Determination of full set material constants of piezoceramics from phase velocities

JOURNAL OF APPLID PHYSICS VOLUM 92, NUMBR 8 15 OCTOBR 2002 Determination of full set material constants of piezoceramics from phase velocities Haifeng Wang and Wenwu Cao Materials Research Institute and National Resource Center for Medical Ultrasonic Transducer ngineering, The Pennsylvania State University, University Park, Pennsylvania 16802 Received 8 April 2002; accepted for publication 15 July 2002 The determination of elastic and piezoelectric constants by means of the resonance method is described in detail in the Institute of lectrical and lectronics ngineers Standards. However, in practice, it is not easy to obtain a self-consistent complete set of elastic and piezoelectric coefficients using the resonance method alone, since the geometric requirements are often too difficult to meet in some materials and property variation from sample to sample could be substantial. In this article, we report on a characterization scheme by regression of ultrasonic velocities in different directions. The method using the Levenberg-Marquardt algorithm is very stable. Only two samples are needed to determine the whole set of the material constants. The method has been applied to doped piezoceramic lead zirconate titanate and material constants obtained at the same frequency show excellent self-consistency. 2002 American Institute of Physics. DOI: 10.1063/1.1505998 I. INTRODUCTION High frequency ultrasonic array transducers have become more popular in medical ultrasonic imaging. 1 3 Traditionally, the design of ultrasonic transducers was often accomplished by using simple analytic one-dimensional models, such as the Mason model or Krimholtz, Leedom, and Matthaei model. However, this design philosophy is no longer adequate for array transducers because some two- and three-dimensional effects must be taken into account in array transducer modeling. These effects, if not controlled, can cause unwanted mode coupling and spurious responses that seriously degrade pulse length and frequency band shape. As a result, two- and three-dimensional finite element analyses are used more often nowadays, for which the complete set of elastic, piezoelectric and dielectric constants of piezoelectric ceramics are needed. 3,4 There is some literature on the measurement of high frequency behavior of piezoceramics. 5 7 The methods presented in the I standard for piezoelectric material characterization have been used in those works. However, all of these works are based on a one-dimensional thickness mode, no report has been given on the full set of material constants at high frequencies. The difficulty in the fabrication of all required geometry of resonance samples is the underlying reason, and it is not possible to take into account the frequency dependence since the resonance frequency of each sample is different. In addition, even with the hybrid of resonance and ultrasonic methods, we still need several samples in order to obtain all of the independent constants for anisotropic materials. Property fluctuations from sample to sample in some systems often make it impossible to get a self-consistent data set. 8 In this article we report an optimization routine that can be used to determine the full set of physical properties of piezoelectric materials by using ultrasonic spectroscopy. This procedure will be applied to study poled piezoceramic lead zirconate titanate PZT-5H. II. WAV PROPAGATION IN PIZOLCTRIC CRAMICS The propagation of a plane wave in piezoelectric ceramics is governed by the well-known Christoffel equation: 9 l ipc pq e pjl j l i e iq l i ij l l j qj u j 2 u i, i, j1,2,3,p,q1,2,3,4,5,6, where c pq, e iq, and S ij are the elastic stiffness constants with zero electric field, piezoelectric strain constants, and dielectric constants of constant strain, respectively; l i are the components of the unit vector l in the wave propagation direction, l iq are the elements of the 36 matrix l 1 0 0 0 l 3 l 2 0 l 2 0 l 3 0 l 1 2 and l pj are the transpose, u i are the particle velocities, is the density, and is the phase velocity in the medium. Three elastic waves, one longitudinal or quasilongitudinal, a fast and a slow shear or quasishear, can propagate along the direction l with three different velocities and their associated polarization vectors. First, we consider the wave propagation in the XZ plane of a poled piezoceramic. We assume lx 1 l 1 x 3 l 3 x 1 cos x 3 sin, then the Christoffel equation becomes where 0 0 l 3 l 2 l 1 0 a 11 0 a 13 0 a 22 0 u 2 a 13 0 a u 33u1 2u1 u 2 3 u 3, 1 3 0021-8979/2002/92(8)/4578/6/$19.00 4578 2002 American Institute of Physics

J. Appl. Phys., Vol. 92, No. 8, 15 October 2002 H. Wang and W. Cao 4579 a 11 c 11 cos 2 c 44 sin 2 e 31 e 15 2 S 11 cos 2 S 33 sin 2 sin2 cos 2, 4a a 22 c 44 sin 2 c 66 cos 2, 4b a 33 c 33 sin 2 c 44 cos 2 e 33 sin 2 e 15 cos 2 2 S 11 cos 2 S 33 sin 2, 4c a 13 c 13 c 44 sin cos e 31e 15 e 33 sin 2 e 15 cos 2 S 11 cos 2 S 33 sin 2 sin cos. 4d Solving for the eigenvalues of q. 3 leads to three solutions for the velocities: 1 a 22 /, 5 2 a 11 a 33 a 11 a 33 2 4a 2 13 /2 1/2, 6 3 a 11 a 33 a 11 a 33 2 4a 13 /2 1/2. 7 Substituting these three solutions back into q. 3, three eigenvectors can be obtained, which provide the particle displacement directions of the waves. We can easily find that solution 1 is a pure shear wave, whose particle displacement direction is along the Y axis. Solutions 2 and 3 are quasilongitudinal and quasishear waves, respectively, and their particle displacement directions are in the XZ plane. Next we consider the wave propagation in the XY plane of a poled piezoceramic. We assume lx 1 l 1 x 2 l 2 x 1 cos x 2 sin, then the Christoffel equation becomes b 11 b 12 0 b 12 b 22 0 u 2 0 0 b u 33u1 u 2 3 u 3, 8 where b 11 c 11 cos 2 c 66 sin 2, 9a b 22 c 11 sin 2 c 66 cos 2, 9b b 33 e 15 / 11, 9c b 12 c 12 c 66 sin cos. 9d Then we can obtain three solutions for the velocities: 1 b 33 /, 2 c 66 /, 10 11 3 c 11 /. 12 Here c 66 (c 11 c 12 )/2. Substituting these three solutions back into q. 8, three eigenvectors can be obtained, which provide the particle displacement directions of the waves. We found that solution 1 is a pure shear wave, whose particle displacement direction is along the Z axis. Solutions 2 and 3 are pure shear and pure longitudinal waves, respectively, and FIG. 1. xperiment setup. their particle displacement directions are in the XY plane. Since the phase velocities of these three waves are independent of propagation direction, the XY plane is an acoustically isotropic plane. III. XPRIMNTAL RSULTS The ultrasonic spectroscopy method is used to measure the phase velocities in the piezoelectric ceramics. The experimental setup used in our investigation is shown in Fig. 1. Two transducers Panametrics V358 with center frequency of 50 MHz, and bandwidth of 80% are aligned properly in a 332015 cm water tank. The distance between the two transducer is 3 cm. The transmitting transducer is driven by a 200 MHz computer-controlled pulser Panametrics 5052PR. The signal from the receiving transducer is sampled by a digital oscilloscope Tektronix TDS 460A. The sampling rate is set to 10 Gs/s. ach sampling window contains 2500 time-domain data points. To reduce the ubiquitous random errors, each measurement is averaged 64 times. The data are transferred to a computer via a general-purpose interface bus for offline storage and analysis. One rotation table Unislide Rotary Table is used to rotate the sample to change angles of incidence through a computer-controlled motor. The phase spectra w is obtained through a fast Fourier transform of the output signal when the sample is absent. With the sample in place, the trigger delay time is adjusted to compensate for the additional delay resulting from the sample path length. The phase of the output signal with the sample in place is obtained. When the wave is incident at an angle other than 0, a shear wave is generated by the mode conversion effect. The phase velocity p of the acoustic wave is calculated using

4580 J. Appl. Phys., Vol. 92, No. 8, 15 October 2002 H. Wang and W. Cao FIG. 3. The phase velocity as a function of propagation direction in the XZ plane of a PZT-5H sample. FIG. 2. Illustration of geometric relation between the transducer and sample. a PZT z and PZT x samples used for measuring phase velocity of the waves in the XZ plane; b PZT x sample used for measuring phase velocities of the waves in the XY plane. wave. The PZT z specimen was used to measure the phase velocities when is greater than 45, while the PZT x specimen was used for less than 45. The results at 30 MHz are shown in Fig. 3, where the measured data are p sin 2 i w w2 f w 2 fd cos i, 13 2 where w is the velocity of the water, d is the thickness of the sample, i is the incident angle, is the trigger delay time, and f is the frequency. The water temperature is maintained at 21.30.1 C. Two PZT-5H piezoelectric ceramic specimens Valpey Fisher with a surface of 2525 mm and thickness of 1 mm are tested. The plate sample with the poling direction perpendicular to its large surface is referred to as PZT z; the other is referred to as PZT x, whose poling direction is parallel to its larger surface. The phase velocity in the XZ plane of PZT-5H ceramics as a function of propagation direction angle was measured by using ultrasonic spectroscopy. Both PZT z and PZT x specimens as shown in Fig. 2a were used in order to increase the measurement accuracy, because the signals of the refractive quasilongitudinal waves are weak when the incident angle is near the critical angle of the longitudinal FIG. 4. The phase velocity as a function of propagation direction in the XY plane of a PZT-5H sample.

J. Appl. Phys., Vol. 92, No. 8, 15 October 2002 H. Wang and W. Cao 4581 TABL I. Full set of elastic, piezoelectric, and dielectric constants of a PZT-5H sample at 30 MHz obtained from our method and data set from resonance technique. c 11 lastic constants (*10 10 N/m 2 ) c 12 Our work 12.5* 6.76* 7.26 10.5 2.06 2.85* Resonance 11.5 6.85 6.89 10.15 2.03 2.33 Piezoelectric constants C/m 2 Our work 17.6 6.83 23.1 Resonance 14.7 5.01 24.0 Dielectric constants ( 0 ) S 11 S 33 1590* 1410* c 66 TABL II. ffect of initial guess on the stability of reconstruction algorithm. lastic constants (*10 10 N/m 2 ) Piezoelectric constants C/m 2 Original value 7.81 11.1 2.21 15.3 5.8 23.8 Initial guess #1 9.2166 3.821 7.157 5.398 4.416 8.569 Converged values 7.81 11.1 2.21 15.3 5.8 23.8 Initial guess #2 2.426 3.620 0.902 7.877 5.565 8.065 Converged values 7.81 11.1 2.21 15.3 5.8 23.8 Initial guess #3 5.1081 9.216 3.821 7.156 5.398 4.416 Converged values 7.81 11.1 2.21 15.3 5.8 23.8 shown as discrete points and the lines are theoretical values calculated from the reconstructed elastic, piezoelectric constants the reconstruction of constants will be discussed later. Note that there are no experimental data for the pure shear wave, whose particle displacement direction is along the Y axis, because it cannot be generated through the mode conversion effect in our experimental arrangement. The results of measurements and calculations for the phase velocity in the XY plane of PZT-5H ceramics at 30 MHz are shown in Fig. 4, where the experimental data are shown as discrete points and the calculated results are shown as lines. Only the PZT x specimen was used in the measurements as shown in Fig. 2b. Similarly, there are no experimental data for the pure shear wave that is polarized along the poling direction and normal to the plane of incidence. To estimate the error bounds of our results, the fractional error equation is derived from the theory of error propagation to first order in the measurement uncertainties 10 2 L 2 L 2 w 2 2 w w 2 w 2 i 2 2 i 1 w 2 L L 2 1 w 2 2 w w w 2 w w 2 w 2 i i 2, 14 where L/L, ( w )/( w ), w / w, and i / i are the relative measurement uncertainties of L, w, w, and i, respectively. In our experiments, they are approximately 110 3,110 4,410 4, and 110 3, respectively. Therefore, the relative measurement error is near 0.8%. Since a piezoelectric material behaves as a capacitor in a frequency range away from resonance, the magnitude of the dielectric constant S may be calculated as d S 2 fz 0 A, 15 where d is the sample thickness, Z is the measured impedance magnitude, A is the sample area, 0 is the permittivity of vacuum (8.8510 12 F/m), and f is the frequency that is away from resonance. 6 All impedance measurements were made with an HP4194A impedance/gain-phase analyzer using the HP16901A spring clip fixture. IV. LASTIC AND PIZOLCTRIC CONSTANTS RCONSTRUCTION From the discussion in Sec. II, we know that the phase velocities of bulk waves in the piezoceramic are related to the elastic, piezoelectric, and dielectric constants through the Christoffel equation. By inverting this equation, these constants can be determined from a suitable set of experimental velocities in various directions. In this situation, there are more data than independent material constants to be determined, which can be treated by a nonlinear least-squares optimization procedure. It has been shown that the phase velocities of the quasilongitudinal and quasishear waves in the XZ plane of piezoceramics are related to the elastic constants c 11, c 13, c 33, and c 44, piezoelectric constants e 15, e 31, and e 33, and dielectric constants 11 and 33 qs. 6 and 7, while the S S phase velocities of the pure longitudinal and shear waves in the XY plane are simply related to the elastic constants c 11 and c 66, respectively qs. 11 and 12. Therefore, c 11 and c 66 are first determined from the measurement in the XY plane. After c 11 and measured S 11 and S 33 are substituted into the Christoffel equation, a C-language program based on the Levenberg-Marquardt algorithm, 11,12 is used to determine c 13, c 33, c 44, e 15, e 31, and e 33. The program is run on a Pentium PC computer and the results can be obtained within a minute. The full set of elastic, piezoelectric, and dielectric constants at 30 MHz is given in Table I. The constants with a star * are the directly measured ones and the others are quantities reconstructed from the phase velocity data. We also list the experimental results in low frequencies by using resonance methods 13 for comparison. In order to check the stability of the reconstruction algorithm based on the Levenberg-Marquardt method, we performed computer simulations to investigate several critical

4582 J. Appl. Phys., Vol. 92, No. 8, 15 October 2002 H. Wang and W. Cao TABL III. ffect of scatter in dielectric constants on the stability of reconstruction algorithm. lastic constants (*10 10 N/m 2 ) Piezoelectric constants C/m 2 Origin value 7.81 11.1 2.21 15.3 5.8 23.8 Scatter in dielectric constants 1% 7.81 11.1 2.21 15.3 5.8 23.8 5% 7.83 11.3 2.2 15.7 6.1 23.1 10% 7.84 11.4 2.2 15.7 6.2 22.9 factors initial guess and scatter in experimental data, which can influence the accuracy of the property reconstruction algorithm. To investigate the effect of an initial guess, a set of elastic, piezoelectric, and dielectric constants was selected given in the first row from Table II IV. From these original data, a set of phase velocities was calculated for given propagation direction using qs. 6 and 7. A discrete set of refraction angles was selected to simulate a typical data set obtained by the ultrasonic spectroscopy. Then, the Levenberg-Marquardt inversion algorithm was applied to calculated phase velocities for different initial guesses the effective digits are random generated between 0 10. The reconstructed sets of elastic and piezoelectric constants are compared to the original set to see the accuracy and the results are given in Table II. The first row in the table lists the true constant values for the material. The other rows show the reconstructed constants. We concluded that the set of reconstructed constants is independent of the initial guess selection so long as the guess is reasonable convergence is guaranteed for initial input randomly generated between 0 50. In other words, the Levenberg-Marquardt method is globally convergent in this case. Next, we investigated the effect of random noise in the input dielectric constants. Similarly, for the selected set of elastic, piezoelectric, and dielectric constants, the phase velocities at different refraction angle were calculated first. Then, random noises were added to the dielectric constants. We used reconstructions with random noises at relative levels of 1%, 5%, and 10%, respectively. The results are summarized in Table III. The first row in the table still lists the true constant values for the material. The other rows show the reconstructed constants, where the first column in these TABL IV. ffect of scatter in phase velocities on the stability of reconstruction algorithm. lastic constants (*10 10 N/m 2 ) Piezoelectric constants N/m 2 Origin value 7.81 11.1 2.21 15.3 5.8 23.8 Scatter in phase velocities 0.1% 7.83 11.2 2.22 15.1 5.9 23.6 1% 7.93 10.7 2.17 14.8 5.5 24.1 5% 8.94 9.1 2.01 11.4 4.5 28.7 rows gives the values of the random noise starting with 1% noise, going up to 10% noise. One can see that all the constants can be reconstructed and the algorithm seems to be more stable for the elastic constants than for the piezoelectric constants. For example, the values of elastic constants change within 3% at 5% noise level while the values of piezoelectric constants change around 5%. Finally, we studied the effect of scatter in phase velocity data. After the phase velocities at a different refraction angle were calculated based on the selected set of elastic, piezoelectric, and dielectric constants, random noises were added to the phase velocities. Random noise relative levels at 0.1%, 1%, and 5% were used. The results are given in Table IV. The first row in the table still lists the true constant values for the material. The other rows show the reconstructed constants, where the first column in the table gives the levels of the random noise starting with 0.1% noise, going up to 5% noise. One can see that the algorithm is sensitive to the uncertainty of phase velocity. Generally speaking, the uncertainty of phase velocity should not exceed 1%. V. CONCLUSION Ultrasonic wave propagation in piezoceramics was studied and the phase velocities of waves in different propagation directions in the XZ and XY planes of piezoelectric ceramic PZT-5H were measured by using ultrasonic spectroscopy. Two directly related elastic constants c 11 and c 66 were determined from pure mode waves. Then quasilongitudinal and quasishear waves were used to reconstruct the full matrix elastic constants and piezoelectric constants by an optimization routine based on the Levenberg-Marquardt algorithm. The improved characterization scheme only needs two PZT-5H specimens. The stability of this method has been evaluated by computer simulation using a synthetic set of input data. The effects of initial guess and experimental data with random noise were studied. It is shown that the Levenberg-Marquardt method is globally convergent in this case and it is more stable to the scatter in dielectric constants than to the scatter in phase velocities. 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