Pure-component surfactant adsorption isotherm

Consider the pure surfactant adsorption isotherms shown in Figure 1. At concentrations between the CAC and the CMC, there is a unique concentration level correspond ng to each adsorption level for each pure component. Since this concentration corresponds to formation of admicelles on specific patchs on the surface, for component i, we call the concentration CACi" the variable CAC (no superscript) will be reserved to refer to the concentration which corresponds to admicelle formation on the most energetic patch on the surface. We will only consider binary mixtures of surfactants, so the subscript i can refer to either component A or B. For a surfactant mixture, the total surfactant concentration required to reach a specified adsorption level is defined as CACm. ... 

When two similarly structured anionic surfactants adsorb on minerals, the mixed admicelle approximately obeys ideal solution theory (jUL - Below the CMC, the total adsorption at any total surfactant concentration is intermediate between the pure component adsorption levels. Adsorption of each surfactant component in these systems can be easily predicted from pure component adsorption isotherms by combining ideal solution theory with an empirical correspond ng states theory approach (Z3). ... 

Scamehorn et. al. (20) also presented a simple, semi—empirical method based on ideal solution theory and the concept of reduced adsorption isotherms to predict the mixed adsorption isotherm and admicellar composition from the pure component isotherms. In this work, we present a more general theory, based only on ideal solution theory, and present detailed mixed system data for a binary mixed surfactant system (two members of a homologous series) and use it to test this model. The thermodynamics of admicelle formation is also compared to that of micelle formation for this same system. 

A previously proposed theory to describe mixed adsorption in these systems (20) depended not only on ideal solution theory, but also on the correspond ng states theory to apply to surfactant mixtures. In that model, it was assumed that the adsorption isotherms for the pure components coincided when plotted against a reduced concentration. This occurs when the ratio CACB E/CACrt is the same at any adsorption level. When true, this simplifies the prediction of mixed adsorption isotherms somewhat, but that model is really a special case of the model presented here. 

Scamehorn et. al. ( ) also developed a reduced adsorption equation to describe the adsorption of mixtures of anionic surfactants, which are members of homologous series. The equations were semi-empirical and were based on ideal solution theory and the theory of corresponding states. To apply these equations, a critical concentration for each pure component in the mixture is chosen, so that when the equilibrium concentrations of the pure component adsorption isotherms are divided by their critical concentrations, the adsorption isotherms would coincide. The advantage of... 

The adsorption of binary mixtures of anionic surfactants in the bilayer region has also been modeled by using just the pure component adsorption isotherms and ideal solution theory to describe the formation of mixed admicelles (3 ). Positive deviation from ideality in the mixed admicelle phase was reported, and the non-ideality was attributed to the planar shape of the admicelle. However, a computational error was made in comparison of the ideal solution theory equations to the experimental data, even though the theoretical equations presented were correct. Thus, the positive deviation from ideal mixed admicelle formation was in error. 

The only information needed to predict the mixture surfactant concentration to attain a specified adsorption level is the pure component adsorption isotherms measured at the same experimental conditions as the mixture isotherms. These isotherms are needed to obtain the pure component standard states. 

The method of predicting the mixture adsorption isotherms is to first select the feed mole fractions of interest and to pick an adsorption level within Region II. The pure component standard states are determined from the total equilibrium concentration that occurs at that set level of adsorption for the pure surfactant component adsorption isotherms. The total equilibrium mixture concentration corresponding to the selected adsorption level is then calculated from Equation 8. This procedure is repeated at different levels of adsorption until enough points are collected to completely descibe the mixture adsorption isotherm curve. 

It is seen that the additional (nonnalised) activity coefficients introduced in Eq. (2.10) to establish the consistency between the standard potentials of the pure components and those at infinite dilution, can be incorporated into the constant Kj in Eq. (2.15). Therefore, if a diluted solution with activity coefficients of unity is taken as the standard state, the form of Eqs. (2.13) and (2.14) remains unchanged. The equations (2.14) and (2.15) are the most general relationships from which meiny well-known isotherms for non-ionic surfactants can be obtained. For further derivation it is necessary to express the surface molar fractions, x-, in terms of their Gibbs adsorption values Tj. For this we introduce the degree of surface coverage, i.e. 9j = TjCOj or 0j = TjCO. Here to is the partial molar area averaged over all components or all... 

The rigorous equation of state for mixed surface layers can be transforms into a simple relationship (3.41) for surfactants possessing different adsorption parameters. To calculate the surface tension of the surfactant mixtures one can use various types of surface tension information of the individual components, either experimental values for a given concentration in the pure solutions, or the parameters of the corresponding adsorption isotherms. 

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. 

In practice, we often have a mixture of surfactants. Even pure surfactants of technical origin will be a mixture because of production impurilies. If two surfactants adsorb simultaneously at an interface, the component of the highest bulk concentration will adsorb fastest, as the transport by diffusion is mainly controlled by the concentration of surfactant. A surfactant of lower concentration adsorbs slower, however, and due to a higher surface activity it can displace another surfactant from the interface. The composition of the surface layer at any moment is in a local equilibrium and governed by a respective adsorption isotherm for the surfactant mixture. 

---