Explaining Giant Apparent $\mathrm{p}{K}_a$ Shifts in Weak Polyelectrolyte Brushes

Recent experiments on weak polyelectrolyte brushes found marked shifts in the effective $\mathrm{p}{K}_a$ that 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 demonstrate that the magnitude of the polyelectrolyte effect is almost invariant with respect to salt concentration but depends on the grafting density of the brush. This invariance is due to a complex cancellation of multiple effects. Based on our results, we show how the experimentally determined $\mathrm{p}{K}_a$ shifts may be used to infer the grafting density of brushes, a parameter that is difficult to measure directly.

Figure 1

(a) Snapshot of the brush model (small ions are not shown for clarity). Black lines bound to the simulation box and additional periodic images in the $x\text{−}y$ plane are shown to illustrate the periodic boundary conditions. The snapshot was produced using the visualization software VMD [35]. (b) Schematic representation of the simulation setup: simulation box containing the brush, coupled to a reservoir at a fixed pH and salt concentration.

Figure 2

Titration curve of the weak polyelectrolyte brush with different grafting densities as a function of the pH and for different salt concentrations. Markers correspond to simulation results, while the dashed lines indicate the SCF results. As a reference, the ideal titration curve is shown as described by the Henderson-Hasselbalch equation (HH). Subfigure (a) shows the data for the higher grafting density and subfigure (b) the data for the lower grafting density.

Figure 3

The $\mathrm{p}{K}_a$ shift of brushes with different grafting densities as a function of ionic strength. Solid symbols represent data from particle-based simulations, whereas empty symbols represent SCF results.

Figure 4

Donnan and polyelectrolyte effect in the brush with a grafting density of $\mathrm{\Gamma }=0.079/{\mathrm{nm}}^2$ as a function of pH at different salt concentrations. (a) Difference between the local pH inside the brush and pH in the bulk. (b) Degree of ionization of the brush in comparison with the ideal Henderson-Hasselbalch equation (HH).