Mathematical modeling of pigment dispersion taking into account the full agglomerate particle size distribution

The purpose of this work is to develop a mathematical model that can quantify the dispersion of pigments, with a focus on the mechanical breakage of pigment agglomerates. The underlying physical mechanism was assumed to be surface erosion of spherical pigment agglomerates. The full agglomerate particle size distribution was simulated. Data from two previous experimental investigations were used for model validation. The first concerns two different yellow organic pigments dispersed in nitrocellulose/ethanol vehicles in a ball mill and the second a red organic pigment dispersed in a solvent-based acrylic vehicle in a three-roll mill. When the linear rate of agglomerate surface erosion was taken to be proportional to the external agglomerate surface area, simulations of the volume-moment mean diameter over time were in good quantitative agreement with experimental data for all three pigments. The only adjustable parameter used was an apparent rate constant for the linear agglomerate erosion rate. Model simulations, at selected values of time, for the full agglomerate particle size distribution were in good qualitative agreement with the measured values. A quantitative match of the experimental particle size distributions could be obtained using time-dependent fragment distributions, but this resulted in a very slight improvement in the simulated transient mean diameter only. The model provides a mechanistic understanding of the agglomerate breakage process that can be used, e.g., in the development of novel dispersion principles and for analysis of dispersion failures. The general applicability of the model, beyond the three pigments considered, needs to be confirmed.