Contribution of small ions to the scattered intensity in the hypernetted chain approximation: Application to micellar solutions

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Abstract
The HNC approximation is applied to the primitive model of asymmetrical electrolytes that mimic charged micellar solutions. The structure factors Sij(q) are calculated for two or three component systems (polyions, counterions, and coions). The scattered intensity I(q) is obtained by two approaches: The first one considers the structural system where the structural charge of the polyion is equal to the aggregation number. The second one defines an effective system with a larger and less charged effective polyion which contains ‘‘condensed counterions.’’ We show the equivalence between both approaches in different cases and particularly in the case where the counterions contribute significantly to the scattered intensity. An application is presented: the investigation of a micellar solution of octyltrimethylammonium bromide with x-ray scattering. In this case, the high scattering length density of bromide ions contributes strongly to the total scattered intensity. The experimental curve which presents a large intensity at high scattering vectors can be fitted with the structural model. The structural charge, or aggregation number, which is the single fit parameter, is equal to 27±4.