Definition of the CMC

In addition, if this definition of the cmc is also incorporated into (4.3), then we obtain the result that for large values of N (e.g. 100)... 

Calculated concentrations, using (4.9), for the various components, surfactant monomers, counter-ions and micelles, for the case of CTAB micellization (with a cmc of 0.9mM), is shown in Figure 4.5. Clearly, the micelle concentration increases rapidly at the cmc, which explains the sharp transition in surfactant solution properties referred to earlier. It is also interesting to note that the law of mass action (in the form of equation 4.9) predicts an increase in counterion (Br ions) concentration and a decrease in free monomer concentration above the cmc. It has been proposed that for ionic surfactants, a useful definition of the cmc would be... 

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... 

This definition of the CMC originates from the qualitative analysis of experimental concentration dependencies of physico-chemical properties, and is not quite strict. Indeed, the limits of the concentration range corresponding to the CMC depend on the error limits of the applied experimental method and on its sensitivity to micellar concentration. For instance, the equivalent conductivity of aqueous solutions of ionic surfactants decreases drastically just above the CMC. Sometimes other properties of surfactant solutions, for example, the intensity... 

All the definitions of the CMC discussed above reflect a general feature of surfactant solutions, namely, a qualitative change in the concentration dependencies of their properties at the CMC. This means that the thermodynamic state of such systems must also differ from the state below the CMC and cannot be described by conventional theories proposed for non-micellar solutions. A brief review of the thermodynamics of micellisation is presented in the following section. 

Substituting (5.44) into (5.41) we obtain the multiplier (KiCtoti)" on the right-hand side. If Kicioti < 1 the whole multiplier is very small and if K Ctoti > 1 the whole multiplier is very large because U exceeds unity essentially. For the case K Ctou = 1, which can be used as another definition of the CMC (ctoti)cMc = 1/Ki. Equation (5.41) is transformed then into the form... 

Note that all these abovementioned expressions for the standard affinity and the standard thermodynamic functions of micellisation were based on the definition of the CMC as an inflection point in the dependence ai(ctoti) given by Eqs. (5.53) and (5.54). If we use other definitions of the CMC, the corresponding relations will be different. 

A simple example is one where the hydrophobic interactions result in < i for all > 1, but the packing constraints on the chains and heads result in a minimum energy for a finite value of N = M (i.e., m < for N Af). (Below, a curvature energy model is discussed this model can also be used to motivate, but not to calculate in detail, a study of micellar sizes and shapes.) If this minimum is deep n rises sharply compared to ksT around N = Af), the distribution of micelles will be nearly monodisperse. In this approximation, one can consider monomers and micelles of aggregation number Af only. At small values of 4>s (or equivalently, at small values of fi). Pi Pm (Af > 1) almost all the surfactant exists as monomers and the number of micelles is exponentially small. The requirement that all amphiphiles have the same chemical potential in equilibrium, Eq. (8.2) and the definition of the CMC (where Pi = Pi, Pm = Pmc ft = = c) allows us to calculate... 

For example, it is difficult to discuss the results of the studies cited in the previous section in a collective manner, because each study (or each group of studies) considered different types of surfactants and oils, different energy parameters, different numbers of neighbors, or different definitions of the cmc or what constitutes an aggregate. Therefore, we restrict ourselves here to a general overview of the results from the literature. 

A consensus definition of a drug-like molecule has been derived from the analysis of the CMC database by defining qualifying (covering 80% of drug molecules) ranges of calculated physical properties such as molecular weight, log P, molar refractivity, and number of atoms ... 

The results of the CMC determinations are given in Table I which also includes the data of Miyamoto (10). As is generally found, the CMC values obtained from the k vs. N plots are slightly, but definitely, higher than those from the A- N plots which show satisfactory agreement with the y - In M derived values at the two temperatures where comparison is possible. Furthermore, agreement with the values of Miyamoto obtained by surface tension and dye solubilization is satisfactory for CaDS but not so for MgDS. 

In terms of micellar models, the cmc value has a precise definition in the pseudo-phase separation model, in which the micelles are treated as a separate phase. The cmc value is defined, in terms of the pseudo-phase model, as the concentration of maximnm solubility of the monomer in that particular solvent. The pseudo-phase model has a number of shortcomings however, the concept of the cmc value, as it is described in terms of this model, is very useful when discussing the association of surfactants into micelles. It is for this reason that the cmc value is, perhaps, the most frequently measnred and discussed micellar parameter [39]. 

There are also surfactants which display monomer h-mer association in nonpolar solvents [12,14] they often contain two hydrocarbon tails. Dialkyl sulfosucci-nates [15-22], dinonylnaphthalenesulfonates [23, 24] and phosphatidylcholines [25-31] are perhaps the most extensively studied. Aggregation numbers typically range from 12 to about 30, but dodecylammonium benzoate forms a trimer in a monomer h-mer association [32]. Even with these surfactants, the definition of a CMC is not without ambiguity. Thus in the case of Aerosol OT (sodium di(2-ethylhexyl) sulfosuccinate, AOT) [33-36], a premicellar linear trimer which undergoes a conformational transition to a closed trimeric unit has been implicated. 

As shown by the above review of the common methods used for the determination of the CMC, a series of measurements of some property of the surfactant solution as a function of surfactant concentration is usually involved. This is followed by the detection of some characteristic point - which is called the CMC. Methodical differences may originate from the choice of the characteristic point, the kind of plot on which this point is chosen, the kind of data which are plotted and the effect of the dye. The CMC is not a very sharply defined point above which properties are qualitatively different from those below. In fact, all properties of a solution in the CMC region vary in a continuous manner and so do all of their derivatives. A micelle is by definition a reversible aggregate of a large but not infinite number of monomers. The micelle formation reaction must obey the laws of chemical equilibrium and, as such, the concentration dependence of the degree of micellization has to change gradually. Consequently, all properties of the... 

Because the aim of this section is to show that the definition of the MTR given above can be used to explain observed deviations in the MTR, the following calculation for estimating the CMT of a surfactant is adequate. From Eq. (4.38), the relation between the CMC and counterion concentration for monodisperse micelle formation is... 

While CMC is assumed to be an observable and definite value in the case of surfactant monomers, there are frequent reports in the literature of the formation of aggregates or micelle-like associations in solutions of organic solutes so dilute as to preclude apparently the formation of micelles [208, 267-269, 272, 275,278]. Work with different types of commercial surfactants has indicated that molecularly non-homogeneous surfactants do not display the sharp inflection in surface tension associated with CMC in molecularly homogeneous monomers, but rather the onset of aggregation is broad and indistinct [253,267,268]. The lack of well-defined CMCs for non-homogeneous surfactants is speculated to result from the successive micellization of the heterogeneous monomers at different stoichiometric concentrations of the surfactant, which results in a breadth of the monomeric-micelle transition zone. 

Table 6.1 Theoretical estimation of the types of structures of protein self-assembly induced by interactions of food proteins (0.5 % w/v) with food-grade surfactants in aqueous solution (pH = 7.2, ionic strength = 0.05 M, 293 K) below the cmc (CITREM cmc = 15 mg/L SSL cmc = 3.5 mg/L PGE cmc = 1.0 mg/L (IFin el al., 2005)). See the text for definitions of k and k2.

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