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.     

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.     

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.     

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.     

Power-law stage of slow relaxation in solutions with spherical micelles

General (independent of models selected for surfactant molecular aggregates) analytical relations are derived to describe the initial stage of slow relaxation in micellar solutions with spherical micelles. This stage precedes the final stage of the relaxation occurring via an exponential decay of disturbances with time. The relations obtained are applicable throughout the interval of micellar solution concentrations from the first to the second critical micellization concentration. It is shown that the initial stage is characterized by power laws of variations in the concentrations of monomers and micelles with time, these laws being different for the relaxation processes proceeding from above and below toward equilibrium values of micellar solution parameters. Relations are derived for the duration of this stage, and the effect of initial conditions is studied. Characteristic times of the power-law stage are determined and compared with the characteristic time of the final exponent-law relaxation stage. The behavior of these times is investigated at surfactant solution concentrations in the vicinity of, and noticeably above, the first critical micellization concentration. On the basis of the droplet and quasi-droplet thermodynamic models of surfactant molecular aggregates, numerical solutions are found for nonlinearized equations of slow relaxation for the time dependence of surfactant monomer concentrations at all stages of the slow relaxation. Numerical results obtained from the models are compared with the results of a general analytical study.     

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.     

Structure and dynamics of concentrated micellar solutions of sodium dodecyl sulfate

In micellar solutions of sodium dodecyl sulfate, as the concentration of surfactants increases, the spheroid shape of the micelles changes from almost spherical to ellipsoidal with increasing ratio of half-axes ratio, and further the transition to cylindrical micelles occurs. The micelles in an aqueous solution can directly contact (compact aggregates) or be separated from one another by layers of intermicellar medium (periodical colloid structures). In the latter case, the thickness of the layer can significantly exceed the micelle size, and then no mutual correlation in micelle arrangement is observed. According to the data of small-angle X-ray scattering, the relationship between the surfactant concentration and formation of “quasi-crystalline” micellar structure is nonlinear, which can be due to both micelle aggregation processes and nonuniformity of their structure. The possible influence of ordered micellar structures on the diffusion mobility of micelles is shown.     

Novel fluorescent probe as aggregation predictor and micro-polarity reporter for micelles and mixed micelles

Aggregational behaviour of micelles sodium dodecyl sulphate (SDS and Triton X-100, TX-100 both in pure and mixed form) and micelle like aggregates such as polymer-surfactant system [polymer poly(vinyl pyrrolidone), PVP]-SDS have been studied by using fluorescence characteristics of a newly synthesized probe. The critical micelle concentration (CMC) values determined at various surfactant compositions are lower than the ideal values indicating a synergistic effect. The value of the interaction parameter for the surfactant mixture has been determined which agrees well with the value calculated according to molecular thermodynamic theory. The total aggregation number of surfactant in mixed micelle shows a drastic variation in the SDS mole fraction range 0 < or = alpha1 < or = 0.3 and beyond the range it remains practically constant. Molar-based partition coefficients for the dye between the micellar and aqueous phase have been determined and a non-linear variation is obtained for the mixed micellar system. Variations of micro-polarity in the mixed micellar region have been investigated as a function of surfactant composition and results have been explained in terms of a suitable realistic model.     

Molecular dynamics simulation and thermodynamic modeling of the self-assembly of the triterpenoids asiatic acid and madecassic acid in aqueous solution

The self-assembly behavior of the triterpenoids asiatic acid (AA) and madecassic acid (MA), both widely studied bioactive phytochemicals that are similar in structure to bile salts, were investigated in aqueous solution through atomistic-level molecular dynamics (MD) simulation. AA and MA molecules initially distributed randomly in solution were observed to aggregate into micelles during 75 ns of MD simulation. A "hydrophobic contact criterion" was developed to identify micellar aggregates from the computer simulation results. From the computer simulation data, the aggregation number of AA and MA micelles, the monomer concentration, the principal moments of the micelle radius of gyration tensor, the one-dimensional growth exhibited by AA and MA micelles as the aggregation number increases, the level of internal ordering within AA and MA micelles (quantified using two different orientational order parameters), the local environment of atoms within AA and MA in the micellar environment, and the total, hydrophilic, and hydrophobic solvent accessible surface areas of the AA and MA micelles were each evaluated. The MD simulations conducted provide insights into the self-assembly behavior of structurally complex, nontraditional surfactants in aqueous solution. Motivated by the high computational cost required to obtain an accurate estimate of the critical micelle concentrations (CMCs) of AA and MA from evaluation of the average monomer concentration present in the AA and MA simulation cells, a modified computer simulation/molecular-thermodynamic model (referred to as the MCS-MT model) was formulated to quantify the free-energy change associated with optimal AA and MA micelle formation in order to predict the CMCs of AA and MA. The predicted CMC of AA was found to be 59 microM, compared with the experimentally measured CMC of 17 microM, and the predicted CMC of MA was found to be 96 microM, compared with the experimentally measured CMC of 62 microM. The AA and MA CMCs predicted using the MCS-MT model are much more accurate than the CMCs inferred from the monomer concentrations of AA and MA present in the simulation cells after micelle self-assembly (2390 microM and 11,300 microM, respectively). The theoretical modeling results obtained for AA and MA indicate that, by combining computer simulation inputs with molecular-thermodynamic models of surfactant self-assembly, reasonably accurate estimates of surfactant CMCs can be obtained with a fraction of the computational expense that would be required by using computer simulations alone.     

Second CMC in surfactant micellar systems

Experimental reports of surfactant systems displaying a second critical micelle concentration (second CMC) have been surveyed. It turns out that surfactant micelles usually show a growth behavior with some typical features. (i) Micelles grow weakly at low surfactant concentrations but may switch to a much stronger growth behavior at higher concentrations. The second CMC is defined as the point of transition from weakly to strongly growing micelles. (ii) Micelles are found to be non-spherically shaped below the second CMC. (iii) At the second CMC micelles are found to be much smaller, with aggregation numbers typically 100–200, than expected for flexible micelles. (iv) Micelles of intermediate size are present in a narrow concentration regime close to the second CMC. (v) Micelles grow much stronger above the second CMC than expected from a sphere-to-rod transition. The conventional spherocylindrical micelle model predicts a smooth growth behavior that contradicts the appearance of a second CMC. Modifying the model by means of including swollen end caps neither account for the presence of micelles with intermediate size, nor the strong growth behavior above the second CMC. Taking into account micelle flexibility is not consistent with the rather low micelle aggregation numbers observed at the second CMC. On the other hand, a recently proposed alternative theoretical approach, the general micelle model, have been demonstrated to take into account basically all features that are typical of experimentally observed micellar growth behaviors.     

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.     

Mesoscopic simulation of self-assembly in surfactant oligomers by dissipative particle dynamics

Self-assembling properties of surfactant oligomers in an aqueous medium is simulated by dissipative particle dynamics (DPD). The critical micellar concentration (CMC) of dimeric (oligomerization = 2) and trimeric (oligomerization = 3) surfactant is much lower than their single-chain counterpart. All surfactants form spherical micelles at the concentration not far above their CMC. The transition from spherical to cylindrical micelles exhibits with increasing surfactant concentration. Lamellar micelles will appear with further increasing the surfactant concentration. For dimeric and trimeric surfactants, cylindrical micelles transform into extremely long “wormlike” or “threadlike” micelles before the transition to lamellar micelles. These results are in qualitative agreement with laboratory experiment. Average aggregation numbers (AN) of micelles increase with a power law of AN  c when the surfactant concentration c CMC. The self-diffusion coefficients will drop with a power law of D  c⁻ when wormlike micelles are formed.     

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.