Numerical method and general discussion of integral equations for the primitive model of the electric interface

We describe an efficient numerical algorithm for solving integral equations commonly used in the theory of the primitive electrode. The method is applied to an approximation obtained from the first Born-Green-Yvon (BGY) equation using a modified Croxton-McQuarrie local neutrality ansatz, with accurate bulk correlations. For 1 M solutions of 1-1 electrolytes, and 0.5 M solutions of 2-2, 2-1, and 1-2 electrolytes, the agreement with computer experiments is good. To apply the method to other integral equations we formulate them as approximation schemes for the closure of the first member of the BGY hierarchy. Many of them are then seen to satisfy the local electroneutrality condition. We also suggest a new approximation which might be accurate even at very high couplings.