Derjaguin’s approximation

As one example we calculate the van der Waals force between two identical spheres from the van der Waals energy per unit area w = -Ah/V2tix2 (Eq. 6.17). Using Derjaguin s approximation we can directly write... 

Derjaguin s approximation allows us to calculate the force (or energy) between bodies of arbitrary shape from the force per unit area (or energy per unit area) between two planar surfaces, provided the curvature of the bodies is small compared to the typical decay lengths of the forces involved. 

Dispersion in aqueous electrolyte. According to Derjaguin s approximation (Eq. 6.36) the force between two particles is... 

Two approaches for the calculation of the double-layer contribution are explored. Hogg el of. (16) linearized the Poisson- -Boltzmann equation to compute the double-layer force between two dissimilar plane surfaces, then used Derjaguin s approximation to extend this result to the interaction of two spheres of different radii. When the radius of one sphere is infinite, their result becomes... 

Bell el al. (17) have found that, when wz > 5 and kx < 2, the error in using Derjaguin s approximation is less than 10%. This was based on the interaction of two spherical particles. For sphere-plane interactions, the error should be even less. To calculate the rates according to Eqs. [l] through [3], one needs accurate estimates of m. near 3 mOXf which is typically of the order of jc-1 hence... 

Using Derjaguin s approximation, the force between two uncharged spheres (ps) of diameter a separated by a center-to-center distance of l can be calculated from the force between two parallel plates (p) using the expression28... 

Using eq 27 and Derjaguin s approximation, one obtains for the force between two spherical particles immersed in a solution of particles of polydisperse sizes the expression... 

We apply the Derjaguin s approximation (Eq. (12.3)) to the low-potential approximate expression for the plate-plate interaction energy, that is, Eqs. (9.53) and (9.65), obtaining the following two formulas for the interaction between two similar spheres 1 and 2 of radius a carrying unperturbed surface potential ij/f, at separation H at constant surface potential, V (H), and that for the constants surface charged density case, V (//) ... 

Arbitrary Potentials Derjaguin s Approximation Combined with tbe Linear Superposition Approximation... 

In Chapter 11, we derived the double-layer interaction energy between two parallel plates with arbitrary surface potentials at large separations compared with the Debye length 1/k with the help of the linear superposition approximation. These results, which do not depend on the type of the double-layer interaction, can be applied both to the constant surface potential and to the constant surface charge density cases as well as their mixed case. In addition, the results obtained on the basis of the linear superposition approximation can be applied not only to hard particles but also to soft particles. We now apply Derjaguin s approximation to these results to obtain the sphere-sphere interaction energy, as shown below. 

FIGURE 12.3 Derjaguin s approximation for the two interacting parallel cylinders 1 and 2 at separation H, having radii ai and A2> respectively. 

DERJAGUIN S APPROXIMATION AT SMALL SEPARATIONS For the special case of two identical crossed cylinders (aj = a2),... 

Consider the validity of Derjaguin s approximation. In this approximation, the interaction energy between two spheres of radii oj and 02 at separation H between their surfaces is obtained by integrating the corresponding interaction energy between two parallel membranes at separation h via Eq. (13.28). We thus obtain... 

We see that the first term of the expansion of Eq. (13.52) indeed agrees with Deijaguin s approximation (Eq. (13.51)). That is, Derjaguin s approximation yields the correct leading order expression for the interaction energy and the next-order correction terms are of the order of 1/Ka, lKa2, and Hl a +a2). 

This chapter deals with a method for obtaining the exact solution to the linearized Poisson-Boltzmann equation on the basis of Schwartz s method [1] without recourse to Derjaguin s approximation [2]. Then we apply this method to derive series expansion representations for the double-layer interaction between spheres [3-13] and those between two parallel cylinders [14, 15]. 

Also we see from Eq. (14.59) that the next-order curvature correction to Deija-guin s approximation [2] is of the order of l/y/ica) (1=1, 2). This has been suggested first by Dukhin and Lyklema [20] and discussed by Kijlstra [21]. In the case of the interaction between particles with constant surface potential, on the other hand, the next-order curvature correction to Derjaguin s approximation is of the order of l/jca, (i = 1, 2), since in this case no electric fields are induced within the interacting particles. 

Consider first the case of two parallel soft cylinders (Fig. 15.7). With the help of Derjaguin s approximation for two parallel cylinders [8,9] (Eq. (12.38)), namely,... 

With the help of Derjaguin s approximation (see Chapter 12), one can derive the van der Waals interaction energy between two crossed cylinders of radii oj and a2 at separation //between their surfaces (Fig. 19.12). By substituting Eq. (19.20) into Eq. (12.48), we obtain... 

For nonspherical, particulate depletants, the orientational degrees of freedom are reduced upon the approach to the colloids surfaces, which like in the case of polymeric depletants leads to a continuous variation of the depletant numher density along the surface normal. In these cases, the depletion potential is usually calculated in a two-step procedure. First, the potential or the force between two flat surfaces is derived, from which the potential between spherical particles can be calculated using Derjaguin s approximation. The latter states that the potential between two curved surfaces can be calculated, if the functional form of the potential energy per surface... 

Figure 2.5 Schematic of Derjaguin s approximation for a rotational symmetric body interacting with a planar surface.

Figure 2.7 Calculating the interaction between two spheres with Derjaguin s approximation.

Using Derjaguin s approximation, we can calculate the van der Waals force between objects of different shape (for an extended list, see Ref [3]). We start with a particularly instructive case, the van der Waals force between two slabs of thickness d (Figure 2.8a). The energy per unit area is... 

In an experiment, we typically measure the force between two spheres or a sphere and a plane. To calculate the force between a sphere and a plane, we use Derjaguin s approximation (2.73) and integrate the force per unit area. The result is [1105]... 

The force for a sphere of radius Ep approaching a planar surface with adsorbed molecules is according to Derjaguin s approximation (2.67) [1307]... 

With the given parameters lo = 15.8 nm. The force between a sphere and a planar surface is F = 2jtFp Va using Derjaguin s approximation. For the bare sphere and the polymer brush, we have... 

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