Anomalous dielectric relaxation

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... 

FRACTIONAL ROTATIONAL DIFFUSION AND ANOMALOUS DIELECTRIC RELAXATION IN DIPOLE SYSTEMS... 

C. Anomalous Dielectric Relaxation in the Context of the Debye Noninertial Rotational Diffusion Model... 

C. Inertial Effects in Anomalous Dielectric Relaxation of Linear and Symmetrical Top Molecules... 

Returning to anomalous dielectric relaxation, it appears that a significant amount of experimental data on disordered systems supports the following empirical expressions for dielectric loss spectra, namely, the Cole-Cole equation... 

Our purpose is to demonstrate how it is possible to describe the anomalous dielectric relaxation from microscopic models of the underlying processes. Moreover, we shall illustrate how the effects of the inertia of the molecules and an external potential arising from crystalline anisotropy or indeed any other mechanism could be included. 

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). 

The principal result of our calculation is that the Debye theory (based on the Smoluchowski equation), when extended to fractional dynamics via a onedimensional noninertial fractional Fourier-Planck equation in configuration space, can explain the Cole-Cole anomalous dielectric relaxation that appears in some complex systems and disordered materials. A further result of our calculation is that the aftereffect solution [Eq. (66)] is, with slight modifications, the moment generating function of the configuration space distribution function. Hence the mean-square angular displacement of a dipole, and so on, may be easily calculated by differentiation. We must remark, however, that the fractional Debye theory can be used only at low frequencies (got < 1) just as... 

Thus we have demonstrated how the empirical Havriliak-Negami equation [Eq. (11)] can be obtained from a microscopic model, namely, the fractional Fokker-Planck equation [Eq. (101)] applied to noninteracting rotators. This model can explain the anomalous relaxation of complex dipolar systems, where the anomalous exponents ct and v differ from unity (corresponding to the classical Debye theory of dielectric relaxation) that is, the relaxation process is characterized by a broad distribution of relaxation times. Hence, the empirical Havriliak-Negami equation of anomalous dielectric relaxation which has been... 

Here, we shall present both the exact and approximate solution for the anomalous dielectric relaxation of an assembly of fixed axis dipoles rotating in a double-well potential ... 

Pal, S., Balasubramanian, S., and Bagchi, B. Anomalous dielectric relaxation of water molecules at the surface of an aqueous micelle, /. Chem. Phys., 120, 1912, 2004. 

N. Nandi and B. Bagchi, Anomalous dielectric relaxation of aqueous protein solutions. J. Phys. Chem. A, 102 (1998), 8217. 

N. Nandi and B. Bagchi, Anomalous dielectric relaxation of aqueous protein solutions. J. Phys. Chem. A, 102 (1998), 8217-8221 N. Nandi and B. Bagchi, Dielectric relaxation of biological water,/. Phys. Chem. B, 101 (1997), 10954-10961. 

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