Macroscopic approach of Lifshitz

In condensed phases such as liquids, two molecules are not isolated but have many other molecules in their vicinity. As we have seen in Section 2.5, the effective polarizability of a molecule changes when other molecules surround it. In Hamaker s treatment, the additivity of pair-potentials is assumed and the influence of the neighboring molecules is ignored, in contrast with interactions in real condensed systems. In addition, the additivity approach cannot be applied to molecules interacting in a third medium, as we have seen 

Since Lifshitz s derivation is too difficult and beyond the scope of this book, the Hamaker constant based on the Lifshitz theory is expressed as 

It is essential to know the relation of e(iv) with v, in order to calculate the Hamaker constant from the sum over many frequencies. The static dielectric constants, , e2 and e3 are the values of this function at zero frequency. The integral in Equation (564) has a lowest value of vl = 2nkTlh = 3.9 x 1013Hz (s-1) at 25°C. This corresponds to a wavelength of 760 nm. If we assume that the major contribution to the Hamaker constant comes from the frequencies in the visible light or UV region, the relation of e(iv) with v can be given as 

When the Hamaker constant is positive, it corresponds to attraction between molecules, and when it is negative, it corresponds to repulsion. By definition, 3 = 1 and n3 = 1 for a vacuum. As we know from McLachlan s equation (Equation (92)), the presence of a solvent medium (3) rather than a free space considerably reduces the magnitude of van der Waals interactions. However, the interaction between identical molecules in a solvent is always attractive due to the square factor in Equation (567). On the other hand, the interaction between two dissimilar molecules can be attractive or repulsive depending on dielectric constant and refractive index values. Repulsive van der Waals interactions occur when n is intermediate between nx and n2 in Equation (566). If two bodies interact across a vacuum (or practically in a gas such as air at low pressure), the van der Waals forces are also attractive. When repulsive forces are present within a liquid film on a surface, the thickness of the film increases, thus favoring its spread on the solid. However, if the attractive forces are present within this film, the thickness decreases and favors contraction as a liquid drop on the solid (see Chapter 9). 

All the above derivations of Lifshitz continuum approach are valid when the materials are electrically non-conductive (insulating), and the interacting surfaces are farther apart than molecular dimensions (D a). However, if we consider conductive materials such as metals, their static e= °° and Equations (566) and (567) are not valid. For this case, it is possible to approximate the metal dielectric constant as 

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