Novel numerical method for calculating initial flux of colloid particle adsorption through an energy barrier

Using variable substitution, we present a general method for the numerical solution of stiff, ordinary, linear, homogeneous differential equations characteristic of colloid particle adsorption/deposition over an energy barrier. For the example of the radial impinging jet system, we demonstrate the application of this method of calculating the colloid concentration profile and initial particle flux in the presence of repulsive electrostatic interactions between the particle and adsorption surface. We show that our method works well in systems with energy barriers up to the order of hundreds of kT, at which point the adsorption flux vanishes. The numerical results obtained with our method are in good agreement with the known limiting analytical approximations for the particle flux through an energy barrier and for a low Péclet number. The developed numerical code is very stable over a wide range of physical parameters, and its accuracy for the most challenging parameter sets is on the order of 10−4. To achieve this stability, we have derived and employed a single formula for the van der Waals dispersion interaction, working at both a small and a large separation distance. We show that this formula converges to the known available analytical expressions for dispersion forces in the limit of small and large separation distance. We also demonstrate that the maximum deviations between our formula and the other equations appear in the intermediate range of the separation distance and do not exceed 10%.

Particle–interface interaction energy (dotted line), colloid concentration (solid line), and normal component of particle flux (dashed line) in a system with a high energy barrier.Figure optionsDownload as PowerPoint slide