Lifshitz Theory A Continuum Approach

As shown above, there have been identified several mechanisms involved in the interactions between atoms and molecules, denominated collectively as the van der Waals forces. In atomic and completely nonpolar molecular systems (hydrocarbons, fluorocarbons, etc.) the London dispersion forces provide the major contribution to the total interaction potential. However, in many molecular systems containing atoms of very different electronegativities and polarizabilities the dipole-dipole (Keesom) and dipole-induced dipole (Debye) forces may also make significant contributions to the total interaction. 

The basic derivations of the van der Waals forces is based on isolated atoms and molecules. However, in many particle calculations or in the condensed state major difficulties arise in calculating the net potential over all possible interactions. The Debye interaction, for example is non additive so that a simple integration of Equation (4.27) over all units will not provide the total dipole-induced dipole interaction. A similar problem is encountered with the dipole-dipole interactions which depend not only on the simple electrostatic interaction analysis, but must include the relative spatial orientation of each interacting pair of dipoles. Additionally, in the condensed state, the calculation must include an average of all rotational motion. In simple electrolyte solutions, the (approximately) symmetric point charge ionic interactions can be handled in terms of a dielectric. The problem of van der Waals forces can, in principle, be approached similarly, however, the mathematical complexity of a complete analysis makes the Keesom force, like the Debye interaction, effectively nonadditive. 

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. 

Even though the two theories of atomic and molecular interaction appear to arise from distinctly different initial premises, the end results of the two are perhaps surprisingly alike. However, the interactions involved are the same, but they are expressed in different ways. For example, the Lifshitz theory predicts the same distance dependence of the overall interactions as that of Hamaker. 

The Lifshitz formulation of the Hamaker constant (or its equivalent) for two like bodies (1) interacting through a second medium (2) is given by 

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