Derjaguin approximation

The Derjaguin approximation expresses the force F(h) between two bodies as a function of the surface separation as[2] where W(h) is the interaction energy per unit area between the two planar walls and Reff the effective radius.

Conversely, when the force profile is known, one can evaluate the interaction energy as When one considers two planar walls, the corresponding quantities are expressed per unit area.

Conversely, one has The main restriction of the Derjaguin approximation is that it is only valid at distances much smaller than the size of the objects involved, namely h ≪ R1 and h ≪ R2.

The surfaces of the two respective spheres are thought to be sliced into infinitesimal disks of width dr and radius r as shown in the figure.

When introducing the equation above, the upper integration limit was replaced by infinity, which is approximately correct as long as h ≪ R. In the general case of two convex bodies, the effective radius can be expressed as follows[6] where R'i and R"i are the principal radii of curvature for the surfaces i = 1 and 2, evaluated at points of closest approach distance, and φ is the angle between the planes spanned by the circles with smaller curvature radii.