Application of the Adam−Gibbs Equation to the Non-Equilibrium Glassy State

Abstract

The Tool−Narayanaswamy−Moynihan (TNM) equation for the temperature (T) and fictive temperature (Tf) dependence of the relaxation time in glassy materials is compared with the usual nonlinear form of the Adam−Gibbs (AG) equation. It is shown that the relationship derived between the Narayanaswamy parameter x and the temperature T2 at which the configurational entropy reduces to zero, namely x ≈ 1 − T2/Tf, leads to unrealistic values of T2 for many polymer glasses. This problem is resolved by expressing the configurational entropy as a function of both T and Tf, with a partitioning parameter xs (0 ≤ xs ≤ 1) controlling their respective contributions. Comparing TNM with this new nonlinear AG expression incorporating Sc(T,Tf) leads to an explicit relationship between x and xs involving T, T2, and Tf, from which a number of predictions may be made. (1) For T ≈ Tf, i.e., for relaxations close to equilibrium, the quantity 1 − T2/Tf is identified as the minimum possible value for x, implying that T2 ≥ Tf (1 − x), by an amount depending on the value of xs. This resolves the apparently anomalous values of T2. (2) For relaxations further from equilibrium, the TNM equation with constant x becomes increasingly inappropriate. (3) With increasing annealing temperature and increasing annealing time, the analysis predicts increasing values of x, as has often been reported in the literature. The origin of the dependence of Sc on T and Tf is considered from the theory of Gibbs and DiMarzio, and it is argued that typical values of x observed experimentally may be associated with the freezing-in of only a certain fraction of either flexed bonds and/or vacant lattice sites (holes) at the glass transition. Thus, it is possible to identify xs, and indirectly x, with the relative contributions of physically meaningful parameters, such as intermolecular and intramolecular bond energies, to the freezing-in process.