Phase Stability in Open Systems

The thermal and mechanical stability criteria (8.1.23) and (8.1.31) apply both to pure fluids and to mixtures however, for homogeneous mixtures, those criteria are not sufficient to identify stable systems because, in addition to energy and volume fluctuations, mixtures have concentration fluctuations. These fluctuations occur in localized regions of a system when material spontaneously aggregates and redisperses. If such fluctuations are not to disturb a system s stability, then the mixture must satisfy a set of conditions known as the material or diffusional stability criteria. These criteria are derived in a manner similar to that given in 8.1.2 for (8.1.23) and (8.1.31), so we only sketch the procedure here. 

Consider a one-phase binary mixture of components 1 and 2 confined to an isolated vessel, and imagine dividing the fluid into parts A and B. But unlike the pure case, region B is open to A, so that a fluctuation occurring in part B disturbs not only its internal energy and volume V, but also the mole numbers Nf and N . Consequently, the concentration in B fluctuates by transfers of material to and from part A. In addition to the constraints on U, V, and S given by (8.1.4)-(8.1.6), the total amoxmts of each component are conserved. 

As in 8.1.2, stable equilibrium occurs when the total entropy is a maximum hence. 

At equilibrium we must have 5S = 0, which requires each bracketed term in (8.3.3) to be zero. Just as in (8.1.11) and (8.1.12), the first term implies and the second 

As in (8.1.16), if the inequality in (8.3.4) is to be obeyed for all possible variations 5li, 8V, 5Ni, and 8N2, then S must be negative definite that is, the principal minors of S must satisfy the following four inequalities  

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