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.     

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.     

Thermodynamic Characteristics of the Micellization in Droplet and Quasi-Droplet Models of Surfactant Molecular Aggregates with Account of Experimental Data on Equilibrium Micelle Distribution

Formuals for the thermodynamic characteristics of micellization in the droplet and quasi-droplet models of surfactant molecular aggregates are derived. These formulas account for the experimental data on the mean size of micelles and average statistical scatter of their sizes in the equilibrium state. These formulas cover critical micellization concentration corresponded to the onset of surfactant accumulation in micelles and higher (than CMC) concentrations at which micelles incorporate noticeable or even the largest portion of surfactant in micellar solution. Analytical dependence of thermodynamic characteristics of micellization on the initial parameters of droplet and quasi-droplet models of molecular aggregates at critical micellization concentration is disclosed.     

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.     

Thermodynamic and Kinetic Foundations of the Micellization Theory: 5. Hierarchy of Kinetic Times

The characteristic kinetic times of micellization in the solution of a nonionic surfactant: the times of establishment of quasi-equilibrium concentrations of molecular aggregates in micellar, subcritical, and overcritical regions, times of establishment of quasi-equilibrium concentrations of molecular aggregates in the near-critical region of their sizes, the average time between two successive acts of emission of surfactant monomers by a micelle, the average value of micelle lifetime, the time of establishment of quasi-stationary mode of matter exchange between the solution and molecular aggregate, as well as the times of fast and slow relaxation in a solution were analyzed. The hierarchy of these times disclosing complex multistage kinetic process of micelle formation and decomposition and the establishment of equilibrium in the micellar solution was revealed. It was shown that this hierarchy is provided by the small parameters of the kinetic theory. The inverse problem of micellization kinetics was discussed; this problem allows us to find the characteristics of the formation work for micellar aggregate from the experimental data on the relaxation time of micellar solution.     

Kinetics of the surface tension of micellar solutions: Comparison of different experimental techniques

The kinetics of the surface tension of micellar solutions of nonionic surfactant Triton X-100 is measured experimentally by means of three different techniques: oscillating jet, maximum bubble pressure and inclined plate. They allow to study the micellization kinetics at various time scales (from a few milliseconds to a few seconds) in fairly large concentration region up to 50 times CMC. The experimental data are satisfactorily explained by a theoretical model accounting for the kinetics of micellization, diffusion of surfactant species and expansion of the bubble interface. By this model are computed the characteristic times of diffusion and micellization, which are of comparable magnitude (about 5 to 200 ms), and the Gibbs' elasticity. The micellization time constant corresponds to the slow relaxation process known to coincide with the disintegration of micelles. Comparing our data with other data from literature one can conclude that more realistic information for the micellization kinetics is obtained by the maximum bubble pressure and the oscillating jet method. The inclined plate seems too slow to measure the relaxation processes in micellar solutions of this surfactant.     

Prediction of conformational characteristics and micellar solution properties of fluorocarbon surfactants

A molecular-thermodynamic theory is developed to model the micellization of fluorocarbon surfactants in aqueous solutions, by combining a molecular model that evaluates the free energy of micellization of fluorocarbon surfactant micelles with a previously developed thermodynamic framework describing the free energy of the micellar solution. In the molecular model of micellization developed, a single-chain mean-field theory is combined with an appropriate rotational isomeric state model of fluorocarbon chains to describe the packing of the fluorocarbon surfactant tails inside the micelle core. Utilizing this single-chain mean-field theory, the packing free energies of fluorocarbon surfactants are evaluated and compared with those of their hydrocarbon analogues. We find that the greater rigidity of the fluorocarbon chain promotes its packing in micellar aggregates of low curvatures, such as bilayers. In addition, the mean-field approach is utilized to predict the average conformational characteristics (specifically, the bond order parameters) of fluorocarbon and hydrocarbon surfactant tails within the micelle core, and the predictions are found to agree well with the available experimental results. The electrostatic effects in fluorocarbon ionic surfactant micelles are modeled by allowing for counterion binding onto the charged micelle surface, which accounts explicitly for the effect of the counterion type on the micellar solution properties. In addition, a theoretical formulation is developed to evaluate the free energy of micellization and the size distribution of finite disklike micelles, which often form in the case of fluorocarbon surfactants. We find that, compared to their hydrocarbon analogues, fluorocarbon surfactants exhibit a greater tendency to form cylindrical or disklike micelles, as a result of their larger molecular volume as well as due to the greater conformational rigidity of the fluorocarbon tails. The molecular-thermodynamic theory developed is then applied to several ionic fluorocarbon surfactant-electrolyte systems, including perfluoroalkanoates and perfluorosulfonates with added LiCl or NH(4)Cl, and various micellar solution properties, including critical micelle concentrations (cmc's), optimal micelle shapes, and average micelle aggregation numbers, are predicted. The predicted micellar solution properties agree reasonably well with the available experimental results.     

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.     

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.     

Kinetics of micellization: its significance to technological processes

The association of many classes of surface active molecules into micellar aggregates is a well-known phenomenon. Micelles are often drawn as static structures of spherical aggregates of oriented molecules. However, micelles are in dynamic equilibrium with surfactant monomers in the bulk solution constantly being exchanged with the surfactant molecules in the micelles. Additionally, the micelles themselves are continuously disintegrating and reforming. The first process is a fast relaxation process typically referred to as τ₁. The latter is a slow relaxation process with relaxation time τ₂. Thus, τ₂ represents the entire process of the formation or disintegration of a micelle. The slow relaxation time is directly correlated with the average lifetime of a micelle, and hence the molecular packing in the micelle, which in turn relates to the stability of a micelle. It was shown earlier by Shah and coworkers that the stability of sodium dodecyl sulfate (SDS) micelles plays an important role in various technological processes involving an increase in interfacial area, such as foaming, wetting, emulsification, solubilization and detergency. The slow relaxation time of SDS micelles, as measured by pressure-jump and temperature-jump techniques was in the range of 10⁻⁴–10¹ s depending on the surfactant concentration. A maximum relaxation time and thus a maximum micellar stability was found at 200 mM SDS, corresponding to the least foaming, largest bubble size, longest wetting time of textile, largest emulsion droplet size and the most rapid solubilization of oil. These results are explained in terms of the flux of surfactant monomers from the bulk to the interface, which determines the dynamic surface tension. The more stable micelles lead to less monomer flux and hence to a higher dynamic surface tension. As the SDS concentration increases, the micelles become more rigid and stable as a result of the decrease in intermicellar distance. The smaller the intermicellar distance, the larger the Coulombic repulsive forces between the micelles leading to enhanced stability of micelles (presumably by increased counterion binding to the micelles). The Center for Surface Science & Engineering at the University of Florida has developed methods using stopped-flow and pressure-jump with optical detection to determine the slow relaxation time of micelles of nonionic surfactants. The results show relaxation times τ₂ in the range of seconds for Triton X-100 to minutes for polyoxyethylene alkyl ethers. The slow relaxation times are much longer for nonionic surfactants than for ionic surfactants, because of the absence of ionic repulsion between the head groups. The observed relaxation time τ₂ was related to dynamic surface tension and foaming experiments. A slow break-up of micelles, (i.e. a long relaxation time τ₂) corresponds to a high dynamic surface tension and low foamability, whereas a fast break-up of micelles, leads to a lower dynamic surface tension and higher foamability. In conclusion, micellar stability and thus the micellar break-up time is a key factor in controlling technological processes involving a rapid increase in interfacial area, such as foaming, wetting, emulsification and oil solubilization. First, the available monomers adsorb onto the freshly created interface. Then, additional monomers must be provided by the break-up of micelles. Especially when the free monomer concentration is low, as indicated by a low CMC, the micellar break-up time is a rate limiting step in the supply of monomers, which is the case for many nonionic surfactant solutions. Therefore, relaxation time data of surfactant solutions enables us to predict the performance of a given surfactant solution. Moreover, the results suggest that one can design appropriate micelles with specific stability or τ₂ by controlling the surfactant structure, concentration and physico-chemical conditions, as well as by mixing anionic/cationic or ionic/nonionic surfactants for a desired technological application.     

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.     

Importance of micellar kinetics in relation to technological processes

The association of many classes of surface-active molecules into micellar aggregates is a well-known phenomenon. Micelles are in dynamic equilibrium, constantly disintegrating and reforming. This relaxation process is characterized by the slow micellar relaxation time constant, tau(2), which is directly related to the micellar stability. Theories of the kinetics of micelle formation and disintegration have been discussed to identify the gaps in our complete understanding of this kinetic process. The micellar stability of sodium dodecyl sulfate micelles has been shown to significantly influence technological processes involving a rapid increase in interfacial area, such as foaming, wetting, emulsification, solubilization, and detergency. First, the available monomers adsorb onto the freshly created interface. Then, additional monomers must be provided by the breakup of micelles. Especially when the free monomer concentration is low, which is the case for many nonionic surfactant solutions, the micellar breakup time is a rate-limiting step in the supply of monomers. The Center for Surface Science & Engineering at the University of Florida has developed methods using stopped flow and pressure jump with optical detection to determine the slow relaxation time of micelles of nonionic surfactants. The results showed that the ionic surfactants such as SDS exhibit slow relaxation times in the range from milliseconds to seconds, whereas nonionic surfactants exhibit slow relaxation times in the range from seconds (for Triton X-100) to minutes (for polyoxyethylene alkyl ethers). The slow relaxation times are much longer for nonionic surfactants than for ionic surfactants, because of the absence of ionic repulsion between the head groups. The observed relaxation times showed a direct correlation with dynamic surface tension and foaming experiments. In conclusion, relaxation time data of surfactant solutions correlate with the dynamic properties of the micellar solutions. Moreover, the results suggest that appropriate micelles with specific stability or tau(2) can be designed by controlling the surfactant structure, concentration, and physicochemical conditions (e.g., salt concentration, temperature, and pressure). One can also tailor micelles by mixing anionic/cationic or ionic/nonionic surfactants for a desired stability to control various technological processes.     

Micellar solutions of ionic surfactants and their mixtures with nonionic surfactants: Theoretical modeling vs. Experiment

Here, we review two recent theoretical models in the field of ionic surfactant micelles and discuss the comparison of their predictions with experimental data. The first approach is based on the analysis of the stepwise thinning (stratification) of liquid films formed from micellar solutions. From the experimental step-wise dependence of the film thickness on time, it is possible to determine the micelle aggregation number and charge. The second approach is based on a complete system of equations (a generalized phase separation model), which describes the chemical and mechanical equilibrium of ionic micelles, including the effects of electrostatic and non-electrostatic interactions, and counterion binding. The parameters of this model can be determined by fitting a given set of experimental data, for example, the dependence of the critical micellization concentration on the salt concentration. The model is generalized to mixed solutions of ionic and nonionic surfactants. It quantitatively describes the dependencies of the critical micellization concentration on the composition of the surfactant mixture and on the electrolyte concentration, and predicts the concentrations of the monomers that are in equilibrium with the micelles, as well as the solution’s electrolytic conductivity; the micelle composition, aggregation number, ionization degree and surface electric potential. These predictions are in very good agreement with experimental data, including data from stratifying films. The model can find applications for the analysis and quantitative interpretation of the properties of various micellar solutions of ionic surfactants and mixed solutions of ionic and nonionic surfactants.     

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.     

Aggregation work at polydisperse micellization: ideal solution and "dressed micelle" models comparing to molecular dynamics simulations

General thermodynamic relations for the work of polydisperse micelle formation in the model of ideal solution of molecular aggregates in nonionic surfactant solution and the model of "dressed micelles" in ionic solution have been considered. In particular, the dependence of the aggregation work on the total concentration of nonionic surfactant has been analyzed. The analogous dependence for the work of formation of ionic aggregates has been examined with regard to existence of two variables of a state of an ionic aggregate, the aggregation numbers of surface active ions and counterions. To verify the thermodynamic models, the molecular dynamics simulations of micellization in nonionic and ionic surfactant solutions at two total surfactant concentrations have been performed. It was shown that for nonionic surfactants, even at relatively high total surfactant concentrations, the shape and behavior of the work of polydisperse micelle formation found within the model of the ideal solution at different total surfactant concentrations agrees fairly well with the numerical experiment. For ionic surfactant solutions, the numerical results indicate a strong screening of ionic aggregates by the bound counterions. This fact as well as independence of the coefficient in the law of mass action for ionic aggregates on total surfactant concentration and predictable behavior of the "waterfall" lines of surfaces of the aggregation work upholds the model of "dressed" ionic aggregates.     

Dynamic surface tension of micellar solutions studied by the maximum bubble pressure method. 1. Experiment

The effect of the micelles on the dynamic surface tension of micellar surfactant solutions is studied experimentally by means of the maximum bubble pressure method. Different frequencies of bubbling ranging approximately between 1 and 30 s^–1 are applied. The time dependence of the surface tension is calculated using a dead time correction. Water solutions of two types of surfactants with different concentrations are investigated: sodium dodecyl sulfate and nonylphenol polyglycol ether. The surface tension relaxes more quickly in the presence of micelles. The characteristic times of relaxation of the surface tension seem to be in the millisecond range. The time constants observed experimentally are explained in terms of the theory of surfactant diffusion affected by micellization kinetics.