Surfactant Frumkin adsorption isotherm

The Frumkin adsorption isotherm for the surfactant ions gives... 

The standard deviation has been determined as ct = j where v is the number of degrees of freedom in the fit. The parameters for the molecular interaction /3, the maximum adsorption Too, the equilibrium constant for adsorption of surfactant ions Ki, and the equilibrium constant for adsorption of counterions K2, are thus obtained. The non-linear equations for the Frumkin adsorption isotherm have been numerically solved by the bisection method. 

A. Surfactant Adsorption. The surface density of surfactant has a relevant role for both the double-layer (via the surface charge density) and the hydration interaction (via the dipole moment density of the ion pairs formed). It will be computed using the Frumkin adsorption isotherm ... 

Adsorption layers of the same kind as at fluid interfaces are also formed at low-energy solid -water surfaces, as it was established on PE, polystyrene, paraffin, carbon black, and other related materials. The classical Langmuir or Frumkin adsorption isotherm is often applicable to describe this behaviour. Studies on surfactant adsorption at various solid surfaces have been summarised in a great number of reviews [2, 7, 8, 54, 98, 101, 111, 121, 126, 141, 144, 145, 177, 186, 190, 194-198]. The adsorption at the solid/liquid interfaces is governed by a number of factors ... 

Surfactant keeps emulsion droplets and latex particles colloidally stable against coalescence/aggregation. The surfactant plays another important role in emulsion polymerisation besides stabilisation. Surfactant is critically involved in the nucleation mechanism (i.e., how the particles are formed) of the polymer latex particles (418,419). The amount of surfactant used is critical in controlling the latex particle size distribution. As surfactant is added to an emulsion, some remains dissolved in the aqueous phase, and some adsorbs onto the surface of the emulsion droplets according to an adsorption isotherm (e.g., Langmuir, Freundhch, or Frumkin adsorption isotherms) (173). 

In Eq. 16, hi is another adsorption constant (independent of surface coverage) and is equal to the product of hi in Eq. 11 and the base of natural logarithm (= 2.718). For systems containing only one surfactant. Pi = Pu = 0, and Eqs. 15 and 16 reduce to the well-known Frumkin equation of state and adsorption isotherm described as... 

The description of a mixed adsorption layer of ionic and nonionic surfactants requires the appropriate adsorption isotherms. For example, the Frumkin isotherm gives... 

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). 

Each surfactant adsorption isotherm (that of Langmuir, Volmer, Frumkin, etc.), and the related expressions for the surface tension and surface chemical potential, can be derived from an expression for the surface free energy, F, which corresponds to a given physical model. This derivation helps us obtain (or identify) the self-consistent system of equations, referring to a given model, which is to be applied to interpret a set of experimental data. Combination of equations corresponding to different models (say, Langmuir adsorption isotherm with Frumkin surface tension isotherm) is incorrect and must be avoided. 

The von Szyszkowski isotherm establishes the connection between the change in surface tension y and the surfactant bulk concentration. Stauff (1957) has evaluated the parameters of this semi-empirical adsorption isotherm and has shown that it is in agreement with interfacial thermodynamics. Frumkin s isotherm has often recently been used to describe the adsorption of different types of surfactants, for example by Lunkenheimer (1983), Miller (1986), Wiisteneck et al. (1993), and others. One of the main aims of this book is to show that in the many... 

The kinetics of adsorption from solutions of surfactant mixtures are described on the basis of a generalised Langmuir isotherm. The simultaneous adsorption leads to the replacement of less surface active compounds by those of higher surface activity, which are usually present in the bulk at much lower concentration. More general descriptions of the process are possible on the basis of the Frumkin and Frumkin-Damaskin isotherms, which include specific interfacial properties of the individual surface active species. Quantitative studies of such very complex models can be performed only numerically. 

On the other hand, results are also presented to show the experimental limitations of methods and measuring procedures. As mentioned above, the knowledge of an adsorption isotherm of a surfactant is of fundamental importance for the study of adsorption dynamics. For the surfactants discussed in this chapter, the parameters of the Frumkin isotherm are summarised in the tables of Appendix 5D. In case the interfacial interaction parameter a is zero, the Frumkin isotherm changes into a Langmuir isotherm. 

It was shown in Chapter 2 that the theoretical models defined by Eqs. (3.1)-(3.10) can be used also to describe the behaviour of the solutions of ionic surfactant RX in absence and presence of inorganic electrolyte XY. In this case, the Frumkin constant, in addition to the Van der Waals interaction, involves also the inter-ion interaction in the surface layer. Now instead of the concentration c the corresponding adsorption isotherms should be a function of the mean ionic products c = f (Crx xy rx > where f is the average activity coefficient of ions in the solution bulk. An equation accurately representing measured values of f. is the Debye-Hiickel euqation corrected for short-range interactions... 

To demonstrate the influence of the adsorption isotherm on the adsorption kinetics of a surfactant, the change in surface tension with time y(t) for a Frumkin isotherm is shown in Fig. 4.5 for three values of the interaction parameter a in Eq. (2.37). The case a = 0 is identical with the Langmuir isotherm. The parameter b was chosen such that the equilibrium surface tension is 25 mN/m. As one can see, the shape of the adsorption isotherm has a significant influence on the course of the adsorption kinetics, here given in terms of dynamic surface pressure as a function of time. 

As the surface layer is electroneutral, and therefore X j = JP y, then from Eqs (12) and (40) for nonideal (Frumkin) surface layers and nonideal bulk solutions of one ionic surfactant, wifli or without additional nonsurface active electrolyte, the adsorption isotherm follows... 

Table 1 lists the six most popular surfactant adsorption isotherms, i.e., those of Henry, Freundlich, Langmuir, Volmer (10), Frumkin (11), and van der Waals (9). For cj— 0 all other isotherms (except that of Freundlich) reduce to the Henry isotherm. The physical difference between the Langmuir and Volmer isotherms is that the former corre-... 

As shown above, various such equations exist, such as the classical ones named Langmuir or Frumkin isotherm. However, it was also shown that peculiarities of surfactants in adsorption layers can be described quantitatively only if special models are used. Their impact on adsorption kinetics was reviewed recently (Fainerman et al. 1998) and found to be significant. While the reorientation of adsorbed molecules mimics an acceleration of the adsorption process, surface aggregation on the contrary apparently decelerates it. 

First of all, we see that the data of the two experimental methods complement each other adequately. The dotted line refers to the diffusion model with a diffusion coefficient of D= 3 10 cm /s, which corresponds to the physically reasonable value for this surfactant. One can see that this line does not fit the experimental data very well, however, the solid line does. This solid line was calculated for D = 1 10 cm /s, a value slightly smaller than expected. The reason could be that we used here a mixed adsorption isotherm based on a Frumkin model, while it was shown that alkyl dimethyl phosphine oxides are better described by the reorientation model. This could explain the lower value of the diffusion coefficients. 

In the extraction systems the surfactant is distributed between the phases. Boguslavsky et al. systematically studied the adsorption of tetra-alkylammonium salts in such systems at the water-nitrobenzene interface [11, 76, 90, 91]. The values of the Volta potential at this interface are consistent with the thermodynamic distribution theory. The adsorption isotherms are formally described by the Frumkin equation with the increasing size of tetra-alkylammonium cation, the repulsion between the adsorbate particles increases. It testifies to the fact that the cations of alkylammonium... 

The adsorption of ionic surfactants creates an adsorption layer of surfactant ions, a Stern layer of counterions and a diffusive layer distributed by the electric field of the charged surface. Every layer has its own contribution to surface tension. For example, the adsorption of dodecyl sulfate (DS") ions from the sodium dodecyl sulfate solution is described by the modified Frumkin isotherm as... 

This idea is a consequent transfer of the three-dimensional van der Waals equation into the interfacial model developed by Cassel and Huckel (cf. Appendix 2B.1). The advantages of Frumkin s position is a more realistic consideration of the real properties of a two-dimensional surface state of the adsorption layer of soluble surfactants. This equation is comparable to a real gas isotherm. This means that the surface molecular area of the adsorbed molecules are taken into consideration. Frumkin (1925) additionally introduced, on the basis of the van der Waals equation, the intermolecular interacting force of adsorbed molecules represented by a . 

The Stem-Martynov isotherm does not take into account the intermolecular interaction of adsorbed molecules, as it is considered in the Frumkin isotherm (cf Eq. (2.43)). The classical version of the Framkin isotherm was derived for nonionic surfactant adsorption layers. The incorporation of electrostatic interaction was proposed by Borwankar Wasan (1986, 1988). 

Thus, deviations from the ideal Langmuir isotherm can be caused both by intermolecular interactions, which result in an enthalpy of mixing, and by area differences between molecules, which produce a non-ideal entropy of mixing [18]. For a simple case where the interactions are of the Frumkin type and the partial molar areas of solvent and surfactant are constant the entropic effect of area differences results in typical features of macromolecular adsorption, e.g., a steep initial increase of adsorption ( high affinity adsorption) and a very slow rise once the surface is approximately half filled [18]. 

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. 

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