Logarithmic Relaxation in a Colloidal System

TL;DR

This paper investigates the logarithmic relaxation behavior in a colloidal system near a higher-order glass transition singularity, revealing detailed asymptotic solutions and identifying conditions for observable logarithmic decay in dynamics.

Contribution

It provides a detailed mode-coupling theory analysis of logarithmic relaxation in colloids, identifying parameter conditions for this behavior and connecting it to subdiffusive mean-squared displacement.

Findings

Logarithmic decay occurs over large time intervals near critical parameters.

Subdiffusive power-law behavior in mean-squared displacement is observed.

Crossover from concave to convex decay functions as parameters vary.

Abstract

The slow dynamics for a colloidal suspension of particles interacting with a hard-core repulsion complemented by a short-ranged attraction is discussed within the frame of mode-coupling theory for ideal glass transitions for parameter points near a higher-order glass-transition singularity. The solutions of the equations of motion for the density correlation functions are solved for the square-well system in quantitative detail by asymptotic expansion using the distance of the three control parameters packing fraction, attraction strength and attraction range from their critical values as small parameters. For given wave vectors, distinguished surfaces in parameter space are identified where the next-to-leading order contributions for the expansion vanish so that the decay functions exhibit a logarithmic decay over large time intervals. For both coherent and tagged particle dynamics the leading-order logarithmic decay is accessible in the liquid regime for wave vectors of several times the principal peak in the structure factor. The logarithmic decay in the correlation function is manifested in the mean-squared displacement as a subdiffusive power law with an exponent varying sensitively with the control parameters. Shifting parameters through the distinguished surfaces, the correlation functions and the logarithm of the mean-squared displacement considered as functions of the logarithm of the time exhibit a crossover from concave to convex behavior, and a similar scenario is obtained when varying the wave vector.