Liquid-glass transition, a free-volume approach

The free-volume model, which has been useful for describing the behavior of the viscosity $\eta$ of dense liquids and glasses, is extended to account for their thermodynamic behavior as well. Experimental results for the heat capacity $C_p$ and the volume $\overline{v}$ show that the system falls out of complete, metastable thermodynamic equilibrium at the glass transition temperature $T_g$. As a first step in understanding these universal phenomena, a theory of the underlying metastable phase, the amorphous phase, is developed. Recent molecular-dynamic calculations demonstrating the existence of a cellular structure in liquids and the properties of the local free energy of the molecular cells permit us to formulate more precisely and justify in more detail the standard free-volume model. In particular, it is possible to define the free volume and distinguish solid-like and liquidlike cells. This leads to the introduction of percolation theory, which is used to describe the gradual development of the communal entropy of the amorphous phase. We then determine the probability distribution of the cellular volume as a function of the fraction of liquidlike cells, $p$. The equilibrium liquid-glass transition is associated with the increase of $p$ with temperature. This occurs via a phase transition which is most probably first order. The results of our theory give a generalized equation for the viscosity which agrees accurately with experimental results at all temperatures. Results for $C_p$ and $\overline{v}$ are also obtained. This equilibrium theory can provide the basis for a relaxation theory of the kinetic effects observed around and below $T_g$. The relationship between the entropy theory and the free-volume model is also clarified.