Mass action model of micellization

The mass action model of micellization assumes that the cationic micelle, MP ", is foraed by an all-or-none process from N monomers, D" ", and N-P firmly bound anions, X . 

The mass-action model should be verified before we discuss micelle thermodynamics. Recent progress in electrochemical techniques makes it possible to measure monomeric concentrations of surfactant ions and counterions, and determination of the micellization constant has become possible. The first equality of (4.24) has three parameters to be determined— K , n, and m, which are the most important factors for the mass-action model of micelle formation. For monodisperse micelles, the following equations result from (4.13) and (4.14), respectively ... 

These relations can be explained perfectly by the mass-action model of micelle formation. ... 

Almost thirty years ago the author began his studies in colloid chemistry at the laboratory of Professor Ryohei Matuura of Kyushu University. His graduate thesis was on the elimination of radioactive species from aqueous solution by foam fractionation. He has, except for a few years of absence, been at the university ever since, and many students have contributed to his subsequent work on micelle formation and related phenomena. Nearly sixty papers have been published thus far. Recently, in search of a new orientation, he decided to assemble his findings and publish them in book form for review and critique. In addition, his use of the mass action model of micelle has received much criticism, especially since the introduction of the phase separation model. Many recent reports have postulated a role for Laplace pressure in micellization. Although such a hypothesis would provide an easy explanation for micelle formation, it neglects the fact that an interfacial tension exists between two macroscopic phases. The present book cautions against too ready an acceptance of the phase separation model of micelle formation. 

According to the mass action model of micellization of ionic surfactants [24], the equilibrium between surfactant monomers and micelles is given by [25)... 

Mass-action model of surfactant micelle formation was used for development of the conceptual retention model in micellar liquid chromatography. The retention model is based upon the analysis of changing of the sorbat microenvironment in going from mobile phase (micellar surfactant solution, containing organic solvent-modifier) to stationary phase (the surfactant covered surface of the alkyl bonded silica gel) according to equation ... 

We see in Section 8.8 that surfactants undergo aggregation in nonaqueous solvents also, but the degree of aggregation is very much less (n < 10), and the threshold for aggregation is far less sharp than in water. The mass action model for micellization seems preferable for nonaqueous systems. 

The mass action model describes micelle formation as an equilibrium process. The micellar aggregation number becomes an important parameter. The solubilization process can be treated as a stepwise addition of solute molecules to the micelles. However, the partition coefficient based on this model requires the aggregation number, which makes it difficult to use in practice. There are several methods of simplification. One is to define the partition coefficient as ... 

In the previous chapters, the dissolution and micellization of surfactants in aqueous solutions were discussed from the standpoint of the degrees of freedom as given by the phase rule. The mass-action model for micelle formation was found to be better for explaining the phenomena of surfactant solutions than the phase-separation model. Two models have similarly been used to explain the Krafft point, one postulating a phase transition at the Krafft point and the other a solubility increase up to the CMC at the Krafft point. The most recent version of the first approach is a melting-point model for a hydrated surfactant solid. The most direct approach to the second model of the Krafft point rests entirely on measurements of the solubility and CMC of surfactants with temperature. From these mesurements the concept of the Krafft point can be made clear. This chapter first reviews the concepts used to relate the dissolution of surfactants to their micellization, and then shows that the concept of a micelle temperature range (MTR) can be used to elucidate various phenomena concerning dissolution... 

Equation (21) is not strictly valid for calculating the heat of micellization because certain assumptions made in its derivation do not hold here. The equation implies that the micelle is at equilibrium near cmc in a standard state [27,54]. However, micelles are not definite stoichiometric entities but aggregates of different sizes that are in dynamic equilibrium with themselves and surfactant monomers. The aggregation number may vary with temperature. An extended mass action model describes micellization as a multiple equilibrium characterized by a series of equilibrium constants (see Section 6.2). Because these equilibrium constants cannot be determined, the micellar equilibrium is usually described by... 

The mass action model (MAM) for binary ionic or nonionic surfactants and the pseudo-phase separation model (PSM) which were developed earlier (I EC Fundamentals 1983, 22, 230 J. Phys. Chem. 1984, 88, 1642) have been extended. The new models include a micelle aggregation number and counterion binding parameter which depend on the mixed micelle composition. Thus, the models can describe mixtures of ionic/nonionic surfactants more realistically. These models generally predict no azeotropic micellization. For the PSM, calculated mixed erne s and especially monomer concentrations can differ significantly from those of the previous models. The results are used to estimate the Redlich-Kister parameters of monomer mixing in the mixed micelles from data on mixed erne s of Lange and Beck (1973), Funasaki and Hada (1979), and others. 

These four equations reduce to the equations of Ref. (11) if N and are fixed. The micelle concentration is then calculated from a mass action model expression as before, where (17)... 

To this point, only models based on the pseudo—phase separation model have been discussed. Mixed micelle models utilizing the mass action model may be necessary for micelles with small aggregation numbers, as demonstrated by Kamrath and Franses ( ). However, even for large micelles, the fundamental basis for the pseudophase separation model needs to be examined. In micelles, how much solvent or how many counterions (bound or in the electrical double layer) should be included in the micellar pseudo-phase is unclear. The difficulty is normally surmounted by assuming that the pseudo—phase consists of only the surfactant components i.e., solvent or counterions are ignored. The validity of treating the micelle on a surfactant—oniy basis has not been verified. Funasaki and Hada (22) have questioned the thermodynamic consistency of such an approach. 

The thermodynamics of micelle formation has been studied extensively. There is for example a mass action model (Wennestrdm and Lindman, 1979) that assumes that micelles can be described by an aggregate Mm with a single aggregation number m, so that the only descriptive equation is mMi Mm. A more complex form assumes the multiple equilibrium model, allowing aggregates of different sizes to be in equilibrium with each other (Tanford, 1978 Wennestrdm and Lindman, 1979 Israelachvili, 1992). 

In the mass action model the micellar system can be described by only one parameter, and despite this simplicity, a good qualitative description of the main physical properties is obtained, for example the onset of cmc (critical micelle concentration), as shown in Figure 9.7. Notice that the formation of micelles becomes appreciable only at the cmc, and after that, by increasing further the surfactant concentration, all added surfactant is transformed directly into micelles, so that the surfactant concentration in solution remains constant at the level of cmc. 

Following this, the thermodynamic arguments needed for determining CMC are discussed (Section 8.5). Here, we describe two approaches, namely, the mass action model (based on treating micellization as a chemical reaction ) and the phase equilibrium model (which treats micellization as a phase separation phenomenon). The entropy change due to micellization and the concept of hydrophobic effect are also described, along with the definition of thermodynamic standard states. 

The Mass Action Model The mass action model represents a very different approach to the interpretation of the thermodynamic properties of a surfactant solution than does the pseudo-phase model presented in the previous section. A chemical equilibrium is assumed to exist between the monomer and the micelle. For this reaction an equilibrium constant can be written to relate the activity (concentrations) of monomer and micelle present. The most comprehensive treatment of this process is due to Burchfield and Woolley.22 We will now describe the procedure followed, although we will not attempt to fill in all the steps of the derivation. The aggregation of an anionic surfactant MA is approximated by a simple equilibrium in which the monomeric anion and cation combine to form one aggregate species (micelle) having an aggregation number n, with a fraction of bound counterions, f3. The reaction isdd... 

Several models have been developed to interpret micellar behavior (Mukerjee, 1967 Lieberman et al., 1996). Two models, the mass-action and phase-separation models are described here in mor detail. In the mass-action model, micelles are in equilibrium with the unassociated surfactant or monomer. For nonionic surfactants with an aggregation numb itbfe mass-action model predicts thatn molecules of monomeric nonionized surfactaStajeact to form a micelleM ... 

Summarizing the statements of these three most commonly used models, it appears that the so-called mass action and phase-separation models simulate a third condition which must be fulfilled with respect to the formation of micelles a size limiting process. The latter is independent of the cooperativity and has to be interpreted by a molecular model. The limitation of the aggregate size in the mass action model is determined by the aggregation number. This is, essentially, the reason that this model has been preferred in the description of micelle forming systems. The multiple equilibrium model as comprised by the Eqs. (10—13) contains no such size limiting features. An improvement in this respect requires a functional relationship between the equilibrium constants and the association number n, i.e.,... 

The physical properties of the micelle, e.g. average aggregation number, shape and formation constant, can be deduced from the chemical-shift data from the mass action model.48 Using the mass action law model for micelle formation, the... 

In the treatment by the mass action model, micellization is considered as an association-dissociation equilibrium of individual molecules or ions with micelles in the concentration range above CMC. 

This two-state model, which assumes that surfactant exists either as monomers or micelles, is almost certainly an oversimplification. The mass action model assumes an equilibrium between monomers, n-mers and micelles, with the proviso that the bulk of the surfactant is present as monomers or micelles. In other words micelliza-tion is considered to occur over a narrow range of surfactant concentration, at least for aqueous micelles [1,2,23]. The thermodynamics of micellization have been discussed in terms of the hydrophobic interactions and the electrostatic interactions of the head groups, and, for ionic micelles, of the counterions with the ionic head groups [18,22,24],... 

Aggregation of surfactants in apolar solvents, e.g., aliphatic or aromatic hydrocarbons, occurs provided that small amounts of water are present [1,126,127], These aggregates are often called reverse micelles, although the solutions do not always appear to have a critical micelle concentration, and surfactant association is often governed by a multiple equilibrium, mass action, model vith a large spread of aggregate sizes [130,131], It has recently been suggested that the existence of a monomer f -mer equilibrium should be used as a criterion of micellization, and that this term should not be applied to self-associated systems which involve multiple equilibria [132],... 

The micelle has too small an aggregation number to be considered as a phase in the usual sense, and yet normally contains too many surfactant molecules to be considered as a chemical species. It is this dichotomy that makes an exact theory of solubilization by micelles difficult. The primary theoretical approaches to the problem are based on either a pseudophase model, mass action model, multiple equilibrium model, or the thermodynamics of small systems [191-196]. Technically, bulk thermodynamics should not apply to solute partitioning into small aggregates, since these solvents are interfacial phases with large surface-to-volume ratios. In contrast to a bulk phase, whose properties are invariant with position, the properties of small aggregates are expected to vary with distance from the interface [195]. The lattice model of solute partitioning concludes that virtually all types of solutes should favor the interface over the interior of a spherical micelle. While for cylindrical micelles, the internal distribution of solutes... 

There are two basic approaches to modeling the thermodynamics of micelle formation. The mass action model views the micelles as reversible complexes of the monomer that are aggregating and predicts the sharp change in tendency of incremental surfactant to form micelles instead of monomer at the CMC this sharp transition is a consequence of the relatively large number of molecules forming the aggregate. The mass action model predicts that micelles are present below the CMC but at very low concentrations. The ocher major model used to describe micelle formation is the pseudophase separation model, which views micelles as a separate thermodynamic phase in equilibrium with monomer. Because micelle formation is a second-order phase transition, micelles are not a true thermodynamic phase, and this model is an approximation. However, the assumption that there are no mi-celies present below the CMC, and that the surfactant activity becomes constant above the CMC. is close to reality. and the mathematical simplicity of the pseudophase... 

As mentioned earlier, studies of simple linear surfactants in a solvent (i.e, those without any third component) allow one to examine the sufficiency of coarse-grained lattice models for predicting the aggregation behavior of micelles and to examine the limits of applicability of analytical lattice approximations such as quasi-chemical theory or self-consistent field theory (in the case of polymers). The results available from the simulations for the structure and shapes of micelles, the polydispersity, and the cmc show that the lattice approach can be used reliably to obtain such information qualitatively as well as quantitatively. The results are generally consistent with what one would expect from mass-action models and other theoretical techniques as well as from experiments. For example. Desplat and Care [31] report micellization results (the cmc and micellar size) for the surfactant h ti (for a temperature of = ksT/tts = /(-ts = 1-18 and... 

In the literature on micelle formation two primary models have gained general acceptance as useful (although not necessarily accurate) models for understanding the energetics of the process of self-association. The two approaches are the mass-action model, in which the micelles and monomeric species are considered to be in a kind of chemical equilibrium... 

The alternative approach to modeling micelle formation is to think in terms of a phase separation model in which, at the cmc, the concentration of the free surfactant molecules becomes constant (like a solubihty limit or Ksp), and all additional molecules go into the formation of micelles. Analysis of the two approaches produces the same general result in terms of the energetics of micelle formation (with some slight differences in detail), so that the choice of model is really a matter of preference and circumstances. There is evidence that the activity of free surfactant molecules does increase above the cmc, which tends to support the mass-action model however, for most purposes, that detail is of little consequence. 

Experimental data indicate that the changes in rates of variation of physical properties near the CMC actually occur over a narrow range of concentrations and not discontinuously at a single concentration. Hraice a chemical reaction or mass action model should be more realistic than the phase separation model for describing the thermodynamics of micellization. That is, we consider that micelle formation occurs as follows ... 

S.2.2.3 Thermodynamic Parameters The CMC value dependence on temperature is used for the determination of thermodynamic parameters applying models [60]. The mass action model, apparent and partial model, and phase separation model [14,59,60] are apphed to estimate the thermodynamic parameters Gibbs energy, enthalpy, and entropy of micelle formation. The enthalpy and entropy change for the miceUization can be determined using the Gibbs-Helmholtz equation [55]. 

The mass action model allows a simple extension to be made to the case of ionic surfactants, in which micelles attract a substantial proportion of counter ions, into an attached layer. For a micelle made of n-surfactant ions, (where n — p) charges are associated with counter ions, i.e. having a net charge of p units and degree of dissociation pjn, the following equilibrium may be established (for an anionic surfactant with Na+ counter ions),... 

An alternative general approach (20) is the mass action model where the transfer of the solute in the micelle is treated as a binding phenomenon through the equilibrium process ... 

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