Derjaguin equation

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 force of cohesion, i.e. the maximum value of attractive force between the particles, may be determined by a direct measurement of force, F required to separate macroscopic (sufficiently large) particles of radius r, brought into a contact with each other. Such a measurement yields the free energy of interaction (cohesion) in a direct contact, A (h0) = Ff n r,. Due to linear dependence of F on r, one can then use F, to evaluate the cohesive force F2 = (r2/r )Fx, acting between particles in real dispersions consisting of particles with the same physico-chemical properties but of much smaller size, e.g. with r2 10 8 m (i.e. in the cases when direct force measurements can not be carried out). At the same time, in agreement with the Derjaguin equation... 

The Derjaguin equation for the force of the molecular adhesion between solid particles in a liquid medium can be written as... 

This equation can be transformed into a formula describing the interaction between curved surfaces, such as that between two spherical double-layers. This is carried out by using the Derjaguin equation. The latter connects the force between two spherical double-layers and the interaction energy per unit area, Vr, of two plane interacting double-layers. It is assumed that both the spherical and the plane double-layers carry the same surface charge density, which leads to the following ... 

Here we consider the total interaction between two charged particles in suspension, surrounded by tlieir counterions and added electrolyte. This is tire celebrated DLVO tlieory, derived independently by Derjaguin and Landau and by Verwey and Overbeek [44]. By combining tlie van der Waals interaction (equation (02.6.4)) witli tlie repulsion due to the electric double layers (equation (C2.6.lOI), we obtain... 

At the junction of the adsorbed film and the liquid meniscus the chemical potential of the adsorbate must be the resultant of the joint action of the wall and the curvature of the meniscus. As Derjaguin pointed out, the conventional treatment involves the tacit assumption that the curvature falls jumpwise from 2/r to zero at the junction, whereas the change must actually be a continuous one. Derjaguin put forward a corrected Kelvin equation to take this state of affairs into account but it contains a term which is difficult to evaluate numerically, and has aroused little practical interest. 

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] ... 

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. 

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.

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... 

FIG. 3 Comparison of the linear Derjaguin approximation with a numerical solution of the linear Poisson-Boltzmann equation for (a) constant potential and (b) constant charge density boundary conditions. (From Ref. 13.)... 

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. 

Differentiation gives the force between the large spheres. Thus, the force between two colloidal spheres can be calculated from Eq. (71) and some equation like Eq. (70) to yield the pressure [Lozada-Cassou uses what is called the Born-Green-Yvon equation, see Eq (97) below] or from Eqs. (72) and (76). The bridge between the two methods is the Derjaguin approximation, Eq. (64). In principle, either scheme is acceptable. The few calculations made so far suggest that Eqs. (72) and (76) give more accurate results. 

Around 1967, Broekhoff and de Boer [16], following Derjaguin [17], pointed out that the supposition introduced in the derivation of the Kelvin equation, viz. the equality of the thermodynamic potential of the adsorbed multilayer to the thermodynamic potential of the liquified gas (see Eqn. 12.28), cannot be correct. This can be seen immediately from an inspection of the common t curve (Fig. 12.5) at each t value lower than, say, 2 run, the relative equilibrium pressure is lower than 1, the equilibrium pressure of the liquefied gas. 

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]. 

From this equation it is dear that the gradient should be directly proportional to the particle radius a and inversely proportional to the square of the valence of the counterion used. A number of experimental studies have been made that do not seem to confirm these predictions (Ottewill and Shaw, 1966) and even more refined treatments of the kinetic process have not removed the discrepancy (Derjaguin and Muller, 1967 Honig et a ., 1971). There is little doubt that as a kinetic process coagulation is rather complicated further discussion on this point will be given later. 

This potential is valid for arbitrarily widely separated spheres, but it breaks down when the Debye double layers begin to overlap significantly (Russel et al. 1989). However, for small separations, kD < 2, the Derjaguin approximation may still be used, as long as D < a-, that is, Ka 2. An analytic expression can then be obtained if the potential is small enough that the Poisson-Boltzmann equation can be linearized (Russel et al. 1989, p. 117). For small separations between particles, a choice must be made between a constant-potential or a constant-charge boundary condition. For a constant-potential boundary condition, one can write the approximate expression... 

The DLVO theory, which was developed independently by Derjaguin and Landau and by Verwey and Overbeek to analyze quantitatively the influence of electrostatic forces on the stability of lyophobic colloidal particles, has been adapted to describe the influence of similar forces on the flocculation and stability of simple model emulsions stabilized by ionic emulsifiers. The charge on the surface of emulsion droplets arises from ionization of the hydrophilic part of the adsorbed surfactant and gives rise to electrical double layers. Theoretical equations, which were originally developed to deal with monodispersed inorganic solids of diameters less than 1 pm, have to be extensively modified when applied to even the simplest of emulsions, because the adsorbed emulsifier is of finite thickness and droplets, unlike solids, can deform and coalesce. Washington has pointed out that in lipid emulsions, an additional repulsive force not considered by the theory due to the solvent at close distances is also important. 

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. 

This entry is organized in the following paragraphs First, the advanced determination of van der Waals interaction between spherical particles is described. Second, the relevant approximate expressions and direct numerical solutions for the double-layer interaction between spherical surfaces are reviewed. Third, the experimental data obtained for AFM tips having nano-sized radii of curvature and the DLVO forces predicted by the Derjaguin approximation and improved predictions are compared. Finally, a summary of the review and recommended equations for determining the DLVO interaction force and energy between colloid and nano-sized particles is included. 

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