Free energy interfacial curvature

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... 

The dominant contribution to the free energy of lengthy (rubbery) polymer chains is entropy. This is known to accoimt for rubber elasticity, which can be satisfactorily modelled by the entropy of the cross-linked pol3rmer chains alone. A simple illustrative model of copolymer self-assembly can be developed by extending rubber elasticity theory to include bending as well as stretching deformations, to calculate chain entropy as a function of interfacial curvatures in diblock aggregates. 

The interfacial free energy associated with the creation of the micellar core-water interface, as well as with the shielding part of that interface. This contribution is obtained from available hydrocarbon-water interfacial tension data, and the interfacial area per monomer. The effect of interfacial curvature on interfacial tension was obtained from the Tolman equation [23]. 

Qualitatively the thermodynamics of microemulsions is well understood as the interplay between a small interfacial free energy and a small entropy of mixing. However, because of these contributions being small, other small effects, such as the influence of curvature on the interfacial tension and the influence of fluctuations, become important, and quantitative understanding still leaves a lot to be desired. 

In Sec. II we discuss the mechanism by which the interfacial tension may become ultralow. After that, in Sec. Ill we mention curvature effects of the oil/water interface. Subsequently, a number of models for thermodynamic calculations are described (Sec. IV). In Secs. V-VII we discuss droplet-type microemulsions in some detail. Section Vdescribes a thermodynamic formalism that incorporates the interfacial free energy (as influenced by the curvature) and the free energy of mixing of droplets and continuous medium and ultimately leads to equations for the size distribution of microemulsion droplets. This size distribution is important because measurable properties can be calculated... 

From the above reasoning we expect that each composition of the interface has its own curvature at which the interface forms most easily and thus has the lowest interfacial tension (this interfacial tension, a, of the droplet interface should not be confused with that of the macroscopic interface, y). This consideration was made more quantitative by Helfrich [13], who presented an expression for the curvature free energy. 

A theoretical basis for different shapes of microemulsions (even for small W/O or O/W volume fractions) has been established on the basis of the relationship between shape and interfacial curvature [350,351]. It is reasonable to expect that the relevant properties of the surfactant film are represented by a bending elasticity with a spontaneous curvature, Co (as was demonstrated for binary systems). If the elastic modulii k, ksT, the fluctuations in curvature of the film are very small, and the entropy associated with them can be neglected. The actual morphology is the result of the competition between the tendency to minimize the bending free energy (which prefers spheres of optimal radius of curvature, = l/c ) and the necessity to use up all of the water, oil, and surfactant... 

The monomer consumption from the interfacial layer (cosurfactant effect, see Sec. II.B) modifies the film curvature energy. The formation of water-swollen spherical polymer particles dispersed in the oily phase corresponds to the minimum free energy of the system. 

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