Revisiting the Phase Diagram of Hard Sphere Dumbbells with Nested Sampling: Known Phases and New Packing Variants

TL;DR

This study extends nested sampling (NS) to non-spherical, rigid molecular models by analyzing hard-dumbbell particles across bond lengths $0\le L\le 1$ and a range of pressures. By treating density as the NS driving variable, the authors map the phase diagram and recover known phases (fluid, plastic crystal, CPx close-packed variants, AP) while uncovering a new $Pnma$ packing arrangement and richer CPx variants. The results align with existing EOS and MC findings but also reveal sampling challenges linked to jamming and multi-basin landscapes, particularly for intermediate to large $L$, underscoring NS’s power to discover novel packing patterns without prior phase knowledge. Overall, the work demonstrates NS as a versatile tool for exploring the phase behavior of non-spherical particles and paves the way for applying NS to more realistic molecular systems.

Abstract

We explore the use of the nested sampling technique to sample the configuration space of non-spherical hard particles. We employ the technique on the hard dumbbell system consisting of two hard spheres connected by a rigid bond, and investigate the phase stability across a wide pressure range and for bond lengths from completely overlapping to tangential hard spheres. Nested sampling recovers all previously identified features of the phase diagram and identifies a family of new packing variants. The fluid phase, plastic crystal, close packed solid phases and aperiodic crystal are all sampled, and the transition points between these are mapped. Our results show good agreement with predictions made by existing equations of state, and former Monte Carlo simulations. Nested sampling also identified a close packed structure with Pnma symmetry which has not previously been considered.

Figures (13)

• Figure 1: Shape of dumbbell particles with placing the hard spheres at different distances, $L$, normalised by the diameter of the constituting spheres. $L=0.0$ represents the simple hard-sphere system, while in case of $L=1.0$ the two spheres has only a single point of contact. Colouring is just for clarity.
• Figure 2: Compressibility factor as a function of packing fraction, calculated by the equation of state (solid lines) from Ref. vega_linear_1994 and from nested sampling (solid circles), for three different bond lengths in the fluid phase.
• Figure 3: Compressibility (solid lines) and density curves (dashed lines) plotted for three parallel nested sampling runs at bond length $L=0.2$, as a function of pressure. The peaks in the compressibility curves, and the changes in gradient for the density observed at the same pressure values signal phase transitions in the system (marked by vertical grey dashed lines). Included are results of a MC simulation of the system under a range of relevant pressures, shown by open circles.
• Figure 4: Compressibility (black solid line), average nematic order parameter (blue dots) and average $Q_6$ bond order parameter (red dots) of configurations generated during NS, plotted as a function of pressure, for the system of dumbbells of bond length 0.1 to 0.4$\sigma$. Vertical dashed lines highlight phase transitions.
• Figure 5: Top panel: Compressibility curve calculated by nested sampling for $L=0.2\sigma$. Example solid configurations generated during the sampling are shown for the plastic crystal (dumbbells in ordered position but random orientation) and the ground state structure of a fully ordered static crystal. Spheres are shown smaller than their true size for clarity. Bottom panel: The mean autocovariances of the vector representing the bond over a MC run. The points on the compressibility curve correspond to the pressures at which the MC runs were performed (pressure values are also shown along some of the autocovariance lines.