The Piezoelectric Phase

Figure 19.11 Phase maps f(x, y) of a 6x4/um region of a fatigued FeCap (108 cycles) after negative (left) and positive (right) poling and evolution map of the piezoelectric phase signal f(x, E) (central picture) under varying (triangular shape) electric field E of the horizontal line indicated by the horizontal arrows. PI, P2, LI and L2 are discussed in detail later.

Also 3-3 composites were reported (Bowen et al. 2001). Such stmctures exhibit better poling properties due to the piezoelectric phase cormected in all 3 dimensions however the technology of making it is more comphcated. 

Thus, in a composite the ME effect is given by the product between the magnetostriction of the magnetic phase and piezoresponse of the piezoelectric phase [50], The two possible ME coefficients in case of composites are... 

Figure 3.39 shows the strain versus field relationship of various types of electromechanical ceramics. Hard piezoelectric materials. Types I to IV, are typically formulated to operate well below the transition region. The piezoelectric phase is highly stable and typically formulated for high coercive force and minimal domain wall movement (minimal... 

Ferroelectric Ceramic—Polymer Composites. The motivation for the development of composite ferroelectric materials arose from the need for a combination of desirable properties that often caimot be obtained in single-phase materials. For example, in an electromechanical transducer, the piezoelectric sensitivity might be maximized and the density minimized to obtain a good acoustic matching with water, and the transducer made mechanically flexible to conform to a curved surface (see COMPOSITE MATERIALS, CERAMiC-MATRix). 

In order to anticipate problems and to interpret observations under the extreme conditions of shock compression, it is necessary to consider structural and electronic characteristics of PVDF. Although the phenomenological piezoelectric properties of PVDF are similar to those of the piezoelectric crystals, the structure of the materials is far more complex due to its ferroelectric nature and a heterogeneous mixture of crystalline and amorphous phases which are strongly dependent on mechanical and electrical history. 

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. 

In particular, blends of PVDF with a series of different polymers (polymethylmethacrylate [100-102], polyethylmethacrylate [101], polyvinyl acetate [101]), for suitable compositions, if quenched from the melt and then annealed above the glass transition temperature, yield the piezoelectric [3 form, rather than the normally obtained a form. The change in the location of the glass transition temperature due to the blending, which would produce changes in the nucleation rates, has been suggested as responsible for this behavior. A second factor which was identified as controlling this behavior is the increase of local /rans-planar conformations in the mixed amorphous phase, due to specific interactions between the polymers [102]. 

Fig. 3. Schematic beam path of a phase-measurement interference microscope (PMIM, Fizeau optics). The beam partially reflected at the reference plane and at the sample surface interfere with each other while the reference plane is moved by the piezoelectric transducer for automatic phase determination. A reflectivity of at least 1% is required for the sample surface...

PVDF is mainly obtained by radical polymerisation of 1,1-difluoroethylene head to tail is the preferred mode of linking between the monomer units, but according to the polymerisation conditions, head to head or tail to tail links may appear. The inversion percentage, which depends upon the polymerisation temperature (3.5% at 20°C, around 6% at 140°C), can be quantified by F or C NMR spectroscopy [30] or FTIR spectroscopy [31], and affects the crystallinity of the polymer and its physical properties. The latter have been extensively summarised by Lovinger [30]. Upon recrystallisation from the melted state, PVDF features a spherulitic structure with a crystalline phase representing 50% of the whole material [32]. Four different crystalline phases (a, jS, y, S) may be identified, but the a phase is the most common as it is the most stable from a thermodynamic point of view. Its helical structure is composed of two antiparallel chains. The other phases may be obtained, as shown by the conversion diagram (Fig. 7), by applying a mechanical or thermal stress or an electrical polarisation. The / phase owns ferroelectric, piezoelectric and pyroelectric properties. 

Fig. 9.4. Dependence of piezoelectric properties of PbZrOj-PbTiOj on composition. The zirconate-rich phase is rhombohedral, whereas the titanate-rich phase is tetrahedral. The piezoelectric coefficients reach a maximum near the morphotropic phase boundary, approximately 45% PbZrOj and 55% PbTiOj. (After Jaffe et al., 1954.)...

The crystallographic and piezoelectric properties of the ceramics depend dramatically on composition. As shown in Fig. 9.4, the zirconate-rich phase is rhombohedral, and the titanate-rich phase is tetragonal. Near the morphotrophic phase boundary, the piezoelectric coefficient reaches its maximum. Various commercial PZT ceramics are made from a solid solution with a zirconate-titanate ratio near this point, plus a few percent of various additives to fine tune the properties for different applications. 

The unique piezoelectric and pyroelectric properties of semicrystalline films of PVDF arise from changes in the polarization imparted to the overall film by the crystalline P-phase. The polar nature of the P-phase is, in turn, a direct result of the parallel alignment of the dipole moment of the repeat units in the unit cell (Figure 11.1). The crystal polarization is defined as the dipole moment density of the crystal ... 

From Eq, (1) it is clear that a model of crystal polarization that is adequate for the description of the piezoelectric and pyroelectric properties of the P-phase of PVDF must include an accurate description of both the dipole moment of the repeat unit and the unit cell volume as functions of temperature and applied mechanical stress or strain. The dipole moment of the repeat unit includes contributions from the intrinsic polarity of chemical bonds (primarily carbon-fluorine) owing to differences in electron affinity, induced dipole moments owing to atomic and electronic polarizability, and attenuation owing to the thermal oscillations of the dipole. Previous modeling efforts have emphasized the importance of one more of these effects electronic polarizability based on continuum dielectric theory" or Lorentz field sums of dipole lattices" static, atomic level modeling of the intrinsic bond polarity" atomic level modeling of bond polarity and electronic and atomic polarizability in the absence of thermal motion. " The unit cell volume is responsive to the effects of temperature and stress and therefore requires a model based on an expression of the free energy of the crystal. 

The high-frequency dielectric constant is determined by the effects of electronic polarization. An accurate estimate of this property lends confidence to the modeling of the electronic polarization contribution in the piezoelectric and pyroelectric responses. The constant strain dielectric constants (k, dimensionless) are computed from the normal modes of the crystal (see Table 11.1). Comparison of the zero- and high-frequency dielectric constants indicates that electronic polarization accounts for 94% of the total dielectric response. Our calculated value for k (experimental value of 1.85 estimated from the index of refraction of the P-phase of PVDF. ... 

The material properties appearing in Eqs. (6)-(9) are defined by the partial derivatives of the dependent variables (P, c, e) with respect to the independent variables. At this point, to maintain consistency with the literature on the P-phase of PVDF, we label c as the 1 axis, a as the 2 axis, and, b as the 3 axis. In evaluating the piezoelectric and pyroelectric responses we consider changes in polarization along the 3 axis only polarization along the 1 and 2 axes remains zero, by symmetry, for all the cases considered here. The direct piezoelectric strain 03 , pC/N) and stress (gaj, C/iiE) coefficients are defined in Eqs. (10) and (11),... 

The piezoelectric constant of polymer films is usually a function of the frequency of the applied strain, and the constant is expressed by a complex quantity. In other words, the open-circuit voltage across the film surfaces is not in phase with the applied strain and the short-circuit current is not in phase with the strain rate. This effect, first pointed out by Fukada, Date and Emura (1968) and designated piezoelectric relaxation or dispersion, will be discussed in this review in terms of irreversible thermodynamics and composite-system theory. 

The piezoelectricity of polymeric materials has in general a relax-ational nature and the piezoelectric stress constant e is a function of the frequency of the applied strain in a similar way to the elastic modulus and dielectric constant. The induced polarization has in-phase and out-of-phase components to the strain and the e-constant is expressed as a complex quantity, as in Eq. (32). 

B) The crystallite has a non-relaxing piezoelectricity but the amorphous phase in the film has a mechanical relaxation and hence the ratio of the strain of the crystallite to that of the film as a whole, S/S in Eq. (62), becomes frequency-dependent. 

In Case (A), as will be discussed in 4.3, the piezoelectric relaxation is described as a coupling of dielectric relaxation and mechanical relaxation. In Case (B), on the other hand, the mechanical relaxation in the amorphous phase plays an important role in the piezoelectric relaxation ( 4.4). 

Piezoelectric polymer film is usually partially crystalline and the crystallites are embedded in the amorphous phase, which exhibits mechanical relaxations. Therefore, the strain of each crystallite, S, may differ in both amplitude and phase from that of the film as a whole, S. In this case the complex piezoelectric constant of the film is written by putting S/S — K (complex quantity) in Eq. (62) as... 

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