Spinodal criterion general criteria

Here / = 1/7 in the standard notation. From our general statements in Section in. A, the spinodal criterion derived from the exact free energy (38) must be identical to this this is shown explicitly in Appendix C. Note that the spinodal condition depends only on the (first-order) moment densities p, and the second-order moment densities py of the distribution p(cr) [given by Eqs. (40) and (41)] it is independent of any other of its properties. This simplification, which has been pointed out by a number of authors [11, 12], is particularly useful for the case of power-law moments (defined by weight functions vt>f(excess free energy only depends on the moments of order 0, 1... K — 1 of the density distribution, the spinodal condition involves only 2K— moments [up to order 2(K — 1)]. 

In conventional polymer notation, this result would read/m = — (a + 1) c< a/( +1) +/.] From this we can now obtain the spinodal condition, for example, which identifies the value of % where the parent becomes unstable. In our case of a single moment density the general criterion (50) simplifies to... 

To conclude this section on the dense random copolymer model, wc briefly discuss the spinodal criterion and ask whether critical or multicritical points can exist for a general parent distribution pW(a) (with p = 1). The criterion (53), applied to our one-moment free energy becomes... 

As expected from the general discussion in Section III. A, the criterion (57) can also be derived from the exact free energy an alternative form involving the spinodal determinant Y is given in Appendix D. Equation (57) shows that the location of critical points depend only on the moment densities p[t py, and pijk [11, 46]. For a system with an excess free energy depending only on power-law moments up to order K - 1, the critical point condition thus involves power-law moments of the parent only up to order 3 (K — 1). 

When this occurs, he suggested permutting the component subscripts or using "another differential coefficient of the same general form." He noted that Eq. (1) is the criterion of the limit of stability (i.e., the spinodal surface). In an earlier discussion on that subject, Gibbs showed that all of the following partial derivatives vanish on the spinodal surface. 

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