Weak Polyelectrolyte Brushes with Mono- and Divalent Counterions

We use state-of-the-art coarse-grained simulations to study weak polyelectrolyte brushes in the presence of mono- and divalent counterions. In the monovalent case, our simulations reproduce the marked pKa shifts found in recent experiments, which are linear in the logarithm of the salt concentration. Comparing explicit-particle simulations with mean-field calculations, we show that for high grafting densities the salt concentration effect can be explained using the ideal Donnan theory, but for low grafting densities the full shift is due to a combination of the Donnan effect and the polyelectrolyte effect. The latter originates from electrostatic correlations that are neglected in the Donnan picture and that are only approximately included in the mean-field theory. Moreover, we show that the magnitude of the polyelectrolyte effect is almost invariant with respect to the salt concentration, but depends on the grafting density of the brush. In the presence of divalent counterions, we observe a complete breakdown of the mean-field picture. We demonstrate the preferred uptake of divalent ions by the brush, which is further enhanced beyond mean-field predictions by electrostatic correlation effects. Finally, our simulations reveal a hitherto unobserved two-stage swelling of the brush as a function of the pH in the presence of divalent salt. This phenomenon arises as a combined effect of charge regulation and ion partitioning. Overall, our theoretical findings provide valuable insights for designing smart pH-responsive interfaces with various technological and biomedical applications.