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.     

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.     

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.     

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 phthalocyanine dendrimer capable of forming spherical micelles

A novel and intrinsically spherical micelle has been prepared by utilizing a silicon phthalocyanine ((WG3)SiPc) that has a thin hydrophobic alkyl chain and a bulky hydrophilic poly(aryl ether) dendrimer with terminal carboxyl groups, as its two axial ligands. Gel-permeation chromatography and cryo-transmission electron microscopic experiments indicate that (WG3)SiPcs self-assemble to form a spherical micelle at very low concentrations in aqueous solution. Depending on the pH of the aqueous phase, (WG3)SiPc shuttles between aqueous and organic phases. In the presence of hydrophobic molecules, this transfer is accompanied by the inclusion of these (guest) molecules, indicating that the micelle acts as a molecular capsule with a nanospace surrounded by functional phthalocyanine planes.     

The preparation and characterization of the cross-linked spherical, cylindrical, and vesicular micelles of poly(styrene-b-isoprene) diblock copolymers

PI cores of the micelles of poly(styrene-b-isoprene) (PS-b-PI) diblock copolymers, in PS selective solvents, were cross-linked with sulfur monochloride (S2Cl2). The cross-linked micellar structure was maintained after dialysis in THF (neutral solvent) and did not change during heating. Cross-linking brought about the opportunity for TEM images in a solution state; otherwise, the micellar structure would be destroyed (or changed) during the evaporation of the solvent on a carbon-coated copper grid. The Flory interaction parameter, chi, between the PI block and the solvent was controlled by mixing two selective solvents (DMP/toluene, DMP/DEP and DEP/DBP) which have different degrees of selectivity for the PS block, as well as heating the solutions. Two block copolymers, PS(7.2K)-b-PI(7.8K) and PS(5.5K)-b-PI(18.8K), were studied in order to clarify the effects of the relative chain length of each block on the micelle structure in the selective solvents. PS(7.2K)-b-PI(7.8K), which is nearly symmetric, showed only spherical micelles in the DMP/toluene mixture. The basic spherical micellar shape of PS(7.2K)-b-PI(7.8K) did not change with chi, while the size and aggregation number of the micelles increased as chi increased until 2.05 and then were saturated after that. PS(5.5K)-b-PI(18.8K), which is asymmetric, showed a structural change from spherical to cylindrical to vesicular micelles with an increase in the selectivity of the DMP/DEP and DEP/DBP mixtures (which was also confirmed by TEM and SAXS studies). Giant vesicular micelles with a diameter of approximately 2.5 microm were observed in high-selectivity solvents. The size of the vesicular micelle seemed to decrease as selectivity decreased. The systematic changes of the micellar structures of PS(5.5K)-b-PI(18.8K), via changes in solvent selectivity, could be demonstrated through TEM images, which were prepared by evaporating the solvent of the cross-linked micellar solution onto the carbon-coated grid after dialysis.     

A multiscale model for kinetics of formation and disintegration of spherical micelles

Dynamics of self-assembly and structural transitions in surfactant systems often involve a large span of length and time scales. A comprehensive understanding of these processes requires development of models connecting phenomena taking place on different scales. In this paper, we develop a multiscale model for formation and disintegration of spherical nonionic micelles. The study is performed under the assumption that the dominant mechanism of micelle formation (disintegration) is a stepwise addition (removal) of single monomers to (from) a surfactant aggregate. Different scales of these processes are investigated using a combination of coarse-grained molecular dynamics simulations, analytical and numerical solution of stochastic differential equations, and a numerical solution of kinetic equations. The removal of a surfactant from an aggregate is modeled by a Langevin equation for a single reaction coordinate, the distance between the centers of mass of the surfactant and the aggregate, with parameters obtained from a series of constrained molecular dynamics simulations. We demonstrate that the reverse process of addition of a surfactant molecule to an aggregate involves at least two additional degrees of freedom, orientation of the surfactant molecule and micellar microstructure. These additional degrees of freedom play an active role in the monomer addition process and neglecting their contribution leads to qualitative discrepancies in predicted surfactant addition rates. We propose a stochastic model for the monomer addition which takes the two additional degrees of freedom into account and extracts the model parameters from molecular dynamics simulations. The surfactant addition rates are determined from Brownian dynamics simulations of this model. The obtained addition and removal rates are then incorporated into the kinetic model of micellar formation and disintegration.     

Effect of mixing on the morphology of cylindrical micelles

Increasing the spontaneous curvature of an amphiphile can lead to a first-order morphology transition from threadlike micelles to a branched network. The two morphologies were linked to entropy-driven topological defects; networks are dominated by Y-junctions, while linear threadlike structures are dominated by spherical end-caps. In this paper we investigate the effect of mixing on the morphological transitions in nonionic amphiphilic systems. We find that mixed equilibrium structures are obtained within seconds; these mixed cylindrical structures display comparable numbers of end-caps and branch points, resulting in a novel 'short armed' branched (SAB) morphology. Quite surprisingly, the probability of either defect (end-caps or branch points) is independent of composition, so that neither a first-order nor a second-order morphological transition is observed. A possible explanation may be local demixing of the two amphiphilic components, which adds a degree of freedom and thus enables the formation of a unique morphology that cannot be obtained in single-component systems. We further find that within a relatively large composition range phase equilibrium exists between vesicles, SAB micelles, and spherical micelles.     

Periodic arrays of interfacial cylindrical reverse micelles

We report an approach for the fabrication of periodic molecular nanostructures on surfaces. The approach involves biomimetic self-organization of synthetic wedge-shaped amphiphilic molecules into multilayer arrays of cylindrical reverse micelles. The films were characterized by atomic force microscopy and X-ray reflectivity. These nanostructured films self-assemble in solution but remain stable upon removal and exposure to ambient conditions, making them potentially suitable for a variety of dry pattern transfer methods. We illustrate the generality of this approach by using two distinct molecular systems that vary in size by a factor of 2.     

Model of cylindrical micelles with different endcaps

The system of alcohol (C₈OH) and hexadecyl trimethyl ammonium bromide (CTAB) was studied by molecular dynamics simulation, and the cylindrical (or rod-like) micelles with different endcaps were obtained. The simulation results presented the different terminal structures of the cylindrical (rod-like) micelles. According to the critical packing parameters of the micellar stability theory, the microscopic structures of cylindrical (or rod-like) micelles with different endcaps were analyzed. The shorter rod-like micelle was constituted by the cylindrical body and two swelling spherical endcaps, while the longer cylindrical (or rod-like) micelle had two spherical crown endcaps less than the hemisphere. The models of the cylindrical (rod-like) micelles were presented in detail based on the simulation results and the micellar stability theory.     

Electroosmosis at spherical and cylindrical metal surfaces

The profiles and velocities of electroosmotic flows at spherical and cylindrical metal surfaces are experimentally studied. Main differences are established between electroosmotic flows at the surfaces of ionite granules and particles with electronic conductivity.     

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.     

Dielectric relaxation in coexisting nematic and smectic B phases

Mixtures of a polar solute 4-n-pentyl-4'-cyanobiphenyl in a non-polar nematic solvent exhibit two separated low frequency dielectric relaxations for concentrations of the solute between 2 mol% and 20 mol% over a limited temperature range. This behaviour is attributed to coexisting nematic and smectic B phases, in which the polar solute probe has different relaxation frequencies. The observed dielectric spectra can be accurately fitted to two Debye-like relaxations, and the strengths of the absorptions are proportional to the amounts of the coexisting phases. A microscopic determination of the phase diagram confirms the assignment of the coexisting phases, and it is concluded that there is a preference of the dipolar probe molecule for the smectic B phase, which is induced as a result of solute-solvent interactions.     

On the intramicellar fluorescence quenching rate constant in cylindrical micelles

A theoretical model developed for the intramicellar fluorescence quenching rate constant in spherical micelles is extended to cylindrical micelles for diffusion- and reaction-controlled processes. The rate constants for prolate ellipsoidal micelles are expected to be intermediate between those of spherical and those of cylindrical micelles of the same volume. The importance of diffusion transients is dicussed as a function of micellar shape and the fluorescence decay time of the excited probe.     

Shear induced relaxation of polymer micelles at the solid-liquid interface

A 20% aqueous solution of (ethylene oxide) 99-(propylene oxide) 65-(ethylene oxide) 99, F127, was investigated by combining rheology in a cone/plate-geometry and surface-sensitive grazing incident neutron scattering. The crystalline structure formed by the polymer micelles becomes less pronounced for low shear rates, but correlations increase for higher shear rates. After stopping shear a slow relaxation of the micelles is found in the vicinity (50 mum thick layer) of a hydrophilic silicon wall (strong micelle-wall interaction), while a fast relaxation is observed in the boundary layer against the hydrophobic silicon wall (weak micelle-wall interaction). The results show that in the vicinity of the interface wall-particle interactions compete heavily with the shear force acting on the liquid.     

Vibrational relaxation of azide in formamide reverse micelles

Static and ultrafast infrared spectroscopy have been used to measure absorption spectra and vibrational energy relaxation (VER) times for the antisymmetric stretching vibrational band of azide, N(3)(-), in formamide-containing reverse micelles (RMs). RMs were formed in n-heptane using the surfactant AOT, sodium bis(2-ethylhexyl) sulfosuccinate. The VER times were found to be significantly longer than in bulk formamide. The VER times became longer as the molar ratio of formamide to AOT, omega(F), was decreased. Decreasing omega(F) also resulted in substantial blue shifts of the azide static absorption band compared to the frequency in bulk formamide. The omega(F) dependent studies are consistent with expected size trends, where a larger RM results in more bulklike polar solvent and faster VER rates. These results are in contrast to aqueous AOT RMs where VER times were indistinguishable from those in the bulk and the static spectral shifts were much smaller. The differences between the static and dynamic behavior in aqueous and formamide RMs are related to differences in structural changes upon confinement in RMs.