Alternate Smoluchowski Equation

My previous post on Electroomosis theory describes the deviation between expected and calculated zeta potential using the Smoluchowski equation. In this post I will explain another method of solving for the zeta potential and I will consider the mistakes made in the previous calculations.

In a paper by Kim(Mukhtar) in J. Membrane Sci. (1996), the zeta potential of PES membranes is found using electroosmosis. Flow rates were measured using constant current. An alternative version of the Smoluchowski Equation was used to calculate the zeta potential. For comparison’s sake, here is the original form of the Smoluchowski equation:

U = Eεζ/η

where U is the velocity, ε is the dielectric constant of the solution * the permittivity of free space, ζ is the zeta potential, E is the electric field, and η the viscosity. We can use the following relationship to find the current density (J):

J = κE,

where κ is the conductivity of the solution. Substituting this equation into the Smoluchowski equation gives

U = Jεζ/ηκ

and if we change velocity to the flow rate (Q=U*Area), we can change current density to current (J=I/Area), so that

Q = Iεζ/ηκ

This is the alternative Smoluchowski equation used by Kim(Muhktar) to solve for zeta potential.

I have measured flow rate and conductivity and know the viscosity and permittitivity of our solutions. While I ran experiments at constant voltage, I wrote down the current and it didn’t change much during the experiments. Here are the calculated zeta potentials of pnc-Si, track etched, PES, and cellulose membranes:

All of these values are in the expected range and can be supported by the papers I mention in the literature column.

So what was the problem? This analysis makes it look like pnc-Si follows EO theory just fine. My guess is that I wasn’t coming up with a good estimate of the electric field, and when I take that out of the Smoluchowski equation, there’s no longer a problem. So can we go back and find the field? To do that, we can use the first form of the Smoluchowski equation and insert the calculated zeta potentials. Since we also need a velocity rather than a flow rate, we need to divide our flow rate by both the active area and the porosity (this part will be tricky for PES and cellulose – still working on it). We can then solve for the field, and by multiplying the field by the distance of the membranes, we can also find the voltage drop.

Now to me, these fields seem high, but the voltage drop seems reasonable. It shows that we don’t need to set the voltage very high to drive EO in our membrane, and the closer we get our electrodes the better. The TE membranes have lower overall fields, and this is what probably results in the lower normalized flow rates for TE compared to pnc-Si.