Crystalline Phases of Polydisperse Spheres

We use specialized Monte Carlo simulation methods and moment free energy calculations to provide conclusive evidence that dense polydisperse spheres at equilibrium demix into coexisting fcc phases, with more phases appearing as the spread of diameters increases. We manage to track up to four coexisting phases. Each of these is fractionated: it contains a narrower distribution of particle sizes than is present in the system overall. We also demonstrate that, surprisingly, demixing transitions can be nearly continuous, accompanied by fluctuations in local particle size correlated over many lattice spacings.

Figure 1

(a) Simulation results for the partial phase diagram of the model (1) with parent distribution (2). Asterisks: points where new solid phases appear; dashed lines: phase boundary slopes found by histogram reweighting. $F=\mathrm{\text{fluid}}$, $S=\mathrm{\text{solid}}$. Colored (or shaded) symbols: state points considered in Fig. 2. (b) MFE calculation of phase diagram of hard spheres with the same parent form. The dashed line shows a trajectory comparable to that followed by the simulations.

Figure 2

Distribution of the overall number density $p(n)$ at the SS and SSSS state points indicated by the colored (or shaded) symbols in Fig. 1a.

Figure 3

(a) Solid: parent distribution at ($n^0=1.73$, $\delta =13.5\%$). Symbols: Simulation results for the four daughter distributions. The associated fractional volumes ${\lambda }_{\alpha }$ are (left to right) 0.209, 0.188, 0.232, 0.373. (b) MFE results at the comparable state point ($n^0=1.232$, $\delta =8.7\%$); fractional volumes are 0.273, 0.162, 0.200, 0.365.

Figure 4

Correlation volume ${\xi }^3$ in the solid phases encountered along the phase diagram trajectories of Fig. 1. (a) Simulations, (b) MFE calculations.