This protocol describes the procedure of measuring the temperature dependence of the full set material constants of piezoelectric materials using resonant ultrasound spectroscopy (RUS).
During the operation of high power electromechanical devices, a temperature rise is unavoidable due to mechanical and electrical losses, causing the degradation of device performance. In order to evaluate such degradations using computer simulations, full matrix material properties at elevated temperatures are needed as inputs. It is extremely difficult to measure such data for ferroelectric materials due to their strong anisotropic nature and property variation among samples of different geometries. Because the degree of depolarization is boundary condition dependent, data obtained by the IEEE (Institute of Electrical and Electronics Engineers) impedance resonance technique, which requires several samples with drastically different geometries, usually lack self-consistency. The resonant ultrasound spectroscopy (RUS) technique allows the full set material constants to be measured using only one sample, which can eliminate errors caused by sample to sample variation. A detailed RUS procedure is demonstrated here using a lead zirconate titanate (PZT-4) piezoceramic sample. In the example, the complete set of material constants was measured from room temperature to 120 °C. Measured free dielectric constants and
were compared with calculated ones based on the measured full set data, and piezoelectric constants d15 and d33 were also calculated using different formulas. Excellent agreement was found in the entire range of temperatures, which confirmed the self-consistency of the data set obtained by the RUS.
Lead zirconate titanate (PZT) piezoelectric ceramics, (1-x)PbZrO3-xPbTiO3, and its derivatives have been widely used in ultrasonic transducers, sensors and actuators since the 1950s1. Many of these electromechanical devices are used at high temperature ranges, such as for space vehicles and underground well logging. Moreover, high power devices, such as therapeutic ultrasonic transducers, piezoelectric transformers and sonar projectors, often heat-up during operation. Such temperature rises will change the resonance frequencies and the focal point of transducers, causing severe performance degradation. High intensity focused ultrasound (HIFU) technology, already used in clinical practice for the treatment of tumors, uses ultrasonic transducers made of PZT ceramics. During operation, the temperature of these transducers will increase, causing a change of the material constants of the PZT resonator, which in turn will change the HIFU focal point as well as the output power2,3. The shift of focal point may lead to serious unwanted results, i.e., healthy tissues being destroyed instead of cancer tissues. On the other hand, if the focal point shift can be predicted, one could use electronic designs to correct such shift. Therefore, measuring the temperature dependence of the full set material properties of piezoelectric materials is very important for the design and evaluation of many electromechanical devices, particularly high power devices.
Poled ferroelectric materials are the best piezoelectric materials known today. In fact, nearly all piezoelectric materials currently in use are ferroelectric materials, including solid solution PZT ceramics and (1-x)Pb(Mg1/3Nb2/3)O3-xPbTiO3 (PMN-PT) single crystals. The IEEE (Institute of Electrical and Electronics Engineers) impedance resonance method requires 5-7 samples with drastically different geometries in order to characterize the full set material constants4. It is nearly impossible to obtain self-consistent full set matrix data using the IEEE impedance resonance method for ferroelectric materials because the degree of poling depends on the sample geometry (boundary conditions), while sample properties depend on the level of poling. To avoid problems caused by sample to sample variations, all constants should be measured from one sample. Li et al. reported the successful measurement of all constants from one sample at room temperature by using a combination of pulse-echo ultrasound and inverse impedance spectroscopy5. Unfortunately, this technique is hard to perform at elevated temperatures because it is not possible to perform ultrasonic measurements directly inside the furnace. There are also no commercially available shear transducers that can work at high temperatures. In addition, the coupling grease that bound the transducer and the sample cannot work at high temperatures.
In principle, the RUS technique has the capability to determine the full set material constants of piezoelectric materials and their temperature dependence using only one sample6,7. But there are several critical steps for proper implementation of the RUS technique. First, the full set of tensor properties at room temperature should be accurately determined using a combination of pulse-echo and RUS techniques. Second, this room temperature data set can be used to predict the resonance frequencies and to match the measured ones in order to identify the corresponding modes. Third, for each small increment of temperature from room temperature up, one needs to perform spectrum reconstruction against the measured resonance spectrum in order to retrieve the full set constants at this new temperature from the measured resonance spectrum. Then, using the new data set as the new starting point, we can increase the temperature by another small temperature step to get the full set constants at the next temperature. Continuing this process will allow us to obtain the temperature dependence of the full set material constants.
Here, a PZT-4 piezoceramic sample is used to illustrate the measurement procedure of the RUS technique. The poled PZT-4 ceramic has ∞m symmetry with 10 independent material constants: 5 elastic constants, 3 piezoelectric constants and 2 dielectric constants. Because the dielectric constants are insensitive to the change of resonance frequencies, they were measured separately using the same sample. The temperature dependence of clamped dielectric constants and
were measured directly from the capacitance measurements, while the free dielectric constants
and
measured at the same time were used as data consistency checks. The temperature dependence of elastic stiffness constants at a constant electric field
,
,
,
and
, and piezoelectric stress constants e15, e31 and e33 were determined by the RUS technique using the same sample.
1. Sample Preparation
Note: PZT-4 ceramic samples of the desired size can be directly ordered from many PZT ceramic manufacturers. One may also cut the sample from a larger PZT ceramic block using a diamond cutting machine, then repole the sample to restore depoling caused by cutting and polishing. Here, the sample shape is a parallelepiped with each dimension between 3 mm and 10 mm. Larger size samples are not necessary but accuracy might be compromised if samples are too small.
2. Pulse-echo Ultrasound Measurement
Note: In this paper, and
represent the ith row jth column element of elastic stiffness tensors at constant electric field and constant electric displacement, respectively;
and
represent the ith row jth column element of elastic compliance tensors at constant electric field and constant electric displacement, respectively; dij represents the ith row jth column element of piezoelectric strain tensor; eij represents the ith row jth column element of piezoelectric stress tensor;
and
represent the ith row jth column element of clamped and free dielectric constants, respectively. All matrix material constants are in Voigt notation.
3. Measure the Temperature Dependence of Dielectric Constants
4. Resonance Frequencies Measurement at Room Temperature and Mode Identification
5. Resonance Spectrum Measurement at Higher Temperatures and the Determination of Temperature Dependence of Full Set Material Constants
The LM algorism used in the inversion is a local minimum finder. Therefore, the initial values of elastic stiffness constants ,
,
,
, and
, and piezoelectric constants, e15, e31 and e33 should be given within a reasonable range from their true values. The constants
,
, and
, at room temperature can be precisely determined by the ultrasonic pulse-echo technique. The piezoelectric constants e15 at room temperature can be determined by the formula:
. Therefore, only values of
,
, e31 and e33 at room temperature need to be estimated in the beginning process. Traditional ultrasonic or resonant methods using several samples may be used to obtain the full set material constants at room temperature. Although results obtained by using several samples may be inconsistent, they are good enough to be used as the initial guess values of
,
, e31 and e33.
Figures 5 and 6 show the measured elastic constant tensor components and piezoelectric coefficient tensor components, respectively, as a function of temperature for the demonstration sample PZT-4 ceramics10. One can see from Figure 5 that the elastic constants ,
, and
increase with temperature while the elastic constants
and
are nearly independent of temperature in the temperature range from 20 to 120 °C. On the other hand, the piezoelectric constants e33, e31 and e15 are strongly temperature dependent as shown in Figure 6.
Figure 7 is the comparison between measured dielectric constants (dots) under stress free condition and the predicted ones (lines) calculated based on the full set material constants obtained by the RUS method10. Excellent agreement was found for both . In Figure 8, the dots represent piezoelectric constants d15 and d33 calculated using one set of formula while the lines represent their values calculated using another set of formula as given in step 5.3.3. Again, excellent agreement was found for both quantities. These results confirmed that the full set material constants obtained for the PZT-4 piezoceramic sample is highly self-consistent for the temperature range from 20 to 120 °C. The estimated relative errors of the constants measured by the RUS method are less than 3%. Note that if the full matrix material constants are not self-consistent, the integrity of the sample and mode identification process must be rechecked.
Figure 1: A rectangular parallelepiped PZT-4 piezoceramic sample. The dimensions measured by a micrometer are: lx = 4.461 mm, ly = 6.073 mm and lz = 4.914 mm. The mass density of this sample is 7,609.2 kg/mm3. Please click here to view a larger version of this figure.
Figure 2: Experimental setup for measuring the resonance frequency spectrum. It consists of a dynamic resonant system and a computer. Please click here to view a larger version of this figure.
Figure 3: Resonant ultrasound spectrum of the sample shown in Figure 1 at 30 °C (red) and 100 °C (blue). The spectrum shifts slowly with the increase of temperature. Modes identified at room temperature can serve as the reference for higher temperature mode identification. The notation convention for resonance modes was given in reference6. Please click here to view a larger version of this figure.
Figure 4: Air furnace with transmitting and receiving transducers inside. LiNbO3 single crystals were used to make the transmitting and receiving transducers to endure high temperatures. A thermocouple was used to measure the temperature of the sample inside the furnace. Please click here to view a larger version of this figure.
Figure 5: Inversion results of elastic stiffness constants ,
,
,
, and
. Overall, the elastic stiffness constants
,
and
, increase with temperature from 20 to 120 °C. Compared with
,
and
, the constants
and
are less sensitive to temperature. The constant
is nearly a linear function of temperature. This figure has been modified from reference10 with permission from AIP Publishing LLC. Please click here to view a larger version of this figure.
Figure 6: Inversion results of piezoelectric stress constants, ,
and
. The piezoelectric stress constants
,
and
increase with temperature from 20 to 120 °C. The constant
is nearly a linear function of temperature. This figure has been modified from reference10 with permission from AIP Publishing LLC. Please click here to view a larger version of this figure.
Figure 7: Comparison between measured and predicted free dielectric constants. Solid line and up-triangles are for ; dashed line and down-triangles are for
. The relative errors
and
are below 1.6% and 2.4%, respectively, in the whole temperature range of 20-120 °C, where
and
are measured and calculated
, respectively, and where
and
are measured and calculated
, respectively. This figure has been modified from reference10 with permission from AIP Publishing LLC. Please click here to view a larger version of this figure.
Figure 8: Comparison between and
values calculated using different formulas. The calculation formulas for
are:
(blue solid line) and
(blue triangle), and for
are:
(red dashed line) and
(red square). The relative errors of
are below 0.8%, and 1.2%, respectively, in the whole temperature range. Please click here to view a larger version of this figure.
Figure 9: A typical resonant ultrasound spectrum of a PZT-5A sample. The quality factor Q of the PZT-5A sample is about seventy-five12 . Generally speaking, the lower the Q-factor of the sample, the more difficult for mode identification. Generally, the RUS method will not give accurate results when the Q-factor is less than 100. Please click here to view a larger version of this figure.
The RUS technique described here can measure the full set material constants using only one sample, which eliminates errors caused by property variation from sample to sample so that self-consistency can be guaranteed. The method can be used for any solid material with a high quality factor Q, no matter if they are piezoelectric or not. All other standard characterization techniques require several samples to get the full set data and are difficult to achieve self-consistent data.
It is important to precisely measure the elastic constants ,
and
by the ultrasonic pulse-echo method at room temperature. Otherwise, the mode identification would be very difficult because calculated resonance frequencies of many modes are sensitive to these constants.
The failure of inversion calculations at the initial temperature will lead to the failure of determining the full set constants at higher temperatures because mode identification at the initial temperature is used as the base for mode identification at higher temperatures.
At room temperature, 6 constants out of the 10 constants to be determined can be obtained from the pulse-echo method and the capacitance measurements. Hence, only 4 unknown constants, ,
, e31 and e33, need to be estimated in the first round of forward calculation in the RUS procedure. The starting values for these 4 unknowns can be guessed based on other constants already known (in the same order of magnitude). Generally speaking, identifying about 20 modes is easy in the RUS forward process. These 20 modes are easily identified because they are well separated in the resonance spectrum, such as Au-3 and Ag-1 modes in Figure 3. Matching these 20 modes by adjusting the input values of these 4 estimated constants will give us a set of more accurate guessed values. Then, more number of modes can be identified by matching the calculated frequencies with those measured ones using better guessed input values. Finally, by using more number of identified modes, more accurate values of
,
, e31 and e33 can be refined by the backward process in the RUS method.
To reduce random fluctuations in the measured data, the temperature dependence of measured resonance frequencies corresponding to each mode was fitted to a polynomial function. Note that there must be an adequate number of modes measured to ensure the accuracy of the inversion results. From experience, the number of resonance frequencies measured should be at least 5 times the number of material constants to be determined13.
This protocol describes the procedure of determining the temperature dependence of the full matrix material constants by the RUS technique, using PZT-4 ceramic as an example. The focus here is on the procedure of the RUS technique, not the measured results of PZT-410.
The temperature range of the setup is limited by the temperature endurance of the electric wires and the transducers inside the furnace. This technique might be used at even higher temperatures if the sample is held by two buffer rods and the acoustic signal is sent and received through the buffer rods. In that case, electric wires and transducers will be outside of the furnace to avoid heating.
In principle, this RUS technique can be used on any type of solid material so long as it has a high mechanical Q-value (>100). For low Q-value materials, there is peak overlapping problem, making it hard to identify the resonance frequencies as shown in Figure 9.
The authors have nothing to disclose.
This work was supported by the National Natural Science Foundation of China (Grant No. 11374245), the NIH under Grant No. P41-EB2182, the Natural Science Foundation of Fujian Province, China (Grant No. 2013J01163), and the Open Research Fund of the State Key Laboratory of Acoustics, Chinese Academy of Science (Grant No. SKLA201306).
PZT-4 | TRS | |
paraffin | MTI Corporation | 8002-74-2 |
conductive silver paint | MG Chemicals | 842-20G |
Al2O3 Powder | MTI Corporation | |
coupling grease | Panametrics |