Ergodicity and slowing down in glass-forming systems with soft potentials: No finite-temperature singularities

The aim of this paper is to discuss some basic notions regarding generic glass-forming systems composed of particles interacting via soft potentials. Excluding explicitly hard-core interaction, we discuss the so-called glass transition in which a supercooled amorphous state is formed, accompanied by a spectacular slowing down of relaxation to equilibrium, when the temperature is changed over a relatively small interval. Using the classical example of a 50-50 binary liquid of $N$ particles with different interaction length scales, we show the following. (i) The system remains ergodic at all temperatures. (ii) The number of topologically distinct configurations can be computed, is temperature independent, and is exponential in $N$. (iii) Any two configurations in phase space can be connected using elementary moves whose number is polynomially bounded in $N$, showing that the graph of configurations has the small world property. (iv) The entropy of the system can be estimated at any temperature (or energy), and there is no Kauzmann crisis at any positive temperature. (v) The mechanism for the super-Arrhenius temperature dependence of the relaxation time is explained, connecting it to an entropic squeeze at the glass transition. (vi) There is no Vogel-Fulcher crisis at any finite temperature $T>0$.

Figure 1

(Color online) Typical correlation functions at different temperatures for the model under discussion. Shown is the shear stress–shear stress correlation function which decays exponentially at high temperatures, and then, at about $T=0.5$, changes character to a stretched exponential form. At temperatures lower than 0.4 the correlations decay to a finite value, which is determined by the mean shear modulus, indicating that the material behaves like a solid for the given measurement time. For further details about these measurements see [20].

Figure 2

(Color online) Upper panel: Voronoi polygon construction in the liquid state 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). Lower panel: Typical Voronoi construction in the glass phase at $T=0.1$. Note the total disappearance of liquidlike defects (large particles in pentagons and small particles in heptagons).

Figure 3

(Color online) T1 process in the Voronoi tessellation (in red lines) and the equivalent flip process in the Delaunay triangulation (dashed curves)

Figure 4

Configuration with four nodes that cannot be flipped, because that would lead to double links. In three dimensions there is more than one corresponding graph, but none of them is the dual of a Voronoi [21].

Figure 5

Christmas tree. Two nodes are at the bottom, the others (only four shown) are in the stem of the tree.

Figure 6

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

Figure 7

Concentrations of all the Voronoi cells as a function of $T$. Note the almost degeneracy of pairs of cells: the upper pair are the glasslike quasispecies, the middle pair the two hexagons, and the lower pair the liquidlike quasispecies. Observe the existence of two typical temperature ranges, one around $T_2$ where the liquidlike quasispecies deplete severely, and the other around $T_1$ where the hexagons deplete rapidly.

Figure 8

(Color online) An example of a stable phase formed by pentagonal and heptagonal quasispecies at very low temperatures.

Figure 9

(Color online) An example of a phase obtained by simulation at low temperatures when hexagons appear in the phase of Fig. 8. Note the tendency of hexagons to clump together, a tendency that is ignored in our approximate estimate of the configurational entropy.