Lattice Model for Colloidal Gels and Glasses

We study a lattice model of attractive colloids. It is exactly solvable on sparse random graphs. As the pressure and temperature are varied, it reproduces many characteristic phenomena of liquids, glasses, and colloidal systems such as ideal gel formation, liquid-glass phase coexistence, jamming, or the reentrance of the glass transition.

Figure 1

High-temperature behavior of the model with $c=3$, $l=1$ and no entropic penalty. ${\rho }_{\mathrm{RCP}}-\rho$ is the difference between the packing fraction, or density, and its maximum achievable value (random close packing); $\mu$ is the chemical potential. The static equilibrium solution (bold/red) undergoes a phase transition at ${\rho }_K=0.5725$ and converges to the densest packing ${\rho }_{\mathrm{RCP}}=0.57574$ at $\mu =\infty$. However, the dynamics falls out of equilibrium at ${\rho }_d=0.5708$ and the packing fraction following a slow annealing can be increased up to at most $0.5731\le {\rho }_J\le 0.5748$. The dotted/blue curves are Monte Carlo annealings for $N={10}^5$ with rates $d\mu ={10}^{-4}$, ${10}^{-5}$, ${10}^{-6}$, and ${10}^{-7}$ (from top to bottom) and the (dashed/black) line ${\rho }_{\mathrm{Dyn}}$ is the isocomplexity approximation for the infinitely slow annealing (see the text). The inset shows the same plot on a logarithmic scale.

Figure 2

Three representative phase diagrams of the model in the temperature ($T$)—packing fraction ($\rho$) plane. The black/continuous curves are the colloid-rich–colloid-poor coexistence lines $T_{\mathrm{LL}}$, and the corresponding spinodal lines are black/dashed curves. The orange/dashed-dotted curves correspond to the percolation of physical clusters, and intercept the critical point of the phase separation. The dynamical $T_d(\rho )$ (blue/dotted curve) and the Kauzmann $T_K(\rho )$ (red/continuous curve) glass transition lines are also shown. The black filled area is beyond the random close packing fraction ${\rho }_{\mathrm{RCP}}$. Left panel: $c=10$, $l=3$ and no entropic penalty. An ideal gel appears at low enough temperature ($T\lesssim 0.1$) and low volume fraction ($0.1\lesssim \rho \lesssim 0.4$). Central panel: $c=6$, $l=4$ and no entropic penalty. A phase coexistence between a colloid-poor and an attractive colloid-rich glass phase occurs for low temperature ($T\approx 0.05$) resulting in an arrested phase separation. Right panel: $c=4$, $l=2$, and $s_p=2$. The glass transition lines $T_d(\rho )$ and $T_K(\rho )$ exhibit a reentrant shape (see inset), resulting in a glass-liquid-glass transition along isochores at densities around $\rho \approx 0.62$.