Introduction

Abstract

By now the study of ion transport and conductivity in ionically conducting materials is a subject of interest to physicists, chemists, materials scientists, and engineers. The interest of physicists is to understand the complex motion of ions leading to steady state diffusion and conductivity. This is a challenging endeavor because of the large number of ions in most ionic conductors of practical interest. The motions of the ions are not independent due to mutual ion interactions as well as interactions with the matrix ions. For this reason the problem of diffusion and electrical conductivity of interacting many-ion system has remained unsolved up to the present time. Historically it was Michael Faraday who discovered ionic transport in electrolytes in the years after 1830 [1]. Following Faraday, it was Kohlrausch [2] in Göttingen, Germany who made the first measurement of electrical relaxation of alkali ions in the Leyden jar (a glass) in 1854. For experimental data, see [3]. He found the relaxation has time-dependence given by

$$ {\phi}_K(t)= \exp \left[-{\left(t/\tau \right)}^{1-n}\right],\ \mathrm{where}\ 0<\left(1-n\right)\le 1, $$

the stretched exponential function, or the Kohlrausch decay function, which continues to be relevant in conductivity relaxation of ionic conductors, structural relaxation of glass-forming liquids, and other research areas. By the way, the stretched exponential function (1.1) was found to describe well the mechanical relaxation in the natural polymer, silk, in 1863 and 1866 by F. Kohlrausch [4, 5], the son of R. Kohlrausch. Nowadays, the function is known to fit well the structural relaxation of glass-forming materials and systems in general. However, since the times Faraday or Kohlrausch started the field, 180 years have gone by and remarkably the problem has not been solved in the physics world. Surprisingly few theoretical attempts have been made in the past years to solve the problem.