Effects of polydispersity on hard sphere crystals

We use simple models and molecular dynamics simulations to determine the effects of polydispersity δ on the equation of state for hard sphere crystals. Experiments show that the osmotic pressure for poly-(methyl methacrylate) (PMMA) spheres with a poly-(12-hydroxy stearic acid) (PHSA) layer with a 5% polydispersity exceeds the value expected for hard spheres as the volume fraction φ increases, particularly for φ>0.60. Mean field theory predicts a higher osmotic pressure with increasing polydispersity, but the effects are only significant for δ>0.10. Molecular dynamics simulations with δ=0.05 bound the equation of state between a metastable disordered upper limit and a crystalline organized polydisperse (possibly) lower limit. The pressure for the PMMA-PHSA spheres lies close to the organized polydisperse limit, indicating a preference for a crystalline ordered arrangement where smaller particles surround larger ones. Thus, the higher osmotic pressure seen in the equation of state of PMMA-PHSA spheres is a direct effect of polydispersity, manifest as a pronounced reduction in the crystalline close packed volume fraction from φ_max(FCC, δ=0)=0.7404 to φ_max(FCC, δ=0.1)=0.665. The random close packing φ_max(RCP) is almost independent of polydispersity. This leads to a crossing of values of φ_max(FCC) and φ_max(RCP) and hence a possible terminal polydispersity of 0.12±0.01, consistent with other simulations, theories, and experiments. Since our results do not include size fractionation of the liquid and solid, the exact meaning of this crossing is unclear and its agreement with previously reported terminal polydispersities may be coincidental.