Fowkes equation

In this version the relationship is called the Girifalco-Good-Fowkes equation. (We use a similar approach again in Chapter 10, e.g., see Equations (10.77) and (10.78), to determine the Hamaker constant for the van der Waals interaction forces.) Although we use y and yd interchangeably, it is important to recognize that yd values are determined by a particular strategy as illustrated in Example 6.5. 

EXAMPLE 6.5 Estimation of Interfacial Tensions Using the Girifalco-Good-Fowkes Equation. The following are the interfacial tensions for the various two-phase surfaces formed by n-octane (O), water (W), and mercury (Hg) for n-octane-water, y = 50.8 mJ m 2 for n-octane-mercury, y = 375 mJ m 2 and for water-mercury, y = 426 mJ m 2. Assuming that only London forces operate between molecules of the hydrocarbon, use Equation (100) to estimate y d for water and mercury. Do the values thus obtained make sense Take y values from Table 6.1 for the interfaces with air of these liquids. 

Finally, we turn our attention to the third contribution to van der Waals attraction, London (or dispersion) forces between a pair of induced dipoles. It will be noted that (at least) one permanent dipole is needed for the preceding sources of attraction to be operative. No such restriction is present for the London component. Therefore this latter quantity is present between molecules of all substances, whether or not they have a permanent dipole. These are the same forces that we considered in Chapter 6 when we discussed the Girifalco-Good-Fowkes equation. 

The same logic that we used to obtain the Girifalco-Good-Fowkes equation in Section 6.10 suggests that the dispersion component of the surface tension yd may be better to use than 7 itself when additional interactions besides London forces operate between the molecules. Also, it has been suggested that intermolecular spacing should be explicitly considered within the bulk phases, especially when the interaction at d = d0 is evaluated. The Hamaker approach, after all, treats matter as continuous, and at small separations the graininess of matter can make a difference in the attraction. The latter has been incorporated into one model, which results in the expression... 

An expression for the Hamaker constant analogous to Equation (67) had been proposed by Fowkes (1964) for the case when only dispersion forces determine the surface tension. The Fowkes equation... 

Laplace equation A thermodynamic derivation Determining surface tension from the Kelvin equation Heat of immersion from surface tension and contact angle Surface tension and the height of a meniscus at a wall Interfacial tensions from the Girifalco-Good-Fowkes equation... 

This ratio is transformed to Fowkes equation when the polar term is negligibly small. Using Eq. (4) and experimental data on interfadal and surface tension of polymers, Wu calculated the polarity of some polymers. He estimates the contribution of the polarity to the surface tension of polymers to be considerable. Thus, for polylvinyl acetate) it takes up 33%, for poly (methyl methacrylate) 28%, and for polychloroprene 11%. 

By combining his equation with the Young equation (Equation (648)), Fowkes obtained the Young-Fowkes equation... 

In 1969, based on the Fowkes equation, Owens and Wendt proposed a new expression by dividing the surface tension into two components, dispersive, yf, and polar, y , using a geometric mean approach to combine their contributions. They assumed that the free energy of adhesion of a polymer in contact with a liquid can be represented by the equation... 

The surface energy can be obtained from the equation of Schultz et al, who derived it from Fowkes equation. 

Fluorocarbon and hydrocarbon modified PDMS surfaces are compared in Table 2. The contact angle data are obtained by the Good-Girifalco-Fowkes equation. It is striking that the hydrocarbon contact angle liquid gives better agreement with the JKR result for the hydrocarbon surface whereas the fluorocarbon liquid data better fit the fluorocarbon surface... 

From a practical applications point of view, both the critical surface tension approach and the use of contact angles with the Good-Girifalco-Fowkes equation represent handy tools for the characterization of the wettability, and therefore something of the chemical nature, of solid surfaces. The choice of technique is basically one of preference and convenience. 

They extended Fowkes equation [liquation (7)] to a more general form as follows ... 

Table 5.3 lists a few approximate values of O for liquid/water interfaces, as obtained by applying Equation 5.30 to experimental values for the interfacial and surface tensions. Alternatively, O may be evaluated theoretically. It is noted that Fowkes equation for the interfacial tension, Equation 5.24, is a special case of Girifalco and Good s approximation, namely, for the condition that the attraction within and between the phases across the interface is governed by dispersion forces. 

Table II shows the list of the surface free energies Ys of untreated fPTFE, plasma-treated fPTFE, NIPAAm-g-fPTFE, PNIPAAm and NIPAAm gel plates at 25 and 40 C calculated on the basis of the modified Fowkes equation (Kitazaki-Hata s method) (19).

This represents the dispersion component of surface free energy, taken from reference 66 Calculated using extended Fowkes equation to two components, taken from reference 68 from reference 64... 

Calculated using extended Fowkes equation to three components, taken from reference 19... 

In order to calculate polymer/filler interaction, or more exactly the reversible work of adhesion characterizing it, the surface tension of the polymer must also be known. This quantity is usually determined by contact angle measurements or occasionally the pendant drop method is used. The former method is based on the Young, Dupre and Fowkes equations (Eqs. 21,8, and 10), but the result is influenced by the surface quality of the substrate. Moreover, the surface (structure, orientation, density) of polymers usuaUy differs from the bulk, which might bias the results. Accuracy of the technique maybe increased by using two or more liquids for the measurements. The use of the pendant drop method is limited due to technical problems (long time to reach equilibrium, stability of the polymer, evaluation problems etc.). Occasionally IGC is also used for the characterization of polymers [30]. 

When polar contributions are neglected, (27) reduces to the Fowkes equation [196]. In terms of the Good-Girifalco equation (21), the interaction parameter is given by ... 

Notice that, unlike the Girifalco-Good equation, the cross term includes a contribution only from the dispersion forces. Thus, the Fowkes equation is based on the fundamental assumption that the crossinteraction term across the interface (work of adhesion) is due to dispersion forces alone. 

In the case of mercury (Hg) the specific contribution includes only the metallic part, while in the case of alcohols, the specific contribution includes both polar and hydrogen bonding contribution. Examples 3.3 and 3.4 illustrate some applications of the Fowkes equation. 

Example 3.3. Use of the Fowkes equation for liquid-Uquid interfaces. 

The liquid-liquid interfacial tension of w-hexane (o) and water (w) is 51.1 mN m. Estimate based on the Fowkes equation the dispersion and specific surface tensions of water. The surface tensions of hexane and water are 18.4 and 72.8 mN m respectively, at room temperature. 

Thus, using the Fowkes equation for the interfacial tension of water/hexane we obtain ... 

The interfacial tension of perfluorohexane-water is then calculated with the Fowkes equation using the dispersion value of water previously estimated ... 

We assume that perfluorohexane only has dispersion contributions (non-polar molecule). We can see that the Fowkes equation indeed predicts higher interfacial tension than for water-hexane, in qualitative agreement with the experimental data. Quantitatively, the agreement is not good as there is about 8% deviation from the experimental value. The Fowkes equation underestimates the interfacial tension significantly and predicts a behaviour much closer to water-aUcanes than the data indicate. 

Then, we apply again the Fowkes equation, this time to the mercury-water interface using the previously estimated dispersion values for the surface tensions of mercury and water ... 

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