Gibbs surface model

The purpose of this chapter is to introduce the effect of surfaces and interfaces on the thermodynamics of materials. While interface is a general term used for solid-solid, solid-liquid, liquid-liquid, solid-gas and liquid-gas boundaries, surface is the term normally used for the two latter types of phase boundary. The thermodynamic theory of interfaces between isotropic phases were first formulated by Gibbs [1], The treatment of such systems is based on the definition of an isotropic surface tension, cr, which is an excess surface stress per unit surface area. The Gibbs surface model for fluid surfaces is presented in Section 6.1 along with the derivation of the equilibrium conditions for curved interfaces, the Laplace equation. 

Gibbs surface model and definition of surface tension... 

This is the definition of the surface tension according to the Gibbs surface model [1], According to this definition, the surface tension is related to an interface, which behaves mechanically as a membrane stretched uniformly and isotropically by a force which is the same at all points and in all directions. The surface tension is given in J m-2. It should be noted that the volumes of both phases involved are defined by the Gibbs dividing surface X that is located at the position which makes the contribution from the curvatures negligible. 

Quantitative determinations of the thicknesses of a multiple - layered sample (for example, two polymer layers in intimate contact) by ATR spectroscopy has been shown to be possible. The attenuation effect on the evanescent wave by the layer in contact with the IRE surface must be taken into account (112). Extension of this idea of a step-type concentration profile for an adsorbed surfactant layer on an IRE surface was made (113). and equations relating the Gibbs surface excess to the absorbance in the infrared spectrum of a sufficiently thin adsorbed surfactant layer were developed. The addition of a thin layer of a viscous hydrocarbon liquid to the IRE surface was investigated as a model of a liquid-liquid interface (114) for studies of metal extraction ( Ni+2, Cu+2) by a hydrophobic chelating agent. The extraction of the metals from an aqueous buffer into the hydrocarbon layer was monitored kinetically by the appearance of bands unique to the complex formed. 

For molecules which differ in size or shape interactions between the surface of the molecules, different Gibbs excess models, such as NRTL [34] or UNIQUAC [35], are recommended, respectively. The predictive group contribution method UNIFAC [36] will fail if several polar groups compose a solvent or solute molecule. As a... 

The 2D dividing surface model was originally proposed by Gibbs [83] (p 219). 

Here and f are the extents of adsorption reactions which imply an independent transport of each constituent to the Gibbs geometrical surface from the bulk phases b and b such that dnf = d + d Gibbs surface system model is used and the surface phase considered also contains parts of the homogeneous phases. Therefore no direct comparison between Eqs. (32) and (42) is possible. The last-mentioned equation cannot, however, be in contradiction to our assumptions because the regarded surface phase also contains homogeneous masses it represents instead an alternative way of extending the theory to nonequilibrium systems. 

Figure 1.20. Plane surface sorption system (PSSS) in a box of total volume (V ) including a certain mass of sorptive gas (m ) part of it being adsorbed on the surface of the sorbent, the sorbate having the absolute mass (m ) (layer model) and the Gibbs surface excess mass (mog). ...

There is, of course, also a generalized form of the ordinary Gibbs surface tension equation, which is broadly consistent with the Helfrich expression (Eq. (3)) above, albeit of an entirely different model-independent nature. At constant temperature, it reads [20] ... 

The most straightforward (and the most developed) approach to multicomponent adsorption is in further development of the thermodynamics of a surface phase, similar to the bulk-phase thermodynamics. In this way, the Gibbs surface thermodynamics should be completed by an equation of state or by an excess model for a proper thermodynamie potential. An extended review of the fundamentals and the history of the development of this approaeh may be found in Refs. 8, 9, and 78. The approach has become espeeially popular and widely used for praetieal modeling of multicomponent adsorption after the works of de Boer [79] and, especially, Myers and Prausnitz [80]. The latter authors made the natural step of introducing the activity coefficients y of the components in an adsorbed phase. In terms of these coefficients, the chemical potentials of the adsorbate may be expressed as... 

Fig. 7.2 Concentration of the gas c as a function of the distance z from the solid surface. Graphic representation of adsorbed amount and Gibbs surface excess amount, a Layer model b Gibbs representation...

Figure 8 The electronic spectral shift relative to the gas phase of a model chromophore located in the bulk of several liquids (solid lines), at the Gibbs surface (dotted lines), and on the organic side of the Gibbs surface (dashed line). (Data taken from Ref. 372.)...

It has been pointed out [138] that algebraically equivalent expressions can be derived without invoking a surface solution model. Instead, surface excess as defined by the procedure of Gibbs is used, the dividing surface always being located so that the sum of the surface excess quantities equals a given constant value. This last is conveniently taken to be the maximum value of F. A somewhat related treatment was made by Handa and Mukeijee for the surface tension of mixtures of fluorocarbons and hydrocarbons [139]. 

The term G T, a,, A/, ) is the Gibbs free energy of the full electrochemical system x < x < X2 in Fig. 5.4). It includes the electrode surface, which is influenced by possible reconstructions, adsorption, and charging, and the part of the electrolyte that deviates from the uniform ion distribution of the bulk electrolyte. The importance of these requirements becomes evident if we consider the theoretical modeling. If the interface model is chosen too small, then the excess charges on the electrode are not fuUy considered and/or, within the interface only part of the total potential drop is included, resulting in an electrostatic potential value at X = X2 that differs from the requited bulk electrolyte value < s-However, if we constrain such a model to reproduce the electrostatic potential... 

Thermodynamics of the ITIES was developed by several authors [2-6] on the basis of the interfacial phase model of Gibbs or Guggenheim. General treatments were outlined by Kakiuchi and Senda [5] and by Girault and Schiffrin [6]. At a constant temperature T and pressure p the change in the surface tension y can be related to the relative surface excess concentrations Tf " of the species i with respect to both solvents [6],... 

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