714 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
A corrector theory for diffusion-homogenization limits of linear transport equations
This paper concerns the diffusion-homogenization of transport equations when
both the adimensionalized scale of the heterogeneities and the
adimensionalized mean-free path \eps converge to 0. When \alpha=\eps, it is
well known that the heterogeneous transport solution converges to a homogenized
diffusion solution. We are interested here in the situation where
0<\eps\ll\alpha\ll1 and in the respective rates of convergences to the
homogenized limit and to the diffusive limit. Our main result is an
approximation to the transport solution with an error term that is negligible
compared to the maximum of and \frac\eps\alpha. After establishing
the diffusion-homogenization limit to the transport solution, we show that the
corrector is dominated by an error to homogenization when \alpha^2\ll\eps and
by an an error to diffusion when \eps\ll\alpha^2.
Our regime of interest involves singular perturbations in the small parameter
\eta=\frac\eps\alpha. Disconnected local equilibria at need to be
reconnected to provide a global equilibrium on the cell of periodicity when
. This reconnection between local and global equilibria is shown to
hold when sufficient {\em no-drift} conditions are satisfied. The Hilbert
expansion methodology followed in this paper builds on corrector theories for
the result developed in \cite{NBAPuVo}.Comment: 25 page
Efficient numerical calculation of drift and diffusion coefficients in the diffusion approximation of kinetic equations
In this paper we study the diffusion approximation of a swarming model given
by a system of interacting Langevin equations with nonlinear friction. The
diffusion approximation requires the calculation of the drift and diffusion
coefficients that are given as averages of solutions to appropriate Poisson
equations. We present a new numerical method for computing these coefficients
that is based on the calculation of the eigenvalues and eigenfunctions of a
Schr\"odinger operator. These theoretical results are supported by numerical
simulations showcasing the efficiency of the method
Extrinsic electromagnetic chirality in all-photodesigned one-dimensional THz metamaterials
We suggest that all-photodesigned metamaterials, sub-wavelength custom
patterns of photo-excited carriers on a semiconductor, can display an exotic
extrinsic electromagnetic chirality in terahertz (THz) frequency range. We
consider a photo-induced pattern exhibiting 1D geometrical chirality, i.e. its
mirror image can not be superposed onto itself by translations without
rotations and, in the long wavelength limit, we evaluate its bianisotropic
response. The photo-induced extrinsic chirality turns out to be fully
reconfigurable by recasting the optical illumination which supports the
photo-excited carriers. The all-photodesigning technique represents a feasible,
easy and powerful method for achieving effective matter functionalization and,
combined with the chiral asymmetry, it could be the platform for a new
generation of reconfigurable devices for THz wave polarization manipulation.Comment: 11 page
Diffusion and guiding center approximation for particle transport in strong magnetic fields
International audienceThe diffusion limit of the linear Boltzmann equation with a strong magnetic field is performed. The giration period of particles around the magnetic field is assumed to be much smaller than the collision relaxation time which is supposed to be much smaller than the macroscopic time. The limiting equation is shown to be a diffusion equation in the parallel direction while in the orthogonal direction, the guiding center motion is obtained. The diffusion constant in the parallel direction is obtained through the study of a new collision operator obtained by averaging the original one. Moreover, a correction to the guiding center motion is derived
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