Energy-stability-area theory

Roughly 60 years ago Derjaguin, Landau, Verwey, and Overbeek developed a theory to explain the aggregation of aqueous dispersions quantitatively [66,157,158], This theory is called DLVO theory. In DLVO theory, coagulation of dispersed particles is explained by the interplay between two forces the attractive van der Waals force and the repulsive electrostatic double-layer force. These forces are sometimes referred to as DLVO forces. Van der Waals forces promote coagulation while the double layer-force stabilizes dispersions. Taking into account both components we can approximate the energy per unit area between two infinitely extended solids which are separated by a gap x ... 

Potentially of equal importance is the relationship between strain and catalyst stability. A calculation of the contribution to the total free energy of a catalyst crystal caused by the presence of strain-inducing microscopic precipitates50 showed that the extra free energy increases with the size of the crystal and inhibits it from sintering. This theory is an interesting one since it provides a mechanism which the catalyst scientist can exploit in his search for stable, high surface-area materials. The theory predicts the equilibrium crystallite size of the iron crystals of an ammonia synthesis catalyst with acceptable accuracy. 

The effect of electrolyte concentration on the transition from common to Newton black films and the stability of both types of films are explained using a model in which the interaction energy for films with planar interfaces is obtained by adding to the classical DLVO forces the hydration force. The theory takes into account the reassociation of the charges of the interface with the counterions as the electrolyte concentration increases and their replacements by ion pairs. This affects both the double layer repulsion, because the charge on the interface is decreased, and the hydration repulsion, because the ion pair density is increased by increasing the ionic strength. The theory also accounts for the thermal fluctuations of the two interfaces. Each of the two interfaces is considered as formed of small planar surfaces with a Boltzmannian distribution of the interdistances across the liquid film. The area of the small planar surfaces is calculated on the basis of a harmonic approximation of the interaction potential. It is shown that the fluctuations decrease the stability of both kinds of black films. 

Upon mixing two immiscible liquids, one of the two liquids (i.e., the dispersed phase) is subdivided into smaller droplets. The surface area and the interfacial free energy increase, and the system is then thermodynamically unstable. Without continuous mixing, the droplets will be stabilized throughout the dispersion medium by dissolving the surface-active agent. There are several theories for the stabilization of emulsions but a single theory cannot account for the stabilization of all emulsions. 

In the interfacial tension theory, the adsorption of a surfactant lowers the interfacial tension between two liquids. A reduction in attractive forces of dispersed liquid for its own molecules lowers the interfacial free energy of the system and prevents the coalescence of the droplets or phase separation. Therefore the surfactant facilitates the stable emulsion system of the large interfacial area by breaking up the liquid into smaller droplets. However, the emulsions prepared with sodium dodecyl (lauryl) sulfate separate into two liquids upon standing even though the interfacial tension is reduced. The lowering of the interfacial tension in the stabilization of emulsions is not the only factor we should consider. 

The theory of topolo cal resonance energy (TR ) represents an important area for chemical applications of matching pdynomials. As this theory will be discussed in more detail elsewhere in this book, we offer here only few brief remarks. As already explained in 4.S.3.5, TRE is the measure of the effect of cyclic conjugation [100,101] on total x-electron energy. TRE therefore also measures the effect of cyclic conjugation on the thermodynamic stability of a conjugated molecule (within the Huckd molecular orbital approximation). This interpretation of TRE is beyond dispute. 

The first term on the right-hand side of equation (8.7) is the contribution of the head group repulsion, while the second is the interfacial energy contribution where Ahg is the total surface area of the head groups and (Tmic is the interfacial tension. Within the framework of the Gouy-Chapmann theory, the dressed micelle model allows the estimation of values, which are for sodium dodecyl sulfate (SDS), sodium octyl sulfate, and teradecyltrimethylammonium bromide, 15-16, 11 and 11-14 mN m , respectively (15). Note that these values are up to a factor of 3 lower than those of the pure monomers (cf. Table 8.2). A further decrease of or is possible in the case of emulsions of organic liquids where the interface is saturated with stabilizer. For example, a value of about 4 mN m was determined for a toluene emulsion stabilized with potassium lau-rate (16). 

The adsorption of polymers at the surface of emulsion droplets is often an important factor, with a great impact on their stability. The particular interaction energy per unit film area, however, depends on the solvent quality with respect to the polymer. Thus, for theta solvents, the steric interaction energy / (A) can be calculated using the theory of Dolan and Edwards ... 

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