Resonant Characteristics of Rectangular Hammerhead Microcantilevers Vibrating Laterally in Viscous Liquid Media Jinjin Zhang1, Fabien Josse1, Stephen Heinrich2, Nicholas Nigro3 1 Electrical and Computer Engineering 2 Civil, Construction and Environmental Engineering 3 Mechanical Engineering Marquette University, Milwaukee, USA
[email protected]
Isabelle Dufour4, Oliver Brand5 4 IMS Laboratory, Université de Bordeaux CNRS, Talence, France 5 School of Electrical and Computer Engineering Georgia Institute of Technology, Atlanta, USA exerted on the stem and head must be evaluated separately due to the difference in dimensions. Since the cross-section of either part is still rectangular, the semi-analytical expression for the hydrodynamic function in [1] can still be applied. However, in Ref. 1, the discrepancy between analytical and numerical results is relatively large for small thicknesses and high Re. As a result, it is necessary to obtain a new analytical expression for the hydrodynamic function to accurately determine the sensing characteristics.
Abstract—The resonant characteristics of laterally vibrating rectangular hammerhead microcantilevers in viscous liquid media are investigated. The rectangular hammerhead microcantilever is modeled as an Euler-Bernoulli beam (stem) and a rigid body (head). A modified semi-analytical expression for the hydrodynamic function in terms of the Reynolds number, Re, and aspect ratio, h/b, is proposed to rapidly evaluate the sensing characteristics. Using this expression, the resonance frequency, quality factor and normalized surface mass sensitivity are investigated as a function of the dimensions of the microcantilever and liquid properties. Guidelines for design of hammerhead microcantilever geometry are proposed to achieve efficient sensing platforms for liquid-phase operation. The improvement in the sensing area and characteristics are expected to yield higher sensitivity of detection and improved signal-tonoise ratio in liquid-phase chemical sensing applications.
I.
In this work, a theoretical model of laterally vibrating rectangular hammerhead microcantilevers in viscous liquids is presented. A modified semi-analytical expression for the hydrodynamic function in terms of Re and h/b is proposed and compared with the numerical results. The sensing characteristics are investigated as a function of the geometrical parameters of the rectangular hammerhead microcantilevers and liquid properties, and compared with those of the rectangular prismatic beams. Guidelines for the design of hammerhead microcantilevers for sensor applications are proposed.
INTRODUCTION
Dynamically driven rectangular prismatic microcantilevers excited in the in-plane direction have been investigated and used in liquid-phase sensing applications due to their relatively high frequency stability and mass sensitivity [1-2]. However, in bio-chemical sensing applications, the performance of rectangular prismatic microcantilever-based sensors is restricted due to their limited surface sensing area. Thus, to increase the surface sensing area and improve sensing characteristics, it is proposed to investigate rectangular hammerhead microcantilevers driven in the in-plane flexural vibration mode in viscous liquid media.
II.
THEORETICAL ANALYSIS
A. Equation of Motion The geometry of a rectangular hammerhead microcantilever, with dimensions on the order of microns, is shown in Fig. 1. To model the stem as an Euler-Bernoulli beam, it is generally assumed that its length is much larger than its width (L1>>b1). The equation of motion for a laterally vibrating stem in viscous liquids is given by
For a rectangular hammerhead microcantilever laterally vibrating in a vacuum, the resonance frequency has been obtained in a close-form analytical expression by assuming the head as a point mass at the tip of the stem [3-4]. However, when the vibrating microcantilevers are immersed in viscous liquids, the liquid will impose hydrodynamic forces on the stem and head, which are not accounted for. Thus, it is no longer appropriate to model the head as a point mass. In this work, the stem and head are modeled as an Euler-Bernoulli beam and a rigid body, respectively. As a result, both translational and rotational motions of the head must be taken into account. In viscous liquids, the hydrodynamic forces
Figure 1: A rectangular hammerhead microcantilever with length and width of the stem being L1, b1, respectively; length and width of the head being L2, b2, respectively and thickness being h.
This work is supported in part by NSF Grant No. ECCS-1128992 and NSF Grant No. ECCS-1128554.
978-1-4799-0342-9/13/$31.00 ©2013 IEEE
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EI stem
h⎞ ⎛ ⎛h⎞ Γ R ⎜ Re, ⎟ = Γ R1 ( Re ) Γ R 2 ⎜ ⎟ , (9a) b ⎝ ⎠ ⎝b⎠ h⎞ ⎛ ⎛h⎞ Γ I ⎜ Re, ⎟ = Γ I 1 ( Re ) Γ I 2 ⎜ ⎟ , (9b) b ⎝ ⎠ ⎝b⎠ In equation (9a-b), ΓR1(Re) are ΓI1(Re) are expected to depend on the boundary layer thickness, and are functions of Re-0.5 [6]. ΓR2(h/b) and ΓI2(h/b) can be expanded into a summation of multiple h/b terms (similar to Taylor series expansion). The proposed real and imaginary parts of the hydrodynamic function are as follows:
∂ 4 vstem ( x, t ) ∂ 2 vstem ( x, t ) + ρb b1h = Fstem,liquid ( x, t ) , (1) 4 ∂x ∂t 2
where E is the Young’s modulus, Istem the second moment of area of the cross-section of the stem, ρb the mass density of the microcantilever; vstem is the displacement in the y-direction. The hydrodynamic force on the stem, Fstem.liquid, is force per unit length, and is given by
Fstem ,liquid ( x, t ) = − g1, stem
∂vstem ( x, t )
− g 2, stem
∂ 2 vstem ( x, t )
, (2) ∂t ∂t 2 In equation (2), g1,stem and g2,stem are coefficients associated with the viscous damping and the effective mass coming from the liquid, respectively [1].
⎡ h⎞ 1 ⎛ Γ R ⎜ Re, ⎟ = ⎢ ⎢ b Re ⎝ ⎠ ⎣
B. Boundary Conditions In order to solve equation (1), four boundary conditions are needed. Two boundary conditions are defined at the fixed end. One of them states that the stem is perfectly fixed at the support end, and is given by
p q ⎡ ⎤ ⎡ ⎤ h ⎞ ⎢ 1 pmax ⎛ h ⎞ 2 ⎥ ⎢ 1 qmax ⎛ h ⎞ 2 ⎥ ⎛ Γ I ⎜ Re, ⎟ = + E F ∑ p⎜ ⎟ ∑ q ⎜ ⎟ (10b) b ⎠ ⎢ Re p = 0 ⎝ b ⎠ ⎥ ⎢ Re q = 0 ⎝ b ⎠ ⎥ ⎝ ⎣ ⎦ ⎣ ⎦ Using the surface fitting tool in Matlab, each coefficient and power index in equation (10a-b) is determined simultaneously to obtain the optimum fitting results. The coefficients (Cm, Dn, Ep, Fq) and power indices (mmax, nmax, pmax, qmax) are determined to minimize the differences between the numerical data and the proposed analytical expression. The power indices and coefficients are determined as mmax=2, nmax=4, pmax=1, qmax=4, C0=0.9003, C1=0.6105, C2=2.1722, D0=0, D1= -0.0021, D2= -0.1459, D3=0.8255, D4= -1.3388, E0=2.5758, E1= -1.3388, F0=0.9003, F1= -0.7121, F2=1.6845, F3=0.8236 and F4=0.4178.
vstem ( 0, t ) = 0 , (3) The exciting force is assumed and modeled by an equivalent, harmonic support rotation [5]. The bending slope at the support end of the hammerhead microcantilever is given by ∂vstem ( x, t )
jω t , (4) x=0 = θ0e ∂x In equation (4), θ0 and ω are the amplitude and angular frequency of the effective support rotation. The remaining two boundary conditions represent the moment and force balances at the junction between the stem and head (x=L1). The width of the head is assumed to be much larger than that of the stem, i.e., b2>>b1. Thus, the head is modeled as a rigid body, and translational motion and rotational motion of the head are taken into account. In viscous liquids, the inertial and damping forces contribute in the moment and force balance equations, which are given by
∑M ∑S
moment
x = L1
force x = L1
=0,
= 0.
D. Frequency Response Based on the boundary conditions, the equation of motion can be solved and the normalized frequency response at the tip of the stem is obtained as
X ( L1 ) L1θ 0
(5) (6)
L1θ 0
θ0 K
sin KL1
.
(11) In equation (11), K, A1 and A2 are functions of the properties of the hammerhead microcantilever and surrounding liquid medium. A1 and A2 also depend on θ0. Using equation (11), the resonance frequency and quality factor can be extracted from the frequency spectrum.
π
g 2 ⎡⎣ Re ( x ) , h b ( x ) ⎤⎦ =
=
A1 ( cosh KL1 − cos KL1 ) + A2 ( sinh KL1 − sin KL1 ) +
C. Hydrodynamic Function The hydrodynamic function (Γ= ΓR+jΓI) is defined as a normalized hydrodynamic force per unit length [1]. It consists of two components: the viscous damping force, which is associated with ΓI, and the inertial force, which is associated with ΓR:
g1 ⎣⎡ Re ( x ) , h b ( x ) ⎦⎤ =
m n ⎤ ⎡n ⎤ ⎛ h ⎞ 2 ⎥ ⎢ max ⎛ h ⎞ 2 ⎥ + C D (10a) ∑ m ⎜⎝ b ⎟⎠ ⎥ ⎢ ∑ n ⎜ ⎟ ⎝b⎠ ⎥ m =0 n=0 ⎦ ⎣ ⎦
mmax
ρ f b 2 ( x ) Γ I ⎣⎡ Re ( x ) , h b ( x ) ⎦⎤ ω , (7) 4
III.
π
ρ f b 2 ( x ) Γ R ⎡⎣ Re ( x ) , h b ( x ) ⎤⎦ , (8) 4 Modified analytical expressions for ΓR and ΓI are developed to minimize the discrepancy between numerical and analytical results. The mathematical forms of ΓR and ΓI are proposed as shown in equation (9a-b).
RESULTS AND DISCUSSIONS
A. Results of the Semi-analytical Expression Real and imaginary parts of the hydrodynamic function obtained analytically and numerically are compared. The ranges of the percent differences between the analytical and numerical results of the real and imaginary parts are [-3.8%,
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microcantilevers are narrower than that of the prismatic beam.
Normalized Amplitude Deflection [dB]
100
C. Resonance Frequency The resonance frequency is investigated as a function of liquid properties and dimensions of the rectangular hammerhead microcantilevers. Fig. 3 (top) shows the resonance frequency as a function of glycerol concentration in water. As viscosity increases, the resonance frequency will decrease. Fig. 3 (bottom) shows the resonance frequency with respect to length (L2) and width (b2) of the head for three different stems at a thickness of 12μm. As the length or width of the head increases, the resonance frequency decreases for hammerhead microcantilevers, and it is due to the increase in the mass of the head. For hammerhead microcantilevers with stiffer stems (smaller L1/b1), the resonance frequency is higher.
50
0 HHMC-1 (air) HHMC-1 (water) HHMC-2 (air) HHMC-2 (water) HHMC-3 (air) HHMC-3 (water) PB (air) PB (water)
-50
-100 0
500
1000
1500
2000
Excitation Frequency [kHz]
Figure 2: Calculated frequency spectra of three hammerhead microcantilevers (HHMC) of dimensions [(200×45×12) + (50×200×12)] µm3 (HHMC-1), [(150×45×12) + (50×200×12)] µm3 (HHMC-2), [(200×90×12) + (50×200×12)] µm3 (HHMC-3) and a rectangular prismatic beam (PB) of dimensions (200×45×12) µm3 laterally vibrating in air and water.
D. Quality Factor The quality factor is investigated as a function of liquid properties and dimensions of the rectangular hammerhead microcantilever. Fig. 4 (top) shows the quality factor as a function of glycerol concentration in water. As viscosity increases, the quality factor will decrease. Fig. 4 (bottom) shows the quality factor with respect to the length (L2) and width (b2) of the head for three different stems at a thickness of 12μm. As the length of the head increases, the quality factor decreases (because the increase in damping exceeds the increase in mechanical energy), and the mass center of the head moves away from the tip of the stem. As the width of the head increases, for shorter heads, the quality factor increases because the increase in mechanical energy exceeds the increase in damping. Note that the distance between the mass center of the head and the tip of the stem remains fixed at L1+L2/2.
6.1%] and [-2.0%, 2.8%], respectively, for Re ~ [10, 10000] and h/b ~ [1/56, 1]. B. Frequency Spectrum Using equation (11), the simulated frequency spectra of four microcantilevers of different dimensions vibrating laterally are shown in Fig. 2. For the investigated geometries, the resonance frequency is highest for the prismatic rectangular beam, because no additional mass is attached at the end of the beam. Percent change of the resonance frequency from air to water for the prismatic beam, 8.8%, is larger than those of rectangular hammerhead microcantilevers, 1.8%~4.9%. It is also found that the 3-dB bandwidths for the hammerhead
Viscosity [cP]
Viscosity [cP] 1.01 1.6
1.31
1.76
2.5
3.72
1.01 50
6
1.31
1.76
2.5
Quality Factor
Resonance Frequency [MHz]
40 Prismatic Beam HHMC-1 HHMC-2 HHMC-3
1
6
Prismatic Beam HHMC-1 HHMC-2 HHMC-3
45 1.4
1.2
3.72
35 30 25
0.8
20 0.6
15 10
0.4 0
10
20
30
40
0
50
10
20
30
40
50
Percent Glycerol [%]
Percent Glycerol [%]
Figure 4: Calculated quality factor as a functions of glycerol concentration in water (top: microcantilever dimensions as in Fig. 2) and of the dimensions of the head (bottom).
Figure 3: Calculated resonance frequency as a functions of glycerol concentration in water (top: microcantilever dimensions as in Fig. 2) and of the dimensions of the head (bottom).
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TABLE I. SENSOR CHARACTERISTICS PREDICTED FOR TWO PRISMATIC BEAMS AND THREE HAMMERHEAD MICROCANTILEVERS [μm3]
fair [MHz]
fwater [MHz]
Percent Change
Qwater
Sn,water [μm2/ng]
200×45×12
1.548
1.412
8.8%
36
14.9
200×90×12
3.095
2.944
4.9%
59
15.8
1.428
1.386
2.9%
49
15.5
0.761
0.738
3.0%
40
16.8
0.564
0.554
1.8%
46
17.2
Geometry
200×90×12+ 50×200×12 150×45×12+ 50×200×12 150×45×12+ 50×300×12
Figure 5: Calculated normalized surface mass sensitivity as a function of the dimensions of the head.
properties of the liquid and compared to those of laterally vibrating prismatic beams. A modified analytical expression for the hydrodynamic function was presented for the purpose of rapid evaluation of the sensing characteristics.
E. Normalized Surface Mass Sensitivity Normalized surface mass sensitivity is defined and used for comparison between the microcantilevers with different resonance frequency and surface sensing area, and is given by
For the range of dimensions investigated, if only the length of the head increases, the mass center of the head will move away from the tip of the stem and the resonance frequency and quality factor will decrease due to the increase in the mass attached at the end of the stem. In contrast, if only the width of the head increases, the mass center of the head will not move and the resonance frequency will decrease, but the normalized surface mass sensitivity will increase; the quality factor will increase rapidly for shorter heads. This is because the increase in mechanical energy exceeds the increase in damping. Such trends can be used to optimize device geometry and maximize the frequency stability in sensing applications. In general, to obtain an efficient hammerhead microcantilever-based sensor platform, the stem of the microcantilever can be designed stiffer and the head can be made wider and shorter. For appropriately designed hammerhead microcantilevers, the improvement in the sensing area and quality factor are expected to yield much lower limits of detection in (bio) chemical sensing applications.
Δf f Sn = res res . (12) Δm A In equation (12), ∆fres is the shift of the resonance frequency due to added mass, ∆m (sorption of target molecules), on the microcantilever surface; A is the surface sensing area of the microcantilever. It is assumed that ∆m is dominated by the change in the effective density. The normalized surface mass sensitivity, Sn, is investigated as a function of the dimensions of the rectangular hammerhead microcantilever. Fig. 5 shows Sn as a function of the length (L2) and width (b2) of the head for three different stems at a thickness of 12μm. As the width of the head increases, Sn increases, because the decrease in ∆fres is smaller than that of the resonance frequency. F. Comparison between the Prismatic Beam and Hammerhead Microcantilever The sensing characteristics of rectangular and prismatic microcantilevers are analyzed and compared in Table I. It is found that the decrease in the resonance frequency from air to water is smaller for the hammerhead microcantilevers. It indicates that a laterally vibrating hammerhead microcantilever will have high chemical sensitivity in both air and viscous liquids. The resonance frequency and quality factor of prismatic beams are larger than those of the hammerhead microcantilevers with the dimensions of the stem being identical to those of the prismatic beams. Although the resonance frequency for a hammerhead microcantilever of dimensions L1×b1×h+L2×b2×h with a wider and shorter head is lower compared to a prismatic beam of dimensions (L1+L2)×b1×h, the quality factor for the hammerhead microcantilever is higher. It indicates that a microcantilever with a lower resonance frequency may yield higher quality factor due to geometrical effects. IV.
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[2]
[3]
[4]
[5]
CONCLUSIONS
[6]
The sensing characteristics of laterally vibrating rectangular hammerhead microcantilevers in a viscous liquid were analyzed in terms of the microcantilever geometry and the
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2013 Joint UFFC, EFTF and PFM Symposium
