A Simple Mean-Field Description of the Viscosity of Glass-Forming Polymers

The low-temperature behavior of glass-forming liquids is still poorly understood. Here, we propose a simple model with no divergence of relaxation time at any finite temperature. We hypothesize a “key degree of freedom” (KDF) associated with the Arrhenius-Andrade-Eyring transition state mechanism of the liquid flow. We then assume that the KDF energy levels are quantized in increments of ε (where ε can be an energy of a gauche’-trans-gauche kink¹ or another low-energy excitation). In that case, the “fictive temperature”, T_f, is proportional to the KDF internal energy, determined by the Planck-Bose-Einstein statistics, T_f = <u >/k_B = (ε/k_B)(exp(βε) – 1)^-1, where β = ( k_BT)^-1. The temperature-dependent viscosity (or the shift factor) is then estimated in the standard fashion, ln(η) = const + (E_a/ k_BT_f) = const + (E_a/ε)(exp(βε) – 1) (where E_a is the Arrhenius activation energy). This expression is similar to the Mauro² equation and is shown to agree very well with multiple experimental data sets for both polymers and organic liquids. The theory can be further extended to describe the behavior of random copolymers and miscible blends.
¹M. Mansfield and R. H. Boyd, Journal of Polymer Science: Polymer Physics Edition, 16, 1227 (1979); ²J. C. Mauro et al., PNAS 106, 19780 (2009).