0 50 100 150 200 250 z -0.004 -0.002 0 0.002 0 20 40 60 80 100 z -6 -5 -4 -3 -2 -1 0 1 R (0.1,100) (0.01,100) (0.1,10) (0.01,10) FIG. 3: (Color online) The rescaled ratio R (Eq.(75)) between the leading-order correction (74) and the Debye-H¨ uckel form (55) of the charge density profile for several choices of pairs ( κ, ξ$_{b}$ ) as indicated in the plot. z is in v 1 / 3 0 units, where v$_{0}$ is the volume per molecule in the close-packed system and we assume that all molecules are of similar size. FIG. 2: (Color online) Charge density profiles in the semi-infinite system. The dashed line is the solution of the linearized equation (55), φ $^{(1)}$( z ), the dash-dotted line is the correction term (74), φ $^{(2)}$( z ), and the solid line is the sum of the two functions φ $^{(2)}$( z ), for κ = 0 . 01, σ = 0 . 5, ξ$_{b}$ = 100, and n$_{1}$ = − 0 . 5. The charge density is in e/v$_{0}$ units where e is the elementary charge, and z is in v 1 / 3 0 units, where v$_{0}$ is the volume per molecule in the close-packed system and we assume that all molecules are of similar size. 21