Hierarchical approach to aggregate equilibria

Hierarchical aggregation is generally viewed as a kinetic phenomenon governed by kinetic growth laws, such as in the Smoluchowski equation, and modeled using diffusion or reaction limited kinetic growth models. Some aggregates, especially those controlled by surface grafting or surfactants, display reversible stability. For these equilibrated aggregates a simple thermodynamic model is proposed to describe the size distribution and the enthalpy and entropy of aggregation. The model uses the average degree of aggregation, z_{i(i−1)}, as the central quantifying parameter. Here i is an index reflecting the hierarchical level of structure in an aggregate, for instance, composed of crystals (i=0), clustered primary particles (i=1), aggregates (i=2), and agglomerates of aggregates (i=3). A change in Gibbs free energy for aggregation is given by ΔG_{i(i−1)}=−RTln(1/z_{i(i−1)}) for each level (i>0). This expression is advantageous since the degree of aggregation is directly determined in small-angle neutron and x-ray scattering, by transmission electron microscopy, simulation, or through spectroscopy. The atomistic hierarchical model enables an understanding of the mechanism of equilibrium aggregation since it provides expressions for entropy and enthalpy of aggregation at each structural/thermodynamic level. The model can be extended to describe pseudoequilibrium for industrially relevant materials such as condensation polymers. Applications in organic pigments and wormlike micelles are also briefly demonstrated.