How to Solve Fick's Second Law for Spherical Diffusion?

[Chris-jap]

Hello every body

I have previously post my question in this topic:
Physics Help and Math Help - Physics Forums > Science Education > Homework & Coursework Questions > Other Sciences > Fick and Cottrell Law

And after Goku suggestion I post my question here.

So my problem deal with the resolution with fick second law of diffusion (in one dimension)

In the case of a planar electrode (one dimension) the current density is proportinnal to the concentration of electroactive species: i=-nFkdC(x=0,t)/dt
From Fick law dC(x,t)/dt=Dd2C(x,t)/d2x

So in the case of initial condition C(x,t=0)=C0
I found this solution (not me, on internet) C(x,t)=C0erf(x/(Dt)1/2)

And so we can deduce Cottrell Law i=-nFAC0(D/Pit)1/2

Now I would like to found the expression of i in the case of spherical electrode and spherical diffusion, which species are inside the sphere (yes inside and not outside) of radius R
With C(R,t)=0 for t>0 and C(r,t=0)=C0
I would like to found the expression of C(r,t)

I think fick law in spherical diffusion is dC(r,t)/dt=D1/r2d/dr(r2d/dr(C(r,t)))

Is it right?

But now how can I find C(r,t) then dC(r,t)/dr for r=R ?

Do you have any suggestion?

Thank you for your attention
Chris

 

[Clausius2]

I think you're going to find a lot of standard methods for solving these equation, including Green Functions. Since there is no characteristic scale in your problem, I propose you to solve your problem in terms of a similarity variable \eta. You should be able to work out, via scalement of the equation, that \eta=r/\sqrt{Dt}. That is, you're scaling the radial coordinate with the diffusion length. Performing the change of variable you would end up with an ordinary equation for \eta, something like:

\frac{-1}{2}\eta\frac{\partial \C}{\partial \eta}=\frac{1}{\eta}\frac{\partial}{\partial \eta}\left(\eta\frac{\partial C}{\partial \eta}\right)

Now it's on your own.

 

[Chris-jap]

Thank you for your help

I have done the change you propose
So now if Iunderstand C(r,t) becomes C(n)

I found a similar differential equation that the one you propose (n2 instead of n)

-1/2 n dC/n = 1/n2 d/dn(n2 dC/dn)

Are you agree with this one?

And then
nd2C/dn2 + (2-1/2*n2)dC/dn=0

Is it a non linear differential equation?

Is it possible to solve it?