Surfactant chemical potential

Assume diat die chemical potential, p., of surfactant in aggregates of size N in equilibrium widi one anodier is unifonii. One may dierefore write... 

The standard chemical potentials are approximately tire same if tire surfactant in each aggregate sees nearly tire same interaction witli tire solvent. This simplifying assumption tlien gives... 

Beyond the CMC, surfactants which are added to the solution thus form micelles which are in equilibrium with the free surfactants. This explains why Xi and level off at that concentration. Note that even though it is called critical, the CMC is not related to a phase transition. Therefore, it is not defined unambiguously. In the simulations, some authors identify it with the concentration where more than half of the surfactants are assembled into aggregates [114] others determine the intersection point of linear fits to the low concentration and the high concentration regime, either plotting the free surfactant concentration vs the total surfactant concentration [115], or plotting the surfactant chemical potential vs ln( ) [119]. 

The function /[0(r)] has three minima by construction and guarantees three-phase coexistence of the oil-rich phase, water-rich phase, and microemulsion. The minima for oil-rich and water-rich phases are of equal depth, which makes the system symmetric, therefore fi is zero. Varying the parameter /o makes the microemulsion more or less stable with respect to the other two bulk uniform phases. Thus /o is related to the chemical potential of the surfactant. The constant g2 depends on go /o and is chosen in such a way that the correlation function G r) = (0(r)0(O)) decays monotonically in the oil-rich and water-rich phases [12,13]. This is the case when gi > 4y/l +/o - go- Here we take, arbitrarily, gj = 4y l +/o - go + 0.01. 

Electrochemical redox studies of electroactive species solubilized in the water core of reverse microemulsions of water, toluene, cosurfactant, and AOT [28,29] have illustrated a percolation phenomenon in faradaic electron transfer. This phenomenon was observed when the cosurfactant used was acrylamide or other primary amide [28,30]. The oxidation or reduction chemistry appeared to switch on when cosurfactant chemical potential was raised above a certain threshold value. This switching phenomenon was later confirmed to coincide with percolation in electrical conductivity [31], as suggested by earlier work from the group of Francoise Candau [32]. The explanations for this amide-cosurfactant-induced percolation center around increases in interfacial flexibility [32] and increased disorder in surfactant chain packing [33]. These increases in flexibility and disorder appear to lead to increased interdroplet attraction, coalescence, and cluster formation. 

Another example of chemical-potential-driven percolation is in the recent report on the use of simple poly(oxyethylene)alkyl ethers, C, ), as cosurfactants in reverse water, alkane, and AOT microemulsions [27]. While studying temperature-driven percolation, Nazario et al. also examined the effects of added C, ) as cosurfactants, and found that these cosurfactants decreased the temperature threshold for percolation. Based on these collective observations one can conclude that linear alcohols as cosurfactants tend to stiffen the surfactant interface, and that amides and poly(oxyethylene) alkyl ethers as cosurfactants tend to make this interface more flexible and enhance clustering, leading to more facile percolation. 

Here scalar order parameter, has the interpretation of a normalized difference between the oil and water concentrations go is the strength of surfactant and /o is the parameter describing the stability of the microemulsion and is proportional to the chemical potential of the surfactant. The constant go is solely responsible for the creation of internal surfaces in the model. The microemulsion or the lamellar phase forms only when go is negative. The function/(<))) is the bulk free energy and describes the coexistence of the pure water phase (4> = —1), pure oil phase (4> = 1), and microemulsion (< ) = 0), provided that/o = 0 (in the mean-held approximation). One can easily calculate the correlation function (4>(r)(0)) — (4>(r) (4>(0)) in various bulk homogeneous phases. In the microemulsion this function oscillates, indicating local correlations between water-rich and oil-rich domains. In the pure water or oil phases it should decay monotonically to zero. This does occur, provided that g2 > 4 /TT/o — go- Because of the < ), —<(> (oil-water) symmetry of the model, the interface between the oil-rich and water-rich domains is given by... 

For obvious reasons, we need to introduce surface contributions in the thermodynamic framework. Typically, in interface thermodynamics, the area in the system, e.g. the area of an air-water interface, is a state variable that can be adjusted by the observer while keeping the intensive variables (such as the temperature, pressure and chemical potentials) fixed. The unique feature in selfassembling systems is that the observer cannot adjust the area of a membrane in the same way, unless the membrane is put in a frame. Systems that have self-assembly characteristics are conveniently handled in a setting of thermodynamics of small systems, developed by Hill [12], and applied to surfactant self-assembly by Hall and Pethica [13]. In this approach, it is not necessary to make assumptions about the structure of the aggregates in order to define exactly the equilibrium conditions. However, for the present purpose, it is convenient to take the bilayer as an example. 

Application of Activity at cmc. The above consideration suggested us to propose a new treatment for ionic micelle formation. According to thermodynamics, the micelle-monomer equilibrium is achieved when the chemical potential of surfactant in the micelle is equal to that in the bulk solution. The free energy of micelle formation can be represented by the use of the critical micelle activity, cma, which is the activity of surfactant at the cmc, as... 

This transition may j-.e. reducing the specific surface energy, f. The reduction of f to sufficiently small values was accounted for by Ruckenstein (15) in terms of the so called dilution effect". Accumulation of surfactant and cosurfactant at the interface not only causes significant reduction in the interfacial tension, but also results in reduction of the chemical potential of surfactant and cosurfactant in bulk solution. The latter reduction may exceed the positive free energy caused by the total interfacial tension and hence the overall Ag of the system may become negative. Further analysis by Ruckenstein and Krishnan (16) have showed that micelle formation encountered with water soluble surfactants reduces the dilution effect as a result of the association of the the surfactants molecules. However, if a cosurfactant is added, it can reduce the interfacial tension by further adsorption and introduces a dilution effect. The treatment of Ruckenstein and Krishnan (16) also highlighted the role of interfacial tension in the formation of microemulsions. When the contribution of surfactant and cosurfactant adsorption is taken into account, the entropy of the drops becomes negligible and the interfacial tension does not need to attain ultralow values before stable microemulsions form. 

We commence with the adsorption of nonionic surfactants, which does not require the consideration of the effect of the electrical double layer on adsorption. The equilibrium distribution of the surfactant molecules and the solvent between the bulk solution (b) and at the surface (s) is determined by the respective chemical potentials. The chemical potential /zf of each component i in the surface layer can be expressed in terms of partial molar fraction, xf, partial molar area a>i, and surface tension y by the Butler equation as [14]... 

The change in a, is caused by the change in bulk solute concentration. This is the Gibbs surface tension equation. Basically, these equations describe the fact that increasing the chemical potential of the adsorbing species reduces the energy required to produce new surface (i.e. y). This, of course, is the principal action of surfactants, which will be discussed in more detail in a later section. 

Equations (4.7) and (4.8) are both very useful when analysing surfactant aggregation behaviour and experimental cmc values. The difference in standard chemical potentials - Pi must contain within it the molecular forces and energetics of formation of micelles, which could be estimated from theory. These will be a function of the surfactant molecule and will determine the value of its cmc. 

Using this approach, a model can be developed by considering the chemical potentials of the individual surfactant components. Here, we consider only the region where the adsorbed monolayer is "saturated" with surfactant (for example, at or above the cmc) and where no "bulk-like" water is present at the interface. Under these conditions the sum of the surface mole fractions of surfactant is assumed to equal unity. This approach diverges from standard treatments of adsorption at interfaces (see ref 28) in that the solvent is not explicitly Included in the treatment. While the "residual" solvent at the interface can clearly effect the surface free energy of the system, we now consider these effects to be accounted for in the standard chemical potentials at the surface and in the nonideal net interaction parameter in the mixed pseudo-phase. 

With these considerations in mind, the chemical potential of the ith free monomeric surfactant component in solution is given by... 

The micellization of the surfactant in an aqueous solution can be regarded as a phase separation process (4,12). At equilibrium, the chemical potential of surfactant component i,/tj, is equal to that in the micelle... 

For the ionic surfactants (1-1 type), we should take account of the electrically charged species and the possibility of doing electrical work. The micelle may be regarded as a charged pseudo-phase, and the chemical potential is replaced by the electrochemical potential (12). The effective electrical work in micelle formation is... 

Adsorption of a Single Surfactant. We denote a quantity valid for surfactant i at its critical micelle concentration (cmc) in a solution of only surfactant i in water by the superscript c(i). Thus, the chemical potential of the surfactant at the cmc in the equilibrium solution or in the surface phase at the onset of micellization in the solution is given by... 

We are interested in the behavior of surfactant molecules in the mixed adsorbed film. The nonideal behavior of a mixed adsorbed film is correlated to activity coefficients of surface-active components with reference to the pure adsorbed film of each component. In the same manner as the previous paper ( ), we can express the chemical potentials of 1-octadecanol and dodecylammonium chloride in the mixed adsorbed film as follows ... 

It is apparent that CMC values can be expressed in a variety of different concentration units. The measured value of cCMC and hence of AG c for a particular system depends on the units chosen, so some uniformity must be established. The issue is ultimately a question of defining the standard state to which the superscript on AG C refers. When mole fractions are used for concentrations, AG c directly measures the free energy difference per mole between surfactant molecules in micelles and in water. To see how this comes about, it is instructive to examine Reaction (A) —this focuses attention on the surfactant and ignores bound counterions — from the point of view of a phase equilibrium. The thermodynamic criterion for a phase equilibrium is that the chemical potential of the surfactant (subscript 5) be the same in the micelle (superscript mic) and in water (superscript W) n = n. In general, pt, = + RTIn ah in which... 

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