Effects of Multivalent Counterions on Cylindrical Polyelectrolyte Brushes

A scaling theory is presented to analyze the effects of the added multivalent counterions on properties of cylindrical polyelectrolyte brushes. The theory demonstrates four possible regimes for the behavior of the brushes, while there are only two regimes for the quenched brushes and three for the annealed brushes with monovalent counterions. In the first regime, the salt concentration is too low to have effects on the brushes. In the next regime, the brush thickness decreases with the increase of the concentration of the added salt. The thickness of quenched polyelectrolyte brushes is independent of the salt concentration in the third regime,while the annealed polyelectrolyte brushes swell with the increase of the salt concentration in the regime and depend on the counterion valence. In the last regime, the brush with added multivalent counterions behaves as the brush with monovalent salt does. These regimes are consequences of the balance of the entropic elasticity of the grafted chains and the osmotic pressure of the mobile ions. The regimes are connected with the exchange of the monovalent and the multivalent counterions in the brush. The boundaries between the neighbour regimes are also discussed. The new findings wait for experiment and computer simulation testing.