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 condensed counterions of any valence, including univalent, is identified for separation distances between a Debye screening length and about a third of a Debye length. The characteristics of the potential are explored as a function of ionic strength, dielectric constant, counterion valence, and charge density of the polyions. The work required to assemble more than two polyions in parallel array is shown not to be pairwise additive. The free energy of assembling many parallel polyions in a hexagonally ordered cluster is calculated 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 theoretical 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 pairwise attractive interactions.