Derjaguin integration

Again a Derjaguin integration can be used to evaluate these expressions for the elastic repulsion for spheres. As demanded by physical considerations. 

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... 

In this equation, S =(1-I-Z-3/2L2+L2/2jI-3- I o(1/jI 2 + 1/ 3) + Ho I2L2L ), where Ho = minimum distance of separation of the surfaces of the particles. Of course, the Derjaguin integration procedure is only valid if a L2, Z-3. Note that if polymers 2 and 3 are identical, equation (14.12) reduces to the result derived previously for homosteric stabilization. 

When the repulsion barrier is large (i.e., < max is about 10 kBTor larger), one can evaluate the integral in the expression for Wusing what are known as asymptotic techniques and obtain the following expression (Derjaguin 1989, p. 162) ... 

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. 

The name, DLYO, originates from the first letter in the surname of the four authors (Derjaguin, Landau, Verwey and Overbeek) from two different groups, which originally published these ideas. The theory is based on the competition between two contributions, a repulsive electric double layer and an attractive van der Waals force [4,5]. The interaction in the electric double layer was originally obtained from mean field calculations via the Poisson-Boltzmann equation [Eq. (4)]. However, the interaction can also be determined by MC simulations (Sec. II. B) and by approximate integral equations like HNC (Sec. II. C). This chapter will focus on the first two possibilities. 

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... 

This result can be used to obtain the force F between two spheres of radius a, with D a, by using the Derjaguin approximation that relates the force between two spheres to the potential between two flat plates that is, F (spheres) = traWe (flat plates). When this is integrated to obtain the potential, one obtains... 

The interaction energy between the particles is then obtained by integrating Eq. (7) from infinity to the separation distance h. The Derjaguin approximation can be applied to van der Waals, double-layer, and many other interactions. For van der Waals interaction, either Eq. (1) or (3) for E h) can be substituted into Eq. (7) to determine the force between two spheres. 

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 ... 

Note that the value of W j is determined mainly by the values of the integrand in the vicinity of the electrostatic maximum (barrier) of f/y (see Eigure 5.13), insofar as U j enters Equation 5.325a as an exponent. By using the method of the saddle point, Derjaguin estimated the integral in Equation 5.325a ... 

Collision efficiency was calculated by the method proposed for the first time by Dukhin Derjaguin (1958). To calculate the integral in Eq. (10.25) it is necessary to know the distribution of the radial velocity of particles whose centre are located at a distance equal to their radius from the bubble surface. The latter is presented as superposition of the rate of particle sedimentation on a bubble surface and radial components of liquid velocity calculated for the position of particle centres. Such an approximation is possibly true for moderate Reynolds numbers until the boundary hydrodynamic layer arises. At a particle size commensurable with the hydrodynamic layer thickness, the differential of the radial liquid velocity at a distance equal to the particle diameter is a double liquid velocity which corresponds to the position of the particle centre. Such a situation radically differs from the situation at Reynolds numbers of the order of unity and less when the velocity in the hydrodynamic field of a bubble varies at a distance of the order ab ap. At a distance of the order of the particle diameter it varies by less than about 10%. Just for such conditions the identification of particle velocity and liquid local velocity was proposed and seems to be sufficiently exact. In situations of commensurability of the size of particle and hydrodynamic boundary layer thickness at strongly retarded surface such identification leads to an error and nothing is known about its magnitude. 

Spheres. One advantage of these analytical formulae is that they allow simple analytic results to be established for spheres by means of the Derjaguin (1934) integration procedure. This permits flat plate potentials to be converted into sphere potentials. The mixing free energy for spheres obtained in this fashion... 

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... 

Joanny et al. (1979) then used a simple Derjaguin-type integration to examine the repulsion between two spherical particles, each of radius a a i ). 

Figure 2. Integral isotherm of wedging stress at the boundary hydrate coating [Derjaguin, Churaev, 1984],...

Using the Derjaguin approximation (2.27) we obtain the interaction between two big spheres with radius / 3> D by integration... 

The numerical integration of Equation 4.110 can be carried out by using the boundary condition [199] z /z = -qK (qr)/Ko(qr) for some appropriately fixed r q (see Equation 4.117). Alternatively, approximate analytical solutions of the problem are available [199,210,213]. In particular, Derjaguin [214] derived an asymptotic formula for the elevation of the contact line at the outer surface of a thin cylinder,... 

The theories developed for calculating the oscillatory force are based on modeling by means of the integral equations of statistical mechanics [449-453] or numerical simulations [454-457]. As a rule, these approaches are related to complicated theoretical expressions or numerical procedures, in contrast with the Derjaguin-Landau-Verwey-Overbeek (DLVO) theory, one of its main advantages being its simplicity [36], To overcome this difficulty, some relatively simple semiempirical expressions have been proposed [458,459] on the basis of fits of theoretical results for hard-sphere fluids. 

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