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.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

Viscometric detection of sphere to cylinder transition and polydispersity in aqueous micellar solutions

Viscosity measurements of aqueous surfactant solutions using a Cannon-Fenske capillary viscometer have been made to examine the conditions under which spherical micelles grow into cylindrical micelles. Surfactants with different polar head groups and hydrocarbon chain lengths have been studied at various solution conditions. The observed transition from spherical to cylindrical micelles is explained in terms of the attractive and repulsive forces associated with the micellization process. Further the viscosity of SDS micellar solutions have been coraputed assuming the size distribution data generated by light scattering measurements. It is found that the relative viscosity calculated for a polydisperse solution is close to that calculated assuming monodispersed, rigid rods having size equal to the weight average aggregation numbers. Also the calculated viscosities compare well with the experimental viscometric data; thus demonstrating the compatibility of micellar size; determined from light scattering and viscometric measurements. Finally, the relative roles of solution polydispersity and micelle flexibility in the interpretation of experimental viscometric data are evaluated.     

Refining of nonionic surfactant micellization theory based on the law of mass action

Two approaches to determining critical micelle concentration (CMC) are assessed, i.e., from the inflection point in the curve for the concentration dependence of the degree of micellization and as K^1/(1–n), where K is the constant of the law of mass action and n is the aggregation number. The latter approach makes the theory simpler, while the former explicitly expresses the critical degree of micellization via the aggregation number. The concentrations of monomers and micelles are analyzed as functions of the overall concentration of a surfactant in a micellar solution. These functions look much simpler in the graphical form as compared with their complex exact analytical representation. This has resulted in derivation of simple analytical approximations for these functions, with these approximations being useful for calculations. The concentration dependence of the surfactant diffusion coefficient has been considered based on these approximations. It turned out that this dependence not only provides the known method for determining the diffusion coefficient of micelles, but also gives the possibility in principle to determine the aggregation number from the slope of the dependence of the diffusion coefficient on the inverse concentration (counted from the CMC in the CMC units). This new method for determining the aggregation number has been tested using the literature data on the diffusion coefficient of penta(ethylene glycol)-1-hexyl ether in an aqueous solution.     

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.     

A Monomer‐Micelle Model for the Formation of Simple Taurocholate Micelles

A kinetic dialysis technique was used to validate a relationship between monomer taurocholate (TC) concentration and total TC concentration in TC solutions containing 0.15 M NaCl and 0.01 M buffer (pH = 7.4). Based on the experimental data and Mukerjee's equations, the number average degree and the weight average degree of TC aggregates were estimated to be nearly the same (～5), indicating that simple TC micelles were the only aggregates. Furthermore, the TC dimer concentration was quantified to be negligible. According to the validated relationship, aggregation number of 5 for simple TC micelles, and the definition of critical micelle concentration (CMC), a modified monomer‐micelle model was used for describing simple TC micelle formation. Moreover, the CMC value was estimated to be ～6.3 mM, which is consistent with the reported value of ～6.0 mM.     

Micellization theory based on the law of mass action with a variable aggregation number

Variations in the aggregation number of spherical micelles are considered within the micellization theory based on the law of mass action. The mechanism of micellization in a polydisperse aggregated system and the transition to a monodisperse model are explained. A relation between aggregation numbers and chemical potentials of molecules or ions is determined using the curve for equilibrium distribution of aggregates over the aggregation numbers. It is shown that the aggregation numbers of nonionic surfactants unambiguously grow with concentration; however, such a conclusion cannot be drawn for ionic surfactants. For the explicit concentration dependence of the aggregation number, two versions of an analog of the Langmuir equation are proposed to be used, i.e., versions comprising the total surfactant concentration and the concentration of monomers. Comparison with experimental data is carried out by the example of conventional surfactants, namely, sodium dodecyl sulfate and hexadecyltrimethylammonium bromide.     

Synergistic sphere-to-rod micelle transition in mixed solutions of sodium dodecyl sulfate and cocoamidopropyl betaine

Static and dynamic light scattering experiments show that the mixed micelles of sodium dodecyl sulfate (SDS) and cocoamidopropyl betaine (CAPB) undergo a sphere-to-rod transition at unexpectedly low total surfactant concentrations, about 10 mM. The lowest transition concentration is observed at molar fraction 0.8 of CAPB in the surfactant mixture. The transition brings about a sharp increase in the viscosity of the respective surfactant solutions due to the growth of rodlike micelles. Parallel experiments with mixed solutions of CAPB and sodium laureth sulfate (sodium dodecyl-trioxyethylene sulfate, SDP3S) showed that the sphere-to-rod transition in SDP3S/CAPB mixtures occurs at higher surfactant concentrations, above 40 mM. The observed difference in the transition concentrations for SDS and SDP3S can be explained by the bulkier SDP3S headgroup. The latter should lead to larger mean area per molecule in the micelles containing SDP3S and, hence, to smaller spontaneous radius of curvature of the micelles (i.e., less favored transition from spherical to rodlike micelles). The static light scattering data are used to determine the mean aggregation number and the effective size of the spherical mixed SDS/CAPB micelles. From the dependence of the aggregation number on the surfactant concentration, the mean energy for transfer of a surfactant molecule from a spherical into a rodlike micelle is estimated.     

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.     

Micelle–monomer equilibria in solutions of ionic surfactants and in ionic–nonionic mixtures: A generalized phase separation model

On the basis of a detailed physicochemical model, a complete system of equations is formulated that describes the equilibrium between micelles and monomers in solutions of ionic surfactants and their mixtures with nonionic surfactants. The equations of the system express mass balances, chemical and mechanical equilibria. Each nonionic surfactant is characterized by a single thermodynamic parameter — its micellization constant. Each ionic surfactant is characterized by three parameters, including the Stern constant that quantifies the counterion binding. In the case of mixed micelles, each pair of surfactants is characterized with an interaction parameter, β, in terms of the regular solution theory. The comparison of the model with experimental data for surfactant binary mixtures shows that β is constant — independent of the micelle composition and electrolyte concentration. The solution of the system of equations gives the concentrations of all monomeric species, the micelle composition, ionization degree, surface potential and mean area per head group. Upon additional assumptions for the micelle shape, the mean aggregation number can be also estimated. The model gives quantitative theoretical interpretation of the dependence of the critical micellization concentration (CMC) of ionic surfactants on the ionic strength; of the CMC of mixed surfactant solutions, and of the electrolytic conductivity of micellar solutions. It turns out, that in the absence of added salt the conductivity is completely dominated by the contribution of the small ions: monomers and counterions. The theoretical predictions are in good agreement with experimental data.