Macroscopic Calculation Lifshitz Theory

In the microscopic calculation pairwise additivity was assumed. We ignored the influence of neighboring molecules on the interaction between any pair of molecules. In reality the van der Waals force between two molecules is changed by the presence of a third molecule. For example, the polarizability can change. This problem of additivity is completely avoided in the macroscopic theory developed by Lifshitz [118,119]. Lifshitz neglects the discrete atomic structure and the solids are treated as continuous materials with bulk properties such as the 

Though the original work is difficult to understand very good reviews about the van der Waals interaction between macroscopic bodies have appeared [114,120], In the macroscopic treatment the molecular polarizability and the ionization frequency are replaced by the static and frequency dependent dielectric permittivity. The Hamaker constant turns out to be the sum over many frequencies. The sum can be converted into an integral. For a material 1 interacting with material 2 across a medium 3, the non-retarded5 Hamaker constant is 

The first term, which contains the the static dielectric permittivities of the three media , 2, and 3, represents the Keesom plus the Debye energy. It plays an important role for forces in water since water molecules have a strong dipole moment. Usually, however, the second term dominates in Eq. (6.23). The dielectric permittivity is not a constant but it depends on the frequency of the electric field. The static dielectric permittivities are the values of this dielectric function at zero frequency. 1 iv), 2 iv), and 3(iv) are the dielectric permittivities at imaginary frequencies iv, and v = 2 KksT/h = 3.9 x 1013 Hz at 25°C. This corresponds to a wavelength of 760 nm, which is the optical regime of the spectrum. The energy is in the order of electronic states of the outer electrons. 

In order to calculate the Hamaker constant the dielectric properties of all three materials need to be known. For frequencies starting in the visible range the dielectric permittivity can be described by 

5 A complete equation of the van der Waals force, which also contains retarded parts, is described in Ref. [119]. 

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Lifshitz et al. [389,390] developed an alternative approach to the calculation of the Hamaker constant Ag in condensed phases, called the macroscopic theory. The latter is not limited by the assumption for pair-wise additivity of the van der Waals interaction (see also Refs. [36,380,391]). The Lifshitz theory treats each phase as a continuous medium characterized by a given nniform dielectric permittivity, which is dependent on the frequency, v, of the propagating electromagnetic waves. For the symmetric configuration of two identical phases i interacting across a medinm j," the macroscopic theory provides the expression [36]... 

Additional complications can arise when two bodies, i.e. the tip and the sample, interact in liquid (Fig. 3c). The interaction energy of two macroscopic phases across a dielectric medium can be calculated based on the Lifshitz continuum theory. In contact, when the distance between the phases corresponds to the nonretarded regime, the Hamaker constant in Eq. (2) is approximated by ... 

Dispersion. Dispersion or London-van der Waals forces are ubiquitous. The most rigorous calculations of such forces are based on an analysis of the macroscopic electrodynamic properties of the interacting media. However, such a full description is exceptionally demanding both computationally and in terms of the physical property data required. For engineering applications there is a need to adopt a procedure for calculation which accurately represents the results of modem theory yet has more modest computational and data needs. An efficient approach is to use an effective Lifshitz-Hamaker constant for flat plates with a Hamaker geometric factor [9]. Effective Lifshitz-Hamaker constants may be calculated from readily available optical and dielectric data [10]. 

Another (more strict) way of calculating the energy of dispersion interactions between the two volumes is based on Lifshitz s macroscopic theory and is briefly summarized in Chapter VII,2. 

A more satisfactory method for calculating the attraction between colloidal particles is the macroscopic continuum theory due to Lifshitz (Lifshitz, 1956 Dzyaloshinskii et al., 1961) and subsequently elaborated by Ninham and coworkers (Mahanty and Ninham, 1976). This expresses the interaction in terms of the bulk dielectric properties of the two colloidal particles. The power of the Lifshitz formalism lies in its ability to encompass all many-body interactions to deal properly with the effects of intermediate substances (here the microscopic method is quite vague) and to include contributions from all resonant electronic and molecular frequencies. Its disadvantage lies in the dramatic increase in the complexity of the calculations, although such computations are readily performed with the aid of a digital computer. 

Lifshitz avoided this problem of additivity by developing a continuum theory of van der Waals forces that used quantum field theory. Simple accounts of the theory are given by Israelachvili and Adamson. Being a continuum theory, it does not involve the distinctions associated with the names of London, Debye and Keesom, which follow from considerations of molecular structure. The expressions for interaction between macroscopic bodies (e.g. Eqns. 2-4) remain valid, except that the Hamaker constant has to be calculated in an entirely different way. 

Needless to say, equation (21.8) is a bit cumbersome and its original derivation is rather lengthy. However, many subsequent treatments of the macroscopic theory are now available which provide both a more readily understandable approach and many useful approximate expressions. In fact, by using the method of Parsegian and Ninham to determine the dielectric response function from absorption data and reflectance measurements, it is now quite straightforward to calculate dispersion forces from Lifshitz s theory. 

The method considered here for determination of these forces is the modified Lifshitz macroscopic approach, considering the microscopic approach results. This approach uses the optical properties of interacting macroscopic bodies to calculate the van der Waals attraction from the imaginary part of the complex dielectric constants. The other possible approach—the microscopic theory—uses the interactions between individual atoms and molecules postulating their additive property. The microscopic approach, which is limited to only a few pairs of atoms or molecules, has problems with the condensation to solids and ignores the charge-carrier motion. The macroscopic approach has great mathematical difficulties, so the interaction between two half-spaces is the only one to be calculated (Krupp, 1967). 

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