Analytical Representation of Solutions to Lattice-Hole Theory

Abstract

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 = hV, 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 = δ2V, 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.

Temperature and Pressure Dependence of the Free Volume in Polyisobutylene from Positron Lifetime and Pressure‐Volume‐Temperature Experiments

“…The analysis of pressure-volume-temperature (PVT) experiments employing the Simha-Somcynsky equation of state (SS EOS [15][16][17] ) can provide reasonable values for V f and V occ and has recently received renewed attention. [17][18][19][20][21][22][23][24] Only limited information about the local structures of the free volume, the hole size and shape distribution, and the hole density is available. During the past decade positron annihilation lifetime spectroscopy (PALS) has developed to be the most important method for studying subnanometersize holes in polymers.…”

“…[17] It is assumed that each mer which occupies a cell of the SS lattice has one external degree of freedom and, in the case of polymers, that the macromolecules are of infinite length. This assumption determines the molecular weight of a SS mer, Figure 1 shows isobars of the specific volume V for selected pressures, the fits of Equation (2) to these data and the calculated occupied volume V occ ¼ yV.…”

“…The SS lattice-hole theory [15][16][17] describes a liquid as a periodic lattice of cells of equal size each having z ¼ 12 nearest neighbors. The disorder is modeled by assuming a statistical mixture of occupied (fraction y) and unoccupied cells (holes or vacancies, fraction h ¼ 1 À y).…”

“…Although v SS is smaller than the mean volume of an isobutylene unit, v m ¼ 102 Å 3 , it is larger than found for other polymers, for which v SS /v m % ½. [15][16][17][21][22][23][24] This large volume of the SS lattice cell correlates with the strong character of the PIB chains mentioned in the introduction.…”

“…polymer chain dynamics are therefore related to a hole free volume, V f ¼ V fh ¼ V À V occ (where the occupied volume V occ has to be defined) rather than to the total free volume V ft ¼ V À V W (V W ¼ van der Waals volume). The analysis of pressure-volume-temperature (PVT) experiments employing the Simha-Somcynsky equation of state (SS EOS [15][16][17] ) can provide reasonable values for V f and V occ and has recently received renewed attention. [17][18][19][20][21][22][23][24] Only limited information about the local structures of the free volume, the hole size and shape distribution, and the hole density is available.…”