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1. A Pore-Scale Model for Electrokinetic In situ Recovery of Copper: The Influence of Mineral Occurrence, Zeta Potential, and Electric Potential

   https://doi.org/10.1007/s11242-023-02023-2  

   Tang, Kunning; Li, Zhe; Da Wang, Ying; McClure, James; Su, Hongli; Mostaghimi, Peyman; Armstrong, Ryan T (December 2023, Transport in Porous Media)

   AbstractElectrokinetic in-situ recovery is an alternative to conventional mining, relying on the application of an electric potential to enhance the subsurface flow of ions. Understanding the pore-scale flow and ion transport under electric potential is essential for petrophysical properties estimation and flow behavior characterization. The governing physics of electrokinetic transport is electromigration and electroosmotic flow, which depend on the electric potential gradient, mineral occurrence, domain morphology (tortuosity and porosity, grain size and distribution, etc.), and electrolyte properties (local pH distribution and lixiviant type and concentration, etc.). Herein, mineral occurrence and its associated zeta potential are investigated for EK transport. The new Ek model which is designed to solve the EK flow in complex porous media in a highly parallelizable manner includes three coupled equations: (1) Poisson equation, (2) Nernst–Planck equation, and (3) Navier–Stokes equation. These equations were solved using the lattice Boltzmann method within X-ray computed microtomography images. The proposed model is validated against COMSOL multiphysics in a two-dimensional microchannel in terms of fluid flow behavior when the electrical double layer is both resolvable and unresolvable. A more complex chalcopyrite-silica system is then obtained by micro-CT scanning to evaluate the model performance. The effects of mineral occurrence, zeta potential, and electric potential on the three-dimensional chalcopyrite-silica system were evaluated. Although the positive zeta potential of chalcopyrite can induce a flow of ferric ion counter to the direction of electromigration, the net effect is dependent on the occurrence of chalcopyrite. However, the ion flux induced by electromigration was the dominant transport mechanism, whereas advection induced by electroosmosis made a lower contribution. Overall, a pore-scale EK model is proposed for direct simulation on pore-scale images. The proposed model can be coupled with other geochemical models for full physicochemical transport simulations. Meanwhile, electrokinetic transport shows promise as a human-controllable technique because the electromigration of ions and the applied electric potential can be easily controlled externally. Graphical abstract 

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2. The Effects of Inductive Electric Field on the Spatial and Temporal Evolution of the Inner Magnetospheric Ring Current

   https://doi.org/10.1029/2020JA028554  

   Liu, Jianghuai; Ilie, Raluca (March 2021, Journal of Geophysical Research: Space Physics)

   Abstract Charged particles are observed to be injected into the inner magnetosphere from the plasma sheet and energized up to high energies over short distance and time, during both geomagnetic storms and substorms. Numerous studies suggest that it is the short‐duration and high‐speed plasma flows, which are closely associated with the global effects of magnetic reconnection and inductive effects, rather than the slow and steady convection that control the earthward transport of plasma and magnetic flux from the magnetotail, especially during geomagnetic activities. In order to include the effect of the inductive electric field produced by the temporal change of magnetic field on the dynamics of ring current, we implemented both theoretical and numerical modifications to an inner magnetosphere kinetic model—Hot Electron‐Ion Drift Integrator. New drift terms associated with the inductive electric field are incorporated into the calculation of bounce‐averaged coefficients for the distribution function, and their numerical implementations and the associated effects on total drift and energization rate are discussed. Numerical simulations show that the local particle drifts are significantly altered by the presence of inductive electric fields, in addition to the changing magnetic gradient‐curvature drift due to the distortion of magnetic field, and at certain locations, the inductive drift dominates both the potential and the magnetic gradient‐curvature drift. The presence of a self‐consistent inductive electric field alters the overall particle trajectories, energization, and pitch angle, resulting in significant changes in the topology and the strength of the ring current. 

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3. A model for the electric field-driven flow and deformation of a drop or vesicle in strong electrolyte solutions

   https://doi.org/10.1017/jfm.2022.469  

   Ma, Manman; Booty, Michael R.; Siegel, Michael (July 2022, Journal of Fluid Mechanics)

   A model is constructed to describe the flow field and arbitrary deformation of a drop or vesicle that contains and is embedded in an electrolyte solution, where the flow and deformation are caused by an applied electric field. The applied field produces an electrokinetic flow, which is set up on the charge-up time scale $$\tau _{*c}=\lambda _{*} a_{*}/D_{*}$$ , where $$\lambda _{*}$$ is the Debye screening length, $$a_{*}$$ is the inclusion length scale and $$D_{*}$$ is an ion diffusion constant. The model is based on the Poisson–Nernst–Planck and Stokes equations. These are reduced or simplified by forming the limit of strong electrolytes, for which dissolved salts are completely ionised in solution, together with the limit of thin Debye layers. Debye layers of opposite polarity form on either side of the drop interface or vesicle membrane, together forming an electrical double layer. Two formulations of the model are given. One utilises an integral equation for the velocity field on the interface or membrane surface together with a pair of integral equations for the electrostatic potential on the outer faces of the double layer. The other utilises a form of the stress-balance boundary condition that incorporates the double layer structure into relations between the dependent variables on the layers’ outer faces. This constitutes an interfacial boundary condition that drives an otherwise unforced Stokes flow outside the double layer. For both formulations relations derived from the transport of ions in each Debye layer give additional boundary conditions for the potential and ion concentrations outside the double layer. 

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4. Instability of a planar fluid interface under a tangential electric field in a stagnation point flow

   https://doi.org/10.1017/jfm.2021.967  

   Firouznia, Mohammadhossein; Miksis, Michael J.; Vlahovska, Petia M.; Saintillan, David (January 2022, Journal of Fluid Mechanics)

   The interface between two immiscible fluids can become unstable under the effect of an imposed tangential electric field along with a stagnation point flow. This canonical situation, which arises in a wide range of electrohydrodynamic systems including at the equator of electrified droplets, can result in unstable interface deflections where the perturbed interface gets drawn along the extensional axis of the flow while experiencing strong charge build-up. Here, we present analytical and numerical analyses of the stability of a planar interface separating two immiscible fluid layers subject to a tangential electric field and a stagnation point flow. The interfacial charge dynamics is captured by a conservation equation accounting for Ohmic conduction, advection by the flow and finite charge relaxation. Using this model, we perform a local linear stability analysis in the vicinity of the stagnation point to study the behaviour of the system in terms of the relevant dimensionless groups of the problem. The local theory is complemented with a numerical normal-mode linear stability analysis based on the full system of equations and boundary conditions using the boundary element method. Our analysis demonstrates the subtle interplay of charge convection and conduction in the dynamics of the system, which oppose one another in the dominant unstable eigenmode. Finally, numerical simulations of the full nonlinear problem demonstrate how the coupling of flow and interfacial charge dynamics can give rise to nonlinear phenomena such as tip formation and the growth of charge density shocks. 

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5. Integral equation method for the 1D steady-state Poisson-Nernst-Planck equations

   https://doi.org/10.1007/s10825-023-02092-y  

   Chao, Zhen; Geng, Weihua; Krasny, Robert (September 2023, Journal of Computational Electronics)

   Abstract An integral equation method is presented for the 1D steady-state Poisson-Nernst-Planck equations modeling ion transport through membrane channels. The differential equations are recast as integral equations using Green’s 3rd identity yielding a fixed-point problem for the electric potential gradient and ion concentrations. The integrals are discretized by a combination of midpoint and trapezoid rules, and the resulting algebraic equations are solved by Gummel iteration. Numerical tests for electroneutral and non-electroneutral systems demonstrate the method’s 2nd order accuracy and ability to resolve sharp boundary layers. The method is applied to a 1D model of the K$$^+$$ ${}^{+}$ ion channel with a fixed charge density that ensures cation selectivity. In these tests, the proposed integral equation method yields potential and concentration profiles in good agreement with published results. 

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