The Derjaguin approximation

The integration runs over the entire surface of the solid. Please note that here A is the cross-sectional area. Often we have to deal with rotational-symmetric configurations. Then it is reasonable to integrate in cylindrical coordinates  

In many cases the following expression is more useful  

The approximation is only valid if the characteristic decay length of the surface force is small in comparison to the curvature of the surfaces. Approximation (6.30) is sometimes called the Derjaguin approximation in honor of Derjaguin s work. He used this approach to calculated the interaction between two ellipsoids [132], 

6 Boris Vladimirovich Derjaguin, 1902-1994. Russian physicochemist, professor in Moscow. 

A special, but nevertheless important, case is the interaction between two identical spheres. It is important to understand the stability of dispersions. For the case of two spheres of equal radius R, the parameters x and r are related by (Fig. 6.5) 

From simple geometry we can easily rearrange this equation in the form 

Using the result given in (6.49), we can then obtain the corresponding interaction energy between spheres  

Once again the interaction energy decays exponentially and is strongly dependent on both the snrface potential and the electrolyte concentration. 

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In Section 2.1, we had calculated the van der Waals force between two spheres and between two planar surfaces. What if the two interacting bodies do not have such a simple geometry We could try to do an integration similar to the one that was carried out for the two spheres. This, however, might be very difficult and lead to long expressions. The Derjaguin approximation is a simple way to overcome this 

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The force-distance profiles presented in the following sections are generally plotted as F(D)/Rv.D, i.e. the force axis is normalized by dividing F(D) by the mean radius R of the mica sheets. In the Derjaguin approximation (11),... 

For spherical particles of radii ax and 02 we could use the Derjaguin approximation (see for example reference 29) to calculate the potential ... 

The so-called Derjaguin equation relates in a general way the force F h) between curved surfaces to the interaction energy per unit area E(h), provided the radius of curvature R is larger than the range of the interactions [17]. Adopting the Derjaguin approximation, one obtains ... 

Figure 6.13 Diagram used to explain the Derjaguin approximation for the interaction between two spheres.

A schematic diagram of the analysis method used is given in Figure 6.13. In this procedure, called the Derjaguin approximation , we consider the interaction of the circnlar annnlns (dx) with an imaginary parallel snrface plane at distance Z. With this assumption, the total interaction energy Vs between the spheres is then given by... 

EXAMPLE 11.4 Interaction Between Spherical Particles The Use of the Derjaguin Approximation. Spherical particles can be approximated by a stack of circular rings with planar faces as shown in Figure 11.8. Use Equation (86) to describe the repulsion between rings separated by a distance z and derive an expression for the repulsion between the two spheres of equal radius Rs. Assume that the strongest interaction occurs along the line of centers and make any approximations consistent with this to obtain the final result. 

The Derjaguin approximation illustrated in the above example is suitable when kR > 10, that is, when the radius of curvature of the surface, denoted by the radius R, is much larger than the thickness of the double layer, denoted by k 1. (Note that for a spherical particle R = Rs, the radius of the particle.) Other approaches are required for thick double layers, and Verwey and Overbeek (1948) have tabulated results for this case. The results can be approximated by the following expression when the Debye-Hiickel approximation holds ... 

Analogous to the planar case considered in Section 13.3, using the Derjaguin approximation, the interaction energy between two spherical particles of radius R, can be written as... 

Interaction between spherical particles The Derjaguin approximation 525... 

At this point it is probably instructive to discuss the use of the symbols D, x, and . D is the shortest distance between two solids of arbitrary geometry. Usually we use x for the thickness of the gap between two infinitely extended solids. For example, it appears in the Derjaguin approximation because there we integrate over many such hypothetical gaps. is a coordinate describing a position within the gap. At a given gap thickness x, the potential changes with (Fig. 6.9). D is the distance between finite, macroscopic bodies. 

Figure 6.10 Electrostatic double-layer force between a sphere of R = 3 /um radius and a flat surface in water containing 1 mM monovalent salt. The force was calculated using the nonlinear Poisson-Boltzmann equation and the Derjaguin approximation for constant potentials (tpi = 80 mV, ip2 = 50 mV) and for constant surface charge (i/2/Ad so that at large distances both lead to the same potential.

For two identical spheres of radius a separated by the distance of closest approach z, the interaction energy is given, in the Derjaguin approximation, by1... 

By use of the Derjaguin approximation, the repulsive interaction energy between two identical spherical particles of radius a is given by... 

If the radius of the spherical colloidal particles a is much larger than the shortest distance between the surfaces of two particles, the repulsive free energy between two identical spherical particles at a distance H0 of shortest approach is given in the Derjaguin approximation by12... 

Detjaguin approximation. For the present comparisons with experiment, the Derjaguin approximation is considered to be applicable, since the curvature radius of the surface (cylindrical surface) is much larger than the separation between the (20) two surfaces (105 times). 

Recently,21,28 two kinds of self-consistent mean field models were developed to calculate the interaction potential between layers of flexible polymer chains grafted to spherical surfaces whose radii are comparable to the separation between the two surfaces. The calculations showed that for small particles the repulsion is less steep than that provided by the Derjaguin approximation, while with increasing radii, the interaction potential becomes close to that given by the... 

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. 

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 ... 

Differentiation gives the force between the large spheres. Thus, the force between two colloidal spheres can be calculated from Eq. (71) and some equation like Eq. (70) to yield the pressure [Lozada-Cassou uses what is called the Born-Green-Yvon equation, see Eq (97) below] or from Eqs. (72) and (76). The bridge between the two methods is the Derjaguin approximation, Eq. (64). In principle, either scheme is acceptable. The few calculations made so far suggest that Eqs. (72) and (76) give more accurate results. 

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