Difference between revisions of "Percolation Model for Slow Dynamics in Glass-Forming Materials"

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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.
 
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.
  
== Theory - Hard spheres ==
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== Basic Idea ==
 
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This paper is fairly dense in math and equations. Here we will only give an outline of the arguments made and key results.
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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.
 
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.
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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.
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[[Image:glass_config_fig1.jpg|thumb|300px| Schematic of allowed regions in configuration space for hard spheres. (a) <math>\phi = \phi_J </math>system is jammed; only jammed states (discrete points in the config space) are allowed. (b) ]]
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== Hard spheres ==
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== Finite energy barriers ==

Revision as of 22:31, 23 November 2010

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.

Schematic of allowed regions in configuration space for hard spheres. (a) <math>\phi = \phi_J </math>system is jammed; only jammed states (discrete points in the config space) are allowed. (b)

Hard spheres

Finite energy barriers