Scattering Functions of Flexible Polyelectrolytes in the Presence of Mixed Valence Counterions: Condensation and Scaling

Abstract

We have investigated by small-angle X-ray and neutron scattering the structure of salt-free aqueous solutions of sulfonated polystyrene with variable fractions of monovalent (sodium, Na) and divalent (calcium, Ca) counterions. We follow the variation of the position of the maximum in the scattered intensity, q*, as a function of the concentration, c, and the fraction of Ca counterions. A relevant parameter is the degree of condensation of Na and Ca counterions in the different mixtures. We survey the literature and check that, in the case of monovalent counterions, the variation of q* with c can be described by the scaling theory of Dobrynin, Colby, and Rubinstein, provided the counterion condensation is included according to the Manning-Oosawa approach. Scattering data from mixed valence systems are still in agreement with this scaling model at lower concentrations, except for pure CaPSS 2 aqueous solutions. For increasing concentration, a discrepancy is enhanced that extends toward higher Ca fractions. This indicates that counterion condensation is not the only process. Another possible one is a retraction of the macroions due to divalent counterions that induce bridging-type correlations between monomers, as already observed on the form factors of CaPSS 2 and NaPSS plus added CaCl 2 salt. Such a retraction modifies the scaling, in particular, via a change in the overlap concentration c*. Furthermore, it is noticeable that q* measurements have to be handle with care. We show that misinterpretation of the characteristic maximum in the small-angle scattering of polyelectrolyte solutions may occur, especially at high concentration, if the contribution of the condensed counterions to the total scattering function is predominant.