Magnetically Assisted Separation of Weakly Magnetic Metal Ions in Porous Media. Part 2: Numerical Simulations

TL;DR

This work develops a two-dimensional multiphysics framework to predict magnetophoresis of weakly magnetic metal ions in porous media under static, nonuniform magnetic fields, focusing on MnCl$_2$ (paramagnetic) and ZnCl$_2$ (diamagnetic) in silica gels. It couples static magnetic-field calculations with drift-diffusion transport and investigates two drag models (Stokes and Brinkman) to capture pore-scale resistance, finding that permeability-driven Brinkman dynamics—calibrated with measured permeability—are essential to reproduce experimental transport trends. The simulations reveal field-induced clustering as a key mechanism, with cluster size evolving in time according to a power-law and dictating transient and steady transport; paramagnetic forces contribute modestly but non-negligibly, while the Kelvin-field gradient force dominates. In binary mixtures, hydrodynamic coupling between paramagnetic and diamagnetic clusters alters transport, indicating collective effects and interspecies interactions that must be accounted for in magnetically assisted separations. Overall, the Brinkman formulation with accurate permeability and dynamically evolving clusters provides a predictive framework for magnetically driven ion transport in complex porous systems, with implications for energy-efficient separation technologies.

Abstract

We present a numerical investigation of the magnetophoresis of metal ions in porous media under static, nonuniform magnetic fields. The multiphysics simulations couple momentum transport, mass diffusion, and magnetic field equations, with the porous medium modeled using two distinct approaches: a Stokes-based formulation incorporating effective diffusivity, and a Brinkman-based formulation that explicitly accounts for permeability and medium-induced drag. Comparison with recent experimental data [Nwachuwku et al. Submitted, 2025] reveals that the Stokes model partially fails to capture key trends, while the Brinkman model, with permeability accurately reproduces observed transport behavior on various porous media. Our simulations predict that both paramagnetic (MnCl2) and diamagnetic (ZnCl2) ions may form field-induced clusters under magnetic gradients over a range of concentrations of 1mM-100mM and magnetic field gradients of up to 100 T2/m. The dominant driving force is found to be the magnetic gradient (Kelvin) force, while the paramagnetic force from concentration gradients contributes minimally. In binary mixtures, hydrodynamic interactions between paramagnetic and diamagnetic clusters significantly alter transport dynamics. Specifically, paramagnetic clusters can pull diamagnetic clusters along the magnetic field gradient, enhancing diamagnetic migration and suppressing the motion of paramagnetic species. These findings highlight the importance of porous media modeling and interspecies interactions in predicting magnetophoretic transport of ionic mixtures.

Figures (16)

• Figure 1: Schematic representation of the experimental set-up used in part (I) of this series (a) and a cross-section of the domain showing the NdFeB permanent magnet and the cell containing porous media along with the representative mesh used in 2D simulations (b).
• Figure 2: (a) Magnetic flux density ($\mathbf{B}$) map calculated by 2D-Simulations. (b) Normalized percentage difference between computed magnetic flux density and the experiments. (c) Calculated magnetic flux density gradient $\mathbf{B}\cdot\nabla \mathbf{B}$ in 2D-simulations. (d) Normalized percentage difference in magnetic field gradients between experiments reported in part (I) and calculations of this study.
• Figure 3: Temporal evolution of metal ion concentration along with the predictions of the model (solid and dashed curves) for MnCl$_2$ (a) and ZnCl$2$ (b) at an initial concentration of c$_\text{0}$ = 100mM. In part (a) and (b), the filled and open symbols denote the concentrations in the near and far sections, respectively. The dashed line shows the model predictions for metal ion hydration size, and the continuous curves denote the model predictions for a constant cluster size. The spatio-temporal evolution of metal ion concentration along with velocity vectors and velocity contours for MnCl$_2$(c) and ZnCl$_2$ (d) for an initial concentration of 100mM after t = 0 (i), t = 1 (ii), t = 12 (iii), and t = 72 (iv) hours. Velocity labels in (c,d) have a unit of nm/s.
• Figure 4: Temporal evolution of metal ion concentration along with the predictions of the model (solid curves) for MnCl$_2$ (a) and ZnCl$2$ (b) at initial concentration c$_\text{0}$ = 100mM. In part (a) and (b), the filled and open symbols denote the concentrations in the near and far halves. The continuous curves denote the model predictions for a dynamically evolving cluster size. The spatio-temporal evolution of metal ion concentration along with velocity vectors and velocity contours for MnCl$_2$(c) and ZnCl$_2$ (d) for an initial concentration of 100mM after t = 0 (i), t = 1 (ii), t = 12 (iii), and t = 72 (iv) hours. Velocity labels in (c,d) have a unit of nm/s.
• Figure 5: Temporal evolution of metal ion concentration for (a) MnCl$_2$ and (b) ZnCl$_2$ at c$_0$ = 100 mM. In (a,b) the dashed line, dotted line and the continuous lines correspond to model predictions by including paramagnetic force, Kelvin force and both forces, respectively. The ratio of the paramagnetic force to that of the Kelvin force as a function of time for MnCl$_2$ (c) and ZnCl$_2$ (d) at various initial concentrations.