Surfactants Gibbs equation

The type of behavior shown by the ethanol-water system reaches an extreme in the case of higher-molecular-weight solutes of the polar-nonpolar type, such as, soaps and detergents [91]. As illustrated in Fig. Ul-9e, the decrease in surface tension now takes place at very low concentrations sometimes showing a point of abrupt change in slope in a y/C plot [92]. The surface tension becomes essentially constant beyond a certain concentration identified with micelle formation (see Section XIII-5). The lines in Fig. III-9e are fits to Eq. III-57. The authors combined this analysis with the Gibbs equation (Section III-SB) to obtain the surface excess of surfactant and an alcohol cosurfactant. 

The Gibbs equation allows the amount of surfactant adsorbed at the interface to be calculated from the interfacial tension values measured with different concentrations of surfactant, but at constant counterion concentration. The amount adsorbed can be converted to the area of a surfactant molecule. The co-areas at the air-water interface are in the range of 4.4-5.9 nm2/molecule [56,57]. A comparison of these values with those from molecular models indicates that all four surfactants are oriented normally to the interface with the carbon chain outstretched and closely packed. The co-areas at the oil-water interface are greater (heptane-water, 4.9-6.6 nm2/molecule benzene-water, 5.9-7.5 nm2/molecule). This relatively small increase of about 10% for the heptane-water and about 30% for the benzene-water interface means that the orientation at the oil-water interface is the same as at the air-water interface, but the a-sulfo fatty acid ester films are more expanded [56]. 

The Gibbs equation relates the extent of adsorption at an interface (reversible equilibrium) to the change in interfacial tension qualitatively, Eq. (4.3) predicts that a substance which reduces the surface (interfacial) tension [(Sy/8 In aj) < 0] will be adsorbed at the surface (interface). Electrolytes have the tendency to increase (slightly) y, but most organic molecules, especially surface active substances (long chain fatty acids, detergents, surfactants) decrease the surface tension (Fig. 4.1). Amphi-pathic molecules (which contain hydrophobic and hydrophilic groups) become oriented at the interface. 

When a surfactant is injected into the liquid beneath an insoluble monolayer, surfactant molecules may adsorb at the surface, penetrating between the monolayer molecules. However it is difficult to determine the extent of this penetration. In principle, equilibrium penetration is described by the Gibbs equation, but the practical application of this equation is complicated by the need to evaluate the dependence of the activity of monolayer substance on surface pressure. There have been several approaches to this problem. In this paper, previously published surface pressure-area Isotherms for cholesterol monolayers on solutions of hexadecy1-trimethyl-ammonium bromide have been analysed by three different methods and the results compared. For this system there is no significant difference between the adsorption calculated by the equation of Pethica and that from the procedure of Alexander and Barnes, but analysis by the method of Motomura, et al. gives results which differ considerably. These differences indicate that an independent experimental measurement of the adsorption should be capable of discriminating between the Motomura method and the other two. 

In principle, the penetration or adsorption of surfactant, Tg is given by the Gibbs equation. For a non-ionic monolayer and an ionised surfactant (as in the system examined), this equation is ... 

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). 

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],... 

The Gibbs equation in this form could be applied to a solution of a non-ionic surfactant. For a solution of an ionic surfactant in the absence of any other electrolyte, Haydon and co-workers3,151 have argued that equations (4.20) and (4.21) should be modified to allow for the fact that both the anions and the cations of the surfactant will adsorb at the solution surface in order to maintain local electrical neutrality (even though not all of these ions are surface-active in the amphiphilic sense). For a solution of a 1 1 ionic surfactant a factor of 2 is required to allow for this simultaneous adsorption of cations and anions, and equation (4.21) must be modified to... 

Adsorption isotherms represent a relationship between the adsorbed amount at an interface and the equilibrium activity of an adsorbed particle (also the concentration of a dissolved substance or partial gas pressure) at a constant temperature. The analysis of adsorption isotherms can yield thermodynamic data for the given adsorption system. Theoretical adsorption isotherms derived from statistical and kinetic data, and using the described assumptions (see 3.1), are known only for the gas-solid interface or for dilute solutions of surfactants (Gibbs). Those for the system gas-solid are of a few basic types that can be thermodynamically predicted81. From temperature relations it is possible to calculate adsorption and activation energies or rate constants for individual isotherms. Since there are no theoretically founded equations of adsorption isotherms for dissolved surfactants on solids, the adsorption of gases on solides can be used as a starting point for an interpretation. 

Hua, X.Y. and Rosen, M.J. (1982) Calculation of the coefficient in the Gibbs equation for the adsorption of ionic surfactants from aqueous binary mixtures with nonionic surfactants. /. Colloid Interface Sci., 87, 469. 

From Gibbs adsorption equation it is possible to determine the area Ao occupied by one surfactant molecule in the adsorption layer which obeys dAa/dlgC = const. One can write Gibbs equation in the form [371-373]... 

A reason to use this form of Gibbs equation is that the electrolyte concentration in the source solution is constant and always much higher than the surfactant concentration. 

Consider now the more complicated case of a double layer with variable surface charge and an ionic surfactant. To be specific, we look at an oxide in a solution containing the anionic surfactant Na A" and NaCl at constant temperature. The Gibbs equations are the following variants of (3.4.10a and 10b]... 

For the interpretation of the slope, the Gibbs equation can be used. Consider first a solution, containing one undisssociated surfactant s plus an undissociated additive A at constant pressure. Then the equation assumes the form... 

As the Gibbs equation remains our primary tool, let us start with item (v). Some time ago this issue has given rise to a lively discussion in the literature but the issue is now resolved. Originally the question was whether for a fully dissociated ionic surfactant such as A Na or C Br" the adsorption term in the Gibbs equation (T dju ) should be written as RTF d In c or 2RTF d In c if the solution is ideal. We shall now analyze the problem thermodynamically. 

Let us take by way of example a surfactant of the A Na type, (abbreviated ANa) such as sodium dodecyl sulphate. For cationic surfactants the reasoning is similar. Let the solution also contain dissolved NaCl. For this system the Gibbs equation [4.6.4] becomes... 

It is important to remember that an equilibrium is established between the surfactant molecules at the surface or interface and those remaining in the bulk of the solution. This equilibrium is expressed in terms of the Gibbs equation. In developing this expression it is necessary to imagine a definite boundary between the bulk of the solution and the interfacial layer (see Fig. 6.3). The real system containing the interfacial layer is then compared with this reference system, in which... 

An interesting effect arises when the surfactant is contaminated with surface-active impurities. A pronounced minimum in the surface tension-log c plot is observed at the cmc, which would seem to be an apparent violation of the Gibbs equation, suggesting a desorption (positive dy/d[logc] value) in the vicinity of the cmc. The minimum in fact arises because of the release below the cmc of the surface-active impurities on the breakup of the surfactant micelles in which they were solubilised. 

The extent of adsorption at the interface can be calculated using the Gibbs equation. The lowering of surface tension increases with increase of surfactant concentration until the critical micelle concentration is reached, i.e. until the surfactant forms micelles at higher concentrations the surface tension remains effectively constant. 

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... 

In addition to the Gibbs equation, three other equations have been suggested that relate concentration of the surface-active agent at the interface, surface or interfacial tension, and equilibrium concentration of the surfactant in a liquid phase. The Langmuir equation (Langmuir, 1917)... 

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... 

We have seen that at some point below but near the CMC the surface becomes essentially saturated with surfactant (F Tm). The relation between y and log C, the Gibbs equation, dy = 23nRT m log C (equation 2.19a), in that region therefore becomes essentially linear. This inear relation continues to the CMC (in fact, it is usually used to determine the CMC). 

Reversibility of Adsorption. Apparently, the data in Figure 10.13 imply that the Gibbs equation (10.2) does not hold for the protein. As we have seen, it is valid for the amphiphile. However, the slopes dll/d In c given in the figure differ only by a factor 2 between the two surfactants, whereas the values of Fm differ by two orders of magnitude. The explanation is not fully clear. Application of the Gibbs equation to polymers is anyway questionable, because it is generally not known what the relation is between concentration (c) and activity (a) of the surfactant. Moreover, proteins and other polymers are virtually always mixtures. 

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. 

Surfactants. Surfactants come in two main types small amphiphilic molecules (for short called amphiphiles ) and polymers, among which are proteins. Small-molecule surfactants readily exchange between surface and solution, and a dynamic equilibrium is thus established, in accordance with the presumptions of the Gibbs equation. Most amphiphiles exhibit a critical micellization concentration (CMC), greatly... 

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. 

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