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and 

where 

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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, 

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 A ξ$_{b}$ 

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