149 research outputs found
Comparison and numerical treatment of generalised Nernst-Planck models
In its most widespread, classical formulation, the
Nernst-Planck-Poisson system for ion transport in electrolytes fails to take
into account finite ion sizes. As a consequence, it predicts unphysically
high ion concentrations near electrode surfaces. Historical and recent
approaches to an approriate modification of the model are able to fix this
problem. Several appropriate formulations are compared in this paper. The
resulting equations are reformulated using absolute activities as basic
variables describing the species amounts. This reformulation allows to
introduce a straightforward generalisation of the Scharfetter-Gummel finite
volume discretization scheme for drift-diffusion equations. It is shown that
it is thermodynamically consistent in the sense that the solution of the
corresponding discretized generalized Poisson-Boltzmann system describing the
thermodynamical equilibrium is a stationary state of the discretized
time-dependent generalized Nerns-Planck system. Numerical examples
demonstrate the improved physical correctness of the generalised models and
the feasibility of the numerical approach
Comparison and numerical treatment of generalized Nernst--Planck models
In its most widespread, classical formulation, the Nernst-Planck-Poisson system for ion transport in electrolytes fails to take into account finite ion sizes. As a consequence, it predicts unphysically high ion concentrations near electrode surfaces. Historical and recent approaches to an approriate modification of the model are able to fix this problem. Several appropriate formulations are compared in this paper. The resulting equations are reformulated using absolute activities as basic variables describing the species amounts. This reformulation allows to introduce a straightforward generalisation of the Scharfetter-Gummel finite volume discretization scheme for drift-diffusion equations. It is shown that it is thermodynamically consistent in the sense that the solution of the corresponding discretized generalized Poisson-Boltzmann system describing the thermodynamic equilibrium is a stationary state of the discretized time-dependent generalized Nernst-Planck system. Numerical examples demonstrate the improved physical correctness of the generalised models and the feasibility of the numerical approach
Error Analysis of Virtual Element Methods for the Time-dependent Poisson-Nernst-Planck Equations
We discuss and analyze the virtual element method on general polygonal meshes
for the time-dependent Poisson-Nernst-Planck equations, which are a nonlinear
coupled system widely used in semiconductors and ion channels. The spatial
discretization is based on the elliptic projection and the projection
operator, and for the temporal discretization, the backward Euler scheme is
employed. After presenting the semi and fully discrete schemes, we derive the a
priori error estimates in the and norms. Finally, a numerical
experiment verifies the theoretical convergence results
Modeling and Simulation of Thermo-Fluid-Electrochemical Ion Flow in Biological Channels
In this article we address the study of ion charge transport in the
biological channels separating the intra and extracellular regions of a cell.
The focus of the investigation is devoted to including thermal driving forces
in the well-known velocity-extended Poisson-Nernst-Planck (vPNP)
electrodiffusion model. Two extensions of the vPNP system are proposed: the
velocity-extended Thermo-Hydrodynamic model (vTHD) and the velocity-extended
Electro-Thermal model (vET). Both formulations are based on the principles of
conservation of mass, momentum and energy, and collapse into the vPNP model
under thermodynamical equilibrium conditions. Upon introducing a suitable
one-dimensional geometrical representation of the channel, we discuss
appropriate boundary conditions that depend only on effectively accessible
measurable quantities. Then, we describe the novel models, the solution map
used to iteratively solve them, and the mixed-hybrid flux-conservative
stabilized finite element scheme used to discretize the linearized equations.
Finally, we successfully apply our computational algorithms to the simulation
of two different realistic biological channels: 1) the Gramicidin-A channel
considered in~\cite{JeromeBPJ}; and 2) the bipolar nanofluidic diode considered
in~\cite{Siwy7}
Nonlinear electrochemical relaxation around conductors
We analyze the simplest problem of electrochemical relaxation in more than
one dimension - the response of an uncharged, ideally polarizable metallic
sphere (or cylinder) in a symmetric, binary electrolyte to a uniform electric
field. In order to go beyond the circuit approximation for thin double layers,
our analysis is based on the Poisson-Nernst-Planck (PNP) equations of dilute
solution theory. Unlike most previous studies, however, we focus on the
nonlinear regime, where the applied voltage across the conductor is larger than
the thermal voltage. In such strong electric fields, the classical model
predicts that the double layer adsorbs enough ions to produce bulk
concentration gradients and surface conduction. Our analysis begins with a
general derivation of surface conservation laws in the thin double-layer limit,
which provide effective boundary conditions on the quasi-neutral bulk. We solve
the resulting nonlinear partial differential equations numerically for strong
fields and also perform a time-dependent asymptotic analysis for weaker fields,
where bulk diffusion and surface conduction arise as first-order corrections.
We also derive various dimensionless parameters comparing surface to bulk
transport processes, which generalize the Bikerman-Dukhin number. Our results
have basic relevance for double-layer charging dynamics and nonlinear
electrokinetics in the ubiquitous PNP approximation.Comment: 25 pages, 17 figures, 4 table
Transient electrohydrodynamic flow with concentration dependent fluid properties: modelling and energy-stable numerical schemes
Transport of electrolytic solutions under influence of electric fields occurs
in phenomena ranging from biology to geophysics. Here, we present a continuum
model for single-phase electrohydrodynamic flow, which can be derived from
fundamental thermodynamic principles. This results in a generalized
Navier-Stokes-Poisson-Nernst-Planck system, where fluid properties such as
density and permittivity depend on the ion concentration fields. We propose
strategies for constructing numerical schemes for this set of equations, where
solving the electrochemical and the hydrodynamic subproblems are decoupled at
each time step. We provide time discretizations of the model that suffice to
satisfy the same energy dissipation law as the continuous model. In particular,
we propose both linear and non-linear discretizations of the electrochemical
subproblem, along with a projection scheme for the fluid flow. The efficiency
of the approach is demonstrated by numerical simulations using several of the
proposed schemes
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