Thermal Activation Relations

Abstract

A material is considered which has one or more thermally activated linear response functions that obey Boltzmann statistics to good approximation. These response functions may, for example, be mechanical or dielectric relaxation times, diffusion coefficient, viscosity, etc. Neither the entropy change, ΔS, nor the enthalpy change, ΔH, in the activation process need be temperature-independent. An expression for ΔG, the Gibbs free-energy change in the thermally activated process, is considered which applies to a system with a nonzero or zero glass transition temperature, Tg, and allows the product c1gc2g of the WLF polymer equation to be interpreted and its variation with material explained by an activation energy model. Explicit expressions for the resulting temperature-dependent ΔS and ΔH are given which involve the material parameter E; this is the enthalpy and activation energy of the process only when Tg=0 and thus an Arrhenius equation applies. The results pertain to mechanical and dielectric dispersion experiments on amorphous polymers and other glass-forming materials. In addition, they apply to such processes as diffusion by a vacancy mechanism, to viscous flow, and, when Tg=0, to intrinsic semiconduction and many other processes. It is found that the Lawson—Keyes relation, ΔS/ΔH≃4α, should not be applied when Tg≠0 and may be very inaccurate even in the case where Tg=0. An improved relation between ΔS and ΔH which holds for Tg=0 is presented; it furnishes a possible explanation for the negative value of ΔS sometimes found experimentally. An expression for ΔG correct to first order in pressure and temperature is given which applies to all situations where an Arrhenius equation is found. On identifying the semiconductor energy gap for thermal activation as a Gibbs free energy difference, the results are illustrated by analyzing pressure and temperature data pertaining to intrinsic semiconduction in Si and Ge. Experimental dielectric dispersion results for isoamyl bromide (Tg≠0) are also analyzed and compared with results of previous work. Finally, temperature-dependent aspects of the transient and frequency response of a distributed, linear activated system with Tg≠0 are examined when either the pre-exponential factor or the activation parameter E is distributed, or when they are simultaneously distributed and are linearly related.