Microemulsion free energy

Abraham, M. H Treiner, C., Roses, M., Rafols, C Ishihama, Y. Linear free energy relationship analysis of microemulsion electrokinetic chromatographic determination of lipophilidty. J. Chromatogr. A 1996, 752, 243-249. 

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... 

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. 

Any surfactant adsorption will lower the oil-water interfacial tension, but these calculations show that effective oil recovery depends on virtually eliminating y. That microemulsion formulations are pertinent to this may be seen by reexamining Figure 8.11. Whether we look at microemulsions from the emulsion or the micellar perspective, we conclude that the oil-water interfacial free energy must be very low in these systems. From the emulsion perspective, we are led to this conclusion from the spontaneous formation and stability of microemulsions. From a micellar point of view, a pseudophase is close to an embryo phase and, as such, has no meaningful y value. 

Surfactants form semiflexible elastic films at interfaces. In general, the Gibbs free energy of a surfactant film depends on its curvature. Here we are not talking about the indirect effect of the Laplace pressure but a real mechanical effect. In fact, the interfacial tension of most microemulsions is very small so that the Laplace pressure is low. Since the curvature plays such an important role, it is useful to introduce two parameters, the principal curvatures... 

Unlike micelles, an emulsion is a liquid system in which one liquid is dispersed in a second, immiscible liquid, usually in droplets, with emulsiLers added to stabilize the dispersed system. Conventional emulsions possess droplet diameters of more than 200 nm, and are therefore optically opaque or milky. Conventional emulsions are thermodynamically unstable, tending to reduce their total free energy by reducing the total area of the two-phase interface. In contrast, microemulsions with droplet diameters less than 100 nm are optically clear and thermodynamically stable. Unlike conventional emulsions that require the input of a substantial amount of energy, microemulsions are easy to prepare and form spontaneously on mixing, with little or no mechanical energy applied (Lawrence and Rees, 2000). 

If the interfacial tension between two liquids is reduced to a sufficiently low value on addition of a surfactant, emulsification will readily take place, because only a relatively small increase in the surface free energy of the system is involved. If tt y0, a microemulsion may form (see page 269). 

Several theories have been proposed to account for the thermodynamic stability of microemulsions. The most recent theories showed that the driving force for microemulsion formation is the ultralow interfacial tension (in the region of 10 4-10 2 mN m 1). This means that the energy required for formation of the interface (the large number of small droplets) A Ay is compensated by the entropy of dispersion —TAS, which means that the free energy of formation of microemulsions AG is zero or negative. 

Emulsions made by agitation of pure immiscible liquids are usually very unstable and break within a short time. Therefore, a surfactant, mostly termed emulsifier, is necessary for stabilisation. Emulsifiers reduce the interfacial tension and, hence, the total free energy of the interface between two immiscible phases. Furthermore, they initiate a steric or an electrostatic repulsion between the droplets and, thus, prevent coalescence. So-called macroemulsions are in general opaque and have a drop size > 400 nm. In specific cases, two immiscible liquids form transparent systems with submicroscopic droplets, and these are termed microemulsions. Generally speaking a microemulsion is formed when a micellar solution is in contact with hydrocarbon or another oil which is spontaneously solubilised. Then the micelles transform into microemulsion droplets which are thermodynamically stable and their typical size lies in the range of 5-50 nm. Furthermore bicontinuous microemulsions are also known and, sometimes, blue-white emulsions with an intermediate drop size are named miniemulsions. In certain cases they can have a quite uniform drop size distribution and only a small content of surfactant. An interesting application of this emulsion type is the encapsulation of active substances after a polymerisation step [25, 26]. 

A predictive molecular thermodynamics approach is developed for microemulsions, to determine their structural and compositional characteristics [3.7]. The theory is built upon a molecular level model for the free energy change. For illustrative purposes, numerical calculations are performed for the system water, cyclohexane, sodium dodecyl sulfate as surfactant, pentanol as cosurfactant and NaCl as electrolyte. The droplet radius, the thickness of the surfactant layer at the interface, the number of molecules of various species in the droplets, and the distribution of the components between droplets and the continuous phase are calculated. The theory also predicts the transition from a mi-... 

Single phase microemulsions are treated in the next section. Two general thermodynamic equations are derived from the condition that the free energy of the system should be a minimum with respect to both the radius r of the globules as well as the volume fraction of the dispersed phase. The first equation can be employed to calculate the radius while the second, a generalized Laplace equation, can be used to explain the instability of the spherical shape of the globules. The two and three phase systems are examined in Sections III and IV of the paper. 

For the sake of simplicity, it will be assumed that the microemu15ion contains spherical globules of a single size. Their dispersion in the continuous phase is accompanied by an increase in the entropy of the system and the corresponding free energy change per unit volume of microemulsion is denoted by if. The Helmholtz free energy f per unit volume of microemulsion is written as the sum... 

The mechanical equilibrium condition between the microemulsion and environment provides a second relationship between the micropressures P2 and pi. The variation of the total free energy F of the microemulsion of volume V can be written as ... 

Similar attempts were made by Likhtman et al. [13] and Reiss [14]. Reference 13 employed the ideal mixture expression for the entropy and Ref. 14 an expression derived previously by Reiss in his nucleation theory These authors added the interfacial free energy contribution to the entropic contribution. However, the free energy expressions of Refs. 13 and 14 do not provide a radius for which the free energy is minimum. An improved thermodynamic treatment was developed by Ruckenstein [15,16] and Overbeek [17] that included the chemical potentials in the expression of the free energy, since those potentials depend on the distribution of the surfactant and cosurfactant among the continuous, dispersed, and interfacial regions of the microemulsion. Ruckenstein and Krishnan [18] could explain, on the basis of the treatment in Refs. 15 and 16, the phase behavior of a three-component oil-water-nonionic surfactant system reported by Shinoda and Saito [19],... 

From the point of view of traditional thermodynamics, a microemulsion is a multicomponent mixture in thermodynamic equilibrium. The change at constant temperature of the Helmholtz free energy of such a system can therefore be written as... 

Equation (15), which is based on a model, must, however, be equivalent to Eq. (12), which is based on the traditional thermodynamics of a multicomponent mixture. For the free energy changes given by Eqs. (12) and (15) to be the same for arbitrary changes in the independent variables V, , r, and Nit the respective coefficients multiplying dV, d, dr, and dNt must be equal. It should be emphasized, however, that depends on the distribution at equilibrium of tire moles Ni of species i between the two media of the microemulsion and their interface,... 

The standard free energy difference A//W in eq 3.1 is due to the transfer of one surfactant molecule and (gAi/gsd alcohol molecules from water and (goi/gsi) oil molecules from pure oil to the interfacial layer of the microemulsion droplet. Expressions for A/ ° are provided in section 7 of the paper. 

The equations developed in previous sections can be used to calculate the structural features of microemulsions, provided explicit expressions for the standard free energies of transfer of surfactant and alcohol molecules from their infinitely dilute states in water and of oil molecules from the pure oil phase to the interfacial layer of the microemulsion droplets are available. Such expressions are given below for spherical layers of O/W droplets and W/O droplets and also for flat layers. The difference in the standard state free energy consists of a number of contributions ... 

For this problem already the simple mean field approximation becomes rather involved [197,213]. Therefore, we describe here only an approach, which is even more simplified, appropriate for wavenumbers q near the characteristic wavenumber q, but strictly correct neither for q—>0 nor for large q the spirit of our approach is similar to the long wavelength approximation encountered in the mean field theory of blends, Eq. (7). That is, we write the effective free energy functional as an expansion in powers of t t and include terms (Vv /)2 as well as (V2 /)2, as in the related problem of lamellar phases of microemulsions [232,233],namely [234]... 

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