The Pseudo-Phase Model

Two principal approaches have been used to describe the thermodynamics of surfactant solutions — the pseudo-phase model and the mass action model. 

We assume that the micelle is a pure surfactant phase with a chemical potential given by fi%. If the chemical potential of the monomer is /i2, then the condition of phase equilibrium is given by 

But //2 is related to the activity a2 of the monomer in the aqueous phase by the equation 

In the pseudo-phase model, ideal solutions are assumed2 so that 72 = 1 and 02 = am. (18.71) 

By using equation (18.72), equations can be obtained for relating the other thermodynamic properties to m. The total Gibbs free energy G of the surfactant solution is the sum of the contributions from the solvent, the monomer, and the micelle. That is, 

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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... 

Figure 18.14 Comparison of the stoichiometric molality m and the molality of the monomer m2 in a surfactant solution, according to the pseudo-phase model.

The pseudo-phase model is an idealization of results that are obtained for real systems. Figure 18.16, for example, shows V and Cp for sodium dodecylsulfate21 graphed as a function of 1/m. The circles represent the experimental results. Note that a sharp break is not obtained at 1/m = 1/CMC, but the intersection of the lines extrapolated from before and after the CMC gives a value of 1/CMC = 120 or CMC = 0.0083bb mol-kg-1. 

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... 

For a more detailed discussion of the pseudo-phase model, see J. E. Desnoyers and G. Perron, Thermodynamic methods , Chapter 1 in Surfactant Solutions New Methods of Investigation, R. Zana, Editor, Marcel Dekker, Inc., New York, 1987. 

The aminolysis and methanolysis of ionized phenyl salicylate (189) have been examined under micellar conditions. The effect of CTABr on the rates of aminolysis of (189) by -butylamine, piperidine, and pyrrolidine is to bring about a rate decrease (up to 17-fold with pyrrolidine). The results are interpreted in terms of binding constants for the amines with CTABr and the pseudo-phase model.The effects of mixed surfactants SDS and CTABr on the methanolysis of (189) and the alkaline hydrolysis of phenyl benzoate suggest that micellar aggregates are involved in the processes.The effects of NaOH and KBr on the intramolecular general base-catalysed methanolysis of (189) in the presence of CTABr has been investigated. Pseudo-first-order rate constants were not affected by either additive but other changes were noted. The effect of mixed MeCN-water solvents on the same reaction has also been probed. 

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]. 

The surfactants used are 1-1 electrolytes, so that, below the cmc, the monomer concentration is equal to the counterion concentration. Above the cmc, any additional surfactant was considered to be incorporated to the micelles, as required by the pseudo-phase model of micelle formation. Experimentally, however, the monomer concentration is known to decrease somewhat with surfactant concentration above the cmc. In this treatment it is assumed that as micelles are formed, the... 

At the level of approximation of the pseudo-phase model, the surfactant tails are viewed as an independent pseudo-phase in which an uniform effective concentration of solute is assumed. In the surfactant monolayer model, the water/oil interface is supposed to be covered by a monolayer of surfactant molecules into which the solute can be adsorbed. In this latter case, the mole fraction of solute at the interface is defined by the interfacial composition k (37), with the assumption that no water and no oil is adsorbed in the surfactant film, given as follows ... 

Figure 9.9. Schematic representation of the different localization states of a hydrophobic solute in a microemulsion. The solute could be solubilized in the oil microdomain or at the interface. The interfacial surfactant area is noted as a and the curvature radius as R. This schematic represents the case of a nonionic surfactant where the hydration of the polar head is temperature (r)-dependant. With the pseudo-phase model, the solute concentration is considered over the volume occupied by the tails of the surfactant, whereas in the surfactant monolayer model, the binding of the solute into the surfactant monolayer is considered...

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