Multiscale Model for the Dielectric Permittivity (original) (raw)
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Microscopic model of a non-Debye dielectric relaxation: The Cole-Cole law and its generalization
Theoretical and Mathematical Physics, 2012
Based on a self-similar spatial-temporal structure of the relaxation process, we construct a microscopic model for a non-Debye (nonexponential) dielectric relaxation in complex systems. In this model, we derive the Cole-Cole expression for the complex dielectric permittivity and show that the exponent α involved in that expression is equal to the fractal dimension of the spatial-temporal self-similar ensemble characterizing the structure of the medium and the relaxation process occurring in it. We find a relation between the macroscopic relaxation time and the micro-and mesoparameters of the system. We obtain a generalized Cole-Cole expression for the complex dielectric permittivity involving log-periodic corrections that occur because of a discrete scaling invariance of the fractal structure generating the relaxation process on the mesoscopic scale. The found expression for the dielectric permittivity can be used to interpret dielectric spectra in disordered dielectrics.
Microscopic model of dielectric α-relaxation in disordered media
Fractional Calculus and Applied Analysis, 2013
The micro/mesoscopic theory of dielectric relaxation has been developed. Based on the fractional kinetics it gives a possibility to obtain the desired expression for the complex dielectric permittivity (CDP) and describe the asymmetric peaks that are created presumably by the so-called "excess wing" located in high-frequency region. The well-known empirical Cole-Davidson expression and its generalization for the CDP were obtained from this theory. This theory is based on self-similar phenomenon and multi-channel organization of relaxation process in disordered dielectrics. The relaxation parameters are connected with the structural parameters of the medium considered.
On some generalizations of the Debye equation for dielectric relaxation
RANA : reports on applied and numerical analysis, 1988
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Journal of Physics: Condensed Matter, 2003
Based on the relationship between the power-law exponent and relaxation time ν(τ) recently established in Ryabov et al (2002 J. Chem. Phys. 116 8610) for non-exponential relaxation in disordered systems and conventional Arrhenius temperature dependence for relaxation time, it becomes possible to derive the empirical Vogel-Fulcher-Tamman (VFT) equation ω p (T) = ω 0 exp[−DT V F /(T − T V F)], connecting the maximum of the loss peak with temperature. The fitting parameters D and T V F of this equation are related accordingly with parameters (ν 0 , τ s τ 0), entering to ν(τ) = ν 0 [ln(τ/τ s)/ ln(τ/τ 0)] and (τ A , E) figuring in the Arrhenius formula τ (T) = τ A exp(E/T). It has been shown that, in order to establish the loss peak VFT dependence, a complex permittivity function should contain at least two relaxation times obeying the Arrhenius formula with two different set of parameters τ A1,A2 and E 1,2. It has been shown that (1) at a certain combination of initial parameters the parameter T V F can be negative or even accept complex valued (2). The temperature dependence of the minimum frequency formed by the two nearest peaks also obeys the VFT equation with another set of fitting parameters. The available experimental data obtained for different substances confirm the validity and specific 'universality' of the VFT equation. It has been shown that the empirical VFT equation is approximate and possible corrections to this equation are found. As a main consequence, which follows from the correct 'reading' of the VFT equation and interpretation of complex permittivity functions with two or more characteristic relaxation times, we suggest a new type of kinetic equation containing non-integer (fractional) integrals and derivatives. We suppose that this kinetic equation describes a wide class of dielectric relaxation phenomena taking place in heterogeneous substances.
Microscopic models for dielectric relaxation in disordered systems
Physical Review E, 2004
It is shown 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 to yield the empirical Havriliak-Negami (HN) equation of anomalous dielectric relaxation from a microscopic model based on a kinetic equation just as in the Debye model. This kinetic equation is obtained by means of a generalization of the noninertial Fokker-Planck equation of conventional Brownian motion (generally known as the Smoluchowski equation) to fractional kinetics governed by the HN relaxation mechanism. For the simple case of noninteracting dipoles it may be solved by Fourier transform techniques to yield the Green function and the complex dielectric susceptibility corresponding to the HN anomalous relaxation mechanism.
Dielectric relaxation of the cowle-cowle type and self-similar relaxation processes
Russian Physics Journal, 1997
In order to describe relaxation processes not obeying an exponential law a model of a self-similar relaxation process is proposed which is described by an equation containing fractional differentiation operators. It turns out that the complex susceptibility corresponding to such a model system has a form which agrees with the known empirical Cowle-Cowle expression.
Non-Debye dielectric relaxation in complex materials
Chemical Physics, 2002
The paper considers several examples of non-Debye dielectric response in complex heterogeneous media. The percolation phenomenon and Cole-Cole relaxation in disordered matter are discussed in detail. The proposed models are illustrated by different sample systems: ionic microemulsions, porous glasses, porous silicon, polymer-water mixtures, and polymer-microcomposite materials. The models enable us to establish the relationship between the parameters of dielectric relaxation broadening, structural properties of the media and transport features of charge carriers in the considered systems. In addition, the origins of ''strange kinetic'' phenomena were discussed based on statistical physics and fractional time evolution ideas.
On generalizations of the Debye equation for dielectric relaxation
Physica A-statistical Mechanics and Its Applications, 1988
In some previous papers one of us (G.A.K.) discussed dielectric relaxation phenomena with the aid of non-equilibrium thermodynamics. In particular the Debye equation for dielectric relaxation in polar liquids was derived. It was also noted that generalizations of the Debye equation may be derived if one assumes that several microscopic phenomena occur which give rise to dielectric relaxation and that the contributions of these microscopic phenomena to the macroscopic polarization may be introduced as vectorial internal degrees of freedom in the entropy. If it is assumed that there are n vectorial internal degrees of freedom an explicit from for the relaxation equation may be derived, provided the developed formalism may be linearized. This relaxation equation has the form of a linear relation among the electric field E, the first n derivatives with respect to time of this field, the polarization vector P and the first n + 1 derivatives with respect to time of P. It is the purpose of the present paper to give full details of the derivations of the above mentioned results. It is also shown in this paper that if a part of the total polarization P is reversible (i.e. if this part does not contribute to the entropy production) the coefficient of the time derivative of order n + 1 of P in the relaxation equation is zero.
Models of dielectric relaxation based on completely monotone functions
Fractional Calculus and Applied Analysis, 2016
The relaxation properties of dielectric materials are described, in the frequency domain, according to one of the several models proposed over the years: Kohlrausch-Williams-Watts, Cole-Cole, Cole-Davidson, Havriliak-Negami (with its modified version) and Excess wing model are among the most famous. Their description in the time domain involves some mathematical functions whose knowledge is of fundamental importance for a full understanding of the models. In this work, we survey the main dielectric models and we illustrate the corresponding time-domain functions. In particular, we stress the attention on the completely monotone character of the relaxation and response functions. We also provide a characterization of the models in terms of differential operators of fractional order.
Dielectric relaxation as a multiplicative stochastic process
Physica A: Statistical Mechanics and its Applications, 1982
A rigorous and general approach is developed to the relaxation of molecular dipoles on the microscopic scale, embodied in the orientational time-autocorrelation function. The usual difficulties of using the stochastic Liouville equation (SLE) are bypassed by replacing the cumulant expansion with a continued fraction. This reduces to that of Sack or Gross in the appropriate limit. The autocorrelation function is formed from approximants of this continued fraction, which is ideally suited for numerical computation, and as a basis for the newly developed technique of semi-stochastic molecular dynamics simulation. The numerical solution automatically produces the spectral moments of interest to order of truncation, so that the number of unknowns is reduced to one at each and every stage of approximation. This concerns the rate of energy dissipation, denoted by fi, a scalar, tensor or super-tensor according to the nature of the diffusion process under consideration. The new continued fraction can be used to describe spatial rotational diffusion of the asymmetric top using the appropriate Fokker-Planck diffusion operator. It is a considerable improvement therefore on a model such as the planar itinerant librator, an approximant of the Mori continued fraction.