Sphere London attractive force between

The London attractive force between a sphere and a plane is given by... 

Only small particles, less than 1 pm in diameter, would show this effect. Krupp explained this in terms of the equations for London-van der Waals attractive forces between rigid spheres, together with the Hertz equations of contacL Because the attraction is proportional to particle diameter, the force at the particle contact decreases with D. However, the elastic area of the contact spot decreases faster, from the Hertz Equation (9.1), with Thus, as the particle gets smaller, the contact pressure must rise to the point at which plastic deformation occurs. 

Also, in the 1930 s London (9) indicated the quantum mechanical origin of dispersion forces between apolar molecules and in subsequent work extended these ideas to interaction between particles (10). It was shown that whereas the force between molecules varied inversely as the seventh power of the separation distance, that between thick flat plates varied inversely as the third power of the distance of surface separation. These ideas lead directly to the concept of a "long range van der Waals attractive force. A similar relationship was found for interaction between spheres (10). 

Hamaker (1937) extended London s treatment (London, 1934) of the dispersion forces between atoms to calculate those between colloidal particles. The result for the attraction between two spheres, each of radius a, is... 

The dispersed species will be attraeted to each other through electric dipole interactions, which may be due to (1) two permanent dipoles, (2) dipole-induced dipole, or (3) indueed dipole-induced dipole. The latter forces between nonpolar moleeules are also called London dispersion forces. Except for quite polar materials, the London dispersion forces are the more signifieant of the three. Whereas for molecules the force varies inversely with the sixth power of the intermolecular distance, the nature of the variation with distance is somewhat different for dispersions. For dispersed droplets (or particles, ete.) the dispersion forces can be approximated by adding up die attraetions between all inter-droplet pairs of molecules. When added this way the dispersion force between two droplets decays less rapidly as a function of separation distance than is the case for individual molecules. For two spheres of radius a, separated by distance H, the attractive energy, can be approximated by. 

As in part II, we shall assume also in this part that the attraction between coUoidal particles is entirely based upon the London-Van der Waals forces. Hamaker showed how the London-Van der Waals interaction between two spherical particles may be found from the interaction b etween the elements of these spheres. His expression for the energy of attraction Va) runs, using our symbols,... 

Figure 4,11. Bradley s frieory of attraction between rigid spheres based on London forces.

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