later, Lange [65] applied the mass action law to ionic micelles and wrote the equilibrium of formation of the micelle as
(1.8)
where K is the counterion, A the surfactant ion, and p and q the stoichiometric coefficients. Although Lange considered the activity coefficients of the different species, for simplicity we will ignore them and write the equilibrium constant as
(1.9)
Writing [A] = c k and [K] = c k + N, where N is the equivalent concentration of added salt, it is finally found that
(1.10)
where L = [K p A q ]. Thus with logc k as ordinate and log(c k + N) as abscissa, this is the equation of a straight line with the slope –p/q, which corresponds to the empirical one found by Corrin and Harkins. This point has been discussed in detail by Hall [66] in his theory for dilute solutions of polyelectrolytes and of ionic surfactants.
The effects of solvents (alkyl alcohols C n H2n+1OH, n = 1–4; HOCH2CH2OH, glycerol, 1,4‐dioxane, and heptanol) on the critical concentration for micelle formation of cationic soaps was studied by Corrin and Harkins [67], Herzfeld et al. [68], and Reichenberg [69]. Klevens [70] found that increasing the temperature causes an apparent decrease in the cmc, as determined by spectral changes in various dyes. However, this same author found the opposite effect when the micelles formation was determined by refraction [71].
Simultaneously, other experimental techniques, mainly spectroscopic ones, were introduced for the determination of the cmc. After a paper published by Sheppard and Geddes [72], in which the authors reported that by the addition of cetyl pyridinium chloride, the absorption spectrum of aqueous pinacyanol chloride was shifted from that exhibited in aqueous solutions to that in non‐polar solvents, Corrin et al. [73] used this property to determine the cmc of laurate and myristate potassium salts, giving values of 6 × 10−3 M and 0.023–0.024 M, respectively. The concentration of soap at which this spectral change occurs was taken as the cmc, proposing that the dye is solubilized in a non‐polar environment within the micelle. Klevens [74] performed a similar work by studying the changes in the spectrum of pinacyanol chloride in solutions of myristate, laurate, caprate and caprylate potassium salts, and sodium lauryl sulfate. These studies were extended to other surfactants [75] and other dyes as p‐dimethylaminoazobenzene [76]. By using suitable dyes (Rhodamine 6G, Fluorescein, Acridine Orange, Acridine Yellow, Acriflavine, and Dichlorofluorescein) fluorescence spectroscopy was soon adopted [77, 78].
In 1950, Klevens [79] studied the solubility of some polycyclic hydrocarbons in water and in solutions of potassium laurate (at 25 °C). For all the polycyclic hydrocarbons, he showed that by increasing the concentration of the surfactant, their solubility also increased. Particularly, for pyrene he measured solubilities of 0.77 × 10−6 and 2.24 × 10−3 M in water and potassium laurate (0.50 M), respectively.
One year later, Ekwall [80] studied the sodium cholate association by measuring the fluorescence intensity, and determined that the lowest concentration at which polycyclic hydrocarbons (3,4‐benzopyrene included) are solubilized is 0.018 M. This corresponds to the beginning of the micelle formation, although “at first relatively small amounts of cholate ion aggregates and the actual micelle formation occurs at about 0.040 to 0.044 M.” Foerster and Selinger [81] observed that in micelles of cetyldimethylbenzylammonium chloride, pyrene forms dimers in excited states (excimers).
In the period 1971–1980, the number of papers on solubilized pyrene in micelle solutions increased very quickly. The fluorescence decay of the excited state of pyrene received an important attention. The aggregation number and microviscosities of the micellar interior [82], the permeability of these micelles with respect to nonionic and ionic quenchers [83], oxygen penetration of micelles [84], or the environmental effects on the vibronic band intensities in pyrene monomer fluorescence in micellar systems [85, 86] were published. Kalyanasundaram and Thomas carefully analyzed the lifetime of the monomer fluorescence and the ratio I 3 /I 1 of the third and first vibronic band intensities of pyrene in sodium lauryl sulfate as a function of its concentration. Both curves have a sigmoidal shape (see Figure 1.3 of the paper). A value of 8 × 10−3 M for the cmc of the surfactant was given.
However, Nakajima [86] plotted the ratio I 1 /I 3 and accepted the cmc as the concentration at which the first break is observed (point A in Figure 1.4). At low concentrations of the surfactants the values of the I 1/I 3 ratio are high, typical of a hydrophilic environment for pyrene, the value in water being 1.96 [87] while at high surfactant concentrations the I 1/I 3 ratio tend to typical values of non‐polar solvents. For instance, at high surfactant concentrations of sodium cholate and sodium deoxycholate, the I 1/I 3 ratio is around 0.7 [88] while the value in cyclohexane is 0.61 [89]. This suggests that the polarity of the microenvironment of pyrene is a lipophilic one. Andersson and Olofsson [90], when performing a calorimetric study of nonionic surfactants, also made use of Nakajima’s approach.
Figure 1.4 Typical plot of a sigmoidal curve. Example ϕ = I 1/I 3 (ratio of the intensities of the first and third vibronic peaks of pyrene) vs increasing concentration of a surfactant (left) and its first derivative (right). The shape of curves from isothermal titration calorimetry are similar in shape (see below for a description).
Other authors have proposed the inflection point of the curve (point B) as cmc [91]. As such it fulfills the condition
where ϕ would be the I 1/I 3 ratio. The expression is also valid for any other property that exhibits a sigmoidal behavior as the obtained enthalpograms from isothermal titration calorimetry (ITC) [92]. The plot of (dϕ/dS t ) vs S t is shown in Figure 1.4 (right) and the cmc is easily obtained from the peak.
Aguiar et al. [93] have analyzed both points (A and B) for several surfactants and proposed an approach for choosing between one or the other point. Occasionally, both A and C points have been accepted as an indication that the system has two cmc values. We consider that this is not correct. These different points of view introduce an important question related to the determination of the cmc from sigmoidal curves, which are frequently found when using some experimental techniques.
By now, some different approaches to determine the cmc have already been introduced. Rusanov [94] has reviewed the definitions of cmc based on the application of the mass action law to the aggregation process in surfactant solutions. Among them, we must mention the definition given by the equation
which