Kinetics Associated with the Change of the Number Density of Micelles in Solution

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.

Surfactant desorption and scission free energies for cylindrical and spherical micelles from umbrella-sampling molecular dynamics simulations

HYPOTHESIS

The free energies associated with adsorption/desorption of individual surfactants from micelles and the fusion/scission of long micelles can be used to estimate the rate constants for micellar kinetics as functions of surfactant and salt concentration.

EXPERIMENTS

We compute the escape free energies △G_esc of surfactant from micelles and the scission free energies △G_sciss of long micelles from coarse-grained molecular dynamics simulations coupled with umbrella sampling, for micelles of both sodium dodecylsulfate (SDS) in sodium chloride (NaCl) and cetyltrimethylammonium chloride (CTAC) in sodium salicylate (NaSal).

FINDINGS

For spherical micelles, △G_esc values have maxima at certain aggregation numbers, and at salt-to-surfactant molar concentration ratios R near unity, consistent with experiments. For cylindrical micelles, SDS/NaCl shows a minimum, and CTAC/NaSal a maximum in △G_esc, both at R ~ 0.7, while △G_sciss of CTAC micelles also peaks at around R ~ 0.7 and that of SDS micelles increases monotonically with R. We explain the non-monotonic dependence of escape and scission free energies on R by a combination of electrostatic screening and the decrease of micelle radius with increasing R. Transitions from predominantly spherical to cylindrical micelles, and between adsorption/desorption and fusion/scission kinetics with changing salt concentration can be inferred from the free energies for CTAC/NaSal.

Mechanism of Micelle Birth and Death

In micellar surfactant solutions, changes in the total number of micelles are rare events that can occur by either of two mechanisms-by stepwise association and dissociation via insertion and expulsion of individual molecules or by fission and fusion of entire micelles. Molecular dynamics simulations are used here to estimate rates of these competing mechanisms in a simple model of block copolymer micelles in homopolymer solvent. This model exhibits a crossover with increasing degree of repulsion between solvent and micelle core components, from a regime dominated by association and dissociation to a regime dominated by fission and fusion.

Transformation from Globular to Cylindrical Mixed Micelles through Molecular Exchange that Induces Micelle Fusion

Transformations between different micellar morphologies in solution induced by changes in composition, salt, or temperature are well-known phenomena; however, the understanding of the associated kinetic pathways is still limited. Especially for mixed surfactant systems, the micelles can take a very wide range of structures, depending on the surfactant packing parameter and other thermodynamic conditions. Synchrotron-based small-angle X-ray scattering (SAXS) in combination with fast mixing using a stopped-flow apparatus can give direct access to the structural kinetics on a millisecond time scale. Here, this approach is used to study the formation of cylindrical micelles after mixing two solutions with globular micelles of the nonionic surfactant dodecyl maltoside (DDM) and the anionic surfactant sodium dodecyl sulfate (SDS), respectively. Two separate processes were identified: (i) a transition in micellar shell structure, interpreted as exchange of surfactant molecules resulting in mixed globular micelles, and subsequently, (ii) fusion into larger, cylindrical structures.

Lack of a unique kinetic pathway in the growth and decay of Pluronic micelles

We report kinetic experiments on dilute brine solutions of P84, P94 and P104 Pluronic copolymer micelles. The growth and the decay of micelles after temperature steps are measured by non-standard time resolved multi-angle photon correlation spectroscopy. Several concurrent mechanisms are at work during the very slow equilibration of solutions, namely insertion/expulsion of unimers, aggregation/dissociation of micellar aggregates, and fusion/budding of micellar aggregates. Their relative rates determine both the kinetic pathways and the morphologies of the micellar assemblies, which depend markedly on modest changes in the copolymer molecular weight. For the typical Pluronic copolymers investigated here, none of these elementary processes can be neglected if the resulting morphology is to be explained. This feature imposes multiple kinetic behaviours where growth and decay of Pluronic micelles become strongly dependent on the thermal history. We point out to some possible shortcomings in the studies of micellar growth kinetics by light scattering techniques. Extensive time-resolved multiangle measurements are a prerequisite for avoiding these pitfalls.

Multi-scale times and modes of fast and slow relaxation in solutions with coexisting spherical and cylindrical micelles according to the difference Becker-Döring kinetic equations

The eigenvalues and eigenvectors of the matrix of coefficients of the linearized kinetic equations applied to aggregation in surfactant solution determine the full spectrum of characteristic times and specific modes of micellar relaxation. The dependence of these relaxation times and modes on the total surfactant concentration has been analyzed for concentrations in the vicinity and well above the second critical micelle concentration (cmc2) for systems with coexisting spherical and cylindrical micelles. The analysis has been done on the basis of a discrete form of the Becker-Döring kinetic equations employing the Smoluchowsky diffusion model for the attachment rates of surfactant monomers to surfactant aggregates with matching the rates for spherical aggregates and the rates for large cylindrical micelles. The equilibrium distribution of surfactant aggregates in solution has been modeled as having one maximum for monomers, another maximum for spherical micelles and wide slowly descending branch for cylindrical micelles. The results of computations have been compared with the analytical ones known in the limiting cases from solutions of the continuous Becker-Döring kinetic equation. They demonstrated a fair agreement even in the vicinity of the cmc2 where the analytical theory looses formally its applicability.

Kinetics of the sphere-to-rod like micelle transition in a pluronic triblock copolymer

The kinetics of the sphere-to-rod transition was studied in aqueous micelle solutions of triblock copolymer poly(ethylene oxide)-poly(propylene oxide)-poly(ethylene oxide) pluronic P103 (PEO(17)PPO(60)PEO(17)). This transition was triggered by a temperature jump from the sphere phase to the rod phase and monitored with dynamic light scattering. The combination of the scattering intensity and the hydrodynamic radius were used to show that the micelles grow steadily as rods throughout the growth process. The transition was found to exhibit a single exponential behavior even in the case of large deviations from equilibrium. The linear increase in the decay rate with increasing copolymer concentration shows that the transition is dominated by a mechanism involving fusion and fragmentation of proper micelles. The decays of the sphere-to-rod transition were simulated for two pathways: random fusion fragmentation and successive addition of spherical micelles to rods. We show that micelle growth most likely occurs via random fusion-fragmentation. The second order rate constant for fusion and the fragmentation rate are calculated for the case of random fusion-fragmentation.

Stopped-flow kinetic studies of sphere-to-rod transitions of sodium alkyl sulfate micelles induced by hydrotropic salt

The kinetics and mechanism of sphere-to-rod transitions of sodium alkyl sulfate micelles induced by hydrotropic salt, p-toluidine hydrochloride (PTHC), were investigated by stopped-flow with light scattering detection. Spherical sodium dodecyl sulfate (SDS) micelles transform into short ellipsoidal shapes at low salt concentrations ([PTHC]/[SDS], chi(PTHC)=0.3 and 0.4). Upon stopped-flow mixing aqueous solutions of spherical SDS micelles with PTHC, the scattered light intensity gradually increases with time. Single exponential fitting of the dynamic traces leads to characteristic relaxation time, tau(g), for the growth process from spherical to ellipsoidal micelles, and it increases with increasing SDS concentrations. This suggests that ellipsoidal micelles might be produced by successive insertion of unimers into spherical micelles, similar to the case of formation of spherical micelles as suggested by Aniansson-Wall (A-W) theory. At chi(PTHC) > or = 0.5, rod-like micelles with much higher axial ratio form. The scattered light intensity exhibits an initially abrupt increase and then levels off. The dynamic curves can be well fitted with single exponential functions, and the obtained tau(g) decreases with increasing SDS concentration. Thus, the growth from spherical to rod-like micelles might proceed via fusion of spherical micelles, in agreement with mechanism proposed by Ikeda et al. At chi(PTHC)=0.3 and 0.6, the apparent activation energies obtained from temperature dependent kinetic studies for the micellar growth are 40.4 and 3.6 kJ/mol, respectively. The large differences between activation energies for the growth from spherical to ellipsoidal micelles at low chi(PTHC) and the sphere-to-rod transition at high chi(PTHC) further indicate that they should follow different mechanisms. Moreover, the sphere-to-rod transition kinetics of sodium alkyl sulfate with varying hydrophobic chain lengths (n=10, 12, 14, and 16) are also studied. The longer the carbon chain lengths, the slower the sphere-to-rod transition.

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.

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.

Importance of Micellar Kinetics in Relation to Technological Processes

The association of many classes of surface-active molecules into micellar aggregates is a well-known phenomenon. Micelles are in dynamic equilibrium, constantly disintegrating and reforming. This relaxation process is characterized by the slow micellar relaxation time constant, tau(2), which is directly related to the micellar stability. Theories of the kinetics of micelle formation and disintegration have been discussed to identify the gaps in our complete understanding of this kinetic process. The micellar stability of sodium dodecyl sulfate micelles has been shown to significantly influence technological processes involving a rapid increase in interfacial area, such as foaming, wetting, emulsification, solubilization, and detergency. First, the available monomers adsorb onto the freshly created interface. Then, additional monomers must be provided by the breakup of micelles. Especially when the free monomer concentration is low, which is the case for many nonionic surfactant solutions, the micellar breakup time is a rate-limiting step in the supply of monomers. 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 showed that the ionic surfactants such as SDS exhibit slow relaxation times in the range from milliseconds to seconds, whereas nonionic surfactants exhibit slow relaxation times in the range from 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 times showed a direct correlation with dynamic surface tension and foaming experiments. In conclusion, relaxation time data of surfactant solutions correlate with the dynamic properties of the micellar solutions. Moreover, the results suggest that appropriate micelles with specific stability or tau(2) can be designed by controlling the surfactant structure, concentration, and physicochemical conditions (e.g., salt concentration, temperature, and pressure). One can also tailor micelles by mixing anionic/cationic or ionic/nonionic surfactants for a desired stability to control various technological processes.

Kinetic and mechanistic studies of signal peptidase I from Escherichia coli

Signal peptidases of prokaryotic organisms reside in the outer leaflet of the cytoplasmic membrane and catalyze the hydrolytic cleavage of a specific peptide bond of membrane-imbedded preproteins to liberate mature proteins for secretion. In this manuscript, we report new and efficient peptide substrates for SPase and their use to explore features of this enzyme's reaction mechanism. The enzyme used in this study was recombinant SPase I of Escherichia coli that had been solubilized with Triton X-100 and purified to near homogeneity. Our new substrates are based on the fluorogenic peptide reported by Zhong and Benkovic [(1998) Anal. Biochem. 255, 66], Y(NO2)FSASALA approximately KIK(Abz)-NH(2) (Y(NO2), 3-nitro-L-tyrosine; K(Abz), epsilon-(2-aminobenzoyl)-L-Lys; hydrolysis at A approximately K). We found that when a signal peptide-like sequence is appended onto the N-terminus of this peptide to produce K(5)-L(10)-Y(NO2)FSASALA approximately KIK(Abz)-NH(2), k(c)/K(m) increases from 85 to 2.5 x 10(6) M(-)(1) s(-)(1). k(c)/K(m) decreases with increasing concentration of Triton X-100 micelles under the condition [Triton X-100](micelle) > [S](0) > [E](0). We explain this apparent inhibition with a model of surface dilution kinetics in which "empty" micelles compete with substrate-containing micelles for micelle-bound enzyme. Fusion of micelle-bound enzyme with a substrate-containing micelle leads to formation of productive E:S substrate complexes while fusion of micelle-bound enzyme with an "empty" micelle is nonproductive and inhibitory. The dependence of steady-state kinetic parameters for the SPase-catalyzed hydrolysis of K(5)-L(10)-Y(NO2)FSASALA approximately KIK(Abz)-NH(2) on [Triton X-100](micelle) supports this model. Product inhibition and solvent isotope effects were also investigated and could be interpreted in the context of this model.