Poisson Nernst Planck Equations Research Articles

Hydration kinetics of Portland cement shifting from silicate to aluminate dominance based on multi-mineral reactions and interactions

A theoretical model for cement hydration was proposed to study multi-mineral reactive transport processes under various mineral compositions and determine the effect of multi-mineral reactions and interactions on cement hydration kinetics shifting from silicate to aluminate dominance. The reaction rates of each mineral dissolution, product precipitation, ionic diffusion, and adsorption were calculated individually through the degrees of undersaturation and supersaturation associated with the ionic concentration, all of which were coupled in the modified Poisson–Nernst–Planck equation. The hydration heat flow was then theoretically calculated by the superposition of the reaction rates of silicate and aluminate phases, which was derived from the calculated ionic concentration. The model was validated by comparison with experimental data obtained under various conditions, showing consistency. The combined effect of multi-mineral reactions and interactions on hydration kinetics was investigated using the model, and the results indicated that (1) faster dissolution of gypsum or a higher ratio of tricalcium aluminate to tricalcium silicate leads to a larger time interval between silicate and aluminate peaks; (2) faster precipitation of calcium silicate hydrate results in a more significant difference between silicate and aluminate peaks; and (3) the sulfate ion retards cement hydration kinetics shifting from silicate to aluminate dominance.

Diffusiophoresis of a soft particle incorporating ion partitioning and hydrophobic core

The diffusiophoresis of a soft particle governed by an imposed ionic concentration gradient is studied based on a modified ion transport equation. The surface of the rigid core of the soft particle is considered to be hydrophobic, which is coated with a charged polyelectrolyte layer (PEL) whose dielectric constant is different from the dispersion medium. This step-change in the dielectric constant of the PEL and the dispersion medium creates an ion partitioning effect. The governing equations based on the coupled Navier–Stokes–Nernst–Planck–Poisson equations are modified to incorporate the ion partitioning effect. A simplified model based on the linear perturbation technique under a small applied concentration gradient in the scale of the particle radius is developed. The simplified model agrees well with the exact numerical solutions for a Debye length smaller than the particle size. We have analyzed the impact of the core slip length and PEL permittivity on the mobility of the soft particle. The interfacial slip enhances the impact of the core charge density on the diffusiophoresis. In this case, a reversal of mobility occurs at a thinner Debye length compared to a hydrophilic core. Ion partitioning attenuates the concentration of mobile counterions in PEL, leading to a reduction in screening of PEL fixed charge. We find that the diffusiophoresis of the soft particle in a non-z:z asymmetric electrolyte strongly depends on the valence of the counterions. Diffusiophoresis of the soft particle in a KCl-NaCl mixture is considered to achieve positive mobility.

Energy dissipative and positivity preserving schemes for large-convection ion transport with steric and solvation effects

Ion transport plays a crucial role in biophysical and electrochemical applications. While being modeled by the classical Poisson–Nernst–Planck (PNP) equations, many ionic features are ignored in the course-graining modeling of ionic interactions. In this work, we study ion transport by modified PNP equations including effects arising from steric interactions and the Born solvation, which both could cause strong convection. The modified NP equations are first rewritten in a gradient-flow reformulation that involves mobility and chemical potential terms. The mobility is then approximated by upwind fluxes with various flux limiters on the half-grid points, and the chemical potential terms are then implicitly approximated by central differencing. The resulting first-order in time scheme is proved to be energy dissipative unconditionally. Furthermore, the unconditional existence of positive numerical concentration is proved by making use of the singularity of the entropy term, which prevents the concentration from approaching zero. The flux with the Minmod or Osher limiter is reduced to a second-order upwind flux in the strong convective limit. Numerical tests are performed to confirm the expected order of accuracy and the properties of the developed scheme at discrete level. In addition, our numerical experiments demonstrate that the developed schemes with limiters can achieve higher accuracy than the Scharfetter-Gummel scheme and classical upwind scheme in tests with strong convection. Finally, the developed schemes are applied to numerically assess the effects of Coulombic interaction, steric interaction, and Born solvation on the rectifying transport of ions through nanopores.

New mixed finite element methods for the coupled Stokes and Poisson–Nernst–Planck equations in Banach spaces

In this paper we employ a Banach spaces-based framework to introduce and analyze new mixed finite element methods for the numerical solution of the coupled Stokes and Poisson–Nernst–Planck equations, which is a nonlinear model describing the dynamics of electrically charged incompressible fluids. The pressure of the fluid is eliminated from the system (though computed afterwardsviaa postprocessing formula) thanks to the incompressibility condition and the incorporation of the fluid pseudostress as an auxiliary unknown. In turn, besides the electrostatic potential and the concentration of ionized particles, we use the electric field (rescaled gradient of the potential) and total ionic fluxes as new unknowns. The resulting fully mixed variational formulation in Banach spaces can be written as a coupled system consisting of two saddle-point problems, each one with nonlinear source terms depending on the remaining unknowns, and a perturbed saddle-point problem with linear source terms, which is in turn additionally perturbed by a bilinear form. The well-posedness of the continuous formulation is a consequence of a fixed-point strategy in combination with the Banach theorem, the Babuška–Brezzi theory, the solvability of abstract perturbed saddle-point problems, and the Banach–Nečas–Babuška theorem. For this we also employ smallness assumptions on the data. An analogous approach, but using now both the Brouwer and Banach theorems, and invoking suitable stability conditions on arbitrary finite element subspaces, is employed to conclude the existence and uniqueness of solution for the associated Galerkin scheme.A priorierror estimates are derived, and examples of discrete spaces that fit the theory, include,e.g., Raviart–Thomas elements of orderkalong with piecewise polynomials of degree ≤k. In addition, the latter yield approximate local conservation of momentum for all three equations involved. Finally, rates of convergence are specified and several numerical experiments confirm the theoretical error bounds. These tests also illustrate the aforementioned balance-preserving properties and the applicability of the proposed family of methods.

Long-Range ACEO Phenomena in Microfluidic Channel

Microfluidic devices are increasingly utilized in numerous industries, including that of medicine, for their abilities to pump and mix fluid at a microscale. Within these devices, microchannels paired with microelectrodes enable the mixing and transportation of ionized fluid. The ionization process charges the microchannel and manipulates the fluid with an electric field. Although complex in operation at the microscale, microchannels within microfluidic devices are easy to produce and economical. This paper uses simulations to convey helpful insights into the analysis of electrokinetic microfluidic device phenomena. The simulations in this paper use the Navier–Stokes and Poisson Nernst–Planck equations solved using COMSOL to determine the maximum attainable fluid velocity with an electric potential applied to the microchannel and the most suitable frequency or voltage to use for transporting the fluid. Alternating current electroosmosis (ACEO) directs and provides velocity to the ionized fluid. ACEO can also mix the fluid at low frequencies for the purpose of dispersing particles. DC electroosmosis (DCEO) applies voltage along the microchannel to create an electric field that ionizes fluid within the microchannel, making it a cost-effective method for transporting fluid. This paper explores a method for an alternate efficient utilization of microfluidic devices for efficient mixing and transportation of ionized fluid and analyzes the electrokinetic phenomena through simulations using the Navier–Stokes and Poisson Nernst–Planck equations. The results provide insights into the parameters at play for transporting the fluid using alternating current electroosmosis (ACEO) and DC electroosmosis (DCEO).

Nano-scale solution of the Poisson-Nernst-Planck (PNP) equations in a fraction of two neighboring cells reveals the magnitude of intercellular electrochemical waves.

The basic building blocks of the electrophysiology of cardiomyocytes are ion channels integrated in the cell membranes. Close to the ion channels there are very strong electrical and chemical gradients. However, these gradients extend for only a few nano-meters and are therefore commonly ignored in mathematical models. The full complexity of the dynamics is modelled by the Poisson-Nernst-Planck (PNP) equations but these equations must be solved using temporal and spatial scales of nano-seconds and nano-meters. Here we report solutions of the PNP equations in a fraction of two abuttal cells separated by a tiny extracellular space. We show that when only the potassium channels of the two cells are open, a stationary solution is reached with the well-known Debye layer close to the membranes. When the sodium channels of one of the cells are opened, a very strong and brief electrochemical wave emanates from the channels. If the extracellular space is sufficiently small and the number of sodium channels is sufficiently high, the wave extends all the way over to the neighboring cell and may therefore explain cardiac conduction even at very low levels of gap junctional coupling.

Impedance-Frequency Response of Closed Electrolytic Cells

The electric AC response of electrolytic cells with DC bias is analyzed solving numerically the Poisson-Nernst-Planck equations and avoiding the commonly used infinite solution approximation. The results show the presence of an additional low-frequency dispersion process associated with the finite spacing of the electrodes. Moreover, we find that the condition of fixed ionic content inside the electrolytic cell has a strong bearing on both the steady-state and the frequency response. For example: the characteristic frequency of the high-frequency dispersion decreases when the DC potential increases and/or the electrode spacing decreases in the closed cell case, while it remains essentially insensitive on these changes for open cells. Finally, approximate analytic expressions for the dependences of the main parameters of both dispersion processes are also presented.

Existence of [formula omitted] singular solutions to Euler–Nernst–Planck–Poisson system on [formula omitted] with free-moving charges

We construct a special C1,α blow up solution to the three dimensional system modeling electro-hydrodynamics, which is strongly coupled with incompressible Euler equation and Nernst–Planck–Poisson equation. Our construction lies on the framework established in Elgindi et al. (2021) and relies on a special solution to variant spherical Laplacian.

Convergence analysis of structure-preserving numerical methods based on Slotboom transformation for the Poisson–Nernst–Planck equations

Convergence analysis of structure-preserving numerical methods based on Slotboom transformation for the Poisson–Nernst–Planck equations

Continuous Submicron Particle Separation Via Vortex-Enhanced Ionic Concentration Polarization: A Numerical Investigation.

Separation and isolation of suspended submicron particles is fundamental to a wide range of applications, including desalination, chemical processing, and medical diagnostics. Ion concentration polarization (ICP), an electrokinetic phenomenon in micro-nano interfaces, has gained attention due to its unique ability to manipulate molecules or particles in suspension and solution. Less well understood, though, is the ability of this phenomenon to generate circulatory fluid flow, and how this enables and enhances continuous particle capture. Here, we perform a comprehensive study of a low-voltage ICP, demonstrating a new electrokinetic method for extracting submicron particles via flow-enhanced particle redirection. To do so, a 2D-FEM model solves the Poisson-Nernst-Planck equation coupled with the Navier-Stokes and continuity equations. Four distinct operational modes (Allowed, Blocked, Captured, and Dodged) were recognized as a function of the particle's charges and sizes, resulting in the capture or release from ICP-induced vortices, with the critical particle dimensions determined by appropriately tuning inlet flow rates (200-800 [µm/s]) and applied voltages (0-2.5 [V]). It is found that vortices are generated above a non-dimensional ICP-induced velocity of U*=1, which represents an equilibrium between ICP velocity and lateral flow velocity. It was also found that in the case of multi-target separation, the surface charge of the particle, rather than a particle's size, is the primary determinant of particle trajectory. These findings contribute to a better understanding of ICP-based particle separation and isolation, as well as laying the foundations for the rational design and optimization of ICP-based sorting systems.

Mathematical Modeling of the Selective Transport of Singly Charged Ions Through Multilayer Composite Ion-Exchange Membrane during Electrodialysis

The deposition of several alternating anion- and cation-exchange surface layers (layer-by-layer method) is a promising technique for the modification of ion-exchange membranes, which makes it possible to essentially increase their selectivity to singly charged ions. This paper presents a one-dimensional model, which is based on the Nernst–Planck–Poisson equations and describes the competitive transfer of singly and doubly charged ions through a multilayer composite ion-exchange membrane. It has been revealed for the first time that, as in the earlier studied case of a bilayer membrane, the dependence of the specific permselectivity coefficient (P1/2) of a multilayer membrane on the electrical current density passes through a maximum left( {P_{{{1 mathord{left/ {vphantom {1 2}} right. kern-0em} 2}}}^{{max }}} right). It has been shown that an increase in the number of nanosized modification bilayers n leads to the growth of P_{{{1 mathord{left/ {vphantom {1 2}} right. kern-0em} 2}}}^{{max }}, but the flux of a preferably transferred ion decreases in this case. It has been established that P_{{{1 mathord{left/ {vphantom {1 2}} right. kern-0em} 2}}}^{{max }} is attained at underlimiting current densities and relatively low potential drop. The simulated dependences P_{{{1 mathord{left/ {vphantom {1 2}} right. kern-0em} 2}}}^{{max }}(n) qualitatively agree with the known literature experimental and theoretical results.

Upscaling the Poisson–Nernst–Planck equations for ion transport in weakly heterogeneous charged porous media

The Poisson–Nernst–Planck (PNP) equations govern the continuum level description of ions in electrolytes and especially the impact of charged surfaces. In numerous applications such surfaces are complex, varying on a small lengthscale compared to the overall scale of the system, often prohibiting the direct prediction of the osmotic swelling pressures induced by ion behaviours in Debye layers near surfaces. With periodicity, upscaling techniques can be readily used to determine the behaviour of the swelling pressure on large lengthscales without solving the PNP equations on the complex domain, though generalising to cases where the periodicity is only approximate is more challenging. Here, we generalise a method by Bruna and Chapman (2015) for upscaling a non-periodic diffusion equation to the PNP equations. After upscaling, we find a rational derivation of the swelling pressure closely resembling the classical, though phenomenological, use of Donnan membrane theory predictions for the swelling pressure in cartilage, together with a novel contribution driven by heterogeneous fixed (surface) charges. The resulting macroscale model is also shown to be thermodynamically consistent, though its comparison with a recent upscaled models for swelling pressure in cartilage mechanics emphasises the need to understand how macroscale models depend on differing upscaling techniques, especially in the absence of perfect periodicity.

Pressure-sensitive ion conduction in a conical channel: Optimal pressure and geometry

Using both analytic and numerical analyses of the Poisson–Nernst–Planck equations, we theoretically investigate the electric conductivity of a conical channel which, in accordance with recent experiments, exhibits a strong non-linear pressure dependence. This mechanosensitive diodic behavior stems from the pressure-sensitive build-up or depletion of salt in the pore. From our analytic results, we find that the optimal geometry for this diodic behavior strongly depends on the flow rate with the ideal ratio of tip-to-base-radii being equal to 0.22 at zero-flow. With increased flow, this optimal ratio becomes smaller and, simultaneously, the diodic performance becomes weaker. Consequently an optimal diode is obtained at zero-flow, which is realized by applying a pressure drop that is proportional to the applied potential and to the inverse square of the tip radius, thereby countering electro-osmotic flow. When the applied pressure deviates from this ideal pressure drop the diodic performance falls sharply, explaining the dramatic mechanosensitivity observed in experiments.

Numerical Simulation Study on the Relationships between Mineralized Structures and Induced Polarization Properties of Seafloor Polymetallic Sulfide Rocks

The induced polarization (IP) method plays an important role in the detection of seafloor polymetallic sulfide deposits. Numerical simulations based on the Poisson–Nernst–Planck equation and the Maxwell equation were performed. The effects of mineralized structures on the IP and electrical conductivity properties of seafloor sulfide-bearing rocks were investigated. The results show that total chargeability increases linearly as the volume content of disseminated metal sulfides increases when the volume content is below 20%. However, total chargeability increases nonlinearly with increasing volume content in vein and massive metal sulfides when the volume content is below 30%. The electrical resistivity of disseminated metal sulfides mainly depends on the conductivity of pore water. The electrical resistivity of vein and massive sulfides mainly depends on the volume content and the length of sulfides. Increase in the aspect ratio (0.36 to 0.93) of seafloor massive sulfides causes relaxation time constants and total chargeability to decrease. Relaxation time constants and total chargeability also decrease with increase in the tortuosity of seafloor vein sulfides from 1.0 to 1.38. This study is of great value for the electrical survey of seafloor polymetallic sulfide deposits.

Charge Regulation and pH Effects on Thermo-Osmotic Conversion.

Thermo-osmotic energy conversion using waste heat is one of the approaches to harvesting sustainable energy and reducing associated environmental impacts simultaneously. In principle, ions transport through a charged nanopore membrane under the effect of a thermal gradient, inducing a different voltage between two sides of the membrane. Recent publications mainly reported novel materials for enhancing the thermoelectric voltage in response to temperature difference, the so-called Seebeck coefficient. However, the effect of the surface charge distribution along nanopores on thermo-osmotic conversion has not been discussed yet. In this paper, a numerical simulation based on the Nernst–Planck–Poisson equations, Navier–Stokes equations, and heat transfer equations is carried out to consider the effect of surface charge-regulation density and pH of KCl solutions on the Seebeck coefficient. The results show that the highest ionic Seebeck coefficient of −0.64 mV/K is obtained at 10−4 M KCl solution and pH 9. The pH level and pore structure also reveal a strong effect on the thermo-osmotic performance. Moreover, the pH level at one reservoir is varied from 5 to 9, while the pH of 5 is fixed at the other reservoir to investigate the pH effect on the thermos-osmosis ion transport. The results confirm the feasibility that using the pH can enhance the thermo-osmotic conversion for harvesting osmotic power from low-grade heat energy.

Global existence and temporal decay of large solutions for the Poisson–Nernst–Planck equations in low regularity spaces

We are concerned with the global existence and decay rates of large solutions for the Poisson–Nernst–Planck equations. Based on careful observation of algebraic structure of the equations and using the weighted Chemin–Lerner‐type norm, we obtain the global existence and optimal decay rates of large solutions without requiring the summation of initial densities of a negatively and positively charged species that is small enough. Moreover, the large solution is obtained for initial densities belonging to the low regularity Besov spaces with different regularity and integral indices, which indicates more specific coupling relations between the difference and the summation of negatively and positively charged densities.

Formation of Ordered Patterns in Electroresponsive Polymer Ionic Liquid Blends

AbstractDirecting reaction‐diffusion (RD) phenomena, through the use of external stimuli has been one of the widely used approaches for designing multifunctional soft materials. Using modeling and simulation, it is demonstrated that the nonuniform electric field can be harnessed to create intricate ordered patterns in polymer ionic liquid (PIL) blends. The investigation begins with the establishment of the equilibrium phase diagrams of electroresponsive PIL blends and subsequently, use the Poisson–Nernst–Planck equations to model the kinetics of pattern formation. The simulations reveal that in the presence of nonuniform electric field the ionic liquid (IL) rich domains self‐aggregate in high electric field regions. Thus, the ordering of the electric field regions effectively dictates the ordering of the IL‐rich phase in the PIL blends. It is also demonstrated that the mechanism of spatiotemporal pattern formation is quite robust and can be dynamically controlled by varying the distribution of electric field. It is believed that the methodology provides a simplistic mechanism for creating ordered patterns in soft materials through RD phenomena that can be exploited for designing other similar stimuli‐responsive systems.

Computational framework for resolving boundary layers in electrochemical systems using weak imposition of Dirichlet boundary conditions

We present a finite element based computational framework to model electrochemical systems. The electrochemical system is represented by the coupled Poisson–Nernst–Planck (PNP) and Navier–Stokes (NS) equations. The key quantity of interest in such simulations is the current (flux) at the system boundaries. Accurately computing the current flux is challenging due to the small critical dimension of the boundary layers (small Debye layer) that require fine mesh resolution at the boundaries. We present a numerical framework which resolves this challenge by utilizing a weak imposition of Dirichlet boundary conditions for Poisson–Nernst–Planck equations. In this numerical framework we utilize a block iterative strategy to solve NS and PNP equations. This allows us to efficiently and easily implement the weak imposition of Dirichlet boundary conditions. The results from our numerical framework shows excellent agreement when compared to strong imposition of boundary conditions (strong imposition requires a much finer mesh). Furthermore, we show that the weak imposition of the boundary conditions allows us to resolve the fluxes in the boundary layers with much coarser meshes compared to strong imposition. We also show that the method converges as we refine the mesh near the boundaries at a much faster rate compared to strong imposition of the boundary layer. We present multiple test cases with varying boundary layer thickness to illustrate the utility of the numerical framework. We illustrate the approach on canonical 3D problems that otherwise would have been computationally intractable to solve accurately. Lastly, we simulate electrokinetic instabilities near a perm-selective membrane with weakly imposed boundary conditions on the membrane. This approach substantially reduces the computational cost of modeling thin boundary layers in electrochemical systems.

Analytical solution to the Poisson–Nernst–Planck equations for the charging of a long electrolyte-filled slit pore

We study the charging dynamics of a long electrolyte-filled slit pore in response to a suddenly applied potential. In particular, we analytically solve the Poisson–Nernst–Planck (PNP) equations for a pore for which λD≪H≪L, with λD the Debye length and H and L the pore’s width and length. For small applied potentials, we find the time-dependent potential drop between the pore’s surface and its center to be in complete agreement with a prediction of the celebrated transmission line model. For moderate to high applied potentials, prior numerical work showed that charging slows down at late times; Our analytical model reproduces and explains such biexponential charge buildup.

A weak Galerkin finite element method for time-dependent Poisson–Nernst–Planck equations

In this paper, we present a weak Galerkin finite element method for solving the time-dependent Poisson–Nernst–Planck equations. The error analysis of semidiscrete in space and fully discrete WG schemes is given. The effect of the nonlinear coupling term in the equations is considered in discrete schemes. Numerical experiments for different shapes of meshes are implemented to verify the theoretical results.