Statistical mechanics of the glass transition as revealed by a Voronoi tesselation

The statistical mechanics of simple glass forming systems in two dimensions is worked out. The glass disorder is encoded via a Voronoi tesselation, and the statistical mechanics is performed directly in this encoding. The theory provides, without free parameters, an explanation of the glass transition phenomenology, including the identification of two different temperatures, $T_g$ and $T_c$, the first associated with jamming and the second associated with crystallization at very low temperatures.

Figure 1

(Color online). Upper panel: a Voronoi polygon construction at $T=3$, with the seven-color code used in this paper. Small particles in pentagons (heptagons) are light green (dark green) and large particles in pentagons (heptagons) are violet (pink). Hexaons are in blue, octagons in black, and squares in yellow. Lower panel: a typical Voronoi construction in the glass phase at $T=0.1$. Note the total disappearance of liquidlike defects.

Figure 2

(Color online). Concentration of defects under slow cooling. Upper panel: the concentration of the liquidlike defects: large particles in pentagons (red dots) and small particles in heptagons (blue dots). Lower panel: the fit of the concentration of liquidlike defects as a function of temperature according to Eq. (4).

Figure 3

(Color online). The average energies of the Voronoi cells as a function of the temperature as measured in the simulations.

Figure 4

(Color online). Upper panel: the concentration of the liquidlike defect (small in heptagon) as a function of temperature, together with the quadratic fit of the type Eq. (4). Lower panel: the concentrations of all the Voronoi cells as a function of $T$. Note the close-to-degeneracy of pairs of cells: the upper pair is the glasslike defects, the middle pair is the two hexagons, and the lower pair the liquidlike defects. Observe the existence of two temperatures, interpreted as $T_g$ and $T_c$.

Figure 5

(Color online). An example of a stable phase formed by glasslike defects at very low temperatures.