Phase transitions, Kauzmann curves, and inverse melting

Walter Kauzmann's classic 1948 review of liquid supercooling and glass formation drew attention to the temperatures at which (by extrapolation) enthalpies and entropies of liquid and crystal phases would appear to become equal. In the temperature-pressure (T, p) plane, the collection of such 'Kauzmann temperatures' generate characteristic curves. The present study examines the connection of those Kauzmann loci to equilibrium inverse melting phenomena, i.e. cases where isobaric heating causes freezing of the liquid. Such cases are associated with local minima or maxima in the melting curve p_m(T), and we point out the possible relevance of melting curve maxima to the thermodynamics of protein folding. Both equal-enthalpy and equal-entropy Kauzmann curves must pass through melting curve extrema. Three thermodynamic identities have been obtained to describe the vicinity of these points; they involve, respectively, the slopes of the two Kauzmann curves, and the second temperature derivative of the melting pressure. The second of these three equations is formally identical to the first Ehrenfest relation for second-order phase transitions, but carries no phase-transition implication. For purposes of specific numerical illustration, the inverse-melting behavior displayed by ³He at low temperature has been analyzed in detail.