Assume that polyoxyethylene sorbitan fatty acid ester (polysorbate) is the surfactant and the T effect on the CMC over a T range is to be evaluated.For such system, as T increases, the CMC initially decreases and then increases. Due to the little chance of hydrogen bond formation at higher Ts, the initial decrease of the CMC with T is a consequence of the decreased hydrophilicity of the surfactant molecules.In other words, the increase in T results in decreasing hydration of the hydrophilic oxyethylene group, which favors micellization. Thus, as T increases the start of micellization occurs at lower concentrations. On the other hand, dissolving the surfactant molecules in water makes the hydrophobic group disfigure the water structure. Additionally, increasing T also causes an increase in the breakdown of water structure surrounding the hydrophobic groups, which disfavors micellization. Therefore, the beginning of micellization tends to occur at higher concentrations as T increases. So, as T increases further, the effect of the hydrophobic groups begins to apply an influence and finally dominates as the CMC reaches a minimum value and finally increases with T.

A detailed answer to your question, see our article

Mirgorod, T.A. Dolenko, Liquid Polyamorphous Transition and Self-Оrganization in Aqueous Solutions of Ionic surfactant, Langmuir 31 (2015) 8535-8547.

in nonioin  Ethylene Oxide Surfactants, CMC first decreses with increasing Temperature then increses. Read This Article:

http://www.sciencedirect.com/science/article/pii/S0927775797002380

regards

Thanks to all for their explanations

You can refer our published papers: (1) J. Surf. Detergent.,2013, 16, 739-749; (2) J. App. Soln. Chem. Modeling, 2012, 1, 65-73

Fundamentally, the explanation is based on on the competition of two processes  concerning the temperature change of water structure that stipulates the hydrophobic hydration of hydrocarbon chain of surfactant (the longer chain the lower CMC) and hydration properties of hydrophilic groups that begin to play more important role usually in the interval about 20-50 degrees when the water structure became less and facilitate the hydrophilic hydration of polar group of surfactant molecule. So, point of temperature minimum should depend from surfactant nature and HLB may be.

The changes in the logarithm of the CMC with the temperature are similar to the way in which the isothermal compressibility of water changes with temperature. The difference between systems in minimum temperature. At water it is equal 45 0С. In ionic surfactants and hydrocarbons, 25 ° C. The presence of surfactants and hydrocarbons in water changes the temperature dependence of the isothermal compressibility of water. This includes all the effects that experts write about. Therefore, the Gibbs energy changes to form a cavity of the same size, which is equal to the volume of the molecule / isothermal compressibility.

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