Ion-pair correlation function in electric double layer theory. Part 1.—Ions in the diffuse layer

Abstract

The pair correlation function g(ij)(12) between two ions 1 and 2, species i and j, situated in a planar electric double layer, should be symmetrical in i and j, when the two ions are equidistant from the interface. An integral equation of the Kirkwood type from the theory of liquids can be set up for g(ij)(12) in the double layer by means of the usual charging process on one of the ions 1 and 2. In the subsequent approximations inherent in the closure procedures required to obtain a solution of the integral equation, the symmetry property is lost. However, by charging ions 1 and 2 at the same rate ξ, the necessary symmetry at equal distances from the interface is retained. When closure is achieved by adding the individual self-atmospheres of the two ions and a size is given to these two ions only, then one obtains g(ij)(12)=g(ij)(12, ξ= 0) exp [–1//kT∫10 dξ{ejϕ(i)1(2, ξ)+eiϕ(j)2(1, ξ)}]. Here ei, ej are the charges of ions 1 and 2, ϕ(i)1(2, ξ) is the potential at the position r2 due to ion 1 situated at r1 in the absence of ion 2 and ϕ(j)2(1, ξ) is defined similarly; both ions carry a fraction ξ of their full charge. This results is also obtained as the leading term in a method of successive approximations based on the work of Outhwaite. An alternative symmetrical expression for g(ij)(12) is obtained by charging each of the ions 1 and 2 separately and then taking the mean of ln g(ij)(12).