Fig. 2: Thermodynamics of icosahedral and decahedral hard sphere clusters from numerical calculations.

a Free energies calculated for icosahedral model clusters (blue, Mackay and anti-Mackay types), decahedral model clusters (yellow), and simulated clusters (red). Data points are free energy values ((anti-)Mackay and decahedral clusters) and averaged over five free energy values (simulated clusters; with standard errors). Lines are guides to the eyes. Free energies fluctuate with the number of particles and exhibit characteristic minima. The number of particles N is controlled by confinement radius in units of the sphere diameter σ. In all free energy calculations and simulations, packing fraction is fixed at 52%. b, c Occurrence statistics of icosahedral (Ih, b) and decahedral (Dh, c) clusters for 21 confinement radii (solid bars). 192 simulations were performed at each confinement radius. The occurrence of Ih and Dh does not add up to 100% because defected icosahedral clusters with weakly broken icosahedral symmetry¹⁵ are not classified as icosahedral clusters. Point group symmetry is quantified at the end of each simulation run and averaged over all simulations at a given system size independent of cluster symmetry. Open symbols show point group order parameters (see “Methods”). Lines are guides to the eye.