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Recently Olesen et al. [96] have published a method for analyzing the ITC curves that allows the determination of the aggregation number. As in previous cases, we will limit the presentation to neutral surfactants. The mass‐action model and monodispersity of micelles are assumed.

Other significant papers for determining the aggregation number of micelles are those by Debye and by Turro and Yekta. After his landmark paper published in 1947 for the molecular weight determination by light scattering, Debye [107] immediately published the first determinations of micellar molecular weights by this technique in 1949 [108], the surfactants being alkyl quaternary ammonium salts and amine hydrochlorides. A few years later, Tartar and Lelong [109] determined the micellar molecular weights of some paraffin chain salts by the same technique. Nowadays, the technique is almost routinely applied in laboratories for determination of molecular weights of polymers, micelles, and so on.

In 1978, Turro and Yekta [110] presented a simple procedure for determination of the mean aggregation number of micelles by measuring the steady‐state fluorescence quenching of a luminescent probe by a hydrophobic quencher. The Poisson statistics to describe the distribution of the luminescent molecule D and the quencher Q in a solution that contains a well‐defined but unknown micelle concentration [M] was accepted. Both D and Q are selected in such a way that they reside exclusively in the micellar phase. D will partition itself both among micelles containing Q and among “empty” micelles. They also assumed that only excited micelles of D, D*, emit in the micelles containing no Q, i.e. D* is completely quenched when it occupies a micelle containing at least one Q. Under these conditions a “very simple expression” for obtaining the aggregation number is deduced.

In 1899 Biltz [111] published the book Practical Methods for Determining Molecular Weights. The book is a summation of the practical methods for determining molecular weights by vapordensity and other methods based on colligative properties, mainly from the measurement of the increase of the boiling point of a solution with respect to a pure solvent, and the freezing‐point method. Both methods, together with the Nernst method (based upon the principle of lowering of solubility) are the only ones that “find practical application in the laboratory.” When the freezing‐point method is applied to some salts, Biltz accepts the Arrhenius dissociation theory of electrolyte solutions. Otherwise “the electrolytic dissociation in aqueous solutions can lead to smaller molecular weights than would be expected from the formula of the substance.” Earlier in 1896, Krafft [112] noticed that the sodium salts of the shorter fatty acids exist in the “molecular state” (meaning that they are in the state of single molecules, and not in that of molecular aggregates) in aqueous solution and that they give twice the normal rise of boiling point which in fact it would correspond with a hydrolytic decomposition into sodium (hydroxide, in the original) and the fatty acid. Following Biltz, the reverse can take place if “… by condensation several simple molecules form a more complex molecule…. The term association has recently been proposed for this kind of condensation.” Similarly, Kahlenberg and Schreiner [113] observed that the reduction of the freezing point of solutions of sodium oleate resulted in a molecular weight, which is nearly twice as large, like the theoretical formula. Botazzi and d'Errico [114] investigated glycogen of different concentrations by viscosity, freezing point, and electrical conductivity and observed that when the concentration of glycogen solutions reaches a certain maximum it appears that the colloidal particles combine with one another to form micelles. McBain et al. [115] measured the freezing point and the conductivity of sodium and potassium salts of saturated fatty acids that remain liquid at 0°. From this paper we must remark the comment that “free ions of charge equal and opposite to that of the charged colloid are present in the sol or gel.” In 1935 McBain and Betz [116] measured the freezing point of undecyl and lauryl sulfonic acids, expressing the results in terms of the osmotic coefficient. They concluded that in dilute solutions they behave as simple moderately weak electrolytes but with increased concentration molecules and ions associate into neutral and ionic micelles. They also considered that micelles “owing to the wide spacing of their charges, have the ionic strength similar to uni‐univalent electrolytes.” McBain and Betz [116] and Johnston and McBain [117] carried out careful freezing point measurements on colloidal electrolytes (potassium and sodium oleate, sodium decyl and dodecyl sulphate, sodium decyl sulfonate and sodium deoxycholate) and, in 1947, Gonick and McBain [118] showed a “cryoscopic” evidence of a micellar association in aqueous solution of nonionic detergents. Following their own words, “since the depression of the freezing point is determined primarily by the number of solute particles per unit weight of solvent, the osmotic coefficient expresses directly, to a first approximation, the ratio of the true number of solute particles to that obtaining at complete dispersion of the solute and thus gives a measure of the average degree of association into micelles or other aggregates.” The critical concentration of micelles formation was evident from an abrupt drop in the coefficient. Furthermore, they noticed that the addition of potassium chloride to a solution of nonaethylene glycol (mono) laurate caused no significant change in the degree of association of the detergent. Herrington and Sahi [119] studied the nonionic surfactants sucrose monolaurate and sucrose monooleate in aqueous solution by the freezing point and vapor pressure methods. For the analysis of the results they accepted that micelles are monodisperse and observed that the aggregation numbers do not increase significantly with temperature.

Starting from a model published by Burchfield and Woolley [120], Coello et al. [121, 122] combined freezing point and sodium ion activities measurements to obtain both the aggregation number and the fraction of bound counterions of aqueous bile salt solutions. The mass action model for an ionic micelle and the fraction of bound counterions to the micelles were considered. The theory of the freezing point depression is very well known [54] and does not need further attention.

While in a monodisperse system (as we have analyzed above) the aggregation number is a single‐value variable [123], in a polydisperse self‐aggregation system the aggregation number can assume all possible values from 2 to ∞. For these systems only the average aggregation number of the aggregates is obtained. However, different experimental techniques lead to aggregation numbers that are not the same. For instance, the average aggregation number measured from colligative properties is the number average aggregation number, , given by

(1.22)

where X _i is the concentration of the ith species. Similarly, the average aggregation numbers obtained from static light scattering and viscosity techniques are the weight average and the z‐average aggregation numbers, and , respectively, given by

(1.23)

(1.24)

For analyzing the experimental results from the experimental techniques, the hypothesis of a constant average aggregation number with concentration is frequently used. Let us examine this hypothesis by following the landmark paper by Israelachvili et al. [124]. In this presentation we are following the notation of this original paper.

Each aggregate in the solution is characterized by its own free energy. They also accepted that the dilute solution theory holds. Equilibrium thermodynamics requires that in a system of molecules that form aggregated structures in solution, the chemical potential of all identical molecules in different aggregates is the same [125]. For an aggregate containing i monomer molecules, this statement is expressed as

(1.25)

where is the standard part of the chemical potential (the mean interaction free energy per molecule) in aggregates of aggregation number i and X _i is the concentration of the ith aggregate. The subscript “1” corresponds to monomers in solution. That is to say, the left‐side term corresponds to the chemical potential of the free monomer and the right‐side term is the chemical potential per monomer incorporated in a micelle.

From the previous equation and the material balance in a polydispersed micellar solution, it follows that the number average aggregation number is [124]

(1.26)

which relates the rate of change of micelle concentration with monomer concentration to the mean micelle aggregation number. It should be noticed that this equation reduces to Eq. (1.19) for a monodisperse system. In such a case, the various average aggregation numbers are identical to n [123].

The standard deviation is given by

(1.27)

which relates it to the rate of change of the mean aggregation number with respect to micelle concentration and means that if the standard deviation is close to zero, the aggregation number must almost be independent of the total surfactant concentration.

Well above cmc, Eq. (1.27) may be approximated by [125]

(1.28)

which shows that if the system is highly polydisperse the average aggregation number will be very sensitive to the total surfactant concentration. Therefore, a rapid change of concentration is evidence of a large distribution of polydispersity in micelle size. Nagarajan [123] remarks that “this is a general thermodynamic result applicable to any self‐assembling system” and that “interpreting any experimental data, one must ensure that this equation is not violated.” Similar comments have been provided by Rusanov [126].

A well‐known approach to study a fast process is to rapidly perturb a system in equilibrium and measure the relaxation time required by the system to adjust to the new equilibrium. Typical techniques are temperature jump, pressure jump, and ultrasonics [127]. The time scale depends on the relaxation technique [128]. The mathematical analysis to study simple A↔B and A+B↔C systems by temperature jump has been didactically published by Finholt [129]. Kresheck et al. [130] applied the temperature‐jump (jump 5.2 °C) technique to study the dissociation of the dodecylpyridinium iodide micelle. Pressure‐jump studies have also been carried out by several authors [131–133].

In the ultrasonic methods (Ultrasonic Absorption Spectrometry) the perturbation of equilibrium is due to the periodic variation of pressure and temperature due to the passage of the sound wave through the system. Relaxation of the equilibrium gives rise to changes in the quantity α /f ² (where α is the sound absorption coefficient at frequency f ) and from its dependence with frequency the relaxation time is obtained [128].

Platz [134] has presented a simplified analysis of the Aniansson and Wall [135] isodesmic model for analyzing the kinetics of micelle association and dissociation in surfactant solutions. The theory is nowadays commonly known as the Teubner–Kahlweit–Aniansson–Wall theory [136, 137], the key equation being Eq. (1.14), characterized by forward ( ) and back kinetic constants, which correspond to the exchange of monomers between micelles. The ratio between the kinetic constants is directly the equilibrium constant. Further developments of the theory are due to Lang et al. [133] and Telgmann and Kaatze [138, 139]. In relaxation experiments two characteristic times are observed. The so‐called “fast process” corresponds to the kinetic analysis of Aniansson and Wall and the “slow process” is assumed to be a change of the total number of micelles.

After several assumptions (micellar distribution is Gaussian‐like, reactions between aggregates do not have to be considered, and the change in the number of the micelles is much slower than the interchange of the monomer) Eq. (1.29) is deduced [140]:

(1.29)

where τ ₁ is the relaxation time (associated to k _b ), σ ² is the variance of the micellar distribution, is the backward rate constant at micelle sizes around the mean micelle size , and is the average monomer concentration. In addition, is identified with the concentration at cmc, s _cmc , at concentrations above the cmc. When , k _b , the aggregation number and the standard deviation (see above) do not appreciably change with the concentration, and the previous equation suggests a linear relationship between and the concentration. This fact has been verified, for instance, for alkyl sulfates [128].

The backward kinetic constant depends on the length of the alkyl chain of the surfactant. For instance, values in the interval 10–0.8 × 10⁹/s have been measured by Kaatze [136] for ammonium chloride surfactants CH₃C _x−1H_2(x−1)NH₃ ⁺Cl⁻ (x = 5–8).