Application of the preisach model to ferroelectric composites

Abstract

The Preisach model [Preisach, 1935] has been successfully employed to study magnetic materials in which the phenomenon of hysteresis needs to be taken into account. Ferroelectric materials, the electric analog of magnetic materials, likewise exhibit hysteresis and field history dependence. The Preisach model is a mathematical model which provides a means for determining the highly nonlinear relationship between polarization and applied field. In this study, the Preisach model was used as a tool for modeling nonlinear behavior in ferroelectrics and ferroelectric composites. In addition, a model of ferroelectric behavior based on a combination of the Preisach model and the Landau theory for the second order phase transition was developed so as to allow discussion on the effect of temperature on physical properties, which the Preisach model could not tackle. The Preisach model considers a material to be a collection of hysteretic units with square hysteresis loops having two polarization states (called Preisach hysterons). A Preisach hysteron is only endowed with electrical properties so that it cannot take into account other effects, such as temperature. On the other hand, the Landau theory [Lines and Glass, 1977] for ferroelectrics has been successful in explaining the thermodynamic properties of ferroelectrics near phase transition temperatures, but it cannot describe minor loops. In this study, a new approach which combines the two models was suggested. In essence, Preisach hysterons were modified to become hysterons whose characteristics were described by the Landau theory, i.e. square-looped hysterons were replaced by hysteresis loops of the Landau theory. Thus, broad features of the Landau theory were included in the new model. This hybrid, Preisach-Landau model, allows discussion on phase transition, major and minor loops, and the effect of temperature on physical properties. Using this model, the D-E major and minor loops at different temperatures can be simulated. Also, the coercive field, remanent polarization and permittivity as a function of temperature may be calculated. In this model, the deletion property and the property of equal vertical chords of the classical Preisach model hold, but not the congruency property [Mayergoyz, 1991], the latter is nevertheless not so commonly observed in many real materials. Some ferroelectric materials, such as triglycine sulfate (TGS), has a finite dielectric constant at the ferroelectric-paraelectric phase transition temperature Tc, which is in contra-distinction to the expectation of the Landau theory of (second order) phase transition. A more refined Preisach-Landau model was proposed to tackle features such as finite dielectric constant at Tc. Comparing with experimental results on TGS [Gaffar et al., 1989], this modification produced a finite reciprocal dielectric constant at phase transition temperature as observed in experiments but the earlier Preisach-Landau model could not. Also, this modification was able to account for finer features of the experimental 1/Er-T curve while the general feature of the remanent polarization versus temperature curve was not affected. We used the Preisach model to study the nonlinear dielectricity of the ferroelectric polymer polyvinylidene fluoride (PVDF) as an example. The electric displacement D in the material when subjected to a sinusoidal electric field of a given frequency was calculated. Both the in-phase and out-of-phase components as well as higher harmonics emerged naturally from the model calculation. D-E loops at different field amplitudes were simulated and Fourier analyzed. The Fourier coefficients obtained were compared with the experimental data of Furukawa et al. [Furukawa et at, 1987]. Almost all the broad experimental features were reproduced by the simulations. A Preisach-Landau model was also used in this work, and the simulated results were compared with the results obtained from the Preisach model. Both simulations gave similar answers. It is remarkable that the power of the Preisach model was preserved in a Preisach-Landau model. In the study of ferroelectric composites, first a multi-layered ferroelectric composite was studied by using the concepts of the Preisach model to describe each constituent material. Under the assumption that the free charge on each interface was constant, the theory for multi-layered ferroelectric composites was analyzed. The results obtained were compared to D-E measurements made in the poling of polyvinylidenefluoride-trifluoroethylene (P(VDF-TrFE)) copolymer film sandwiched between ferroelectric triglycine sulfate crystal (TGS) electrodes [Ploss and Ploss, 1996]. In general, the computer simulations were in good agreement with experimental results. The D-E histories of the copolymer and the electrode material during poling were also obtained. A Preisach-Landau model was also applied to tackle the 0-3 composite problem. We considered a composite comprising spherical ferroelectric inclusions in a linear dielectric matrix. A D.C. electric field Ep was applied to polarize a virgin TGS/polymer composite. Under this poling process, we considered different conditions to discuss the variation of physical behavior of the composite and its constituents. These included the effect of the poling field Ep, poling temperature Tp and conductivity Qm of the matrix. We showed that the remanent polarization of the composite could be enhanced by increasing Ep, Tp and Qm.