Structure of Polyelectrolytes with Mixed Monovalent and Divalent Counterions: SAXS Measurements and Poisson-Boltzmann Analysis

[GRAPHICS] . We have studied by small angle X-ray scattering (SAXS) the structure of salt-free polyelectrolytes solutions containing monovalent and divalent counterions. We have considered mixtures of sulfonated polystyrene with monovalent (Na+) and divalent (Ca2+) counterions and measured the position of the scattering peak, q*, as a function of the monomer concentration c(p) and the monovalent/divalent content. The aim is to understand the variations observed in q* position when the valence of the counterions is gradually increased. This work is a continuation of a previous study in which first measurements were performed on a rather small number of sodium-PSS/calcium-PSS mixtures. In the present work, we used synchrotron radiation to improve the quality of the data and vary the monovalent/divalent ratio with a much finer step. Indeed this gives new interesting results in the ranges of low and large divalent content. We analyzed SAXS results through the isotropic model and scaling approach description introduced by de Gennes et al. and developed by Dobrynin et al.. In this model, one key parameter is the chemical charge and/or the effective charge fraction f(eff) of the polyions. Although the chemical charge fraction f of sodium-PSS and calcium-PSS polyelectrolyte is fixed by the synthesis, the effective charge fraction in mixtures varies with the monovalent/divalent ratio. This quantity has been calculated solving the Poisson-Boltzmann equation in the frame of the cell model for various monovalent/divalent contents and different concentrations. Severe deviations are found in the effective charge values of mixtures at finite concentrations compared to the classical Manning-Oosawa prediction (infinite dilution limiting law). We demonstrate that the evolution of q* is still compatible with the isotropic model and the scaling approach in the low concentration range provided that the divalent content is not too high. In particular, a power law relation q* proportional to f(eff) (similar to 0.3) can be found which looks very close to the one observed for weakly charged polyelectrolytes (q* proportional to f(2/7) in good solvent or q* proportional to f(1/3) in Theta solvent). Mixtures finally provide a way to adjust the effective charge fraction without changing the chemical nature of the polyions. However this procedure gives improvements to data prediction only in a limited range; it is still not able to fully explain the high concentration range, as well as the high divalent content mixtures. This is certainly due to the fact that the Poisson-Boltzmann theory is not able to take into account local interactions between monomers and divalent counterions, which goes beyond the mean field approach.