Formation of loose clusters in polyelectrolyte solutions

report further development of a pair potential for two identically charged rodlike polyions oriented in parallel. An attractive farce mediated by cond ensed counterions of any valence, including univalent, is identified for se paration distances between a Debye screening length and about a third of a Debye length. The characteristics of the potential are explored as a functi on of ionic strength, dielectric constant, counterion valence, and charge d ensity of the polyions. The work required to assemble more than two polyion s in parallel array is shown not to be pairwise additive. The free energy o f assembling many parallel polyions in a hexagonally ordered cluster is cal culated as a function of the number N of polyions in the cluster. The free energy has a minimum at a finite cluster size. Discrepancies between the th eoretical structure of the stable cluster and experimental observations on polyelectrolyte clusters (with univalent counterions) are discussed, and it is tentatively concluded that the apparently loose and disordered clusters observed in solution may be understood as stabilized by relatively few pai rwise attractive interactions.