Quantification of Mono- and Multivalent Counterion-Mediated Bridging in Polyelectrolyte Brushes

Multivalent counterion-induced bridging interactions have been identified as the key mechanism of drastic collapse of the height of polyelectrolyte (PE) brushes. In this article, we employ all-atom molecular dynamics simulations to quantify the bridging interactions in PE brushes for counterions of different sizes and valences. We identify that unlike the current notion, bridging interactions are not the sole function of the counterion valence. Rather the bridging interactions depend on the fraction of counterions (of a given type) that get physically condensed on the PE backbone as well as the size of the counterion solvation shell. These mechanisms ensure that certain monovalent counterions demonstrate much stronger bridging interactions than those witnessed for certain divalent and trivalent counterions, while certain counterions of identical valences show drastically different bridging interactions. We argue that these counterion-specific bridging interactions eventually enable not only the significant reduction of the PE brush height in the presence of certain multivalent screening counterions but may also give rise to scenarios where the brush height reduction for certain monovalent counterions is larger than that observed for certain divalent and trivalent counterions. The latter observation contradicts the experimental findings where the multivalent counterions invariably led to a larger decrease in the height of the PE brushes; we argue that this discrepancy stems from the fact that in our simulations we only consider densely grafted and short (and hence less flexible) PE brushes that hinder the formation of different laterally inhomogeneous structures (such as pinned micelles and cylindrical bundles) that would have led to a larger brush height reduction (in experiments, which invariably consider longer and less densely grafted brushes, the formation of such inhomogeneous structures is primarily responsible for the larger brush height reduction in the presence of multivalent counterions). Finally, we also probe the dynamic properties of the counterions (i.e., their time-dependent displacements) and their bridging interactions (i.e., lifetime of bridging interactions).