A. Solution of the linearized EL equations Let us first consider the linearized equations (41)- (43). For the linearized equations we shall simplify the notation, ϑ (1) ≡ ϑ, η (1) ≡ η, φ (1) ≡ φ . For the electrostatic part we obtain the linearized equations of the well known form (the spatial dependence of permittivity leads to nonlinear contributions to the EL equations) which together give where Note that κ is the inverse Debye length in units of the molecular size. Solution of Eq. (53) with the boundary condition Eq. (46) has the well known form The equations for ( η, ϑ ) are formally the same as in the Landau theory for a mixture near the demixing critical point, and can be written in the form where the matrix M = ( M$_{ij}$ ) with the indices 1 and 2 corresponding to s and ρ respectively, is defined in (33), and the vector v is defined below Eq.(26). In the semi-infinite systems in the one-phase state ϑ ( z ) , η ( z ) → 0 for z → ∞ , and from the Ansatz: 16 ϑ = t$_{1}$ exp( − λ$_{1}$z ) + t$_{2}$ exp( − λ$_{2}$z ) (57) η = n$_{1}$ exp( − λ$_{1}$z ) + n$_{2}$ exp( − λ$_{2}$z ) (58) v ′′ = Mv (56) φ ( z ) = − κσe − $^{κz}$. (55) κ 2 = 4 πe $^{2}$¯ ρ$_{c}$ kT ¯ ϵ . (54) φ ( z ) ′′ = κ $^{2}$φ ( z ) , (53) eψ ( z ) + kT φ ( z ) ¯ ρ$_{c}$ = 0 (52) ¯ ϵ 4 π d $^{2}$ψ ( z ) dz 2 + eφ ( z ) = 0 (51)