Gibbs’ dividing surface

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. 

We suppose that the Gibbs dividing surface (see Section III-5) is located at the surface of the solid (with the implication that the solid itself is not soluble). It follows that the surface excess F, according to this definition, is given by (see Problem XI-9)... 

Surface Excess With a Gibbs dividing surface placed at the surface of the solid, the surface excess of component i, F (moVm"), is the amount per unit area of solid contained in the region near the surface, above that contained at the fluid-phase concentration far from the surface. This is depicted in two ways in Fig. 16-4. The quantity adsorbed per unit mass of adsorbent is... 

Xgds location of the Gibbs dividing surface between the main... 

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

Recall that the Gibbs dividing surface is only a geometrical surface with no thickness and thus has no volume ... 

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... 

The advantage of this expression is that although the adsorption of each component depends on the Gibbs dividing surface, the right-hand side is independent of its position. We can thus define the relative adsorption of component B with respect to component A ... 

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. 

Gibbs approach, for treating surfactant adsorption, 24 134-138 Gibbs dividing surface, 24 135 Gibbs-Duhem equation, 3 744 24 134, 135, 672, 677... 

The second concept that has to be considered is that of absolute adsorption or adsorption of an individual component. This can be considered as the true adsorption isotherm for a given component that refers to the actual quantity of that component present in the adsorbed phase as opposed to its relative excess relative to the bulk liquid. It is a surface concentration. From a practical point of view, the main interest lies in resolving the composite isotherm into individual isotherms. To do this, the introduction of the concept of a Gibbs dividing surface is necessary. Figure 10.6 shows the concept of the surface phase model. 

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... 

Whatever may be said for the mathematical utility of this procedure, its pedagogic results have been disastrous, and I know of no other widely used area of thermodynamics where so much confusion exists. Even to this day the literature is full of errors and imprecisely defined capillary quantities, all of them going back ultimately to a too literal interpretation of the Gibbs dividing surfaces. Guggenheim (3) and Defay (I) have partly alleviated this situation by emphasizing that experimentally mea-... 

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). 

Here y is a generalized interfacial tension, Ci and C2 are bending stresses associated with the curvatures ci and eg, respectively A is the internal interfacial area per unit volume of microemulsion ui and ni are the chemical potentials per molecule and the number of molecules of species i, respectively 0 is the volume fraction of the dispersed phase and P2 and pi are the pressures inside the globules and in the continuous phase in the space between the globules. Here the actual physical surface of the globule (to the extent to which it can be defined) of radius r is selected as the Gibbs dividing surface. 

The thermodynamic work done by the system, 8W (2), follows from Equations 1 and 2. For a Gibbs dividing surface, bounded by the closed curve C, which divides the total volume into the volumes u and vp, it is found that (5)... 

In general the values of rA and rB depend on the position chosen for the Gibbs dividing surface. However, two quantities, TB(A) and rB(n) (and correspondingly wBa(A) and nB°(n)), may be defined in a way that is invariant to this choice (see [l.e]). TB(A) is called the relative surface excess concentration of B with respect to A, or more simply the relative adsorption of B it is the value of rB when the surface is chosen to make rA = 0. rB(n) is called the reduced surface excess concentration of B, or more simply the reduced adsorption of B it is the value of rB when the surface is chosen to make the total excess r = rt = 0. 

To overcome this problem, Gibbs (1877) proposed an alternative approach. This makes use of the concept of surface excess to quantify the amount adsorbed. Comparison is made with a reference system, which is divided into two zones (A, of volume K3,0 and B, of volume V8,0) by an imaginary surface - the Gibbs dividing surface (or GDS) - which is placed parallel to the adsorbent surface. The reference system occupies the same volume V as the real system, so that ... 

However, this is not always true. Complications arise, for example, if the adsorbent undergoes some form of elastic deformation or if the pore structure is modified as a result of the adsorption process. We adopt this convention in order to simplify the thermodynamic treatment. Similarly, we assume that the area of the Gibbs dividing surface is equal to the constant surface area of the adsorbent. We must not forget that we have made these simplifying assumptions when we come to interpret experimental data - especially if there is any indication of low pressure hysteresis. 

It follows from the discussion of the quantitative expression of adsorption in Chapter 2 that the most appropriate demarcation between the gas and the adsorbed phase is the Gibbs dividing surface (GDS). This enables us to express the adsorption data in terms of the surface excess and avoids having to determine (or assume) the absolute thickness of the adsorbed layer. 

The next question is where should we locate the Gibbs dividing surface For several reasons it is most convenient to locate the GDS as closely as possible to the solid surface. By doing so we can minimize the effect of operational temperature and facilitate the comparison of adsorption data. 

The indirect determination of the buoyancy is obtained by the assessment of the sample volume from its density or by pycnometry - as in the previous section and with the same implications for the location of the Gibbs dividing surface. 

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