Equilibrium micellisation

The first of these theories applies the law of mass action to the equilibrium between unassociated molecules or ions and micelles, as illustrated by the following simplified calculation for the micellisation of non-ionic surfactants. If c is the stoichiometric concentration of the solution, x is the fraction of monomer units aggregated and m is the number of monomer units per micelle,... 

Since the equilibrium constant, 07C, in equation (4.23) and the standard free energy change, AG°, for the micellisation of 1 mole of surfactant are related by... 

The older pseudophase, or ion-binding model had characterised ionic micellisation through a "chemical" equilibrium reaction... 

We have already seen in Chapter 9 that the aggregation of particles can be discussed in terms of a sequence of addition reactions, shown in equation (9.7). Exactly the same arguments can he applied to the case of micellisation, but we have to discuss the problem in a little more detail since, unlike floes, micelles do not grow to a macroscopic size. In particular we have to take account of the fact that the successive equilibrium constants K(i, i + 1) [equation (9.8) depend on i. 

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. 

Both problems, changes of the equilibrium adsorption with micellar concentration and the influence of micelles on the adsorption rate, are the subjects of this review. Various definitions of the CMC are represented at the outset. Nowadays the thermodynamics of micellisation is the most developed part of modem theories of micellar systems. Two main approaches ("quasichemical and "pseudophase") are discussed in the second section of this chapter. In section 3 the thermodynamic equations for the Gibbs adsorption of surfactants in the micellar region are considered together with corresponding experimental data. The subsequent sections are devoted to non-equilibrium micellar systems. First, section 4 delineates briefly the theory of... 

As shown above, the quasichemical approach to the micellisation is based on the condition of aggregation equilibrium (5.21), which allows us to obtain the mass action law (5.22) for the micellisation process. The simplest situation arises when monodisperse micelles (only with aggregation number ni), composed only of the non-ionic surfactant molecules (component 1), are formed. The corresponding reaction can be represented by the following equation... 

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]... 

Simultaneously, the equilibrium between micelles and monomers in the subsurface layer is violated. If the diffusion rate of monomers is comparable with or less than the rate of micellisation, the deficiency of monomers can partly be compensated at the expense of micelles. In this way the concentration of micelles in the solution becomes non-uniform too. 

If the relationship between ci and Cm is known, the set of Eqs. (5.229), (5.230) can be reduced to a single diffusion equation. Joos and Van Hunsel assumed that the diffusion proceeds essentially slower than the second (slow) step of micellisation so that there is equilibrium between micelles and monomers at any moment [84]... 

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 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. 

With solubilisates having significant water solubility, it is of interest to know both the distribution ratio of solubilisate between micelles and water under saturation and unsaturation conditions. To measure the distribution ratio under unsaturation conditions, a dialysis technique can be employed, using membranes that are permeable to solubilisate but not to micelles. Ultrafiltration and gel filtration techniques can be applied to obtain the above information. The data are treated using the phase-separation model of micellisation (micelles are considered to be a separate phase in equilibrium with monomers). 

It is well established that when an amphiphilic block copolymer is dissolved in a selective solvent at a fixed temperature, above a specific concentration called the critical micelle concentration (cmc), micellisation occurs. Below the cmc, only molecularly dissolved copolymer chains (unimers) are present in the solution, while above the cmc multimolecular micelles are in thermodynamic equilibrium with the unimers. This process is in analogy to classical low molecular weight surfactants, differing in that the cmc is much lower in the case of block copolymers macrosurfactants. The self-assembly arises from the need of the copolymer chains to minimise energetically unfavourable solvophobic interactions. Therefore, micelle formation is dictated by two opposite forces, the attractive force between the insoluble blocks, which leads to aggregation, and the repulsive one between the soluble blocks preventing unlimited growth of the micelle. At the same time, the interaction of the soluble blocks and the solvent is responsible for the stabilisation of the micelles [1, 10]. 

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