Thermodynamic properties of a simple model of like-charged attracting rods

We study the thermodynamic properties of a simple model for the possible mechanism of attraction between like charged rodlike polyions inside a polyelectrolyte solution. We consider two polyions in parallel planes, with $Z$ charges each, in a solution containing multivalent counterions of valence $\alpha$. The model is solved exactly for $Z \le 13$ for a general angle $\theta$ between the rods and supposing that $n$ counterions are condensed on each polyion. The free energy has two minima, one at $\theta=0$ (parallel rods) and another at $\theta=\pi/2$ (perpendicular rods). In general, in situations where an attractive force develops at small distances between the centers of the polyions, the perpendicular configuration has the lowest free energy at large distances, while at small distances the parallel configuration minimizes the free energy of the model. However, at low temperatures, a reentrant behavior is observed, such that the perpendicular configuration is the global minimum for both large and small distances, while the parallel configuration minimizes the free energy at intermediate distances.