801 research outputs found
A Conservative Finite Difference Scheme for Poisson-Nernst-Planck Equations
A macroscopic model to describe the dynamics of ion transport in ion channels
is the Poisson-Nernst-Planck(PNP) equations. In this paper, we develop a
finite-difference method for solving PNP equations, which is second-order
accurate in both space and time. We use the physical parameters specifically
suited toward the modelling of ion channels. We present a simple iterative
scheme to solve the system of nonlinear equations resulting from discretizing
the equations implicitly in time, which is demonstrated to converge in a few
iterations. We place emphasis on ensuring numerical methods to have the same
physical properties that the PNP equations themselves also possess, namely
conservation of total ions and correct rates of energy dissipation. We describe
in detail an approach to derive a finite-difference method that preserves the
total concentration of ions exactly in time. Further, we illustrate that, using
realistic values of the physical parameters, the conservation property is
critical in obtaining correct numerical solutions over long time scales
A study of electrochemical transport and diffuse charge dynamics using a Langevin equation
This thesis aims to develop new numerical and computational tools to study electrochemical transport and diffuse charge dynamics at small scales. Previous efforts at modeling electrokinetic phenomena at scales where the noncontinuum effects become significant have included continuum models based on the Poisson-Nernst-Planck equations and atomic simulations using molecular dynamics algorithms. Neither of them is easy to use or conducive to electrokinetic transport modeling in strong confinement or over long time scales. This work introduces a new approach based on a Langevin equation for diffuse charge dynamics in nanofluidic devices, which incorporates features from both continuum and atomistic methods. The model is then extended to include steric effects resulting from finite ion size, and applied to the phenomenon of double layer charging in a symmetric binary electrolyte between parallel-plate blocking electrodes, between which a voltage is applied. Finally, the results of this approach are compared to those of the continuum model based on the Poisson-Nernst-Planck equations
Modeling transport of charged species in pore networks: solution of the Nernst-Planck equations coupled with fluid flow and charge conservation equations
A pore network modeling (PNM) framework for the simulation of transport of
charged species, such as ions, in porous media is presented. It includes the
Nernst-Planck (NP) equations for each charged species in the electrolytic
solution in addition to a charge conservation equation which relates the
species concentration to each other. Moreover, momentum and mass conservation
equations are adopted and there solution allows for the calculation of the
advective contribution to the transport in the NP equations.
The proposed framework is developed by first deriving the numerical model
equations (NMEs) corresponding to the partial differential equations (PDEs)
based on several different time and space discretization schemes, which are
compared to assess solutions accuracy. The derivation also considers various
charge conservation scenarios, which also have pros and cons in terms of speed
and accuracy. Ion transport problems in arbitrary pore networks were considered
and solved using both PNM and finite element method (FEM) solvers. Comparisons
showed an average deviation, in terms of ions concentration, between PNM and
FEM below with the PNM simulations being over times faster
than the FEM ones for a medium including about pores. The improved
accuracy is achieved by utilizing more accurate discretization schemes for both
the advective and migrative terms, adopted from the CFD literature. The NMEs
were implemented within the open-source package OpenPNM based on the iterative
Gummel algorithm with relaxation.
This work presents a comprehensive approach to modeling charged species
transport suitable for a wide range of applications from electrochemical
devices to nanoparticle movement in the subsurface
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