On the Kinetics of the Formation of Small Micelles. 1. Broadband Ultrasonic Absorption Spectrometry

Nonlinear dynamics in micellar surfactant solutions. I. Kinetics

This is the first of a pair of articles that present the theory of kinetic and transport phenomena in micelle-forming surfactant solutions in a form that facilitates discussion of large deviations from equilibrium. Our goal is to construct approximate but robust reduced models for both homogeneous and inhomogeneous systems as differential equations for unimer concentration c_{1}, micelle number concentration c_{m}, average micelle aggregation number q and (optionally) aggregation number variance σ_{m}^{2}. This first article discusses kinetics in homogeneous solutions. We focus particularly on developing models that can describe both weakly perturbed states and states in which c_{1} is suppressed significantly below the critical micelle concentration, which leads to rapid shrinkage and dissociation of any remaining micelles. This focus is motivated by the strong local suppression of c_{1} that is predicted to occur near interfaces during some adsorption processes that are considered in the second article. Toward this end, we develop a general nonlinear theory of fast stepwise processes for systems that may be subjected to large changes in q and c_{1}. This is combined with the existing nonlinear theory of slow association and dissociation processes to construct a general model for systems governed by stepwise reaction kinetics. We also consider situations in which the slow process of micelle creation and destruction instead occurs primarily by micelle fission and fusion, and analyze the dependencies of micelle lifetime and the slow relaxation time upon surfactant concentration in systems controlled by either association-dissociation or fission-fusion mechanisms.

Universal Scaling for the Exit Dynamics of Block Copolymers from Micelles at Short and Long Time Scales

The correlation function for the exit of poloxamer copolymers from equilibrated micelles is found to show up to four regimes depending on the chain flexibility: an initial fast reorganization, a logarithmic intermediate regime, followed by an exponential intermediate regime, and a final exponential decay. The logarithmic intermediate regime has been observed experimentally and attributed to the polydispersity of the polymer samples. However, we present dynamic single-chain mean-field theory simulations with chains of variable flexibility which show the same logarithmic relaxation but with strictly monodisperse systems. In agreement with our previous studies, we propose that this logarithmic response arises from a degeneracy of energy states of the hydrophobic block in the micelle core. For this to occur, a sufficiently large number of degenerate conformational states are required, which depend on the polymer flexibility and therefore should not be present for rigid polymers. Experimental results for monodisperse polymeric samples claiming the absence of such a logarithmic response may also lack a sufficient number of hydrophobic blocks for the required number of configurational states for this type of response to be seen. The insight gained from analyzing the simulation results allows us to propose a modified Eyring equation capable of reproducing the observed dynamic behavior. On scaling experimental results from different sources and systems according to this equation, we find a unique master curve showing a universal nature of the intermediate regimes: the logarithmic regime together with the secondary exponential decay. The terminal exponential regime at long times proposed by the standard Halperin and Alexander model is beyond the range of the data analyzed in this article. The universality observed suggests an entropic origin of the short-time dynamic response of this class of systems rather than the polydispersity.

Simulation of diblock copolymer surfactants. II. Micelle kinetics

Molecular dynamics (MD) simulations are used to measure dynamical properties of a simple bead-spring model of A-B diblock copolymer molecules, and to characterize rates and mechanisms of several dynamical processes. Dynamical properties are analyzed within the context of a kinetic population model that allows for both stepwise insertion and expulsion of individual free molecules and occasional fission and fusion of micelles. Kinetic coefficients for stepwise processes and micelle fission have been extracted from MD simulations of individual micelles. Insertion of a free surfactant molecule into a preexisting micelle is shown to be a completely diffusion-controlled process for the model studied here. Estimates are given for rates of rare events that create and destroy entire micelles by competing mechanisms involving stepwise association and dissociation or fission and fusion. Both mechanisms are shown to be relevant over the range of parameters studied here, with association and dissociation dominating in systems with more soluble surfactants and fission and fusion dominating in systems with less soluble surfactants.

Anomalous water dynamics at surfaces and interfaces: synergistic effects of confinement and surface interactions

In nature, water is often found in contact with surfaces that are extended on the scale of molecule size but small on a macroscopic scale. Examples include lipid bilayers and reverse micelles as well as biomolecules like proteins, DNA and zeolites, to name a few. While the presence of surfaces and interfaces interrupts the continuous hydrogen bond network of liquid water, confinement on a mesoscopic scale introduces new features. Even when extended on a molecular scale, natural and biological surfaces often have features (like charge, hydrophobicity) that vary on the scale of the molecular diameter of water. As a result, many new and exotic features, which are not seen in the bulk, appear in the dynamics of water close to the surface. These different behaviors bear the signature of both water-surface interactions and of confinement. In other words, the altered properties are the result of the synergistic effects of surface-water interactions and confinement. Ultrafast spectroscopy, theoretical modeling and computer simulations together form powerful synergistic approaches towards an understanding of the properties of confined water in such systems as nanocavities, reverse micelles (RMs), water inside and outside biomolecules like proteins and DNA, and also between two hydrophobic walls. We shall review the experimental results and place them in the context of theory and simulations. For water confined within RMs, we discuss the possible interference effects propagating from opposite surfaces. Similar interference is found to give rise to an effective attractive force between two hydrophobic surfaces immersed and kept fixed at a separation of d, with the force showing an exponential dependence on this distance. For protein and DNA hydration, we shall examine a multitude of timescales that arise from frustration effects due to the inherent heterogeneity of these surfaces. We pay particular attention to the role of orientational correlations and modification of the same due to interaction with the surfaces.

Spatio-temporal correlations in aqueous systems: computational studies of static and dynamic heterogeneity by 2D-IR spectroscopy

Local heterogeneity is ubiquitous in natural aqueous systems. It can be caused locally by external biomolecular subsystems like proteins, DNA, micelles and reverse micelles, nanoscopic materials etc., but can also be intrinsic to the thermodynamic nature of the aqueous solution itself (like binary mixtures or at the gas-liquid interface). The altered dynamics of water in the presence of such diverse surfaces has attracted considerable attention in recent years. As these interfaces are quite narrow, only a few molecular layers thick, they are hard to study by conventional methods. The recent development of two dimensional infra-red (2D-IR) spectroscopy allows us to estimate length and time scales of such dynamics fairly accurately. In this work, we present a series of interesting studies employing two dimensional infra-red spectroscopy (2D-IR) to investigate (i) the heterogeneous dynamics of water inside reverse micelles of varying sizes, (ii) supercritical water near the Widom line that is known to exhibit pronounced density fluctuations and also study (iii) the collective and local polarization fluctuation of water molecules in the presence of several different proteins. The spatio-temporal correlation of confined water molecules inside reverse micelles of varying sizes is well captured through the spectral diffusion of corresponding 2D-IR spectra. In the case of supercritical water also, we observe a strong signature of dynamic heterogeneity from the elongated nature of the 2D-IR spectra. In this case the relaxation is ultrafast. We find remarkable agreement between the different tools employed to study the relaxation of density heterogeneity. For aqueous protein solutions, we find that the calculated dielectric constant of the respective systems unanimously shows a noticeable increment compared to that of neat water. However, the 'effective' dielectric constant for successive layers shows significant variation, with the layer adjacent to the protein having a much lower value. Relaxation is also slowest at the surface. We find that the dielectric constant achieves the bulk value at distances more than 3 nm from the surface of the protein.

Monomer Exchange Kinetics, Dynamics of Concentration Fluctuations, and Chain Isomerization of Nonionic Surfactant/Water Systems. Evidence from Broadband Ultrasonic Spectra

Ultrasonic absorption spectra, measured between 0.1 and 2000 MHz, are discussed for a variety of poly(ethylene glycol) monoalkyl ether/water (CiEj=H2O) mixtures. Depending on the temperature, the surfactant concentration, and on the length of the hydrophobic (Ci) as well as the hydrophilic part (Ej) of the surfactant molecules, the spectra reveal a multitude of shapes. The set of spectra, however, can be consistently described considering (i) a relaxation term representing the monomer exchange of the micellar solutions, (ii) another one that reflects the local fluctuations in the surfactant concentration, and, with several systems, (iii) additional terms due to CiEj associations or conformational isomerizations. The parameters of these terms are discussed in the light of relevant models. Evidence is presented for a more general view of a fluctuation controlled monomer exchange mechanism that combines aspects of both theoretical models, the micelle formation/decay kinetics and dynamics of local concentration fluctuations.

Fluctuations near the critical micelle concentration. II. Ultrasonic attenuation spectra and scaling

Based on a thermodynamic model of amphiphile solutions derived in the first part of the paper, the ultrasonic attenuation of such systems has been considered theoretically, including fluctuations of local concentrations and micelle sizes. At amphiphile concentrations smaller than the critical micelle concentration (cmc), scaling behavior in terms of the concentration distance to the cmc is predicted by theory, in fair agreement with experimental evidence. The scaling function in the sound attenuation below the cmc reveals the unsymmetric broadening in the spectra that clearly emerges from measurements when approaching the cmc. The shape of the scaling function corresponds to the experimental spectra of solutions with comparatively large cmc as well as with the relaxation spectral function of the unifying model of non-critical concentration fluctuations. Above the cmc, an additional relaxation term is predicted in correspondence with the Landau-Khalatnikov term in the sound attenuation of superfluid helium. This term is difficult to verify by measurements because, in the relevant frequency range, other processes may also contribute the ultrasonic attenuation spectra.

Octylglucopyranoside and cyclodextrin in water. Self-aggregation and complex formation

At 25 degrees C ultrasonic attenuation spectra between 100 kHz and 400 MHz as well as sound velocities and densities of aqueous solutions of the surfactant n-octyl-beta-d-glucopyranoside and of the cage compound alpha-cyclodextrin have been measured. The liquid system reveals a critical association concentration (cac) exceeding the cmc of the surfactant almost by the cyclodextrin concentration and thus indicating a significant formation of cyclodextrin-surfactant inclusion complexes. The ultrasonic spectra display altogether four relaxation regions. The one with largest relaxation time (0.27 mus < or = tau(1) < or = 1.6 mus) exhibits a noticeable amplitude at surfactant concentrations larger than the cac only. It is assigned to the monomer exchange between the micelles and the suspending phase. A term with relaxation time tau(2) (33 ns < or = tau(2) < or = 135 ns) is characteristic for solutions containing both solutes. It is assumed to reflect the inclusion complex formation. Complexes with 1:1, 2:1, and 1:2 stoichiometry appear to exist. The terms at higher frequencies (4.8 ns < or = tau(3) < or = 9.8 ns; 0.8 ns <or = tau(4) < or = 2 ns) are due to fluctuations of the carbohydrate residues around the glucosidic bond angles and to the rotational isomerization of the exocyclic hydroxymethyl groups, respectively.

Monomer Exchange Kinetics, Radial Diffusion, and Hydrocarbon Chain Isomerization of Sodium Dodecylsulfate Micelles in Water

Molecular relaxation properties of sodium dodecylsulfate in aqueous solutions with surfactant concentrations between 0.009 and 0.4 mol/L have been studied using broadband ultrasonic spectrometry in the frequency range 0.1-2000 MHz. Ultrasonic excess attenuation characteristics were found that could be well represented by a sum of three relaxation terms, each one characterized by a discrete relaxation time. The low-frequency term with concentration-dependent relaxation time, tau1, between 0.06 and 3.5 micros is discussed in terms of the surfactant monomer exchange. The noticeable effect from the incomplete dissociation of the surfactant counter ions and the variation of the monomer concentration following thereby is discussed. The second relaxation term (0.9 <or= tau2 <or= 2.5 ns) is assigned to the limited radial diffusion of monomers within the micelles, yielding a mean diffusion length of 0.5 nm, in correspondence to protrusions by four methyl groups. The high-frequency relaxation term (0.1 <or= tau3 <or= 0.2 ns) reflects the structural isomerizations of the hydrocarbon chains in the micellar cores, largely resembling those of liquid alkanes.

Critical fluctuations of the micellar triethylene glycol monoheptyl ether-water system

Using the equal volume criterion and also the pseudospinodal conception the critical demixing point of the triethylene glycol monoheptyl ether/water system (C7E3H2O) has been determined as Ycrit=0.1 and Tcrit=296.46 K (Y, mass fraction of surfactant). From density measurements the critical micelle concentration (cmc) followed as Ycmc=0.007 at 288.15 K and Ycmc=0.0066 at 298.15 K. The (static) shear viscosity etas and the mutual diffusion coefficient D of the C7E3H2O mixture of critical composition have been evaluated to yield their singular and background parts. From a combined treatment of both quantities the relaxation rate Gamma of order parameter fluctuations has been derived. Gamma follows power law with universal critical exponent and amplitude Gamma0=3.1 x 10(9) s(-1). Broadband ultrasonic spectra of C7E3H2O mixtures exhibit a noncritical relaxation, reflecting the monomer exchange between micelles and the suspending phase, and a critical term due to concentration fluctuations. The former is subject to a relaxation time distribution that broadens when approaching the critical temperature. The latter can be well represented with the aid of the dynamic scaling model by Bhattacharjee and Ferrell (BF) [Phys. Rev. A. 31, 1788 (1985)]. The half-attenuation frequency in the scaling function of the latter model is noticeably smaller (Omega12 (BF) approximately 1) than the theoretically predicted value Omega12 (BF)=2.1. This result has been taken as an indication of a coupling between the fluctuations in the local concentration and the kinetics of micelle formation, in correspondence with the idea of a fluctuation controlled monomer exchange [T. Telgmann and U. Kaatze, Langmuir 18, 3068 (2002)].

Crossover behavior in micellar solutions with lower critical demixing point: broadband ultrasonic spectrometry of the isobutoxyethanol-water system

The aggregation kinetics of isobutoxyethanol-water mixtures with lower critical demixing point has been investigated. Two types of kinetics have been observed, a diffusion-controlled formation of micellar species and the formation of a microheterogeneous liquid structure, governed by fluctuations in the local concentration. Ultrasonic attenuation spectra of isobutoxyethanol-water mixtures have been measured between 100 kHz and 2 GHz at 25 degrees C and at several concentrations, covering the complete composition range. With the mixture of critical composition measurements have been performed at some temperatures near the critical temperature (T(c)=299.51 K). In addition to the asymptotic high frequency background contribution, the broadband spectra reveal a Bhattacharjee-Ferrell relaxation term due to critical concentration fluctuations, a restricted Hill term reflecting the monomer exchange between micelles and the suspending phase, and two Debye-type relaxation terms that are assigned to chemical relaxations. The relaxation rates of the Bhattacharjee-Ferrell term exceed those from static and dynamic light scattering (amplitude Gamma(0)=5.3 x 10(9) s(-1)), likely due to the effect of a second parallel pathway of relaxation in the ultrasonic field. The adiabatic coupling constant following from the amplitude in the ultrasonic spectrum agrees with that from a thermodynamic relation (g=1.3). The restricted Hill term displays the features of an extended Teubner-Kahlweit-Aniansson-Wall model of the micelle formation and decay kinetics in surfactant solutions with high critical micelle concentration (C=0.6 mol/l). The idea of a fluctuation controlled monomer exchange in aqueous solutions of poly(ethylene glycol) monoalkyl ether-water mixtures near the critical point is briefly discussed.

Kinetics of adsorption from micellar solutions

Previous studies on surfactant adsorption mostly deal with dilute systems without aggregation in the bulk phase. At the same time, micellar solutions can be more important from the point of view of applications. If one attempts to estimate the equilibrium adsorption, neglecting the influence of micelles can lead to reasonable results. The situation differs for non-equilibrium systems when the adsorption rate can increase by an order of magnitude at the increase of the surfactant concentration beyond the CMC. A critical survey of various models describing the influence of micelles on adsorption kinetics at the liquid-gas interface is given and the theoretical results are compared with existing experimental data. The theories proposed for the case of large deviations from the equilibrium are usually based on some unjustifiable assumptions and can describe the kinetic dependencies of adsorption in only a limited number of situations. Consequently, only rough estimates of the kinetic coefficients of micellization can be obtained from experimental data on dynamic surface tension. More rigorous equations can be derived if the system only deviates slightly from equilibrium. In the latter case, the agreement between theoretical and experimental results is essentially better and measurements of the dynamic surface elasticity of micellar solutions allow us to study the micellization kinetics.

Particulate characterization by ultrasound

Ultrasound is a technique that is capable of tremendous accuracy and precision, whether in the laboratory or on-line in the factory. Data acquisition rates of the order of kHz transform process monitoring, providing unprecedented information of high statistical value. Ultrasound is complementary to other measurement modalities, probing the elastic and thermal properties of materials. In the absence of air bubbles, ultrasound travels through concentrated systems of particles, providing information about size, compressibility, structure, distribution, chemical and phase state. It is safe, can be non-invasive and can be economical. In these respects, it is unique.