Piezoelectricity piezoelectric strain coefficient

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 ... 

From tensor algebra, the tensor property relating two associated tensor quantities, of rank / and rank g, is of rank (/-b g). Hence, the physical property connecting /, and aj is the third-rank tensor known as the piezoelectric effect, and it contains 3 = 27 piezoelectric strain coefficients, dyk. The piezoelectric coefficients are products of electrostriction constants, the electric polarization, and components of the dielectric tensor. 

The piezoelectric strain coefficients now take the form d and the relation between the applied stress, o- , and induced polarization, P,-, can be expressed in final matrix-like form as ... 

A centrosymmetric stress cannot produce a noncentrosymmetric polarization in a centrosymmetric crystal. Electric dipoles cannot form in crystals with an inversion center. Hence, only the twenty noncentrosymmetric point groups are associated with piezoelectricity (the noncentrosymmetric cubic class 432 has a combination of other symmetry elements which preclude piezoelectricity). The piezoelectric strain coefficients, dj for these point groups are given in Table 8.7, where, as expected, crystal symmetry dictates the number of independent coefficients. For example, triclinic crystals require the full set of 18 coefficients to describe their piezoelectric properties, but mono-chnic crystals require only 8 or 10, depending on the point group. 

TABLE 8.7. Piezoelectric Strain Coefficients in the Noncentrosymmetric Point Groups... 

Class Point Group Piezoelectric Strain Coefficients... 

Here, die = e26/Gq = 3.1 x 10 mV is the piezoelectric strain coefficient. Note that the derivation assumes laterally infinite resonators it does not account for energy trapping. Equation 120 is therefore expected to miss a numerical factor of order unity. Inputting values ( 26 = 3.1 pm V ), we arrive at ... 

Pyro- and Piezoelectric Properties The electric field application on a ferroelectric nanoceramic/polymer composite creates a macroscopic polarization in the sample, responsible for the piezo- and pyroelectricity of the composite. It is possible to induce ferroelectric behavior in an inert matrix [Huang et al., 2004] or to improve the piezo-and pyroelectricity of polymers. Lam and Chan [2005] studied the influence of lead magnesium niobate-lead titanate (PMN-PT) particles on the ferroelectric properties of a PVDF-TrFE matrix. The piezoelectric and pyroelectric coefficients were measured in the electrical field direction. The Curie point of PVDF-TrFE and PMN-PT is around 105 and 120°C, respectively. Different polarization procedures are possible. As the signs of piezoelectric coefficients of ceramic and copolymer are opposite, the poling conditions modify the piezoelectric properties of the sample. In all cases, the increase in the longitudinal piezoelectric strain coefficient, 33, with ceramic phase poled) at < / = 0.4, the piezoelectric coefficient increases up to 15 pC/N. The decrease in da for parallel polarization is due primarily to the increase in piezoelectric activity of the ceramic phase with the volume fraction of PMN-PT. The maximum piezoelectric coefficient was obtained for antiparallel polarization, and at < / = 0.4 of PMN-PT, it reached 30pC/N. 

When written in matrix form these equations relate the properties to the crystallographic directions. For ceramics and other crystals the piezoelectric constants are anisotropic. For this reason, they are expressed in tensor form. The directional properties are defined by the use of subscripts. For example, d i is the piezoelectric strain coefficient where the stress or strain direction is along the 1 axis and the dielectric displacement or electric field direction is along the 3 axis (i.e., the electrodes are perpendicular to the 3 axis). The notation can be understood by looking at Figure 31.19. 

We estimated the elastic constants and piezoelectric coefficients numerically from changes in the free energy and polarization of the crystal at finite applied strains. The results are given in Table 11.1. Contributions to the direct piezoelectric stress coefficients, gs (/= 1, 2, 3), were estimated from numerical derivatives of Ap and of (cos cp). The direct piezoelectric strain coefficients were calculated from Eq. (12). 

T,. r, and Tft, by the piezoelectric strain coefficients Some of the piezoelectric coefficients such as and 33 are null as a result of symmetry [15]. 

If a piezoelectric transducer is simply being used as a displacement or strain indicator, then the relationships described above are sufficient to allow the calculation of the electrical signal which will be generated upon perturbation of a device. All that must be known are the stiffness constants (elastic properties), the piezoelectric strain coefficients (piezoelectric properties) of the transducer material, and the strain induced. If however, acoustic waves are being employed for mass sensing applications, then one must consider the dynamics of particle motion and how wave propagation is affected by the finite boundaries of a transducer. 

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. 

Fig. 5 Temperature dependence of piezoelectric strain coefficient, dsi, of the poled and annealed nylon 11 and nylon 7 in comparison with that of poled and annealed PVDF (Adapted from Y. Takashi et al.)...

Fig. 6 (a) Curves of electric field displacement, D, versus applied electric field, E, (D-E) and (b) temperature dependence of the piezoelectric strain coefficient, dsi, for (a) PVDF-nylon 11 bilaminate, (b) PVDF, and (c) nylon 11 films (Adapted fi om J. Su et al.)... 

Here the symbols po, L, y, and e(x) denote the density, the sample thickness, the electrostriction coefficient, and the piezoelectric strain coefficient. From Eqs. 24 and 25, one sees that from the current response J t) of the pressure pulse experiment, the direct image of the space-charge distribution p x) or the gradient of the piezoelectric coefficient can be deduced by using the simple transformation x = ct. 

The occurrence of piezoelectric behaviour in LB films has been known for some time [57,58], and a 30 X-type layer LB film of (37) was found to give opposite signs of the piezoelectric strain coefficients d i and d [59], the latter having a value of 1.5 pC which is approximately an order of magnitude lower than that of the well-documented polymer poly(vinylidene fluoride) (PVDF). Values for 31 of 0.023 and 0.170 pC N have also been obtained for alternate-layer structures of 22-tricosenoic acid with docosylamine, and a ruthenium complex with docosanoic acid respectively [60]. As the use of pyroelectric materials in detector applications requires that the materials possess only low levels of piezoelectricity (high levels introduce problems of microphony), this suggests that the former materials would be better suited for pyroelectric detector applications, while the latter system would be more appropriate for piezoelectric-based applications. 

Fig. 5.2 Piezoelectric strain coefficient as a function of poling field for 70 30 VDF TrFE copolymer. All film samples were poled at 100 2°C.

In hydrostatic mode the piezoelectric strain coefficient is deter-... 

Composite piezoelectric materials may be represented by the so-called simple series, simple parallel and the modified cubes diphasic models (Fig. 6.4). The modified cubes model was developed as a generalization of the series, parallel and cubes models. It is adapted for the representation of 0-3 composite sheet materials. Estimated values of the average longitudinal piezoelectric strain coefficient 33 and the average piezoelectric voltage coefficient 33 for the composite may be evaluated in terms of these models. References to the piezoelectric ceramic and the polymer phase will be indicated by superscripts 1 and 2 respectively. 

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