The Derjaguin transform for interactions between oppositely curved surfaces

The Derjaguin transform for interactions between oppositely curved surfaces 

In 1934, B. V. Derjaguin showed how the interaction between two spheres or between a sphere and a plane near contact could be derived from the interaction between facing plane-parallel surfaces.1 There were two conditions  

Schematically the transform is seen as a series of steps on a curved surface (see Fig. L2.7). The distance between facing patches grows from its minimum Z by a rate that depends on the radius of curvature (see Fig. L2.8). 

Specifically, write the distance between patches as h = l + Ri(l — cos 0t) + R2(l - cos 62). Because the radii are much greater than Z, and because the interaction between planar surfaces decays at a rate greater than or equal to 1/Z2, there will be important contributions to the interaction only for small 0, 02. The operative range of the cosine functions is such that they can be approximated by cos 0 = 1- 0z/2. 

The integration over oppositely curved spherical surfaces is the per-unit-area interaction energy Gvv h) between planes weighted by the area 

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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... 

L2.3.C. The Derjaguin transform for interactions between oppositely curved surfaces... 

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