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{e_jϕ⁽ⁱ⁾₁(2, ξ)+e_iϕ^(j)₂(1, ξ)}]. Here e_i, e_j 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).