Derjaguin transform

The Derjaguin idea, a mainstay in colloid science since its 1934 publication, was rediscovered by nuclear physicists in the 1970s. In the physics literature one speaks of "proximity forces," surface forces that fit the criteria already given. The "Derjaguin transformation" or "Derjaguin approximation" of colloid science, to convert parallel-surface interaction into that between oppositely curved surfaces, becomes the physicists "proximity force theorem" used in nuclear physics and in the transformation of Casimir forces.23... 

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. 

Close up, when separation Z R and Z R2, the interaction is dominated by interactions of the closest parts of the two spheres. Using the remarkable procedure derived by Derjaguin, we can express the force FSS(Z Ri, R2) between two spheres in terms of the energy of interaction Gpp(Z) between two parallel planes whose separation is also Z [see the Prelude and Level 2, Subsection L2.3.C on the Derjaguin transform see also Eq. (L2.106) and Table S.l.a in Level 2] ... 

L2.3.A. Interactions between two semi-infinite media, 182 L2.3.B. Layered systems, 190 L2.3.C. The Derjaguin transform for interactions between oppositely curved surfaces, 204 L2.3.D. Hamaker approximation Hybridization to modern theory, 208 L2.3.E. Point particles in dilute gases and suspensions, 214 L2.3.F. Point particles and a planar substrate, 228 L2.3.G. Line particles in dilute suspension, 232... 

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. 

Table S.l. Spheres at separations small compared with radius, Derjaguin transform from Lifshitz planar result, including retardation and all higher-order interactions...

When R l, the first term in [ ] dominates to give [-(Aim/2m/6)](R//). This limit can also be extracted from the Derjaguin transform result for small-differences-in-e and neglected retardation. 

Table C.2. Perpendicular cylinders, Rx = R2 = R, Derjaguin transform from full Lifshitz planar result, including retardation...

L2.3.C. The Derjaguin transform for interactions between oppositely curved surfaces... 

Derjaguin transform from full Lifshitz result, including retardation C.l.a. Force per unit length C.l.b. Free energy of interaction per unit length C.l.c.l. Nonretarded (infinite light velocity) limit C.l.c.2. Cylinders of equal radii C.l.c.3. Cylinder with a plane... 

The interaction between two spherical colloids can be transformed by the Derjaguin approximation [29] to the interaction between two flat surfaces (see Appendix A). The net osmotic pressure in an electric double layer is the difference between the internal force, F n, and the external or bulk force, Fex, and is related to the force between two colloids Posm = F n — Fex/a, where a is the area. 

In 1934 Derjaguin [29] published a method that transformed an additive force, F, between two curved bodies at a separation, b0, to the free energy per area between two flat surfaces, W(b0). The free energy per area, is calculated as the work to bring the two bodies from infinite separation to the actual separation, where the separation refers to the closest distance between the... 

According to the Derjaguin approximation (see Appendix B), the force between the surfaces is related to the free energy per area between two flat surfaces. Then, standard thermodynamics can be used to transform the free energy into the osmotic pressure ... 

Papers by Frumkin Levich (1947) and by Derjaguin Dukhin (1961) are apparently only of methodical nature, since they do not take into account the bubble surface effect which causes retardation at a < 0.03 cm. Indeed if at a < 0.03 cm a strong but not complete retardation of the bubble surface takes place, the discussed theories cannot be applied directly, but can be transformed to the hypothesis of incomplete retardation. 

Some results obtained by Doroszkowski and Lamboume for the distance dependence of the steric repulsion for polystyrene stabilizing moieties in toluene are shown in Fig. 13.1. Also shown are the predictions of the theory of Hesselink et al. (1971) including the individual osmotic and elastic components. These were obtained by a numerical Derjaguin-like integration procedure that transformed flat plate potentials into potentials for spheres. It... 

The general equation (13.4), which was first derived by Evans and Napper (1978), can be simplified for those systems where the Deijaguin integration transforms a flat plate potential into a sphere potential. If "Fttotal potential energy per unit area between two parallel flat plates separated by a distance h, then the Derjaguin integration can be approximated by... 

This equation can be transformed into a formula describing the interaction between curved surfaces, such as that between two spherical double-layers. This is carried out by using the Derjaguin equation. The latter connects the force between two spherical double-layers and the interaction energy per unit area, Vr, of two plane interacting double-layers. It is assumed that both the spherical and the plane double-layers carry the same surface charge density, which leads to the following ... 

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