A population balance equation model to predict regimes of controlled nanoparticle aggregation

• Formation of stable finite-sized clusters observed in charged colloidal dispersions.
• A population balance equation (PBE) model was developed to capture this phenomenon.
• Model successfully predicts aggregation regimes and final aggregate size.

Forming stable clusters or aggregates of nanoparticles is of interest in a number of emerging applications. While formation of unstable fractal aggregates and flocs has been well-studied with both experiments and theory, the conditions that lead to stable, finite-sized clusters is not as well understood. Here, we present an integrated experimental and modeling study to explore aggregation in concentrated attractive colloidal suspensions. A population balance equation (PBE) model is used to predict the aggregation dynamics of quiescent colloidal suspensions. A DLVO (Derjaguin–Landau–Verwey–Overbeek) type potential is used to describe the interparticle potential, with attractive interactions arising from van der Waals forces and long-range repulsive interactions caused by electrostatics. The PBE model includes a full calculation of stability ratio variations as a function of aggregate size, such that the energy barrier increases with increasing size. As the ionic strength is decreased, the model predicts three regimes of behavior: uncontrolled aggregation into large flocs, controlled aggregation into stable clusters, and no aggregation. The model is tested experimentally using latex particles at different salt concentrations and particle concentrations. When the Hamaker constant and surface potential are fit to aggregate size measurements collected at one salt concentration, the model accurately predicts the final mean aggregate size and regimes of aggregation at other salt concentrations and the same particle concentration. This result suggests that van der Waals and electrostatic forces are the dominant particle interactions in determining the final aggregate state. The mean aggregate size and aggregation regimes at different particle concentrations could be accurately predicted by adjusting the surface potential. This parameter adjustment is consistent with the expectation that increasing colloid weight fractions cause aggregates to have a more fractal nature and hence have a lower effective repulsion. However, the model predicts much faster aggregation rates than what are observed experimentally. This discrepancy may be due to hydrodynamic effects or another slow dynamical process which is not accounted for in the model. Nevertheless, this study presents the first PBE model that can successfully predict stable aggregate size and aggregation regimes of charged colloidal particles over a range of salt concentrations and particle concentrations.

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