# Critical micelle concentration

In colloidal and surface chemistry, the critical micelle concentration (CMC) is defined as the concentration of surfactants above which micelles form and all additional surfactants added to the system will form micelles.[1]

The CMC is an important characteristic of a surfactant. Before reaching the CMC, the surface tension changes strongly with the concentration of the surfactant. After reaching the CMC, the surface tension remains relatively constant or changes with a lower slope. The value of the CMC for a given dispersant in a given medium depends on temperature, pressure, and (sometimes strongly) on the presence and concentration of other surface active substances and electrolytes. Micelles only form above critical micelle temperature.

For example, the value of CMC for sodium dodecyl sulfate in water (without other additives or salts) at 25 °C, atmospheric pressure, is 8x10−3 mol/L.[2]

## Description

Top to Bottom: Increasing concentration of surfactant in water first leads to the formation of a layer on the surface. After reaching the CMC micelles begin forming. Notice that the existence of micelles does not preclude the existence of individual surfactant molecules in solution.

Upon introducing surfactants (or any surface active materials) into a system, they will initially partition into the interface, reducing the system free energy by:

1. lowering the energy of the interface (calculated as area times surface tension), and
2. removing the hydrophobic parts of the surfactant from contact with water.

Subsequently, when the surface coverage by the surfactants increases, the surface free energy (surface tension) decreases and the surfactants start aggregating into micelles, thus again decreasing the system's free energy by decreasing the contact area of hydrophobic parts of the surfactant with water.[3] Upon reaching CMC, any further addition of surfactants will just increase the number of micelles (in the ideal case).

There are several theoretical definitions of CMC. One well-known definition is that CMC is the total concentration of surfactants under the conditions:[4]

if C = CMC, (d3${\displaystyle \phi }$/dCt3) = 0
${\displaystyle \phi }$ = A[Cs] + B[Cm]; i.e., in words Cs = [single surfactant ion] , Cm = [micelles] and A and B are proportionality constants
Ct = Cs + NCm; i.e., N = represents the number of detergent ions per micelle

The CMC generally depends on the method of measuring the samples, since A and B depend on the properties of the solution such as conductance and photochemical characteristics. When the degree of aggregation is monodisperse, then the CMC is not related to the method of measurement. On the other hand, when the degree of aggregation is polydisperse, then CMC is related to both the method of measurement and the dispersion.

The common procedure to determine the CMC from experimental data is to look for the intersection of two straight lines traced through plots of the measured property versus the surfactant concentration. This visual data analysis method is highly subjective and can lead to very different CMC values depending on the type of representation, the quality of the data and the chosen interval around the CMC.[5] A preferred method is the fit of the experimental data with a model of the measured property. Fit functions for properties such as electrical conductivity, surface tension, NMR chemical shifts, absorption, self-diffusion coefficients, fluorescence intensity and mean translational diffusion coefficient of fluorescent dyes in surfactant solutions have been presented.[6][7][8] These fit functions are based on a model for the concentrations of monomeric and micellised surfactants in solution, which establishes a well-defined analytical definition of the CMC, independent from the technique.

The CMC is the concentration of surfactants in the bulk at which micelles start forming. The word bulk is important because surfactants partition between the bulk and interface and CMC is independent of interface and is therefore a characteristic of the surfactant molecule. In most situations, such as surface tension measurements or conductivity measurements, the amount of surfactant at the interface is negligible compared to that in the bulk and CMC can be approximated by the total concentration.

There are important situations where interfacial areas are large and the amount of surfactant at the interface cannot be neglected. If, for example, air bubbles are introduced into a solution of a surfactant above CMC, these bubbles, as they rise to the surface, pull out the surfactants from the bulk to the top of the solution creating a foam column and thus reducing the concentration in bulk to below CMC. This is one of the easiest methods to remove surfactants from effluents (foam flotation). Thus in foams with sufficient interfacial area there will not be any micelles. Similar reasoning holds for emulsions.

CMC is most typically measured by plotting surface tension versus surfactant concentration in an automated measurement.

The other situation arises in detergents. One initially starts off with concentrations greater than CMC in water and on adding fabric with large interfacial area and waiting for equilibrium, the surfactant concentration goes below CMC and no micelles are left. Therefore, the solubilization plays a minor role in detergents. Removal of oily soil occurs by modification of the contact angles and release of oil in the form of emulsion.

In petroleum industry, CMC is considered prior to injecting surfactant in reservoir regarding enhanced oil recovery (EOR) application. Below the CMC point, interfacial tension between oil and water phase is no longer effectively reduced.[9] If the concentration of the surfactant is kept a little above the CMC, the additional amount covers the dissolution with existing brine in the reservoir. It is desired that the surfactant will work at the lowest interfacial tension (IFT).

## Simulations and Data sets

The formation of micelles take place only above a particular temperature and above a particular concentration called critical micelles concentration.[10][11] The reliability of some experimental data has been questioned by some authors who have proposed standard data sets.[12] Attempts have also been made to use machine learning and other statistical models to predict CMC. Some of these statistical prediction attempts have also curated data sets from literature.[13][14]

## References

1. ^ IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version:  (2006–) "critical micelle concentration". doi:10.1351/goldbook.C01395
2. ^ Ana Domínguez, Aurora Fernández, Noemí González, Emilia Iglesias, and Luis Montenegro "Determination of Critical Micelle Concentration of Some Surfactants by Three Techniques", Journal of Chemical Education, Vol. 74 No. 10 October 1997, pp. 1227–1231 (pdf)
3. ^ Hakiki, F., Maharsi, D.A. and Marhaendrajana, T. (2016). Surfactant-Polymer Coreflood Simulation and Uncertainty Analysis Derived from Laboratory Study. Journal of Engineering and Technological Sciences. 47(6):706-724. doi: 10.5614/j.eng.technol.sci.2015.47.6.9
4. ^ Phillips J. The energetics of micelle formation. Transactions of the Faraday Society 1955;51:561-9
5. ^ Mukerjee, P.; Mysels, K. J. In Critical Micelle Concentrations of Aqueous Surfactant Systems; NIST National Institute of Standards and Technology: Washington D.C. USA, 1971; Vol. NSRDS-NBS 36
6. ^ Al-Soufi W, Piñeiro L, Novo M. A model for monomer and micellar concentrations in surfactant solutions: Application to conductivity, NMR, diffusion, and surface tension data. J.Colloid Interface Sci. 2012;370:102-10,DOI: 10.1016/j.jcis.2011.12.037
7. ^ Lucas Piñeiro, Sonia Freire, Jorge Bordello, Mercedes Novo, and Wajih Al-Soufi, Dye Exchange in Micellar Solutions. Quantitative Analysis of Bulk and Single Molecule Fluorescence Titrations. Soft Matter, 2013,9, 10779-10790, DOI: 10.1039/c3sm52092g
8. ^ www.usc.es/fotofqm/en/units/single-molecule-fluorescence/concentration-model-surfactants-near-cmc
9. ^ Hakiki, Farizal. A Critical Review of Microbial Enhanced Oil Recovery Using Artificial Sandstone Core: A Mathematical Mode l. Paper IPA14-SE-119. Proceeding of The 38th IPA Conference and Exhibition, Jakarta, Indonesia, May 2014.
10. ^ William C, Jordan, Kirk E, Warren, Patrick B, Noro, Massimo G, Bray, David J, Anderson, Richard L, Toward a standard protocol for micelle simulation, The Journal of Physical Chemistry B, 2016, [1]
11. ^ Vishnyakov, Aleksey, Lee, Ming-Tsung, Neimark, Alexander V, Prediction of the critical micelle concentration of nonionic surfactants by dissipative particle dynamics simulations, The journal of physical chemistry letters 2013, [2]
12. ^ Swope, William C, Johnston, Michael, Andrew, Duff, Andrew Ian, McDonagh, James L, Anderson, Richard L, Alva, Gabriela, Tek, Andy T, Maschino, Alexander P The Challenge to Reconcile Experimental Micellar Properties of the CnEm Nonionic Surfactant Family. The Journal of Physical Chemistry B 2019. [3]
13. ^ McDonagh, J., Swope, W., Anderson, R. L., Johnson, M. A., & Bray, D. J. What Can Digitization Do For Formulated Product Innovation and Development?. ChemRxiv 2019. [4]
14. ^ Huibers, P. D., Lobanov, V. S., Katritzky, A. R., Shah, D. O., & Karelson, M. Prediction of critical micelle concentration using a quantitative structure− property relationship approach. 1. Nonionic surfactants. Langmuir, 1996. [5]