Dielectric relaxation continued

The DISPA analysis described above is based on comparison of an experimental curve to a reference circle. Although that display is qualitatively useful in identifying the correct line-broadening mechanism, the analogous Cole-Cole plotS in dielectric relaxation continues as a popular display mode after nearly 40 years), a display in vyhich the same experimental DISPA data is compared to a straight line could better help to determine quantitatively the line-broaoen-ing parameter(s) of that mechanism. 

The dielectric relaxation of bulk mixtures of poly(2jS-di-methylphenylene oxide) and atactic polystyrene has been measured as a function of sample composition, frequency, and temperature. The results are compared with earlier dynamic mechanical and (differential scanning) calorimetric studies of the same samples. It is concluded that the polymers are miscible but probably not at a segmental level. A detailed analysis suggests that the particular samples investigated may be considered in terms of a continuous phase-dispersed phase concept, in which the former is a PS-rich and the latter a PPO-rich material, except for the sample containing 75% PPO-25% PS in which the converse is postulated. 

Chapter 8 by W. T. Coffey, Y. P. Kalmykov, and S. V. Titov, entitled Fractional Rotational Diffusion and Anomalous Dielectric Relaxation in Dipole Systems, provides an introduction to the theory of fractional rotational Brownian motion and microscopic models for dielectric relaxation in disordered systems. The authors indicate how anomalous relaxation has its origins in anomalous diffusion and that a physical explanation of anomalous diffusion may be given via the continuous time random walk model. It is demonstrated how this model may be used to justify the fractional diffusion equation. In particular, the Debye theory of dielectric relaxation of an assembly of polar molecules is reformulated using a fractional noninertial Fokker-Planck equation for the purpose of extending that theory to explain anomalous dielectric relaxation. Thus, the authors show how the Debye rotational diffusion model of dielectric relaxation of polar molecules (which may be described in microscopic fashion as the diffusion limit of a discrete time random walk on the surface of the unit sphere) may be extended via the continuous-time random walk to yield the empirical Cole-Cole, Cole-Davidson, and Havriliak-Negami equations of anomalous dielectric relaxation from a microscopic model based on a... 

Another most important question in anomalous dielectric relaxation is the physical interpretation of the parameters a and v in the various relaxation formulas and what are the physical conditions that give rise to these parameters. Here we shall give a reasonably convincing derivation of the fractional Smoluckowski equation from the discrete orientation model of dielectric relaxation. In the continuum limit of the orientation sites, such an approach provides a justification for the fractional diffusion equation used in the explanation of the Cole-Cole equation. Moreover, the fundamental solution of that equation for the free rotator will, by appealing to self-similarity, provide some justification for the neglect of spatial derivatives of higher order than the second in the Kramers-Moyal expansion. In order to accomplish this, it is first necessary to explain the concept of the continuous-time random walk (CTRW). 

We remark that if the jump length distance is also a Levy process, the mean-square displacement does not exist which has led to conceptual difficulties in applying this process to dielectric relaxation. Using these simplifications, one can identify two specialized forms of a continuous time random walk ... 

We shall now almost exclusively concentrate on the fractal time random walk excluding inertial effects and the discrete orientation model of dielectric relaxation. We shall demonstrate how in the diffusion limit this walk will yield a fractional generalization of the Debye-Frohlich model. Just as in the conventional Debye relaxation, a fractional generalization of the Debye-Frohlich model may be derived from a number of very different models of the relaxation process (compare the approach of Refs. 22, 23, 28 and 34—36). The advantage of using an approach based on a kinetic equation such as the fractional Fokker-Planck equation (FFPE) however is that such a method may easily be extended to include the effects of the inertia of the dipoles, external potentials, and so on. Moreover, the FFPE (by use of a theorem of operational calculus generalized to fractional exponents and continued fraction methods) clearly indicates how many existing results of the classical theory of the Brownian motion may be extended to include fractional dynamics. 

---