Derjaguin theory

FIGURE 6.8 Schematic view of the potential profile evolution (a, dashed lines) and resulting interaction force (b) for two dissimilar plane double layers. The upper pane in (b) shows the signs of surface charges and interaction force for different values of the separation distance d. In (a), the closed circles indicate the potential value of the second surface and the continuous line, the profile for an isolated surface. (Adapted from Adv. Colloid Interface ScL, 42, Kihira, H., E. Matijevic, An assessment of heterocoagulation theories, 1-31. Copyright 1992, with permission from Elsevier.) 

Extensive numerical results were presented for the case of plane surfaces by Devereux and De Bruyn (1963) however, some results have been challenged (Lyklema 2006). 

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B. Y. Derjaguin, Theory of Stability of Colloids and Thin Films, Consultants Bureau, New York, 1989. 

B. V. Derjaguin, Theory of Stability of Thin Films and Colloids. New York Consultants Bureau/Plenum Press (1989). 

Classic nucleation theory must be modified for nucleation near a critical point. Observed supercooling and superheating far exceeds that predicted by conventional theory and McGraw and Reiss [36] pointed out that if a usually neglected excluded volume term is retained the free energy of the critical nucleus increases considerably. As noted by Derjaguin [37], a similar problem occurs in the theory of cavitation. In binary systems the composition of the nuclei will differ from that of the bulk... 

The combined effect of van der Waals and electrostatic forces acting together was considered by Derjaguin and Landau (5) and independently by Vervey and Overbeek (6), and is therefore called DLVO theory. It predicts that the total interaction energy per unit area, also known as the effective interface potential, is given by V(f) = ( ) + dl ( )- absence of externally imposed forces, the equiHbrium thickness of the Hquid film... 

DLVO Theory. The overall stabiUty of a particle dispersion depends on the sum of the attractive and repulsive forces as a function of the distance separating the particles. DLVO theory, named for Derjaguin and Landau (11) and Verwey and Overbeek (12), encompasses van der Waals attraction and electrostatic repulsion between particles, but does not consider steric stabilization. The net energy, AGp between two particles at a given distance is the sum of the repulsive and attractive forces ... 

Two kinds of barriers are important for two-phase emulsions the electric double layer and steric repulsion from adsorbed polymers. An ionic surfactant adsorbed at the interface of an oil droplet in water orients the polar group toward the water. The counterions of the surfactant form a diffuse cloud reaching out into the continuous phase, the electric double layer. When the counterions start overlapping at the approach of two droplets, a repulsion force is experienced. The repulsion from the electric double layer is famous because it played a decisive role in the theory for colloidal stabiUty that is called DLVO, after its originators Derjaguin, Landau, Vervey, and Overbeek (14,15). The theory provided substantial progress in the understanding of colloidal stabihty, and its treatment dominated the colloid science Hterature for several decades. 

To account for some of the shortcomings of the JKR theory, Derjaguin and coworkers [19] developed an alternative theory, known as the DMT theory. According to the DMT theory, the attractive force between the surfaces has a finite range and acts outside the contact zone, where the surface shape is assumed to be Hertzian and not deformed by the effect of the interfacial forces. The predictions of the DMT theory are significantly different compared to the JKR theory. 

By combining Hertz s contact theory (Eq. 1) and with Hamaker s functional form for the attractive force (Eq. 17), the Derjaguin model takes the form... 

Johnson, Kendall and Roberts used an energy-based contact mechanics approach to understand particle adhesion. In their theory, they deviated from the earlier Derjaguin and Krupp models by assuming that tensile stresses are present... 

As indicated, an implicit assumption of the JKR theory is that there are no interactions outside the contact radius. More specifically, the energy arguments used in the development of the JKR theory do not allow specific locations of the adhesion forces to be determined except that they must be associated with the contact line where the two surfaces of the particle and substrate become joined. Adhesion-induced stresses act at the surface and not a result of action-at-a-distance interatomic forces. This results in a stress singularity at the circumference of the contact radius [41]. The validity of this assumption was first questioned by Derjaguin et al. [42], who proposed an alternative model of adhesion (commonly referred to as the DMT theory ). Needless to say, the predictions of the JKR and DMT models are vastly different, as discussed by Tabor [41]. 

Hertzian mechanics alone cannot be used to evaluate the force-distance curves, since adhesive contributions to the contact are not considered. Several theories, namely the JKR [4] model and the Derjaguin, Muller and Torporov (DMT) model [20], can be used to describe adhesion between a sphere and a flat. Briefly, the JKR model balances the elastic Hertzian pressure with attractive forces acting only within the contact area in the DMT theory attractive interactions are assumed to act outside the contact area. In both theories, the adhesive force is predicted to be a linear function of probe radius, R, and the work of adhesion, VFa, and is given by Eqs. 1 and 2 below. 

An issue, at present unresolved, is that Derjaguin, Muller and Toporov [24,25] have put forward a different analysis of the contact mechanics from JKR. Maugis has described a theory which comprehends both the theories as special cases [26]. 

The Yukawa potential is of interest in another connection. According to the Derjaguin-Landau-Verwey-Overbeek (DLVO) theory, colloidal... 

Hence, for two similarly charged surfaces in electrolyte, interactions are determined by both electrostatic doublelayer and van der Waals forces. The consequent phenomena have been described quantitatively by the DLVO theory [6], named after Derjaguin and Landau, and Verwey and Over-beek. The interaction energy, due to combined actions of double-layer and van der Waals forces are schematically given in Fig. 3 as a function of distance D, from which one can see that the interplay of double-layer and van der Waals forces may affect the stability of a particle suspension system. 

At short interparticle distances, the van der Walls forces show that two metallic particles will be mutually attracted. In the absence of repulsive forces opposed to the van der Walls forces the colloidal metal particles will aggregate. Consequently, the use of a protective agent able to induce a repulsive force opposed to the van der Walls forces is necessary to provide stable nanoparticles in solution. The general stabihzation mechanisms of colloidal materials have been described in Derjaguin-Landau-Verway-Overbeck (DLVO) theory. [40,41] Stabilization of colloids is usually discussed... 

Derjaguin, B. V. and Landau, L. (1941) Theory of the stability of strongly charged particles in solutions of electrolytes. ActaPhysiochim., URSS., 14, 633—662. 

BA Matthews, CT Rhodes. Use of the Derjaguin, Landau, Verwey and Overbeek theory to interpret pharmaceutical suspension stability. J Pharm Sci 59 521-525, 1970. 

Comparison of the proposed dynamic stability theory for the critical capillary pressure shows acceptable agreement to experimental data on 100-/im permeability sandpacks at reservoir rates and with a commercial a-olefin sulfonate surfactant. The importance of the conjoining/disjoining pressure isotherm and its implications on surfactant formulation (i.e., chemical structure, concentration, and physical properties) is discussed in terms of the Derjaguin-Landau-Verwey-Overbeek (DLVO) theory of classic colloid science. 

The stabilization mechanisms of colloidal materials have been described in Derjaguin-Landau-Verway-Overbeek (DLVO) theory [8, 9]. Colloids stabilization is usually discussed in terms of two main categories, namely charge stabilization and steric stabilization. 

The pair potential of colloidal particles, i.e. the potential energy of interaction between a pair of colloidal particles as a function of separation distance, is calculated from the linear superposition of the individual energy curves. When this was done using the attractive potential calculated from London dispersion forces, Fa, and electrostatic repulsion, Ve, the theory was called the DLVO Theory (from Derjaguin, Landau, Verwey and Overbeek). Here we will use the term to include other potentials, such as those arising from depletion interactions, Kd, and steric repulsion, Vs, and so we may write the total potential energy of interaction as... 

The interplay of forces between particles in lyophobic sols may be interpreted in terms of the theory of Derjaguin and Landau(20) and Verwey and Overbeek(14). Their theory... 

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