A Relation Between Fragility, Specific Heat, and Shear Modulus Temperature Dependence for Simple Supercooled Liquids

Abstract

It is often said that the fragility and specific heat discontinuity at the glass transition are related characteristic features of glasses. We give a relation between these quantities for simple liquids, and also find that the temperature dependence of the shear modulus should be closely related to these quantities. According to the Interstitialcy Theory, the liquid state shear modulus G = G_o, exp (-βc) = G_o exp {-γ[(T/T_o)-l ]}, where G_ois the shear modulus at a reference temperature T_o, which can be taken to be the glass temperature. In these relations, βis the diaelastic shear susceptibility, c is the interstitialcy concentration, T is the temperature, and dc/dT = γ/βT_o. It has been proposed by Dyre, Olsen and Christensen that U in the viscosity η = η_o exp (U/kT) be given as the work done in shoving aside particles during a diffusion step, and therefore be proportional to the (liquid state) shear modulus. If so, then combining the above relations, the fragility F = [d log η/d (T_g/T]T=T_g becomes F=(1+γ) log [η (T_g)/η_o], where γ = - (Tg/G)(dG/dT). Then, since δC_v, = U_F(dc/dT), where U_F is the interstitialcy formation energy, one obtains further with reasonable simplifying approximations δC_v/C_v~γG(T)/G_oo, where G_oo is the zero temperature crystalline shear modulus. For a typical fragile glass γ ~ 2 (F ~ 50) with G(T_g)/G_oo ~ 1/2, then δC_v/C_v, ~ 1, ut close to zero for strong glasses, in fair agreement with available experimental results. The perspective given by the Interstititialcy Theory is then that the fragility is given phenomenologically by the temperature dependence of the shear modulus, or microscopically by the rate of increase of the equilibrium interstitialcy concentration with temperature.