Derjaguin approximation

Some of the above results also follow directly from the so called Deijaguin approximation. Deijaguin [10] showed that there exists a simple (approximate) relation for the foree between curved objects and the interaction potential between two flat plates. In the Deijaguin approximation the spherical surface is replaced by a collection of flat rings. Consider two spheres with radius 1 at a center-to-center 

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. 

With respect to the depletion interaction the Derjaguin approximation becomes accurate when considering a depletion agent which is small compared to the radius of the colloidal spheres. For example, applying the Derjaguin approximation to (2.3), the case of penetrable hard spheres, using (2.28) 

Applying the Deijaguin approximation to the interaction between a sphere and 

This is an important relation as it allows one to obtain the interaction potential between two parallel plates from the measured force between a sphere and a wall (see Sect. 2.6) 

If we repeat this procedure for surface-surface interaction, we can write for unit area from Equation (550) 

For sphere-sphere interaction for spheres having two different radii, from Equation (543) we have 

However, Equation (557) can also be related to the interaction potential energy between surfaces 

The Derjaguin approximation can be applied to any type of force law, such as attraction, repulsion, or oscillation, if D is much less than the radii of the spheres. This has been verified experimentally and is very useful for interpreting experimental force data. The Derjaguin approximation shows that, even though the same pair-potential force is operating, the distance dependence of the force between two curved surfaces is guite different from that between two flat surfaces. 

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Surface forces measurement directly determines interaction forces between two surfaces as a function of the surface separation (D) using a simple spring balance. Instruments employed are a surface forces apparatus (SFA), developed by Israelachivili and Tabor [17], and a colloidal probe atomic force microscope introduced by Ducker et al. [18] (Fig. 1). The former utilizes crossed cylinder geometry, and the latter uses the sphere-plate geometry. For both geometries, the measured force (F) normalized by the mean radius (R) of cylinders or a sphere, F/R, is known to be proportional to the interaction energy, Gf, between flat plates (Derjaguin approximation). 

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 ... 

The geometry. It is clear that the geometry of the system is much simplified in the slab model. Another possibility is to model the protein as a sphere and the stationary phase as a planar surface. For such systems, numerical solutions of the Poisson-Boltzmann equations are required [33]. However, by using the Equation 15.67 in combination with a Derjaguin approximation, it is possible to find an approximate expression for the interaction energy at the point where it has a minimum. The following expression is obtained [31] ... 

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... 

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],... 

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.

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... 

Solution To think specifically, consider a sphere of radius R and a flat (or its Derjaguin-approximation equivalent, two perpendicular cylinders of radius R), and neglect all but nonretarded van der Waals forces. The force is the negative derivative of the free energy — ( Ham / 6) (R/ Z),... 

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... 

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