Poisson-Nemst-Planck model

Corry B, Kuyucak S, -Chun SHg. Dielectric self-energy in Poisson-Boltzmann and Poisson-Nemst-Planck models of ion channels. Biophys J 2003 84(6) 3594—606. 

Cervera, J. Schiedt, B. Ramirez, R A Poisson/Nemst-Planck Model for ionic transport through synthetic... 

Constantin, D. Siwy, Z. S. Poisson-Nemst-Planck model of ion current rectification through a nanofluidic diode. Phys Rev E 2007, 76, 041202. 

These basic equations form the so-called Poisson-Nemst-Planck (PNP) model for IPMCs and describe the fundamental physics wifliin the polymer membrane. A number of aufliors have developed electromechanical (actuator) and mechanoe-lectrical (sensor) models based on the PNP model as well as modified PNP models (Nemat-Nasser 2002 Nemat-Nasser and Zamani 2006 Wallmersperger et al. 2007 Zhang and Yang 2007 Porfiri 2008 Chen and Tan 2008 Aureli et al. 2009). This model will be described further in subsequent chapters of this book. 

Equations 1 and 2 make up what is commonly called the Poisson-Nemst-Planck (PNP) model for IPMCs and describes the fundamental physics within the polymer membrane. 

Numerical simulations on the basis of the Poisson equation combined with the Nemst-Planck model offer another possibility of describing such systems. In the approaches by these groups, the finite-difference method was used to simultaneously solve the Nemst-Planck and the Poisson equations. Due to the complex mathematical procedure, the computing times are very large. So far, this approach has not yet been used for describing the response of thin membranes. Morf proposed a simpler finite-difference approach by combining the Poisson and Nemst-Planck equations in a stepwise way. The Poisson model was used to numerically evaluate the potential profile (Equation 22.23) after each iteration step of the concentration profile (Equations 22.21 and 22.22). The updated potential was then used in the next step ... 

The above set of partial differential equatimis are highly coupled to each other. As an example, the solution of the Poisson equation (Eq. 1) is affected by the ionic mass transfer (c,) and the electric potential ( ). The Nemst-Planck equation (Eq. 5) is a fimction of the ionic mass transfer (c,), the velocity ( m ), and the electric potential ((f)). The Stokes equation (Eq. 8) is also a function of all these variable (C,-, u, and ). Therefore, these equations must be solved simultaneously with the corresponding boundary conditions to model the electrokinetics in the nanochannels. Here, it should be mentioned again that the transient response of the electrokinetics in the nanochannels is negligible, and in the simulations, the steady-state forms of the governing equations are usually solved. 

In the small nanochaimels (from a few to about 100 nm), the electric double layer (EDL) thickness becomes larger or at least comparable with the nanochaimels lateral dimensions. It affects the balance of bulk ionic concentrations of co-ions and counterions in the nanochannels. Thus, many conventional approaches such as the Poisson—Boltzmann equation and the Helmholtz-Smoluchowski slip velocity, which are based on the thin EDL assumption and equal number of co-ions and counterions, lose their credibility and cannot be utilized to model the electrokinetic effects through these nanoscale channels. The Poisson equation, the Navier-Stokes equations, and the Nemst-Planck equation should be solved directly to model the electrokinetic effects and find the electric... 

Sah [1970] introduced the use of networks of electrical elements of infinitesimal size to describe charge carrier motion and generation/recombination in semiconductors. Barker [1975] noted that the Nemst-Planck-Poisson equation system for an unsupported binary electrolyte could be represented by a three-rail transmission line (Figure 2.2.8fl), in which a central conductor with a fixed capacitive reactance per unit length is connected by shunt capacitances to two resistive rails representing the individual ion conductivities. Electrical potentials measured between points on the central rail correspond to electrostatic potential differences between the corresponding points in the cell while potentials computed for the resistive rails correspond to differences in electrochemical potential. This idea was further developed by Brumleve and Buck [1978], and by Franceschetti [1994] who noted that nothing in principle prevents extension of the model to two or three dimensional systems. 

This section describes the numerical techniques used for solving the set of differential equations that model the electrodiffusion of ions in solution. The method has historically been called the Poisson-Nernst-Planck (PNP) method because it is based on the coupHng of the Poisson equation with the Nernst-Planck equation. The basic equations used in the PNP method include the Poisson equation (Eq. [18]), the charge continuity equation (Eq. [55]), and the current density of the Nemst-Planck equation (Eq. [54]). 

Using commercial software (COMSOL Multiphysics), the Nemst-Planck, Poisson, and Navier-Stokes equations are generally solved in a 2D axial symmetric geometry to mimic the 3D structure of conical-shaped nanopores. The more complicated 3D model generates results with similar accuracy but requires additional computational resource. To approximate the semi-infinite solution, the exterior boundary of bulk solution in the model is extended to a distance r=20 pm and z=20 pm away from the pore membrane. The parameters Dj and P for the ionic species are chosen to reflect the electrolyte identity. ... 

Finally, for very thin phases or nanopores, that is, for nanopotentiometry, as weU as for high-frequency experiments, electroneutrality cannot be assumed any more so that the Nemst-Planck-Poisson model must be used. - ... 

A more in-depth elucidation of the mechanism of permselective membranes can be done by using the Nemst-Planck/Poisson equations. As a model system, the gold nanoporous membranes introduced by Martin et al. were used however, the surface charge was established by their chemical modification with self-assembled monolayers (SAMs) of chemisorbed electrically charged thiol derivatives. The average pore diameters of such membranes can be very precisely determined with gas permeation experiments based on the kinetic theory of gases ... 

Sokalski, T., W. Kucza, M. Danielewski, and A. Lewenstam. 2009. Time-dependent phenomena in the potential response of ion-selective electrodes treated by the Nemst-Planck-Poisson model. Part 2 Transmembrane processes and detection limit. Anal. Chem. 81 5016-5022. 

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