Longitudinal piezoelectric coefficient

Figure 13.7 Relaxation of the longitudinal piezoelectric coefficient in Sm-modified lead titanate ceramics at different temperatures.

The coefficient is called the longitudinal piezoelectric coefficient. If the faces over which the polarisation is measured remain the same, but the stress is applied in a perpendicular direction (Figure 6.13c), the relationship is... 

Figure 18.6 Orientation dependence of the longitudinal piezoelectric coefficient djj in the tetragonal and rhombohedral phases of PZT. From Ref [35].

Figure 18.7 Effective longitudinal piezoelectric coefficient dss and dielectric permittivity 33 for various PZT compositions. The curves are calculated for a single crystal using a phenomenological approach, and shown in comparison with experimental data for a PZTceramic sample. From Ref [36. ...

Davis et al. [42] have recently introduced the terms rotator and extender for a variety of ferroelectrics based on oxygen octahedra, in order to classify them with respect to whether the shear or the collinear effect dominates in the piezoelectric response. In extenders, the dominant polarization extension is directly related to the collinear piezoelectric effect, whereas in rotators the dominant contribution to the piezoelectric effect is the polarization rotation, that is directly related to the shear piezoelectric effect. Thus, extenders are ferroelectrics with a large longitudinal piezoelectric coefficient 33 that is related to a large relative dielectric susceptibihty %33, while rotators are ferroelectrics with large shear coefficients dis and 24. which are related to transverse susceptibilities and %2z. correspondingly. Electrostrictive... 

The condition for the maximum twinniug suppression is fulfilled for example for the quartz bar of the orientation (XYa) 150° for the transversal piezoelectric effect. The mechanical load is applied in the direction of rotated length axis at this time. Electrodes cover the surfaces perpendicular to the xi -axis. Condition Eq. (7.8) cannot be fiilfilled for the quartz elements for the longitudinal piezoelectric effect at all. The longitudinal piezoelectric coefficient is equal to zero for such orientations. Practical apphcation must be designed in order to get the twiiming suppression as high as possible simultaneously with sufficient piezoelectric coefficient. 

In bulk material, the resistivity is independent of crystal orientation because silicon is cubic. However, if the carriers are constrained to travel in a very thin sheet, eg, in an inversion layer, the mobility, and thus the resistivity, become anisotropic (18). Mobility is also sensitive to both hydrostatic pressure and uniaxial tension and compression, which gives rise to a substantial piezoresistive effect. Because of crystal symmetry, however, there is no piezoelectric effect. The resistivity gradually decreases as hydrostatic pressure is increased, and then abrupdy drops several orders of magnitude at ca 11 GPa (160,000 psi), where a phase transformation occurs and silicon becomes a metal (35). The longitudinal piezoresistive coefficient varies with the direction of stress, the impurity concentration, and the temperature. At about 25°C, given stress in a (100) direction and resistivities of a few hundredths of an O-cm, the coefficient values are 500—600 m2/N (50—60 cm2/dyn). 

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... 

The dynamic press allows measurements of the longitudinal, transverse and shear piezoelectric coefficients in the frequency range from 0.01 Hz to about 100 Hz. The lower limit is determined by the insulation resistance of the sample and cables, and charge drifts associated... 

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. 

Figure 18.9 Longitudinal c/33 piezoelectric coefficients for the PZT 60/40 thin films with (100), (111), and random orientation as a function of the driving field amplitude. From Ref [38].

The orientational dependence of 22 (the longitudinal strain along the y-axis) and 33 (the longitudinal strain along the z-axis) possess the same mathematical form, while the details of the properties are summarized on the anisotropy ratio. Ay, a function of the piezoelectric coefficients. 

Coefficients d, d2i and d-ij, describe the longitudinal piezoelectric effect (see symbol L in Table 5.1). The normal mechanical stress component causes piezoelectric polarization parallel to it in such case. Second possibihty is the piezoelectric polarization perpendicular to the applied normal mechanical stress. Such piezoelectric effect is so called transversal effect (see symbol T in Table 5.1) and it is characterized by one of the coefficients di2, d -i, J21, dj2, dn or J32. Application of shear mechanical stress might result in the piezoelectric polarization perpendicular to the plane of applied shear. Such shear piezoelectric effect is called longitudinal shear (see symbol 5l in Table 5.1) and it is characterized by one of the piezoelectric coefficients du, d25 or d e- Second possibility of shear piezoelectric effect is the piezoelectric polarization parallel to the plane of the applied shear stress. Such effect is called transversal shear (see symbol in Table 5.1 and in Fig. 5.2). This effect is related to one of the piezoelectric coefficients J15, di, d24, d26, d- orc 35. 

The best today s composite structure with 2-2-0 (air cavity) connectivity - Moonie - uses flextensional principle (Dogan et al. 1997 Fernandez et al. 1998 Xu et al. 1991 Zhang et al. 1999). Composite is made from the ceramic disc (PZT ceramics) and from two metallic plates (e.g. brass) with small air cavities (Fig. 7.29) glued together. By means of the metallic caps the hydrostatic pressure on major transducer surface is transferred to the longitudinal tension of the ceramic disc. This stress transfer will improve significantly the effective hydrostatic piezoelectric coefficient of the structure due to the different signs of ( 33 and d-n coefficients in PZT ceramics. The effective hydrostatic coefficient d could be further tuned in the value by the... 

The theoretical fitting curves seen in figures 6 to 8 were all fitted simultaneously to the first and second harmonic terms in equation (4), and the longitudinal piezoelectric and electrostiictive material coefficients were determined for PLZT (9.5/65/35) and PLZT (9.0/65/35), as seen in table 1. It can be noted that the experimental results and the theory are generally in good agreement. The small discrepancy between theory and experiment can be associated with random experimental xmcertainties. 

The above experimental methods were devoted to the study of the piezoeleetric coefficient as this one is flie relevant parameter of ferroelectrets. Ferroelectric polymers Uke PVDF and its copolymers show transversal and longitudinal piezoelectric activity depending on the degree and type of stretching (uniaxial vs. biaxial) causing molecular orientation. 

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. 

If all the coefficients of equation (2) are known, one can accurately predict the longitudinal strain under a varying electric field for a given piezoelectric or electrostrictive material, and even for a material exhibiting both piezoelectric and electrostrictive effects, such as irreversible electrostrictive materials. For ideal reversible electrostrictive materials, which possess no remnant polarization at zero electric field, the odd power term of the electric field in equation (2) vanishes. However, we will consider the relaxor PLZT ceramics studied in this chapter as irreversible electrostrictives, to account for any ferroelectric behaviour under dc bias fields, and we will therefore include both terms of the electric field in equation (2). 

Equation 6 is an approximate expression for die piezoelectric thickness coefficient 33 = dP I Fii — / At)NqulD/YM + / At)Nq[)lD/YD that describes the longitudinal direct piezoelectricity in die diickness direction of a slab or film. If the combined charge Nqp on all dipoles per area A is replaced by the bipolar interfacial charge density o, die second line of Eq. 6 yields the expression 33 ([Pg.496]

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