In a solution, if a surfactant molecule can exist in two states, that is being a monomer or being inside a micelle, then it means that both states are equally possible. However, both states are entropically and energetically different. So molecules will either go to micelles or remain as monomers, since they are indistinguishable. There cannot be any 'equillibrium' between monomers and micelles, I think. If there is one, then it can only be a non-equllibrium state, kinetically trapped due to energetic reasons.

'It seems in practice, we just measure a certain property of the system (e.g. surface tension) at different surfactant concentrations and look for the point with maximum change in gradient (e.g. slope of surface tension vs concentration)' - yes that it is how it is done. It supposes that there is a coincidence of reached surface saturation and start of micellization. It is often forgotten to check whether indeed micelles are formed or not. However, it may coincide, e.g. with the point of saturation.

Micelles are thermodynamically stable - the nowadays often misused term 'colloidal solution' (one phase system) applies. There is a dynamic equilibrium between micellar dissociation and formation, i.e. the monomers are part of the micelles only for a distinct moment. To form these thermodynamically stable micelles a critical concentration of monomers must be present.

The cmc is well defined in case of pure one component surfactants. However, in case of the various applications this usually does not apply. Then saturation at the air-water interface with the most surface actives molecules, coincides with formation of only a few micelles. Increasing surfactant concentration, amount and composition of micelles changes, as well as that at the surface because now the most surface active compounds will partition into the micelles. This can be confirmed by the observation of a minimum in surface tension - concentration curve.

Hi Yun,

Using light scattering techniques, the presence of micelles as a function of concentration can be readily determined based on both the scattering intensity, which is proportional to the 6th power of particle radius and is therefore highly sensitive to the onset of micelle formation, as well as the size determined by DLS, where the measured size can be seen to increase dramatically at the CMC.

https://www.malvern.com/en/support/events-and-training/webinars/W160623SizeAggregationMicellesDLS.html

From your question it is not clear what interests you. The article you refer to is old in 1955. You need to look at our review article on micelle formation, and then ask a specific question. On the subject of micelle formation, there are many problems and the answers to them may not be unambiguous.

Correct: Brownian dynamics will (almost!) always lead to some micelles, however, in equilibrium, at low monomeric concentrations, there may be extremely few or a very long wait for the assembly of at least one. The CMC is the moment where properties change most (the derivative exhibits an extreme) because, microscopically, the probabilites of a newly added monomer to be either part of a micelle or not, change substantially. This is because the average free energy tilts towards the formation of micelles above the CMC. At the CMC (or close to it, depending on definition) the entropy loss by forming a micelle is compensated by the gain of micelle formation. However, there are fluctuations on top of an average free energy value: even if the energy minimum favours monomers, higher energy states (micelles) are always reached since a Boltzmann factor never completely vanishes for finite energies (but it can be arbitrarily small). The transition @CMC is not as sharp as for instance a first order phase transition is. However, micelle formation is still a cooperative phenomenon, but with a limited number of molecules - but far from infinity. Since it is cooperative (although not with infinite numbers), it exhibits relatively steep changes in properties and probabilities. That explains why the CMC remains a useful and meaningful quantity - although there is no infinitely sharp transiton. A rough equivalent would be finite length DNA melting where the melting temperature is meaningful although the transition is not strictly sharp.

For micellization there are several models., And hence the definition of CMC.

1. Phase transition of the first kind. CMС is the concentration of a phase transition of the crystallization type. The principle of everything or nothing works here. The surfactant is either in solution or in micelles.

2. Chemical reaction of associate formation. CMC is the concentration of associate formation. In this model, one can use the Le Chatelier principle. Micelles and monomers are in equilibrium.

3. Our model. Stretched in concentration, a polyamorphous transition in an ensemble of small water systems. CMС is the average concentration of the transition. The surfactant is either in solution or in micelles. There is an article in my profile.

4. Our model. A quantum transition. We are working on this article. The idea is published in our questions and patented in the Russian Federation.

There are other models.

Probably, Yun is interested in CMC as a critical phenomenon of a reversible transition from true solution to micellar one. In the case of an individual surfactant, this transition is critical (namely, sharp and cooperative) one only for surfactants having a sufficiently long alkyl chain and forming large micelles in a narrow concentration range corresponding to the CMC. Short-chain surfactants are combined into small micelles not cooperatively, but gradually and in a wide concentration range, so strictly speaking, the word "critical" for items is not applicable. As a rule, a low-cooperative association of surfactants takes place in organic solvents and then it is better to talk about the critical association concentration.

A new look at CMC and micelle formation can be seen in our preprint "Quantum entanglement in micellar solutions of ionic surfactants".