Queen Mary University of London
Existing textbook expressions for the energy and heat capacity of gases and solids are widely taught in physics courses. However, no such expression exists for a liquid. The reason for this was summarized by Landau as "liquids have no small parameter", and discussed in detail in Landau&Lifshitz Statistical Physics textbook. Based on the old idea of J Frenkel, I formulate the problem in the language of phonons, and calculate liquid energy and heat capacity for both classical and quantum cases. The resulting equation relates liquid heat capacity to its viscosity with no fitting parameters, and is compared with the experimental data of mercury.
I subsequently address the old and very controversial problem of glass transition, and propose the solution that is based on elastic interactions in a liquid. Central to this discussion is the range of propagation of high-frequency elastic waves in a liquid, which I call "liquid elasticity length d". d measures the range over which local relaxation events in a liquid elastically interact with each other via
the elastic waves they induce. The non-trivial point is that d increases with liquid relaxation time tau (or viscosity), contrary to the usual decrease of d with viscosity for commonly discussed hydrodynamic waves. Hence, d is small at high temperature but increases on lowering the temperature. This sets the cooperativity of molecular relaxation in a liquid, the phenomenon whose physical origin has remained not understood. A self-consistent calculation shows that the increase of d on lowering the temperature gives the famous Vogel-Fulcher-Tammann law for tau and viscosity. In this theory, I also discuss other central properties of glass transition: the absence of divergence at a finite temperature, the origins of two dynamic crossovers and the origin of liquid "fragility" (Angell plot).
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