Ionic transport in mixed-alkali glasses: hop through the distinctly different conduction pathways of low dimensionality

Abstract

A feature common to various solids, including single-alkali (SA) and mixed-alkali (MA) glasses, is a frequency-dependent ionic conductivity that shows the power law and the linear behavior with frequency. In spite of the advances made, the origin of this behavior continues to be controversial. We report our measurements of the conductivity of a series of MA borate glasses (Li1−xAx)2B4O7 (A = Na, K, Rb, Cs; 0 ⩽ x ⩽ 1.0) in the frequency range of 100 Hz–15 MHz and in the temperature range from 300 K to less than the glass transition temperature Tg . Using a self-similar spatial structure model, we show that the real process of ionic transport in the SA and the MA glass systems can be described by the fractional kinetic equations containing non-integer integration/differentiation operators. In the procedure of a systematic deduction of the ionic transport in glass systems, we obtained two important insights. Firstly, the time-dependent conductivity σ(t) ∼ exp (t/τc)α reproduces the empirical expression of mean square displacement of the mobile ions 〈r2 (t)〉 ∼tα as a first approximation of ions moving through the fractal pathway and leads to the universal power-law behavior at frequency scales. Secondly, the modified fractional Rayleigh equation with a repulsive interaction provides a quantitative explanation for experimental findings on the SA and the MA glasses. Investigations on the power-law exponent β in the SA and the MA borate glasses indicate that the ions move through the different branches of the fractal structured conduction pathways due to the structural character, associated with a site mismatch effect, and Coulomb blockade by the randomly distributed ions.