Surface tension from direct correlation function

The surface tension at the surface of tension, that is can be calculated from (4.228) and (4.231), which are the extension to a spherical surface of the calculation of plane surface via the direct correlation function. For this model in the mean-field approximation these equations give ... 

The surface tension from the two-density trajectory is calculated from (5.144) and is shown in Fig. 5.10 for s = 1. It is tetter than the original van der Waals result, but the improvement is obtained only after difficult computations. Finally we could, in principle, combine both methods of improvement suggested in this section that is, use a two-density trajectory but with m a density-dependent parameter determined from the second moment of the local value of the direct correlation function. Such a theory would be a very sophisticated form of the van der Waals theory but it may be doubted whether its accuracy would justify its computational difficulties. 

We conclude this chapter by comparing the surface tension of a Lennard-Jones liquid, as calculated from the theories above, with that found by computer simulation. Figure 7.4 shows the computer results of Fig. 6.4 as a best single line. The four theoretical curves are one solution of the YBG equation, one approximation based on the direct correlation function, one modified van der Waals approximation, and one... 

Fio. 7.4. The surface tension of a Lennaid-Jones liquid from line 1, a solution of the YBG equation line 2, an appraxiniation for the direct correlation function line 3, perturbation dieory line 4, perturbation dteory. The heavy line is the result of conq>uter simulation (S 6.4), and the dashed line from a modified van der Waals equation. ... 

Finally, we return to the case of antisymmetric surfaces, i.e. a situation where one surface of the thin film favors the A-rich phase and the other the B-rich phase (Fig. Id). Simulations were recently carried out [266] in order to test the predictions Eq. (127) on the anomalous interfacial broadening (Sect. 2.5). Figure 24 demonstrates that this phenomenon can indeed be readily observed. Using the interfacial tension a that has been independently measured [215], o= 0.015, and the correlation length 3.6 lattice spacings from a direct study of the bulk correlation function, one can evaluate Eq. (127) quantitatively, using... 

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