By making use of the potential distribution in the electric double layers of two identical spherical colloidal particles, obtained numerically in the preceding paper (Hoskin 1955), the free energy of interaction of the two particles is calculated on the Manchester University Electronic Computer. Various equivalent formulae for both the interaction energy and the repulsive force are applied and compared. It is demonstrated that for the mesh used here, which is based on dipolar co-ordinates, the most accurate method is that which expresses the force in terms of the potential distribution on the median plane. The method of Derjaguin (1934, 1939) for determining the free energy, which treats two spherical particles as consisting of sections of two infinite parallel plates, is shown to yield a good approximation over a wide range of the relevant parameters. Three convenient methods of evaluating the free energy, which are based on the Derjaguin formula, are developed. These are suitable at (i) large particle separations, (ii) small surface potentials and (iii) large surface potentials.

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The potential distribution in the electric double layers of two identical spherical colloidal particles is obtained numerically for cases covering wide ranges of the particle radius, particle separation and surface potential. The method of solution of the Poisson-Boltzmann equation by finite differences on an electronic computer is described, and the distributions obtained are used to discuss the accuracy of the approximate methods that have been developed. It is shown that a series of spherical harmonics (which is only valid for low surface potential) is practical only for small particles and large separation, and unless these conditions hold, accurate results may only be obtained by retaining many terms in the series.

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