Micellization phase equilibrium model

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. 

In summary, whether a reaction equilibrium or a phase equilibrium approach is adopted depends on the size of the micelles formed. In aqueous systems the phase equilibrium model is generally used. In Section 8.5 we see that thermodynamic analyses based on either model merge as n increases. Since a degree of approximation is introduced by using the phase equilibrium model to describe micellization, micelles are sometimes called pseudophases. 

An illustration of how both the reaction equilibrium and phase equilibrium models can be applied to micellization is provided by Example 8.1. 

EXAMPLE 8.1 Reaction Equilibrium and Phase Equilibrium Models of Micellization. Research in which the CMC of an ionic surfactant M+ S is studied as a function of added salt, say M+X, ... 

Under what conditions can the formation of micelles be described in terms of reaction-equilibrium formalism When is the phase equilibrium model appropriate ... 

Reaction equilibrium and phase equilibrium models of micellization 361... 

Blandamer Ml, Cullis PM, Soldi LG et al (1995) Thermodynamics of micellar systems comparison of mass action and phase equilibrium models for the calculation of standard Gibbs energies of micelle formation. Adv Colloid Interface Sci 58 171-209. doi I0.I0I6/... 

Taking Simultaneous Micellizadon and Adsorption Phenomena into Consideration In the presence of an adsorbent in contact with the surfactant solution, monomers of each species will be adsorbed at the solid/ liquid interface until the dual monomer/micelle, monomer/adsorbed-phase equilibrium is reached. A simplified model for calculating these equilibria has been built for the pseudo-binary systems investigated, based on the RST theory and the following assumptions ... 

The equilibrium in these systems above the cloud point then involves monomer-micelle equilibrium in the dilute phase and monomer in the dilute phase in equilibrium with the coacervate phase. Prediction o-f the distribution of surfactant component between phases involves modeling of both of these equilibrium processes (98). It should be kept in mind that the region under discussion here involves only a small fraction of the total phase space in the nonionic surfactant—water system (105). Other compositions may involve more than two equilibrium phases, liquid crystals, or other structures. As the temperature or surfactant composition or concentration is varied, these regions may be encroached upon, something that the surfactant technologist must be wary of when working with nonionic surfactant systems. 

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 ... 

In the phase separation model of micelle formation (cf. Sect. 3.1) it is also possible to include the counterions specifically. One has made the distinction between the uncharged phase and the charged micellar pseudophase295). These models can, for example, be used to predict how the CMC varies with salt concentration46, but as used they are open to the same kind of criticism as is the equilibrium model. 

The phase separation model follows exactly the description of a two-phase equilibrium, i.e., equating the respective chemical potentials of the particular surfactant in both phases (i.e., monomers in the nonpolar solvent and the micelles) at the critical concentration (CMC). Thus, (assuming ideal condition)... 

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.,... 

Various approaches have been employed to tackle the problem of micelle formation. The most simple approach treats micelles as a single phase, and this is referred to as the phase-separation model. In this model, micelle formation is considered as a phase-separation phenomenon, and the cmc is then taken as the saturation concentration of the amphiphile in the monomeric state, whereas the micelles constitute the separated pseudophase. Above the cmc, a phase equilibrium exists with a constant activity of the surfactant in the micellar phase. The Krafft point is viewed as the temperature at which a solid-hydrated surfactant, the micelles, and a solution saturated with undissociated surfactant molecules are in equiUbrium at a given pressure. 

The simplest thermodynamic model for solubilization is the pseudophase or phase separation model. The micelles are treated as a separate phase consisting of surfactant and the solubilized molecules. Solubilization is regarded as a simple distribution or equilibrium of the solute between the aqueous and the miceUar phases, i.e.. 

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... 

The pseudophase equilibrium model considers solutes to be dissolved in continuous and discrete (micelles or droplets) phases in a way similar to the partition in conventional two-phase systems of oil and water [3, 4j. The equilibrium distribution constants are... 

If co-crystals are to solve solubility problems one must assess their true or thermodynamic solubility so that development strategies are guided by the fundamental properties of co-crystals. Measuring the solubility of co-crystals that generate supersaturation of the parent drug is often experimentally impossible due to conversion. Eutectic points, described in Section 11.4, provide a measure of co-crystal solubility under thermodynamic equilibrium conditions. The solution at the eutectic point is saturated with co-crystal and solution concentrations represent experimentally accessible thermodynamic solubility values. Once co-crystal solubility is determined at the eutectic, the solubility under different solution conditions (pH, co-former, micelle concentration) can be obtained from solubility models that consider the appropriate solution phase equilibrium expressions. 

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). 

Two main approaches to the thermodynamic analysis of the micellization process have gained wide acceptance. In the phase separation approach the micelles are considered to form a separate phase at the CMC, whilst in the mass-action approach micelles and unassociated monomers are considered to be in association-dissociation equilibrium. In both of these treatments the micellization phenomenon is described in terms o.f the classical system of thermodynamics. Theories of micelle formation based on statistical mechanics have also been proposed [16Q-162] but will not be considered further. The application of the mass-action and phase-separation models to both ionic and non-ionic micellar systems will be briefly outlined and their limitations discussed. More recent developments in this field will be presented. 

Experimentally determined rate constants for various micellar-mediated reactions show either a monotonic decrease (i.e., micellar rate inhibition) or increase (i.e., micellar rate acceleration) with increase in [Suifl CMC, where [Surf]T represents total micelle-forming surfactant concentration (Figure 3.1). Menger and Portnoy obtained rate constants — [SurfJx plots — for hydrolysis of a few esters in the presence of anionic and cationic surfactants, which are almost similar to those plots shown in Figure 3.1. These authors explained their observations in terms of a proposed reaction mechanism as shown in Scheme 3.1 which is now called Menger s phase-separation model, enzyme-kinetic-type model, or preequilibrium kinetic (PEK) model for micellar-mediated reactions. In Scheme 3.1, Kj is the equilibrium... 

In the literature on micelle formation, two primaiy models have gained general acceptance as useful (although not necessarily accurate) for understanding the energetic basis of the process. 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, and the phase separation model, in which the micelles are considered to constitute a new phase formed in the system at and above the critical micelle concentration. In each case, classical thermodynamic approaches are used to describe the overall process of micellization. 

Rahaman and Hatton [152] developed a thermodynamic model for the prediction of the sizes of the protein filled and unfilled RMs as a function of system parameters such as ionic strength, protein charge, and size, Wq and protein concentration for both phase transfer and injection techniques. The important assumptions considered include (i) reverse micellar population is bidisperse, (ii) charge distribution is uniform, (iii) electrostatic interactions within a micelle and between a protein and micellar interface are represented by nonlinear Poisson-Boltzmann equation, (iv) the equilibrium micellar radii are assumed to be those that minimize the system free energy, and (v) water transferred between the two phases is too small to change chemical potential. 

The Pseudo-Phase Model Consider a process in which surfactant is added to water that is acting as a solvent. Initially the surfactant dissolves as monomer species, either as molecules for a non-ionic surfactant or as monomeric ions for an ionic surfactant. When the concentration of surfactant reaches the CMC, a micelle separates from solution. In the pseudo-phase model,20 the assumption is made that this micelle is a separate pure phase that is in equilibrium with the dissolved monomeric surfactant. To maintain equilibrium, continued addition of surfactant causes the micellar phase to grow, with the concentration of the monomer staying constant at the CMC value. This relationship is shown in Figure 18.14 in which we plot m, the stoichiometric molality,y against mj, the molality of the monomer in the solution. Below the CMC, m = m2, while above the CMC, m2 = CMC and the fraction a of the surfactant present as monomer... 

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... 

Binary surfactant mixtures have traditionally been modeled using the pseudophase-separation approach, in which the micelles are treated as a separate, inLnite phase in equilibrium with the... 

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