content_image/373867.jpg content_image/373868.jpg content_image/373869.jpg
where H [ { ˆ s, ˆ φ } ] = H $^{SR}$[ { ˆ s, ˆ φ $^{2}$} ] + E $^{C}$[ { ˆ φ } ]. The grand potential in the above statistical- mechanical model is given by 

<img src='content_image/373867.jpg'>

The advantage of such kind of modeling with the assumption (12) is the possibility for investigating the effect of fluctuations of the composition (for example by means of computer simulations), and the disadvantage is neglecting the coupling between ϵ (and hence the electrostatic energy of states { ˆ o$_{i}$ } ) and the composition fluctuations. Which effect, the fluctuations of the composition and their coupling to the density of ions or the spatial variation of ϵ , plays the dominant role in determining the charge distribution near the charged wall in a system exhibiting critical fluctuations of concentration, remains an open question. 

## B. Mean-field approximation and continuous Landau-type model 

In the mean-field (MF) approximation the grand potential is assumed to correspond to the minimum of the functional: 

<img src='content_image/373868.jpg'>

where the electrostatic energy density e$_{e}$ [ s, ρ$_{c}$, φ, ψ ] is given by (9), and k$_{B}$W [ s, ρ$_{c}$, φ ] is the entropy density. In this work we focus on the semi-infinite system, and assume the lattice-gas form (ideal mixing entropy) 

<img src='content_image/373869.jpg'>

where we take into account the dependence of the fields on the distance z from the planar wall. 

We are interested in thermodynamic conditions corresponding to stability of the uniform fluid close to the demixing transition. The equilibrium values of s and ρ$_{c}$ in the bulk are the uniform solutions of the Euler-Lagrange (EL) equations obtained from the minimization of the bulk part of the functional (17); they are denoted by ¯ s and ¯ ρ$_{c}$ . From now on we focus 

