## CMC Saturation

### Quick Start

Most people think that they understand what CMC and what has happened before and after. Experience shows that most people are wrong. This app shows more realistically what is happening. There are just two parameters: K_{ads} which controls how much the surfactant likes to be at the water/air interface, and n, the micellar aggregation number which controls how much the surfactant likes to be with itself. [The CMC is implicit via normalizing the plote.] From these two numbers, you can find out the concentration of monomer, adsorbed species and micelles. It makes for interesting viewing!

### CMC Saturation

_{ads}

It is often said that the CMC is the point at which the surface can absorb no more surfactant which then starts to form micelles. This is obviously wrong. CMC can be determined from bulk properties such as conductivity, diffusion coefficient or turbidity independent of any surface effects.

It is equally tempting to say that the minimum surface tension observed at CMC corresponds to the surface being fully saturated with surfactant. This again is wrong. As Rosen has pointed out, by the time the surface tension has diminished by ~20 dyne/cm the surfactant at the surface is already locally close-packed, without fully covering the surface and there are plenty of surfactants that are only, say, 85% saturated at the CMC. How can this be?

Using the approach of Durbut^{1} it is possible to get a clearer picture of what is going on.

The concentration of micelles, M, is governed by an equilibrium constant, K_{M} which depends on the free monomer concentration, m, and the micelle aggregation number n.

K_{M}=M/m^{n}

The concentration of free absorption sites, A, at the air interface is governed by the Langmuir equation and the adsoporption constant K_{ads}

K_{ads}=(m-A)/(m.A)

Finally, if we ignore the tiny amount of surfactant at the surface, the monomer concentration is related to the total surfactant concentration, S by:

S=m+nK_{M}m^{n}

From these three equations it is possible to plot the free monomer concentration, the fraction of surfactant held in micelles and the fraction of the surface saturated by monomer. For convenience K_{M} is set to 1; so to convert the surfactant concentration scale to real units you multiply by the experimental CMC.

What this tells us is that above a micelle aggregation number of ~20, n makes little difference to the situation. The big effect is from K_{ads}. When this is small, although a standard surface tension versus concentration plot will show a typical CMC behaviour, at the CMC the surface is anything but saturated with surfactant. If K_{ads}=1 then the surface is only 50% saturated. More typical is a value of 10 when the surface is 90% saturated.

As Durbut makes clear, the model is a simplification. But it conveys some key ideas very clearly and captures more closely what is going on at the CMC which, in surface tension terms is "nothing much - there's simply very little spare surfactant to do anything more."

^{1}Patrick Durbut, Surface Activity, Chapter 3 in Handbook of Detergents, Part A,Surfactant Science Series 82, ed. Guy Broze, Marcel Decker, 1999