Curvature Correction to Derjaguin Approximation

The next-order correction terms to Derjaguin s formula and HHF formula can be derived as follows [13] Consider two spherical particles 1 and 2 in an electrolyte solution, having radii oi and 02 and surface potentials i/ oi and 1/ 02, respectively, at a closest distance, H, between their surfaces (Fig. 12.2). We assume that i/ oi and i//q2 are constant, independent of H, and are small enough to apply the linear Debye-Hiickel linearization approximation. The electrostatic interaction free energy (H) of two spheres at constant surface potential in the Debye-Hlickel approximation is given by 

We choose a spherical polar coordinate system (r, 6, cp) in which the origin O is located at the center of sphere 1, the 6 = 0 hne coincides with the hne joining the centers of the two spheres, and (p is the azimuthal angle about the 6 = 0 line. By symmetry, the electric potential ij/ in the electrolyte solution does not depend on the angle (p. In the Debye-Hilckel approximation, il/ r,6) satisfies 

FIGURE 12.2 Interaction between two spheres 1 and 2 at a closest separation H, each having radii ai and 02, respectively. 

With the help of the above method, we finally obtain from Eq. (12.24) the following expression for the interaction energy V H) 

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