerances and material inhomogeneities. For this vibration mode, the evaluation of amplitude and phase signal is equally possible, however the impedance changes depend on the frequency characteristics of the primary coil(s). As described above, simultaneous single-mode excitation can be achieved by connecting multiple coils in series or parallel, or by a single large coil driving all elements. We have investigated several combinations of resonator elements and excitation layout in air and with a liquid load of a droplet of DI water. Shown in Fig. 3.23 is the simultaneous measurement of similar

63

3.4.4. Array Multi-Mode Excitation

0.20

in air DI water

0.06

in air DI water

Al − flexural mode

0.04 0.02

∆f ∆f

0

−0.02

Impedance Phase

Impedance Phase

0.15 0.10

∆f

0.05 0

Si − FSM diagonal

−0.05

(100) Si

(111) Si −0.10

480

500

520

540

Frequency / kHz

560

210

220

230

240

250

Frequency / kHz

260

270

Figure 3.23: Simultaneous excitation of a circular face shear mode in (100) silicon and a radial flexural mode in aluminum around ≈250 kHz, and different frequency shifts with DI water loading due to different sensitivities.

circular face shear vibration modes in different (100) and (111) Si resonator elements, and the similar behavior of these resonances can be seen, resulting in simultaneous, multiple measurements for increased redundancy. Of greater interest is the excitation of different modes of vibration exhibiting different sensitivities. For example, the circular face shear mode in silicon elements is sensitive to the density-viscosity product of a liquid load. On the other hand, a radial flexural mode in an aluminum plate radiates compressional waves into the liquid, which are reflected at the surface boundary, and thus it is sensitive to the liquid height. This volume sensitivity results in much larger frequency shifts than those due to other liquid properties. A combined setup of a circular shear mode resonator and a flexural mode resonator can thus simultaneously measure density, viscosity, and liquid volume (Fig. 3.23). Other properties which can influence acoustic resonators are for example temperature, compressibility, conductivity, and permittivity. With an electromagnetically excited sensor array it is possibly to remotely measure these physical parameters in a single experiment at the same time by simply evaluating the complete impedance spectrum of the excitation circuit.

64

3.5

3. Basic Resonator Operation

Summary

Resonator design and experimental setup are tied very closely to the FEM and equivalent circuit models. By analyzing the eigenmodes of a resonator geometry, we have qualitatively determined the necessary force distribution for a successful excitation and designed our experimental setup accordingly. We have focused on three primary resonator devices, inexpensive aluminum and brass plates, simple micromachined silicon membranes, and thin, suspended metal resonator arrays. For composite resonators with conductive layers, such as our silicon membranes, we have shown how to modify the elements to be suitable in different frequency ranges. A large variety of different vibration modes, harmonics and superposition of modes have been successfully excited and detected. The suitability of specific modes for different sensor applications will be shown in the following chapters. We have demonstrated the feasibility of electromagnetic excitation of acoustic sensor array devices. As shown, it is possible to excite similar and different modes of vibration in these array elements simultaneously leading to increased redundancy and measurement of multiple physical properties with a single sensor. A miniaturized (silicon) resonator array can be a new alternative to established acoustic sensor devices taking advantage of the remote nature of electromagnetic excitation and the simultaneous measurement of multiple physical properties such as added mass, density, viscosity, temperature, and liquid volume. The proposed silicon sensor arrays can be fabricated in a standard micromachining process on an SOI wafer.

4 Mass Microbalance Application The primary sensor application of bulk wave resonators has traditionally been mass sensing. Added mass on the resonator surface results in changes in the elastic and geometric properties, leading to a shift of the resonant frequency. With a high frequency resolution a very high mass resolution down to a few pico- and femtograms can be achieved. Due to this sensitivity bulk resonators, especially quartz discs, are in use as thickness monitors for various deposition techniques, e.g. evaporation, sputtering, etc. [27]. More advanced applications include the detection of particles and (bio-)molecules adsorbed on the resonator surface to measure concentration, surface coverage, selective absorption, process speed, or many other parameters. In order to validate the suitability of electromagnetic excitation compared to piezoelectric or capacitive excitation, we have also investigated our resonator elements in terms of a simple mass microbalance. While liquid phase applications have been the focus of our study, we can show that our technology can also be easily applied to this kind of measurements. Additionally, the mass changes can be exploited with thickness shear modes as one independent parameter of a multi-mode sensor array.

4.1

Analytical Description

The electromagnetic-acoustic transducers can be used as resonant sensors similar to quartz resonators. The acoustic part of the transduction scheme is similar to QCR sensors, only the transformation of the mechanical resonance into an electrical signal is different. Acoustic loads put into contact with the sensor surface change the properties of wave propagation and result in a sensor response in form of a frequency shift. In the most simple case, the acoustic sensor acts as a mass balance, which is well described for thickness shear motion. A TSM resonator can be described by the general wave equation, which

66

4. Mass Microbalance Application

can be simplified for high-Q materials, where damping is negligible. The wave propagation is limited to the thickness dimension z and displacement, e.g. for a diagonal mode, is limited to a lateral dimension x. From the equations of motion 2.10 one obtains a wave equation for displacement ux with mass density and shear stiffness of the resonator, ρR and µR respectively. ∂ 2 ux (z, t) ∂ 2 ux (z, t) = ρ (4.1) R ∂z 2 ∂t2 Solving this equation yields a wave function with the wave number k and the boundary condition of stress-free surfaces ∂ux /∂z = 0, i.e. the resonator is operated without any restoring forces. µR

ux (z, t) = ux0 cos (kz) exp (jωt)

(4.2)

The TSM resonance frequency fR of harmonic n can be calculated as a function of resonator thickness hR r µR n fR = , (4.3) 2hR ρR and the shear wave phase velocity vs is given by r µR . vs = ρR

(4.4)

In resonance, thickness shear mode resonators have their displacement maxima at the surfaces, making them very sensitive to changes to the surface properties. Different models exist to describe these TSM resonators, e.g., very basic physical models and equivalent circuit models, delivering similar results [24, 25, 26, 31, 35, 77, 78]. Following the energy balance approach as described in chapter 2, we have a simple analytical description of the resonance behavior. According to the Rayleigh hypothesis, mechanical resonance is achieved when the kinetic energy is balanced by the potential energy of the system [7]. Energy is periodically exchanged between both forms at resonance, similar to other types of oscillators. Added mass in form of a thin, rigid layer at the surface causes an increase in the kinetic energy (displacement maximum), with no changes to the potential energy. Therefore, the resonance frequency changes to rebalance kinetic and potential energy.

67

4.1. Analytical Description

The kinetic energy density is defined by the particle velocity in all directions. Following from equation 2.11, we can sum the kinetic energies from infinitesimal slices across the resonator thickness. The peak kinetic energy density wk is therefore given by: ! Z hR ω 2 u2x0 ρR hR ω2 2 ux (z) dz = ρl ux0 + ρR ρl + , (4.5) wk = 2 2 2 0 where ρl is the areal mass density (mass per area) of the acoustic load on the surface of the TSM resonator. The peak potential energy density on the other hand occurs at the oscillation point of maximum displacement and zero particle velocity. From equation 2.12 and integrating over the thickness it can be therefore written as Z hR µR k 2 u2x0 hR 1 2 2 sin2 (kz) dz = . (4.6) wp = µR k ux0 2 4 0 Balancing peak kinetic and peak potential energy density results in a relationship between resonant frequency and acoustic mass load. We obtain ω 2 2ρl R , (4.7) =1+ ω hR ρR

where the unperturbed resonant frequency ωR , i.e. for no acoustic load, is given as r nπ µR . (4.8) ωR = hR ρR Equation 4.7 can be seen as a universal description of TSM resonator mass sensitivity. For weak mass loading, i.e. ρl ≪ hR ρR , the universal solution can be approximated with a Taylor series expansion as a linear relationship between frequency shift ∆f and added mass. ρl ∆f =− fR hR ρR

(4.9)

This equation recover’s the Sauerbrey formula [27], who first defined the resulting frequency shift as follows: ∆f = −nfR2 √

2 ∆m . ρ R µR A

(4.10)

68

4. Mass Microbalance Application

Here, the frequency shift ∆f is a function of the fundamental thickness shear mode frequency fR divided by the mass density and shear stiffness of the resonator, ρR and µR respectively, and the surface mass density ∆m/A acting as the acoustic load, operated at the n-th harmonic. The Sauerbrey equation is only valid if the additional mass can be treated as an extended thickness of the resonator, i.e. the layer must be sufficiently thin, rigid and homogeneous. This equation reflects a direct proportional relation between frequency shift and mass loading of a sensor element. According to Sauerbrey’s investigation any material capable of transmitting acoustic waves and resonating in a thickness shear mode experiences these frequency shifts, with the strength of the shift mainly influenced by the elastic properties and the intrinsic loss of the material. Silicon was also chosen for its higher elasticity and lower temperature coefficients of the elastic stiffness than for example quartz glass. The thin metallic layer also acts as an acoustic load for the silicon resonator, changing its fundamental frequency. The mechanical resonance of the metallic layer itself has no influence on the sensor resonance though, due to its minimal thickness. As the behavior and properties of mass loading are well known for piezoelectric TSM sensors, this vibration mode has been the primary focus of initial studies for the concept of electromagnetic-acoustic resonators.

4.2

Suitable Modes of Vibration

Added mass on the surface of the resonator influences every mode of vibration. As a linear relationship for thin mass layers has been derived for TSM vibrations, as outlined above, we have also utilized these modes for mass sensing. For electromagnetic coupling the planar spiral coil exciting radial modes has been most efficient and these modes are therefore used in this application. Face shear motion and flexural motion follow different relations as the kinetic and potential energies are differently distributed, the Sauerbrey equation is only valid for thickness shear motion. The (100) silicon membranes operate in the fundamental thickness shear modes around 15 MHz. At that frequency the penetration depth of the eddy currents is only a few microns. As discussed in the previous chapter, the

4.3. Equivalent Circuit Model

69

Figure 4.1: Diagonal, circular and radial TSM (left to right) for round (100) silicon membrane at ≈ 15 MHz.

optimal copper thickness for our elements is between 1 and 2 microns. Because a rigid mass layer only contributes a frequency shift of the resonance and no damping, we can utilize the radial TSM of our high-Q silicon membranes with no additional copper coating for our most precise measurements. Circular and diagonal modes are equally suited but result in a lower sensor signal due to the reduced coupling efficiency (Fig. 4.1). The radial TSM displays an out-of-plane component at the center of the element due to conservation of mass, which however does not contribute a considerable error to the frequency shift for solid, homogeneous mass loads in vacuum and gaseous environments.

4.3

Equivalent Circuit Model

To account for additional surface mass the basic equivalent circuit has to be extended by an additional lumped mass element. To use a lumped element and not assume a variable mass distribution, the added mass has to be considered as a homogeneous surface layer across the active resonator region, similar to the assumptions for the analytical Sauerbrey equation. The modified equivalent circuit is shown in figure 4.2. The lumped elements equivalent circuit cannot account for randomly distributed mass loads, since the position of the added mass in regards to the resonance mode shape strongly influences the resulting frequency shift. A more detailed finite element approach is required in such situations.

70

4. Mass Microbalance Application

T Ri AC

Zi

T1

Rc

Re

Ct Lc

Le

T3

i uv

k

FL {X, Y} T2

v

h

n

m

m

T4

XT

Excitation circuit

Eddy currents

Acoustic resonator

Mass load

Figure 4.2: Electromagnetic-acoustic equivalent circuit with added ideal mass element.

4.4

Results and Discussion

We have utilized our resonators as mass sensors in the process of electroplating the silicon membranes for application as face shear mode elements, as explained in chapter 3. Besides the frequency shift due to the added mass, the efficiency of mode excitation and corresponding quality factors are also affected by an increase of copper layer thickness and changes to the copper surface. Figure 3.3 shows the changes to a radial face shear mode at ≈ 250 kHz with increasing copper layer thickness. From these measurements we can calculate the added mass, or of more interest, the average added copper layer thickness (Tab. 4.1). To verify the use of silicon resonators as acoustic sensors, several elements have been coated with a monolayer of 300 nm large, spherical polystyrene particles [91, 92]. Polystyrene particles have been used due to rigid, very densely packed deposition as a monolayer, with thus a pre-defined thickness. Table 4.1: Copper layer thickness calculated from frequency shift of radial TSM. Sample (111) Si 296µm quad

(100) Si 185µm quad,

f0 / MHz

f1 / MHz

∆f / kHz

dCu /µm

8.511

8.191

320.528

3.8

8.511

8.044

467.468

5.0

15.890

15.570

319.585

2.0

15.890

15.152

737.617

3.3

71

4.4. Results and Discussion

Normalized Impedance Zi /Ω

5 4

unloaded 300 nm PS

∆f

3 2 1 0 −1 −2

15.410

15.415

15.420

Frequency f / MHz

15.425

Figure 4.3: Shift of unloaded fundamental TSM frequency after coating with a 300 nm polystyrene monolayer.

With appropriate deposition techniques this layer also represents an acoustic load for the sensor corresponding with the Sauerbrey equation 4.5. The frequency shift attained by a QCM sensor with the same coating was -4.65 kHz. Sauerbrey predicts a shift of -4.83 kHz for such a quartz crystal resonator. Deviations from theory are primarily down to not ideal polystyrene layer homogeneity, which cannot be avoided with the current coating methodology. The confidence level of polystyrene layer thickness and deposited mass is 90% for this process, i.e. at least 90% of the surface area is covered with a monolayer of tightly packed polysterene particles. Deviations result from holes, packing mismatch, and particle stacking. For a quadratic (1 0 0) silicon membrane with a thickness of 190 µm the measured frequency shift resulted in -6.6 kHz, whereas theory predicts a change of -6.95 kHz (Fig. 4.3). These measured values lie within 95% of the theoretical values and verify the Sauerbrey like characteristics also for electromagnetic-acoustic resonances in silicon membranes. Damping is minimal, which also is an indicator for a smooth coating of the elements. Silicon resonators are equally well suited for film thickness monitoring as QCM sensors.

72

4.5

4. Mass Microbalance Application

Summary

Sensors based on the resonance principle have found a large variety of applications as mass microbalance. We have demonstrated that electromagnetically excited resonators can also serve as microbalance in a gaseous environment. Here, the most effective mode is the radial TSM. The well described mass sensitivity relations can be applied in a similar way. The measurement yields information for example on the thickness of a deposited layer, a metal or a chemically sensitive film, the latter pointing at the mass increase due to selective adsorption of molecules when the resonator is working as a chemical sensor. As the primary focus of this research has been liquid phase applications, we did not pursue this idea much further. However, since the microbalance application is very attractive in liquid environment as well, vibration modes without out-of-plane vibration components are favorable. We will utilize face shear modes for the determination of liquid material properties which exist at lower frequencies. In terms of a multi-mode device, it is simply a matter of increasing the operating frequency with no changes to the measurement setup to excite the thickness shear mode and to achieve a sufficient mass sensitivity.

5 Liquid Phase Density-Viscosity Sensing The remote, non-contact nature of excitation and detection truly begins to shine with measurements in liquids. Other transducer methods always introduce design challenges, e.g. electric contacts and insulation, capacitive coupling, and leakage. With electromagnetic-acoustic transducers, the sensor element can be simply placed inside the measurement chamber, while the electric circuitry is spatially separated on the outside. Therefore, the main focus of this thesis has been the application of these resonator devices for liquid phase sensing. Since acoustic sensors such as quartz crystal resonators have proven to be sensitive to density and viscosity, our primary interest has been to validate the suitability of our devices for this application as well. In-plane shear motion of a surface loaded with a liquid layer results in a frequency shift and damping of the mechanical resonance proportional to the square root of density and viscosity. Out-of-plane vibration must be avoided. Both thickness shear modes (TSM) and face shear modes (FSM) of our aluminum and silicon resonator elements can be suitable for these measurements, however the strong viscous damping of acoustic shear waves in liquids has to be taken into account. Therefore, the low transduction efficiency of electromagnetic resonators needs to be carefully addressed in terms of resonator design, vibration mode, and experimental setup.

5.1

Analytical Description

In a Newtonian liquid for 1D shear excitation in the x-y plane, the shear velocity gradient ∂vx /∂z and the shear stress Txz are related by a constant parameter, the shear viscosity η. Txz = −η

∂vx ∂z

(5.1)

74

5. Liquid Phase Density-Viscosity Sensing

Viscous Liquid

ρ, η

z

Penetration Depth

Viscous Liquid

δ

ρ, η

x

Resonator ρ R, µR

z

Penetration Depth

δ

x

Resonator ρR , cR

Displacement Profile ux(z)

Displacement Profile ux (x,y)

Figure 5.1: Displacement profiles in a viscous liquid for a diagonal thickness shear mode (left) and a diagonal face shear mode (right).

Therefore, to measure viscosity a shearing surface an evanescent shear wave in the liquid can be used. In an incompressible liquid, the resulting velocity field vx can be calculated by solving the Navier-Stokes equation for onedimensional flow: ∂vx ∂ 2 vx . (5.2) η 2 =ρ ∂z ∂t With a periodic driving force at the solid-liquid interface we can solve this wave equation as: z z (5.3) vx (z, t) = vx0 cos − ωt exp − , δ δ with the penetration depth as:

δ=

r

2η . ωρ

(5.4)

The complex shear stress acting on the surface to generate the shear wave is thus given by: ηvx0 ∂vx = Txz = −η (1 + j) . (5.5) ∂z z=0 δ

The resulting displacement profiles of diagonal shear motion is shown in figure 5.1. Face shear and thickness shear modes differ in terms of the energy stored in the resonator, but evanescent shear waves in the liquid follow the same equations for a similar surface motion.

5.2. Suitable Modes of Vibration

75

Stored and dissipated energy can be described by a complex acoustic impedance Za as the ratio of shear stress and velocity at the boundary of a semi-infinite viscous liquid. r ωρη Txz (1 + j) (5.6) Za = − = vx z=0 2

Since the impedance is related to the additional kinetic and potential energy, √ these are also related to the square root of density and viscosity, ρη. As shown in the previous chapter for a simple mass loading, the energy balance is directly related to the resulting frequency changes. Consequently, the frequency shift and damping due to a viscous load will be a function of this density-viscosity product as well. √ ∆f ∝ ρη (5.7) √ (5.8) ∆Z ∝ ρη

From energy balance or using an equivalent circuit method, an approximation of the frequency shift of thickness shear modes is possible, yielding the equation of Kanazawa and Gordon [28, 93]: s fR3 ρη 1 ∼ . (5.9) ∆f = − n πρR µR Here, the frequency shift ∆f is a function of the thickness shear mode frequency fR divided by the mass density and shear stiffness of the resonator, ρR and µR respectively, and the density-viscosity product ρη acting as the acoustic load, operated at the n-th harmonic. For face shear modes no simple analytical relationship can be derived, especially due to the dependence on the specific two-dimensional shape of vibration. The general relation to the density-viscosity product stays applicable, but the proportionality factors have to be evaluated for each specific resonator element and mode shape.

5.2

Suitable Modes of Vibration

For pure liquid property sensing as described above, we require a vibration mode with ideally only shear vibration at the surface. Any out-of-plane

76

5. Liquid Phase Density-Viscosity Sensing

component radiates compressional waves and results in secondary frequency shifts and damping due to liquid level, compressibility and other parameters. In case of a semi-infinite liquid these compressional waves just decrease the acoustic energy in the resonator, thereby increasing the insertion loss. In case of finite dimensions of the liquid the situation becomes more involved. Since damping of compressional waves is quite small in many pure liquids and hence the penetration depth much larger than geometric dimensions, the liquid must be understood as liquid cavity with acoustic properties dependent on frequency, liquid level and material properties like compressibility. This cross sensitivity may result in secondary frequency shifts or additional damping. Here we focus on suitable face shear eigenmodes for the circular aluminum resonator with a central mesa structure and the modified silicon membranes. The optimal modes of vibration for these elements are shown in figure 5.2. The circular FSM of the aluminum plate shows no out-of-plane vibration both in the eigenmode and in the excitation analysis. The acoustic energy is well trapped in the central mesa structure and this is ideal for sensing the liquid properties with shear waves. There is primarily only viscous damping of this resonance, additional damping due to dissipated energy is minimal. The total volume and mass of the liquid on the resonator has no significant influence on the sensor response, and a direct relationship between the density-viscosity product and the frequency shift of the resonance can be es-

Figure 5.2: Optimal modes of vibration in viscous liquids: Circular FSM of aluminum resonator (left) and diagonal FSM of Si membrane.

5.3. Equivalent Circuit Model

77

tablished. Circular FSM or torsional plate resonators are thus suitable sensor devices for measuring liquid properties and distinguishing between different fluids. Radial and diagonal modes of the plate resonator exhibit considerable out-of-plane motion. In our experiments with the modified silicon transducers we have focused on circular and round membranes in (100) and (111) silicon. Here, both the circular and diagonal modes are primarily shear motion with negligible out-ofplane components, while the significant out-of-plane vibration of the radial modes results in additional damping due to compressional waves radiated into the liquid. Due to lower electro-mechanical transduction efficiency of thickness shear modes, the diagonal FSM has been primarily used in our experiments. Compared to the excitation of a circular mode, the eddy current coupling is considerably stronger for a diagonal mode, because the primary coil is placed directly centered to the resonator and not out-of-center, see chapter 3. Theoretically, thickness shear modes are equally suited for liquid property sensing. For our method and devices however, the coupling efficiency is too low to overcome the strong damping of high frequency evanescent shear waves. With our current experimental setups, we are not able to measure frequency shift and damping of TSM resonances in a truly reproducible fashion. The measured resonance peak strengths and the signal to noise ratio are simply too small for exact calculation of frequency shift and impedance amplitude. Therefore our experiments focus on the face shear modes described above.

5.3

Equivalent Circuit Model

Acoustic loading by a viscous liquid can be modeled by adding additional mass ml and damping hl elements to the equivalent mechanical circuit (Fig. 5.3). For more complex liquids, this can be replaced by a general complex impedance, e.g. to also account for elastic energy storage in the liquid. The additional mass and damping elements are both functions of the densityviscosity product ρη, as well as dependent on the geometry of the mode

78

5. Liquid Phase Density-Viscosity Sensing

T Ri AC

Zi

T1

Rc

Re

Ct Lc

Le

T3

i uv

k

{X, Y} T2

FL v

h

n

m

hl

ml

T4

XT

Excitation circuit

Eddy currents

Acoustic resonator

Viscous Load

Figure 5.3: Electromagnetic-acoustic equivalent circuit with added ideal mass element.

shape. These parameters have to be fitted for different resonator devices, and the density and viscosity can then be extracted from the fitting function. Furthermore, this is again a one-dimensional model. Therefore, the different mode shapes of circular or diagonal shear vibrations also has to be taken into account. Out-of-plane components of the vibration must be negligible to be able to use this simple model for comparison with measurement data. The next chapter 6 explains the additional elements necessary to account for compressional waves radiated into the liquid. That extension of the model can also be used to include the effect of small flexural motion on the measurement of liquid properties. Ideally, multiple sensor elements are used in different vibration modes to independently measure, e.g., density, viscosity and liquid level, and with further data processing we can then eliminate the parasitic influences on the individual measurement parameters. Alternatively, for a similar densities and equal measurement volumina, the flexural components result in a constant offset of the frequency shift and damping, which can thus be easily subtracted.

5.4 5.4.1

Results and Discussion Metal Plate Elements

The metal mesa plate elements have been our first devices for experimental validation of sensing in the liquid phase, due to their higher electro-

79

5.4.1. Metal Plate Elements Table 5.1: Frequency shift and damping behavior of circular FSM at 142 kHz in aluminum due to liquid loading. ρ / kg/m3

η / mPas

Zi / mΩ

∆f / Hz

Ethanol

775

1.12

122.07

−31.3

Propanol

785

1.51

112.21

−62.5

Glycerine 1:10

815

3.65

88.55

−77.5

Glycerine 1:5

930

10.92

52.37

−136.3

Liquid

mechanical coupling factors. The stronger coupling is required to overcome the high damping in viscous liquids. For these measurements, the mesa plates were combined with plastic tubular cylinders to form a liquid measurement cell with the resonators forming the bottom. Depending on the mode of vibration, the plastic cylinders were attached on the outer edge of the mesa groove or the outer edge of the metal plate. E.g., for the circular FSM the resonator motion is minimal at the outer edge of the mesa groove, but larger at the outer edge of the plate, therefore the plastic cylinders were attached to the region of minimal vibration to minimize the interference and damping of the desired circular mode shape. The height of the measurement liquids has been kept constant to minimize interference due to spurious out-of-plane vibration components. For the metal plate elements, the circular and diagonal face shear mode shape have given the best results [94, 95]. In both cases, the vibration is effectively trapped in the mesa structure of the resonator element. Table 5.1 and figure 5.4 show the measurement results of the circular face shear mode for different liquids, in terms of frequency shift ∆f and damping. The normalized impedance Zi (eq. 3.1) is given as measure of damping and changes to the Q-factor of the resonance. Similar results have been obtained for the diagonal FSM, where the excitation and detection efficiency is not as high (Table 5.2). The results show a nearly linear relationship between √ frequency shift ∆f and square root of density-viscosity ρη, however the measurement error remains too large to serve as proof of this dependency. Both modes are nevertheless well suited to distinguish between different fluids

80

Normalized Impedance Zi /Ω

5. Liquid Phase Density-Viscosity Sensing

0.14 0.12 0.1

in air Ethanol Propanol Glycerine 1:10 Glycerine 1:5

increasing viscosity

0.08 0.06 0.04 0.02 0 −0.02

141.65

141.70

141.75

141.80

141.85

Frequency f / kHz

141.90

141.95

Figure 5.4: Normalized impedance spectrum of primary coil for liquid loading of circular FSM at 142 kHz in aluminum resonator yielding frequency shifts and damping as summarized in Table 5.1.

and to determine the density-viscosity product. The influence of sample mass and volume is negligible and parasitic effects influencing the sensor response have not been observed. The metal plates work very well, are easy to integrate in a measurement cell and easy to clean and replace. They are best suited for larger liquid volumina and lower frequency vibration modes. The exact relationship between density-viscosity product and frequency shift and damping has to be experimentally derived for each specific setup, as the vibration mode shape strongly influences the proportionality factor for frequency shift and damping. The boundary conditions of the specific setup (e.g. clamped, simply supported or Table 5.2: Frequency shift and damping behavior of diagonal FSM at 217 kHz in aluminum due to liquid loading. ρ / kg/m3

η / mPas

Zi / mΩ

∆f / Hz

Ethanol

775

1.12

13.47

−47.5

Propanol

785

1.51

12.14

−57.0

Glycerine 1:10

815

3.65

13.18

−100.0

Glycerine 1:5

930

10.92

9.32

−152.5

Liquid

81

60 40 20 0 −20 −40

in air

1-Propanol

0.6 0.5 0.4 0.3 0.2 0.1 0

Phase shift Θ/◦

Normalized Impedance Zi /mΩ

5.4.2. Silicon Membrane Resonators

221.0 221.2 221.4 221.6 221.8 222.0 222.2 222.4 222.6

Frequency f / kHz

Figure 5.5: Measurement results for diagonal FSM of (111) silicon membrane for different alcohols in terms of density-viscosity product vs. frequency shift and normalized impedance.

free edges) also influence the mode shape considerably, therefore a specific calibration is necessary for each specific resonator elements and measurement cell.

5.4.2

Silicon Membrane Resonators

After modifying multiple silicon elements with increased copper layer thickness as described in chapter 3, they became suitable for face shear mode excitation and thus liquid phase sensing. Excitation was achieved with a quadratic planar spiral coil (15x15 mm) and two alternated NdFeB permanent magnets (measured flux density of 0.4 T), resulting in linear in-plane forces (see Fig. 3.14). The resonance experiences a distinct and reproducible frequency shift and damping characteristics when put in contact with a fluid. We have achieved reproducible results for the investigation of small volumes of different alcohols. The measurement cell was a quadratic, 20x20 mm, plastic cylinder (PMMA) with a thin plastic foil at the bottom to minimize distance to the excitation coil, into which the silicon resonator element was placed. Excitation and detection was performed remotely on the outside of the liquid reservoir, i.e. there was no direct contact between the electric

82

5. Liquid Phase Density-Viscosity Sensing

Table 5.3: Influence of density and viscosity on resonant response of diagonal FSM of (111) Si membrane. ρ / kg/m3

η / mPas

Zi / mΩ

Θ / m◦

∆f / Hz

1-Propanol

803.5

1.898

13.2

94.20

−262.5

1-Butanol

809.8

2.524

12.2

87.93

−337.5

1-Pentanol

814.4

3.441

10.4

77.54

−350.0

1-Heptanol

821.9

5.942

8.5

61.56

−425.0

1-Octanol

827.0

7.368

7.8

56.56

−450.0

1-Decanol

829.7

10.974

6.9

47.36

−562.5

Liquid

parts and the analyte. This again demonstrates a strong advantage of electromagnetic excitation, where difficulties of electrical insulation, crosstalk, parasitic capacitances, etc. are circumvented. Flexural components of the vibration result in an offset corresponding to the liquid volume, which can be considered constant for the same volume of the different test liquids, though. For our experimental results shown here, we have utilized the first diagonal face shear mode at 222 kHz of a 300 µm thick (111) Si membrane, where the electromechanical coupling efficiency was strongest and only a minimal amount of flexural movement could perturb the measurements [84]. In figure 5.5 we have plotted the normalized impedance and phase curves for a single measurement. We have investigated six different primary alcohols from 1Propanol to 1-Decanol. Table 5.3 lists the density and viscosity of these alcohols at a temperature of 25◦ C, as well as the resulting frequency shift compared to the resonance in air and the normalized impedance magnitude and phase peaks. The liquid volume was kept constant at 100 µl put inside the custom designed liquid reservoir. The silicon membrane and the reservoir were emptied, cleaned with isopropanol, and dried after each measurement to remove all traces of the previous liquid. The measurement time was kept the same for all liquids to minimize effects due to evaporation. Figure 5.6 visualizes the resulting frequency shift and additional viscous damping for increasing dynamic viscosity and density. As the characteristic acoustic impedance of a shear wave in a viscous liquid depends on the density-

83

5.5. Summary

700

Impedance damping ∆Z/mΩ

650 600

12

550 500

10

450 400

8

350

Frequency shift ∆f / Hz

Impedance Damping Frequency Shift

14

300 6

250 1

1.5

Density-viscosity

2

√

2.5

ρη / Pa s

3/2

3

m

200

−1

Figure 5.6: Measurement results for diagonal FSM of (111) silicon membrane for different alcohols in terms of density-viscosity product vs. frequency shift and impedance damping.

visosity product ρη, it is a reasonable assumption that the shown parameters are also functions of ρη. A linear correlation of the frequency shift can be fitted to the square root of density and viscosity, as suggested by theory. The Q-factor, or the damping, is characterized in terms of the normalized impedance peak ∆Z in the frequency response of Zi , which also shows a nearly linear relationship.

5.5

Summary

By utilizing a different set of vibration modes compared to mass sensing, namely pure face shear vibrations, we have demonstrated the use of these sensor devices for liquid phase measurements. Different technologies for the resonator element have been considered (metallic disks, polymer foils, micromachined membranes) and found suitable for this application at different operating frequencies and mode shapes. Similar to other acoustic resonators,

84

5. Liquid Phase Density-Viscosity Sensing

the electromagnetic-acoustic transducer is in principle a viable sensor for measuring liquid properties as density and viscosity. These transducers can take full advantage of the remote nature of excitation and detection, as well as the inexpensive fabrication. The resonator element can be fully immersed into the liquid analyte, while the more expensive excitation and detection setup is placed on the outside. At the same time, the detection efficiency is considerably lower than that of conventional wired piezoelectric transducers. Therefore, dedicated sensor electronics are required. Using single crystalline silicon elements as electromagnetic-acoustic resonators yields the advantage of higher Q-factors than other suitable materials. However, for efficient excitation the conductive layer of the substrate needs to be adjusted to the specific frequency range of operation. We have demonstrated that additional electroplating on our silicon membranes improves the coupling efficiency for face shear modes considerably, while resulting in a negligible reduction of Q. These modified elements are therefore also suitable for liquid phase sensing at higher sensitivity and accuracy than polymer foils or metal plates. As proof of concept we validated the theoretical relationship of resonance frequency and impedance amplitude to the density-viscosity product of different alcohols.

6 Liquid Level Measurement Liquid level measurement is a key aspect for many applications in the processing industry, in analytical chemistry, and the pharmaceutical industry to determine, e.g., the exact liquid volume in a reaction chamber. A large variety of measurement principles exist, ranging from mechanical, electrical, and optical methods to ultrasonic and radar measurements [96, 97, 98]. However, many of these principles are not suitable for sensing very small amounts or changes of liquid volume and liquid level. Ultrasonic transducers allow for very precise liquid level measurements, where the sensitivity can be improved by increasing the operating frequency. If the liquid level is a known property, we can in turn use the measurement results to evaluate acoustic wave velocity and acoustic impedance in the liquid [18]. Ultrasonic liquid level sensor devices are primarily based on piezoelectric or capacitive excitation methods. In general, the geometric electrode configuration limits each transducer to a single mode of vibration and therefore limits the sensitivity of the sensor element. In this chapter we introduce the application of our electromagnetic-acoustic resonators as liquid level sensors. Here, out-of-plane flexural vibrations are generated to radiate compressional, longitudinal waves into the liquid. In contrast to acoustic shear waves, these display a very large penetration depth and are highly sensitive to the liquid level. We measure the frequency shifts and changed damping characteristics due to interference effects instead of the time-of-flight measurements usually employed by ultrasonic liquid level sensors. Our experiments show, that by using continuous, resonant excitation the achievable resolution is not limited by the wavelength of the ultrasonic waves radiated into the liquid. We again employ an experimental setup consisting of a simple resonating element placed at the bottom of a fluidic chamber, which is remotely excited by a planar coil. We have found that while out-of-plane motion is an unwanted component for mass and viscosity sensors, with specific flexural modes we can utilize the

86

6. Liquid Level Measurement

same experimental setup as liquid level sensors as well. This presents us with a unique and novel advantage, the possibility to measure multiple properties with the same, single sensor element. With sensor arrays instead of single elements, we can also measure these properties simultaneously by exciting different vibration modes in each element.

6.1

Suitable Modes of Vibration

As described in chapter 2, depending on setup of primary coil, magnets, and resonator, a multitude of Lorentz force distributions can be realized. Flexural, out-of-plane motion generates longitudinal, compressional waves which are radiated into a liquid. Compared to ultrasonic shear waves, the much larger penetration depth in viscous liquids causes the compressional wave to be reflected at boundaries of the liquid to return and interfere with the resonator motion. We employ continuous excitation generating standing acoustic waves and taking advantage of this interference behavior. As the generated mode of vibration is depending on external parameters such as the alignment of coil and magnets, and the driving frequency, the actual sensor element does not have to be exchanged for sensing different properties. The frequency f of the eigenmodes of the resonator is determined by the material properties, elasticity coefficients and density, and the geometry of the element. The ultrasonic velocity c of compressional waves in the liquid analyte is determined by the square root of compressibility κ and density p ρ: c = 1/(κρ) and we can calculate the wavelength λ of the longitudinal wave with λ = c/f . Due to interference effects of longitudinal waves reflected from the liquid surface, it can be expected that the resonance behavior cycles through a repetitive pattern whenever the level increases by λ/2. Within this cycle constructive and destructive interference are alternating. The minimum and maximum interference correspond to a level difference of λ/4. Due to this repetitive pattern, a single measurement is ambiguous in terms of absolute height, however, depending on mode shape, frequency shift and amplitude damping can be evaluated as two distinct parameters. The most efficiently excited flexural vibration eigenmode with our ex-

6.2. Equivalent Circuit Model

87

Figure 6.1: Experimental setup for liquid level measurements with resonator element integrated into liquid reservoir and excited by planar coil and external magnet.

perimental setup is a radial flexural face shear mode. This mode features radial in-plane vibration with a strong flexural component at the center of the resonator, due to conservation of mass. Therefore, a planar spiral coil and perpendicular magnetic field is the ideal excitation setup and the experiments also show that the radial flexure mode is one of the best coupled modes of vibration. For the aluminum mesa plates the radial flexure mode has an eigenfrequency of 289 kHz (Fig. 6.1).

6.2

Equivalent Circuit Model

To model the behavior of standing compressional waves radiated into a liquid and reflected back to the resonator surface we have amended our equivalent circuit from chapter 2. The liquid level sensitivity and interference behavior is given by a lossy transmission line, where the characteristic wave impedance Za corresponds to the acoustic impedance of the liquid and line length l corresponds to the liquid level [99]. The surface boundary condition for the acoustic wave at the liquid/air boundary is included in the load resistance Rr . Due to the complexity of the vibration mode shape, we also take into account the viscous shear load as shown in chapter 5, for clarity these

88

6. Liquid Level Measurement

T Ri AC

Zi

T1

Rc

Re

Ct Lc

Le k

T3

i uv

Za

FL {X, Y} T2

v

h

n

T4

l

XT

Excitation circuit

Eddy currents

Rr(l)

m

Acoustic resonator

Acoustic wave

Figure 6.2: Electromagnetic-acoustic equivalent circuit, with eddy current induction described by electromagnetic transformer T , coupling between current i, voltage u and force F , velocity v in the resonator modeled by electro-mechanical transformer XT , and the standing longitudinal wave in a liquid above the resonator represented by a lossy transmission line with a wave impedance Za .

elements have been omitted from figure 6.2, but included in the simulation. Strictly speaking, the mechanical shear and normal velocities are different variables depending on the 3D mode shape, however, for a particular dominant mode we can assume that they are linked by an approximate constant factor and thus both can be represented by one lumped variable in the same 1D equivalent circuit. The transmission line represents a one-dimensional approximation for the interference of compressional waves and resonator movement. Effects not included in this simple demonstration of the transmission line modeling are multiple transmission paths depending on the three-dimensional mode shape of the resonator vibration. Other effects include mode conversion at the liquid surface boundary or resonator surface. These effects are partially approximated by introducing a dependence of the reflection coefficients on the length l, i.e. Rr (l). The simulation procedure consists of calculating resonator mass and compliance from density, elasticity and geometry and fitting the damping parameter to measurement data in air. Then we calculate the acoustic impedance and wave velocity of the liquid from density, compressibility and geometry and enter these as the transmission line parameters, including the liquid level as its length. Finally we fit the reflection and coupling coefficients to measurement data at this liquid level.

89

6.3. Results and Discussion

Normalized Impedance Zi /Ω

0.05 0µl

0.04 increasing fluid volume

0.03

1500µl 1400µl

0.02 0.01

900µl

1100µl 1200µl

800µl

1000µl

1300µl

0 265

270

275

280

Frequency f / kHz

285

Figure 6.3: Preliminary results during density-viscosity experiments showing a strong dependence on volume in liquid measurement cell.

6.3 6.3.1

Results and Discussion Preliminary Results

While analyzing the efficiency of different modes of vibration for densityviscosity measurements, we experienced much larger frequency shifts than anticipated for viscous loads for specific vibration modes. Further investigation revealed these frequency shifts to be dependent on the liquid volume inserted into our measurement cells. Figure 6.3 shows such combined measurement results. Upon detailed analysis of the employed mode shape, we concluded the dominance of radiated compressional waves as described above for this vibration [100].

6.3.2

Measurement Setup

To utilize an electromagnetic-acoustic resonating element as a liquid sensor, we can simply place the element inside a fluidic chamber and remotely excite it from the outside as outlined above. The resonator can also be integrated into or already be part of the chamber. The only requirement is that the resonating element must be conductive and that the eddy current induction is

90

6. Liquid Level Measurement

Figure 6.4: Experimental setup for liquid level measurements with resonator element integrated into liquid reservoir and excited by planar coil and external magnet.

not shielded by the chamber walls. Figure 6.4 shows the measurement principle with a resonator placed inside the liquid chamber. The photo depicts a measurement setup, where the liquid chamber is part of the plate resonator and placed above primary induction coil (0.12–0.19 mm diameter, Roadrunner Electronics Ltd., UK) and permanent NdFeB magnet with a measured static magnetic flux density at the sensor element of 0.375 T. The liquid volume has been inserted with microliter pipettes (Eppendorf epResearch3 Pack ) in steps of 25±0.25 µl or 50±0.35 µl. The fluidic chamber for the aluminum resonator was a round plastic cylinder with a diameter of 20.5 mm. All geometrical parameters were measured with a micrometer gauge. This results in a liquid level increase of about 76 µm per 25 µl added water volume. Uncertainties in liquid level are derived from uncertainties in added liquid volume and geometrical parameters of the fluid chamber, which has to be taken into account for calibration of the device. According to the calculations from above, the resonance pattern should cycle with every 900 µl added volume, corresponding to a liquid level change of λ/2. For precise measurements a temperature stable setup is required, primarily due to the temperature coefficients of density and ultrasonic velocity. The experiments shown here have been carried out at a temperature of 25±0.5◦ C.

91

6.3.3. Measurement and Simulation Comparison

0.06

Normalized Impedance Zi /Ω

Normalized Impedance Zi /Ω

0.06 0.05 0.04 0.03 0.02 0.01 0 −0.01

271

272

273

274

275

Frequency f / kHz

276

277

0.05 0.04 0.03 0.02 0.01 0 −0.01

271

272

273

274

275

Frequency f / kHz

276

277

Figure 6.5: Impedance measurement (left) and simulation (right) for flexural resonance at 275 kHz with liquid volume at maximum around 900 µl, corresponding to a height at maximum of 2.75 mm = λ/2. Each curve corresponds to a change of 25 µl water or 76 µm height, respectively.

In standard literature, a volumetric temperature expansion coefficient of DIwater of 0.256 ml · l−1 K−1 can be found, leading to a height deviation of ±0.384 µm/ml, or 0.013 % for this setup. On the other hand, from literature values we can estimate a temperature coefficient for the ultrasonic velocity of 2.67 ms−1 K−1 around 25◦ C, leading to an error of ±1.34 ms−1 , or 0.089 %, in the velocity. These primary effects partially compensate each other, leading to a systematic frequency error of ±209 Hz, or 0.076 % at our compressional mode operating frequency of 275 kHz. To further minimize the influence of the temperature dependence, we have also been developing a thermostat based, temperature stabilized experimental setup operated at ±0.05 K for simultaneous measurements with different acoustic resonator sensors.

6.3.3

Measurement and Simulation Comparison

As our sensor elements we utilize aluminum plate resonators and silicon membranes with copper coating as the conductive layer. For liquid level measurement we excited the resonators at a radially symmetric flexural mode of vibration at 275 kHz. This mode of vibration is efficiently excited with a planar spiral coil and a perpendicular magnetic field in the normal direction, resulting in radial Lorentz forces. Figure 6.5 shows the comparison between

92

6. Liquid Level Measurement

0.12

Normalized Impedance Zi /Ω

Normalized Impedance Zi /Ω

0.12 0.1 0.08 0.06 0.04 0.02 0 −0.02

270

272

274

276

Frequency f / kHz

278

0.1 0.08 0.06 0.04 0.02 0 −0.02

270

272

274

276

Frequency f / kHz

278

Figure 6.6: Impedance measurement (left) and simulation (right) for flexural resonance at 275 kHz with liquid volume at maximum around 2700 µl, corresponding to a height at maximum of 8.25 mm = 3/2λ. Here, each curve corresponds to a change of 50 µl water or 154 µm height.

measured and simulated impedance curves for an aluminum resonator and a liquid loading around 900 µl, with 25 µl difference between the different spectra. For both, measurement and equivalent circuit simulation a maximum impedance is achieved for a liquid level of λ/2, or 2.75 mm. The frequency shifts for each 25 µl step are around 150 Hz. At maximum impedance the resonance curves achieve Q-factors of about 300-400. At these measurement parameters and with a temperature deviation of 209 Hz, equipment accuracy of 5 Hz, and measurement points every 12.5 Hz (depending on frequency span), we can calculate a liquid level resolution of 6.3 µm at an accuracy of about 100 µm. With increasing liquid level, distortion due to secondary propagation paths is reduced. Only the normal components of the compressional wave pattern are reflected to the resonator surface, other components are increasingly dampened by multiple reflections and longer propagation paths. The resulting frequency spectrum is shown in figure 6.6 for the impedance amplitude and figure 6.7 for the impedance phase. Here, the maximum of the envelope is around 2700 µl, corresponding to a height of 3λ/2 = 8.25 mm. The correlation between measurement and simulation is improved compared to lower liquid levels, most likely due to the decrease of the influence of multiple,

93

6.3.3. Measurement and Simulation Comparison

Normalized Phase Θ/◦

Normalized Phase Θ/◦

0.2 0.1 0 −0.1 −0.2

268

270

272

274

276

Frequency f / kHz

278

280

0.1 0 −0.1 −0.2

268

270

272

274

276

Frequency f / kHz

278

280

Figure 6.7: Comparison of phase measurement (left) and simulation (right) corresponding to figure 6.6.

reflected propagation paths. The frequency shifts increase to 750 Hz for a 50 µl step at envelope maximum, and the resonance curve shows a Q-factor up to 1000. With a frequency step size of 15 Hz, the liquid level resolution therefore improves to 3 µm, at an accuracy of about 40 µm, the latter dominated by temperature effects. This corresponds to a liquid volume resolution of 1000 nl for our experimental setup. The measurements in figures 6.5 and 6.6 show a cyclic interference behavior as predicted. There are however some noticeable differences in the envelope around each interference cycle, which are primarily the result of the 1D-approximation of the three-dimensional radiation pattern of the compressional wave. Mainly the width of the envelope and the frequency shifting between different liquid levels cannot be exactly reproduced. However, the general behavior is successfully modeled. The equivalent circuit simulation can be used to predict the cyclic interference pattern accurately and to model the primary acoustic wave behavior. Silicon elements have been successfully excited to a similar mode of vibration and have shown the same cyclic interference behavior. However, the coupling coefficients for flexural and face shear modes are weaker than those obtained with aluminum resonators. A modified design and experimental setup is required to fully utilize the better elastic properties and thus the larger intrinsic mechanical Q-factor of these devices. Other modes of flexural

94

6. Liquid Level Measurement

vibration have also been investigated for both silicon and aluminum resonators. The resonant vibration of face shear modes in our plate resonators is much stronger though, purely flexural modes induce only a weak secondary voltage. Finally, we would like to note another possible application of the sensor. As described by Hauptmann et al. for different piezoelectric transducers [18], if the liquid level is a known property, we can in turn use the measurement results to evaluate acoustic wave velocity and acoustic impedance in the liquid. Acoustic velocity and impedance primarily depend on the product and fraction of density and compressibility, respectively. From our liquid level resolution we can estimate an acoustic velocity resolution down to 0.4 m/s, or 270 ppm. From measurements of the in-plane circular face shear mode we can extract the density-viscosity product. Therefore, by analyzing frequency shift and damping of two vibration modes in the same resonator element, we can successfully establish an accurate description of the liquid analyte in terms of three fundamental physical properties without requiring multiple sensor elements, literature values, or complex algorithms.

6.4

Summary

In this chapter we have presented a novel approach to ultrasonic liquid level measurements utilizing electromagnetic-acoustic resonator sensors. Instead of pulsed ultrasonic waves, where the resolution is limited by the wavelength, we continuously excite a standing compressional wave in the liquid and measure frequency shifts and damping due to interference effects of the reflected wave at the resonator surface. With this method we have achieved a liquid level resolution of 3 µm at a height of 10 mm. Our experiments show the suitability of this method to measure and control very small amounts of liquids. To further improve level resolution and accuracy, a better temperature stability of the measurement setup and specialized sensor electronics for smaller frequency steps around resonance are required. Our equivalent circuit model from chapter 2 has been refined with a onedimensional approximation of the compressional wave radiated into the liquid

6.4. Summary

95

to predict the cyclic interference pattern seen in the measurements. A lossy transmission line with a length corresponding to the liquid level sufficiently models this behavior. The elements of the complete equivalent circuit are based on the material properties of excitation coil, resonator element, and liquid analyte, while the coupling coefficients and acoustic damping elements were used as fit parameters. We have successfully demonstrated a suitable correlation of experimental and simulated results.

7 Conclusions Electromagnetic-acoustic resonators have been established as the third major family of resonant acoustic sensors. In the work presented in this thesis, we have shown how to take advantage of the unique features offered by this method of excitation. Of major interest, especially for liquid phase sensing, is the ability to spatially separate resonator element from excitation and readout electronics. We have demonstrated how this remote excitation can be exploited and shown that it greatly reduces the packaging complexity and problematics, e.g. by removing problems with electric insulation, parasitic elements, or other undesired effects. Resonator elements can be simply placed inside measurement cells or even be part of them, no further is packaging required. Since electromagnetic-acoustic excitation is also possible for many kinds of resonator materials, including low cost metal plates or plastic foils, this opens the door to treating the vibrating element as an easily replaceable part of the measurement setup. Experimenting with complex, aggressive, and polluted analytes becomes more feasible, as rigorous cleaning and preparation methods are no longer necessary. Furthermore, for measurements with higher precision, which is basically limited by the elastic quality and intrinsic loss of the resonator, we have demonstrated the use of single crystalline silicon membranes, which exhibit better elastic properties than even quartz, and are easier and cheaper to fabricate and modify. Another advantage that ties into the spatial separation is the ability to work with a bare resonator surface, which is not perturbed by metalization or passivation layers. This adds a great deal of freedom for the possible surface chemistry, especially on inert materials like silicon. However, from our point of view, the major and most unique advantage lies with the ability to excite a wide variety of fundamentally different vibration modes in the same resonator element. Due to the different physical mechanisms at work and the spatial separation of the excitation sources, i.e. resonator, coils and magnets, we have several degrees of freedom that

98

7. Conclusions

influence the distribution of currents, forces and displacement. The geometry of the coil and its alignment to the conductive surface define the eddy current distribution, which, together with the direction of the external magnetic fields, defines the distribution of Lorentz forces. The forces then again have to match the desired vibration mode shape both in terms of direction and driving frequency. Changing these degrees of freedom determines which vibration mode is suppressed, and which is successfully excited. With this flexibility, the same vibrating element and the same excitation setup can be used for fundamentally different vibrations, e.g. pure shear movement or dominant flexural motion. Since the surface motion is dependent on different physical properties, e.g. density, viscosity, level, and compressibility in this case, one and the same sensor is suitable to measure multiple physical properties. That is very much unlike established ultrasonic devices, where for multiple measurands, multiple sensors with different designs are generally required. As proof of concept, we have demonstrated the suitability of our resonator designs to act as mass-microbalance, density-viscosity sensors, and liquid level measurement devices. The designs and experimental setups were based on theoretical and finite element analysis of suitable eigenmodes and the requisite Lorentz force distribution. For prediction and verification of experimental results, we have developed an equivalent circuit model based on the physical properties of the electrical and mechanical components and the acoustic loads. The circuit is based on electro-mechanical analogies and can thus be easily analyzed and expanded with the powerful network theory tools of electrical engineering. Our colleagues and other groups have also demonstrated different approaches to electromagnetic-acoustic resonators, e.g. by impressing a current into cantilevers, suspended bridges and membranes [23, 63, 101, 102, 103]. While losing the non-contact advantage and some degrees of freedom, these methods can display improved coupling efficiency and also be operated in non-resonant modes.

7.1. Perspectives

7.1

99

Perspectives

Future work on this topic should focus on establishing calibration curves for the different resonators and modes of operation. Establishing analytical relationships between surface mass or density-viscosity and resonance frequency shift similar to Sauerbrey’s and Kanazawa’s equation for quartz crystal resonators requires the analytical description of the complete sensing mechanism. Since we recovered the known linear relation of frequency shift to added mass and the square root of the liquid density-viscosity product, a constant factor of proportionality, or transduction factor can be expected. The same applies to liquid level sensing. It also relies on changes in the acoustic load, here due to constructive and destructive interference of compressional acoustic waves in the liquid cavity. Consequently, the acoustic load is frequency dependent, just in contrast to density-viscosity measurements where the frequency dependence of the acoustic load is rather weak. Although frequency shift measurement is the prevailing measurement technique, the impedance maximum carries independent information of the system under investigation, first of all information about acoustic energy dissipation. Besides material related absorption this parameter also has the capability to be used for estimation of other attenuation mechanisms, e.g., amount of acoustic wave radiation or acoustic wave scattering. Again, an analytical relation between acoustic properties and its electrical representation is required. This part is more challenging, since frequency measurement does not rely on the height of the resonance peak. The analytical deviation of the transduction factor or transduction function requires respective relations dealing with mutual inductance of the planar coil and its counterpart formed in the resonator as well as analytical approximations of the mechanical displacement (pattern) of the resonator. On the other hand, the liquid density-viscosity product also shows the same square root dependence with regard to change in the coil impedance maximum which coincides with the behavior of the acoustic load. With knowledge of the transduction parameters, analytically derived from calibration curves, absolute values of mass, density, viscosity, or compressibil-

100

7. Conclusions

ity (changes) can be determined. With a clever design of the measurement cell the above products can be separated, e.g. [104]. Once a complete set of calibration curves, or transfer functions, has been created, we can directly analyze multiple physical measurands of an unknown liquid. This furthermore allows for a new class of applications. As described before, if one measurand is a known property, we can in turn use the measurement results to evaluate other paramaters, e.g., acoustic wave velocity and acoustic impedance instead of liquid level. Therefore, by analyzing frequency shift and damping of multiple vibration modes in the same resonator element, we can successfully establish an accurate description of the liquid analyte in terms of multiple fundamental physical properties.

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A Appendix A.1

Additional tables Table A.1: Resonator material properties. Variable c11 c12 c44 EAl ECu EN iAg ESi νAl νCu νN iAg νSi ρAl ρCu ρN iAg ρSi σAl σCu

Value 166 GPa 64 GPa 80 GPa 72 GPa 110 GPa 125 GPa 170 GPa 0.34 0.343 0.30 0.26 2710 kg/m3 8960 kg/m3 8670 kg/m3 2329 kg/m3 3.550 · 107 S/m 5.998 · 107 S/m

Description elastic stiffness coefficient of silicon elastic stiffness coefficient of silicon elastic stiffness coefficient of silicon elastic modulus of aluminum elastic modulus of copper elastic modulus of nickel-silver elastic modulus of silicon (isotropic) Poisson ratio of aluminum Poisson ratio of copper Poisson ratio of nickel-silver Poisson ratio of silicon (isotropic) density of aluminum density of copper density of nickel-silver density of silicon conductivity of aluminum conductivity of copper

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Table A.2: Planar excitation coil parameters (Ct → 0). Geometry Lc / µH Cp / pF Rc / mΩ org spiral N63 23.33 7.3 7990 org spiral N63 1.0 5.1 124.8 new spiral round 3.3 7.1 5200 new spiral rect 3.2 7.6 5000 new linear N44 0.1185 4.5 15.7 new linear N44 0.1109 4.3 139 new radial 1 0.1060 6.3 37.5 new radial 2 0.1192 5.3 14.9 new radial 3 0.1113 6.1 13.6

Table A.3: Successfully excited and detected flexural and face shear eigenmodes of simple 15x15 mm aluminum plate, with radial, diagonal and circular excitation setup.

radial 143 204 284 416 517 530 545

Flex, FSM / kHz diagonal circular 1 circular 2 22.9 9.2 202 40.4 50.9 339 64.8 92.5 369 138 132 388 143 210 487 280 245 518 339 288 571 369 386 388 462 435 571 487 710 497 575 643 697

117

A.1. Additional tables

Table A.4: Successfully excited and detected TSM eigenmodes of simple 15x15 mm aluminum plate, with radial, diagonal and circular excitation setup.

radial 2.35 2.375 2.38 2.385

TSM / MHz diagonal circular 1 circular 2 2.32 2.34 2.32 2.42 2.36 2.415 2.43 2.48

Table A.5: Successfully excited and detected flexural and face shear eigenmodes of round mesa aluminum plate, with radial, diagonal and circular excitation setup.

radial 15.5 48 101 168 217 264 287 457 621

Flex, FSM / kHz diagonal circular 1 circular 2 14.7 64.3 46.0 30.9 142 144 48.0 536 218 81.9 699 265 144 286 169 359 217 398 223 435 275 476 370 494 398 513 465 609 493 513 699

Table A.6: Successfully excited and detected TSM eigenmodes of round mesa aluminum plate, with radial, diagonal and circular excitation setup. TSM / MHz radial diagonal circular 1 circular 2 1.57 1.52 1.525 1.55 1.60 1.63 1.71

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A. Appendix

Table A.7: Successfully excited and detected flexural and face shear eigenmodes of round mesa brass (CuZn) plate, with radial, diagonal and circular excitation setup.

radial 68.5 172 192 305 417

Flex, FSM / kHz diagonal circular 1 circular 2 30.9 30.9 68.5 55.4 41.8 88.4 112.5 55.5 112.4 145 95.6 145 266 112.4 193 344 318 266 403 291 468 318

Table A.8: Successfully excited and detected TSM eigenmodes of round mesa brass (CuZn) plate, with radial, diagonal and circular excitation setup. TSM / MHz radial diagonal circular 1 circular 2 1.105 1.099 1.130 1.146

Table A.9: Electroplating settings at 22◦ C and results for additional copper layer thickness and smoothness on Si membranes. Si sample (100) 190 µm (111) 300 µm (100) 390 µm (100) 190 µm (100) 190 µm (111) 300 µm (100) 190 µm (111) 300 µm (100) 190 µm (100) 190 µm

Q Q Q Q Q Q Q Q Q R

electroplating settings Cu thickness TSM freq. / MHz 1 V, 1.6 A, 10 min ≈ 20 − 25 µm 16.05 →n/a 0.5 V, 0.5 A, 1 min ≈ 4 − 6 µm 8.51 → 8.19 2 V, 2.6 A, 1 min ≈ 5 µm 7.45 →n/a 0.5 V, 0.6 A, 2 min ≈ 6 µm 15.89 → 15.57 1 V, 1.4 A, 2 min ≈ 13 µm 16.06 →n/a 1 V, 1.6 A, 1 min ≈ 5 µm 8.19 → 8.05 1 V, 1.5 A, 1 min ≈ 4 µm 15.57 → 15.15 0.4 V, 0.35 A, 10 min 8.05 → 7.99 0.4 V, 0.38 A, 10 min 0.4 V, 0.42 A, 10 min

quality bad good bad good bad good good good medium medium

A.2. MATLAB routines

A.2

MATLAB routines

Save impedance analyzer measurement - ZA save.m % function [ZA] = ZA_save (title, shorttitle) title = input(’Title: ’, ’s’); shorttitle = input(’Short Title: ’, ’s’); % Create a tcpip object. ZA = instrfind(’Type’, ’tcpip’, ’RemoteHost’, ’140.78.120.12’, ’RemotePort’, 5025, ’Tag’, ’’); if isempty(ZA) ZA = tcpip(’140.78.120.12’, 5025); else fclose(ZA); ZA = ZA(1) end % Connect to ZA. fopen(ZA); % Communicating with ZA. % data1 = query(ZA, ’*IDN?’); % fprintf(ZA, ’BWFACT 3’); fprintf(ZA, ’BEEPDONE on’); fprintf(ZA, fprintf(ZA, fprintf(ZA, fprintf(ZA, fprintf(ZA, fprintf(ZA, fprintf(ZA, fprintf(ZA,

’SING’); ’*WAI’); ’TRAC A’); [’TITL "’, title, ’"’]); ’AUTO’); ’TRAC B’); ’AUTO’); ’*WAI’);

fprintf(ZA, [’SAVDASC "’, shorttitle, ’"’]); fprintf(ZA, [’SAVDTIF "’, shorttitle, ’"’]); fprintf(ZA, ’*WAI’); input(’Save in progress’); % Disconnect from ZA. fclose(ZA);

Save impedance analyzer sweep - ZA sweep.m % function [ZA] = ZA_sweep (title, shorttitle, start, stop, span) title = input(’Title: ’, ’s’); shorttitle = input(’Short Title: ’, ’s’); start = input(’Start Freq: ’); stop = input(’Stop Freq: ’); span = input(’Span Freq: ’); % Create a tcpip object. ZA = instrfind(’Type’, ’tcpip’, ’RemoteHost’, ’140.78.120.12’, ’RemotePort’, 5025, ’Tag’, ’’); if isempty(ZA) ZA = tcpip(’140.78.120.12’, 5025); else fclose(ZA); ZA = ZA(1) end step = (stop-start)/span; % Connect to ZA.

119

120

A. Appendix

fopen(ZA); % Communicating with ZA. % data1 = query(ZA, ’*IDN?’); % fprintf(ZA, ’BWFACT 3’); fprintf(ZA, ’BEEPDONE off’); i=0; while (i 0) freq(k)=d{1}; data(k)=d{2} + i*d{3}; else break; end; k=k+1; end; fclose(f);

Read measuremed phase data - readdata p.m function [freq,data] = readdata_p (fnam) f=fopen(fnam,’rt’); s=fgets(f);

A.2. MATLAB routines

while (length(findstr(’TRACE: B’,s))==0) s=fgets(f); end; while (length(findstr(’Frequency’,s))==0) s=fgets(f); end; k=1; while (~feof(f) ) d=textscan(f,’%n %n %n\n\r’); if (length(d{1}) > 0) freq(k)=d{1}; data(k)=d{2} + i*d{3}; else break; end; k=k+1; end; fclose(f);

Normalize data - norm ZA.m function [norm_data] = norm_ZA (data) s = data(1); e = data(801); rise = (e-s)/801; i=1; while(i