Effect of counterion valence and polymer charge density on the pair potential of two polyions

A theory of attractive forces between identical highly charged polyions in a 1:1 supporting electrolyte, generated by the interaction of the layers of condensed counterions, is extended to a Z:Z' electrolyte and to rodlike polymer segments of any charge density. If the combined charge density of a polyion pair does not exceed the charge density critical for counterion condensation, then there is no attraction. But an attraction arises if the combined charge density does exceed the condensation threshold, even if the individual polymer charge densities lie below the critical value. The intensity of the attraction at a fixed ionic strength decreases with higher values of counterion valence. The effect of increased ionic strength is to decrease both the range and the intensity of the attractive force.