Ion migration by diffusion in an electric field

[somasimple]

TL;DR Summary

Difference between force of diffusion and force of an electric field?

Hi,
A solution contains some ions (charged particles). We are only interested in my exemple to positive ions.
It is assumed that these ions acquired some mobility under a concentration gradient. Their direction is A to B.
Then these ions encounter/cross an electric field which is oriented from B to A.

How is it possible to calculate the travel of these ions?
Is the electric force ever stronger than the diffusive one?

 

[Borek]

Google Wien effect, should be a good starting point to learn more.

 

[somasimple]

This Wien effect largely exceeds my knowledge.
All, I want to know is the behaviour of the charged ion (red at left). It has a speed, v.
(I suppose the charge slows or stops/reverses its direction?)
The electric field is E. Flux is directed normally from + to minus.

Then what happen when the charge is at right of the minus side same distance from the electrode?
I understand that a charge betwwen the electrode is under the influence of an unifiorm electric field.

 

[somasimple]

Here is the second initial condition

 

[TeethWhitener]

There's a lot going on here, and I see a lot of confusion.

For one thing, diffusion isn't a force. It's a statistical property. So a single ion doesn't experience the type of diffusive motion you'd see in a collection of a mole of ions. At the level of individual atoms and molecules, diffusion arises from Brownian motion (Gaussian-distributed random motion). So if you just want to look at the (Newtonian) motion of a single ion in an electric field, the most straightforward thing to do is use the Lorentz force with an extra Brownian term. If you want to get fancy, but you don't want to model the solvent molecules explicitly, you can add a viscous drag force and you can adjust the Lorentz force based on the dielectric constant of your solvent.

Regarding the diffusion behavior of an electrolyte in an external electric field (or even without an external electric field), this is a much more difficult problem in general. The governing equation is known as the Poisson-Boltzmann equation and it is nonlinear:
$$\nabla^2\psi \propto \exp{\psi}$$
where $\psi$ is the electric potential. (I'm too lazy to look up all the proportionality constants right now.) The point is that you equate the electric potential on the left side of the equation with a Boltzmann distribution describing the ions in solution on the right side of the equation. In the limit of small potential (low ion concentration), you can expand the exponential in a Taylor series. Truncating to first order gives the Debye-Huckel equation, which is linear and much easier to work with. The theory of concentrated electrolytes is an active area of research.