1,576 research outputs found
Allen-Cahn and Cahn-Hilliard-like equations for dissipative dynamics of saturated porous media
We consider a saturated porous medium in the regime of solid-fluid
segregation under an applied pressure on the solid constituent. We prove that,
depending on the dissipation mechanism, the dynamics is described either by a
Cahn-Hilliard or by an Allen-Cahn-like equation. More precisely, when the
dissipation is modeled via the Darcy law we find that, for small deformation of
the solid and small variations of the fluid density, the evolution equation is
very similar to the Cahn-Hilliard equation. On the other hand, when only the
Stokes dissipation term is considered, we find that the evolution is governed
by an Allen-Cahn-like equation. We use this theory to describe the formation of
interfaces inside porous media. We consider a recently developed model proposed
to study the solid-liquid segregation in consolidation and we are able to fully
describe the formation of an interface between the fluid-rich and the
fluid-poor phase
Phase Separation Dynamics in Isotropic Ion-Intercalation Particles
Lithium-ion batteries exhibit complex nonlinear dynamics, resulting from
diffusion and phase transformations coupled to ion intercalation reactions.
Using the recently developed Cahn-Hilliard reaction (CHR) theory, we
investigate a simple mathematical model of ion intercalation in a spherical
solid nanoparticle, which predicts transitions from solid-solution radial
diffusion to two-phase shrinking-core dynamics. This general approach extends
previous Li-ion battery models, which either neglect phase separation or
postulate a spherical shrinking-core phase boundary, by predicting phase
separation only under appropriate circumstances. The effect of the applied
current is captured by generalized Butler-Volmer kinetics, formulated in terms
of diffusional chemical potentials, and the model consistently links the
evolving concentration profile to the battery voltage. We examine sources of
charge/discharge asymmetry, such as asymmetric charge transfer and surface
"wetting" by ions within the solid, which can lead to three distinct phase
regions. In order to solve the fourth-order nonlinear CHR
initial-boundary-value problem, a control-volume discretization is developed in
spherical coordinates. The basic physics are illustrated by simulating many
representative cases, including a simple model of the popular cathode material,
lithium iron phosphate (neglecting crystal anisotropy and coherency strain).
Analytical approximations are also derived for the voltage plateau as a
function of the applied current
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