Ising model of a glass transition

Numerical simulations by Tanaka and co-workers indicate that glass-forming systems of moderately polydisperse hard-core particles, in both two and three dimensions, exhibit 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 fluids contain twofold degenerate, locally 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 agrees accurately with the simulation data. I also propose a rationale for the observed relation between the Ising-like correlation lengths and the Vogel-Fulcher-Tamann formula.

Figure 1

Pressure as a function of packing fraction for polydispersties $\Delta =9\%$ (red open squares) and $13\%$ (blue open triangles). The critical points are indicated by the corresponding solid symbols. The blue theoretical curve shows the $J=0$ limit according to Eqs. (6.3) and (6.4). The pressure is expressed in dimensionless units defined in [2].

Figure 2

Pressure as a function of packing fraction, from bottom to top, for polydispersties $\Delta =0\%$ (red circles), $5\%$ (green triangles), $7\%$ (black circles), and $13\%$ (blue triangles). The corresponding solid curves indicate the theory described in the text.

Figure 3

Dimensionless coupling coefficient $J$ as a function of percentage polydispersity $\Delta .$