II. DERIVATION OF THE LANDAU THEORY A. Construction of the lattice model Let us consider a two-component mixture approaching the upper critical point of the demixing transition from the one-phase side and investigate the effect of the presence of a small amount of positively and negatively charged ionic solutes. We are interested in such a system in contact with a charged wall, such that the total charge of the system neutralizes the charge at the wall. Let us first consider this system with the ions replaced by the corresponding neutral molecules. The electrostatic contribution to the free energy will be included in the next step. The four-component mixture close to the demixing transition can be conveniently studied in the framework of the lattice-gas model. In the simplest lattice-gas model of a one- component system the space is divided into cubic cells of a volume comparable to the volume of a molecule. The cells can be either occupied or unoccupied by a center of a single molecule, i.e., multiple cell occupancy is forbidden. For a mixture of molecules of different sizes we consider a semi-microscopic description of cells occupied either by the larger molecule or by a cluster of smaller molecules, and we denote the volume of the lattice cell of a simple cubic lattice (SC) by v$_{0}$ . Here we limit ourselves to a mixture of molecules of similar sizes. The cell-occupancy operators are ˆ o$_{i}$ ( x ) = 1 if the cell x is occupied by the i -th component and ˆ o$_{i}$ ( x ) = 0 otherwise. The two components of the solvent are denoted by i = 1 , 2, and two additional species that can be ionized to carry positive or negative charges are indicated by i = 3 , 4, respectively. We restrict ourselves to thermodynamic conditions corresponding to the stability of a liquid, where total density fluctuations can be neglected. The single-state occupancy and close-packing lead to the constraint We consider an open system with the probability of the semi-microscopic state { ˆ o$_{i}$ ( x ) } given by where β = ( k$_{B}$T ) − $^{1}$, k$_{B}$ is the Boltzmann constant and T the temperature. Ξ is the par- tition function. We include the activity factors in the Boltzmann factor, by assuming the 5 p [ { ˆ o$_{i}$ ( x ) } ] = exp( − βH $^{SR}$[ { ˆ o$_{i}$ ( x ) } ]) ∏ $_{x}$δ $^{Kr}$( ∑ 4 i =1 ˆ o$_{i}$ ( x ) − 1) Ξ , (2) 4 ∑ i =1 ˆ o$_{i}$ ( x ) = 1 . (1)