1301.7019 (J. S. Langer)

Ising Model of a Glass Transition    [PDF]

J. S. Langer

Numerical simulations indicate that several different glass transitions, for polydisperse systems in both two and three dimensions, are characterized by diverging correlation lengths. These correlations are described by Ising-like critical exponents, and are associated with diverging, Vogel-Fulcher-Tamann, structural relaxation times. Related simulations of thermalized hard-disks indicate that the curves of pressure versus packing fraction for different polydispersities exhibit a sequence of transition points, starting with a liquid-hexatic transition for the monodisperse case, and crossing over with increasing polydispersity to glassy, Ising-like critical points. I propose to explain these observations by assuming that glass-forming materials contain twofold degenerate, topologically ordered clusters of particles, similar to the two-state systems that have been invoked to explain other glassy phenomena. This paper starts with a brief statistical derivation of the thermodynamics of thermalized, hard-core particles. It then discusses how a two-state, Ising-like model can be described within that framework in terms of a small number of statistically relevant, internal state variables. The resulting theory is remarkably consistent with the simulation data, but fails near the transition points where Ising symmetry must cross over to rotational symmetry. I also suggest a possible relation between Ising correlation lengths and the Vogel-Fulcher-Tamann formula.

View original: http://arxiv.org/abs/1301.7019