Derjaguin’s equation

The adhesion between the IA L (its hydrophobic part) and an adjacent nonpolar phase can be modeled by the adhesion, in the same Uquids, of the modified soUd surfaces simulating the hydrophobic parts of corresponding surfactants. In this method, experimental complications coimected with the mutual solubility of components are prevented. For the solid/liquid interface, the principal quantitative characteristic of interaction, the free energy of interaction F (mj m , in the plain-parallel gap or film) can be established experimentally from Derjaguin s equation p = kRF, where p is the cohesive force in the immediate contact between two spherical particles immersed in the corresponding liquid medium [29, 30]. 

We have shown that for methylated surfaces Gfiim is small with resped to a, and Derjaguin s equation gives F/2 p/2tti [15,16, 21]. For methylated surfaces... 

The requirement (1) results in the Euler s equation (Equation 2.23), which for the first time has been suggested by Derjaguin [1] and should be referred to as Derjaguin s equation, where disjoining pressure is introduced as... 

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... 

Figure 11.3 is a plot of reduced thermophoretic velocity as a function of Knudsen number showing some experimental data along with curves for Brock s and Derjaguin and Yalamov s equations. It can be seen that although these equations all predict the form of the data set, there appears to be still much room for improvement in both data analysis and theory. 

Using Derjaguin and Yalamov s equation (Eq. 11.17), determine the thermophoretic force on a l-pm-diameter sodium chloride particle. For this calculation use Ca = 1.147, Ct = 2.20, and Cm = 1.146. How does this estimate of thermal force compare with the estimate made by using Epstein s equation (Eq. 11.11) and Brock s equation (Eq. 11.14) ... 

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]. 

As a result, the equilibrium contact angle 6 may be calculated from the known equation of Frumkin-Derjaguin s theory (Frumkin 1938, Derjaguin 1940),... 

In the following sections, however, we shall introduce the farther-reaching London — Van Der Waals attraction, when it will be most Important to have at our disposal an accurate repulsion curve, the advantage of a simpler equation being far less than in Derjaguin s case. 

As Langmuir s paper will be considered separately and the paper by Bergmann c.s. runs on somewhat different lines (explanation of the formation of Schiller layers in sols of blade-shaped particles), we shall here summarize only briefly the way in which the force equation is involved inr Derjaguin s theory of stability. A drawback of this Work is that the double layer repulsion i only considered in the linear approximation (v fe kT), excluding all possibility... 

Equation 9 can be conveniently utilized to derive the interaction potential between two spherical EDLs, following the Derjaguin s method [2]. This is done by utilizing the fact that the EDL interactions between two thin parallel slices of the spherical surfaces are essentially governed by Eq. 9. For the sake of analysis, let us consider two spherical surfaces at a minimum distance of separation of Hq, as depicted in Fig. 2. From Fig. 2, the total potential energy of interaction between two infinitesimally small parallel rings of width dx and radius x can be described as... 

Because this equation was so similar to Bradley s equation, F = —itWD/2, for the adhesion of rigid spheres, Derjaguin concluded from this analysis that the elastic contact deformation made little difference to the adhesion force. 

The above discussion of emulsion stability is further supported by the stability factors calculated with the help of equations 11, 12, and 14. The results are given in Table VI. All values are greater than 10 KT, the condition derived by Derjaguin for stable dispersions. These values are higher than the corresponding values of V/KT, because in this estimation, the effect of van der Waal s interaction is not taken into account. 

The AYL equation that forms the basis for Philip s (1977a) approach was modified by adsorption terms proposed by Derjaguin et al. (1987) and Iwamatsu and Horii (1996) ... 

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. 

From Waldmann s derivation of thermodynamically consistent boundary conditions for higher-order constitutive equations (valid for small Kn ) [2.103], it appears that the theoretical developments just cited, including those of DERJAGUIN and co-workers, have proceeded from either thermodynamically inconsistent boundary conditions or inaccurate (e.g.. Maxwell s) boundary conditions. Therefore, the apparent agreement suggested, for example by SPRINGER [2.134], PHILLIPS [2.128], or ANNIS and MASON [2.135] may be fortuitous, particularly for the slip regime. 

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