Kinetics micellisation

Interestingly, at very low concentrations of micellised Qi(DS)2, the rate of the reaction of 5.1a with 5.2 was observed to be zero-order in 5.1 a and only depending on the concentration of Cu(DS)2 and 5.2. This is akin to the turn-over and saturation kinetics exhibited by enzymes. The acceleration relative to the reaction in organic media in the absence of catalyst, also approaches enzyme-like magnitudes compared to the process in acetonitrile (Chapter 2), Cu(DS)2 micelles accelerate the Diels-Alder reaction between 5.1a and 5.2 by a factor of 1.8710 . This extremely high catalytic efficiency shows how a combination of a beneficial aqueous solvent effect, Lewis-acid catalysis and micellar catalysis can lead to tremendous accelerations. 

Most of the traditional adsorption studies of surfactants correspond to dilute systems without aggregation in the bulk phase. At the same time micellar solutions are much more important from a practical point of view. To estimate the equilibrium adsorption, neglecting the effect of micelles can usually lead to reasonable results. However, the situation changes for nonequilibrium systems when the adsorption rate can increase by orders of magnitude when the of surfactant concentration is increased beyond the CMC. Current interest in the adsorption from micellar solutions is mainly caused by recent observations that the stability of foams and emulsions depends strongly on the concentration in the micellar region [1]. This effect can be explained by the influence of the micellisation rate on the adsorption kinetics. 

It was already mentioned above that the condition of monodispersity of micelles means that only one kind of aggregates with a fixed aggregation number nj is formed. From the point of view of chemical kinetics the reaction (5.39) is a reaction of ni order. Because typical micelles consist of some tens or hundreds molecules the probability of this elementary step is zero. Therefore, Eq. (5.39) presents only the final result of nj-l stepwise reactions of first order. The corresponding equilibrium constant is then a product of n -l constants for each step of the micellisation process. In our simplest case we can consider that all these constants are the same and we get [ 12]... 

The main factors determining the influence of micelles on the adsorption kinetics are the rates of formation and disintegration of micelles. This leads immediately to the consequence that the treatment of the adsorption kinetics at c > CMC is impossible without preliminary investigations of the micellisation kinetics. For this reason the main results of kinetic studies of micellar systems will be presented briefly below. 

The number of kinetic equation corresponds to the number of different aggregates in the system and is usually of the order of hundred. In the general case the same number of relaxation times corresponds to a system of ordinary linear differential equations (5.152) and (5.153). However, the assumptions made by Aniansson and Wall [114] allow us to consider the time interval corresponding to the fast and slow micellisation steps separately, and to reduce the system of Eqs. (5.152) and (5.153) to a single differential equation for each step. 

This gives the following kinetic equation for the slow step of the micellisation process [119] d8c ... 

It is obvious that the kinetic theory of micellisation outlined above, cannot be applied to living polymers. For wormlike micelles one has to distinguish between several processes leading to disintegration and formation of micelles and to determine the corresponding characteristic times [141 - 145] ... 

Concluding the discussion of micellisation kinetics it is necessary to note that a new theory based on the ideas of nucleation kinetics have been proposed recently by Kuni et al. [174,175]. The nucleation theory allows to study in detail the size distribution of aggregates on the basis of a thermodynamic analysis and to obtain more general kinetic equations. 

The first attempt to take into account the two-step kinetic theory of micellisation was made by Fainerman [147]. With that end in view two pairs of diffusion equations (for micelles and monomers) were written down for two situations eorresponding to the fast and slow proeesses. Approximate solutions of the boundary problems for these equations were used subsequently in the course of analysis of experimental data on the adsorption kinetics from micellar solutions [77, 85, 87, 88]. However, as it has been shown by Dushkin et al. [137], this approaeh is equivalent to the PFOR model for the slow proeess and probably eannot be applied to the description of the adsorption kinetics for the fast process. 

A more rigorous approach to the description of the colloid surfactant diffusion to the interfaee was proposed by Noskov [133]. The reduced diffusion equations for micelles and monomers, which take into account the multistep nature of micellisation and the polydispersity of micelles, were derived for time intervals corresponding to the fast and slow processes using the method applied initially by Aniansson and Wall to uniform systems. Analogous equations have been derived later by Johner and Joanny [135] and also by Dushkin et al. [137]. Recently Dushkin has studied also the adsorption kinetics in the framework of a simplified model of quasi-monodisperse micelles. In this case the assumption of the existence of two kinds of micelles permits to study the main features of the surface tension relaxation in real micellar solution [138]. The main steps of the derivation of surfactant diffusion equations in micellar solutions are presented below [133, 134]. 

In conclusion of this section let us consider the surfactant diffusion in a concentrated solution, where the micellisation kinetics is described by the model of Kahlweit et al. (5.185). If we neglect all other routes of the micellisation process, the following diffusion equation corresponds to the mechanism (5.185) ... 

One of the reasons of the insufficient reliability of micellisation kinetics data determined from dynamic surface tensions, consists in the insufficient precision of the calculation methods for the adsorption kinetics from micellar solutions. It has been already noted that the assumption of a small deviation from equilibrium used at the derivation of Eq. (5.248) is not fulfilled by experiments. The assumptions of aggregation equilibrium or equal diffusion rates of micelles and monomers allow to obtain only rough estimates of the dynamic surface tension. An additional cause of these difficulties consists in the lack of reliable methods for surface tension measurements at small surface ages. The recent hydrodynamic analysis of the theoretical foundations of the oscillating jet and maximum bubble pressure methods has shown that using these techniques for measurements in the millisecond time scale requires to account for numerous hydrodynamic effects [105, 158, 159]. These effects are usually not taken into account by experimentalists, in particular in studies of micellar solutions. A detailed analysis of... 

An analysis of the maximum bubble pressure method including all known theoretical approaches was given only recently so that data from literature are only of approximate character [160]. Therefore, the current level of kinetic theories of adsorption from micellar solutions and the corresponding experimental technique is still insufficient for investigations of the micellisation kinetics with a precision comparable to that of bulk relaxation methods. This pessimistic conclusion, however, relates to a less extent to methods based on small (mainly periodic) perturbations of the adsorption equilibrium. 

The main relations between the complex dynamic surface elasticity and the kinetic characteristics of micellisation are presented below [103, 133, 134]. Let a small surface element of the area S be subjected to a small periodic surface dilation 5 S... 

This relation describes not only periodic deformations of a liquid surface. Using methods of integral transformations it is possible to show that the dynamic surface elasticity is a fundamental surface property and its value determines the system response to a small arbitrary surface dilation [161]. With this method it is also possible to determine the dynamic elasticity of liquid-liquid interfaces where the surfactant is soluble in both adjacent phases [133]. Moreover, similar transformations lead to an expression for the dynamic surface elasticity for the case when the mechanism of the slow step of micellisation is determined by scheme (5.185) or for frequencies corresponding to the fast step of micellisation [133,134]. However, as stated above, it is the slow process which mainly influences the adsorption kinetics from micellar solutions. 

At sufficiently high frequencies (co p Dj independent of the relationship between co and tj ) the diffusion in the bulk, and consequently the micellisation, does not influence the surface elasticity. In the opposite case of low frequencies the micellisation kinetics... 

Note that relation (5.281) does not contain any kinetic characteristics of micellisation. If the micellar concentration is low and the formation (disintegration) of micelles is sufficiently fast (tj o), the adsorption rate and, consequently, the dynamic surface elasticity depend only on the efficiency of the surfactant transfer by micelles from the bulk to the surface, and, therefore, on the diffusion coefficient of micelles and the mean aggregation number. This means that the micellar size can be determined from dynamic surface properties. Really, if approximation (5.231) for the diffusion coefficient of micelles is used, it follows from Eq. (5.281)... 

This inequality agrees with the results of direct measurements of the formation and disintegration rate of micelles for solutions of DSN, DACh and CTACh [115]. To the best of our knowledge the micellisation kinetics in solutions of DPO has not been studied so far by relaxation spectrometry of the bulk phase. 

The influence of micellisation on the propagation of capillary waves has been discovered only for solutions of the nonionic surfactant - DePO. The determined values of Z2 are comparable with the results for solutions of DePB but they decrease monotonously with concentration. Therefore, the obtained results evidence that relations (5.284) and (5.285) describe the concentration dependence of the dynamic surface elasticity well. Hence, the method of transverse capillary waves can be used for studies of micellisation kinetics of surfactants with relatively low surface activity. For surfactants with higher surface activity where the formation and disintegration of micelles proceed slower the method of longitudinal surface waves can be used [102, 103]. The characteristics of longitudinal waves are more sensitive to the dynamic surface elasticity, and this allows one to study the micellisation kinetics under the condition... 

The equilibrium and dynamics of adsorption processes from micellar surfactant solutions are considered in Chapter 5. Different approaches (quasichemical and pseudophase) used to describe the micelle formation in equilibrium conditions are analysed. From this analysis relations are derived for the description of the micelle characteristics and equilibrium surface and interfacial tension of micellar solutions. Large attention is paid to the complicated problem, the micellation in surfactant mixtures. It is shown that in the transcritical concentration region the behaviour of surface tension can be quite diverse. The adsorption process in micellar systems is accompanied by the dissolution or formation of micelles. Therefore the kinetics of micelle formation and dissociation is analysed in detail. The considered models assume a fast process of monomer exchange and a slow variation of the micelle size. Examples of experimental dynamic surface tension and interface elasticity studies of micellar solutions are presented. It is shown that from these results one can conclude about the kinetics of dissociation of micelles. The problems and goals of capillary wave spectroscopy of micellar solutions are extensively discussed. This method is very efficient in the analysis of micellar systems, because the characteristic micellisation frequency is quite close to the frequency of capillary waves. 

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