Lattice-switch Monte Carlo simulation for binary hard-sphere crystals

We show how to generalize the lattice-switch Monte Carlo method to calculate the phase diagram of a binary system. A global coordinate transformation is combined with a modification of particle diameters, enabling the multicomponent system in question to be explored and directly compared to a suitable reference state in a single Monte Carlo simulation. We use the method to evaluate the free energies of binary hard-sphere crystals. Calculations at moderate size ratios $\alpha =0.58$ and 0.73 are in agreement with previous results, and confirm $AB2$ and $AB13$ as stable structures. We also find that the $AB\left(\mathrm{Cs}\mathrm{Cl}\right)$ structure is not entropically stable at the size ratio and volume where it has been reported experimentally, and therefore that those observations cannot be explained by packing effects alone.

Figure 1

(Color online) The three binary structures considered in this work, with the $A$ particles shown as pale spheres, $B$ particles shown as darker spheres, and the simulation cell shown as a translucent box. (a) The $AB13$ structure consists of a simple cubic lattice of large spheres with icosahedral clusters of 13 $B$ particles in the body center of each cell. The 112-particle unit cell contains eight $B$ clusters arranged in alternating orientations, where the clusters are rotated by 90° between adjacent subcells. (b) In $AB2$, hexagonal close-packed layers of $A$ particles are stacked directly on top of each other, with a honeycomb pattern of $B$ particles places in the interstitial sites between the $A$ layers. (c) The $AB\left(\mathrm{Cs}\mathrm{Cl}\right)$ structure, consisting of two interlaced simple cubic lattices, with the $B$ particles at the body center of the $A$ cube.

Figure 2

Equations of state determined in the constant pressure ensemble, measuring the density as a function of the pressure. Errors in the density estimates are smaller than the symbol size. The Alder equation of state for the pure crystal [43] is shown as a solid line, and the equation of state for $AB13$ as observed in Ref. [7] is shown as a solid gray line.

Figure 3

Observed $c∕a$ ratios for the target phases.

Figure 4

Raw Helmholtz free energy difference per particle between the reference fcc and target $AB2$ and $AB13$ crystals at $\alpha =0.58$, and $AB\left(\mathrm{Cs}\mathrm{Cl}\right)$ at $\alpha =0.73$, as measured by the lattice-switch simulations. The final data points at low $\varphi$ indicate the lowest density at which the structure was determined to be stable. The points at the high-density end indicate the upper limit of the chosen pressure range explored by the simulations. Some simulations were performed at larger system sizes, as shown here, and indicated that the overall form of the free energy difference does not depend strongly on the system size.

Figure 5

Gibbs free energies as a function of pressure for $AB\left(\mathrm{Cs}\mathrm{Cl}\right)$ $\alpha =0.72$ at a fixed composition of $X=0.5$. The Gibbs free energies of the Mansoori fluid, the pure $A$ fcc crystal coexisting with a Mansoori fluid, and the dual fcc $A$ and $B$ crystal phases are also shown. Finite-size effects cannot be resolved on this plot.

Figure 6

Excess (i.e., with respect to an ideal gas) Helmholtz free energies of $AB2$, $AB13$, and fcc at structures $\alpha =0.58$. Results from the literature for $AB2$ [6] and $AB13$ [7] are also shown for comparison.

Figure 7

Gibbs free energies for $AB2$ and $AB13$ at $\alpha =0.58$. The Gibbs free energies of the Mansoori fluid and of the $A$ fcc crystal coexisting with a $B$-rich Mansoori fluid (at the appropriate compositions) are also shown. Statistical errors and finite-size effects cannot be resolved on this plot.

Figure 8

$P\text{−}X$ phase diagram at $\alpha =0.58$, determined from the lattice-switch results using the methodology described in Sec. 3B.

Figure 9

Partial volume fraction phase diagram, built from the $P\text{−}X$ results shown in Fig. 8 using the equation of state data to determine the densities of coexisting phases along each $P\text{−}X$ coexistence line.