Kinetics of Aggregation and Relaxation in Micellar Surfactant Solutions

Theoretical results published in the last 17 years on the kinetics of aggregation and relaxation in micellar surfactant solutions have been reviewed. The results obtained by the analytical and direct numerical solution of the Becker–Döring kinetic equations and the Smoluchowski generalized equations, which describe different possible mechanisms of aggregation and relaxation on all time scales from ultrafast relaxation while reaching the quasi-equilibrium in the region of subcritical molecular aggregates to the last stage of slow relaxation of micelles to the final aggregated state, have been considered in detail. The droplet model and the model linear with respect to aggregation numbers have been used for the work of aggregation to describe the dynamics of the rearrangement of micellar systems consisting of only spherical, only cylindrical, and coexisting spherical and cylindrical aggregates, with the dynamics being both linear and nonlinear with respect to deviations from equilibrium. The results of molecular simulation of the rearrangement kinetics of micellar systems subjected to initial disturbance have been reviewed.     

Micellization kinetics with allowance for fussion and fission of spherical and cylindrical micelles: 1. Set of nonlinear equations describing slow relaxation

Based on the general kinetic equation that describes the aggregation and fragmentation of surfactant molecular aggregates, a closed set of nonlinear equations is derived for the slow relaxation of surfactant monomer concentration and the total concentrations of coexisting spherical and cylindrical micelles to the equilibrium state of a micellar solution. Both the transitions accompanied by the emission and capture of surfactant monomers by micelles and the transitions resulting from the fussion and fission of micelles, are taken into account. The derived set of equations describes all stages of the slow relaxation from the initial perturbance to the final equilibrium state of a micellar solution.     

Concentrations of Monomers and Cylindrical Micelles above the Second CMC

Based on thermodynamically substantiated linear dependence of the work of cylindrical micelle formation on the aggregation number within a wide range of aggregation numbers where the cylindrical micelles are accumulated in a surfactant solution, the second critical micellization concentration (CMC) is introduced as an overall surfactant concentration at which the ratio of the total amount of substance in cylindrical micelles to the amount of substance in monomers is equal to 0.1, i.e., it is already noticeable. It is shown that this ratio increases rather rapidly with a monomer concentration. The coefficient of the linear dependence of the work of cylindrical micelle formation on the aggregation number in the important practical situation where the ratios of the total concentration of cylindrical micelles and total amount of substance in these micelles to the monomer concentration are equal by the order of magnitude to 1 and 10⁵, respectively, while disc micelles and extended bilayers are still not appeared. In the same situation, the ratios of the total concentration of spherical micelles and total amount of substance in these micelles to the monomer concentration are equal by the order of magnitude to 1 and 10², respectively. The relationship between the overall surfactant concentration and monomer concentration is found. It is shown that the second CMC exceeds by two orders of magnitude the first CMC corresponding to the onset of the noticeable accumulation of surfactant in spherical micelles. The distribution of cylindrical micelles over the aggregation numbers is analyzed. It is demonstrated that, in agreement with the experiment, the distribution is almost uniform in the considerable part of the wide range of aggregation numbers and drops exponentially in the remaining (right-hand) part of this range. Experimental result is confirmed that the total concentration of cylindrical micelles, the mean value, and the mean statistical scatter of aggregation numbers in a cylindrical micelle is proportional to the square root of the overall surfactant concentration. The balance equation of surfactant amount in the vicinity of the final equilibrium state of a materially isolated solution is linearized. This linearization makes it possible to express the deviations of monomer and aggregate concentrations from their equilibrium values at the lower boundary of the region of the linear dependence of the work of cylindrical micelle formation on the aggregation numbers via the deviations of experimentally observed total concentrations of spherical and cylindrical micelles from their equilibrium values. The case of the solutions of such surfactants, for which spherical shape appeared to be unrealizable due to their molecular structure and packing conditions, is considered separately.     

Boltzmann distributions and slow relaxation in systems with spherical and cylindrical micelles

Equilibrium and nonequilibrium distributions of molecular aggregates in a solution of a nonionic surfactant are investigated at the total surfactant concentration above the second critical micelle concentration (CMC2). The investigation is not limited by the choice of a specific micellar model. Expressions for the direct and reverse fluxes of molecular aggregates over the potential humps of the aggregation work are derived. These aggregation work humps set up activation barriers for the formation of spherical and cylindrical micelles. With the aid of the expressions for molecular aggregate fluxes, a set of two kinetic equations of micellization is derived. This set, along with the material balance equation, describes the molecular mechanism of the slow relaxation of micellar solution above the CMC2. A realistic situation has been analyzed when the CMC2 exceeds the first critical micelle concentration, CMC1, by an order of magnitude, and the total surfactant concentration varies within the range lying markedly above the CMC2 but not by more than 2 orders of magnitude. For such conditions, an equation relating the parameters of the aggregation work of a cylindrical micelle to the observable ratio of the total surfactant concentration and the monomer concentration is found for an equilibrium solution. For the same conditions, but in the nonequilibrium state of materially isolated surfactant solution, a closed set of linearized relaxation equations for total concentrations of spherical and cylindrical micelles is derived. These equations determine the time development of two modes of slow relaxation in micellar solutions markedly above the CMC2. Solving the set of equations yields two rates and two times of slow relaxation.     

System of relaxation equations for materially isolated surfactant solution with spherical and cylindrical micelles

Analytical expressions for the direct and reverse fluxes of molecular aggregates over the first and second potential barriers of the aggregation work in the presence of spherical and cylindrical micelles in non-ionic surfactant solution were derived. Expressions for the sum (entering into kinetic equations of micellization) of direct and reverse fluxes of molecular aggregates over the first and second potential barriers of the aggregation work in the vicinity of the final equilibrium state of materially isolated surfactant solution were linearized. In the experimentally important range of the values of overall surfactant concentration in solution where the predominant contribution to the total surfactant amount is introduced by cylindrical micelles, we derived a closed system of two linearized relaxation equations determining the buildup (with time) of experimentally observed total concentrations of spherical and cylindrical micelles in the vicinity of the final equilibrium state of materially isolated surfactant solution. The case of the solutions of such surfactants, for which the spherical shape of a micelle appeared to be unrealizable due to the structure and packing conditions of molecules, was considered separately.Translated from Kolloidnyi Zhurnal, Vol. 67, No. 1, 2005, pp. 38–46. Original Russian Text Copyright © 2005 by Kuni, Shchekin, Rusanov, Grinin.     

Thermodynamic and Kinetic Foundations of the Micellization Theory: 4. Kinetics of Establishment of Equilibrium in a Micellar Solution

A system of the kinetic equations of the material balance for the concentrations of surfactant monomers and micelles in a micellar nonionic surfactant solution was formulated. The equilibrium state of a materially isolated micellar solution was analyzed. The system of the kinetic equations of the material balance of a micellar solution was solved. The total time of the establishment of equilibrium in a micellar solution was determined. It was shown that this time increases or (typically) decreases with an increase in micelle concentration, depending on the degree of micellization.     

The “fine structure” of the slow micellar relaxation mode and the aggregation rates in the range between a potential hump and well in the work of aggregation

An analytical expression has been derived for the quasi-stationary size distribution of surfactant aggregates in a micellar system approaching the final equilibrium state. In contrast to previously known relations, the derived expression takes into account variations in the concentration of monomers during the slow relaxation and enables one to determine the previously unknown fine structure of the linearized mode of slow relaxation, i.e., its dependence on the aggregation numbers in the range between the maximum and minimum of the work of aggregation. This dependence has been reliably confirmed by the numerical solution of the set of linearized Becker–Döering difference equations, which describe the molecular mechanism of the kinetics of micellization and micellar relaxation. In turn, the expression found for the relaxation mode makes it possible to refine the analogous “fine structure” of aggregation rates at different points of the same range between the maximum and minimum of the work of aggregation, in which the aggregation rates appear to be low but exhibit a nonmonotonic behavior. This behavior is also confirmed by the numerical solution of the Becker–Döering difference kinetic equations.     

A simple rule for describing the composition dependence of the aggregation number of mixed micelles

The mean aggregation numbers of mixed micelles composed of hydrocarbon surfactants (nonionic/nonionic and ionic/nonionic surfactants) have been determined by the intensity light-scattering method, in order to compare them with the values calculated by using the equations derived. The equations have been derived for representative micellar shapes (disk-like, rod-like, and spherical shapes) by making the assumptions that (i) the surface area of the hydrocarbon core of a mixed micelle is built up by independent contributions from each surfactant monomer, and (ii) the dimension of the hydrocarbon core is determined by the number of carbon atoms of a surfactant. The closest agreement of the observed aggregation numbers with the calculated ones has been obtained for the mixed micelle of an oblate ellipsoidal shape as a geometrical model for a disk-like micelle. This suggests that an oblate ellipsoidal shape may be more probable for a micelle formed at a moderate range of surfactant concentration than a prolate ellipsoidal (a rod-like) and a spherical shape if the assumptions (i) and (ii) hold. The equations presented here are useful, since they make it possible to calculate an accurate aggregation number of the mixed micelle of any composition from the aggregation numbers of the pure micelles of the components and the number of carbon atoms of component surfactants as long as there is no highly specific interaction between different surfactant components.     

Kinetic description of the relaxation of surfactant solutions containing spherical and cylindrical micelles

Monotonically decaying relaxation of a materially isolated nonionic surfactant solution containing spherical and cylindrical micelles at the arbitrary heights of the first and second potential barriers of aggregation work is kinetically substantiated. The realistic situation, where the height of second potential barrier is at least slightly higher (by the relative value) than that of the first barrier, is studied. Analytical expressions for two relaxation times of materially isolated surfactant solution are calculated. The shortest of these times corresponds to the relatively fast establishment of the mutual quasi-equilibrium of spherical and cylindrical micelles, beginning with relatively small cylindrical micelles. The longest of relaxation times corresponds to the relatively slow establishment of the total equilibrium of surfactant solution. It is shown that this time (the only significant for the establishment of the final equilibrium of materially isolated surfactant solution) is determined by the height of the first potential barrier of aggregation work and is by no means dependent on the height of the second potential barrier about which not much is known. Variations (with time) of the total concentrations of spherical and cylindrical micelles, surfactant monomer concentration, and the total amount of the substance in cylindrical micelles in the approach of solution to the final equilibrium state are described analytically. It is shown that theoretically admitted small relative deviations of the concentrations of spherical and cylindrical micelles from their values in the final equilibrium state are fully measurable in experiment. Calculated relaxation time of surfactant solution can also be measured experimentally together with the aforementioned values. It is elucidated that this time is approximately proportional to the overall solution concentration, if the second critical micellization concentration (CMC2) by the order of magnitude exceeds the first critical micellization concentration (CMC1), and is virtually independent of the overall solution concentration, if the CMC2 exceeds the CMC1 by two orders of magnitude. The characteristic time of the establishment of quasi-equilibrium distribution of cylindrical micelles throughout the region of their sizes is estimated, thus allowing us to establish the lower limit of the height of the first barrier of aggregation work.Translated from Kolloidnyi Zhurnal, Vol. 67, No. 1, 2005, pp. 47–56.Original Russian Text Copyright © 2005 by Kuni, Shchekin, Grinin, Rusanov.     

Extension of the ladder model of self-assembly from cylindrical to disclike surfactant micelles

The ladder model of growth of cylindrical micelles gives expressions for the micellar size distribution and for the mean aggregation number, which are in good agreement with the experiment. Here, we consider this model and its extension to the case of disclike micelles. In analogy with the modeling of elongated micelles as sphero-cylinders, the disclike micelles can be modeled as toro-discs. Upon micelle growth, the hemispherical caps of a cylindrical aggregate remain unchanged, whereas the semitoroidal periphery of a disclike micelle expands. This effect can be taken into account in the expression for the size distribution of the disclike micelles, which predicts the dependence of the micelle mean aggregation number on the surfactant concentration. It turns out that disclike micelles could form in a limited range of surfactant concentrations, and that their mean aggregation number cannot exceed a certain maximal value. Large disclike micelles can exist only near the border with the domain of cylindrical micelles. Then, small variations in the experimental conditions could induce a transformation of the disclike micelles into cylindrical ones.     

Parametric equations of the theory of formation of spherical micelles

Using the notion of aggregation work, we construct a system of differential equations for the aggregation number of micelles which is a function of the parameters of micellization (parametric equations). There are explicit solutions for two important models of spherical micelles. Based on these solutions, we obtain an analytical expression for the equilibrium concentration of surfactant monomers and consequently for the whole spectrum of equilibrium concentrations of molecular aggregates in this framework. Accuracy of these expressions is discussed, and they are applied to an example of micelles formed by sodium dodecyl sulfate.     

The Aggregation Work and Shape of Molecular Aggregates upon the Transition from Spherical to Globular and Cylindrical Micelles

The transition from spherical to globular and cylindrical equilibrium modifications of micelles in solutions of nonionic surfactants is numerically studied within the framework of the droplet model of molecular aggregates. Two branches of the curve of micelle aggregation work are plotted as a function of aggregation numbers. One of these curves corresponds to the globular micelles; the other, to spherocylindrical micelles. At aggregation numbers corresponding to the limiting spherical packing, both the globules and spherocylinders are transformed into the limiting sphere. It is shown that the ratio between the branches depends on the dimensionless parameter characterizing the ratio of electrostatic and surface contributions to the aggregation work. It is elucidated that, at certain values of this parameter and surfactant monomer concentration in solution, in addition to the maximum in the region of submicellar aggregates for spherical micelles, the second maximum arises on the curve of aggregation work as a function of aggregation numbers in the region of transition to spherocylindrical micelles. The appearance of an additional maximum is shown to be caused by the sum of surface, electrostatic, and concentration contributions to the aggregation work and is not directly related to the conformational contribution to the aggregation work.__________Translated from Kolloidnyi Zhurnal, Vol. 67, No. 3, 2005, pp. 363–376.Original Russian Text Copyright © 2005 by Kshevetskiy, Shchekin.     

Self-assembly of symmetrical and dissymmetrical dicationic surfactants in the solid phase and in solution

The effect of the spacer structure (linear, cyclic, bicyclic) and dissymmetry of alkyl fragment in a series of dicationic gemini surfactants on the ability to form thermotropic liquid crystals and on the micellar properties was studied. Transitions crystal-thermotropic liquid crystals-isotropic solution were studied using differential scanning calorimetry and optical microscopy. The self-diffusion coefficients of micelles and monomers of the dicationic dialkyl derivatives of 1,4-diazabicyclo[2.2.2]octane in water were determined by pulsed field gradient NMR spectroscopy. The effect of the surfactant structure on the values of critical micelle concentrations, hydrodynamic radii, and aggregation numbers of micelles was analyzed.     

The interaction of isomeric hexanediols with sodium dodecyl sulfate and dodecyltrimethylammonium bromide micelles

The micelle formation process for a typical anionic surfactant, sodium dodecyl sulfate, and a typical cationic surfactant, dodecyltrimethylammonium bromide, has been investigated in a series of mixed solvents consisting of different concentrations of isomeric hexanediols (1,2-hexanediol and 1,6-hexanediol) in water. The critical micelle concentrations and the degrees of counterion dissociation of the mixed micelles were obtained from conductance experiments. Luminescence probing experiments have been used to determine the concentration of micelles in solution and, hence, the micellar aggregation numbers of the surfactants in the mixed solvent systems. The alcohol aggregation numbers were determined by combining the partition coefficients (obtained using NMR paramagnetic relaxation enhancement experiments) with the micellar concentrations from the luminescence probing experiments. All these results are interpreted in terms of the difference in the interaction of the isomeric hexanediols with the surfactant as a function of the position of the hydroxyl groups on the six-carbon chain of the alcohol. Received: 28 June 2000/Accepted: 5 July 2000