Diffusion, Migration and Einstein Equation

[Dario56]

In the textbook: Electrochemical Systems by Newman and Alyea, 3rd edition, chapter 11.9: Moderately Dilute Solutions, equation for the mole flux of the component $i$ is given by: $$ N_i = - \frac {u_i c_i} {z_i F} \nabla \bar\mu_i\ + c_i v \tag {1}$$

where $u_i$ is the ionic mobility, $c_i$ is the concentration, $\bar \mu_i$ is the electrochemical potential and $v$ is the velocity of the streaming fluid (assuming a low concentration of the component $i$ as situation becomes more complicated at higher concentrations).

One thing to note is that I wrote ionic mobility $u_i$ a bit differently compared to the textbook. I defined it as a terminal velocity of the ion in the unitary electric field, so its measuring unit is $[\frac {m^2}{Vs}]$ in my case. In the textbook they defined it also as a terminal velocity in the unitary electric field, but divided by $z_i F$. This is something to keep in mind to avoid confusion.

Basically, equation (1) states that the mole flux of the component has three contributions: diffusion, migration and convection. Diffusion and migration are accounted for in the first term (electrochemical potential gradient) and convection is accounted for in the second term.

My problem is with the first term $- \frac {u_i c_i } {z_i F}\nabla \mu_i$ which I know how to derive and I will do this here as I don't understand something about the equation (1).

To start, we define diffusion mole flux from as the 1st Fick's law: $$N_{i,dif} = -\frac {D_i c_i} {RT} \nabla \mu_i \tag {2}$$

where $D_i$is the diffusion coefficient and $\mu_i$ is the chemical potential of the component $i$.

Migration mole flux is given by: $$N_{i,mig} = -u_i c_i \nabla \phi \tag {3}$$

where $\phi$ is the electric potential.

If there is no convection, we can write total mole flux of the component as: $$N_i = -(\frac {D_i c_i} {RT} \nabla \mu_i + u_i c_i \nabla \phi) \tag {4}$$

We recall the definition of electrochemical potential $\bar \mu_i$: $$ \bar \mu_i = \mu_i + z_i F \phi \tag {5} $$

If we assume that system is in thermodynamic equilibrium than electrochemical potential gradient and net mole flux vanish: $$ \nabla \bar \mu_i = 0 \tag {6} $$ $$ N_i = 0 \tag {7}$$

From equations (5) and (6), we can write: $$\frac {\nabla \mu_i}{\nabla \phi} = -z_i F \tag {8}$$

From equations (4) and (7), we can write: $$ \frac {\nabla \mu_i}{\nabla \phi} = - \frac {u_i RT}{D_i} \tag {9}$$

By equating (8) and (9), we get the Einstein equation, which relates diffusion coefficient $D_i$ and ionic mobility $u_i$ in the thermodynamic equilibrium: $$D_i = \frac {u_i RT}{z_i F} \tag {10}$$

If we know substitute the equation (10) into equation (4), we can write: $$N_i = -u_i c_i(\frac {1}{z_i F} \nabla \mu_i + \nabla \phi) = - \frac {u_i c_i} {z_i F} \nabla \bar\mu_i\ \tag {11}$$

After the derivation of the first term in the equation (1), I can get to my point. Namely, that the first term in the equation must be equal to zero since it is impossible to write this term in that form (via gradient of the electrochemical potential) without using Einstein equation. We saw earlier that Einstein equation is only valid in the thermodynamic equilibrium and therefore equation (6) applies.

 

[TeethWhitener]

This is a good question, and I'm not sure I'll be able to give you a great answer without seeing how your book carried out the exact derivation--that is, I'm familiar with a different derivation of the Einstein Smoluchowski equation. But I think the logic is a little backwards here. I'm not sure it's completely valid to consider $\frac{\nabla\mu}{\nabla\phi}$ because what happens when your electric potential gradient vanishes?

If you start by substituting in $\nabla\mu_i=\nabla\overline{\mu}_i-z_iF\nabla\phi$ to equation 4, you get
$$N_i=-\left(c_i\nabla\phi\left(u_i-\frac{D_iz_iF}{RT}\right)+\frac{D_ic_i}{RT}\nabla\overline{\mu}_i\right) $$

When the electrochemical potential gradient vanishes, there should be no net mole flux. This leaves

$$ 0=c_i\nabla\phi\left(u_i-\frac{D_iz_iF}{RT}\right) $$
and the only way to satisfy this equation for all $\nabla\phi$ is by letting $D_i=\frac{u_iRT}{z_iF}$.

Again, this is just a guess at what might be a bit of problematic rigor, but what this essentially says is that the Einstein Smoluchowski relation holds because if it didn't, the equilibrium behavior wouldn't be correct.

 

[Dario56]

This is a good question, and I'm not sure I'll be able to give you a great answer without seeing how your book carried out the exact derivation--that is, I'm familiar with a different derivation of the Einstein Smoluchowski equation. But I think the logic is a little backwards here. I'm not sure it's completely valid to consider $\frac{\nabla\mu}{\nabla\phi}$ because what happens when your electric potential gradient vanishes?

If you start by substituting in $\nabla\mu_i=\nabla\overline{\mu}_i-z_iF\nabla\phi$ to equation 4, you get
$$N_i=-\left(c_i\nabla\phi\left(u_i-\frac{D_iz_iF}{RT}\right)+\frac{D_ic_i}{RT}\nabla\overline{\mu}_i\right) $$

When the electrochemical potential gradient vanishes, there should be no net mole flux. This leaves

$$ 0=c_i\nabla\phi\left(u_i-\frac{D_iz_iF}{RT}\right) $$
and the only way to satisfy this equation for all $\nabla\phi$ is by letting $D_i=\frac{u_iRT}{z_iF}$.

Again, this is just a guess at what might be a bit of problematic rigor, but what this essentially says is that the Einstein Smoluchowski relation holds because if it didn't, the equilibrium behavior wouldn't be correct.

Thank you for your feedback. Your derivation is the same as mine as you derived the Einstein equation with the assumption of the thermodynamic equilibrium. I think problems regarding electric potential gradient you mentioned are valid, but not too important as you can do the derivation with the same assumptions and still get the correct results without needing to divide by $\nabla \phi$ (which is what you did).

Since we agree that Einstein equation is only valid in the thermodynamic equilibrium, this begs the question. How can we use it outside of the equilibrium when net flux isn't zero? And by the equation (1), we can see that it is in fact used in that condition since I'm sure no one would write the term $- \frac {u_i c_i}{z_i F} \nabla \bar \mu_i$ if that term is always meant to be equal to zero.

What comes to my mind is that this equation is the approximation which in most cases is good enough and allows us to express both diffusion and migration flux in a more succinct form (by the electrochemical potential gradient). While that may be the case, I have yet to find that explanation in some textbook.

 

[TeethWhitener]

Since we agree that Einstein equation is only valid in the thermodynamic equilibrium, this begs the question.

I think you misunderstood my post. My point was that the Einstein equation has to be valid at least at equilibrium. You seem to want to claim that it is only valid at equilibrium, which I don’t think you can infer from my derivation.

 

[Dario56]

I think you misunderstood my post. My point was that the Einstein equation has to be valid at least at equilibrium. You seem to want to claim that it is only valid at equilibrium, which I don’t think you can infer from my derivation.

I think you can since you can't prove that $u_i = \frac {z_i D_i F}{RT}$ without taking that $N_i = 0$ which is an equilibrium condition. At least, I don't see that proof.

To expand upon my point, I would like to refer you to the Wikipedia article: Einstein relation: https://en.wikipedia.org/wiki/Einstein_relation_(kinetic_theory)#Proof_of_the_general_case.

If you go to the section: "Proof in general case", you can see that this equation is derived exactly from the equilibrium condition: $J(Drift) + J(Diffusion) = 0$

 

[TeethWhitener]

I think you can since you can't prove that $u_i = \frac {z_i D_i F}{RT}$ without taking that $N_i = 0$ which is an equilibrium condition. At least, I don't see that proof.

To expand upon my point, I would like to refer you to the Wikipedia article: Einstein relation: https://en.wikipedia.org/wiki/Einstein_relation_(kinetic_theory)#Proof_of_the_general_case.

If you go to the section: "Proof in general case", you can see that this equation is derived exactly from the equilibrium condition: $J(Drift) + J(Diffusion) = 0$

Yes I know that was your point, but that’s not what you concluded. You concluded that $u_i \neq \frac {z_i D_i F}{RT}$ when $N_i\neq0$. That is not a valid inference, or at least I don’t see how it is.

 

[Dario56]

Yes I know that was your point, but that’s not what you concluded. You concluded that $u_i \neq \frac {z_i D_i F}{RT}$ when $N_i\neq0$. That is not a valid inference, or at least I don’t see how it is.

You are correct that I didn't prove that Einstein equation isn't valid outside the equilibrium. Whether this inference is valid or not isn't really the main point.

Main point is that if we want to use Einstein equation outside the equilibrium, we need to prove that it does hold outside the equilibrium. Since Einstein equation is derived from the equilibrium condition, we have no proof that it is valid outside of it and so we shouldn't use it in such conditions. At least I haven't found such a proof.

This may come down to the fact that usage of Einstein equation outside of equilibrium gives a good match with the experimental results, makes equations nicer and experiments easier. For example, we only need to measure diffusion coefficient or ionic mobility and can calculate the other one with a reasonable accuracy without needing to measure both.

Why this match is in fact good, isn't supported with proofs, but experiments say it's valid.

 

[TeethWhitener]

Main point is that if we want to use Einstein equation outside the equilibrium, we need to prove that it does hold outside the equilibrium. Since Einstein equation is derived from the equilibrium condition, we have no proof that it is valid outside of it and so we shouldn't use it in such conditions. At least I haven't found such a proof.

The proof is basically that there are 5 independent terms in the Einstein equation (ion charge, ion mobility, temperature, Faraday constant, gas constant) and all of them are constant as you move away from (electrochemical) equilibrium.

 

[Dario56]

The proof is basically that there are 5 independent terms in the Einstein equation (ion charge, ion mobility, temperature, Faraday constant, gas constant) and all of them are constant as you move away from (electrochemical) equilibrium.

That ain't proof. Just because these are constants, that doesn't prove that relation between ionic mobility and diffusion coefficient is given by Einstein equation outside the equilibrium. To prove something in science means starting from some basic scientific principles and than using logic to derive a conclusion. This is in fact what derivations provide.

I think that this relation can't be proven in general case with the concept of thermodynamic equilibrium. Different approach is needed. As far as I can tell from searching on the internet, Einstein used the random walk concept to derive this relation which I'll check out.

 

[TeethWhitener]

That ain't proof. Just because these are constants, that doesn't prove that relation between ionic mobility and diffusion coefficient is given by Einstein equation outside the equilibrium.

The relationship doesn't change because it is a relationship between constants. I'm not sure what other proof you want.

 

[Dario56]

The relationship doesn't change because it is a relationship between constants. I'm not sure what other proof you want.

I've already said what constitutes a proof in science and this isn't a proof. Proof is what we both did earlier. There, we proved this relation, but not outside the equilibrium and so we can't claim that it holds outside of it. That isn't scientific.

Also, to repeat myself, Einstein used a different approach (which I don't currently know) to derive this equation which may explain why this one isn't going to give us a general proof.

I rewrite your general flux equation: $$N_i=-\left(c_i\nabla\phi\left(u_i-\frac{D_iz_iF}{RT}\right)+\frac{D_ic_i}{RT}\nabla\overline{\mu}_i\right) $$

Now, we can rearrange it as:

$$ N_i + \frac {D_i c_i}{RT} \nabla \bar \mu_i = c_i \nabla \phi\left (u_i - \frac {z_iD_iF}{RT} \right) $$

To prove Einstein equation validity outside of equilibrium you would need to show that sum on the left side is always zero. This is proven easily for the equilibrium as both terms are zero by the definition of the equilibrium, but I don't know why would this sum be zero in general. What principle can justify this?

 

[TeethWhitener]

But the left side of the equation is Fick's first law using the electrochemical potential as the force, so it has to be zero.

 

[Dario56]

But the left side of the equation is Fick's first law using the electrochemical potential as the force, so it has to be zero.

Fick's 1st law and electrochemical potential are two different stories. We can relate chemical potential, but not electrochemical potential to it in this direct way.

 

[TeethWhitener]

Fick's 1st law and electrochemical potential are two different stories. We can relate chemical potential, but not electrochemical potential to it in this direct way.

Of course we can. Why do you think mole flux vanishes when electrochemical potential gradient is zero? (absent convection)