Coexisting phases of the lattice fluid

At T = 0, u is a linear function of /x regardless of the nature of a specific morphologies as Table 4.1 shows. This implies that the equality in Eq. (1.82) holds where, how ever, on account of the S3niunetry of the external field, K has to be replaced by Kyy defined in Eqs. (5.75) and (5.76). Because the 

Thble 4.1 Possible lattice fluid morphologies M and assodated grand potential n (p) for T = 0. 

Yet another way of looking at Eq. (1.82) is to conclude that at T = 0 density fluctuatioas are completely suppressed. To realize this, consider 

We are thus in a position to calculate the set of chemical potentials at which phases n and (1 coexist at T = 0 from Elq. (1.76a) and entries for H in Table 4.1. 

In the limit of nonvanishing temperatures, simple analytic forms for the u 8, such as the ones compiled in Table 4.1, do not exist. Hence we need to resort to a numerical scheme to solve the Euler-Lagrange equations [see Eq. (4.86)]. This can be accompUshed by an approach detailed in Appendix D.2.1 that starts from the set of (exact) morphologies M compiled in Table 4.1 at T = 0 as starting solutions for a temperature T 0. Once convergence has been attained, the algorithm yields new morphologies for this sufficiently 

Moreover, a comparison between Figs. 4.10(a) and 4.10(b) sliows that, as the strong portion of the substrate becomes more attractive, layering transitions become more pronounced. For instance, the gas droplet coexistence line (T) becomes detached from the remainder of /i (T), thereby causing the one-phase region of droplet phases to increase in size substantially. 

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