by Phillips [95] in 1955 for determining the cmc for an ideal measured property ( ϕ )‐concentration (S t ) relationship. Phillips pretended that Eq. (1.12) corresponds to the point of maximum curvature, but this is not the case. Nakajima's approach fulfills this condition as well as the methodology proposed by Olesen et al. [96] for determining the aggregation number of aggregates from ITC curves. However, the definition of cmc as corresponding to the inflection point in the ϕ vs S t curve has been recommended [97] for the determination of the cmc from ITC curves. For large absolute values of the slope at the inflection point (= ( ϕ C − ϕ A )/(S C − S A )) in previous sigmoidal curves, the difference in the values obtained from any of the two previous equations may be considered negligible for practical purposes. If the cmc is fairly sharp, Hall [98] has proposed that it can be regarded approximately as a second order phase transition.
Among the other definitions for cmc analyzed by Rusanov we would like to remark on the following one. The focus is on a system in which micelles are composed of a single sort of particle. For further details and the analysis of more complex systems, the two papers by Rusanov [94] are recommended.
Let us redefine the Grindley and Bury [45] equilibrium constant as
The equilibrium constant K o would correspond to a hypothetical single step in which a virtual aggregate m j is formed by the binding of an additional monomer to a virtual aggregate of size m j‐1 according to
its equilibrium constant being
(1.15)
The isodesmic model accepts that all K j constants are equal to K o . The difference with the Grindley and Bury equilibrium constant comes from the fact that only (n – 1) steps are required to form a micelle with n monomers. Interestingly, in 1935 Goodeve [99] have pointed out that forming micelles of, say, about 20 molecules must pass through all the intermediate stages of association. The formation of the micelles from the monomer in one stage is, of course, highly improbable as it requires “a collision of 20 molecules at one time.” Goodeve presented Eq. (1.14) as representing the equilibrium according to this point of view.
Equation (1.13) is better understood in the form
where K o and s are both positive, n is usually large, and, independently of the value of n, for K o × s = 1, the concentration of micelles and monomers are the same. Deviations of the product K o × s from that value lead to either m n < s or m n > s. For instance, for n = 50, the ratio m n /s changes by a factor 1.86 × 104 when K o × s varies from 0.9 to 1.1. This is in fact the analysis by Grindley and Bury [45].
This suggests a definition of cmc by the condition
(1.17)
and from the conservation of material (S t = s + n × m n ) it follows that at cmc S t,cmc = (n + 1)s cmc = (n + 1)m n .
From Eq. (1.16) it also follows that
(1.18)
or
In a monodisperse system, this equation may be simplified to
(1.20)
since at cmc, m n = s and n is constant for the whole range of surfactant concentrations. Thus, the larger the rate of change of the micelle concentration is with respect to the change in the monomer concentration, the higher the aggregation number will be.
Once the equilibrium constant and the aggregation number are known, all the thermodynamic functions may be obtained. These thermodynamic quantities have traditionally been determined from the measurement of cmc at different temperatures, the range of temperatures being around 40 °C (or less). The problem is that the dependence of the cmc with temperature is usually low for most of the surfactants and, as the dependence of ΔG o with the concentration is logarithmic, the range of experimental values is even shorter. This introduces an error in the determination of the thermodynamic amounts, which is necessarily rather high.
The commercial introduction of high quality isothermal titration calorimeters has provided a routine way for the determination of previous amounts, which have a much higher precision. In a typical measurement a sample cell is filled with water (or any other appropriate solvent). A surfactant solution is placed in a syringe, which allows the injection of small aliquots into the sample cell at different intervals of time. The solvent of this solution and the one filling the sample cell must be identical to prevent some effects as the dilution heat of inert salts or buffers. The concentration of the surfactant ranges from 10 to 30 times the cmc value. Each injection increases the surfactant concentration in the sample cell from zero to a concentration clearly above the cmc. The heat involved in the process (the concentration in the syringe is always higher than in the sample cell) after each injection is measured and plotted vs the increasing concentration in the sample cell. Figure 1.4 imitates a typical enthalpogram and its derivative. In this case, ϕ = ΔH (in kJ/mol of injectant) is the involved heat after each injection.
An ideal enthalpogram can be subdivided into three concentration ranges. In region I (first injections, till point A in Figure 1.4) the increasing concentration in the cell is still below the cmc. Here, the large enthalpic effects observed are mainly due to breaking micelles into monomers (demicellization process) and dilution of monomers [92]. In region III (final injections, after point C in Figure 1.4), the increasing concentration in the cell is above the cmc. Here, the low enthalpic effects observed are mainly due to dilution of micelles. In the central region (between A and C in Figure 1.4), a sharp decrease is observed and corresponds to a transition from reaching the cmc and exceeding it. Therefore, the cmc corresponds to the inflection point of the curve, which can easily be determined as the first derivative of the curve (