A relation between fragility, specific heat, and shear modulus temperature dependence for simple supercooled liquids

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(0) exp (-beta c) = G(0) exp {-gamma[(T/T-0)-1]}, where G(0) is the shear modulus at a reference temperature T-0, which can be taken to be the glass temperature. In these relations, beta is the diaelastic shear susceptibility, c is the interstitialcy concentration, T is the temperature, and dc/dT = gamma/beta T-0. It has been proposed by Dyre, Olsen and Christensen that U in the viscosity eta = eta(0) 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 eta/d (T-g/T](T=Tg) becomes F=(1+gamma) log [eta (T-g)/eta(0)], where gamma = -(Tg/G)(dG/dT). Then, since delta C-v = U-F(dc/dT), where U-F is the interstitialcy formation energy, one obtains further with reasonable simplifying approximations delta C-v/C(v)similar to gamma G(T)/G(00), where G, is the zero temperature crystalline shear modulus. For a typical fragile glass gamma similar to 2 (F similar to 50) with G(T-g)/G(infinity) similar to 1/2, then delta C-v/C-v similar to 1, but 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.