Gibbs adsorption isotherm surfactants

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. 

Derivation of the Gibbs adsorption isotherm. Determination of the adsorption of surfactants at liquid interfaces. Laboratory project to determine the surface area of the common adsorbent, powdered activated charcoal. 

The surface tension data given in Figure 3.5 was obtained for aqueous solutions of a trivalent cationic surfactant (C0RCI3) in both water and in 150 mM NaCl solution. Use the data and the Gibbs adsorption isotherm to obtain estimates of the minimum surface area per molecule adsorbed at the air/water interface. 

Use the Gibbs adsorption isotherm to describe the type of surfactant adsorption occurring at the air/water interface at points A, B, C and D in Figure 3.6. 

Beside the theoretically derived Gibbs adsorption isotherm, a large number of models have been developed that empirically describe a relationship between the interfacial coverage, the surface tension, and the surfactant concentration in the bulk phase. These adsorption isotherms are known under the names of the authors that first described them—i.e., the Fangmuir, Frumkin, or Volmer isotherms. A complete mathematical description of these isotherms is beyond the scope of this unit and the reader is encouraged to consult the appropriate literature instead (e.g., Dukhin et al., 1995). 

The interfacial tension decreases with increasing amount of surface potential. The reason is the increased interfacial excess of counterions in the electric double layer. In accordance with the Gibbs adsorption isotherms, the interfacial tension must decrease with increasing interfacial excess. At charged interfaces ions have an effect similarly to surfactants at liquid surfaces. 

For ionic surfactants another effect often dominates and usually salt tends to stabilize emulsions. Reason without salt the distance between surfactants in the interface is large because the molecules electrostatically repel each other. This prevents a high surface excess. The addition of salt reduces this lateral repulsion and more surfactant molecules can adsorb at the interface. Then, according to the Gibbs adsorption isotherm Eq. (3.52), the surface tension is reduced and the emulsion is stabilized. 

If we compress a surfactant film on water we observe that the surface tension decreases and the surface pressure increases. What is the reason for this decrease in surface tension We can explain it by use of the Gibbs adsorption isotherm (Eq. (3.52)). On compression, the surface excess increases and hence the surface tension has to decrease. This, however, is relatively abstract. A more illustrative explanation is that the surface tension decreases because the highly polar water surface (high surface tension) is more and more converted into a nonpolar hydrocarbon surface (low surface tension). 

With surfactant the surface tension is reduced according to the Gibbs adsorption isotherm Eq. (3.52). To apply Eq. (3.52) we need to know the surface excess ... 

S. Ikeda, M. Tsunoda and H. Maeda, Application of Gibbs adsorption isotherm to aqueous solutions of a nonionic cationic surfactant, J. Colloid Interface Sci. 67 (1978) 336-348. 

Surfactant surface activity is most completely presented in the form of the Gibbs adsorption isotherm, the plot of solution surface tension versus the logarithm of surfactant concentration. For many pure surfactants, the critical micelle concentration (CMC) defines the limit above which surface tension does not change with concentration, because at this stage, the surface is saturated with surfactant molecules. The CMC is a measure of surfactant efficiency, and the surface tension at or above the CMC (the low-surface-tension plateau) is an index of surfactant effectiveness (Table XIII). A surfactant concentration of 1% was chosen where possible from these various dissimilar studies to ensure a surface tension value above the CMC. Surfactants with hydrophobes based on methylsiloxanes can achieve a low surface tension plateau for aqueous solutions of —21-22 mN/m. There is ample confirmation of this fact in the literature (86, 87). 

From the slope of the linear portion of the y-log C curve (just below the cmc), the surface excess (number of moles of surfactant per unit area at the Uquid/air interface) can be obtained. Then, using the Gibbs adsorption isotherm, dy... 

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. 

In the case where E is used to describe purely the elasticity, then E can be termed the film elasticity of compression modulus . In the general case where the surface behaviour has both an elastic and viscous component, then E can be termed the surface dilational modulus . Basically, E is the measure of the ability of a film to adjust its surface tension in an instant of stress and should be relatively large for the film to remain stable. By combining equation (2.2) with the Gibbs adsorption isotherm equation, it can be shown that E is proportional to (dy/dc), where c is the concentration of the surfactant in the thin film. 

We have considered the equilibrium distribution of a nonelectrolytic surfactant solute between the hulk liquid phase and the interfacial phase in a gas-Uquid system via relation (3.3.106). We have thereby iUustrated the appUca-tion of the Gibbs adsorption isotherm (3.3.40a) to a single nonionic surface-active solute. Chemical reactions can influence such adsorption isotherms in a number of ways. If the surface-active solutes are ionic, the adsorption equilibria are affected. In other cases, the solute to be removed (the colligend) is not surface active but it reacts with or is... 

Concerning the behaviour of ions near the air-water interface (and in principle at all interfaces), the Gibbs adsorption isotherm suggests that an increase in surface tension is an indication for a depletion of ions at this interface. In contrast, as in the case of surfactants, the surface tension decreases, when ions are adsorbed or when they have at least a certain propensity to the interface. While this is rigorously correct in the thermodynamic limit (provided that the activity coefficients are taken into account correctly), caution must be taken if molecular interpretation is attempted. It is one of the great merits of Jungwirth and Tobias work to show... 

In the next section, we discuss hydrodynamic forces between fluid interfaces. The interaction between fluid interfaces is strongly influenced by surfactants and contaminants at the interfaces. Therefore, we first need to introduce a fundamental relation between the amount of substance adsorbed at a fluid interfaces and the surface tension. This is quantitatively expressed in the Gibbs adsorption isotherm. We only introduce the Gibbs adsorption isotherm for a two-component system, that is, a liquid and one dissolved substance. It is... 

Gibbs adsorption isotherm is valid in thermodynamic equilibrium. In equilibrium, the surface tension should not depend on the total surface area if new surface is produced, surfactant from the bulk diffuses to the surface and the same surface tension is estabhshed as before. If, however, the system is not given enough time to equilibrate, the local surface tension changes with an expansion or shrinkage of the geometric surface area A This is characterized with the surface elasticity E, also called surface dilatational modulus [684]. The surface elasticity is defined as... 

Equations D3.5.30 and D3.5.32 are both very valuable. They state that the rate of adsorption can be obtained from plots of the interfacial tension versus either tA- (for t—>0) or lth (for the long-term solution f— >). With these two equations the tool to extract the adsorption rate from experimentally obtained surface tension-time curves is at hand. It should be noted that instead of the Gibbs model, one could use one of the previously mentioned adsorption isotherms such as the Langmuir adsorption isotherm to convert interfacial tension to interfacial coverage data. The adsorption isotherms may be obtained by fitting equilibrium surface tension data versus surfactant concentration. 

From thermodynamics, the lowering of surface free energy due to surfactant adsorption is given by the Gibbs adsorption equation for a binary, isothermal system containing excess electrolyte ... 

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. 

The Gibbs equation contains three independent variables T, a, and p (defined either via concentration or pressure, c or p, respectively), and is a typical thermodynamic relationship. Therefore, it is not possible to retrieve any particular (quantitative) data without having additional information. In order to establish a direct relationship between any two of these three variables, it is necessary to have an independent expression relating them. The latter may be in a form of an empirical relationship, based on experimental studies of the interfacial phenomena (or the experimental data themselves). In such cases the Gibbs equation allows one to establish the dependencies that are difficult to obtain from experiments by using other experimentally determined relationships. For example, the surface tension is relatively easy to measure at mobile interfaces, such as liquid - gas and liquid - liquid ones (see Chapter I). For water soluble surfactants these measurements yield the surface tension as a function of concentration (i.e., the surface tension isotherm). The Gibbs equation allows one then to convert the surface tension isotherm to the adsorption isotherm, T (c), which is difficult to obtain experimentally. 

When adsorption takes place at the surface of a highly porous solid adsorbent, the surface excess can be readily measured, e.g. by measuring the increase in the adsorbent weight in the case of adsorption from vapor, or by following the decrease in the adsorbate concentration during adsorption from solutions. Studies of the adsorption dependence on vapor pressure (or solution concentration) reveal T(p) (or T(c)) adsorption isotherms. In both cases the two-dimensional pressure isotherm can be established from the Gibbs equation (see Chapter II, 2, and Chapter VII, 4). Therefore, it is as a rule possible to establish the dependence between the two of three variables present in the Gibbs equation the surface tension isotherm, a(c), for mobile interfaces and soluble surfactants, the two-dimensional pressure, tt(c), isotherm for insoluble... 

The comparison of the empirical Szyszkowski equation (II. 18) with the Gibbs equation (II.5) indicates that Langmuir adsorption isotherm (11.22) is well suited also for the description of adsorption at the air - surfactant solution interface. It is interesting to point out that at the gas - solid interface, for which eq. (11.22) was originally derived various deviations from Langmuirian behavior are often observed. 

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),... 

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