Gibbs interfacial excess concentration

The choice of the ideal interface in the Gibbs adsorption isotherm (3.52) for a two-component system is, in a certain view, arbitrary. It is, however, convenient. There are two reasons First, on the right side there are physically measurable quantities (a, 7, T), which are related in a simple way to the interfacial excess. Any other choice of the interface would lead to a more complicated expression. Second, the choice of the interface is intuitively evident, at least for ci > C2. One should, however, keep in mind that different spatial distributions of the solute can lead to the same T. Figure 3.6 shows two examples of the same interfacial excess concentration In the first case the distribution of molecules 2 stretches out beyond the interface, but the concentration is nowhere increased. In the second case, the concentration of the molecules 2 is actually increased. 

Reduction of surface or interfacial tension is one of the most commonly measured properties of surfactants in solution. Since it depends directly on the replacement of molecules of solvent at the interface by molecules of surfactant, and therefore on the surface (or interfacial) excess concentration of the surfactant, as shown by the Gibbs equation... 

The adsorption of the anionic surfactant sodium dodecyl sulphate (SDS), probably the most frequently studied surfactant and often used as model substance at the air/water and at the decane /interface is given in Fig. 1.5. The surface and interfacial tension have been plotted as a function of SDS concentration in the aqueous phase. From the slope of the tangents to the curves in Fig. 1.5 the interfacial excess concentration (adsorption density) F at different interfacial tensions can be calculated directly using Gibbs fundamental adsorption isotherm (see section 2.4.1),... 

Investigations of the effects of oil-soluble surfactants on the emulsification of paraffins in aqueous surfactant solutions led to the proposal that the formation of interfacial complexes at the oil-water interface could increase the ease with which emulsions could be formed and, possibly, explain the enhanced stability often found in such systems (Figure 9.9). By definition, an interfacial complex is an association of two or more amphiphilic molecules at an interface in a relationship that will not exist in either of the bulk phases. Each bulk phase must contain at least one component of the complex, although the presence of both in any one phase is not ruled out. The complex can be distinguished from such species as mixed micelles by the fact that micelles (and therefore mixed micelles) are not adsorbed at interfaces. According to the Le Chatelier principle, the formation of an interfacial complex will increase the Gibbs interfacial excess F/ [Eq. (9.2)] for each individual solute involved, and consequently, the interfacial tension of the system will decrease more rapidly with increasing concentration of either component. 

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],... 

In Section 6.4.2 we will find that T represents the Gibbs-surface excess, i.e., T=N/A -N°/A, where is the number of molecules that would have been there if there had been no double layer, and N is the actual number of molecules in the interfacial region. However, when the bulk concentration of the spedes is small, i.e., tfi — 0, then the number of adsorbed molecules tends to f, i.e., f — N/A. 

An adsorption isotherm is a graph of the amount adsorbed versus the pressure of the vapor phase (or concentration in the case of adsorption from solution). The amounts adsorbed can be described by different variables. The first one is the surface excess I in mol/m2. We use the Gibbs convention (interfacial excess volume Va = 0). For a solid surface the Gibbs dividing plane is localized directly at the solid surface. Then we can convert the number of moles adsorbed Na to the surface excess by... 

In treating interfacial (if) regions, we will follow the method of Gibbs and replace the nonuniform interfacial region by a two-dimensional Gibbs surface phase with uniform properties. Properties of this phase are called surface excess properties and their calculation is illustrated for the surface excess concentration of component i in Fig. 8. Here, the actual interfacial region, the region where properties vary, extends from zj to z2 and is replaced by the surface phase located at position z0, with the uniform bulk a and (1 phases extended up to this position. 

For surface-active solutes the surface excess concentration, p can be considered to be equal to the actual surface concentration without significant error. The concentration of surfactant at the interface may therefore be calculated from surface or interfacial tension data by use of the appropriate Gibbs equation. Thus, for dilute solutions of a nonionic surfactant, or for a 1 1 ionic surfactant in the presence of a... 

The preceding discussion of the Gibbs adsorption equation was referenced to a fluid-fluid interface in which the surface excess, T, is calculated based on a measured quantity, a, the interfacial tension. For a sohd-fluid interface, the interfacial tension cannot be measured directly, but the surface excess concentration of the adsorbed species can be, so that the equation is equally useful. In the latter case. Equation (9.16) provides a method for determining the surface tension of the interface based on experimentally accessible data. 

In this manner, the surface excess of ions can be found from the experimental values of the interfacial tension determined for a number of electrolyte concentrations. These measurements require high precision and are often experimentally difficult. Thus, it is preferable to determine the surface excess from the dependence of the differential capacity on the concentration. By differentiating Eq. (4.2.30) with respect to EA and using Eqs (4.2.24) and (4.2.25) in turn we obtain the Gibbs-Lippmann equation... 

The surface concentrations T depend on the thickness of the interfacial region, and we would like to express them through quantities which are independent of it. This can be done for those species which occur both at the interface and in the solution. Usually one of the components of the solution, the solvent, has a much higher concentration then the others. We denote it by the index 0 , and introduce surface excesses with respect to the solvent in the following way In the bulk of the solution the Gibbs-Duhem equation (at constant T and p) is simply E Ni dfri = 0, or ... 

Equation 9 states that the surface excess of solute, T, is proportional to the concentration of solute, C, multiplied by the rate of change of surface tension, with respect to solute concentration, d m,/dCThe concentration of a surfactant in a G—L interface can be calculated from the linear segment of a plot of surface tension versus concentration and similarly for the concentration in an L—L interface from a plot of interfacial tension. In typical applications, the approximate form of the Gibbs equation was employed to calculate the area occupied by a series of sulfosuccinic ester molecules at the air—water interface (8) and the energies of adsorption at the air-water interface for a series of commercial nonionic surfactants (9). 

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... 

The Gibbs adsorption isotherm is a relationship between the surface tension and the excess interfacial concentrations. To derive it we start with Eqs. (3.27) and (3.28). Differentiation of... 

The strict thermodynamic analysis of an interfacial region (also called an -> interphase) [ii] is based on data available from the bulk phases (concentration variables) and the total amount of material involved in the whole system yielding relations expressing the relative surface excess of suitably chosen (charged or not charged) components of the system. In addition, the - Gibbs equation for a polarizable interfacial region contains a factor related to the potential difference between one of the phases (metal) and a suitably chosen - reference electrode immersed in the other phase (solution) and attached to a piece of the same metal that forms one of the phases. 

When surfactant molecules concentrate at the interface, some solvent molecules are displaced, so the surface solvent concentration is lower than the bulk solvent concentration. The Gibbs convention defines the dividing line between the two phases so that the (negative) surface excess of solvent equals zero. Then equation 4 gives the surface excess of (say) laurylsulfonic acid at the air-water interface. When the actual interfacial concentration of surfactant is needed, the situation is more complicated. Methods for handling these complications have been discussed (1,7). 

Adsorption. Some substances tend to adsorb onto an interface, thereby lowering the interfacial tension the amount by which it is lowered is called the surface pressure. The Gibbs equation gives the relation between three variables surface pressure, surface excess (i.e., the excess amount of surfactant in the interface per unit area), and concentration—or, more precisely, thermodynamic activity—of the surfactant in solution. This relation only holds for thermodynamic equilibrium, and the interfacial tension in the Gibbs equation is thus an equilibrium property. Nevertheless, also under nonequilibrium conditions, a tension can be measured at a liquid interface. 

Polymeric surfactants are generally (far) more surface active, but they give lower surface pressures than most amphiphiles. At the plateau value of the surface excess they are not very tightly packed (most amphiphiles are), but they extend fairly far into the solution. The exchange between solution and interface may be very slow, and the Gibbs equation does not seem to hold. Most amphiphiles can displace polymers from the interface, if present in sufficient concentration, since they give a lower interfacial tension. Mixed surface layers can also be formed. 

In a two-phase system consisting of two or more components the composition of the discontinuity surface (see Chapter I) may significantly differ from that of a bulk of both phases in contact. Primarily the components that lower the system s free surface energy are expected to accumulate within the discontinuity surface this spontaneous concentration of substances is referred to as adsorption. The quantitative measure of the adsorption of the /-th component, T was introduced by Gibbs, and is also referred to as the adsorption, or the surface excess ofthe amount of substance. This measure has a meaning of the molar excess of a particular component per unit interfacial area ... 

So far the Gibbs adsorption isotherm represents the best foimded theoretical backgroimd for the calculation of the adsorption excess densities of surfactants. Statistical thermodynamics may enable us in future to calculate adsorption densities by accounting for the chemical structure of a surfactant. Beside the direct calculation of excess adsorption densities F with the help of r - log c-plots, relationships of F and the interfacial tension y as functions of the surfactant bulk concentration are very helpful. 

This notion is complex and a more detailed presentation would need to go beyond the scope of this document. In particular, in order to describe the thermodynamics of interfaces one would need to refer to the notion of surface excess as defined by the Gibbs model. Ne can still say that TJ is the integral of the volume concentration over a distance equal to the thickness of the interfacial zone... 

J. Willard Gibbs showed by thermodynamics that if a mixture of vapours, or a solution, is in contact with a surface, a change of concentration of a component occurs at the interface if the interfacial tension (surface tension) a is altered by such a change. If F is the excess of concentration at the interface above that in the body of the solution, and fi is the chemical potential of the component concerned F= - dajdfM (7). If the solute obeys the gas laws, and r and the concentration c are in mols per unit volume, (7) becomes ... 

---