Moment free energy

C. Relation Between the Two Methods HI. Properties of the Moment Free Energy... 

In previous work, the authors originally arrived independently at two definitions of a reduced free energy in terms of moment densities (moment free energy for short) [31, 32]. Though based on distinctly different principles, the two approaches led to very similar results. We explain this somewhat surprising fact in the present work at the same time, we describe the two methods in more detail and explore the relationship between them. We also discuss issues related to practical applications and give a number of example results for simple model systems. 

We show the equivalence of the two approaches in Section HC. There, we first demonstrate that the general form of the entropy of mixing obtained by the combinatorial method can be transformed to the standard expression — Jdon(o) In n a). Second, we show that the moment free energies arrived at by the two methods are in fact equal, with the projection method being slightly more generally applicable. 

H. DERIVATION OF MOMENT FREE ENERGY A. Projection Method... 

As explained in the previous section, truncatable here means that the excess free energy / depends only on K moment densities p,. Note that, in the first (ideal) term of (6), we have included a dimensional factor R(a) inside the logarithm. This is equivalent to subtracting T dap a) In R(a) from the free energy. Since this term is linear in densities, it has no effect on the exact thermodynamics it contributes harmless additive constants to the chemical potentials p a). However, in the projection route to a moment free energy, it will play a central role. 

This expression, which only depends on the particle size distribution through the generalized mean size m, is our desired moment free energy. 

To end this section, let us state in full the analogue of Eqs. (29) and (32) for the case of several moment densities, restoring the notation used in the previous sections. The square bracket on the right-hand side of Eq. (32) is the moment expression for the entropy of an ideal mixture. If, as in Section II. A, we measure this entropy per unit volume (rather than per particle, as previously in the current section) and generalize to several moment densities, we find by the combinatorial approach the following moment free energy ... 

This is identical to the projected entropy spr, Eq. (9), except for the last term. But by construction, the combinatorial entropy assumes that p0—the overall density—is among the moment densities retained in the moment free energy. The difference scomb — spt = —p0 In p is then linear in this density, and the combinatorial and projection methods therefore predict exactly the same phase behavior. 

In summary, we have shown that the projection and combinatorial methods for obtaining moment free energies give equivalent results. The only difference between the two approaches is that within the projection approach, one need not necessarily retain the zeroth moment, which is the overall density p0 = p, as one of the moment densities on which the moment free energy depends. If p0 does not appear in the excess free energy, this reduces the minimum number of independent variables of the moment free energy by one (see Section V for an example). 

In the previous sections, we have derived by two different routes [namely, Eqs. (9) and (33)] our moment free energy for truncatable polydisperse sys-... 

Let us first collect a few simple properties of the moment free energy (37) which will be useful later. Recall that Eq. (37) faithfully represents the free energy density of any phase with density distribution in the family... 

Equations (42)-(44) are all calculated via the moment free energy. The three corresponding quantities obtained from the exact free energy (38) are, first, the chemical potentials conjugate to p([Pg.287]

The last of these is identical to the result (44) derived from the moment free energy. 

Let us first recap the general criteria for spinodals and critical points in multicomponent systems. In order to treat the criteria derived from the exact and moment free energies simultaneously, we use the common notation p for the vector of densities specifying the system For the exact free energy, the components of p are the values p ) for the moment free energy, they are the reduced set pt. We write the corresponding vector of chemical potentials as... 

The first part of this is simply the spinodal criterion, as expected. To evaluate the second part for the moment free energy (37), we need the third derivative of sm with respect to the moment densities pt. Writing (41) as... 

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... 

It is now easy to see, however, that—as stated in Section H.A—the exact solution can be approached to arbitrary precision by including extra moment densities in the moment free energy. (This leaves the exactness of spinodals, critical points, cloud points, and shadows unaffected, because none of our arguments excluded a null dependence of / on certain of the pt.) Indeed, by adding further moment densities, one can indefinitely extend the family (39) of density distributions, thereby approaching with increasing precision the actual distributions in all phases present this yields phase diagrams of ever-refined accuracy. 

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