Derjaguin approximation
The Derjaguin approximation (or sometimes also referred to as the proximity approximation) due to the Russian scientist Boris Derjaguin expresses the force profile acting between finite size bodies in terms of the force profile between two planar semi-infinite walls. This approximation is widely used to estimate forces between colloidal particles, as forces between two planar bodies are often much easier to calculate. The Derjaguin approximation expresses the force F(h) between two bodies as a function of the surface separation as
where W(h) is the interaction energy per unit area between the two planar walls and Reff the effective radius. When the two bodies are two spheres of radii R1 and R2, respectively, the effective radius is given by
Experimental force profiles between macroscopic bodies as measured with the surface forces apparatus (SFA) or colloidal probe technique are often reported as the ratio F(h)/Reff.
The force F(h) between two bodies is related to the interaction free energy U(h) as
where h is the surface-to-surface separation. 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. The disjoining pressure is the force per unit area and can be expressed by the derivative
where W(h) is the surface free energy 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. Furthermore, it is a continuum approximation and thus valid at distances larger than the molecular length scale. Even when rough surfaces are involved, this approximation has been shown to be valid in many situations. Its range of validity is restricted to distances larger than the characteristic size of the surface roughness features (e.g., root mean square roughness).
Frequent geometries considered involve the interaction between two identical spheres of radius R where the effective radius becomes
In the case of interaction between a sphere of radius R and a planar surface, one has
The above two relations can be obtained as special cases of the expression for Reff given further above. For the situation of perpendicularly crossing cylinders as used in the surface forces apparatus, one has
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