Locating Dividing Surfaces

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]. 

Two alternative means around the difficulty have been used. One, due to Pethica [267] (but see also Alexander and Barnes [268]), is as follows. The Gibbs equation, Eq. III-80, for a three-component system at constant temperature and locating the dividing surface so that Fi is zero becomes... 

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)... 

The basic chemical description of rare events can be written in terms of a set of phenomenological equations of motion for the time dependence of the populations of the reactant and product species [6-9]. Suppose that we are interested in the dynamics of a conformational rearrangement in a small peptide. The concentration of reactant states at time t is N-n(t), and the concentration of product states is N-pU). We assume that we can define the reactants and products as distinct macrostates that are separated by a transition state dividing surface. The transition state surface is typically the location of a significant energy barrier (see Fig. 1). 

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

The meaning of the surface excess is illustrated in Fig. 1, in which the solid line represents the actual concentration profile of an adsorbate i, when the bulk concentration of i in the phase a (a = O or W) is c . The hatched area corresponds to be the surface excess of i, T,. This quantity depends on the location of the dividing surface. On the other hand, the experimentally accessible quantity should not depend on the location of the artificially introduced dividing surface. The relative surface excess, which is independent of the location of the dividing surface, is defined by relativizing it with respect to those of certain reference components. In oil water interfaces, the mutual solubility of solvents can be significant. The relative surface excess in Eq. (3) is then related to the surface excesses through... 

This is exactly the autonomous linearized Hamiltonian (7), the dynamics of which was discussed in detail in Section II. One therefore finds the TS dividing surface and the full set of invariant manifolds described earlier one-dimensional stable and unstable manifolds corresponding to the dynamics of the variables A<2i and APt, respectively, and a central manifold of dimension 2N — 2 that itself decomposes into two-dimensional invariant subspaces spanned by APj and AQj. However, all these manifolds are now moving manifolds that are attached to the TS trajectory. Their actual location in phase space at any given time is obtained from their description in terms of relative coordinates by the time-dependent shift of origin, Eq. (42). 

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. 

In both solvents, the variational transition state (associated with the free energy maximum) corresponds, within the numerical errors, to the dividing surface located at rc = 0. It has to be underlined that this fact is not a previous hypothesis (which would rather correspond to the Conventional Transition State Theory), but it arises, in this particular case, from the Umbrella Sampling calculations. However, there is no information about which is the location of the actual transition state structure in solution. Anyway, the definition of this saddle point has no relevance at all, because the Monte Carlo simulation provides directly the free energy barrier, the determination of the transition state structure requiring additional work and being unnecessary and unuseful. 

Determination of adsorption of any component is made under the condition of a special location of the dividing surface, which corresponds to This surface is referred to as equimolecular for component i. 

One must recognize that TST is much simpler conceptually than VTST. Thus, there is one transition state in TST and that is located at the maximum energy on the MEP (the saddle point). In VTST the dividing surface is temperature dependent since the partition functions and consequently the free energy of activation are temperature dependent. 

The hquid-hquid interface is not a sharply defined surface, but rather an ill-defined region the boundaries of which extend above and below the layer of the interfaciaUy adsorbed exttactant molecules. Nevertheless, it is often convenient to assume a mathematical dividing surface located where the physicochemical properties of the two-phase system experience the sharpest discontinuity. This imaginary surface of zero thickness, which we refer to as the interface, is useful for defining the concenttations of interfaciaUy adsorbed species, which can be expressed in moles per square centimeter (mol cm ). On the other hand. 

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... 

As explained above, surface excess concentrations are defined relative to an arbitrarily chosen dividing surface. A convenient (and seemingly realistic) choice of location of this surface for a binary solution is that at which the surface excess concentration of the solvent (rA) is zero. The above expression then simplifies to... 

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 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. 

Both have the advantage of giving a sample volume (and therefore a location of the dividing surface) which is, by definition, perfectly reproducible from one adsorption bulb to another and from one laboratory to another. Even if not always realistic, it is a sound convention, if the aim is to obtain reproducible measurements and calculations and is consistent with the spirit of the Gibbs representation. It is, for these reasons, certainly well suited for the study of reference materials. Of course, this approach would replace Step 3 in the procedure described above, whereas Steps 1 and 2 would remain necessary. 

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. 

---