The Thermal and Statistical Entropy of Glasses

Abstract

It is not clear if thermodynamic reasoning can be applied to non-ergodic materials such as glasses, which have some degrees of freedom frozen in. The thermal entropy of a glass, measured calorimetrically, varies continuously through the glass transition because there is no latent heat associated with the freezing in of the degrees of freedom and the entropy tends to a finite value as the absolute zero of temperature is approached. Consequently, glasses require special qualification to exclude them from the third law of thermodynamics. The modern view of a glass is that below the glass transition the system has become trapped in a basin on the potential energy surface of the liquid. The statistical entropy of a glass is calculated from the configuration space accessible to the glass. Because there are many more basins accessible to the supercooled liquid above the glass transition than are accessible to the glass, the statistical entropy decreases at Tg and tends to zero as the glass finds its way to a single ground state at the bottom of the basin on approaching 0K. The thermal and statistical perspectives of the entropy of glasses are examined by the computer simulation of three models: hard spheres, five discs in a box and glassy crystals of dimers. The application of the third law of thermodynamics to glasses is discussed. The geometric free volumes are calculated for a hard sphere crystal and glass. The average free volume is approximately equal to the thermodynamic free volume, defined in terms of the classical configuration integral, and is used to estimate the entropy of the glass and crystal. The average free volume correctly predicts the entropy of the crystal at all densities and predicts the statistical entropy of the glass in the high density limit. A free volume distribution function is proposed and used to calculate the average cavity volume. The average cavity volume is found to be 18%-40% greater than the values calculated directly from computer simulation. The free volume distribution of the glass shows the presence of rattler spheres. Five discs in a box exhibit the fluid, glass and crystalline phases of a much larger hard sphere system. There is a glass transition when the discs become trapped in one of the four amorphous basins. The statistical entropy of the glass, measured using the tether method, decreases by kln(4) at the glass transition density. The statistical entropy of the glass, relative to the entropy of the crystal, tends to zero at the absolute zero of temperature. The thermal entropy of the glass is continuous at the glass transition and tends to kln(4) as T → 0K. Glassy crystals of dimers are non-ergodic because when the dimers cannot rotate the crystal is trapped in one of the many equivalent orientational packings. The melting and freezing regions of two and three dimensional dimers are compared to the coexistence lines predicted thermodynamically. It is found that the entropy associated with the frozen in orientational packings must be added to the entropy of a single packing to correctly predict the equilibrium vapour pressure of the glassy crystals. It is concluded that the thermal entropy of the glass determines the stability of the glass relative to another phase. The thermal and statistical entropy of these systems are discussed in terms of the potential energy landscape of liquids and their relevance to statements of the third law of thermodynamics.