Gibbs Dividing Interface

Thermodynamics of Gibbs dividing interface and surface excess functions... 

An approach developed by Guggenheim [106] avoids the somewhat artificial concept of the Gibbs dividing surface by treating the surface region as a bulk phase whose upper and lower limits lie somewhere in the bulk phases not far from the interface. 

Figure 6.2 (a) Illustration of a real physical interface between two homogeneous phases a and p. (b) The hypothetical Gibbs dividing surface X. 

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. 

The adsorption T depends on the position of the Gibbs dividing surface and it is therefore convenient to define a new function, the relative adsorption, that is not dependent on the dividing surface. The absorption of component i at the interface is defined by eq. (6.3) as... 

Figure 6.21 Schematic illustration of the concentration of component A across an interface. The Gibbs dividing surface is positioned such that it gives zero adsorption of component A since the algebraic sum of the two shaded areas with opposite sign is zero.

Since r -1 is independent of the position of X, we can choose the position of X to correspond to rA = 0, as illustrated for a schematic two-dimensional interface in Figure 6.21. The two shaded areas, above and below the interface, are equal and give zero adsorption of A. Recall that it is only for planar surfaces that the position of the Gibbs dividing surface is arbitrary, and in the following we will restrict our treatment to planar surfaces only. 

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. 

The standard approach for describing surfactant adsorption at the gas-liquid interface is based on the Gibbs methodology [16]. The Gibbs dividing surface was introduced and is mathematically defined by the interface line that divides the surface excess of the solvent into two equal parts with opposite signs, and the total surface excess of the solvent is, therefore, equal... 

Another interpretation of the electrocapillary curve is easily obtained from Equation (89). We wish to investigate the effect of changes in the concentration of the aqueous phase on the interfacial tension at constant applied potential. Several assumptions are made at this point to simplify the desired result. More comprehensive treatments of this subject may be consulted for additional details (e.g., Overbeek 1952). We assume that (a) the aqueous phase contains only 1 1 electrolyte, (b) the solution is sufficiently dilute to neglect activity coefficients, (c) the composition of the metallic phase (and therefore jt,Hg) is constant, (d) only the potential drop at the mercury-solution interface is affected by the composition of the solution, and (e) the Gibbs dividing surface can be located in such a way as to make the surface excess equal to zero for all uncharged components (T, = 0). With these assumptions, Equation (89) becomes... 

Example 3.1. To show how our choice of the position of the Gibbs dividing plane influences the surface excess, we consider an equimolar mixture of ethanol and water (p. 25 of Ref. [40]). If the position of the ideal interface is such that TH2o = 0, one finds experimentally that TEthanol = 9-5 x 10 7 mol/m2. If the interface is placed 1 nm outward, then we obtain YEthanol = -130 x 10 7 mol/m2. 

If the interface is chosen to be at a radius r, then the corresponding value for dV13/dA is r /2. The pressure difference T>f) — Pa can in principle be measured. This implies that pp pa 2-y/r and l,f) — Pa = Pf /r are both valid at the same time. This is only possible if, dependent on the radius, one accepts a different interfacial tension. Therefore we used 7 in the second equation. In the case of a curved surface, the interfacial tension depends on the location of the Gibbs dividing plane In the case of flat surfaces this problem does not occur. There, the pressure difference is zero and the surface tension is independent of the location of the ideal interface. 

For solutions the Gibbs dividing plane is conveniently positioned so that the surface excess of the solvent is zero. Then the Gibbs adsorption isotherm (Eq. 3.52) relates the surface tension to the amount of solute adsorbed at the interface ... 

A more comprehensive introduction is Ref. [399], We restrict ourselves to uncharged species and dilute solutions (not binary mixtures). The important subject of polymer adsorption is described in Ref. [400], Adsorption of surfactants is discussed in Ref. [401], Adsorption of ions and formation of surface charges was treated in Chapter 5. In dilute solutions there is no problem in positioning the Gibbs dividing plane, and the analytical surface access is equal to the thermodynamic one, as occurs in the Gibbs equation. For a thorough introduction into this important field of interface science see Ref. [8],... 

See, for example, Chap. 2 in G. Sposito, The Surface Chemistry of Soils, Oxford University Press, New York, 1984. The location of an interface is a molecular-scale concept that macroscopic definitions like Eq. 4.1 cannot make precise. That the interface is likely to be located within three molecular diameters of the periphery of an adsorbent solid is sufficient detail for the application of the concepts in the present section. See D. H. Everett, op. cit.,1 for additional discussion of the interface to which Eq. 4.1 applies (known technically as a Gibbs dividing surface). 

The form eqn (3.1) is to hold strictly for planar surfaces. We have already alluded to curvature corrections to the effective surface tension contribution. If the micellar surface is spherical, the effective surface tension y for both a drop and a hole in the liquid is less than that for a planar interface by a factor (1 —d/R), where d is of the order of one or two molecular radii and R is the position of the Gibbs dividing surface. Corresponding corrections to the electrostatic contributions are expected to be of much more importance, and can be handled within the framework of a capacitance description. Thus for a spherical capacitance the energy per unit area is from electrostatics... 

Two more results support the hypothesis of the electrostatic potential at the ice/water interface the existence of the well-pronounced free energy minimum for the Cl- ion close to the Gibbs dividing surface (Fig. 13), and the behaviour of the unconstrained Na+ and Cl- ions on the liquid side of the ice/water interface shown in Fig. 9, when for long times the Cl- ion showed a tendency to be attracted to the ice/water interface, while Na+ ion at different positions with respect to the ice/water interface ended up on the liquid side of ice/water interface. Calculations of the surface potential of the water/vapor interface arise from work by Wilson, Pohorille and Pratt [57, 58], but for the ice/water interface no such reports have appeared, even though the existence of such an interfacial potential would be very helpful in addressing the Workman-Reynolds effect [21, 23],... 

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