Counterion adsorption on flexible polyelectrolytes: comparison of theories

Counterion adsorption on a flexible polyelectrolyte chain in a spherical cavity is considered by taking a "permuted" charge distribution on the chain so that the "adsorbed" counterions are allowed to move along the backbone. We compute the degree of ionization by using self-consistent field theory (SCFT) and compare with the previously developed variational theory. Analysis of various contributions to the free energy in both theories reveals that the equilibrium degree of ionization is attained mainly as an interplay of the adsorption energy of counterions on the backbone, the translational entropy of the small ions, and their correlated density fluctuations. Degree of ionization computed from SCFT is significantly lower than that from the variational formalism. The difference is entirely due to the density fluctuations of the small ions in the system, which are accounted for in the variational procedure. When these fluctuations are deliberately suppressed in the truncated variational procedure, there emerges a remarkable quantitative agreement in the various contributing factors to the equilibrium degree of ionization, in spite of the fundamental differences in the approximations and computational procedures used in these two schemes. Furthermore, it is found that the total free energies from the truncated variational procedure and the SCFT are in quantitative agreement at low monomer densities and differ from each other at higher monomer densities. The disagreement at higher monomer densities is due to the inability of the variational calculation to accurately compute the solvent entropy at higher concentrations. A comparison of electrostatic energies (which are relatively small) reveals that the Debye-Hückel estimate used in the variational theory is an overestimation of electrostatic energy as compared to the Poisson-Boltzmann estimate. Nevertheless, since the significant effects from density fluctuations of small ions are not captured by the SCFT, and due to the close agreement between SCFT and the other contributing factors in the more transparent variational procedure, the latter is a better computational tool for obtaining the degree of ionization.