Electrostriction coefficient

The strain j Hes along the axis of the electric field, E, or most often along the axis of the induced polari2ation, P. The electrostrictive coefficients for the electric field and polari2ation are M and respectively. Electrostriction is a small effect. In contrast to pie2oeIectric materials, electrostrictive materials... 

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Electrostriction. As distinct from inverse piezoelectric effect, electrostriction is a phenomenon in which the strain and the electrical field inducing the strain are related by Sy = where My are electrostriction coefficients. Several relaxor... 

The electrostriction coefficient is a fourth-rank tensor because it relates a strain tensor (second rank) to the various cross-products of the components of E or D in the. v, y and z directions. 

Ferroelectric materials above their Curie point behave electrostrictively and comparison of the electrostriction coefficient with d j2eP shows them to be of similar magnitude. This suggests that the large -coefficients shown by some ferroelectric materials are due to a combination of large electrostriction coefficients and large spontaneous polarization and permittivity values. 

Although the polycrystalline relaxor-based compositions have useful piezoelectric characteristics, taking their properties overall into account they do not offer significant advantages over the well established PZT system. Their high electrostriction coefficients make them attractive for certain actuator applications (see Section 6.5.2), but the potentially important advance has been the production of single crystals. 

Electrostriction in solids is important as the origin of piezoelectric effects. Von Sterkenberg has measured the electrostrictive coefficients in alkaline (earth) fluorides and found electrostriction there to be anisotropic. 

The ferroelectric Pb(Mgy3Nb2/3)03 (PMN) ceramic has been the snbject of extensive investigations due to its high dielectric coefficient and high electrostrictive coefficient, which renders it suitable for use in capacitors and electrostrictive actuators. However, the successful exploitation of this material is limited by the difficulty of producing a single-phase material with the perovskite structnre. Conventional solid state synthesis techniques invariably resnlt in the formation of one or more pyrochlore phases, which exhibit poor dielectric properties. 

In equations (5)-(8), i is the molecule s moment of Inertia, v the flow velocity, K is the appropriate elastic constant, e the dielectric anisotropy, 8 is the angle between the optical field and the nematic liquid crystal director axis y the viscosity coefficient, the tensorial order parameter (for isotropic phase), the optical electric field, T the nematic-isotropic phase transition temperature, S the order parameter (for liquid-crystal phase), the thermal conductivity, a the absorption constant, pj the density, C the specific heat, B the bulk modulus, v, the velocity of sound, y the electrostrictive coefficient. Table 1 summarizes these optical nonlinearities, their magnitudes and typical relaxation time constants. Also included in Table 1 is the extraordinary large optical nonlinearity we recently observed in excited dye-molecules doped liquid... 

It follows from Eq. (3.20), that the second term in the brackets represents the contribution of stress Um and is independent on film thickness, while the thickness dependence of transition temperature is described by the third term originating from surface contribution, polarization gradient and depolarization field. Keeping in mind that for perovskite structure, which is characteristic for SrTiOs and KTaOs, the electrostriction coefficient Qu < 0, one can see, that 7/(/) increases for compressive misfit strain < 0 as well as for Xs < 0. The dependence of 7 (/) calculated on the basis ofEq. (3.20) for KTaOs is reported in Fig. 3.2. It follows from the Fig. 3.2, that Tf depends essentially on Xs value. The second term in the brackets of Eq. (3.20) is independent of the film thickness and determines the shift of phase transition temperature (approximately 50 K) at / -> 00. Note that only this contribution has been taken into account in Ref. [17]. With a film thickness decrease the third term contribution prevails so that in sufficiently thin films (/ < 50 nm) the appearance of ferroelectric phase at T < 100 K can be expected. The appearance of ferroelectricity in the films of several nm thicknesses even at room temperature (see Fig. 3.2) cannot also be excluded. 

Here Oy are strain tensor components, qtj and % are electrostriction coefficients. 

The second category of actuators is based on electrostriction as exhibited by PMN [Pb(Mg 3Nb2/3)03] based ceramics. Although it is a second-order phenomenon of electromechanical coupling (x = ME, where M is called the electrostrictive coefficient), the induced strain can be extraordinarily large (more than 0.1%) [33], An attractive feature of these materials is the near absence of hysteresis (Fig. 4.1.19b). The superiority of PMN to PZT was demonstrated in a scanning tunneling microscope (STM) [34]. The STM probe was... 

Fig. 16. Electrostriction of a ferroelectric LC-elastomer (43). Big diagram Thickness variation Ah as a function of the applied ac voltage (/ac- Interferometric data were obtained at the fundamental frequency of the electric field (piezoelectricity, first harmonic -t) and at twice the frequency (electrostriction, second harmonic o). Sample temperature 60°C. Inset Electrostrictive coefficient a (-I-) versus temperature. At the temperature where the non-cross-linked polymer would have its phase transition Sc -Sa (about 62.5 0, the tilt angle of 0° is unstable. That is why the electroclinic effect is most effective at this temperature. An electric field of only 1.5 MV/m is sufficient to induce lateral strains of more than 4%.

The symbol A has the same meaning as before and stands for either constant finite strain V or stress t. It describes the field dependence of the conventional linear permittivity. Usually, the factor sq is extracted, and the third-order permittivity written as sqkkim but this would involve yet additional symbols, also for the impermittivities, and really makes only limited sense as it fails to render kklm dimensionless. Similar considerations also hold for the notation of all sorts of electrostriction coefficients. 

Here Sx are the elastic strain components in Voigt notation, d x are the piezoelectric coefficients and M-,jx are the electrostrictive coefficients. 

The electrostrictive coefficient Q links the field-induced strain x and the polarization P by a parabolic relationship, and should not be confused with the general order parameter Q in Landau theory. 

Based on Equations (16.9) and (16.15), the Maxwell effect and electrostrictive effect result in the same relationship between the strain and electric field and they therefore share some common features. For instance, an apparent piezoelectric effect can be observed when a DC bias is applied the strain response can be enhanced by the nonuniformity of the electric field, which can be created either by employing nonuniform materials or by the presence of the space (trapping) charge. Due to the electrostrictive effect and the appearance of the space charge, an insulation material can exhibit piezoelectricity and is known as an electret [9, 10]. The piezoelectric constant of an electret depends on the space charge and its distribution as well as the nonuniformity in the elastic properties and electrostrictive coefficient of the materials. 

For an E-M device, there are various considerations regarding the material properties. Besides their piezoelectric constants and electrostrictive coefficients, many other properties are also critical to the E-M performance. 

Starting with the Landau theory, Trebin and coworkers [122], [124] have calculated the electrostriction coefficients by allowing both the wave vectors and scalar amplitudes of the Fourier components to distort. Good agreement with the data is achieved, except for the case of anomalous electrostriction in BPI. The authors therefore conclude that the explanation for this behavior is beyond the capability of the Landau theory. As described earlier, the same group has also proposed a model of the blue phases incorporating bond-orientational order [45], [46]. However, a calculation of the anomalous electrostriction from this model [123] has had only limited success. 

Ferroelectric polymer materials like PVDF or its derivatives are mentioned, since they behave as ferroelectric materials (see Fig. 2.2) - They have crystallinity and the crystals show polymorphism by controlling the preparation method. Much detailed work has been carried out on piezoelectric and/or pyroelectric properties, together with their characteristics as electroactive actuators. These materials have long been mentioned as typical electroactive polymers. Through these materials, it is considered that the strain induced in the polymer materials is not large. The electrostrictive coefficient is known to be small for polymers. These are non-ionic polymers and the induced strain originates from the reorientation or the deformation of polarized crystallites in the solid materials. 

The electrostrictive behavior of P(VDF-CTFE) copolymers was investigated (Li et al. 2004 Li et al. 2006). A high electromechanical response was obtained in these copolymer films. As other PVDF-based polymers, the processing condition plays a very critical role on its properties. For a well-stretched and annealed P (VDF-CTFE) 88/12 copolymer film, a longitudinal electrostrictive strain as much as 5.5 % was obtained. A linear relationship between the strain response and the was observed, which indicates the electrostrictive nature of the electromechanical response in P(VDF-CTFE) copolymers. The corresponding electric field-related electrostrictive coefficient for the copolymer film is obtained as Mss = —1.23 0.02 X 10 (m /V ) (Li et al. 2006). 

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. 

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