Calculation of the force between planar electrical double layers containing counterion mixtures

The distribution of ions about a planar surface of fixed charge density can be solved within the hypernetted chain (HNC)/mean spherical approximation (MSA) by a simplest yet very accurate method based on the calculus of variations. Here we present an application of that work, the calculation of the potential of mean force and the net pressure for two charged planar surfaces immersed in a salt solution. The expression for the potential of mean force is derived from the Ornstein-Zernike relation for a homogenous fluid by taking the planar geometry limit and employing the conventional HNC approximation. This method requires as input only the distribution of ions about an isolated planar double layer. Though this simple method of pressure calculation is more approximate than some other schemes utilizing the HNC approximation, under many circumstances the agreement among the different theories is excellent. We present results for 1 : 1 and 1 : 2 salts as well as mixtures of monovalent and divalent counterions for planar double layers with fixed surface charge densities. Under identical conditions of charge density and bulk coion concentration, the interaction between the surfaces is repulsive at all distances for the case of a monovalent counterion, while regions of attraction are observed for the case of an electrolyte with divalent counterions. It is found that salt solutions with a mixture of monovalent and divalent counterions show properties intermediate to the pure monovalent and divalent cases but the present of only a small amount of divalent ion is sufficient to cause large changes in the interaction between the surfaces. Systems containing a moderate fraction of divalent ions show behaviour nearly identical to the pure divalent counterion case.