Frumkin model parameters

The data in Table 3.3 obtained from the Frumkin model parameters for diols show, that the molar area of diols is roughly two times larger than that of the normal alcohols. This means that the two polar OH groups are localised in the surface layer. 

To start with this system, the initial guesses for the Frumkin model parameters should be chosen. The van der Waalsian cross-section of the alkyl chain is... 

These two equations present the extension of the Frumkin model to the adsorption of one-surfactant system with two orientational states at the interface. The model equations now contain four free parameters, including cou co2, and b. The equations are highly nonUnear, and regression used in the analysis of surface tension data involves special combinations of Eqs. 23 and 24, which produces a special model fimction used in the least-square minimization with measured surface tension data. Since the model function also contains surface... 

The Frumkin equation of state and adsorption isotherm (2.37) - (2.38) involve one extra parameter a. Thus, the Frumkin model can better fit experimental data. The effect of the parameter a for fixed co values is illustrated by Fig. 2.2. 

Fig. 3.1. The dependence of equilibrium surface tension on concentration for the solutions of normal alcohols in the range of 1-propanol to I-decanol (the number of carbon atoms is shown by figures at the curves) (O), data from [18], 20°C (A), data from [19], 20°C (A), data from [20], 25°C (O), data from [21], 25°C ( ), data from [22], 20°C ( ), data from [23,24], 25 C, theoretical curves were calculated from Frumkin s model, model parameters are listed in Table 3.1.

The temperature of the solutions in all the experiments presented was almost the same, 20 - 25°C. The correspondence between the data reported by different authors is remarkably well. For the processing of experimental data the Frumkin model was employed and the resulting model parameters are presented in Table 3.1. Here the value e characterises the deviation between the theoretical isotherm and the experimental values. Figures 3.2-3.4... 

Table 3.4. Frumkin s model parameters, Eqs. (3.1), (3.2), obtained for fatty acids.

The surface tension isotherms for alkyl dimethyl phosphine oxides (C DMPO) in the interval from Cj to Ci6 at 25 °C are shown in Fig. 3.11. It should be noted that the experimental data reported in [36] for Cj, Ciq, Cp and C[4 are in a good agreement with the data presented in [37, 38], and therefore the results for these experimental data are also shown in Fig. 3.11. The parameters of the Frumkin and reorientation models are summarised in Tables 3.6 and 3.7. Both isotherms agree well with the experimental data. Small differences between the calculated isotherms exist only for nc > 13, while for lower nc the curves for the two models perfectly coincide. It follows then that neither of the two models can be preferred if one takes into account only the agreement between the experimental and theoretical data. However, the negative values of the Frumkin constant a for lower homologues, and the unusual shape of this dependence on nc (cf Fig. 3.12) indicate that for the Frumkin model the coincidence with the experiment is only formal. 

A satisfactory agreement with the experiment can also be formally achieved in the framework of the Frumkin model. This, however, results in physically unrealistic values of intermolecular interaction constant for CnEOj a = -(4 6) for the water/air interface, and a = -10.8 for the water/hexane interface as shown in [57]. A negative value of the constant a corresponds to a repulsion between adsorbed molecules, which is characteristic to solutions of ionic surfactants, where the parameter a compensates the Coulomb interaction. In these cases, however, the value of this parameter is usually much lower than that estimated for CnEOg. We therefore conclude that for the non-ionic surfactant, the Frumkin constant a should be regarded to as a pure fitting parameter, which has no physical meaning. 

We compare now the adsorption behaviour of oxyethylated alcohols (with C,oEOg as an example) at the water/air and water/hexane interfaces, with reference to the data reported in [57]. The experimental and theoretic isotherms at the water/hexane interface are shown in Fig. 3.33. It was mentioned above that the experimental data agree satisfactorily with the Frumkin model for a physically unrealistic value of a = -10.8. Comparing the reorientation model parameters for CiqEOj at the two interfaces (cf. Fig. 3.33 and Table 3.13), one can see that the molar areas are almost the same, while the value of a for the water/hexane interface is 2.5 times higher than that for the water/air interface. Thus the adsorption activity of the oxyethylene groups at the water/hexane interface is significantly higher than that at the water/air interface. 

Fig. 3.41. Dependence of the equilibrium surface tension on c in the solutions of sodium salts of fatty acids, data reported in ( ), [65] and (<>), [33] at 25 C and sodium ions concentration 0.1 M, theoretical curves calculated from the Frumkin model using the model parameters listed in Table 3.17.

Fig. 3.46. Dependence of surface tension for C S04Na solutions on the mean ionic activity c, numbers correspond to the carbon atoms number, without salt ( ) - [68] (O) - [67] ( ) - [69] ( ) - [75] and with additions of inorganic salt (O) - 0.03 M NaCl [70] (A) - 0.1 M NaCI and (A) - 0.5 M NaCl [68], theoretical curves are calculated from the Frumkin model with the parameters listed in Table 3.18.

Fig. 3.52. Dependence of surface tension for C TAB solutions as a function of the mean ionic activity c ( .0,A, 0) at 20°C [80] ( ) at 20 C [81] (A) at 25 >C [82] ( ) at 30°C [82], numbers denote the number of carbon atoms, the theoretical curves were calculated from the Frumkin model using the parameters listed in Table 3.21.

It is seen that the calculated deviation for the reorientation model is two times lower than for the Frumkin model. The dependencies of the isotherm parameters of C TAB on n are similar to those obtained for other surfactants the Frumkin constant a increases with n (cf Table 3.21), the minimum area of the surface active ion C0 2 is almost independent of n, and the molar area in the unfolded state [Pg.246]

C12ACI (A without and A with NaCl to give total Cci=20 mmol/I) Cncr5 mmol/1, at 20 °C [33], the theoretical curves were calculated from the Frumkin model using the parameters listed in Table 3.23. 

It should be noted first that the Frumkin model is the most general one with respect to its application to surfactants of different nature. In spite of the fact that, e.g., for oxyethylated nonionic or ionic surfactants this model is essentially biased, in the majority of practical cases it can be recommended irrespectively of the nature of the surfactant. In the Frumkin model, three parameters are necessary to describe the adsorption and surface tension isotherm. Leaving aside the molar area co which can be estimated from the molecular geometry [16, 84], we concentrate on the results which follow from our development for the parameters a and b for surfactant molecules with linear hydrocarbon chain. Figure 3.59 illustrates the dependence of the Frumkin constant a on the molar area co of various surfactants at n<- = 10. Note that for ionic surfactants the co values are equal to the doubled values of co, from corresponding tables. 

Here H is the surface pressure, F is the adsorption, c is the concentration, b is the adsorption equilibrium constant, co is the area per molecule in the surface layer, and F and G are some functions dependent on F, co and other model parameters denoted here as a, 02,. .. a . For the simplest models considered, namely Langmuir and Frumkin models, co is the model parameter, while for more advanced models this is a property which is defined via model equations. It is essential that in each case F enters the equations via the surface coverage coefficient, 8 = Fco. Also, for each model there exists a parameter, say co, which has the dimension of the area per molecule, and, being introduced into Eq. (7.1), enables one to reduce this equation to a dimensionless form... 

Sorption isotherms at constant pH (most often used types are Langmuir, Freimdlich, and Frumkin) are a step toward a more realistic description of surface phenomena, expressing the Kj as a function of sorbent concentration. Their to the best semi-empirical foimdations allow them to be rather simple in terms of defining equations and amoimt of model parameters. Of course this often does not reflect reality with the necessary accuracy, so their usefulness is restricted to only a few cases. 

Before the proposal of Fainerman et al. was published [6], the adsorption of surfactants at water/oil interfaces has been described only by theories developed for the water/air interface. Such a simply adaptation, of course, does not allow considering precisely the effect of the oil molecnles. Hence, the interaction between the surfactant and oil molecules could only be specified indirectly summarizing different effects in so-called interaction parameters, such as when using in the Frumkin model [7-9], In [6] a new thermodynamic picture was proposed which assumes that the oil molecules are part of the adsorption layer and compete with the adsorbing surfactant molecules. 

Together with Eq. (4) we get the so-called Frumkin Compressibility model (FC model) which can also be used for data analysis supposed the condition C2 c is fulfilled. As it becomes clear from the set of equations discussed above, no specific consideration of the oil phase was made. Any particular interaction of the oil molecules are somehow hidden in the model parameters and a quantitative understanding is impossible. 

All the quantities that appear in the exponent of Eq. (13.23) can be measured, at least in principle, thus permitting the isotherm to be tested by comparison with experiment. This is a great advantage of any theory, inasmuch as it does not contain any adjustable parameters. The disadvantage of the Frumkin model is that it makes no attempt to explain the observed phenomena on the molecular level. Thus, the values of (Co—Cl) and Fn are taken as such it is not explained why they have their observed values or how these values depend on molecular size, on the orientation of the molecules at the interface and on the nature of the interactions with the surface. 

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