## Zeta Potential

### Quick Start

For DLVO theory it is important to know the charge on a particle which will keep (we hope) particles apart. It turns out that the meaning of "charge" on a particle is rather blurred, and so we define it as a charge some distance from the surface of the particle. This Zeta potential, ζ, depends partly on the "real" charge ψ0, and the molar concentration of ions in the solution that "dilute" the effect of the real charge. By sliding the sliders and seeing where ζ is measured, you rapidly get a good feel for what is going on.

### Zeta Potential

Molar Conc.
ψ0 mV
Z1 and Z2
k-1Å
Stern mV
ζ mv

Within DLVO theory charge stabilization is of great importance for aqueous particles. The "potential", ψ, of a particle is actually meaningless as it varies from its "real" potential at the surface, ψ0, all the way to a neutral zone relatively far from the particle. Close to the particle is the "double layer" of charge+opposite charge. By convention, what is entered into DLVO is the Zeta (ζ) potential which is imagined as the potential at the point around the particle where the tight cluster of charges at the surface ends and the solution begins, in other words, the slip plane between particle and solution.

This app is a visual guide to what's going on. You enter a "real" surface charge, ψ0, in mV, plus the concentrations/charges of the ions in solution. From this is immediately calculated k-1, the Debye length which is the distance over which the potential falls by 1/e. [The original theory seems to be in terms of an exponential factor, k, which turns out to be the inverse of a distance value. Hence we talk of the Debye distance as k-1, a somewhat bizarre nomenclature which has stuck]. There is much handwaving about what exactly the double layer is and how k-1 relates to the end of the Stern layer or to the exact spot where the zeta potential is measured. Some even say "Stern=Zeta" yet graphically show the zeta point approximately twice the distance of the Stern, presumably because it makes for graphical convenience. Some define the double layer strictly as the layer very close to the particle comprising the charge on the particle and the counter-charge in the next layer. Others define it as the combination of these first two layers and the "diffuse" layer beyond it. Other say that the zeta potential starts inside the diffuse layer, others say that it defines the end of the diffuse layer and the start of the general solution. The confusion here links to an important point.

The calculation makes the point that as the ionic strength goes up, k-1 goes down, so k goes up and "therefore" the zeta potential goes down. This "therefore" is not obvious. The general assumption is of a Gouy-Chapman formula where ψ=ψ0.exp(-k.x) where x is distance from the particle. There is no reason for the zeta potential to diminish if the x-position of the zeta border happened to reduce with ionic strength. In order to give a specific image for the app, it is assumed that the Stern potential sets in at 5Å and the zeta potential at 10Å for no reason other than it makes the numbers and the graphics look reasonable. Note that the decrease in k-1 is shown graphically by the second charge layer getting closer to (and starting to intermingle with) the first layer. How this should truly be captured graphically I do not know. I've ploughed through many articles on zeta potential and found no clarity on these issues - everyone seems to sweep them under the carpet. If someone has better suggestions, please let me know.

The calculation assumes a dielectric constant of 80 for water, and a temperature of 298K. What value should you choose for ψ0? No-one knows as it's not a really meaningful concept. Just choose a value that gives you a reasonable value for zeta. For the Smoluchowski approximation to be used, the double layer should be much thinner than the particle radius. For graphical purposes, however, the particle is shown with a rather small radius. The graph shows ψ varying with distance. The points where the graph crosses the Stern (S) and Zeta (ζ) lines correspond to the Stern and Zeta potentials shown in the calculated output. Positive ions are red, negative are blue. No attempt is made to show polyvalent ions.

Although the app image is only a crude cartoon I have found it very useful for myself as it has forced me to think harder about something I assumed (wrongly) that I understood. I look forward to making it more insightful if others can suggest improvements.

It is generally accepted that for a zeta value greater than 30mV or less than -30mV then the particles will tend to be stable. But because zeta depends on ionic strength, if you happen (deliberately or accidentally) to put your stable particles into an environment with higher ionic strength, the magnitude of the zeta potential might dip and the particles will coagulate. This happens distressingly often in real-world formulations. The point about zeta is to not take it too seriously. Any single value is going to be a complex brew of effects that are impossible to untangle. Tiny levels of impurities can switch the sign of zeta. And, of course, pH can have a profound effect on it. In the old days, measuring zeta was hard so any single value tended to be regarded with reverence. These days you get zeta "free" with your particle sizing, so you should never measure a single value, but a bunch of them at different ionic strengths, pH, levels of suspected impurities. It is in the broad picture of your zeta measurements that wisdom lies - not in any single value.