Conventional Hypernetted Chain Force Calculations for Charged Plates with Adsorbing Counterions

Abstract

By including counterion adsorption in the form of a sticky potential, the distribution of ions about a charged plate can be solved using the hypernetted chain/mean spherical approximation from the integral equation theories of liquids. The computational effort for such a procedure is simplified by the use of a variational technique. These results for a single plate can then be used to calculate the potential of mean force and net pressure between two charged plates. We present results using the conventional hypernetted chain theory for two charged plates. This theory requires only the ion distributions about a single plate. Differences in the double layer structure about a single plate due to counterion adsorption result in a range of double layer forces between two plates. For a given set of conditions, the case of little or no adsorption creates the largest pressure, and an increase in the adsorption tendency of the counterions results in lower pressures. This method of pressure calculation is more approximate than other integral equation methods which use the ion distribution about two plates to calculate the pressure, but agreement between the theories improves as the thickness of the double layer decreases. The presence of counterion adsorption produces a thinner double layer, so we are confident in the accuracy of our calculations. We present results for monovalent electrolyte and a range of counterion adsorption parameters. While the surface charge on the plates is fixed throughout, the net charge acting on the diffuse layer varies according to the bulk ion concentration and adsorption strength. We also compare these results with the point-ion limit so that ion size effects can be seen. We find that ion adsorption has a significant effect on the net pressure between two plates, and ion size effects depend more on the bulk ion concentration than on the adsorption strength.