Szyszkowski adsorption equation

The quantities determined from surface tension measurements at 25°C are summarized in Table 11.1. The data are compared with those previously reported for TX-lOO in water [21]. In addition to CMC and surface tension, we are able to determine the area per molecule, a, at the air/solution interface when it is satmated with surfactant molecules. It is calculated from a linear fit to the data below the CMC by using the Langmuir-Szyszkowski adsorption equation. We can notice that both ILs display very high CMC valnes. Indeed, there is a factor of 1000 between these valnes and the CMC measured in water. 

When water soluble surfactants adsorb at the interface between a liquid hydrocarbon and water, the trends in adsorption are very similar to those established for the air - solution interface (see Chapter II). The Traube rule remains valid, and the dependence of the surface tension on concentration can be described by Szyszkowski s equation (11.18). Moreover, at identical surfactant concentrations, the absolute values by which the surface tension is lowered at water - air and water - hydrocarbon interfaces are not that different. The surface tension isotherms for these interfaces are parallel to each other (Fig. III-6). That is due to the fact that the work of adsorption per CH2 group, given by eq. (II. 14), is determined mostly by the change in the standard part of the chemical potential of the solution bulk, q0. Similar to the air-water interface, the energy of surfactant adsorption from an aqueous solution at an... 

The adsorption equation (2.38) of von Szyszkowski (1908) is originally an empirical relationship. Stauff (1957) later discussed the physical backgroimd of the constants a and B. 

For ideal (with respect to the enthalpy) surface layers of a surfactant capable of adsorbing in two states (1 and 2) with different partial molar areas cOj (coi > 2) and different adsorption equilibrium constants, Eqs. (2.26) and (2.27) can be transformed into a generalised von Szyszkowski-Langmuir equation of state [25]... 

Let us consider now the dependence of the shape of surface pressure isotherms on the parameters of the reorientation model. The dependence of surface pressure on the maximum area C0 is illustrated in Fig. 2.5. Here Eqs. (2.84)-(2.88) are employed with (02 = const and a = 0. All calculated curves are normalised in such a way that for the concentration 1 O " mol/1, the surface pressure is 30 mN/m. One can see in Fig. 2.5 that with the increase of (Oj the inflection of the isotherm becomes more pronounced, however, for the ratio a)i/( 2 = 4 the calculated curve almost coincides with the one calculated from the von Szyszkowski-Langmuir equation (2.41) which assumes only one adsorption state with (Oo = < = const. 

The above analysis of the viscoelastic behaviour for adsorption layers of a reorientable surfactant leads to important conclusions. It is seen that the most important prerequisite for a realistic prediction of the elastic properties is the adequacy of the theoretical model used to describe the equilibrium adsorption of the surfactant. For example, when we use the von Szyszkowski-Langmuir equation instead of the reorientation model to describe the interfacial tension isotherm, this rather minor difference drastically affects the elasticity modulus of the surface layer. The elasticity modulus, therefore, can be regarded to as a much more sensitive parameter to find the correct equation of state and adsorption isotherm, rather than the surface or interfacial tension. Therefore the study of viscoelastic properties can give much more insight into the nature of subtle phenomena, like reorientation, aggregation etc. 

When the formation of large aggregates of an insoluble surfactant is stimulated by the adsorption of a soluble surfactant, the von Szyszkowski-Langmuir equation of state and adsorption isotherm for the soluble surfaetant read [157]... 

As aggregation of the insoluble component occurs only when its surface concentration is sufficiently high, the description of the two components based on Volmer s equation seems to be more appropriate than that based on the Szyszkowski-Langmuir equation. If a first-order phase transition does not occur in the monolayer, i.e. no aggregates are formed, then the simultaneous solution of Volmer s equation (2.159) for the components 1 and 2, and Pethica s equation (2.152) yields the adsorption isotherm for the soluble component 2 (see [156])... 

An estimation of Ay obtained from Eq. (4.86) shows that it is essential to this non-equilibrium effect into account when Cq> S-IO" mol/cm, which coincides very good with the criterion for the non-diffusional adsorption kinetics in the Langmuir model discussed above. If we consider the von Szyszkowski-Langmuir equation, then the adsorption layer in equilibrium obeys... 

Thus, for any time t the value of a can be calculated from Eqs (71) and (72). If we assume a Langmuir-Szyszkowski adsorption isotherm and interfacial tension equations, the parameters % and can be expressed via the values of the dynamic and equilibrium interfacial pressures, Yl(t) and... 

The adsorption-desorption energies, AG, were calculated by means of and AG =-RTlnCk /k ). They are listed in Table 1. The obtained values, however, cannot be referred to the literature ones by the static methods, since the surface excess near the CMC increases appreciably with c. Then, the effective adsorption-desorption energy near the CMC was computed by means of Szyszkowski s equation it is listed in Table 1. The values of AG obtained are in good agreement with the calculated ones. This also suggests that the proposed mechanism is reasonable. 

We have considered the surface tension behavior of several types of systems, and now it is desirable to discuss in slightly more detail the very important case of aqueous mixtures. If the surface tensions of the separate pure liquids differ appreciably, as in the case of alcohol-water mixtures, then the addition of small amounts of the second component generally results in a marked decrease in surface tension from that of the pure water. The case of ethanol and water is shown in Fig. III-9c. As seen in Section III-5, this effect may be accounted for in terms of selective adsorption of the alcohol at the interface. Dilute aqueous solutions of organic substances can be treated with a semiempirical equation attributed to von Szyszkowski [89,90]... 

These two equations represent the generalized Szyszkowski-Langmuir adsorption model. 

Equations 21 and 22 present the useful extension of the Szyszkowski-Langmuir model to the adsorption with two orientational states at the interface. If the molecular interactions are considered, a similar simphfied model with P = 2 = P and b = b2 = b can be obtained from Eqs. 10 and 11, giving... 

Equations 27 and 28 present the extension of the Szyszkowski-Langmuir model to the adsorption of one-surfactant systems with aggregation at the interface. For the formation of dimmers on the surface, n = 2 and Eqs. 27 and 28 can be expanded to obtain the Frumkin equation of adsorption state. In general, the surface aggregation model described by Eqs. 27 and 28 contains four free parameters, including coi, n, b and Fc, which can be obtained by regression analysis of the data for surface tension versus surfactant concentration in the solution. 

The Szyszkowski equation has proved useful for describing the relatively simple adsorptions under consideration. However, its foundation is at issue because the Langmuir equation was not derived for mobile but for localized adsorbates the translational entropy is not properly accounted for. We return to this issue below, in the mean time accepting [4.3.13] as a useful empirical expression. 

Measurement of the high frequency modulus, c0, as a function of the equilibrium surface pressure, tt, should provide a sensitive criterion for interaction for monolayers that are quite soluble by normal standards, which involve much longer time spans than the inverse frequency of the compression/expansion experiment. A numerical example of the greater sensitivity of an e0 vs. tt plot, compared with that of the ir vs. log c relationship is shown in Figure 1 for a hypothetical case. The specific defini-nition of surface interactions used here to arrive at numerical values includes all mechanisms that produce deviations from Szyszkowski-Langmuir adsorption behavior. Ideal behavior, with zero surface interactions, then is represented by zero values of In fis in the equation of state ... 

This example emphasizes the danger of using the ideal Langmuir-Szyszkowski equation of state in converting surface tension-time data into adsorption—time data even for very dilute monolayers. Also it clearly shows that any conclusion about the existence or non-existence of an... 

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 applicability of eq. (11.22) to a successful description of adsorption from a solution was established by Langmuir himself, when he compared his adsorption isotherm to the Gibbs equation and ended up with the Szyszkowski equation as a result. The transition from localized to non-localized adsorption (which can be viewed as the transition from fixed adsorption sites to moving ones) does not, therefore, change general trends in the adsorption in the cases described. One should also keep in mind that the liquid interface is more uniform in terms of energy than the solid interface, which contains active sites with different interaction potentials.4 The latter is probably the reason why... 

Many adsorption experiments on long chain fatty acids and other amphiphiles at the liquid/air interface and the close agreement with the von Szyszkowski equation is logically one proof of the validity of Langmuir s adsorption isotherm for the interpretation of y - log c -plots of typical surfactants in aqueous solutions (cf. Appendix 5D). This evidence is also justification for use of the kinetic adsorption/desorption mechanism based on the Langmuir model for interpreting the kinetics and dynamics of surface active molecules. 

For most of the conventional amphiphiles it was demonstrated by Rosen [141] that at a surface pressure H = 20 mN/m the surface excess concentration reaches 84-100 % of its saturation value. Then, the (l/c)n=2o value can be related to the change in free energy of adsorption at infinite dilution AG , the saturation adsorption F and temperature T using the Langmuir and von Szyszkowski equations. The negative logarithm of the amphiphile concentration in the bulk phase required for a 20 mN/m reduction in the surface or interfacial tension can be used as a measure of the efficiency of the adsorbed surfactant ... 

The thermodynamics and dynamics of interfacial layers have gained large interest in interfacial research. An accurate description of the thermodynamics of adsorption layers at liquid interfaces is the vital prerequisite for a quantitative understandings of the equilibrium or any non-equilibrium processes going on at the surface of liquids or at the interface between two liquids. The thermodynamic analysis of adsorption layers at liquid/fluid interfaces can provide the equation of state which expresses the surface pressure as the function of surface layer composition, and the adsorption isotherm, which determines the dependence of the adsorption of each dissolved component on their bulk concentrations. From these equations, the surface tension (pressure) isotherm can also be calculated and compared with experimental data. The description of experimental data by the Langmuir adsorption isotherm or the corresponding von Szyszkowski surface tension equation often shows significant deviations. These equations can be derived for a surface layer model where the molecules of the surfactant and the solvent from which the molecules adsorb obey two conditions ... 

It is seen from the von Szyszkowski-Langmuir surface tension isotherm, Eq. (2.41), that at a given temperature the shape of the surface tension isotherm is determined by only one parameter cOg =cO =cd. The other parameter b enters this equation as a dimensionless variable be, in combination with the concentration. Therefore, the value of b does not affect the shape of surface tension isotherm, and only scales this curve with respect to the concentration axis. It should be noted that this dependence on b is characteristic to all the equations presented above. The dependence of the surface pressure isotherm on the molar area co is illustrated by Fig. 2.1. It is seen, that the lower ro is, hence the higher the limiting adsorption T = 1/co, the steeper is the slope of the n(c)-curve. 

Fig. 2.9 Equilibrium surface pressure for BHBCi solutions symbols - experimental data from [13,25], curves -theoretical calculations curve 1 - Langmuir-Szyszkowski equation curves 2 - 5 - reorientation model for 2, 3, 7 and over 50 adsorption states of the BHBCie molecule, respectively (n j =2.52 lO mVmol,...

The general principles of the adsorption of surfactants at liquid/gas interfaces and the structure of adsorption layers are well described in detail for example in [10] as well as in Chapter 2. The main effect of adsorption is the substantial change of the interfacial pressure n of a given interface. This property is described in a first approximation by the Langmuir- von Szyszkowski equation (2.16)... 

FIGURE 2.8 The Szyszkowski isotherms, Gibbs equation, allow for the transition to the adsorption isotherms, F(c). The figure also shows a comparison of characteristic regions of these isotherms at low and high concentrations of adsorbents. 

Derivation of the dependency of surface tension of solutions with the concentration if a specific theory of adsorption is assumed, e.g. it can be shown that the Langmuir theory (discussed in Chapter 7) yields the Szyszkowski equation. 

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