Point particles in dilute gases and suspensions

Point particles in dilute gases and suspensions DIPOLAR INTERACTIONS, REDUCTION OF THE LIFSHITZ RESULT 

It is of practical as well as ideological importance that the modern theory of van der Waals forces reduces to the older forms derived for the interaction of individual small molecules in dilute gases. The modern approach can in fact be used to derive new expressions for the interaction between pairs of solutes in dilute solutions. The essential property of e in the dilute-gas or dilute-solution limit is that the dielectric response is strictly proportional to the number density of gas or solute molecules. That is, an electric field applied to a dilute gas or solution acts on each dilute species without distortion of the field by other gas or solute molecules. 

Imagine media A and B as dilute suspensions or gases (em = 1) whose particle number densities Na, Nb and polarizabilities are so small that susceptibilities can be written 

Because GAmB(Z) is an energy of interaction per unit area, the variables of integration sweep over all positions (yy, zB) on one side but sweep only in the zA direction on the other side. The factor 2nye is there to include all positions that are equally distant from zA at a given xB and ye. 

To extract the function gap(r) buried under the three integrals, take three corresponding derivatives. There are a couple of neat maneuvers to do this  

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L2.3.A. Interactions between two semi-infinite media, 182 L2.3.B. Layered systems, 190 L2.3.C. The Derjaguin transform for interactions between oppositely curved surfaces, 204 L2.3.D. Hamaker approximation Hybridization to modern theory, 208 L2.3.E. Point particles in dilute gases and suspensions, 214 L2.3.F. Point particles and a planar substrate, 228 L2.3.G. Line particles in dilute suspension, 232... 

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