Accurate Determination of the Quantity and Spatial Distribution of Counterions around a Spherical Macroion

The accurate distribution of countercations (Rb⁺ and Sr²⁺ ) around a rigid, spherical, 2.9-nm size polyoxometalate cluster, {Mo₁₃₂ }^42- , is determined by anomalous small-angle X-ray scattering. Both Rb⁺ and Sr²⁺ ions lead to shorter diffuse lengths for {Mo₁₃₂ } than prediction. Most Rb⁺ ions are closely associated with {Mo₁₃₂ } by staying near the skeleton of {Mo₁₃₂ } or in the Stern layer, whereas more Sr²⁺ ions loosely associate with {Mo₁₃₂ } in the diffuse layer. The stronger affinity of Rb⁺ ions towards {Mo₁₃₂ } than that of Sr²⁺ ions explains the anomalous lower critical coagulation concentration of {Mo₁₃₂ } with Rb⁺ compared to Sr²⁺ . The anomalous behavior of {Mo₁₃₂ } can be attributed to majority of negative charges being located at the inner surface of its cavity. The longer anion-cation distance weakens the Coulomb interaction, making the enthalpy change owing to the breakage of hydration layers of cations more important in regulating the counterion-{Mo₁₃₂ } interaction.