  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.
