Entropy crisis, ideal glass transition, and polymer melting: exact solution on a Husimi cactus.

We investigate an extension of the lattice model of melting of semiflexible polymers originally proposed by Flory. Along with a bending penalty epsilon, present in the original model and involving three sites of the lattice, we introduce an interaction energy epsilon (p), corresponding to the presence of a pair of parallel bonds and an interaction energy epsilon (h), associated with a hairpin turn. Both these new terms represent four-site interactions. The model is solved exactly on a Husimi cactus, which approximates a square lattice. We study the phase diagram of the system as a function of the energies. For a proper choice of the interaction energies, the model exhibits a first-order melting transition between a liquid and a crystalline phase at a temperature T(M). The continuation of the liquid phase below T(M) gives rise to a supercooled liquid, which turns continuously into a new low-temperature phase, called metastable liquid, at T(MC)<T(M). This liquid-liquid transition seems to have some features that are characteristic of the critical transition predicted by the mode-coupling theory. The exact calculation provides a thermodynamic justification for the entropy crisis (entropy becoming negative), generally known as the Kauzmann paradox, caused by the rapid drop of the entropy near the Kauzmann temperature. It occurs not in the supercooled liquid, but in the metastable liquid phase since its Helmholtz free energy equals the absolute zero equilibrium free energy at a positive temperature. A continuous ideal glass transition occurs to avoid the crisis when the metastable liquid entropy, and not the excess entropy, goes to zero. The melting transition in the original Flory model, corresponding to the vanishing of the four-site interactions, appears as a tricritical point of the model.