Hamaker approach

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... 

For the interaction of planar slabs, the Hamaker approach entails integration over finite ranges of zA or zB. For the interaction between a half-space A and a parallel slab of B of finite thickness b, this procedure is equivalent to subtracting from E(l) = — (AHam/12 /2) an amount — [AHam/12 (Z +b)2] (see Fig. L2.10). This subtraction yields a form equivalent to the equation of Table P.2.b.3 (see Fig. L2.ll) ... 

Lei us first address some quantitative issues, using the more simple Hamaker approach. If one assumes pairwise additivity, the interaction between a sphere of kind (1) immersed in a fluid of kind (3) and a planar half-space of kind (2) is given by [27] ... 

Furthermore, these van der Waals interactions are important only near the interface, where it is unlikely that either Lifshitz or Hamaker approaches are accurate for spheres of molecular sizes. For example, the magnitude of the interaction for Na+ ions at. 5 A from the interface is only approximately 0.02kT (the values of B used in the calculation, Z Na = — 1X10 50 J m3 was obtained from fit by Bostrom et al. [17] and ZJNa= +0.8X10 511 J m3 was calculated by Karraker and Radke [18]). Eq. (8) might provide a convenient way to account for the interfacial interactions, if suitable values for Bt (not related to the macroscopic Hamaker constants) would be selected. 

The expression van der Waals attraction is widely used and is here defined as the sum of dispersion forces [9], Debye forces [17] and the Keesom forces [18]. Debye forces are Boltzmann-averaged dipole-induced dipole forces, while Keesom forces are Boltzmann-averaged dipole-dipole forces. The interaction for all three terms decays as 1 /r6, where r is the separation between the interacting particles, and they are combined into one term with the proportionality constant denoted the Hamaker constant. In order to determine the van der Waals force there are at least two approaches, either to calculate the force between two particles assuming that the interaction is additive, (this is usually called the Hamaker approach) or to use a variant of Lifshitz theory. 

The description of the van der Waals interaction based on the Lifshitz approach is now sufficiently advanced to provide accurate predictions for the complete interaction energy. For the geometry of two half-spaces, the exact theory is available in a formulation suited for computational purposes. " In parallel with work on planar systems, there has been a focus on the interaction between spheres. " These developed theories have been used as the exact solutions in the validation of the approximate predictions using the Hamaker approach. The significant contribution of the continuum approach to our understanding of the van der Waals interaction lies in the reliable prediction of the Hamaker constant. The interaction energy for two half-spaces and two spheres is summarized below. 

The simple additive approach of Hamaker has been criticised as being inaccurate [Kitchener 1973] and Oliveira [1992] states that it is not valid for condensed media interactions. Visser [1988] has explained the reason why the original Hamaker approach was inaccurate being due to the false assumption that the molecular forces are additive. It cannot be so, since the interactions between molecules with larger separations than near or direct contact, are screened by any molecules at a closer distance. As a consequence, layers of adsorbed materials for instance can influence the interaction between the two interacting macroscopic bodies. Langbein [1969] has demonstrated that when the thicknesses of the surface layers on microscopic bodies are larger than the separation distance, their interaction is... 

The Hamaker approach to calculating the London attraction between colloidal particles is based upon the pairwise additivity of interatomic attractions. Moreover, it is customary to evaluate the Hamaker constant 4 for a substance at a single frequency, usually the electronic correlation frequency for an electron in the ground state of a hydrogen-like atom. 

These expressions based on the Hamaker approach are very useful to gain technical insight, but are based on die same limitations as those discussed in Section 3.3.4. 

The problem was eventually solved (in so far as a theory can be considered a solution) by Lifshitz and co-workers by employing a continuum electrodynamics approach in which each unit or medium is described by its frequency-dependent dielectric permittivity e co). Because of the nature of the beast, an extensive derivation of the Lifshitz theory lies well beyond the scope of this book. However, a brief discussion will aid the reader in seeing the differences and similarities between it and the Hamaker approach. 

The Hamaker approach of pairwise addition of London dispersion forces is approximate because multi-body intermolecular interactions are neglected. In addition, it is implicitly assumed in the London equation that induced dipole-induced dipole interactions are not retarded by the finite time taken for one dipole to reorient in response to instantaneous fluctuations in the other. Because of these approximations an alternative approach was introduced by Lifshitz. This method assumes that the interacting particles and the dispersion medium are all continuous i.e. it is not a molecular theory. The theory involves quantum mechanical calculations of the dielectric permittivity of the continuous media. These calculations are complex, and are not detailed further here. 

The main finding of the Lifshitz theory was that the equations derived with the Hamaker approach are essentially still valid ... 

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