Phase transitions in interfaces. The Cahn transition

We shall first review very briefly the thermodynamics of first-order phase transitions between ordinary bulk phases at equilibrium. We shall then be able to deseribe in similar terms the closely analogous phase transitions in interfaces. Among these is the Cahn transition, our present subject. 

In first-order transitions between bulk phases at given values of the fields (fi T) l, the thermodynamic potential p( s), which is the 

The Gibbs-Duhem equation specifies how p varies with the c-l-1 independent fields p of which it is a function in the notation of 2.3, 

At two-phase equilibrium those c +1 fields p are not all independent, for their possible variations are then restricted by the Clapeyron equation 

If we specify a set of values of the c-l-1 fields appropriate to the coexistence of two phases, a and fi, it still leaves indeterminate the relative amounts of the two phases that are present. Hiat depends on otherwise irrelevant factors sudi as the nature of the walls of the containing vessel. If the walls favour one of the phases, only that phase will be present, for there is then no reason for the system to tolerate the excess free energy r A of an interface of area A between coexisting a and p phases. If, on the other hand, the c-l-1 quantities we specify include at least one density, then some of both phases will be present, with an interface between them (provided the vidue we specify for any such density p, is between pf and f ). 

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