Charge Regulation as a Stabilization Mechanism for Shell-Like Assemblies of Polyoxometalates

We show that the equilibrium size of single-layer shells composed of polyoxometalate macroions is inversely proportional to the dielectric constant of the medium in which they are dispersed. This behavior is consistent with a stabilization mechanism based on Coulomb repulsion combined with charge regulation. We estimate the cohesive energy per bond between macroions on the shells to be approximately $-6kT$. This number is extracted from analysis based on a charge regulation model in combination with a model for defects on a sphere. The value of the cohesive bond energy is in agreement with the model-independent critical aggregate concentration. This observation points to a new class of thermodynamically stable shell-like objects. We point out the possible relevance our findings have for certain surfactant systems.

Figure 1

Polyoxometalates (POMs) ${\mathrm{Mo}}_{72}{\mathrm{Fe}}_{30}$ (top, left) and ${\mathrm{Mo}}_{132}$ (bottom, left) and a cartoon of their shell-like superstructure where the size range refers to the situation in aqueous solutions. The 12 fivefold symmetrical units (with their bipyramid centers of ${\mathrm{MoO}}_7$) are linked by 30 iron (${\mathrm{Mo}}_{72}{\mathrm{Fe}}_{30}$) or ${\mathrm{Mo}}_2^{\mathrm{V}}$ groups (${\mathrm{Mo}}_{132}$). The ${\mathrm{Mo}}_2^{\mathrm{V}}$ groups are in turn stabilized by bidentate acetate ligants (not shown). Note the binuclear linkers of ${\mathrm{Mo}}_{132}$ in comparison to the mononuclear linkers of ${\mathrm{Mo}}_{72}{\mathrm{Fe}}_{30}$. The cartoons in this figure are not to the scale.

Figure 2

Shell radius as determined by dynamic light scattering as a function of the inverse relative dielectric constant of the solvent for two different polyoxometalates: ${\mathrm{Mo}}_{72}{\mathrm{Fe}}_{30}$ and ${\mathrm{Mo}}_{132}$. The theory of this Letter suggests a linear relation between the radius and the inverse dielectric constant. Dielectric constants were varied by using acetone–water mixtures (10–70% in volume) in case of ${\mathrm{Mo}}_{132}$. For the samples with (${\mathrm{Mo}}_{72}{\mathrm{Fe}}_{30}$), the pure solvents water, methanol, ethanol, propanol, and acetone were used. Except for two points corresponding to (${\mathrm{Mo}}_{72}{\mathrm{Fe}}_{30}$) in pure propanol and acetone (as indicated; also see text), data correspond to thermodynamically stable shells. The slope of the line is given by $3824\pm 266\mathrm{nm}$.