Attractive forces between circular polyions of the same charge

We study two models of ring-like polyions which are two-dimensional versions of simple models for colloidal particles (model A) and for rod-like segments of DNA (model B), both in solution with counterions. The counterions may condense on Z sites of the polyions, and we suppose the number of condensed counterions on each polyion n to be fixed. The exact free energy of a pair of polyions is calculated for not too large values of Z, and for both models we find that attractive forces appear between the rings even when the condensed counterions do not neutralize the total charge of the polyions. This force is due to correlations between the condensed counterions and, in general, becomes smaller as the temperature is increased. For model A divergent force may appear as the separation between the rings vanishes, and this makes an analytical study possible for this model and for vanishing separation, showing a universal behavior in this limit. Attractive forces are found for model A if the valence of the counterions is larger than one. For model B, no such divergences are present, and attractive forces are found for a finite range of values of the counterion valence, which depends of Z,n, and the temperature.