Nested sampling for materials: The case of hard spheres

The recently introduced nested sampling algorithm allows the direct and efficient calculation of the partition function of atomistic systems. We demonstrate its applicability to condensed phase systems with periodic boundary conditions by studying the three-dimensional hard-sphere model. Having obtained the partition function, we show how easy it is to calculate the compressibility and the free energy as functions of the packing fraction and local order, verifying that the transition to crystallinity has a very small barrier, and that the entropic contribution of jammed states to the free energy is negligible for packing fractions above the phase transition. We quantify the previously proposed schematic phase diagram and estimate the extent of the region of jammed states. We find that within our samples, the maximally random jammed configuration is surprisingly disordered.

Figure 1

Schematic plot of an arbitrary order parameter, $\Psi$, of hard sphere configurations as a function of the packing fraction, $\varphi$. The grey area represents the inaccessible region, and the black squares show the maximally random jammed state (MRJ), the jammed state with the lowest possible packing fraction (A) and the perfect crystalline structure (B). (Reprint of Fig. 1 of Ref. [40], with the permission of the authors. Copyright 2000 by The American Physical Society.)

Figure 2

Compressibility factor (top panel) and the compressibility itself (in units of $\beta {\sigma }^3$, bottom panel) as a function of packing fraction from our nested sampling runs, for 32 (red, dash-dotted line), 64 (green, dashed line), and 128 (blue, solid line) hard-sphere particles. Five well-known equations of state for hard spheres are shown for comparison with black curves and different symbols. The vertical dashed lines indicate the packing fraction values of freezing and melting. The error of the compressibility factor is on the order of the line thickness, and for the compressibility the standard errors are shown by colored dotted lines.

Figure 3

Gibbs free energy per particle of a periodic lattice of hard spheres with 128 particles in the unit cell as a function of the $Q_6$ order parameter at different pressure values ($\beta p$ in ${\sigma }^{-3}$ units) calculated by Eq. (18). To show the curves for different pressure values in the same plot, $-p\langle V\rangle$ was added to the free energy. The corresponding packing fraction ($\varphi$) values are also indicated. The bottom three panels show the free-energy curves in a larger scale around the phase transition, and the error is shown by dashed lines.

Figure 4

Order parameter $Q_6$ as a function of the packing fraction for $N=32$, 64, and 128 hard spheres. The black dots represent the configurations generated during the nested sampling runs, and the colored symbols represent the quenched jammed configurations (see text). Panel (d) shows the area occupied by the jammed states of 32 (red), 64 (green), and 128 (blue) using a contour line after a little Gaussian smearing. Perfect fcc and hcp structures are indicated by filled black circles, and the points MRJ (maximally random jammed state found) and $A$ (jammed state with the lowest possible packing fraction found) are also shown in the last panel for the 128-particle system.

Figure 5

Convergence toward jammed states during the quenching process. The difference between the farthest and closest first-neighbor distances (the latter is, by definition, the particle diameter) is calculated within every configuration and plotted against the number of MC steps. The black line is the average of several quenching processes. The gray dashed curves represent the slowest and the fastest process from our sample. Both axes are in the logarithmic scale.

Figure 6

The $Q_6$ order parameter as a function of the packing fraction in the case of an individual run with 64 hard spheres (top panels) and two runs with 128 hard spheres (bottom panels). The red dots represent the configurations generated during the nested sampling runs. The vertical lines indicate the packing fraction levels at which the sample sets were saved and subsequently quenched in order to obtain the jammed configurations. The quenched jammed configurations are shown by symbols with matching colors. For easier identification, at the top of the vertical lines, the corresponding symbols are also shown. In the top right panel, some of the initial configurations and their corresponding jammed structures are connected by green lines to show how the quenching changes the order parameter. The points corresponding to perfect fcc and hcp structures are shown by black circles.

Figure 7

Packing fraction of the quenched jammed configurations as a function of their initial packing fraction for the 32, 64, and 128 hard-sphere systems.