JOURNAL OF APPLIED PHYSICS 107, 094905 共2010兲
Broadband optoacoustic measurements of ultrasound attenuation in severely plastically deformed nickel Victor V. Kozhushko,1,a兲 Günther Paltauf,1 Heinz Krenn,1 Stephan Scheriau,2 and Reinhard Pippan2 1
Institute of Physics, Karl-Franzens-University of Graz, Universitätsplatz 5, 8010 Graz, Austria Erich Schmid Institute of Materials Science, Austrian Academy of Science, Jahnstr. 12, 8700 Leoben, Austria
2
共Received 15 October 2009; accepted 28 February 2010; published online 7 May 2010兲 Laser optoacoustics and immersion techniques allowed a broadband ultrasound spectroscopy which was used for measuring the attenuation of severely plastically deformed nickel. A disk shaped specimen of nickel of about 33 mm diameter and 2.5 mm thickness was prepared by the high pressure torsion method. The produced equivalent shear strain linearly increased from a minimum at the center up to 1000% at the edge, gradually refining the grain size distribution down to 200 nm. The metal water interface was illuminated by 5 ns laser pulses, generating longitudinal ultrasound pulses with a pronounced compression phase and a smooth spectrum covering the range from 0.1 up to 150 MHz. The laser beam spot diameter was 6 mm, yielding a maximum power density below 15 MW/ cm2. The ultrasound passed through the sample thickness and a 2 mm layer of coupling water. The pulse was detected by a 25 m thick piezoelectric foil transducer with a diameter of the sensitive area of 2 mm. The transient signals were locally measured at different radii of the specimen. The attenuation almost linearly increases with frequency while its absolute value decreases from the center to the edge of the specimen. © 2010 American Institute of Physics. 关doi:10.1063/1.3371685兴 I. INTRODUCTION
The refinement of the microstructure of polycrystalline 共pc兲 materials leads to a new behavior of the properties of bulk specimens due to size and boundary effects.1 The nanocrystalline 共nc兲 microstructure is very promising for a variety of applications. Compared to other available techniques that produce nc thin films and foils, specimens of metals and composites of macroscopic size in the range of several centimeters, with grains below 1 m diameter can be obtained by means of severe plastic deformation. High pressure torsion 共HPT兲 is a recent technique used for the severe plastic deformation of disk shaped specimens which undergo compression and shear stress between anvils. The method produces huge equivalent shear strain v, which is proportional to the number of rotations n and can be calculated as v = 2nr / 共冑3d兲, where r is the distance from the center and d is the thickness of the sample.2 The increasing of the strain initially multiplies dislocations and then reduces the size of the grains down to about hundred nanometers, after which a saturation in the microstructural refinement is reached.3,4 The fine grain structure increases the hardness of the materials and significantly influences their mechanical properties, which makes them attractive for various applications.5 High chemical purity and dense packing of the specimen highlight severe plastic deformation from the other methods.6 The ultra fine microstructure is characterized by high angle grain boundaries containing a high density of dislocations. The nondestructive methods are challenging for the evaluation of these special materials in terms of their elastic properties. a兲
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Different ultrasonic methods are employed in material science for the correlation of the changes of the mechanical properties and the attenuation with the elastic constants and the microstructure. These are, for example, piezoelectric transducers,7–12 acoustical electromagnetic transducers,13,14 and noncontact laser methods both for the excitation and for the detection.15–19 This paper considers broadband laser optoacoustic spectroscopy of a HPT nickel sample immersed in water. The transformation of laser pulse radiation to heat provides an excitation of ultrasound at the interface between metal and water. The presence of water increases the efficiency of generation. Water also provides acoustical coupling between the specimen under investigation and a piezoelectric detector. This arrangement is used for the broadband measurements of the frequency dependence of the ultrasound attenuation in a HPT deformed nickel specimen. The paper is organized as follows: first the excitation of ultrasound is discussed, the second part considers the damping of ultrasound in pc media and the third part regards the detection of ultrasound; the fourth part is experimental; the general considerations of previous sections are applied to a particular case in final parts, which include the experimental results, a discussion and a conclusion. II. OPTOACOUSTIC PRESSURE SOURCE
Laser ultrasound or optoacoustics 共also known as photoacoustics兲 deals with the conversion of pulsed laser radiation to an elastic disturbance of a medium.20 The short transient form and the resulting smooth broadband spectrum, as well as the high intensity of the localized laser induced ultrasound source are some of the advantages of the technique
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20 mJ Laser 5 ns
Transducer
pulse
Aluminium 25 m Water tank
PVDF Backing PMMA
Thin absorbing layer Sample
Oscilloscope 8 bit, 1 GS/s
FIG. 1. 共Color online兲 Sketch of the experimental arrangement and layered structure of the transducer.
for the nondestructive testing and evaluation of materials. The estimation of ultrasound attenuation requires a spectral analysis of the detected transient signal by Fourier transformation. It is convenient to consider the ultrasound excitation in frequency domain. The transfer function method was introduced for the analysis of the efficiency of optoacoustic conversion.21 In general, the spectrum of the induced acoustic pulse depends on the optical, thermophysical, and mechanical properties of the absorbing and contacting media and on parameters of the laser radiation, such as pulse duration, intensity envelope, and power density. The transfer function method separates materials properties and laser pulse parameters. In a one-dimensional “ansatz” the spectrum of the induced pressure pulse can be written as P共f兲 = K共f兲I0L共f兲,
共1兲
where K共f兲 is a conversion function depending on the properties of the materials, I0 is the intensity of the laser pulse, and L共f兲 the spectrum of its intensity envelope.20,22 Let us consider the excitation of ultrasound at a metalwater interface assuming a semi-infinite medium. A laser pulse of 5 ns duration illuminates the surface of the metal according to the sketch of the experimental setup in Fig. 1. It is assumed that there is either no absorption of the optical radiation in water or it is negligible. The absorption of radiation occurs in metals in a depth range on the order of nanometers, producing a heat source at the interface. The nonstationary temperature changes can be described in terms of heat diffusion. The solution of this task can be obtained by applying the boundary conditions for the temperature and the heat flow. The detailed expressions of the transfer function for the case under consideration are obtained in Refs. 20, 22, and 23. The heat diffusivity of nickel is 2.3⫻ 10−5 m2 / s. The laser pulse duration of 5 ns defines the heat diffusion length as 冑, which is ⬃0.3 m into nickel but only ⬃0.02 m into water. The heat diffusion length is significantly shorter than the traveling distance of acoustical waves in metal during the same time, amounting to ⬃25 m. Therefore, the transient changes of the temperature and the following thermal expansion of the heated layers of materials define the boundary and starting conditions for excitation of a pressure pulse. The ratio between thermal expansion and product of the density and heat capacity was introduced for characterization of the efficiency of excitation.22 As the ther-
mal expansion coefficient of liquids is at least one order of magnitude larger than that of metals, liquid has the main contribution in the excitation of ultrasound. The heat diffusion length is much smaller than the acoustic wavelength in the frequency range up to 1 GHz where the efficiency of optoacoustic conversion is ⬃1 Pa/ 共W / cm2兲 for a metalwater interface.20,22 Therefore, the spectrum of the pressure is proportional to the spectrum of the laser pulse envelope, giving a broadband longitudinal pulse with a pronounced compression phase that makes it very useful in nondestructive testing.24 The probe pulse is launched into the investigated medium where it is attenuated during its propagation through the thickness of the specimen. Commonly, the ratio of the probe pulse spectrum and the spectrum of its reverberations is used for the determination of the frequency dependent attenuation. III. ATTENUATION OF THE PROBE PULSE
Material science distinguishes nc materials with a grain size ⬍100 nm and ultrafine 共uf兲 materials with grains in the range from 100 nm up to 1 m, whereas pc materials have a microstructure coarser than 10 m. The discussions of observations provided in this study rely more on the known theoretical models and experimental results obtained for the pc metals. However, the applicability of these models to uf materials has certainly limitations. Attenuation of an ultrasound pulse is partly due to damping, which relates to the irreversible conversion of the elastic wave energy to heat, and to scattering on the borders between crystallites. As the pulse echo technique measures the total attenuation through a certain distance it is not simple to separate these components in the experiment. For pc materials damping is attributed to several phenomena such as elastic hysteresis and thermoelastic damping, and internal friction. Generally, the contribution of these components increases with frequency but the individual terms demonstrate different frequency dependences and dominate in different frequency ranges. The elastic hysteresis arises from the inelastic deformation of the polycrystals for loading and unloading stress. The contribution due to hysteresis linearly increases with the frequency and it is relatively low as ⬃10−2 dB/ 共cm MHz兲 as it is reported for magnesium.8 For the pc metals the thermoelastic damping is due to temperature fluctuation between randomly orientated anisotropic grains. The maximum contribution to the attenuation is observed in the frequency range ⬍1 MHz while for the higher frequency range the elastic scattering is more important.25 The ultrasound pulse is scattered while propagating through the borders of the randomly oriented crystallites due to the elastic anisotropy of the crystalline cell. The decay of the high frequency part of the ultrasound spectrum strongly depends on the local microstructure of the specimen since the scattering is a function of the mean value of the grain size and the wavelength. The development of the theory of ultrasound scattering in pc materials was initiated more than fifty years ago by several scientific groups, see Refs. 7, 8, and 26. Recent advances in the theory have led to the consideration
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of multiple scattering; also composite materials and alloys as well as textured structures with oriented and deformed grains are included.11 It is noteworthy that in our experimental configuration the direction of the propagation of the longitudinal waves is normal to the direction of the grain orientation in the HPT specimen. Due to the large angle grain boundary the influence of the texture is negligible.27 The reported results agreed with the theory and proved the strong influence of the grain size distribution on ultrasound scattering. There are three types of scattering: Rayleigh, stochastic and diffusion. These types are determined by the ratio of the wavelength of the ultrasound to the mean value of the grain size distribution. Rayleigh scattering takes place in the low frequency range if 2D ⬍ , where D is a mean value of the grain size distribution and is the wavelength. The attenuation is given as ␣R = SRD3 f 4, where SR is a scattering constant and f is the ultrasound frequency. If the wavelength is ⬇2D, the stochastic scattering is described via the expression ␣s ⬀ Df 2. Diffusive scattering is assigned to the range Ⰶ D and does not depend on the frequency, ␣s ⬀ 1 / D. The absolute value of the attenuation increases from the Rayleigh to the stochastic regime showing deviations of the power law dependences in the transition zones. The elastic anisotropy of a material strongly influences the mode conversion of the longitudinal probe pulse to shear waves. The relative inhomogeneity of pc material is defined 0 2 0 兲 where = C11 − C12 − 2C44, C11 = C11 as 2l = 共4 / 525兲2 / 共C11 − 2 / 5, and C11, C12, C44 are elastic constants of the single crystal. The materials with a stronger anisotropy of crystalline cells show stronger inhomogeneity. Literature references give different elastic constants of nickel and vary the inhomogeneity factor in the range l = 0.0317– 0.0416.28,29 The elastic constants C11 = 248.1 GPa, C12 = 154.9 GPa, and C44 = 124.2 GPa 共Ref. 29兲 yield l = 0.0416 which is close to the case of iron with l = 0.0434 that has been considered by Stanke and Kino as an example of a strongly anisotropic medium.26 The stochastic region is defined as 1 ⬍ 2D / ⬍ 1 / l. The asymptotic expressions of the attenuation in the Rayleigh and stochastic modes are defined as
冋 冉 冊册
5 VL ␣R = 共l兲2 2 + 3 12 VS
␣S = 共l兲2共2/兲2D,
quency dependence it should be mentioned that the original work considered the decrement ⌬ which is the ratio of the energy lost per cycle to the total amount of the elastic energy. It is connected with the attenuation as ⌬共兲 = 2a共兲V / where V is the phase velocity and is the cyclic frequency. The second mechanism is frequency dependent due to dynamic interaction. The applied stress induces motion of the dislocations and has a resonance nature of the frequency dependence. The effective loop length Le for the distribution of dislocation loops within the crystal defines the resonance frequency as30
20 = 2G/共1 − 兲L2e ,
共4兲
where G is the shear modulus, is the density, the Poisson ratio, and L is a dislocation loop length or segment of dislocation between atoms of impurities. Due to the damping mechanism there is a phase lag for an oscillating stress and hence a decay of the wave amplitude. The interaction of ultrasound with dislocations changes the elastic modules, decreasing the phase velocity. For a strain amplitude low enough so that there is no breakaway of dislocations, and for all frequencies the attenuation of the longitudinal waves and the variation in the Young’s modulus can be presented as31
2d ⍀⌬0⌳2 , 2V 共20 − 2兲2 + 共d兲2
共5兲
20 − 2 ⌬E ⍀⌬0⌳2 = , E 共20 − 2兲2 + 共d兲2
共6兲
a=
where ⍀ is an orientation factor averaging the applied stress for the possible dislocation slip planes, ⌬0 = 4共1 − 兲 / 2, ⌳ is the dislocation density, d = B / 共b2兲, B defines frictional damping, b is a Burgers vector, and 2 = 2G / 共1 − 兲. An analysis of the contributions of the separate components to attenuation is considered later in connection within the obtained experimental results. IV. DETECTION OF THE PROBE PRESSURE PULSE
5
共2/兲 D , 4
3
共2兲 共3兲
where 共VL / VS兲 is the ratio of the longitudinal and shear phase velocities. The imperfections of the crystal structure such as vacancies, dislocations and interstitial planes are sources of damping in single crystals or separate crystallites. HPT is a coldworking process that increases the number of dislocations in metals by a Frank–Read mechanism. The detailed analysis of the ultrasound interaction with dislocations was considered by Granato and Lücke.30 Considering a Corttell pinning mechanism, two types of losses were shown: the first one is the hysteresis like attenuation which is due to the fact that during the unloading part of the stress the long loops collapse in another way they were formed. It was shown that this mechanism is strain-dependent and frequency independent within the kilocycle frequency range. Regarding the fre-
The advantages of noncontact optical detection for the estimation of the grain size distribution in pc materials are emphasized in several papers.15–19 The first experiments with HPT nickel were carried out with an actively stabilized Michelson interferometer setup which provides a bandwidth up to 125 MHz and about 20 m of lateral spatial resolution for scanning along the radius of the disk shaped sample. The results show a maximum detected frequency up to 100 MHz within the ⬃40 dB dynamic range of the measurements for the region of the nickel specimen where a saturated distribution with about 0.2 m mean grain size is achieved. Electromagnetic acoustical transducers 共EMATs兲 are an example of another non contact method employed both for research and for industrial applications. The bandwidth of the EMATs is usually below 50 MHz, which is not enough for this case.13,14 Piezoelectric detectors, finally, have higher sensitivity and signal to noise ratio in comparison with optical methods. For instance, the estimated Nyquist noise of a typical polyvinylidenefluoride 共PVDF兲 detector represented by a
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capacitor of 12 pF is about 20 V. Therefore, such detectors require a smaller number of signal averages to achieve the potential dynamic range of the signal acquisition device. However, the coupling layer or acoustical contact is a main problem of conventional, piezoelectric ultrasound transducers for quantitative measurements. The immersion technique is an alternative method and is used in this work. Water has well defined acoustical properties and lower ultrasound attenuation in comparison with the majority of liquids.32 A polarized foil of PVDF possesses piezoelectric properties and is used for construction of broadband hydrophones. The acoustical impedance of PVDF is closer to that of water in comparison with other piezoelectric materials. The simplicity of the construction of foil detectors allows production of wideband focused detectors for the solution of different tasks.33,34 The operating frequency range of the detector is usually limited by the first thickness resonance where the wavelength is equal to the thickness of the piezoelectric foil. The goal of our work is the detection of frequencies up to 100 MHz, requiring the thickness of the foil being less than 24 m since the velocity of the longitudinal wave is about 2.4 km/s.35 The value of the longitudinal velocity in PVDF is used only for the estimation and it may differ between different manufactures. A commercially available foil of 25 m thickness was used in the homemade detector construction that gives the highest sensitivity at a frequency of about 48 MHz and the first thickness resonance at about 96 MHz. A sketch of the construction is presented in Fig. 1. A metalized PVDF foil is glued on a backing of polymethylmethacrylate 共PMMA兲 and covered by ⬃25 m protecting aluminum foil. A small amount of conductive glue provided electrical contact between the top electrode and the aluminum foil, which was the electrical ground. The wire carries the electrical signal from the bottom electrode to the amplifier. The displaced charge is proportional to the instantaneous mean stress inside the foil36 Q共t兲 =
e33A h
冕
h
共7兲
P共x,t兲dx,
0
where e33 is the piezoelectric stress constant, A is the area of the transducer, h is the thickness of the foil, P共x , t兲 is the pressure field in the general case including counter propagating waves. The transducer operates in the current detection mode where the induced charge flows through the low impedance resistor R, defining a time response constant as T = 2RC, where C is the capacity of the transducer. The high frequency border of the operating bandwidth should satisfy the condition f ⬍ 1 / T. The measured signal is proportional to the voltage drop and, therefore, to the derivative of the pressure pulse with respect to time as36 U共t兲 = R
e33A dQ =R dt h
冕
h
0
d P共x,t兲dx. dt
共8兲
As the foil has defined thickness the sensitivity depends on frequency. A relative sensitivity is introduced to investigate the dependence of the signal magnitude induced by a pressure wave of unit amplitude on the frequency. The analysis
TABLE I. Acoustical properties of materials.
Water Aluminum PVDF PMMA Nickel
Density 共kg/ m3兲
Velocity 共m/s兲
Impedance 共Mrayl兲
1000 2700 1800 1200 8900
1500 6200 2400 2500 5600
1.5 16.72 4.32 3 49.84
of the spectral sensitivity was carried out numerically. Appling the Fourier transformation, Eq. 共8兲 is converted into frequency domain: U共f兲 = R
− i2 fe33A h
冕
h
P共x, f兲dx.
共9兲
0
It is assumed that the plane wave front is normally incident from water onto the layered system: aluminum—PVDF foil—backing material 共in our case PMMA兲, as it is presented in Fig. 1. There are also thin layers of glue in the real construction. These glue layers are acoustically thin in the considered frequency range and are not included in the model. The relative spectral sensitivity is obtained by linear solution of the problem of propagation of the plane, longitudinal wave front through the multilayer structure, where the boundary conditions for pressure and particle velocity are applied at each interface. This model was employed in previous research for calculation of the transition spectra of periodical structures37 and it is also valid for the calculation of the spectral sensitivity.38 From the solution follows that there are two counter propagating waves in the aluminum layer and the PVDF foil. Reflected waves in the backing material can be neglected due to its thickness. Ultrasound attenuation was not included in the calculation. The integral of the pressure field over the thickness of the PVDF foil was used for the calculation of the relative spectral sensitivity. The properties of the materials used for the calculation are presented in Table I. Velocities and exact thicknesses of the material layers used in our calculations may slightly differ from the value of the real device. The results of the calculation of the sensitivity are presented in Fig. 2共a兲. Three particular cases of the ideal transducer construction are considered. The case of the ‘ideally damped’ transducer implies the absence of the covering aluminum foil and the match of the acoustical impedance of the backing material to the impedance of the PVDF foil. The relative sensitivity shows local minima at zero frequency and at the thickness resonance frequencies according to the rule nc / h where n is an integer, c is the velocity of the longitudinal wave in PVDF and h = 25 m is thickness of the film. The first thickness resonance is at 96 MHz for the velocity of 2.4 km/s. The mismatch of the acoustical impedance of the backing leads to ⬍1 dB increase in the sensitivity at the central frequency corresponding to a wavelength of twice the thickness. The relative transient signals of the voltage drop were calculated under the assumption of a Gaussian time profile of the pressure pulse incident from water, P0共x , t兲 ⬀ exp共−2共t / 兲2兲 / 共冑兲, where is the pulse duration. The
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(a) Spectral sensitivity, a.u.
1
Spectrum of 6 ns Gaussian pulse 0.1
’Ideally damped’ PVDF without aluminium foil backign material is PMMA Aluminum foil of 25 µm and backing material of PMMA
0
20
40
60
Transient signal, a.u.
1.0
80 100 MHz
(b)
0.5
0.0
0.0
-0.5
-0.5 6 ns Gaussian pressure pulse 0
20
40
60
Time, ns
80
140
160
U共f兲 ⬀ Amp共f兲Ss共f兲Att共f兲K共f兲I0L共f兲,
1.0
’Ideally damped’ Influence of the 0.5 PMMA backing
-1.0
120
(c)
6 ns pulse 12 ns pulse
-1.0 0
foil functions as a resonator for the frequencies in the range of 130 MHz producing the series of peaks which appears on the tail of the transient signal for the case of a 6 ns pulse. The amplitudes of these peaks are negligible in the case of 12 ns pulses due to their narrower bandwidth. It is noteworthy that the presence of the peaks can be used as a marker of the frequencies in the range 100–130 MHz in spite of the presence of a local minimum at 100 MHz. The transient form of the detected signal is a convolution of the laser induced pressure pulse, the attenuation in the sample and the spectral sensitivity of the transducer. The spectral analysis allows an estimation of the spectrum of the detected signal and the local attenuation of the ultrasound according to the formula
20
40
60
80
Time, ns
FIG. 2. 共Color online兲 Calculated relative sensitivity of a 25 m PVDF foil transducer 共a兲 in particular cases described in the figure. Simulated transient signals induced by Gaussian pressure pulses propagating from water for 共b兲 the ideally damped and PMMA backing cases without aluminum foil and for 共c兲 layered structure: aluminum foil—PVDF—PMMA backing. The curve in 共c兲 is scaled by a factor of 2 in comparison with 共b兲.
simulation of transient signals of the voltage drop are obtained by means of the inverse Fourier transformation of the product of spectral sensitivity and pressure spectrum, see Figs. 2共b兲 and 2共c兲. The transient signal possesses a bipolar form and is similar to a derivative of the Gaussian pulse in the case of the “ideally damped” transducer. The mismatch of acoustical impedance of the PVDF foil and the backing leads to a partial reflection of the pulse and the transient signal gets an extra hump in time domain, see Fig. 2共b兲. As the impedance of the backing material is lower, reflection changes the phase to and the negative peak goes down in comparison with the ‘ideally damped’ construction. The following reverberations have smaller amplitude. The simulation results also show the influence of the aluminum foil thickness on the spectral sensitivity of a transducer that mimics the construction used in our experiments. Including of the protecting Al foil of 25 m thickness reduces the sensitivity by 10 dB in the range below the first thickness resonance but the sensitivity is nearly the same as for the ideally damped detector in the range 100–130 MHz. The aluminum foil of 25 m is acoustically thin for the frequencies below 100 MHz but the whole construction has a local maximum of the sensitivity at about 130 MHz due to the half wavelength resonance in aluminum. As the impedance of the aluminum is much higher than that of the PVDF and water the metal
共10兲
here Amp共f兲 is the spectral function of the amplifier, Ss共f兲 is the spectral sensitivity of the foil detector, Att共f兲 is the ultrasound attenuation that is proportional to exp共−a共f兲x兲, K共f兲 is the transfer function of the laser radiation to pressure conversion, I0 is the intensity of laser pulse radiation, and L共f兲 is the spectrum of the laser pulse envelope. The amplitude spectrum of a 5 ns laser pulse can be considered as constant in the frequency range up to 150 MHz. The transfer function K共f兲 is nearly constant in this range.20,22 A single noninverting operational amplifier with low input impedance is used for an amplification by 20 dB in the 3 dB bandwidth of 100 MHz. The root mean square of the voltage noise of the employed operational amplifier is ⬃2 mV. The induced ultrasound pulse passes through the specimen and 2 mm of the coupling water layer in our experiment. Note that the pure absorption of water increases according to a second power law of the frequency as ␣ = 2.17⫻ 10−3 dB/ MHz−2 / cm, giving an attenuation of about 4 dB at 100 MHz on a distance of 2 mm.32 Certainly, the real system possesses some losses which are not included in Eq. 共10兲 but it is assumed that they are independent of the frequency or constant and, obviously, they do not depend on the local changes of the sample elastic properties. The influence of diffraction is considered below. All these assumptions simplify the Eq. 共10兲 in the practical application to U共f兲 ⬀ Ss⬘共f兲Att共f兲.
共11兲
The spectrum of the detected transient signal is proportional to the spectral function of the setup and the attenuation of ultrasound. V. EXPERIMENTAL
The HPT nickel sample of 2.50⫾ 0.01 mm in thickness and ⬃33 mm in diameter was deformed to an equivalent strain of about 10 共1000%兲 at a radius of 14 mm. An example of the typical microstructural changes during HPT of nickel is presented in Fig. 3. The large strain value of 10 corresponds to the saturation region where no further grain refinement is observed. At the beginning of severe plastic deformation 共v = 1, about 1.5 mm from the center for our specimen兲 the microstructure consists of big grains with large disorientations in their interior. Due to the plastic de-
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FIG. 3. 共Color online兲 Electron back scattered diffraction micrographs of the axial surface of HPT nickel. The equivalent strains v are marked in the upper left corner of each micrograph.
formation the dislocation density rises immediately. The dislocations do not arrange randomly. By contrast, they form dislocations cells with very small disorientations. With preceding deformation the disorientations between neighboring cells increase leading to the occurrence of high angle grain boundaries. At this point new smaller grains appear in the initially coarse structure.3 The further increasing of the shear strain v ⬎ 2 induces the fragmentation of the small grains that were just developed before. This fragmentation process scales the grain size of HPT deformed nickel down to several hundred nanometers. The saturation of the grain fragmentation process starts at a strain v of about eight. The saturation is given by the balance of the restoration and grain fragmentation process. High chemical purity and dense packing of the HPT method reduce the presence of voids and impurities which can significantly influence the mechanical properties of the specimen. A pumping pulse of a Nd:yttrium aluminum garnet laser of 5 ns duration, 1064 nm wavelength and with a pulse energy of 20 mJ illuminated the water metal interface. A layer of less than 100 m of black acrylic paint was placed on the surface of the metal. The layer absorbed most of the laser radiation and passed it to water, which increased the amplitude of the detected signal by about one order of magnitude compared to a dry interface. The diameter of the pumping laser spot was about 6 mm and the maximum power density on the sample surface was below 15 MW/ cm2. A pressure pulse magnitude of 15 MPa can be estimated from the linear model, which does not take into account any nonlinear effects as, for instance, the dependence of the thermal expansion coefficient of water on the temperature. This effect would tend to increase the efficiency of ultrasound excitation.20 As the transient form of the induced pressure pulse approximates the envelope of the laser pulse intensity, which is very short with respect to the transmitted ultrasound pulse, it can be considered as a ␦-function with a wide bandwidth.
The single scattering model considers only the losses of the coherent part of the pulse. The theory points out that the scattering for shear waves is stronger than the scattering of the longitudinal waves and the absolute attenuation grows from the Rayleigh to the stochastic regime.26 The experimental arrangement with the relatively large excitation spot size of 6 mm and the smaller detection spot size of 2 mm diameter favors the generation and detection of a plane, longitudinal wave. Shear waves are not excited but they appear due to mode conversion on the borders of the grains. They form a non coherent, scattered part of the elastic disturbance, together with scattered longitudinal waves. After mode conversion of the shear waves at the metal water interface, the scattered field arrives generally later at the detector than the primary longitudinal wave. Furthermore, the primary wave is registered with much higher amplitude by the PVDF detector compared to the scattered field because it is coherently summed over the large detector area. The detected wave forms, including the reverberations of the primary pulse, can, therefore, be expected to contain negligible contribution of scattered waves. The surfaces of HPT nickel are mirror like polished and the planes are parallel within 0.01 mm/cm over the entire specimen. The spot size of the laser beam of 6 mm provides an initially plane wave front of the pressure pulse. The influence of diffraction can be estimated from the expression Dd = x / r2, where Dd is a dimensionless factor, x is the traveling distance and r is the radius of the pumping laser spot. The influence of diffraction is negligible if the condition Dd ⬍ 1 is fulfilled. The path of the primary ultrasound pulse in the specimens and the diameter of the laser spot define a lower measured frequency of ⬃5 MHz due to diffraction losses. For higher frequencies the amplitude loss is less than 2 dB. The next reverberations travel a distance equal to 2n + 1 times the specimen thickness and, therefore, they undergo stronger diffraction losses leading to 2n + 1 times shift in the lower frequency limit. The estimation of the attenuation in the lower range requires a diffraction correction procedure that can be done according to Ref. 39. The scanning experiment was carried out by means of translation of the samples from the edge to the center with a step of 4 mm. The sampling rate of the oscilloscope was 1 GS/s and the transient signals were averaged 255 times at each detection spot to reduce electronic noise. VI. RESULTS
Examples of the signals measured at different points of the HPT nickel sample are presented in Fig. 4. The primary pulse arrives at about 1.8 s, which includes the time to pass through 2 mm of water and the sample thickness of 2.5 mm. The major part of the primary pressure pulse is reflected back into the nickel and the first reverberation or echo arrives at about 2.7 s. The acoustical impedance of water is ⬃1.5 Mrayl and that of nickel is ⬃50 Mrayl, see Table I. As the impedances differ by more than 30 times the double reflection of the primary pulse reduces its amplitude by about 1 dB. The dispersion of phase velocity is neglected due to the random orientation of the crystallites, see Fig. 3. The phase
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100
Center 4 mm 8 mm 12 mm
U(t), mV
60 40 20 0
-40
Primary pulse 1.7
30
Attenuation, dB/cm
80
-20
Center 4 mm 8 mm 12 mm
2 thickness / velocity
10
3
1st echo
1 1.8
1.9
2.6
2.7
2.8
5
velocity of the longitudinal pulses can be obtained from the time interval between zero crossing points of the first pulse and that of the second pulse. These measurements gave a lower value of about 5.59⫾ 0.02 km/ s at the center increasing up to 5.62⫾ 0.02 km/ s at the edge. Increasing the distance from the center the amplitude of the signal becomes about 2 times higher and the falling slope is getting steeper. The transient signals measured at the edge of the sample possess extra peaks on the tail due to the discussed properties of the transducer. As verified in the simulations, this indicates frequency components in the signal exceeding the local minimum of spectral sensitivity at about 100 MHz. The spectra of the measured primary pulses are presented in Fig. 5. The transient signals measured at distance of 8 and 12 mm from the center have the predicted spectral features, showing the position of the local minimum of the sensitivity at about 100 MHz and the presence of frequencies above 100 MHz. The spectrum of the transient signal obtained at 12 mm has higher amplitude in the range above 100 MHz since the amplitude of peaks in the tail of the signal is
10-3
Centre 4 mm 8 mm 12 mm
Amplitude, a.u.
10-4
50
100
MHz
µs
FIG. 4. 共Color online兲 Transient signals measured at different points of the HPT nickel sample.
30
10
FIG. 6. 共Color online兲 Attenuation of the ultrasound obtained at different points of the HPT nickel specimen.
larger, see Fig. 4. The spectra of transient signals obtained closer to the center do not demonstrate this feature. It is noteworthy that a dynamic range of more than 60 dB was approached whereas the estimation gives 66 dB for an oscilloscope with eight bit vertical resolution according to the formula 20 log共2NR冑N兲 where NR is the number of effective bits that is equal to seven, N is the number of the averaged signals. The absolute attenuation, which can be obtained from the ratio of the spectral amplitude of the pulses, is limited by the digital noise of the second pulse with the narrower band. The calculation was done by means of the formula
␣共f兲 = 20 log关U p共f兲/U1共f兲兴/2d,
共12兲
where U p共f兲, U1共f兲 are the spectral amplitudes of the primary pulse and its first reverberation, respectively, 2d is twice the thickness. The results of absolute attenuation at different points of the HPT specimen as well as the attenuation of the pc nickel specimen are presented in Fig. 6. The diffraction correction procedure was applied, which in fact was only important for the frequency range below 10 MHz. The absolute value of the attenuation is more than one of magnitude higher at the center in comparison with the edge of the HPT specimen. Note that the attenuation increases almost linearly with frequency. As the dynamic range of the measurements is limited, the reasonable range of the attenuation is ⬍60 dB/ cm. VII. DISCUSSION
10
-5
10-6
10-7 0
20
40
60
80
100
120
140
160
MHz
FIG. 5. 共Color online兲 Spectra of the primary pressure pulses detected at different points of the HPT nickel sample.
Due to the changes of microstructure, scattering could be an explanation of the decreasing attenuation from the center to the edge of the HPT nickel specimen. According to the average grain sizes seen in the micrographs, scattering in the coarse grain structure at the center is within the stochastic regime and Rayleigh scattering dominates for other detection points. The calculated stochastic scattering constant is of the order of 0.1 dB/ cm/ 共MHz2 mm兲, at least one order of magnitude lower in comparison with the obtained experimental attenuation results. The calculated value of the scattering constants for the Rayleigh regime are ⬍10 dB/ cm/
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共MHz4 mm3兲, yielding negligibly small attenuation for uf microstructure with a grain size below 1 m due to the third power law of the grain size dependence. Therefore, scattering does not explain the high value of the attenuation in our specimen. For polycrystals the contribution of the thermoelastic damping is even lower. Thus the main contribution is due to the increased number of dislocations. For well annealed polycrystals the resonance frequency is in the range of about 100 MHz.25 An estimation of the resonance frequency for pc nickel for the ¯ = 84 GPa and the averaged values of the shear modulus G Poisson ration ¯ = 0.3, density 8909 kg/ m3 and dislocation segment of 10−5 m yields a frequency of 80 MHz. For the frequency range ⬍ 0 expressions in Eqs. 共5兲 and 共6兲 are often used in a simplified form for calculating the attenuation and the variation in the phase velocity, see for instance Ref. 18 a ⬀ ⌳2L4e ,
共13兲
V − V0 ⬀ − ⌳L2e , V0
共14兲
V0 is the velocity without dislocation effect. Although a direct application of the Granato–Lücke model to our experimental results with HPT nickel has limitations, it will be discussed in the following. The attenuation is linearly proportional to the dislocation density and it has a fourth power law dependence on the dislocation loop length. The Frank–Reed mechanism is responsible for the dislocation multiplication as well as for the changes of the dislocation loop distribution. The distribution of dislocation loops is getting broader while the effective loop length increases, shifting the resonance to a lower frequency. The attenuation has a maximum near the resonance and it should quickly decrease with frequency above the resonance. The fragmentation of the grains starts from the center of the HPT nickel, reducing the number of dislocations as well as the length of dislocation loops. The dislocations are pinned by grain boundaries and the absolute attenuation becomes lower as it is seen from our experimental results. The refinement of the grain structure achieves saturation with a narrow grain size distribution at the edge of the specimen. It is assumed that the dislocation loop length is comparable with the grain size; in fact loop sizes lower than 10 nm are not observed. The value of the dislocation loop of about 10 nm shifts the resonance frequency to the range of 100 GHz and relative to this value our operational bandwidth is in the low frequency range. The increase in the phase velocity from the center to the edge is an additional argument supporting this assumption. According to Eq. 共14兲 a material with a lower dislocation loop length approaches the phase velocity of materials with low dislocation density. A contradiction between the experimental results and the Granato–Lücke theory arises from the frequency dependence of the ultrasound attenuation in the MHz range. Among the known experimental work done for nc materials the linear frequency dependence was reported for nc-Cu and nc-Pd obtained by pulsed electrodeposition and inert gas condensa-
tion, see Ref. 14. The results showed also more than 10% lower phase velocity in nc materials in comparison with pc materials, which was explained by the porosity of the samples. The analysis of known models does not give an explanation of the hysteresis like behavior of attenuation, which must be strain amplitude dependent. Recent experimental work done with an atomic force microscope shows the increase in contact damping at a specific applied static load for nc nickel.40 The inelastic dislocation motion and the energy lost in generation of dislocation loops was suggested for the explanation. Increasing the number of grain boundary atoms in nc structures influences the grain diffusion, migration and rotation. It is generally accepted that conventional dislocation sources such as the Frank–Read sources cannot operate in nc metals, and that the grain boundaries become potential sources and sinks of dislocations.6 Numerical simulations considering dynamics of nanostructures under hydrostatic pressure ⬎100 MPa and relatively high temperature above 1000 K predicted changes of the microstructure during a timescale of a few nanoseconds.6,41 In our experiment an ultrasound pulse with a pronounced compression phase is induced. The ultrasound pulse of about 10 ns duration with a rising front of a few nanoseconds occupies a region of about 50 m of metal, which undergoes a local strain of ⬃10−4 with a strain rate of ⬃106 s−1. This induces a strong gradient of the stress field within single grains near the center of the HPT specimen and may involve a strong nonlinear recombination of the distribution of dislocation loops. Thus, the broad network of dislocations may be responsible for the amplitude dependent nonlinear effects in pc at high frequencies that is shown by solution of the nonlinear equation of dislocation motion.42 On the other side the grains are nearly homogeneously stressed during the pressure pulse at the edge of the HPT specimen where the dislocations are nucleated and sink during the nanosecond time scale at the grain boundaries. In that case the results obtained for HPT specimen can be explained by the shift in the resonance frequency from the 100 MHz range to the 100 GHz range owing to decreasing of the effective dislocation loop. The assumption of a connection between uf microstructure and high dislocation density in metals requires measurements considering the dependence of the attenuation on the pressure pulse magnitude. Such experiments will be presented in the near future. A high strain magnitude is necessary for a highly dislocation dense metal but the range of the strain amplitude variation is limited since a decrease in laser pulse power reduces the dynamic range of the measurements, whereas, the damaging threshold of the absorbing layer limits a possible increase in the laser power.
VIII. CONCLUSION
An immersion technique using pulsed, laser induced ultrasound is proposed in our experimental arrangement for the evaluation of the attenuation and phase velocity of materials. The experimental setup with the piezoelectric transducer possesses an operational bandwidth from 5 up to 90 MHz
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that is comparable with non contact optical methods but it has higher signal to noise ratio and, therefore, a lower detectable pressure. The absolute attenuation and velocities at different points of the HPT nickel are defined by the dislocation arrangement. The experimental results reveal hysteresis like attenuation in pc nickel with high dislocation density as well as in uf nickel within the MHz frequency range, as it has been previously reported for nc metals. This indicates that the mechanism of dislocation nucleation and collapse of the broad dislocation network in pc metals under conditions of a short pressure pulse can be applied also to the case of increased numbers of grain boundary atoms in nc metals. ACKNOWLEDGMENTS
Financial support of this work by the Austrian Science Fund FWF 共Project Nos. S10402-16 and S10407-N16兲 is gratefully acknowledged. The authors are very much obliged to the referee for numerous corrections and suggestions. Y. T. Zhu and T. G. Langdon, Mater. Sci. Eng. A 409, 234 共2005兲. R. Z. Valiev, R. K. Islamgaliev, and I. V. Alexandrov, Prog. Mater. Sci. 45, 103 共2000兲. 3 R. Pippan, F. Wetscher, M. Hafok, A. Vorhauer, and I. Sabirov, Adv. Eng. Mater. 8, 1046 共2006兲. 4 A. Vorhauer and R. Pippan, Scr. Mater. 51, 921 共2004兲. 5 A. P. Zhilyaev and T. G. Langdon, Prog. Mater. Sci. 53, 893 共2008兲. 6 K. S. Kumar, H. Van Swygenhoven, and S. Suresh, Acta Mater. 51, 5743 共2003兲. 7 W. P. Mason and H. J. McSkimin, J. Acoust. Soc. Am. 19, 466 共1947兲. 8 L. G. Merkulov, Sov. Phys. Tech. Phys. 1, 59 共1956兲. 9 J. Szilard and G. Scruton, Ultrasonics 11, 114 共1973兲. 10 B. Kopec, Ultrasonics 13, 267 共1975兲. 11 R. B. Thompson, F. J. Margetan, P. Haldipur, L. Yu, A. Li, P. Panetta, and H. Wasan, Wave Motion 45, 655 共2008兲. 12 D. Nicoletti and A. Anderson, J. Acoust. Soc. Am. 101, 686 共1997兲. 13 H. Ogi, M. Hirao, and T. Honda, J. Acoust. Soc. Am. 98, 458 共1995兲. 14 M. J. Lang, M. Duarte-Dominguez, R. Birringer, R. Hempelmann, H. Natter, and W. Arnold, Nanostruct. Mater. 12, 5 共1999兲. 1 2
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