Morphometric Approach to Many-Body Correlations in Hard Spheres

We model the thermodynamics of local structures within the hard sphere liquid at arbitrary volume fractions through the morphometric calculation of $n$-body correlations. We calculate absolute free energies of local geometric motifs in excellent quantitative agreement with molecular dynamics simulations across the liquid and supercooled liquid regimes. We find a bimodality in the density library of states where fivefold symmetric structures appear lower in free energy than fourfold symmetric structures and from a single reaction path predict a dynamical barrier which scales linearly in the compressibility factor. The method provides a new route to assess changes in the free energy landscape at volume fractions dynamically inaccessible to conventional techniques.

Figure 1

The system considered showing (a) the local particles surrounded by the remaining liquid acting as a thermal reservoir at fixed chemical potential and temperature, and (b) partition of space into the local $\mathcal{L}$ and remaining $\mathcal{R}$ components with dividing surface $\partial \mathcal{L}$. In this Letter, $\mathcal{L}$ is chosen as the space inaccessible to the center of a test particle (shown faded) representing the remaining liquid.

Figure 2

Static many-body structure in the hard sphere liquid. Left: populations of small local structures in the hard sphere liquid determined from molecular dynamics simulations of 1372 monodisperse (open circles) and 8% polydisperse (solid triangles) hard spheres against the theoretical prediction of this work (lines). Variations against volume fraction $\eta$ and compressibility $Z=\beta p/\rho$ shown. The hard sphere freezing and melting volume fractions are indicated by vertical dashed lines. Right: theoretical free energy distribution for the $n=12$ local library of states at several volume fractions. The distribution is shifted to lower energies at higher volume fractions and develops an increasingly bimodal structure. Populations are decomposed into those structures containing pentagonal bipyramids without octahedra (light fill) and the remaining structures (dark fill).

Figure 3

Reaction for transition between tripyramid and octahedron $n=6$ structures. Stationary points are indicated by markers: there is a discontinuity in free energy at the end points due to the additional integration over the reaction coordinate and symmetry in the case of the octahedron. Inset: variation of activation barrier with volume fraction $\eta$ and compressibility $Z=\beta p/\rho$ from this theoretical reaction path (dashed line) and measured $\alpha$-relaxation times in bulk molecular dynamics simulations (solid line), where $\eta =0.45$ is indicated with a vertical dotted line.