Disordered systems 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... 

Stretched exponential relaxation is a fascinating phenomenon, because it describes the equilibration of a very wide class of disordered materials. The form was first observed by Kohlrausch in 1847, in the time-dependent decay of the electric charge stored on a glass surface, which is caused by the dielectric relaxation of the glass. The same decay is observed below the glass transition temperature of many oxide and polymeric glasses, as well as spin glasses and other disordered systems. 

In the following section the power of the fractional derivative technique is demonstrated using as example the derivation of all three known patterns of anomalous, nonexponential dielectric relaxation of an inhomogeneous medium in the time domain. It is explicitly assumed that the fractional derivative is related to the dimension of a temporal fractal ensemble (in the sense that the relaxation times are distributed over a self-similar fractal system). The proposed fractal model of the microstructure of disordered media exhibiting nonexponential dielectric relaxation is constructed by selecting groups of hierarchically subordinated ensembles (subclusters, clusters, superclusters, etc.) from the entire statistical set available. 

II. Microscopic Models for Dielectric Relaxation in Disordered Systems... 

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... 

II. MICROSCOPIC MODELS FOR DIELECTRIC RELAXATION IN DISORDERED SYSTEMS... 

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... 

Note, that Vogel-Fulcher law in the form of Eq. (1.26) was established firstly empirically (see e.g. Ref. [26]) while general theoretical description of its physical nature is still absent. The consideration of hierarchy of relaxational processes allowed to obtain the law in the form (1.26) for T = Tm only, where is the temperature of dielectric susceptibility maximum. The influence of random electric fields on relaxation barriers and hence on relaxation processes also permits to describe the disordered system by Vogel-Fulcher law in supposition of independent (parallel) relaxation processes [34]. 

The approach presented here was first developed for the dielectric relaxation of regular mesh-like polymer networks built from macromolecules with longitudinal dipole moments [30], and was later applied to disordered polymer networks [31,32]. Its key assiunption, namely the absence of any correlations in the orientations of the dipole moments of the different GGS bonds is obviously rather simphfied. However, it leads, as shown above, to simple analytical expressions for the dielectric susceptibility, a very important dynamical quantity in experimental studies of polymers we can now analyze it in great detail for particular GGS systems of interest. Another advantage of this model arises from the fact that one has a straightforward correspondence between the mechanical and the dielectric relaxation forms. From the expressions for the storage and loss modulus, Eqs. 20 and 21, and from those for the dielectric susceptibility Ae, Eqs. 41 and Eq. 42, one sees readily that [31]... 

Similarly to above glassy systems, the disordered ferroelectrics, polymers and composites are also characterized by slow relaxation processes. Their quantitative measure is complex dielectric permittivity, which can be described by generalized Debye law [29-31] in the form of the following empirical formulas ... 

It is, of course, natural from many points of view that aqueous solutions have been in the foreground for studies of electrolyte solutions, while studies of halide ion quadrupole relaxation in non-aque-ous solvents are quite few. However, studies of non-aqueous and mixed solvent systems are in certain respects highly relevant. For example, in order to test relaxation theories the possibility of making marked changes in solvent dipole moment, molecular size, dielectric constant, solvation number etc. should be very helpful. Also, the elucidation of certain general aspects of interactions and particle distributions in electrolyte solutions may be more easily achieved for non-aqueous systems. One such point is ion-pair formation, which for simple salts is not of great importance in water. Finally, of course, the quadrupole relaxation method may, as for aqueous solutions, be applied to more special problems such as ion solvation, complex formation etc. In studies of preferential solvation phenomena disorder effects in the first sphere may in certain cases be expected to lead to dramatic changes in the quadrupole relaxation rate. 

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