Tangent plane distance TPD

To define the tangent plane distance (TPD), note first that by virtue of the coexistence conditions, all phases pW jie on a tangent plane to the free energy surface. Points (p,/) on this tangent plane obey the equation / — p p + II = 0, with p and II the chemical potentials and pressure common to all phases. For a generic phase with density distribution p and free energy/(p), the same expression will have a nonzero value that measures how much below or above the tangent plane it lies. This defines the TPD... 

The criteria of the stability of system I in Fig. 4.6 now can be stated in terms of the tangent plane distance. TPD(x) should be positive over the whole range of x. Note that in Fig. 4.7 the TPD x) from Zj becomes negative for 0.80 < x < 0.88 and, therefore, the system of overall composition Zj is not stable. One the other hand, the overall composition z has a tangent T (xi) that is below the curve and does not have a TPD xi) of less than zero therefore, it is stable said the system cannot split into two phases. 

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