Past exam of the MPhil in micro- and nanotechnology enterprise course of the University of Cambridge / 2023 / Paper 2 / Section B / 23 / iii

For ideal hard spheres, potential energy is zero for nonoverlapping configurations and infinite for overlaps. At fixed temperature and volume, the accessible fluid and crystal configurations therefore have no energetic preference of the ordinary bonding kind: the phase with larger entropy has lower Helmholtz free energy.

Ordering particle centers costs some configurational freedom, but at a high packing fraction it can give each particle more usable cage/free volume than a disordered crowded arrangement. The total number of allowed vibrational and positional configurations can then be greater in a crystal despite its visible long-range order. This is entropy-driven hard-sphere crystallization.

The crystal need not be at close packing. At the stated volume fraction near $0.55$, substantial free space remains for thermal motion. Close packing near $0.74$ is the geometric upper bound for equal spheres, not the fraction at which the crystal first becomes thermodynamically favorable. Treating crystallization as an inevitable decrease in total entropy would confuse visible positional order with the full phase-space count.