Analytical representation of solutions to lattice-hole theory

The Simha and Somcynsky (S-S) lattice-hole theory has been shown to represe nt accurately the pressure-volume-temperature (PVT) surface of chain molecu lar melts and their mixtures. Proceeding beyond its original intent, it has led to correlations with other properties and extension into the steady st ate and relaxing glass. The equilibrium results appear as the solutions of two coupled equations, involving the variables of state and the hole fracti on, h =h(V,T) - a kind of free volume quantity. These are to be solved nume rically. Notwithstanding the theory's quantitative success, its implicit fo rm has on some occasions been a practical limitation, We remedy this situat ion by fitting the scaled and thus general solutions of the coupled equatio ns to accurate algebraic equations, (V) over tilde = (V) over tilde((T) ove r tilde, (P) over tilde) and h = h((V) over tilde, (T) over tilde). In this manner, explicit analytical expressions for configurational thermodynamic functions and their derivatives are now available. The new expressions for (V) over tilde and h are simple to employ; the convergence of the non-linea r least-squares fit is obtained in seconds. The numerical values of the sca ling parameters so derived are nearly identical to those computed from the original coupled equations. Having h and V from the original theory, the co hesive energy density [CED=delta (2)((V) over tilde, (T) over tilde)] was a lso considered. The results are again well represented by a simple algebrai c expression. An expression for the reduced solubility parameter <(<delta>) over tilde> = delta ((T) over tilde, (P) over tilde) is also given. The use fulness of these solutions is further illustrated by an application to the PVT surfaces of polystyrene and polyphenylene ether blends.