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,
(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,
(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
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
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 (
