# Percolation Model for Slow Dynamics in Glass-Forming Materials

Entry: Chia Wei Hsu, AP 225, Fall 2010

G. Lois, J. Blawzdziewicz, and C. S. O'Hern, "Percolation Model for Slow Dynamics in Glass-Forming Materials", Phys. Rev. Lett. **102**, 015702 (2009).

## Summary

In this work, the authors propose an alternate approach to understand the glass transition. Instead of looking at the real space, they focus on the configuration space of the system. There are mobility regions in the configuration space, and the percolation of these regions corresponds to a glass-to-liquid transition. With a mean-field description of such percolation, they show that the stretched-exponential response functions typical of glassy systems can be explained.

## Background

Glassy systems exhibit several unique properties. During a glass transition, the structural relaxation time increases by several orders of magnitude. Also, the structural correlations display an anomalous stretched-exponential time decay: <math>exp(-t/\tau_{\alpha})^{\beta}</math>, where <math>\beta</math> is called the stretching exponent, and <math>\tau_{\alpha}</math> is called the <math>\alpha</math>-relaxation time. Although the stretched-exponential relaxation is universal among glass-forming materials, <math>\beta</math> and <math>\tau_{\alpha}</math> are not. They depend on temperature and density, and they vary from one material to another.

## Basic Idea

The basic idea is that, instead of focusing on the heterogeneous dynamics and percolation in real space (the traditional approach), the authors focus on the *configuration space* and its connection to the anomalous dynamics.

For complete relaxation, the system must be able to diffuse over the whole configuration space. That is, there has to be a path that percolates the configuration space. Thus, we can think of a "percolation transition in the configuration space," which corresponds to the glass transition in real space.

## Hard spheres

Hard spheres interact with infinite repulsion upon contact. At small volume fraction <math>\phi</math>, hard spheres behave like fluids. As <math>\phi</math> increases to <math>\phi_J \approx 0.64</math>, the system becomes collectively jammed. In such state no motion can occur, because any particle displacement will lead to overlap. Therefore only discrete set of points in the configuration space are allowed (fig 1a).

At slightly lower <math>\phi</math>, particles can move around a little bit. Therefore, the discrete points become *mobility domains* in the configuration space (fig 1b). Further decrease of <math>\phi</math> lead to connection between these mobility domains (fig 1c), and eventually to a percolation between these domains (fig 1d) at <math>\phi=\phi_P</math>.

Denote the volume fraction of mobility domains in the configuration by <math>\Pi</math>. Percolation occurs at a critical <math>\Pi_P</math>. When <math>\Pi > \Pi_P</math>, the system can explore the whole configuration space, and it is a metastable liquid. When <math>\Pi < \Pi_P</math>, the system can only diffuse in finite regions of the configuration space, and it is a glass. The distance it can explore is set by the percolation correlation length <math>\xi</math>, which diverges to infinite at <math>\Pi_P</math>.

The authors then describe the percolation with mean-field theory. Following several known scaling laws, they show that the stretching exponent varies with time and satisfies <math>1/3 \leq \beta \leq 1</math>, agreeing with experimental observations.

With a few more assumptions, they further predicts the scaling of <math>\alpha</math>-relaxation time to be

<math>q^2 \tau_{\alpha} \propto \left\{ \begin{array}{ll} \exp\left(\frac{A \phi_J}{\phi_J-\phi}\right)(\phi_P-\phi)^{-2} & \textrm{for } \phi_P-\phi \ll \phi_J-\phi_P\\ \exp\left(\frac{B \phi_J}{\phi_J-\phi}\right) & \textrm{for } \phi_P-\phi \gg \phi_J-\phi_P \end{array} \right. </math>,

where <math>q</math> is the scattering wave number, and <math>A</math>, <math>B</math> are positive constants.

## Finite energy barriers

For systems with a finite energy barrier...

These mobility domains are hyperspherical near the jammed state points. Further away they become filamentary, and the time it takes to explore these regions is large.