Ionic Conductance of Polyelectrolyte-Modified Nanochannels: Nanoconfinement Effects on the Coupled Protonation Equilibria of Polyprotic Brushes

A theoretical methodology is introduced to calculate the low-bias conductance, structure, and composition of long polyelectrolyte-modified nanochannels of arbitrary geometry. This methodology is applied to explore the coupling between acid-base equilibrium and geometry in cylindrical, conical, and trumpet-shaped nanochannels modified by end-grafted layers of poly(2-(methacryloyloxy)ethyl-phosphate) (PMEP), a diprotic polyacid. The ionic conductance and speciation curves (i.e., the fraction of deprotonated, monoprotonated, and diprotonated acid segments) for this system were predicted as a function of the solution pH. The apparent pK_a's and widths of the transitions between the different acid-base states determined from the speciation curves depend on the diameter and shape of the nanochannel and the bulk salt concentration. In the limit of wide channels, the apparent pK_a's and widths can be estimated by a simplified analytical model derived from the more general molecular theory. Both the general and the simplified theory predicts that, due to charge-regulation effects, the first acid-base transition (0/-1 transition) is wider than the second one (-1/-2), and both transitions are wider than the ideal one expected for an isolated acid-base group in the bulk. It is also shown that the inflection points of the conductance versus pH curves provide a very good estimation of the apparent pK_a's of the polyelectrolyte for cylindrical channels, but the quality of the estimation decreases for noncylindrical geometries.