  The equilibrium phase behaviour of hard spheres with size polydispersity is
studied theoretically. We solve numerically the exact phase equilibrium
equations that result from accurate free energy expressions for the fluid and
solid phases, while accounting fully for size fractionation between coexisting
phases. Fluids up to the largest polydispersities that we can study (around
14%) can phase separate by splitting off a solid with a much narrower size
distribution. This shows that experimentally observed terminal polydispersities
above which phase separation no longer occurs must be due to non-equilibrium
effects. We find no evidence of re-entrant melting; instead, sufficiently
compressed solids phase separate into two or more solid phases. Under
appropriate conditions, coexistence of multiple solids with a fluid phase is
also predicted. The solids have smaller polydispersities than the parent phase
as expected, while the reverse is true for the fluid phase, which contains
predominantly smaller particles but also residual amounts of the larger ones.
The properties of the coexisting phases are studied in detail; mean diameter,
polydispersity and volume fraction of the phases all reveal marked
fractionation. We also propose a method for constructing quantities that
optimally distinguish between the coexisting phases, using Principal Component
Analysis in the space of density distributions. We conclude by comparing our
predictions to perturbative theories for near-monodisperse systems and to Monte
Carlo simulations at imposed chemical potential distribution, and find
excellent agreement.
