Complex piezoelectric constant

In this section, we will derive a thermodynamic expression of the complex piezoelectric constant e for a single-phase system with a single relaxation process. 

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... 

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. 

As will be shown in the theory, the electrostriction effect plays an important role in the piezoelectric effect of polymer films. Moreover, a knowledge of the complex electrostriction constant as a function of frequency reveals a new aspect of the relaxational behavior of polymers. In this review a new method for measuring complex electrostriction constant with varying frequency will be presented with some results for poly(vinylidene fluoride). 

Fig. 4. Block diagram of the apparatus for measuring complex piezoelectric stress and strain constants of polymer films with varying frequency (Furukawa and...

However, in contrast to the cases of complex elastic modulus G and dielectric constant e, the imaginary part of the piezoelectric constant, e", does not necessarily imply an energy loss (Holland, 1967). In the former two, G"/G and e"/e express the ratio of energy dissipation per cycle to the total stored energy, but e"/e does not have such a meaning because the piezoelectric effect is a cross-coupling effect between elastic and electric freedoms. As a consequence, e" is not a positive definite quantity in contrast to G" and e". In a similar way to e, however, the Kramers-Kronig relations (Landau and Lifshitz, 1958) hold for e ... 

Fig. 11. Complex piezoelectric strain constant (20 Hz), complex Young s modulus (30 Hz), and complex dielectric constant (1kHz) of uniaxially drawn poly(D-propylene oxide) film plotted against temperature. Draw-ratio = 1.5. Degree of crystallinity=40%. Drawn after Furukawa and Fukada [Nature 221,1235 (1969)] by permission of Macmillan (Journals) Ltd.

Fig. 12. Complex piezoelectric strain constant of uniaxially drawn cellulose triacetate film plotted against temperature. Draw-ratio = 2. Plasticizer content = 10%. Frequency = 20 Hz. Drawn after Fukada, Date, and Emura [J. Soc. Mat Sci. Japan 17,335 (1968)] by permission of the Society of Materials Science, Japan...

Fig. 29. Temperature dependence of complex piezoelectric strain constant of composite film of polyester resin and powdered PZT (50% of the volume) polarized at room temperature under a d.c. field of 100 kV/cm. Reproduced from Fukada and Date [Polymer Journal, 1,410 (1970)] by permission of the Society of Polymer...

The piezoelectric effect has been shown to exhibit a relaxational nature and the complex piezoeletric constant is a function of frequency and temperature. In Group (A), the relaxation is ascribed to either the nature of the crystallite itself or the viscoelasticity of the amorphous phase in which the crystallites are embedded. In the former case, the piezoelectric relaxation is a cross-coupling phenomenon of dielectric relaxation and mechanical relaxation of the crystallite. In the latter, on the other hand, the relaxation is governed by the mechanical relaxation of the amorphous phase. 

Given the functional form of the term related to piezoelectric stiffening, one can define a new complex spring constant, /Cp, taking piezoelectric stiffening into account. Remembering that = (j / icoCo), one writes ... 

Kunkel, H. A. Locke, S. Pikeroen, B. (1990). Finite-Element Analysis of Vibrational Modes in Piezoelectric Ceramics Disks, IEEE Transactions on Ultrasonics Ferroelectrics and Frequency Control, Vol. 37, No. 4, pp.316-328, ISSN 0885-3010 Masaki, M. Hashimoto, H. Masahiko, W. Suzuki, I. (2008). Measurements of Complex Materials Constants of Piezoelectric Ceramics Radial Vibrational Mode of a Ceramic Disk, Journal of the European Ceramic Society, Vol.28, Issue.l, pp.133-138, ISSN 0955-2219... 

Using these calculated three-dimensioiial d values, we can cakulate the macroscopic piezoelectric constants of the bulk PVDF film on the basis of a complex model consistiog of crystalline and amorphous phases (256.260). The macroscopic piezoelectric stress constant ti" is expressed as... 

It was first found in 1968 that the piezoelectric constant also exhibits relaxation and should be represented as a complex quantity such as d > d — id (14J. Other piezoelectric constants are also represented by complex quantities —... 

In the following description, the piezoelectric constant d indicates the shear piezoelectric constant -d, or d . For symmetry C. and D the relation -d, d d holds. In most cases, a sinusoidal stress at 10 Hz is given to the sample and both in-phase component and v/2 out-of-phase component of the resulting sinusoidal polarization are detected. The ratio of polarization to stress is the conqdex piezoelectric strain constant d d - id, and the ratio of polarization to strain is the complex stress constant e t - id. ... 

Rectangular coordinates are assigned to the film so that the 1 axis is in the plane of the film and the 3 axis is normal to the surfime of the film. The complex piezoelectric stress constant is given by -t- ku. ... 

When the induced voltage Fopen is not completely in phase with the applied strain, or the induced current is not completely in phase with the applied strain rate (piezoelectric relaxation), the e-constant becomes a complex quantity as follows ... 

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). 

Finally, while the piezoelectric d, e, g, and h constants are typically reported as real numbers, there is increasing use of the fact that the material response is not always in phase with the applied field. This can be due to a variety of factors, including domain wall motion in ferroelectrics [5]. Thus, coefficients can be described as complex quantities. Discussions of how to measure these constants are given in [6-10],... 

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],... 

The spatial arrangement of the alternating CH2 and CF2 groups along the polymer backbone creates a high dipole moment that accounts for the unique polarity, unusually high dielectric constant, complex polymorphism, and high piezoelectric and pyroelectric activity of the polymer. Because of its polar nature, PVDF is permeation resistant to chlorine and bromine. Both chlorine and bromine are known to permeate most commonly available commercial polymers. 

The photoswitchable complexation/dissociation properties of n donor-acceptor complexes between xanthene dyes and photoisomerizable bipyrid-inium salts have been used to generate an optoelectronic interface [97] (Fig. 28). Eosin isothiocyanate (52) was covalently linked to an electrode surface via a thiourea bond (Fig. 28A). The electron acceptor 3, 3 -bis(N-methylpyridinium) azobenzene 53 was used as the photoisomerizable component. The association constants of the n donor-acceptor complexes generated between eosin and 53a or 53b in solution correspond to Ka — 8.3 x 103 M-1 and Ka — 3.4 x 103 M 1, respectively. The analysis of complexation on the functionalized surface was accomplished by quartz crystal microbalance measurements. The frequency change (Af) of a piezoelectric quartz crystal on which a mass change Am occurs is given by the Sauerbrey equation (Eqn. 1) ... 

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