Half-space interaction layered surfaces

Interactions among lamellar particles and surfaces are important in many contexts. Most surfaces have a contaminant or oxide layer, while particles can be built up from a primary nucleus which subsequently undergoes a variety of thermodynamic conditions. In this fashion, one or more layers of potentially different composition from the nucleus can be added to form a lamellar particle. Interactions of multilayer systems of any geometry resemble the interactions between half-spaces for very small separations. DZYALOSHINSKII et al. [5.581 gave the expression for the nonretarded, zero temperature interaction force between two half-spaces of frequency-dependent dielectric susceptibilities e (x) and e2(x) which were separated by a slab of susceptibility e ix) and thickness l as... 

Equation (2.74) tells us how far the dispersion interaction effectively reaches into the material for the case D -C d, the second and third terms in the square brackets will vanish and the interaction will be the same as if the slabs had infinite thickness. This means that the van der Waals interaction between two bodies with parallel planar surface will occur essentially between surface layers with a thickness of the order of the separation D. So for very small values of D, only a very thin surface layer will contribute. As a consequence, the van der Waals interaction between layered materials will have complex dependence on distance. As an example, we will take the interaction between two half-spaces of material 1 coated with a thin layer of material 2, separated by a gap filled with material 3. The corresponding Hamaker constant Au-m, which should better be called Hamaker function, will depend on the gap thickness D. For a very small gap width D d, the van der Waals interaction will essentially occur between the two layers of material 2 and Aujn (D 0) will be equal to A232, just as if we had the interaction of two half-spaces of material 2 across material 3. For distances much larger than the film thickness of material 2, the interaction will reach far into the material 1 and A12321 (D 0) will approach the value of Am, just as if the layers of material 2 would not exist (Figure 2.9). 

When charged colloidal particles in a dispersion approach each other such that the double layers begin to overlap (when particle separation becomes less than twice the double layer extension), then repulsion will occur. The individual double layers can no longer develop unrestrictedly, as the limited space does not allow complete potential decay [10, 11]. The potential v j2 half-way between the plates is no longer zero (as would be the case for isolated particles at 00). For two spherical particles of radius R and surface potential and condition x i <3 (where k is the reciprocal Debye length), the expression for the electrical double layer repulsive interaction is given by Deryaguin and Landau [10] and Verwey and Overbeek [11],... 

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