# Viscosity at CMC

What happens to viscosity at critical micelle concentration? Does it increase or decrease abruptly, and if so, what's the reason behind it?

I'm aware of changes in other properties such as conductivity, concentration etc, but not viscosity.

• Do you mean as the concentration of the monomer increases (at constant T)? Commented Mar 23, 2021 at 12:28
• No, I mean at the critical micelle concentration, when the individual ions of the soap molecules cluster together Commented Mar 23, 2021 at 12:37
• Yes, sorry, never mind my comment. It turns out the viscosity often increases. Commented Mar 23, 2021 at 13:42
• whats the reason behind it? i tried making a comparison with gases, where the viscosity increases with the radius of the gas, and also with density... here, the "radius" is increasing, due to micelle formation and the "density" (which i think is analogous here to concentration) is decreasing... how do we decide which factor dominates? Commented Mar 23, 2021 at 15:06

## 1 Answer

One of the more fundamental theories of solution viscosity (by Einstein) predicts the following for dilute solutions of perfect spheres in a Newtonian fluid:

$$\frac{\eta}{\eta_0} = 1 +2.5\phi \tag{1}$$

where $$\eta_0$$ is the viscosity of the pure solvent, $$\phi$$ is the volume fraction of solute (here: aggregating detergent) Note we are ignoring higher order terms that describe the dependence on micelle-micelle interactions (e.g. on the second virial coefficient).

The important point of Eq. 1 is that the viscosity is independent of the size of the spheres, or the composition (provided they are rigid, not flexible). So according to this theory, if the volume occupied by the micelles is not different from that occupied by the monomers, you should not see a change in viscosity. But typically you do, and that means the volume fraction is changing, at least according to this theory. How? Maybe through hydration. The theory does not say whether hydration plays a role, but hydration can alter the effective volume occupied by the spheres, if you include the hydration layer as part of the micelle volume.

Most likely the theory is not really accurate and molecular shape also plays a role. However that would require a more complex theory to address.

It turns out that viscosity often goes up at the CMC (with formation of micelles).