and where Note that at the second order in the perturbation expansion we need to calculate the function N$_{φ}$ to the quadratic order in ν . The first term in Eq.(66) is of the order O ( ν $^{2}$κ$^{2}$) (recall that we limit ourselves to boundary conditions such that the fields ϑ, η, φ obtained from linearized equations are all of the order of O ( ν )). The perturbation expansion (50) is justified when κσ = O ( ν ) (see (55)), i.e., for small surface charge and/or weak screening. When the additional condition, namely κ 2 ≪ 1 is satisfied (weak screening), the first term in (66) is negligible compared to the second one. In this case of weak screening the position- dependence of the dielectric constant does not play a dominant role for the leading-order correction to the solution of the linearized EL equations, and we may assume ϵ = ¯ ϵ . From the forms of the fields ϑ, η, φ obtained from linearized equations we obtain the explicit expression for N$_{φ}$ ( z ), and Eq.(65) simplifies to The solution is where and A$_{0}$ can be determined from the electroneutrality condition, A$_{0}$ = − κ A ξ$_{b}$ ( κ + ξ − 1 b ) . (71) ∫ ∞ 0 dz ( φ $^{(1)}$( z ) + φ $^{(2)}$( z )) + σ = 0 . (70) The above condition is satisfied by the solution (55) of the linearized equation, hence for the correction term we have the condition ∫ ∞ 0 dzφ $^{(2)}$( z ) = 0 that yields − ( φ $^{(2)}$) ′′ + κ $^{2}$φ(2) + N$_{φ}$ ( z ) = 0 (65) N$_{φ}$ ( z ) = κ 2 4 πe [ ( ϵ$_{ϑ}$ϑ $^{′}$( z ) − ϵ$_{η}$η $^{′}$( z )) ψ $^{′}$( z ) + ( ϵ$_{ϑ}$ϑ ( z ) − ϵ$_{η}$η ( z )) ψ $^{′′}$( z ) ] + ( φ ( z ) η ( z )) ′′ (66) − ( φ $^{(2)}$) ′′ + κ $^{2}$φ(2) − κσe − κz [ (1 /ξ$_{b}$ + κ ) $^{2}$n$_{1}$ e − z/ξ b ] = 0 . (67) φ $^{(2)}$( z ) = A$_{0}$ e − κz + A ξ$_{b}$e − ( κ +1 /ξ b ) $^{z}$, (68) A = − κσn$_{1}$ ( ξ − 1 b + κ ) 2 ( ξ − 1 b + 2 κ ) (69) 18