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Analytical Representation of Solutions to Lattice‐Hole Theory
Author(s) -
Utracki Leszek A.,
Simha Robert
Publication year - 2001
Publication title -
macromolecular theory and simulations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.37
H-Index - 56
eISSN - 1521-3919
pISSN - 1022-1344
DOI - 10.1002/1521-3919(20010101)10:1<17::aid-mats17>3.0.co;2-b
Subject(s) - algebraic number , thermodynamics , scaling , algebraic equation , simple (philosophy) , mathematics , lattice (music) , algebraic expression , expression (computer science) , statistical physics , physics , mathematical analysis , quantum mechanics , nonlinear system , acoustics , philosophy , geometry , epistemology , computer science , programming language
The Simha and Somcynsky (S‐S) lattice‐hole theory has been shown to represent accurately the pressure‐volume‐temperature (PVT) surface of chain molecular melts and their mixtures. Proceeding beyond its original intent, it has led to correlations with other properties and extension into the steady state and relaxing glass. The equilibrium results appear as the solutions of two coupled equations, involving the variables of state and the hole fraction, h = h ( V, T ) – a kind of free volume quantity. These are to be solved numerically. Notwithstanding the theory's quantitative success, its implicit form has on some occasions been a practical limitation. We remedy this situation by fitting the scaled and thus general solutions of the coupled equations to accurate algebraic equations, V = V ( T , P ) and h = h V , T ). In this manner, explicit analytical expressions for configurational thermodynamic functions and their derivatives are now available. The new expressions for V and h are simple to employ; the convergence of the non‐linear least‐squares fit is obtained in seconds. The numerical values of the scaling parameters so derived are nearly identical to those computed from the original coupled equations. Having h and V from the original theory, the cohesive energy density [ CED = δ 2 V , T )] was also considered. The results are again well represented by a simple algebraic expression. An expression for the reduced solubility parameter δ = δ ( T , P ) is also given. The usefulness of these solutions is further illustrated by an application to the PVT surfaces of polystyrene and polyphenylene ether blends.