Converse piezoelectric coefficient

The piezoelectric coefficients discussed so far are the direct piezoelectric ooefficieats. When an electric field is applied it can produce a strain, and the ooefficieots which describe the strain are refen to as converse piezoelectric coefficients. 

A more complete description of these definitions is provided in a book by Nye (8). With these definitions it is now possible to describe the converse piezoelectric coefficient which relates the strain induced by an electric field lo the magnitude of the electric field E at constant stress ... 

Also, it is importsnl to point out that the direct and convene piezoekctric coefficients are not equal [101], as is usually assumed. The difference tentk to be small. - 10%, in hard and brittle fenoekcirics, but as will be demonstrated later in this chapter, the correction term can dominate in soft, pliable polymcn. The relatioa between tlie direct and converse piezoelectric coefficient is typically determined by deriving a Maxwell relation from a thermodynamic potential. A thorough description of Maxwell relations and thermodynamic potentials is presented in a book by Cailen (90. ... 

Thking the second partial derivatives with respect to the opposite variables In Eq. (44) and making use of the fact that the order of differentiation makes no difference, the relationship of the direct and converse piezoelectric coefficients is found to be... 

The results of measurements by Kepler and Anderson [81] using the static technique are presented in Thble 1. All the data presented in thb and other tables in thb chapter are the experimentally determined values, uncorrccted for changes in electrode area, and therefore arc not the true piezoelectric coeffidents. As we showed earlier in thb chapter, it b the experimentally determined value of the direct piezoelectric coefficient that should be equal to the converse piezoelectric coefficient and. in general, it appears that it b the experimentally determined values of other coeffidenb that should be used for comparison and calculations. 

Ferroelectrics. Among the 32 crystal classes, 11 possess a centre of symmetry and are centrosymmetric and therefore do not possess polar properties. Of the 21 noncentrosymmetric classes, 20 of them exhibit electric polarity when subjected to a stress and are called piezoelectric one of the noncentrosymmetric classes (cubic 432) has other symmetry elements which combine to exclude piezoelectric character. Piezoelectric crystals obey a linear relationship P,- = gijFj between polarization P and force F, where is the piezoelectric coefficient. An inverse piezoelectric effect leads to mechanical deformation or strain under the influence of an electric field. Ten of the 20 piezoelectric classes possess a unique polar axis. In nonconducting crystals, a change in polarization can be observed by a change in temperature, and they are referred to as pyroelectric crystals. If the polarity of a pyroelectric crystal can be reversed by the application on an electric field, we call such a crystal a ferroelectric. A knowledge of the crystal class is therefore sufficient to establish the piezoelectric or the pyroelectric nature of a solid, but reversible polarization is a necessary condition for ferroelectricity. While all ferroelectric materials are also piezoelectric, the converse is not true for example, quartz is piezoelectric, but not ferroelectric. 

Equations (6) and (7) express these relationships. are the elastic compliance constants OC are the linear thermal expansion coefficients 4 and d jj,are the direct and converse piezoelectric strain coefficients, respectively Pk are the pyroelectric coefficients and X are the dielectric susceptibility constants. The superscript a on Pk, Pk, and %ki indicates that these quantities are defined under the conditions of constant stress. If is taken to be the independent variable, then O and are the dependent quantities ... 

The piezoelectric effect entails a linear coupling between electrical and mechanical energies. Numerous piezoelectric coefficients are in use, depending on the electrical and mechanical boundary conditions imposed on the part under test. Each of the piezoelectric d, e, g, and h coefficients can be defined in terms of a direct and a converse effect the two sets of coefficients are related by thermodynamics. For example, the piezoelectric charge coefficient, dkjk, can be defined via [1] ... 

This paper described a number of the means for measuring the piezoelectric coefficients of bulk materials and thin films. In bulk materials, excellent references are available. Numerous means have been used over the years to measure the piezoelectric coefficients, which can be loosely grouped as charge-based and displacement-based. Accurate data can be obtained by many of the techniques, and agreement between measurement types is usually reasonable, provided that comparable excitation levels are utilized. In contrast, for thin films attached to substrates, the mechanical boundary conditions differ in charge and displacement based techniques. As a result, the direct and converse coefficients are not identical. In addition, perhaps because of the relative immaturity of the field, the numerous possible artifacts are not always accounted for, which can lead to erroneous results in thin film measurements. 

The term piezoelectric nonlinearity is used here to describe relationship between mechanical and electrical fields (charge density D vs. stress a, strain x vs. electric field E) in which the proportionality constant d, is dependent on the driving field, Figure 13.1. Thus, for the direct piezoelectric effect one may write D = d(a)a and for the converse effect x = d(E)E. Similar relationships may be defined for other piezoelectric coefficients (g, h, and e) and combination of electro-mechanical variables. The piezoelectric nonlinearity is usually accompanied by the electro-mechanical (D vs. a or x vs. E) hysteresis, as shown in Figure 13.2. By hysteresis we shall simply mean, in the first approximation, that there is a phase lag between the driving field and the response. This phase lag may be accompanied by complex nonlinear processes leading to a more general definition of the hysteresis [2],... 

Figure 13.1 Examples of the field dependence of piezoelectric coefficients (a) direct effect in ferroelectric ceramics, measured with a dynamic press (b) converse effect in rhombohedral 60/40 pzt thin films with different orientations, measured with an optical interferometer [1], correspond to pseudocubic axes.

In the converse piezoelectric effect one usually applies voltage V or electric field E on the sample and measures displacement AZ or strain A///. From relation Al = 0Z33 V for the longitudinal effect, we see that even for materials with exceptionally high piezoelectric coefficient (do3 = 2000pm/V in pzn-pt) the displacement Al is only around 2 nm if 1 V is applied on the sample. For the same voltage the displacement is reduced to 0.2 nm in a typical pzt composition and to only tn 2 pm in quartz. The displacement can be increased by application... 

S]). The direct piezoelectric effect is the production of electric displacement by the application of a mechanical stress the converse piezoelectric effect results in the production of a strain when an electric field is applied to a piezoelectric crystal. The relation between stress and strain, expressed by Equation 2.7, is indicated by the term Elasticity. Numbers in square brackets show the ranks of the crystal property tensors the piezoelectric coefficients are 3rd-rank tensors, and the elastic stiffnesses are 4th-rank tensors. Numbers in parentheses identify Ist-rank tensors (vectors, such as electric field and electric displacement), and 2nd-rai tensors (stress and strain). Note that one could expand this representation to include thermal variables (see [5]) and magnetic variables. 

The coefficient is called the transverse piezoelectric coefficient. The converse piezoelectric effect, relating strain, e to the applied electric field E is similarly simplified to... 

Direct methods for measuring the strain that results from applying a field or vice versa, applying a strain, and measming the accumulated charge are ahim-dant. Interferometers, dilatometers, fiher-optic sensors, optical levers, linear variable displacement transducers, and optical methods are employed to evaluate the piezoelectric strain (converse effect) (69-72). The out-of-plane or thickness piezoelectric coefficient dss can he ascertained as a function of the driving field and frequency. The coefficient is measmed based on the equation... 

Let us now look at the interrelations holding between the material coefficients listed in Table 4.2. In the context of coupling effects it has already been shown that the direct and the converse piezoelectric effects are described by identical material coefficients. This follows immediately from their definition as second derivatives of the associated thermodynamic potential, recognizing the fact that the order of differentiations may be reversed. Direct and converse effects are described by relations between different pairs of variables. The equality of the coefficients governing the direct and the converse effects reduces the number of independent material coefficients for each selection of the triple of independent variables. It does not, however, represent a relation between material coefficients derived from different thermodynamic potentials. 

The direct effect coefficients are defined by the derivatives (5D/SX) = d (piezoelectric strain coefficient), (5D/5x) = e, -(5E/5X) = g (piezoelectric voltage constant) and -(5E/5x) = h. The converse-effect coefficients are defined by the derivatives (8x/5E) = d, (5x/5D) = g, -(5X/5E) = e, and -(5X/5D) = h. As the piezoelectric coefficients are higher-rank tensors, their mathematical treatment is rather tedious. Fortunately, in higher symmetric crystals the number of tensorial components will be drastically reduced due to symmetry constraints. An example is shown below. 

The thermodynamical derivation of piezoelectricity includes two steps (1) The relevant mechanical or electrical quantities are calculated as partial derivatives of the Gibbs free energy with respect either to one of the two mechanical or to one of the two electrical observables, respectively. (2) The second partial derivative of the Gibbs free energy with respect to the other domain (electrical or mechanical, respectively) yields one of the piezoelectric coefficients. Because there is one intensive (force-like or voltage-like) observable, namely, mechanical stress and electrical field, and one extensive (displacement-like) observable, namely, mechanical strain and electrical displacement, in each of the two domains, we have four possible combinations of one mechanical and one electrical observable in total. Thus, we obtain four different piezoelectric coefficients that are usually abbreviated as d, e, g, and h. As the sequence of the two partial derivations can be reversed, we arrive at two different expressions for each coefficient one for direct piezoelectricity (mechanical stimulus leads to an electrical response) and one for inverse or converse piezoelectricity (electrical stimulus leads to a mechanical response). For example, the piezoelectric d coefficient is given by the two alternative terms ... 

We now consider the converse of the above, namely the distortion suffered by an anisotropic crystal as a response to an externally applied electric field , which is known as the converse piezoelectric effect. Here the imposition of an electric field creates a unique alignment direction for the dipoles that eliminates the cancellation effect described earlier. As a result, the crystal deforms. The coefficients relating the strain to the electric field are the same as those that connect the polarization to the stress in the direct effect. The proof will be provided later by noncircular arguments. We thus characterize the present linear dependence by the relation... 

These Maxwell relations show that the matrix elements in Eq. (5.11.2) are symmetric about the diagonals. Specifically, the coefficients of the direct and converse piezoelectric effects are equal—a conclusion that verifies an earlier statement in Section 5.10. Also, the coefficients for the thermal expansion and for the piezocaloric, as well as for the pyroelectric and electrocaloric effects, respectively, are identical. Again, bear in mind that Eq. (5.11.2) represents six relations for e,-, three relations for Dj, and one equation for d5, each of which contains six variables Uy, three variables ,-, and Tas the independent quantities. 

We see from Figure 8.13 that a number of liquid crystal phases, SmC of chiral rod-shape and the tilted columnar phase of chiral disc shape molecules, as well as the SmCP of achiral bent-core, and the tilted bowl-shape molecules all have C2 symmetry with eight independent piezoelectric coefficients. The direct and converse i- piezoelectric effects have been mostly studied in the fluid SmC liquid crystals. 

The magnitude of thb discrepancy can become very large. Using the static technique to determine the direct and converse piezoelectric coeffidenb in PVDF, Kepler and Anderson [81] found that the experimentally determined direct and converse coefficients, the values found before the correction for the change in electrode area was applied, were equal to within a few percent If Eq. (45) was correct, the experimentally determined value of the direct pieimelectric coefficient iQ hF should be corrected by subtracting (P/AyidAf9elastic compliances s and were 4 X 10 Pa and -1.6 x 10 Pa, respectively. Therefore tte correction ta -12 X 10 CVN, a value almost three... 

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