The irreversible approach to equilibrium in glasses

In liquids the molecular configuration is a function of temperature. If in a given substance super-cooling of the liquid is possible to such a low temperature-the 'transformation temperature' (T$_{g}$)-that the viscosity reaches about 10$^{13}$ poises, the required changes in configuration become impossible and at all lower temperatures the structure remains frozen. This causes apparent discontinuities in derivative properties (such as the specific heat) and departures from the Nernst heat theorem. Below T$_{g}$ the substance is said to be in the glassy state. In the neighbourhood of T$_{g}$ relaxation towards the appropriate equilibrium structure can be easily observed. An approximate thermodynamic description of the behaviour can be given by introducing an ordering parameter, z, to represent the configuration. The value of z remains fixed for rapid changes in temperature and pressure. This leads to a number of relations between certain properties of the equilibrium liquid and of the glass. One of these (given by Prigogine & Defay) is $\Delta \kappa \Delta $C$_{p}$ = TV($\Delta \alpha $)$^{2}$, where $\kappa $, C$_{p}$ and $\alpha $ are the compressibility, heat capacity and expansivity, and $\Delta \alpha =\alpha _{\text{liquid}}-\alpha _{\text{glass}}$, etc. New results obtained with glucose and analysis of existing data suggest that the relations are obeyed only very approximately and that the one-parameter model is therefore inadequate. Near equilibrium it is convenient to use-instead of z-a 'fictive temperature', $\overline{T}$, defined so that with fixed p and z the glass would be in equilibrium at T = $\overline{T}$. A fictive pressure, $\overline{p}$, can similarly be defined. To a first approximation, the approach to equilibrium is described by means of a volume viscosity, $\eta _{v}$, such that (at constant p and T) $\frac{1}{V}\frac{\text{d}V}{\text{d}t}=-\frac{p-\overline{p}}{\eta _{v}}$. The relaxation time for observations of the approach to equilibrium is then given by $\tau $ = const. $\eta _{v}\Delta \kappa $, where the constant depends on the experimental arrangement. $\tau $ has been measured for glycerol and glucose near their transformation temperatures and the results used to find $\eta _{v}$ and its dependence on temperature. It was found that: (i) The activation energies of volume viscosity for glycerol (25 kcal/mole) and glucose (130 kcal/mole) are equal to those of the shear viscosity ($\eta $). Thus $\eta _{v}/\eta $ is independent of temperature. (ii) $\eta _{v}/\eta $ is of the order 10 for glycerol and is about 200 for glucose. (iii) dV/dt is proportional to (p - $\overline{p}$) for only very small displacements from equilibrium. It follows that the concept of volume viscosity must be used with caution.