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Counterion distributions and effective electrostatic interactions betweenspherical macroions in polyelectrolyte solutions are calculated viasecond-order perturbation (linear response) theory. By modelling the macroionsas continuous charge distributions that are permeable to counterions,analytical expressions are obtained for counterion profiles and effective pairinteractions in solutions of star-branched and microgel macroions. Thecounterions are found to penetrate stars more easily than microgels, withimportant implications for screening of bare macroion interactions. Theeffective pair interactions are Yukawa in form for separated macroions, but aresoftly repulsive and bounded for overlapping macroions. A one-body volumeenergy, which depends on the average macroion concentration, emerges naturallyin the theory and contributes to the total free energy.Comment: 15 pages, 5 figure

Counterion penetration and effective electrostatic interactions in solutions of polyelectrolyte stars and microgels