Effective charges of polyelectrolytes in a salt-free solution based on counterion chemical potential

The phenomenon of counterion condensation around a flexible polyelectrolyte chain with N monomers is investigated by Monte Carlo simulations in terms of the degree of ionization α, which is proportional to the effective charge. It is operationally defined as the ratio of observed to intrinsic counterion concentration, a = C_o/C_i. The observed counterion concentration in the dilute polyelectrolyte solution is equivalent to an electrolyte solution of concentration c_o with the same counterion chemical potential. It can be determined directly by thermodynamic experiments such as ion-selective electrode. With the polyelectrolyte fixed at the center of the spherical Wigner-Seitz cell, the polymer conformation, counterion distribution, and chemical potential can be obtained. Our simulation shows that the degree of ionization rises as the polymer concentration decreases. This behavior is opposite to that calculated from the infinitely long charged rod model, which is often used to study counterion condensation. Moreover, we find that, for a specified line charge density, a decreases with an increment in chain length and chain flexibility. In fact, the degree of ionization is found to decline with increasing polymer fractal dimension, which can be tuned by varying bending modulus and solvent quality. Those results can be qualitatively explained by a simple model of two-phase approximation.