Logarithmic Relaxation in Glass-Forming Systems

TL;DR

This paper analyzes the asymptotic behavior of correlation functions near higher-order glass-transition singularities within mode-coupling theory, revealing a logarithmic decay law that explains various relaxation phenomena in glass-forming systems.

Contribution

It introduces a polynomial expansion of correlation functions in logarithmic time, providing a unified description of different relaxation regimes near higher-order singularities.

Findings

Logarithmic decay law dominates structural relaxation.

The theory explains the stretching of alpha-relaxation.

Susceptibility spectra can exhibit two peaks.

Abstract

Within the mode-coupling theory for ideal glass transitions, an analysis for the correlation functions of glass-forming systems for states near higher-order glass-transition singularities is presented. It is shown that the solutions of the equations of motion can be asymptotically expanded in polynomials of the logarithm of time t. In leading order, an ln(t)-law is obtained, and the leading corrections are given by a fourth-order polynomial. The correlators interpolate between three scenarios. First, there are planes in parameter space where the dominant corrections to the ln(t)-law vanish, so that the logarithmic decay governs the structural relaxation process. Second, the dynamics due to the higher-order singularity can describe the initial and intermediate part of the alpha-process thereby reducing the range of validity of von Schweidler's law and leading to strong alpha-relaxation stretching. Third, the ln(t)-law can replace the critical decay law of the beta-process leading to a particularly large crossover interval between the end of the transient and the beginning of the alpha-process. This may lead to susceptibility spectra below the band of microscopic excitations exhibiting two peaks. Typical results of the theory are demonstrated for models dealing with one and two correlation functions.