Deijaguin approximation

Fig. VI-2. Schematic representation of the geometry used for the Deijaguin approximation.

From the discussion about the Deijaguin approximation for spheres and Eq. VI-26, show that the approximation for two crossed cylinders of radius R is... 

There are, however, colloidal systems, for which the above requirements are not fulfilled. For example, the surfaces of the proteins can have small radii, and their acidic and basic groups are not completely dissociated. In this case, the Verwey—Overbeek approach cannot be employed and the Langmuir—Deijaguin approximation is not accurate. 

The interaction between two identical spherical particles of radius a, separated by a distance r between their centers, is calculated using a variant of the Deijaguin approximation (see Appendix)... 

Another difficulty in using the Deijaguin approximation arises when the charge is related to the surface potential via various ionic equilibria. Indeed, in the linear approximation, the surface potential is related to the surface charge density, o, of a single planar surface by... 

The free energy of the system due to repulsion was calculated using the Deijaguin approximation, hence by expressing the repulsive free energy between two spherical particles in terms of that between two parallel plates. 

The interaction forcep between two crossed cylinders of radius R, for which experimental data are available,34 can be obtained from the interaction free energy F between two parallel plates using the Deijaguin approximation... 

Equation (102) can be combined with the Deijaguin approximation. HLC obtained the integral numerically. They fit the results to an analytic expression. Proceeding in this manner, they obtained... 

The two primary features of the phenomena are the layer thickness necessary to provide stability and the conditions at which the dispersions flocculate. The first can be quantified by generalizing the potential for terminally anchored chains to interactions between spheres via the Deijaguin approximation, adding the attractive dispersion potential, and then assessing the layer thickness necessary to maintain —fl>mi /fcT < 1 — 2. To illustrate this, consider the small overlap limit of Eq. (122), which transforms into... 

The main features of the Deijaguin approximation are the following (1) It is applicable to any type of force law (attractive, repulsive, oscillatory), if the range of the forces is much smaller than the particles radii, and (2) it reduces the problem for interactions between particles to the simpler problem for interactions in plane-parallel films. 

The effects of solvency. Figure 13.13 displays Klein s results for the interactional energy per unit area, in the Deijaguin approximation, as a function of the reduced distance D/Rg) for two polystyrene samples at two temperatures below the 0-temperature (26-5 °C and 24 °C). It is apparent that, as expected intuitively, the depth of the potential energy well increases as the solvency of the dispersion medium for the stabilizing segments becomes worse. 

The force measured between crossed cylinders (F ), as in the SEA, and between spheres (Fg), as in the MASIF, a distance D apart is normalized by the local geometric mean radius (R). This quantity is related to die fiee energy of interaction per unit area between flat surfaces (W) according to the Deijaguin approximation (30) ... 

The interaction energy potential is generally easier to derive for flat surfaces on the other hand, it is usually more accurate to measure the force-distance relation (F(Z))) between two curved surfaces, because the interaction area is precisely definable. The relationship between U D) per unit area for two flat surfaces and F D) between two curved surfaces is given by the Deijaguin approximation (Deijaguin 1934) which is derived for the close approach of two spheres of radii R and Ri ... 

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... 

Applying the Deijaguin approximation to the interaction between a sphere and... 

The limitation of the Deijaguin approximation is that it only provides reliable results for R > Rg. To obtain results for the interaction potential between spheres for arbitrary q = Rg/R we use the extended Gibbs adsorption equation. Taniguchi et al. [33] and, independently, Eisenriegler et al. [34] found the concentration profile of Gaussian ideal polymer chains around a single hard sphere with radius R which reads... 

Using the Deijaguin approximation (2.27) we obtain the interaction between two big spheres due to the small spheres by integration ... 

In [62] an analysis of the accuracy of the Deijaguin approximation for depletion potentials is presented. From this analysis it follows that the depletion potential of large spheres due to small spheres is underestimated by the Dejjaguin approximation, is surprisingly accurate for disks and is overestimated for rod-like depletion agents. A statistical mechanical analysis of the Deijaguin approximation applied to depletion interactions in colloidal fluids is presented by Henderson [63]. 

The calculated force, F(D), is normalized by the effective local radius, 1 o 18(2) mm of curvature. The mica surfaces showed only minor elastic flattening during compression of the polymer film, as seen from an analysis of the interference fringes. Using the Deijaguin approximation, one can thus... 

Equation 4.185 can be also derived by combining Equation 4.179 with the Deijaguin approximation (Equation 4.174). It is worthwhile noting that the logarithmic term in Equation 4.183 can be neglected only if jc 1. For example, even when jc = 5 x 10", the eontribution of the logarithmic term amounts to about 10% of the result (for y = 1) consequently, for larger values of x this term must be retained. 

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