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Статті в журналах з теми "Équations de Poisson-Nernst Planck"

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Автор: Grafiati

Опубліковано: 28 вересня 2024

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1

Zheng, Qiong, and Guo-Wei Wei. "Poisson–Boltzmann–Nernst–Planck model." Journal of Chemical Physics 134, no. 19 (May 21, 2011): 194101. http://dx.doi.org/10.1063/1.3581031.

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2

Xie, Yan, Jie Cheng, Benzhuo Lu, and Linbo Zhang. "Parallel Adaptive Finite Element Algorithms for Solving the Coupled Electro-diffusion Equations." Computational and Mathematical Biophysics 1 (April 24, 2013): 90–108. http://dx.doi.org/10.2478/mlbmb-2013-0005.

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Анотація:

Abstractrithms for solving the 3D electro-diffusion equations such as the Poisson-Nernst-Planck equations and the size-modified Poisson-Nernst-Planck equations in simulations of biomolecular systems in ionic liquid. A set of transformation methods based on the generalized Slotboom variables is used to solve the coupled equations. Calculations of the diffusion-reaction rate coefficients, electrostatic potential and ion concentrations for various systems verify the method’s validity and stability. The iterations between the Poisson equation and the Nernst- Planck equations in the primitive method and in the transformation method are compared to illustrate how the new method accelerates the convergence of the solution. To speed up the convergence, we introduce the DIIS (direct inversion of the iterative subspace) method including Simple Mixing and Anderson Mixing as under-relaxation techniques, the effectiveness of which on acceleration is shown by numerical tests. It is worth noting that the primitive method fails to solve the size-modified Poisson-Nernst-Planck equations for real protein systems but the transformation method succeeds in the simulations of the ACh-AChE reaction system and the DNA fragment. To improve the accuracy of the solution, we introduce high order elements and mesh adaptation based on an a posteriori error estimator. Numerical results indicate that our mesh adaptation process leads to quasi-optimal convergence. We implement our algorithms using the parallel adaptive finite element package PHG [53] and high parallel efficiency is obtained.

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3

Yang, Ying, and Benzhuo Lu. "An Error Analysis for the Finite Element Approximation to the Steady-State Poisson-Nernst-Planck Equations." Advances in Applied Mathematics and Mechanics 5, no. 1 (February 2013): 113–30. http://dx.doi.org/10.4208/aamm.11-m11184.

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AbstractPoisson-Nernst-Planck equations are a coupled system of nonlinear partial differential equations consisting of the Nernst-Planck equation and the electrostatic Poisson equation with delta distribution sources, which describe the electrodiffusion of ions in a solvated biomolecular system. In this paper, some error bounds for a piecewise finite element approximation to this problem are derived. Several numerical examples including biomolecular problems are shown to support our analysis.

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4

Hineman, Jay L., and Rolf J. Ryham. "Very weak solutions for Poisson–Nernst–Planck system." Nonlinear Analysis: Theory, Methods & Applications 115 (March 2015): 12–24. http://dx.doi.org/10.1016/j.na.2014.11.018.

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5

Eisenberg, Bob, Tzyy-Leng Horng, Tai-Chia Lin, and Chun Liu. "Steric PNP (Poisson-Nernst-Planck): Ions in Channels." Biophysical Journal 104, no. 2 (January 2013): 509a. http://dx.doi.org/10.1016/j.bpj.2012.11.2809.

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6

González Granada, José Rodrigo, and Victor A. Kovtunenko. "Entropy method for generalized Poisson–Nernst–Planck equations." Analysis and Mathematical Physics 8, no. 4 (November 2018): 603–19. http://dx.doi.org/10.1007/s13324-018-0257-1.

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7

Prohl, Andreas, and Markus Schmuck. "Convergent discretizations for the Nernst–Planck–Poisson system." Numerische Mathematik 111, no. 4 (November 26, 2008): 591–630. http://dx.doi.org/10.1007/s00211-008-0194-2.

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8

Kato, Masayuki. "Numerical analysis of the Nernst-Planck-Poisson system." Journal of Theoretical Biology 177, no. 3 (December 1995): 299–304. http://dx.doi.org/10.1006/jtbi.1995.0247.

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9

Meng, Da, Bin Zheng, Guang Lin, and Maria L. Sushko. "Numerical Solution of 3D Poisson-Nernst-Planck Equations Coupled with Classical Density Functional Theory for Modeling Ion and Electron Transport in a Confined Environment." Communications in Computational Physics 16, no. 5 (November 2014): 1298–322. http://dx.doi.org/10.4208/cicp.040913.120514a.

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Анотація:

AbstractWe have developed efficient numerical algorithms for solving 3D steady-state Poisson-Nernst-Planck (PNP) equations with excess chemical potentials described by the classical density functional theory (cDFT). The coupled PNP equations are discretized by a finite difference scheme and solved iteratively using the Gummel method with relaxation. The Nernst-Planck equations are transformed into Laplace equations through the Slotboom transformation. Then, the algebraic multigrid method is applied to efficiently solve the Poisson equation and the transformed Nernst-Planck equations. A novel strategy for calculating excess chemical potentials through fast Fourier transforms is proposed, which reduces computational complexity from O(N2) to O(NlogN), where N is the number of grid points. Integrals involving the Dirac delta function are evaluated directly by coordinate transformation, which yields more accurate results compared to applying numerical quadrature to an approximated delta function. Numerical results for ion and electron transport in solid electrolyte for lithiumion (Li-ion) batteries are shown to be in good agreement with the experimental data and the results from previous studies.

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10

Chaudhry, Jehanzeb Hameed, Jeffrey Comer, Aleksei Aksimentiev, and Luke N. Olson. "A Stabilized Finite Element Method for Modified Poisson-Nernst-Planck Equations to Determine Ion Flow Through a Nanopore." Communications in Computational Physics 15, no. 1 (January 2014): 93–125. http://dx.doi.org/10.4208/cicp.101112.100413a.

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AbstractThe conventional Poisson-Nernst-Planck equations do not account for the finite size of ions explicitly. This leads to solutions featuring unrealistically high ionic concentrations in the regions subject to external potentials, in particular, near highly charged surfaces. A modified form of the Poisson-Nernst-Planck equations accounts for steric effects and results in solutions with finite ion concentrations. Here, we evaluate numerical methods for solving the modified Poisson-Nernst-Planck equations by modeling electric field-driven transport of ions through a nanopore. We describe a novel, robust finite element solver that combines the applications of the Newton’s method to the nonlinear Galerkin form of the equations, augmented with stabilization terms to appropriately handle the drift-diffusion processes.To make direct comparison with particle-based simulations possible, our method is specifically designed to produce solutions under periodic boundary conditions and to conserve the number of ions in the solution domain. We test our finite element solver on a set of challenging numerical experiments that include calculations of the ion distribution in a volume confined between two charged plates, calculations of the ionic current though a nanopore subject to an external electric field, and modeling the effect of a DNA molecule on the ion concentration and nanopore current.

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11

Hashemi Amrei, S. M. H., Gregory H. Miller, Kyle J. M. Bishop, and William D. Ristenpart. "A perturbation solution to the full Poisson–Nernst–Planck equations yields an asymmetric rectified electric field." Soft Matter 16, no. 30 (2020): 7052–62. http://dx.doi.org/10.1039/d0sm00417k.

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12

Ostilla-Mónico, Rodolfo, and Alpha A. Lee. "Controlling turbulent drag across electrolytes using electric fields." Faraday Discussions 199 (2017): 159–73. http://dx.doi.org/10.1039/c6fd00247a.

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Анотація:

Reversible in operando control of friction is an unsolved challenge that is crucial to industrial tribology. Recent studies show that at low sliding velocities, this control can be achieved by applying an electric field across electrolyte lubricants. However, the phenomenology at high sliding velocities is yet unknown. In this paper, we investigate the hydrodynamic friction across electrolytes under shear beyond the transition to turbulence. We develop a novel, highly parallelised numerical method for solving the coupled Navier–Stokes Poisson–Nernst–Planck equation. Our results show that turbulent drag cannot be controlled across dilute electrolytes using static electric fields alone. The limitations of the Poisson–Nernst–Planck formalism hint at ways in which turbulent drag could be controlled using electric fields.

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13

Sun, Ning, and Dilip Gersappe. "Simulation of diffuse-charge capacitance in electric double layer capacitors." Modern Physics Letters B 31, no. 01 (January 10, 2017): 1650431. http://dx.doi.org/10.1142/s0217984916504315.

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We use a Lattice Boltzmann Model (LBM) in order to simulate diffuse-charge dynamics in Electric Double Layer Capacitors (EDLCs). Simulations are carried out for both the charge and the discharge processes on 2D systems of complex random electrode geometries (pure random, random spheres and random fibers). The steric effect of concentrated solutions is considered by using a Modified Poisson–Nernst–Planck (MPNP) equations and compared with regular Poisson–Nernst–Planck (PNP) systems. The effects of electrode microstructures (electrode density, electrode filler morphology, filler size, etc.) on the net charge distribution and charge/discharge time are studied in detail. The influence of applied potential during discharging process is also discussed. Our studies show how electrode morphology can be used to tailor the properties of supercapacitors.

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14

Queralt-Martín, María, Carlos Peiró-González, Marcel Aguilella-Arzo, and Antonio Alcaraz. "Effects of extreme pH on ionic transport through protein nanopores: the role of ion diffusion and charge exclusion." Physical Chemistry Chemical Physics 18, no. 31 (2016): 21668–75. http://dx.doi.org/10.1039/c6cp04180a.

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We combine electrophysiological experiments with the structure-based Poisson–Nernst–Planck 3D calculations to investigate the transport properties of the bacterial porin OmpF under large pH gradients and particularly low salt concentrations.

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15

Gwecho, Abidha Monica, Wang Shu, Onyango Thomas Mboya, and Sudheer Khan. "Solutions of Poisson-Nernst Planck Equations with Ion Interaction." Applied Mathematics 13, no. 03 (2022): 263–81. http://dx.doi.org/10.4236/am.2022.133020.

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16

Schönke, Johannes. "Unsteady analytical solutions to the Poisson–Nernst–Planck equations." Journal of Physics A: Mathematical and Theoretical 45, no. 45 (October 29, 2012): 455204. http://dx.doi.org/10.1088/1751-8113/45/45/455204.

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17

Filipek, Robert, Piotr Kalita, Lucjan Sapa, and Krzysztof Szyszkiewicz. "On local weak solutions to Nernst–Planck–Poisson system." Applicable Analysis 96, no. 13 (September 6, 2016): 2316–32. http://dx.doi.org/10.1080/00036811.2016.1221941.

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18

Golovnev, A., and S. Trimper. "Steady state solution of the Poisson–Nernst–Planck equations." Physics Letters A 374, no. 28 (June 2010): 2886–89. http://dx.doi.org/10.1016/j.physleta.2010.05.004.

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19

SCHMUCK, MARKUS. "ANALYSIS OF THE NAVIER–STOKES–NERNST–PLANCK–POISSON SYSTEM." Mathematical Models and Methods in Applied Sciences 19, no. 06 (June 2009): 993–1014. http://dx.doi.org/10.1142/s0218202509003693.

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Анотація:

We study a fluid-dynamical model based on a coupled Navier–Stokes–Nernst–Planck–Poisson system. Of special interest are the fluid velocity, concentrations of charged particles ranging from colloidal to nano size and the induced quasi-electrostatic potential, which all depend on an externally applied electrical field. For d ≤ 3, we prove existence and in some cases uniqueness of weak solutions. Moreover, we characterize solutions via energy laws, mass conservation, non-negativity and pointwise bounds. Furthermore, the system enjoys an entropy law. Existence of locally strong solutions is verified.

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20

Gavish, Nir. "Poisson–Nernst–Planck equations with high-order steric effects." Physica D: Nonlinear Phenomena 411 (October 2020): 132536. http://dx.doi.org/10.1016/j.physd.2020.132536.

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21

Mathur, Sanjay R., and Jayathi Y. Murthy. "A multigrid method for the Poisson–Nernst–Planck equations." International Journal of Heat and Mass Transfer 52, no. 17-18 (August 2009): 4031–39. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2009.03.040.

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22

Frank, Florian, Nadja Ray, and Peter Knabner. "Numerical investigation of homogenized Stokes–Nernst–Planck–Poisson systems." Computing and Visualization in Science 14, no. 8 (December 2011): 385–400. http://dx.doi.org/10.1007/s00791-013-0189-0.

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23

Lin, Guojian. "A NOTE ON THE CLASSICAL POISSON-NERNST-PLANCK MODELS." Far East Journal of Applied Mathematics 95, no. 1 (September 6, 2016): 13–26. http://dx.doi.org/10.17654/am095010013.

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24

Ray, N., A. Muntean, and P. Knabner. "Rigorous homogenization of a Stokes–Nernst–Planck–Poisson system." Journal of Mathematical Analysis and Applications 390, no. 1 (June 2012): 374–93. http://dx.doi.org/10.1016/j.jmaa.2012.01.052.

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25

Zheng, Qiong, Duan Chen, and Guo-Wei Wei. "Second-order Poisson–Nernst–Planck solver for ion transport." Journal of Computational Physics 230, no. 13 (June 2011): 5239–62. http://dx.doi.org/10.1016/j.jcp.2011.03.020.

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26

Moya, A. A. "Theory of the formation of the electric double layer at the ion exchange membrane–solution interface." Physical Chemistry Chemical Physics 17, no. 7 (2015): 5207–18. http://dx.doi.org/10.1039/c4cp05702c.

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Анотація:

The study of the formation of the electric double layer at the membrane–solution interface based on the Nernst–Planck and Poisson equations including different diffusion coefficient and dielectric constant values in the solution and membrane phases.

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27

Jasielec, Jerzy J. "Electrodiffusion Phenomena in Neuroscience and the Nernst–Planck–Poisson Equations." Electrochem 2, no. 2 (April 5, 2021): 197–215. http://dx.doi.org/10.3390/electrochem2020014.

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Анотація:

This work is aimed to give an electrochemical insight into the ionic transport phenomena in the cellular environment of organized brain tissue. The Nernst–Planck–Poisson (NPP) model is presented, and its applications in the description of electrodiffusion phenomena relevant in nanoscale neurophysiology are reviewed. These phenomena include: the signal propagation in neurons, the liquid junction potential in extracellular space, electrochemical transport in ion channels, the electrical potential distortions invisible to patch-clamp technique, and calcium transport through mitochondrial membrane. The limitations, as well as the extensions of the NPP model that allow us to overcome these limitations, are also discussed.

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28

Bedin, L., and Mark Thompson. "Existence theory for a Poisson-Nernst-Planck model of electrophoresis." Communications on Pure and Applied Analysis 12, no. 1 (September 2012): 157–206. http://dx.doi.org/10.3934/cpaa.2013.12.157.

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29

Nanninga, Paul Marlon. "A computational neuron model based on Poisson-Nernst-Planck theory." ANZIAM Journal 49 (September 21, 2008): 46. http://dx.doi.org/10.21914/anziamj.v50i0.1390.

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30

Shi, Xiangyu, and Linzhang Lu. "Nonconforming finite element method for coupled Poisson–Nernst–Planck equations." Numerical Methods for Partial Differential Equations 37, no. 3 (January 21, 2021): 2714–29. http://dx.doi.org/10.1002/num.22764.

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31

Kinderlehrer, David, Léonard Monsaingeon, and Xiang Xu. "A Wasserstein gradient flow approach to Poisson−Nernst−Planck equations." ESAIM: Control, Optimisation and Calculus of Variations 23, no. 1 (October 10, 2016): 137–64. http://dx.doi.org/10.1051/cocv/2015043.

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32

Liu, Jinn-Liang, and Bob Eisenberg. "Poisson-Nernst-Planck-Fermi theory for modeling biological ion channels." Journal of Chemical Physics 141, no. 22 (December 14, 2014): 22D532. http://dx.doi.org/10.1063/1.4902973.

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33

Lenzi, E. K., L. R. Evangelista, L. Taghizadeh, D. Pasterk, R. S. Zola, T. Sandev, C. Heitzinger, and I. Petreska. "Reliability of Poisson–Nernst–Planck Anomalous Models for Impedance Spectroscopy." Journal of Physical Chemistry B 123, no. 37 (August 12, 2019): 7885–92. http://dx.doi.org/10.1021/acs.jpcb.9b06263.

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34

Liu, Weishi, and Bixiang Wang. "Poisson–Nernst–Planck Systems for Narrow Tubular-Like Membrane Channels." Journal of Dynamics and Differential Equations 22, no. 3 (August 10, 2010): 413–37. http://dx.doi.org/10.1007/s10884-010-9186-x.

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35

Flavell, Allen, Michael Machen, Bob Eisenberg, Julienne Kabre, Chun Liu, and Xiaofan Li. "A conservative finite difference scheme for Poisson–Nernst–Planck equations." Journal of Computational Electronics 13, no. 1 (September 25, 2013): 235–49. http://dx.doi.org/10.1007/s10825-013-0506-3.

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36

Flavell, Allen, Julienne Kabre, and Xiaofan Li. "An energy-preserving discretization for the Poisson–Nernst–Planck equations." Journal of Computational Electronics 16, no. 2 (March 6, 2017): 431–41. http://dx.doi.org/10.1007/s10825-017-0969-8.

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37

Eisenberg, Bob, and Weishi Liu. "Poisson–Nernst–Planck Systems for Ion Channels with Permanent Charges." SIAM Journal on Mathematical Analysis 38, no. 6 (January 2007): 1932–66. http://dx.doi.org/10.1137/060657480.

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38

Horng, Tzyy-Leng, Ping-Hsuan Tsai, and Tai-Chia Lin. "Modification of Bikerman model with specific ion sizes." Computational and Mathematical Biophysics 5, no. 1 (December 20, 2017): 142–49. http://dx.doi.org/10.1515/mlbmb-2017-0010.

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Анотація:

Abstract Classical Poisson-Boltzman and Poisson-Nernst-Planck models can only work when ion concentrations are very dilute, which often mismatches experiments. Researchers have been working on the modification to include finite-size effect of ions, which is non-negelible when ion concentrations are not dilute. One of modified models with steric effect is Bikerman model, which is rather popular nowadays. It is based on the consideration of ion size by putting additional entropy term for solvent in free energy. However, ion size is non-specific in original Bikerman model, which did not consider specific ion sizes. Many researchers have worked on the extension of Bikerman model to have specific ion sizes. A direct extension of original Bikerman model by simply replacing the non-specific ion size to specific ones seems natural and has been acceptable to many researchers in this field.Herewe prove this straight forward extension, in some limiting situations, fails to uphold the basic requirement that ion occupation sites must be identical. This requirement is necessary when computing entropy via particle distribution on occupation sites.We derived a new modified Bikerman model for using specific ion sizes by fixing this problem, and obtained its modified Poisson-Boltzmann and Poisson-Nernst-Planck equations.

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39

Mádai, Eszter, Bartłomiej Matejczyk, András Dallos, Mónika Valiskó, and Dezső Boda. "Controlling ion transport through nanopores: modeling transistor behavior." Physical Chemistry Chemical Physics 20, no. 37 (2018): 24156–67. http://dx.doi.org/10.1039/c8cp03918f.

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Анотація:

We present a modeling study of a nanopore-based transistor computed by a mean-field continuum theory (Poisson–Nernst–Planck, PNP) and a hybrid method including particle simulation (Local Equilibrium Monte Carlo, LEMC) that is able to take ionic correlations into account including the finite size of ions.

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40

null, null. "Some Random Batch Particle Methods for the Poisson-Nernst-Planck and Poisson-Boltzmann Equations." Communications in Computational Physics 32, no. 1 (June 2022): 41–82. http://dx.doi.org/10.4208/cicp.oa-2021-0159.

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41

Corry, Ben, Serdar Kuyucak, and Shin-Ho Chung. "Dielectric Self-Energy in Poisson-Boltzmann and Poisson-Nernst-Planck Models of Ion Channels." Biophysical Journal 84, no. 6 (June 2003): 3594–606. http://dx.doi.org/10.1016/s0006-3495(03)75091-7.

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42

Filipek, Robert, Krzysztof Szyszkiewicz, Bogusław Bożek, Marek Danielewski, and A. Lewenstam. "Diffusion Transport in Electrochemical Systems: A New Approach to Determining of the Membrane Potential at Steady State." Defect and Diffusion Forum 283-286 (March 2009): 487–93. http://dx.doi.org/10.4028/www.scientific.net/ddf.283-286.487.

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Анотація:

Ionic concentrations and electric field space profiles in one dimensional membrane are described using Nernst-Planck-Poisson (NPP) equations. The usual assumptions for the steady state NPP problem requires knowledge of the boundary values of the concentrations and electrical potential difference. In analytical chemistry the potential difference may not be known and its theoretical prediction from the model is desirable. The effective methods of the solution to the NPP equations are presented. The Poisson equation is solved without widely used simplifications such as the constant field or the electroneutrality assumptions. The first method uses a steady state formulation of NPP problem. The original system of ODEs is turned into the system of non-linear algebraic equations with unknowns fluxes of the components and electrical potential difference. The second method uses the time-dependent form of the Nernst-Planck-Poisson equations. Steady-state solution has been obtained by starting from an initial profiles, and letting the numerical system evolve until a stationary solution is reached. The methods have been tested for different electrochemical systems: liquid junction and ion selective electrodes (ISEs). The results for the liquid junction case have been also verified with the approximate solutions leading to a good agreement. Comparison with the experimental results for ISEs has been carried out.

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43

Zhao, Jihong, Chao Deng, and Shangbin Cui. "Global Existence and Asymptotic Behavior of Self-Similar Solutions for the Navier-Stokes-Nernst-Planck-Poisson System in." International Journal of Differential Equations 2011 (2011): 1–19. http://dx.doi.org/10.1155/2011/329014.

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Анотація:

We study the Navier-Stokes-Nernst-Planck-Poisson system modeling the flow of electrohydrodynamics. For small initial data, the global existence, uniqueness, and asymptotic stability as time goes to infinity of self-similar solutions to the Cauchy problem of this system posed in the whole three dimensional space are proved in the function spaces of pseudomeasure type.

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44

Bożek, Bogusław, H. Leszczyński, Katarzyna Tkacz-Śmiech, and Marek Danielewski. "Electrochemistry of Symmetrical Ion Channel: Three-Dimensional Nernst-Planck-Poisson Model." Defect and Diffusion Forum 363 (May 2015): 68–78. http://dx.doi.org/10.4028/www.scientific.net/ddf.363.68.

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Анотація:

The paper provides a physical description of ionic transport through the rigid symmetrical channel. A three-dimensional mathematical model, in which the ionic transport is treated as the electrodiffusion of ions, is presented. The model bases on the solution of the 3D Nernst-Planck-Poisson system for cylindrical geometry. The total flux includes drift (convection) and diffusion terms. It allows simulating the transport characteristics at the steady-state and time evolution of the system. The numerical solutions of the coupled differential diffusion equation system are obtained by finite element method. Examples are presented in which the flow characteristics at the stationary state and during time evolution are compared. It is shown that the stationary state is achieved after about 2×10 -8 s since the process beginning. Various initial conditions (channel charging and dimensions) are considered as the key parameters controlling the selectivity of the channel. The model allows determining the flow characteristic, calculating the local concentration and potential across the channel. The model can be extended to simulate transport in polymer membranes and nanopores which might be useful in designing biosensors and nanodevices.

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45

Liu, Hailiang, Zhongming Wang, Peimeng Yin, and Hui Yu. "Positivity-preserving third order DG schemes for Poisson–Nernst–Planck equations." Journal of Computational Physics 452 (March 2022): 110777. http://dx.doi.org/10.1016/j.jcp.2021.110777.

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46

Ma, Manman, Zhenli Xu, and Liwei Zhang. "Modified Poisson--Nernst--Planck Model with Coulomb and Hard-sphere Correlations." SIAM Journal on Applied Mathematics 81, no. 4 (January 2021): 1645–67. http://dx.doi.org/10.1137/19m1310098.

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47

Delavari, Najmeh, Klas Tybrandt, Magnus Berggren, Benoît Piro, Vincent Noël, Giorgio Mattana, and Igor Zozoulenko. "Nernst–Planck–Poisson analysis of electrolyte-gated organic field-effect transistors." Journal of Physics D: Applied Physics 54, no. 41 (July 29, 2021): 415101. http://dx.doi.org/10.1088/1361-6463/ac14f3.

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48

Burger, M., B. Schlake, and M.-T. Wolfram. "Nonlinear Poisson–Nernst–Planck equations for ion flux through confined geometries." Nonlinearity 25, no. 4 (March 6, 2012): 961–90. http://dx.doi.org/10.1088/0951-7715/25/4/961.

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49

Gao, Huadong, and Dongdong He. "Linearized Conservative Finite Element Methods for the Nernst–Planck–Poisson Equations." Journal of Scientific Computing 72, no. 3 (February 28, 2017): 1269–89. http://dx.doi.org/10.1007/s10915-017-0400-4.

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50

Shobukhov, Andrey V., and Dmitry S. Maximov. "Exact steady state solutions in symmetrical Nernst–Planck–Poisson electrodiffusive models." Journal of Mathematical Chemistry 52, no. 5 (February 2, 2014): 1338–49. http://dx.doi.org/10.1007/s10910-014-0313-5.

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