Point particles and a planar substrate

Begin with the general form for interaction energy per unit area between half-spaces A and B  

Treat one of the spaces, A, as occupied by a condensed material of susceptibility eA while taking the other space, B, to be occupied by a dilute gas (or solution) whose dielectric susceptibility is sB = m + Z ZB/S (see Fig. L2.15). 

The interaction GAmB between the solid or liquid A and the dilute cloud B is now a sum of individual interactions gp(r) between molecules on the right and the substrate on the left. 

If we imagine increasing the separation Z by an incremental amount dZ, it is equivalent to removing an interaction N d/y per unit area, that is, effecting a change —Ndlgv(l) in the energy of interaction per unit area. 

Because N is a particle density per volume, Ndl is a number of particles per area at the distance Z in the slice dl. If the interaction is attractive, gv(l) is negative and the change in energy from increasing Z by +dZ is positive. Hence the minus sign. At the same time, in 

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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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