Energy Gap and the Ideal Glass as a Defective Crystal: A Lattice Model of Monatomic Systems

We use the cell model to justify the use of a lattice model to study the ideal glass transition. Based on empirical evidence and several previous exact calculations, we hypothesize that there exists an energy gap between the lowest possible energy of a glass (the ideal glass IG) and the crystal (CR). The gap is due to the presence of strongly correlated excitations with respect to the ideal CR; thus, one can treat IG as a highly defective crystal. We argue that an excitation in IG requires energy that increases logarithmically with the size of the system; as a consequence, we prove that IG must emerge at a positive temperature T_{K}. We propose an antiferromagnetic Ising model on a lattice to model liquid-crystal transition in a simple fluid or a binary mixture, which is then solved exactly on a recursive (Husimi) lattice to investigate the ideal glass transition, the nature of defects in the supercooled liquid and CR analytically, and the effects of competing interactions on the glass transition. The calculation establishes the gap. The lattice entropy of the supercooled liquid vanishes at a positive temperature T_{K}>0, where IG emerges but where CR has a positive entropy. The macrostate IG is in a particular and unique disordered microstate at T_{K}, just as the ideal CR is in a perfectly ordered microstate at absolute zero. This explains why it is possible for CR to have a higher entropy at T_{K} than IG. The demonstration here of an entropy crisis in monatomic systems along with previously known results strongly suggests that the entropy crisis first noted by Kauzmann and demonstrated by Gibbs and DiMarzio in long polymers appears to be ubiquitous in all supercooled liquids.