Von Szyszkowski equation

Bianco and Marmur [143] have developed a means to measure the surface elasticity of soap bubbles. Their results are well modeled by the von Szyszkowski equation (Eq. III-57) and Eq. Ill-118. They find that the elasticity increases with the size of the bubble for small bubbles but that it may go through a maximum for larger bubbles. Li and Neumann [144] have shown the effects of surface elasticity on wetting and capillary rise phenomena, with important implications for measurement of surface tension. 

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 ... 

For the solution of a single surfactant the last expressions (2.39) and (2.40) transform into the usual von Szyszkowski equation [50]... 

The von Szyszkowski equation (2.41) and Frumkin equations (2.37)-(2.38) have been used for the description of experimental surface tension isotherms of ionic surfactants [40, 58]. Thus the constant a in the Eqs. (2.37)-(2.38) reflects simultaneously intermolecular attractive (van der Waals) and interionic repulsive interactions. As a result, for the ionic surfactants the constant a can have either a positive or negative sign. 

Fig. 2.17. Surface pressure isotherm for 1-decanol solutions O -experimental data [36, 37] calculations from the von Szyszkowski equation (1), Frumkin equation (2), and aggregation model (3).

From Eq. (3.18), the usual von Szyszkowski equation follows for the case of very high surfactant concentration, bc. , 1,... 

First, an exact expression should be derived which gives a relationship between the surface pressure of a surfactant mixture and the surface pressure of the individual solutions. For an ideal (ai = a2 = a = 0) mixture of homologues (co = (0 = C02), Eq. (3.27) can be rewritten in the form of a generalised von Szyszkowski equation... 

The Eq. (3.41) can be easily generalised to mixtures of any number of different components (ideal or non-ideal), including ionic surfactants in presence of electrolytes and surfactants with different molar area. Starting from the generalised von Szyszkowski equation (2.39) one an equation for the mixture of n surfactants results... 

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)... 

Equation (57) combined with Eq. (58) yields a kind of generalised von Szyszkowski equation... 

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]... 

Equation [4.3.13] is the Szyszkowski equation, after B. von Szyszkowski who proposed... 

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. 

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 ... 

Finally, for an ideal surface layer of a n-component ideal bulk solution, Eqs. (2.26) and (2.27) transform into a generalised von Szyszkowski-Langmuir equation of state... 

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. 

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. 

Fig.2.10. Dependence of the Gibbs elasticity modulus CioEOg solutions on surface pressure reorientation model water/air interface (1) reorientation model water/hexane interface (2) von Szyszkowski-Langmuir equation for both interfaces (3).

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]... 

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... 

For a solution that contains only one surfactant, and assuming cob = (O, the Eqs. (7) and (8) transform into the well-known equations of von Szyszkowski (1908) and Langmuir (1907)... 

It is seen from the von Szyszkowski-Langmuir surface tension isotherm, that at a given temperature the shape of the surface tension isotherm is determined only by the parameter co, while parameter b enters this equation as a dimensionless variable be. The dependence of the surface pressure isotherm on the molar area CO is illustrated by Fig. 2. For lower co steeper curves 11(c) are obtained. 

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