Mathematical Modeling of Electrophoretic Motion in Colloidal Particles: Nonlinear Effects and Environmental Implications

Abstract

Electrophoresis plays a crucial role in microfluidics, environmental science, and biomedical engineering, enabling the precise manipulation of charged particles in fluidic systems. Traditional models, such as those by Smoluchowski and Hückel, assume a linear relationship between electrophoretic mobility and electric field strength. However, in high zeta potential (ζ) regimes and strong electric fields, these models fail due to nonlinear electrostatic and hydrodynamic effects. This study develops an advanced mathematical and computational framework that integrates finite Debye length corrections, ion crowding effects, and shear-dependent viscosity to more accurately describe electrophoretic motion in nonlinear regimes. The governing equations, including the Poisson equation for electrostatic potential, Navier-Stokes equations for fluid flow, and the Nernst-Planck equation for ion transport, are extended to incorporate nonlinearities. Asymptotic methods confirm that at low ζ, mobility trends align with classical predictions, but at high ζ, mobility saturates due to charge condensation and ion accumulation effects. Numerical solutions using Finite Element Method (FEM), spectral techniques, and Monte Carlo simulations validate these theoretical insights, revealing that nonlinear mobility suppression significantly deviates from classical models. Our findings demonstrate that nonlinear electrophoresis influences colloidal stability and aggregation, where weakened electrophoretic repulsion under strong fields enhances particle clustering. These insights have critical implications for pollutant dispersion in aqueous systems, microfluidic separation techniques, and nanoparticle transport in biological environments. Future work should focus on extending this model to non-spherical particles, incorporating turbulent flow effects, and further refining experimental validation. This study bridges the gap between classical electrokinetic theory and practical applications, contributing to nanotechnology, environmental remediation, and advanced biomolecular separation