Kauzmann temperature, glass transition entropy model

Fig. 21. Entropy versus log-temperature diagram for the hard-sphere model. The solid curves give the computer simulation values for the supercooled fluid, glass, and crystal. The dashed curves have the following bases (a) a calculation from the virial equation using the known first seven coefficients and higher coefficients obtained from the conjectured closure (the plot corresponds quite closely with that calculated from the Camahan-Starling equation ) and (i>) an extrapolation of higher temperature behavior such as that used by Gordon et al., which implies a maximum in the series of virial coefficients. The entropy is defined in excess of that for the ideal gas at the same temperature and pressure. Some characteristic temperatures are identified 7, fusion point 7 , upper glass transition temperature T/, Kauzmann isoentropic point according to closure virial equation.

The calculation of the partition function can be done by the standard Flory-Huggins lattice method. The lattice model predicts the existence of a true second-order transition at a temperature T2. This is shown schematically in Figure 13 for the entropy-pressure-temperature equation of state. As can be seen, the transition occurs at a critical value of the entropy (zero configurational entropy) and the Kauzmann paradox is resolved for thermodynamic reasons rather than kinetic ones, i.e. one is simply not permitted to extrapolate high temperature behavior through the glass transition. Rather, as the material is cooled, a break in the S-T (or V-T) curves occurs because of a second-order transition. 

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