Derjaguin approximation, limits

It is customarily assumed that the overall particle-particle interaction can be quantified by a net surface force, which is the sum of a number of independent forces. The most often considered force components are those due to the electrodynamic or van der Waals interactions, the electrostatic double-layer interaction, and other non-DLVO interactions. The first two interactions form the basis of the celebrated Derjaguin-Landau-Verwey-Overbeek (DLVO) theory on colloid stability and coagulation. The non-DLVO forces are usually determined by subtracting the DLVO forces from the experimental data. Therefore, precise prediction of DLVO forces is also critical to the determination of the non-DLVO forces. The surface force apparatus and atomic force microscopy (AFM) have been used to successfully quantify these interaction forces and have revealed important information about the surface force components. This chapter focuses on improved predictions for DLVO forces between colloid and nano-sized particles. The force data obtained with AFM tips are used to illustrate limits of the renowned Derjaguin approximation when applied to surfaces with nano-sized radii of curvature. 

LIMITS OF THE DERJAGUIN APPROXIMATION PROBED WITH AFM TIPS... 

The limits of the celebrated Derjaguin approximation for predicting forces between submicron-sized particles have been argued for some time. Now the approximation can be validated using the force data obtained for the interaction between the AFM tips on microfabri-cated cantilevers and the flat surfaces. The radius of curvature of the AFM tips is about 10 nm and provides the ideal geometry with small interaction forces. Fig. 5 shows an example for the forces measured with the graphite (HOPG) flat surfaces and the silicon nitride tips with the radius of curvature of about 7 nm in solution with different pH. 

The surface element integration (SEI) method provides an improvement on the interaction force between a spherical particle and a flat surface. The SEI improves the Derjaguin approximation by replacing infinity in the integration of the Derjaguin approximation by a finite upper limit, leading to the following prediction for the interaction force, Esei, between the tip and the surface ... 

Todd, B.A. Eppell, S.J. Probing the limits of the Derjaguin approximation with scanning force microscopy. Langmuir 2004, 20, 4892-4897. 

It was pointed out by Gotzclmann ct al. that the Derjaguin approximation is exact in the limit-of a macroscopic sphere, which is the only case of interest here [155]. A rigorous proof can be found in the Appendix of Ref. 156. A similar Derjaguin approximation for shear forces exerted on curved substrate surfaces has been proposed by Klein and Kumacheva [150]. 

Here W h)is the interaction potential between two flat plates at distance h. Clearly this approximate relation between the force for spheres and the interaction potential for plates is more accurate the larger the radius of the spheres compared to the range of the interaction. In this chapter we shall frequently use this Derjaguin approximation. It is a useful tool which, under the right conditions (see above), is very accurate but one has to be careful and be aware of its limitations. 

Because for the ideal chain result higher-order h/Rg terms are not available the /(I) =0 limit can not be accessed. In Fig. 2.33 we present the functions/ for ideal chains (small h), spheres, rods and plates. It is clear that the dependence on the interparticle separation/(/z/.() is similar for greatly different depletants. The results for depletion interaction between big spheres discussed here are based on the Derjaguin approximation valid for R = a, L, D for spheres, rods and disks). 

The depletion interaction between two large spheres induced by oblate spheroids was first investigated in the low density limit by Kech and Walz. " Applying the Derjaguin approximation, they found for the limiting case of infinitely thin circular discs... 

The Derjaguin transform or approximation converts the interaction between plane-parallel surfaces into the interaction between oppositely curved surfaces such as spheres. This procedure and its reverse are allowed in the limit in which the closest separation is much smaller than radii of curvature. 

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