Thermodynamic model for dispersion in nanocomposites.

Dispersion of nanoparticles can involve thermally driven diffusion of particles or kinetically driven mixing. On the nano- to colloidal scale a virial expansion of osmotic pressure can be used to quantify dispersion. Scattering can be used to measure the osmotic compressibility associated with concentration fluctuations which are related to the second virial coefficient. These fluctuations might be created thermally or by kinetic mixing. For multi-hierarchical materials such as aggregates which cluster to form a macroscopic network, each level of structure contributes to the free energy. This can be quantified in terms of the total number of free particles contributed by each of the levels of structure. For different levels in a hierarchy the energy of aggregation can be associated with an aggregation equilibrium or an apparent equilibrium. The resulting number of particles can be related to the energy of aggregation. This approach is demonstrated for three systems, pigments dispersed with nonionic surfactants; worm-like micelles; and condensation polymerization.

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