Extracting free energies of counterion binding to polyelectrolytes by molecular dynamics simulations

We use all-atom molecular dynamics simulations to extract ΔG^eff, the free energy of binding of potassium ions K⁺ to the partially charged polyelectrolyte poly(acrylic acid), or PAA, in dilute regimes. Upon increasing the charge fraction of PAA, the chains adopt more extended conformations, and simultaneously, potassium ions bind more strongly (i.e., with more negative ΔG^eff) to the highly charged chains to relieve electrostatic repulsions between charged monomers along the chains. We compare the simulation results with the predictions of a model that describes potassium binding to PAA chains as a reversible reaction whose binding free energy (ΔG^eff) is adjusted from its intrinsic value (ΔG) by electrostatic correlations, captured by a random phase approximation. The bare or intrinsic binding free energy ΔG, which is an input in the model, depends on the binding species and is obtained from the radial distribution function of K⁺ around the charged monomer of a singly charged, short PAA chain in dilute solutions. We find that the model yields semi-quantitative predictions for ΔG^eff and the degree of potassium binding to PAA chains, α, as a function of PAA charge fraction without using fitting parameters.