Piezoelectric response

Due to their high piezoelectric response, electrostriction in ferroelectrics, induced by an applied electric field, can be used as strain-inducing components (just as ferromagnetic materials can be exploited for their magnetostriction). Thus barium... 

Perhaps the most dramatic exception to the perfectly elastic, perfectly plastic materials response is encountered in several brittle, refractory materials that show behaviors indicative of an isotropic compression state above their Hugoniot elastic limits. Upon yielding, these materials exhibit a loss of shear strength. Such behavior was first observed from piezoelectric response measurements of quartz by Neilson and Benedick [62N01]. The electrical response observations were later confirmed in mechanical response measurements of Waekerle [62W01] and Fowles [61F01]. 

Piezoelectric Response Uncoupled Short-Circuit Solution... 

Chen et al. [76C02] and Lawrence and Davison [77L01] have placed the fully coupled nonlinear theory of uniaxial piezoelectric response in a form that is convenient for numerical solution of problems and have simulated a number of experiments in terms of this theory. An example of the results obtained is given below. 

The determination of piezoelectric constants from current pulses is based on interpretation of wave shapes in the weak-coupling approximation. It is of interest to use the wave shapes to evaluate the degree of approximation involved in the various models of piezoelectric response. Such an evaluation is shown in Fig. 4.5, in which normalized current-time wave forms calculated from various models are shown for x-cut quartz and z-cut lithium niobate. In both cases the differences between the fully coupled and weakly coupled solutions are observed to be about 1%, which is within the accuracy limits of the calculations. Hence, for both quartz and lithium niobate, weakly coupled solutions appear adequate for interpretation of observed current-time waveforms. On the other hand, the adequacy of the uncoupled solution is significantly different for the two materials. For x-cut quartz the maximum error of about 1%-1.5% for the nonlinear-uncoupled solution is suitable for all but the most precise interpretation. For z-cut lithium niobate the maximum error of about 8% for the nonlinear-uncoupled solution is greater than that considered acceptable for most cases. The linear-uncoupled solution is seriously in error in each case as it neglects both strain and coupling. 

The piezoelectric response investigation also provides direct evidence that significant inelastic deformation and defect generation can occur well within the elastic range as determined by the Hugoniot elastic limit. In quartz, the Hugoniot elastic limit is 6 GPa, but there is clear evidence for strong nonideal mechanical and electrical effects between 2.5 and 6 GPa. The unusual dielectric breakdown phenomenon that occurs at 800 MPa under certain... 

Piezoelectric Responses of Crystals in the Elastic-Plastic Range... 

Characteristic responses are readily obtained at pressures higher than 10 GPa, but differences have been observed with different loading arrangements. Piezoelectric responses at higher pressures are currently under study [92B01]. Dielectric relaxation and shock-induced conductivity may be involved. 

In this chapter studies of physical effects within the elastic deformation range were extended into stress regions where there are substantial contributions to physical processes from both elastic and inelastic deformation. Those studies include the piezoelectric responses of the piezoelectric crystals, quartz and lithium niobate, similar work on the piezoelectric polymer PVDF, ferroelectric solids, and ferromagnetic alloys which exhibit second- and first-order phase transformations. The resistance of metals has been investigated along with the distinctive shock phenomenon, shock-induced polarization. 

Ferroelectrics Poly(vynidilene fluoride) undergoes electrostriction when subjected to high ac fields, thus can be made into actuators applied pressure produces a piezoelectric response useful in sensors. 

However, a giant piezoelectric effect has been found in the Pb-based complex pervoskite oxides. In particular, the morphotropic boundary relaxor and PbTio3 complex exhibits huge piezoelectric response, so that an effective piezolysis is expected. 

In situations where absorption of the incident radiation by the transducing gas is troublesome a piezoelectric transducer (made from barium titanate, for example) can be attached to the sample (or sample cuvette in the case of liquids) to detect the thermal wave generated in the sample by the modulated light (8,9). The low frequency, critically damped thermal wave bends the sample and transducer thus producing the piezoelectric response. The piezoelectric transducer will also respond to a sound wave in the solid or liquid but only efficiently at a resonant frequency of the transducer typically of the order of 10 to 100 KHz (see Figure 4). Thus neither in the case of microphonic nor piezoelectric detection is the PA effect strictly an acoustic phenomenon but rather a thermal diffusion phenomenon, and the term "photoacoustic" is a now well established misnomer. 

Fig. 9.13. Double piezoelectric response of a tube scanner with symmetric connections. (A) The two y quadrants are connected to an ac voltage source. The two x quadrants are connected to the ground through an ac ammeter. (B) The stress in the x quadrants of the piezoelectric ceramics is equal in magnitude and opposite in sign to the y quadrants. (Reproduced from Chen, 1992a, with permission.)...

The ac current is generated by a combination of piezoelectric effect and inverse piezoelectric effect. In other words, it is a double piezoelectric response. 

The surface charge density is d iu(Q). By integrating over 9, we find the double piezoelectric response on one of the x quadrants to be... 

Fig. 9.15. Measuring circuit for the double piezoelectric response of a tube...

From Fig. 9.16, we obtain c/33 1.05 kN, a value consistent with the value listed in the catalog (1.27 kN). The value might be somewhat lower than the true value because the bonding of the tube ends is not perfectly rigid. If one end of the tube is free, or both ends are free, the deformation pattern varies significantly at the end(s). The net end effect is to reduce the value of the double piezoelectric response. Even if the end-bonding condition is unknown, an accurate measurement of the temperature or time variation of the piezoelectric constant can still be achieved. In other words, if the piezoelectric scanner is calibrated by a direct mechanical measurement or by the scale of images at one temperature, then its variation can be precisely determined by the electrical measurements based on double piezoelectric responses. 

Actual measurements of the double piezoelectric response also indicated that the double piezoelectric responses from individual quadrants vary significantly. As shown by the example in Fig. 9.16, the currents from the two x quadrants differ by about 40%. The currents from two y quadrants differ by about 16%. Therefore, the double piezoelectric response provides a sensitive method for testing the tube scanner. 

The magnitude of the piezoelectric response owing to applications of stress or strain transverse to the chain axis is much greater than the response owing to application of stress or strain parallel to the chain axis. This anisotropy in electrical response reflects the mechanical anisotropy of extended chain polymer... 

The total pyroelectric response at constant pressure (or stress) is the sum of two contributions. Primary pyroelectricity, which is due to changes in the magnitude of the dipole oscillation with temperature, accounts for only about 9% of the total response of the crystal at 300 K. The remaining overwhelming contribution is due to secondary pyroelectricity—the coupling of the piezoelectric response and thermal expansion. 

B) When an alternating voltage (frequency = co) is applied to the film with a c. bias voltage Vo, the film is strained with a frequency co. The induced strain consists of two parts, one independent of V0 and the other proportional to V0. The former is the piezoelectric response and the latter the electrostrictive one, if the piezoelectric constant is assumed to be independent of V0 (Kawai (1), 1969). 

The piezoelectric coefficients are third rank tensors, hence the piezoelectric response is anisotropic. A two subscript matrix notation is also widely used. The number of non-zero coefficients is governed by crystal symmetry, as described by Nye [2], In most single crystals, the piezoelectric coefficients are defined in terms of the crystallographic axes in polycrystalline ceramics, by convention the poling axis is referred to as the 3 axis. 

Because the piezoelectric coefficients can each be expressed in two ways, there are in general two different approaches to measuring the piezoelectric response approaches based on measurement of charge (or current), and those based on measurements of displacement (or strain). Choice of which coefficient to measure is often a matter of convenience. 

Figure 2.1 Schematic illustrations of intrinsic and extrinsic contributions to the piezoelectric constant of perovskite ferroelectrics. (a) and (b) correspond to the intrinsic unit cell shape (a) without and (b) with applied electric field, (c) and (d) correspond to the extrinsic response associated with the change in position of a non-180° domain wall (shown as a black line) (c) before and (d) after an electric field is applied. Note that both intrinsic and extrinsic responses lead to a change in shape of the material due to application of an electric field (and hence to a piezoelectric response). In both cases, the actual distortions are significantly exaggerated to make visualization easier.

It is also important to realize that piezoelectricity implies a linear coupling between dielectric displacement and strain, for example. However, in many ferroelectric materials, this response is linear only over a relatively limited field range (See for example, Figure 2.2). Non-linearity is especially important in ferroelectric materials which show a strong extrinsic contribution to the piezoelectric response [5], In addition, it is quite common for the response to be hysteretic. The amount of hysteresis that is observed depends strongly on the measurement conditions. Larger amplitude excitations often result in larger extrinsic contributions to the coefficients, and more non-linearity and hysteresis in the response. 

Piezoelectric coefficients are also temperature dependent quantities. This is true for both the intrinsic and the extrinsic contributions. Typically, the piezoelectric response of a ferroelectric material increases as the transition temperature is approached from below (See Figure 2.3) [3], Where appropriate thermodynamic data are available, the increase in intrinsic dijk coefficients can be calculated on the basis of phenomenology, and reflects the higher polarizability of the lattice near the transition temperature. The extrinsic contributions are also temperature dependent because domain wall motion is a thermally activated process. Thus, extrinsic contributions are lost as the temperature approaches OK [4], As a note, while the temperature dependence of the intrinsic piezoelectric response can be calculated on the basis of phenomenology, there is currently no complete model describing the temperature dependence of the extrinsic contribution to the piezoelectric coefficients. 

It should be noted that in practice, the piezoelectric response will typically not continue to rise all the way to the transition temperature, as elevated temperatures induce depoling of the ferroelectric, unless appropriate care is taken to insure that the material remains polarized (e. g. by application of a bias electric field). Depoling of this type is often important at temperatures of 1/2 of the Curie temperature, making high transition temperature materials interesting both for the decreased temperature dependence in the response, and the wider use range that can be achieved. 

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