Memory kernel formalism with fractional exponents and its application to dielectric relaxation

Abstract

A fractal Fokker–Planck formalism applied to the dielectric relaxation in glass forming liquids is proposed. This formalism is a modality of the generalized equation of Langevin on the use of fractional-time derivatives simultaneously with the memory function to describe the dynamics of the dipolar-moment autocorrelation function. The goal is to get the description of the complex autocorrelation function numerically, and the real and imaginary parts of the second-order memory function, related to the kernel of the integral hierarchy representation of this autocorrelation function. The results exhibit the memory effect associates with α-dielectric relaxation mode. From the analysis, it is shown the existence of a maximum and the appropriated frequency limit in the imaginary and real parts, respectively, of the second-order memory function. That is required to describe experimental well the complex shear viscosity of the material.The theoretical model was tested with experimental data from dielectric spectroscopy of biphenyl-2-yl isobutyrate (OBPI), a glass forming liquid, in measurements in the frequency domain from 10−2 to 106 Hz. This ester has been selected in this study for its predominant α-relaxation when compared to the β relaxation. The real and imaginary parts of the second-order memory function were evaluated by means of the proposed model and the aid of the parameters of the Havriliak–Negami relationship.