Particle charging bounds, symmetry relations, and an analytic charging rate model for the continuum regime

For many years, ionic particle charging has been described in terms of field or diffusion charging because of their simplicity as models. Important advances in charging theories have come with publication of numerical models of the process. Still, numerical models exact a computational penalty that has led to investigations of approximations for them. In this paper, we present new analytical upper and lower charging bounds for continuum charging, useful for checking the accuracy of charging models. The best of the numerical models predicts charging within these bounds over much of its range, but exceeds the upper bound at moderate fields and long charging times. The standard models and the charging bounds exhibit a symmetry between positive and negative charges which may provide another test for numerical theories. We formulate an analytic charging rate model that remains within the bounds and gives good agreement with unipolar and bipolar charging data. The charging rate symmetry makes bipolar charging calculations quite simple. The adequacy of existing data for testing particle charging models is also discussed.