Scaling properties of the capillary-wave model with interfacial bending rigidity

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Abstract
The scaling properties of interfacial correlations in the extended capillary-wave model (Romero-Rochín, V., Varea, C., and Robledo, A., 1992, Physica A, 184, 367) are analysed and the procedure developed previously by Weeks and collaborators for studying interfacial correlations is confirmed to apply to the extended case and to reproduce most of the earlier features. In addition to this it is found that: (i) the extended model is consistent with the expression for the bending rigidity κ given by the fourth moment of the direct correlation function; (ii) the scaling properties of the extended model differ from those of the ordinary model and we find that for d = 3 dimensions the correlations obey scaling on length scales less than the capillary length L c. The consequences for κ and the interfacial width W when scaling occurs at all length scales are examined, and it is suggested that their limiting values as L c → ∞ depend on the range of the molecular interactions. When these are sufficiently long ranged, κ diverges but W remains finite for d = 3. It is pointed out that multicomponent systems with long-range interactions allow for a finite κ. Finally, the case is studied of small or vanishing interfacial tension γ, in which case W scales with gravity g as W ∼g -1/4.