In this paper, we investigate effects of counterion connectivity (i.e., association of the counterions into a chain molecule) on the electrostatic potential of mean force (EPMF) between two similarly charged cylinder rods in a primitive model electrolyte solution by solving a classical density functional theory. The main findings include the following: (i) The counterion connectivity helps in inducing a like-charge-attractionlike (LCA-like) phenomenology even in a monovalent counterion solution wherein the LCA-like observation generally does not occur without the counterion connectivity. (ii) For divalent counterion solutions, the counterion connectivity can reinforce or weaken the LCA-like observation depending on the chain length $N$, and simply increases the equilibrium nearest surface separation of the rods corresponding to the minimum EPMF to nearly three times the counterion site diameter, whether $N$ is large or small. (iii) If $N$ is large enough, the LCA-like strength tends to be negatively correlated with the electrolyte concentration $c$ over the entire range of the rod surface charge magnitude $|{\sigma }^{*}|$ considered; whereas if $N$ drops, the correlation tends to become positive with decrease of the $|{\sigma }^{*}|$ value, and particularly for modest $|{\sigma }^{*}|$ values, the correlation relationship exhibits an extreme value phenomenon. (iv) In the case of a 1:1 electrolyte, the EPMF effects of the diameters of counterion and coion sites are similar in both situations with and without the counterion connectivity. All of these findings can be explained self-consistently by a recently proposed hydrogen-bonding style mechanism reinforced by one additional concept: flexibility of the counterion chain and the factors affecting it, like $N$ and counterion site valence.

• Received 17 July 2015
• Revised 26 September 2015

DOI:https://doi.org/10.1103/PhysRevE.92.052317