Fragility of liquids using percolation-based transport theories: Correlation between limiting slope of the viscosity and non-exponentiality of relaxation

Transport in super-cooled liquids becomes increasingly heterogeneous with reduction in temperature. This transport is assumed to occur by thermally activated hopping over barriers. At a temperature defined here to be Tc, transport becomes percolative rather than diffusive, and, especially for mechanical relaxation, larger energy barriers must be surmounted in order to establish steady-state response. Use of the same distribution of energy barriers at high and low temperatures, but different theoretical approaches to calculate transport properties, relates expressions for the viscosity at high and low temperatures. Such an approach also allows calculation here of an expression relating the limiting slope, m, of the viscosity to the parameters of an assumed log-normal distribution of energy barriers. Careful calculation of the frequency dependence of transport properties in terms of this distribution of energy barriers also allows expression of the non-exponentiality parameter, β, in terms of the same parameters. Analytical calculations of both (within established approximation schemes) allows quantitative comparison between m and β. The analytical results are evaluated using typical frequencies and times from experiment. Agreement between experiment and calculations is fair. Nevertheless, sensitivity of the calculations to uncertainty in the values of the experimentally derived parameters makes it necessary to view the correspondence with caution. It is tentatively concluded that the major component of the curvature of the viscosity with decreasing temperature is indeed the change in transport type, and not changes in structure. Advances in theory as well as application of numerical methods are needed to help to reduce uncertainty in the conclusions.