Critical point condition, determinant form

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). 

Inserting this into (Dl) gives the determinant form of the critical point condition... 

A critical point must satisfy two conditions. The first defines the limit of stability. In the procedure of Heidemann and Khalil, this condition takes the form that the determinant of matrix Q is zero, where the elements of Q are... 

In fact, the condition just described holds whenever all but one of a set of coexisting phases are of infinitesimal volume compared to the majority phase. This is because the density distribution, p (cr), of the majority phase is negligibly perturbed, whereas that in each minority phase differs from this by a Gibbs-Boltzmann factor, of exactly the form required for (10) we show this formally in Section III. Accordingly, our projection method yields exact cloud point and shadow curves. By the same argument, critical points (which in fact lie at the intersection of these two curves) are exactly determined the same is true for tricritical and higher-order critical points. Finally, spino-dals are also found exactly. We defer explicit proofs of these statements to Section III. 

The particular boundary conditions to be satisfied at the top and bottom surfaces of the fluid layer determine the form of the z dependence of the solution, i.e., the functions T z), n (z), etc. These are then used to locate the marginal state and thereby the critical point of the system, by recalling that the parametric space (defined by the parameters of the differential equations) contains both stable and imstable regions and hence a boundary between these two domains which is termed the marginal state or the state of neutral stability. Furthermore, in most stability problems, all the parameters of the system are fixed a priori except for one which is allowed to vary continuously over some range this one parameter then possesses some critical value that separates the stable from the unstable portions of the range. 

Q(Af, Ap) is a crossover function which has the value of unity at the critical temperature and density and zero under conditions far removed from the critical conditions. Since the critical enhancement is present in a large range of temperatures and densities around the critical point, it was necessary initially to predict the background thermal conductivity in this range and to subtract this firom the experimental values. The amplitude Rd of the critical enhancement was then determined, and parameters in the equation for the background were adjusted. The equation selected by Basu Sengers (1977) for the crossover function was written in the form... 

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