Thermodynamic stability of hard sphere crystals in dimensions 3 through 10

Abstract

Although much is known about the metastable liquid branch of hard spheres—from low dimension d up to ${d\rightarrow \infty }$—its crystal counterpart remains largely unexplored for $d>3$. In particular, it is unclear whether the crystal phase is thermodynamically stable in high dimensions and thus whether a mean-field theory of crystals can ever be exact. In order to determine the stability range of hard sphere crystals, their equation of state is here estimated from numerical simulations, and fluid-crystal coexistence conditions are determined using a generalized Frenkel–Ladd scheme to compute absolute crystal free energies. The results show that the crystal phase is stable at least up to $d={10}$, and the dimensional trends suggest that crystal stability likely persists well beyond that point.

Free energy of a single particle in the periodic reference crystal field

A minimal model of the periodic potential crystal reference calculation is a single particle evolving in the field given by Eq. (18) (see Fig. 5). Because the problem can then be solved analytically, it provides a robust benchmark for our numerical implementation and its optimization.

Fig. 7

Integrand of Eq. (19) for a single particle in the periodic $d=3$ reference crystal from direct integration (black line), direct simulation of the periodic potential (red points), and simulation of the umbrella potential of Eq. (21) (blue points). The different results are indistinguishable below $\lambda \approx 10$, but for $\lambda \gtrsim 10$ results from the umbrella potential deviate from the correct solution due to poor sampling of the bottom of the potential well, wherein the particle is then largely confined

The integrand of Eq. (19) can then be written explicitly as

$$\begin{aligned} \langle U({\mathbf {r}},\lambda ')\rangle _{\lambda '} = \frac{\int U({\mathbf {r}},\lambda ') \exp [-\beta U({\mathbf {r}},\lambda ')]d^d{\mathbf {r}}}{\int \exp [-\beta U({\mathbf {r}},\lambda ')] d^d{\mathbf {r}}}, \end{aligned}$$

(A1)

and thus the system free energy is $\beta F = -\beta \int _0^\lambda \langle U({\mathbf {r}},\lambda ')\rangle _{\lambda '}$. Although the integrand lacks a closed form expression for $d>1$, it can be evaluated numerically with very high accuracy for any $\lambda$ and d. The results for $d=3$ serve as an analytical reference in Fig. 7. Equivalently, the free energy can be computed directly from the standard statistical mechanics expression, $\beta F = -\ln Z$, for the partition function Z, and hence

$$\begin{aligned} \beta F = -\ln \int \exp [-\beta U({\mathbf {r}},\lambda )] d^d {\mathbf {r}}. \end{aligned}$$

(A2)

This expression also does not have a closed form, but can be calculated numerically with high accuracy.

These quantities can be used to benchmark the standard Monte Carlo sampling of the periodic potential as well as by the umbrella sampling scheme described in Sect. 4.3. For $\lambda \gg 1$ standard Monte Carlo sampling leaves the particle trapped at the bottom of one of the wells, but as $\lambda$ decreases the particle regularly explores barriers and crosses over into neighboring wells. Given sufficient sampling, the intermediate $\lambda$ regime is recovered (see Fig. 7), and thus the various features of the direct integration are recapitulated.

This outcome contrasts with Monte Carlo simulations that use the umbrella sampling approach of Eq. (22). In this case, the particle samples all points in the box with equal probability, even though the actual contribution to the partition function of each point is proportional to its Boltzmann weight, $\exp [-\lambda \beta U]$. In the single particle case, both simple Monte Carlo and umbrella sampling work well for $\lambda \lesssim 10$. However, for $N>1$ and $\lambda \sim 1/N$ standard Monte Carlo sampling fails because it then becomes rare for simulations to produce particles which are near the top of energy barriers, whose contributions remain important for accurately calculating the free energy.

For $\lambda \gg 1$, the Boltzmann weight $\exp [-\lambda \beta U]$ concentrates, and the only significant contributions to Z come from a small collection of (degenerate) points in phase space, i.e., the bottom of each well. Both the numerator and the denominator of Eq. (21) diverge in this limit, making the evaluation numerically unstable. Physically, this instability corresponds to $\langle U \rangle$ being vastly undersampled compared with other $\lambda$ regimes, thus leading to marked deviation from the exact result (Fig. 7).

This analysis motivates the high $\lambda$ cutoff used in the umbrella sampling for $N>1$. Because the Boltzmann weight $\exp [-\lambda \beta \sum _i U_i]$ acts on the total energy, the same cutoff of $\lambda =1$ can be used. However, as N grows, it is important to sum these contributions carefully, because $\theta$ then also grows.