Attraction between like-charged surfaces induced by orientational ordering of divalent rigid rod-like counterions : theory and simulations

We consider the interaction between two equally charged surfaces in an electrolyte solution composed of long divalent rigid rod-like counterions of arbitrary length. Further, we study the influence of orientational ordering of rigid rod-like counterions on the interaction between charged surfaces. Density functional theory is introduced, where the spatial distribution of charge within the divalent rod-like counterions is represented by two effective charges at a fixed distance. The result of variational procedure gives an integral differential equation for the electrostatic potential which was solved numerically. From the electrostatic potential and the concentration of counterions, the free energy of two charged surfaces interacting in a solution of rod-like counterions is calculated. For large surface charge densities and for long enough divalent rod-like counterions the minimum of the free energy is obtained at a distance between the surfaces which equals the counterion length. This indicates that a bridging mechanism might be responsible for the attraction between like-charged surfaces. The analysis of the orientational distribution function confirms that, at the minimum of the free energy, the rod-like counterions are oriented perpendicularly and thus connect the like-charged surfaces. Finally, canonical Monte-Carlo simulations confirm the theoretical calculations of the osmotic pressure between like-charged surfaces for long enough rod-like counterions.