Derjaguin’s formula

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... 

The force of molecular attraction between two touching spherical drops is determined by Derjaguin s formula [101]... 

The Derjaguin s formula is applicable to any type of force law (attractive, repulsive, oscillatory) if only (1) the range of the forces, and (2) the surface-to-surface distance are much smaller than the surface curvature radii. This formula is applicable to any kind of surface force, irrespective of its physical origin van der Waals, electrostatic, steric, oscillatory-structural, depletion, etc. It reduces the two-particle interaction problem to the simpler problem for interactions in plane-parallel films. 

The optical properties of quartz cranked into Lifshitz s formula, for plane-parallel surfaces but modified by the Derjaguin transform for a sphere and a flat, gave an attraction that fit neatly with experiments. 

We apply the Derjaguin s approximation (Eq. (12.3)) to the low-potential approximate expression for the plate-plate interaction energy, that is, Eqs. (9.53) and (9.65), obtaining the following two formulas for the interaction between two similar spheres 1 and 2 of radius a carrying unperturbed surface potential ij/f, at separation H at constant surface potential, V (H), and that for the constants surface charged density case, V (//) ... 

A noticeable deviation of sedimentation potentials from Smoluchowski s formula takes place at large siuface concentration variation along the bubble surface. Before considering experimental data, it has to be pointed out that the validity of Smoluchowski s formula for the description of the Dorn effect at large Peclet numbers applies only to solid spherical particles. In particular, the correctness of conclusions of some papers (Dukhin, 1964 Dukhin Buikov, 1965 Derjaguin Dukhin, 1967, 1971) is experimentally confirmed by Usui et al. (1980). Sedimentation potential for four sizes of glass balls appears to be the same. Since the radii of the particles under consideration are approximately 50, 150, 250, and 350 pm, the absence of any effect of Peclet and Reynolds numbers on the sedimentation potential could be demonstrated. 

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