Tension Thermodynamic Approach

As we have seen, the interfadal region is, in fact, three dimensional (i.e., it has a finite thickness). It is very convenient, however, to represent an interface as a mathematical surface of zero thickness because such properties as area and curvature are well defined and because the differential geometry of surfaces is well understood. How can a thermodynamic analysis be developed that reconciles the use of mathematical surfaces with die actual three-dimensional character of the interface  

FIGURE 1.2 Inteifadal region bounded by parallel surfaces. S is the reference surface. 

Similarly, surface excess values of other thermodynamic properties can be defined, as can the surface excess number of moles nf of species i  

Here n, represents the actual number of moles of i in the region between and Sg, represents the moles of i that would be present in the region between 

It is clear from Equations 1.1 and 1.2 that surface excess quantities do take into account the variation of composition and propalies across an interfacial region of finite thickness. As we shall see shortly, they can be used to define interfacial tension. Moreover, since all surface excess properties are assigned to the reference surface S, the area and curvature of S can be identified as the corresponding properties of the interface and used, for example, to describe interfacial deformation. 

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The deviations from the Szyszkowski-Langmuir adsorption theory have led to the proposal of a munber of models for the equihbrium adsorption of surfactants at the gas-Uquid interface. The aim of this paper is to critically analyze the theories and assess their applicabihty to the adsorption of both ionic and nonionic surfactants at the gas-hquid interface. The thermodynamic approach of Butler [14] and the Lucassen-Reynders dividing surface [15] will be used to describe the adsorption layer state and adsorption isotherm as a function of partial molecular area for adsorbed nonionic surfactants. The traditional approach with the Gibbs dividing surface and Gibbs adsorption isotherm, and the Gouy-Chapman electrical double layer electrostatics will be used to describe the adsorption of ionic surfactants and ionic-nonionic surfactant mixtures. The fimdamental modeling of the adsorption processes and the molecular interactions in the adsorption layers will be developed to predict the parameters of the proposed models and improve the adsorption models for ionic surfactants. Finally, experimental data for surface tension will be used to validate the proposed adsorption models. 

Equation (42) provides a thermodynamically valid way to determine y for an interface involving a solid. The thermodynamic approach makes it clear that curvature has an effect on activity for any curved surface. The surface free energy interpretation of y is more plausible for solids than the surface tension interpretation, which is so useful for liquid surfaces. Either interpretation is valid in both cases, and there are situations in which both are useful. From solubility studies on a particle of known size, y5 can be determined by the method of Example 6.2. 

In this paper, a molecular thermodynamic approach is developed to predict the structural and compositional characteristics of microemulsions. The theory can be applied not only to oil-in-water and water-in-cil droplet-type microemulsions but also to bicontinuous microemulsions. This treatment constitutes an extension of our earlier approaches to micelles, mixed micelles, and solubilization but also takes into account the self-association of alcohol in the oil phase and the excluded-volume interactions among the droplets. Illustrative results are presented for an anionic surfactant (SDS) pentanol cyclohexane water NaCl system. Microstructur al features including the droplet radius, the thickness of the surfactant layer at the interface, the number of molecules of various species in a droplet, the size and composition dispersions of the droplets, and the distribution of the surfactant, oil, alcohol, and water molecules in the various microdomains are calculated. Further, the model allows the identification of the transition from a two-phase droplet-type microemulsion system to a three-phase microemulsion system involving a bicontinuous microemulsion. The persistence length of the bicontinuous microemulsion is also predicted by the model. Finally, the model permits the calculation of the interfacial tension between a microemulsion and the coexisting phase. 

We finally contrast the preceding de ivation with the quasi thermodynamic approach initiated by Tolman (8). In his work, the surface tension is assumed to be given by... 

The formation of a surfactant film around droplets facilitates the emulsification process and also tends to minimize the coalescence of droplets. Macroemulsion stability in terms of short and long range interactions has been discussed. For surfactant stabilized macroemulsions, the energy barrier obtained experimentally is very high, which prevents the occurrence of flocculation in primary minimum. Several mechanisms of microemulsion formation have been described. Based on thermodynamic approach to these systems, it has been shown that interfacial tension between oil and water of the order of 10- dynes/cm is needed for spontaneous formation of microemulsions. The distinction between the cosolubilized and microemulsion systems has been emphasized. 

To describe the process of an oil phase emulsifying into foam lamellae, one could adopt an equilibrium thermodynamic approach analogous to that employed to obtain S and E. Such an approach produces the result that emulsification is only favored (negative AG) for negative values of the interfacial tension. In an alternative approach, the tendency of an oil phase to become emulsified and imbibed into foam lamellae has been described through a simplified balance of forces by the lamella number, L (37). For foam lamellae flowing in porous media, it is predicted that oil will be drawn in and pinched off to produce emulsified drops inside foam lamellae when L > 1, where... 

The above kinetic equations, developed based on the thermodynamic approach of Gibbs (1928), Volmer and Weber (1926), and Becker and Doring (1935), belong to the so-called classical nucleation theories. They have been criticized for the use of surface energy (interfacial tension), cr, which is probably of little physical significance when applied to small molecular assemblies of the size of critical nucleus. 

The nucleation process has been discussed above in terms of the so-called classical theories stemming from the thermodynamic approach of Gibbs and Volmer, with the modifications of Becker, Doring and later workers. The main criticism of these theories is their dependence on the interfacial tension (surface energy), 7, e.g. in the Gibbs-Thomson equation, and this term is probably meaningless when applied to clusters of near critical nucleus size. 

According to van Oss et al. s surface thermodynamic approach, these non-electrostatic forces must be related to the acid-ba.se component of the interfacia) tension between materials I and 2. The exjHession for the AB term in Eq. 143) is (tnally (between two spheres of radius a) (13. p. 80) ... 

Equation 2.101 can be used for the calculation of interfacial tension as long as we know the local value of pressure P at the height z. In the frame of the local thermodynamic approach, we, now, assume that the aforementioned EoS model(s) can be used to provide with the local quantities p,(z). / (z), P (z), and V z) connected by the equation... 

The mechanical and thermodynamical approaches are two complementary routes for the description of interfaces and membranes of arbitrary shape. It is very important to find the connection between them that is, to establish relations between the thermodynamically defined tensions and moments, 7, B, and [Eq. (96)], and the mechanical tensors of stresses and moments, a and M. This was done in Ref. 208 by direct calculation of 8m/ in terms of purely mechanical quantities. The following results were obtained [208] ... 

Using surface and interfacial tension data for some members of the homologous series of cationic surfactants we want to demonstrate to suitability of the thermodynamic approach of competitive adsorption for the formation of adsorption layers at different water/fluid interfaces, including those to alkane vapor and liquid alkane. We will restrict ourselves here to hexane as the oil or vapor phase. The particular effects of the alkane chain length have been discussed for example in [9]. For oils different from alkanes less systematic data exist, however, a specific impact of the molecular structure can be expected and the molecular characteristics might be rather different from those we obtained for alkanes. 

An alternative interpretation of the electrochemical double layer comes from a more thermodynamic approach. As an initial point, considering the Gibbs adsorption equation proved useful. This equation originally describes the dependence of the surface tension on the two-dimensional surface concentration (the surface excess F) of adsorbed particles as well as on their chemical potential p. The equation can be extended by introducing an electric term which considers the potential dependence of the surface tension. The Gibbs adsorption equation in its complete form is as follows ... 

The surface tension of a liquid is a measure of its tendency to minimize its surface area. The models for surface tension found in the literature are built on thermodynamic approaches, and they relate surface tension to a number of other physical properties, or to combinations of them. However, the literature contains little concerning the effect of specific molecular features on surface tension and provides no method to calculate surface tension from molecular structure. A set of 146 values for surface tension at 30 °C was extracted from a paper by Jasper that reports values for more than 2200 pure compounds with diverse structures, often at several different temperatures and with an experimental error of approximately 0.10 dyn cm . The compounds were encoded using a variety of topological, geometric, and electronic descriptors. A model was developed for a combined set of alkanes, alkyl esters, and alkyl alcohols which utilized 10 descriptors and had s = 0.4 dyn cm (1.8% of the mean). This model was then used to predict the surface tensions for 20 compounds not used in model development. 

The validity of the thermodynamic approach has been a controversial subject. Conceptually, Wa is the free energy of adhesion between the liquid and the surface divided by the wetted area. Its unit is mN/m. The unit for the three surface tensions in the right hand side of Eq. (3.5) is also mN/m. Although the unit between them is the same, they are different fundamentally. In 1965, Gray [32] wrote fr is clear from these definitions that surface tension and surface-free energy are quite distinct quantities. Surface tension is a tensor, which act perpendicularly to a line in a surface. It is the quantity involved in contact angle equilibrium. Surface-free energy is a scalar quantity without directional properties and it is a property of an area of the surface. It is the quantity involved in thermodynamic properties of surfaces."... 

Good, van Oss, and Caudhury [208-210] generalized this approach to include three different surface tension components from Lifshitz-van der Waals (dispersion) and electron-donor/electron-acceptor polar interactions. They have tested this model on several materials to find these surface tension components [29, 138, 211, 212]. These approaches have recently been disputed on thermodynamic grounds [213] and based on experimental measurements [214, 215]. 

The derivations of the foregoing equations have been based on the principles of thermodynamics and the macroscopic concepts of density, surface tension, and radius of curvature. They may therefore cease to be appropriate as the mean radius of curvature approaches molecular dimensions. 

Now the relationship between the interfacial tension and the composition of the two phases in contact will be analysed thermodynamically by using the approach of J. W. Gibbs. 

R. Although expressions for this parameter exist, they are derived by a hybrid of molecular mechanical and thermodynamic arguments which are not at present known to be consistent as droplet size decreases (8). An analysis of the size limitation of the validity of these arguments has, to our knowledge, never been attempted. Here we evaluate these expressions and others which are thought to be only asymptotically correct. Ve conclude, from the consistency of these apparently independent approaches, that the surface of tension, and, therefore, the surface tension, can be defined with sufficient certainty in the size regime of the critical embryo of classical nucleation theory. 

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