# Migration of ion-exchange particles under the action of a uniformly applied<br/><br/>electric field

Tuesday, December 8, 2009 - 9:00am - 9:40am

EE/CS 3-180

Ehud Yariv (Technion-Israel Institute of Technology)

An ideally polarizable cation-selective solid particle is suspended in an

electrolyte solution and is exposed to a uniformly applied ambient electric

field. The electrokinetic transport processes are described in a closed

mathematical model, consisting of differential equations, representing the

physical balance laws, as well as boundary conditions and integral constraints,

representing the physicochemical condition on the particle boundary and at

large

distances away from it. Solving this model would in principle provide the

electro-kinetic flow about the particle and the concomitant particle drift

relative to the otherwise quiescent fluid.

Using matched asymptotic expansions, the model is analyzed in the

thin-Debye-layer limit. An effective `macroscale' description is extracted,

whereby effective boundary conditions represent appropriate asymptotic matching

with the Debye-scale fields. The macroscale description significantly differs

from that corresponding to a chemically inert ideally polarizable particle.

Thus, ion selectivity on the particle surface results in a macroscale salt

concentration polarization, whereby the electric potential is rendered

non-harmonic. Moreover, the uniform Dirichlet condition governing this

potential on the particle surface is transformed into a non-uniform Dirichlet

condition on the macroscale particle boundary. The Dukhin--Derjaguin slip

formula still holds, but with a non-uniform zeta potential that depends upon

the salt concentration distribution.

For weakly applied fields, an approximate solution is obtained as a

perturbation

to an equilibrium state. The linearized solution corresponds to a uniform zeta

potential; it predicts a particle velocity which is proportional to the applied

field. The associated electrokinetic flow differs however from that in the

comparable electrophoresis of an inert particle surface, since it is driven by

two different agents, electric field and salinity gradients, which are of

comparable magnitude. The velocity field, specifically, is rotational.

electrolyte solution and is exposed to a uniformly applied ambient electric

field. The electrokinetic transport processes are described in a closed

mathematical model, consisting of differential equations, representing the

physical balance laws, as well as boundary conditions and integral constraints,

representing the physicochemical condition on the particle boundary and at

large

distances away from it. Solving this model would in principle provide the

electro-kinetic flow about the particle and the concomitant particle drift

relative to the otherwise quiescent fluid.

Using matched asymptotic expansions, the model is analyzed in the

thin-Debye-layer limit. An effective `macroscale' description is extracted,

whereby effective boundary conditions represent appropriate asymptotic matching

with the Debye-scale fields. The macroscale description significantly differs

from that corresponding to a chemically inert ideally polarizable particle.

Thus, ion selectivity on the particle surface results in a macroscale salt

concentration polarization, whereby the electric potential is rendered

non-harmonic. Moreover, the uniform Dirichlet condition governing this

potential on the particle surface is transformed into a non-uniform Dirichlet

condition on the macroscale particle boundary. The Dukhin--Derjaguin slip

formula still holds, but with a non-uniform zeta potential that depends upon

the salt concentration distribution.

For weakly applied fields, an approximate solution is obtained as a

perturbation

to an equilibrium state. The linearized solution corresponds to a uniform zeta

potential; it predicts a particle velocity which is proportional to the applied

field. The associated electrokinetic flow differs however from that in the

comparable electrophoresis of an inert particle surface, since it is driven by

two different agents, electric field and salinity gradients, which are of

comparable magnitude. The velocity field, specifically, is rotational.

MSC Code:

97Mxx

Keywords: