Osmotic Pressure
One of the reviewers for the diffusion based separations paper was concerned that there would be osmotic pressure which would cause convection across the membrane. Previously we have been aware of osmosis in the DNA experiments by Paul Black and Bill Bernhard, although their results were due to a salt imbalance if I recall correctly. In this post I hope to show that our protein concentrations were too low to cause streaming of fluid across the membrane due to an osmotic pressure.
Deen (AIChe J. 1987) writes that the volume flux (Jv) through a membrane can be represented by:
Lp is the hydraulic permeability and is a function of the porosity (γ), pore size (ro), fluid viscosity (η), and pore length (L). ΔP and ΔΠ are the hydraulic and osmotic pressure differences across the membrane, and σ0 is the osmotic reflection coefficient, which equals 1 in the case where the protein cannot cross the membrane. We can determine the osmotic pressure using the following equation by van’t Hoff,
where R is the gas constant, T the temperature, and C0-CL the concentration difference of the protein across the membrane.
In the following table I’ve calculated a few of these items. The hydraulic pressure, ΔP, is held at zero and the osmotic pressure is calculated for a 10 mg/mL solution (the highest I use in my experiments) of a 50 kD protein that cannot cross the membrane.
This treatment shows that the osmotic flow rate in this case is very small and can be neglected in my theory.
Note: Deen goes on to say that the osmosis reflection coefficient is less than 1 and has a very similar form the steric hindrance equation. We need to keep this in mind if we wish to do more than a superficial treatment of osmotic pressure.