A Mathematical Model for Colloidal Aggregation

The characterization of fine particles is an area of immense significance to many industrial endeavors. It has been estimated that 70% of all industrial processes deal with fine particles at some point in the process. A natural phenomenon occurring in these processes is colloidal aggregation. This study examines aggregation in colloidal systems in order to characterize, examine, and control this occurrence in industrial processes. The study of particle aggregation has been broken into many different areas, such as collision mechanisms, interaction energy etc, but a complete model that integrates these different aspects has never been fully realized. A new model is required to accurately predict the aggregation behavior of colloidal particles. In this work, a new model is developed that integrates Smoluchowski kinetics, total interaction energy between particles, and stability ratios for perikinetic and orthokinetic collision mechanisms. The total particle interaction energy necessary for the calculation of stability ratios is represented by the summation of electrostatic and van der Waals interactions. The electrostatic interactions are modeled using DLVO theory, the linear Poisson-Boltzmann equation, and a numerical solution for the non-linear Poisson-Boltzmann Equation, while the van der Waals interactions are represented by Hamaker theory. The mathematical model is solved using an adjustable discretion technique, which is tested against a specific analytic solution, and yields an assessment of the error intrinsic in the discretization method. The basis of the mathematical model is a population balance framework. The model developed in this study is general in many respects, but could be readily applied to many different aggregation systems with minor modification. A comparison of the mathematical model with previous experiments conducted by Scott Fisher (1998) is carried out for the perikinetic and orthokinetic transport-limited aggregation regimes. The fractal nature of solid-sphere aggregates is considered when comparing the mathematical model predictions with experimental measurements. The previous experiments that are used for comparison utilized polystyrene particles ranging from 100 nm to 500 nm in initial diameter, several initial particle concentrations, and various stirring rates. Zeta potential measurements are presented in order to set the range of transport-limited aggregation. An assessment of the results of the mathematical model with the experimental results show good agreement for transport-limited aggregation within the perikinetic and orthokinetic transport-limited aggregation, with average particle sizes ranging from 100 nm to well over 2 microns