Effect of the polydispersity of diffusing micelles on the surface elasticity

Diffusion of surfactant from a micellar solution to a bare interface. 1. Absorbing boundary

We analyze dynamic adsorption of surfactant from a micellar solution to a rapidly created surface that acts as an absorbing boundary for surfactant monomers (single molecules), along which the monomer concentration vanishes, with no direct micelle adsorption. This somewhat idealized situation is analyzed as a prototype for situations in which strong suppression of monomer concentration accelerates micelle dissociation, and will be used as a starting point for analysis of more realistic boundary conditions in subsequent work. We present scaling arguments and approximate models for particular time and parameter regimes and compare the resulting predictions to numerical simulations of the reaction-diffusion equations for a polydisperse system containing surfactant monomers and clusters of arbitrary aggregation number. The model considered here exhibits an initial period of rapid shrinkage and ultimate dissociation of micelles within a narrow region near the interface. This opens a micelle-free region near the interface after some time τ_e, the width of which increases as t^1/2 at times t≫τ_e. In systems that exhibit disparate fast and slow bulk relaxation times τ₁ and τ₂ in response to small perturbations, τ_e is usually comparable to or greater than τ₁ but much less than τ₂. Such systems exhibit a wide intermediate time regime τ_e<t<τ₂ in which the remaining micellar region reaches a state of partial local equilibrium, followed by a final stage t≫τ₂ in which full local equilibrium is established.

Nonlinear dynamics in micellar surfactant solutions. II. Diffusion

We discuss diffusion in micellar surfactant solutions in a form appropriate for analyzing experiments that involve large deviations from equilibrium. A general nonlinear dynamical model for inhomogeneous systems is developed that describes the effects of diffusion and micelle kinetics as a set of coupled partial differential equations for unimer concentration, micelle number concentration, average micelle aggregation number, and, optionally, the variance of the micelle aggregation number. More specialized models are developed to describe slow dynamics in situations in which the system stays in a state of partial local equilibrium or full local equilibrium. As an illustrative example of a nonlinear transport phenomenon, we discuss a simple model of diffusion from an initially homogeneous micellar solution to a rapidly created absorbing interface with fast unimer adsorption.

Visualization of an adsorption model for surfactant transport from micelle solutions to a clean air/water interface using fluorescence microscopy

This work pertains to visualizing a transport model for adsorption of surfactants from micelle solutions onto a clean air/water interface. Under the condition of surfactant adsorption from very dilute solutions, the time scale for diffusion of a surfactant monomer is much slower than the time scale for kinetic breakdown of the aggregates. A theoretical model predicts two regimes for the adsorption dynamics. We visualize these two regimes under the mechanism of solubilization using fluorescence microscopy, in which an insoluble fluorescent probe, NBD-HAD (4-(hexadecylamino)-7-nitrobenz-2-oxa-1,3-diazole), is used to illuminate the micelles. The dye fluoresces in the microenvironment of micelles but is quenched in the aqueous solution on laser excitation. The region containing micelles is illuminated, but the region which does not contain micelles appears dark. For surfactant solution of C(14)E(6) at concentration just above the critical micelle concentration (C(CMC)), C(CMC)=4.4 mg/L, a dark region between the bright image of the air/water interface and the micelle-containing zone is observed. This dark region becomes smaller with time and finally disappears once equilibrium is reached. For a surfactant solution of C(14)E(6) at the concentration of 4.74C(CMC), which is higher than a critical total surfactant concentration (C(T)(c)) of 4.25C(CMC), we observe bright images through surfactant solutions during the adsorption process. Fluorescence images validate the theoretical model.

Fluorescence visualization and modeling of a micelle-free zone formed at the interface between an oil and an aqueous micellar phase during interfacial surfactant transport

This study examines the transport of surfactant that occurs when an aqueous micellar phase is placed in contact with a clean oil phase in which the surfactant is soluble. Upon contact with oil, surfactant monomer on the aqueous side of the interface adsorbs onto the oil/water interface and subsequently desorbs into the oil and diffuses away from the surface. The depletion of aqueous monomer underneath the interface disturbs the monomer-micelle equilibrium, and aggregates break down to replenish the monomer concentration and accelerate the interfacial transport. The depletion of monomer and micelles drives the diffusive flux of these species toward the surface, and the combined effects of diffusion and aggregate kinetic disassembly, alongside kinetic adsorption and desorption at the interface and diffusion away from the interface into the oil, determine the interfacial transport rate. This interfacial transport is examined here in the quasi-static limit in which the diffusion of monomer and micelles in the aqueous phase is much slower than the time scale for micelle disassembly. In this limit, when the initial bulk concentration of micelles in the aqueous solution is small, the micelle diffusive flux to the surface cannot keep up with the micelle breakdown under the interface, and a micelle-free zone forms. This zone extends from the surface into the aqueous phase up to a boundary that demarcates the beginning of a zone, containing micelles, that extends further into the aqueous phase. Micelles diffuse from the micelle zone to the boundary, where they break down, causing the boundary to retreat. Released monomer diffuses through the micelle-free zone and partitions into the oil phase. The focus of this study is to verify this transport picture by visualizing the micelle-free zone and comparing the movement of the zone to predictions obtained from a transport model based on this two-zone picture. A small hydrophobic dye molecule (Nile red) is incorporated into the micelles; the dye fluoresces only in the hydrophobic environment of the micelles, providing visual contrast between the two zones. Through spatial mapping of the fluorescence using confocal microscopy, the movement of the micelle-free zone boundary can be measured and is shown to compare favorably with simulations of the transport model.

Surface dilational viscoelasticity of C14EO8 micellar solution studied by bubble profile analysis tensiometry

The experimental dependences of viscoelasticity modulus and phase angle as a function of frequency for various C 14EO8 concentrations at the critical micelle concentration (cmc) of 7 micromol/L and far above the cmc (up to 70 x cmc) were studied using the buoyant bubble profile analysis method. With increasing C14EO8 concentration the viscoelasticity modulus decreases and the phase angle increases. At the highest surfactant concentrations, the phase angle was more than 45 degrees . For the theoretical description of the equilibrium surface tension isotherm and the limiting elasticity modulus, a combined theoretical model was used considering surface reorientation and molecular compression. To analyze the experimental dependencies of the viscoelasticity modulus and phase angle on frequency, a model proposed by Joos for fast micellar kinetics was applied. This theory agrees well with the experimental data of the viscoelasticity modulus obtained for all concentrations of the studied nonionic surfactant C14EO8.

Dynamic surface tension of micellar solutions in the millisecond and submillisecond time range

The analysis of the available bubble life times and dead times for the bubble pressure tensiometer BPA-1S shows that dynamic surface tensions can be measured also for surfactant solutions at concentrations many times higher than the corresponding CMC. For the three nonionic surfactants Triton X-100, Triton X-45, and C14EO8 experiments are performed for solutions with a concentration of up to 200 times the CMC (C14EO8). Comparison of the experimental data with micelle kinetics models yields rate constants for the fast micelle dissolution process, which are in a good agreement with values obtained by other experimental methodologies.

Mass transport in micellar surfactant solutions: 1. Relaxation of micelle concentration, aggregation number and polydispersity

The surfactant transfer in micellar solutions includes transport of all types of aggregates and the exchange of monomers between them. Such processes are theoretically described by a system containing tens of kinetic equations, which is practically inapplicable. For this reason, one of the basic problems of micellar kinetics is to simplify the general set of equations without loosing the adequacy and correctness of the theoretical description. Here, we propose a model, which generalizes previous models in the following aspects. First, we do not use the simplifying assumption that the width of the micellar peak is constant under dynamic conditions. Second, we avoid the use of the quasi-equilibrium approximation (local chemical equilibrium between micelles and monomers). Third, we reduce the problem to a self-consistent system of four nonlinear differential equations. Its solution gives the concentration of surfactant monomers, total micelle concentration, mean aggregation number, and halfwidth of the micellar peak as functions of the spatial coordinates and time. Further, we check the predictions of the model for the case of spatially uniform bulk perturbations (such as jumps in temperature, pressure or concentration). The theoretical analysis implies that the relaxations of the three basic parameters (micelle concentration, mean aggregation number, and polydispersity) are characterized by three different characteristic relaxation times. Two of them coincide with the slow and fast micellar relaxation times, which are known in the literature. The third time characterizes the relaxation of the width of the micellar peak (i.e. of the micelle polydispersity). It is intermediate between the slow and fast relaxation times, in the case of not-too-low micellar concentrations. For low micelle concentrations, the third characteristic time is close to the fast relaxation time. Procedure for obtaining the exact numerical solution of the problem is formulated. In addition, asymptotic analytical expressions are derived, which compare very well with the exact numerical solution. In the second part of this study, the obtained set of equations is applied for theoretical modeling of surfactant adsorption from micellar solutions under various dynamic conditions, corresponding to specific experimental methods.

Mass transport in micellar surfactant solutions: 2. Theoretical modeling of adsorption at a quiescent interface

Here, we apply the detailed theoretical model of micellar kinetics from part 1 of this study to the case of surfactant adsorption at a quiescent interface, i.e., to the relaxation of surface tension and adsorption after a small initial perturbation. Our goal is to understand why for some surfactant solutions the surface tension relaxes as inverse-square-root of time, 1/t(1/2), but two different expressions for the characteristic relaxation time are applicable to different cases. In addition, our aim is to clarify why for other surfactant solutions the surface tension relaxes exponentially. For this goal, we carried out a computer modeling of the adsorption process, based on the general system of equations derived in part 1. This analysis reveals the existence of four different consecutive relaxation regimes (stages) for a given micellar solution: two exponential regimes and two inverse-square-root regimes, following one after another in alternating order. Experimentally, depending on the specific surfactant and method, one usually registers only one of these regimes. Therefore, to interpret properly the data, one has to identify which of these four kinetic regimes is observed in the given experiment. Our numerical results for the relaxation of the surface tension, micelle concentration and aggregation number are presented in the form of kinetic diagrams, which reveal the stages of the relaxation process. At low micelle concentrations, "rudimentary" kinetic diagrams could be observed, which are characterized by merging of some stages. Thus, the theoretical modeling reveals a general and physically rich picture of the adsorption process. To facilitate the interpretation of experimental data, we have derived convenient theoretical expressions for the time dependence of surface tension and adsorption in each of the four regimes.

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.