  We use polydispersity in size as a control parameter to explore certain
aspects of melting and freezing transitions in a system of Lennard-Jones
spheres. Both analytical theory and computer simulations are employed to
establish a potentially interesting relationship between observed terminal
polydispersity in Lennard-Jones polydisperse spheres and prediction of the same
in the integral equation based theoretical analysis of liquid-solid transition.
As we increase polydispersity, solid becomes inherently unstable because of the
strain built up due to the size disparity. This aspect is studied here by
calculating the inherent structure (IS) calculation. With polydispersity at
constant volume fraction we find initially a sharp rise of the average IS
energy of the crystalline solid until transition polydispersity, followed by a
cross over to a weaker dependence of IS energy on polydispersity in the
amorphous state. This cross over from FCC to amorphous state predicted by IS
analysis agrees remarkably well with the solid-liquid phase diagram (with
extension into the metastable phase) generated by non-linear integral equation
theories of freezing. Two other well-known criteria of freezing/melting
transitions, the Hansen-Verlet rule of freezing and the Lindemann criterion of
melting are both shown to be remarkably in good agreement with the above two
estimates. Together they seem to indicate a small range of metastability in the
liquid-solid transition in polydisperse solids.
