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on the deviations from the equilibrium fields 

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On the MF level it is possible to take into account the dependence of the dielectric constant on the composition and we assume that ϵ given by (11) depends on the distance z from the planar wall according to 

where 

and we introduced the notation 

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For very small fields ϑ and η , the second term in Eq.(21) is negligible compared to the first one. We shall keep this term in general formulas, valid for arbitrary deviations from bulk equilibrium densities. 

The continuum-space Landau-Ginzburg (LG) functional for the fields ϑ ( z ) and η ( z ) is defined as 

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where A is the area of the confining surface in the semi-infinite geometry. L [ ϑ, η, φ, ψ ] can be derived from the lattice model in a standard way. Here we follow the method desribed in detail in Ref.[18]. In short, in the first step EL equations on the lattice are derived, which contain contributions of the form ∇ $^{2}$f = f ( z + 1) + f ( z − 1) − 2 f ( z ) for f = s, ρ$_{c}$ . For the boundary layer the effect of the missing neighbors is taken into account, and in this way surface EL equations are obtained. At the boundary layer both the bulk and the surface EL equations must be satisfied, and the difference of the two equations gives the corresponding boundary condition. From the continuous version of the EL equations ( ∇ $^{2}$f → d $^{2}$f /dz$^{2}$) we derive the corresponding functional with the appropriate surface term that leads to the boundary conditions. We require that the boundary conditions contain 

