Anisotropic spatially heterogeneous dynamics on the $\alpha$ and $\beta$ relaxation time scales studied via a four-point correlation function

TL;DR

This study investigates the anisotropic nature of four-point correlation functions in glass-forming systems through simulations, revealing strong anisotropy at specific time scales and implications for measuring dynamic correlation lengths.

Contribution

It provides the first detailed analysis of anisotropy in four-point correlation functions and introduces methods to extract anisotropic dynamic correlation lengths from simulation data.

Findings

Anisotropy peaks at nearest neighbor distances during the non-Gaussian parameter peak.

Structure factor remains anisotropic at the smallest wave vectors accessible.

Dynamic correlation length extraction is affected by anisotropy, requiring consideration of directional dependence.

Abstract

We examine the anisotropy of a four-point correlation function $G_4(\vec{k},\vec{r};t)$ and it's associated structure factor $S_4(\vec{k},\vec{q};t)$ calculated using Brownian Dynamics computer simulations of a model glass forming system. These correlation functions measure the spatial correlations of the relaxation of different particles, and we examine the time and temperature dependence of the anisotropy. We find that the anisotropy is strongest at nearest neighbor distances at time scales corresponding to the peak of the non-Gaussian parameter $\alpha_2(t) = 3 < \delta r^4(t) >/[ 5 < \delta r^2(t) >^2] - 1$, but is still pronounced around the $\alpha$ relaxation time. We find that the structure factor $S_4(\vec{k},\vec{q};t)$ is anisotropic even for the smallest wave vector accessible in our simulation suggesting that our system (and other systems commonly used in computer simulations) may be too small to extract the $\vec{q} \to 0$ limit of the structure factor. We find that the determination of a dynamic correlation length from $S_4(\vec{k},\vec{q};t)$ is influenced by the anisotropy. We extract an effective anisotropic dynamic correlation length from the small $q$ behavior of $S_4(\vec{k},\vec{q};t)$.