Phase coexistence, moment free energy

So far in this section, we have shown that the moment free energy gives exact results for spinodals and (multi-) critical points. Now we consider the onset of phase coexistence, where (on varying the temperature, for example) a parent... 

As stated in Section H.A, the moment free energy does not give exact results beyond the onset of phase coexistence—that is, in the regime where the coexisting phases occupy comparable fractions of the total system volume. As shown in Section III.A, the calculated phases will still be in exact thermal equilibrium but the lever rule will now be violated for the transverse degrees of freedom of the density distributions. This is clear from Eq. (11) In general, no linear combination of distributions from this family can match the parent p (a) exactly. 

A more detailed understanding of the failure of the moment free energy beyond phase coexistence can be gained by comparing with the formal solution of the exact phase coexistence problem. Assume that the parent p (phases numbered by a = 1... p. The condition (48), which follows from equality of the chemical potentials p(cr) in all phases, implies that we can write their density distributions pM(cr) as... 

The application of the moment free energy method to the calculation of spi-nodal and critical points is straighforward using conditions (55), and (57) or (D3), respectively, and is further illustrated in Section V below. Therefore in this section we focus on phase coexistence calculations. 

As a concrete example, we now consider phase separation from parent distributions of the form p o) oc exp(ycr) (for — 1 < a < 1, otherwise zero). The shape parameter y is then a fixed function of the parental first moment density p, = J do op(° o). Figure 9 shows the exact coexistence curve for pf = 0.2, along with the predictions from our moment free energy with n moment densities (pf — J do oip(o), i = 1... n) retained. Comparable results are found for other p, . Even for the minimal set of moment densities (n = 1)... 

Figure 9. Coexistence curves for a parent distribution with pf = 0.2. Shown are the values of p, of the coexisting phases horizontal lines guide the eye where new phases appear. Curves are labeled by n, the number of moment densities retained in the moment free energy. Predictions for n = 10 are indistinguishable from an exact calculation (in bold).

The exactness statements in Section HI can also be directly translated to the constant pressure case. The arguments above imply directly that the onset of phase coexistence is found exactly from the moment Gibbs free energy All phases are in the family (A3), because one of them (the parent) is, and the requirement of equal chemical potentials is satisfied. Spinodals and (multi-) critical points are also found exactly. Arguing as in Section HI. A and using the vector notation of Eq. (53), the criterion for such points is found as... 

---