82 research outputs found
Homogenization of the Poisson-Nernst-Planck Equations for Ion Transport in Charged Porous Media
Effective Poisson-Nernst-Planck (PNP) equations are derived for macroscopic
ion transport in charged porous media under periodic fluid flow by an
asymptotic multi-scale expansion with drift. The microscopic setting is a
two-component periodic composite consisting of a dilute electrolyte continuum
(described by standard PNP equations) and a continuous dielectric matrix, which
is impermeable to the ions and carries a given surface charge. Four new
features arise in the upscaled equations: (i) the effective ionic diffusivities
and mobilities become tensors, related to the microstructure; (ii) the
effective permittivity is also a tensor, depending on the electrolyte/matrix
permittivity ratio and the ratio of the Debye screening length to the
macroscopic length of the porous medium; (iii) the microscopic fluidic
convection is replaced by a diffusion-dispersion correction in the effective
diffusion tensor; and (iv) the surface charge per volume appears as a
continuous "background charge density", as in classical membrane models. The
coefficient tensors in the upscaled PNP equations can be calculated from
periodic reference cell problems. For an insulating solid matrix, all gradients
are corrected by the same tensor, and the Einstein relation holds at the
macroscopic scale, which is not generally the case for a polarizable matrix,
unless the permittivity and electric field are suitably defined. In the limit
of thin double layers, Poisson's equation is replaced by macroscopic
electroneutrality (balancing ionic and surface charges). The general form of
the macroscopic PNP equations may also hold for concentrated solution theories,
based on the local-density and mean-field approximations. These results have
broad applicability to ion transport in porous electrodes, separators,
membranes, ion-exchange resins, soils, porous rocks, and biological tissues
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
Unconditionally positivity preserving and energy dissipative schemes for Poisson--Nernst--Planck equations
We develop a set of numerical schemes for the Poisson--Nernst--Planck
equations. We prove that our schemes are mass conservative, uniquely solvable
and keep positivity unconditionally. Furthermore, the first-order scheme is
proven to be unconditionally energy dissipative. These properties hold for
various spatial discretizations. Numerical results are presented to validate
these properties. Moreover, numerical results indicate that the second-order
scheme is also energy dissipative, and both the first- and second-order schemes
preserve the maximum principle for cases where the equation satisfies the
maximum principle.Comment: 24 pages, 10 figure
Models and numerical methods for electrolyte flows
The most common mathematical models for electrolyte flows are based on the dilute solution assumption, leading to a coupled system of the Nernst--Planck--Poisson drift-diffusion equations for ion transport and the Stokes resp. Navier--Stokes equations for fluid flow. This contribution discusses historical and recent model developments beyond the dilute solution assumption and focuses on the effects of finite ion sizes and solvation. A novel numerical solution approach is presented and verified here which aims at preserving on the discrete level consistency with basic thermodynamic principles and structural properties like independence of flow velocities from gradient contributions to external forces
Existence of solution to a system of PDEs modeling the crystal growth inside lithium batteries
The life-cycle of electric batteries depends on a complex system of
interacting electrochemical and growth phenomena that produce dendritic
structures during the discharge cycle. We study herein a system of 3 partial
differential equations combining an Allen--Cahn phase-field model (simulating
the dendrite-electrolyte interface) with the Poisson--Nernst--Planck systems
simulating the electrodynamics and leading to the formation of such dendritic
structures. We prove novel existence, uniqueness and stability results for this
system and use it to produce simulations based on a finite element code.Comment: 27 pages, 22 figures, free software and open source code availabl
- …