Waals Interaction Between Two Particles

In paper [2.12] it is shown that the van der Waals interactions between two particles can be decreased if they are covered with layers that have a Hamaker constant which is near to that of the suspending liquid. It is also suggested that, at sufficiently high concentrations, the collective behavior of the colloidal particles can generate repulsion when the pairwise interactions are attractive. These two effects are suggested to be responsible for the kinetic stability of the system, and a methodology for achieving kinetic stability is provided. 

An intervening medium always diminishes the van der Waals interaction between two particles. The degree of decrease can be estimated from following simple equation ... 

Similarly, van der Waals forces operate between any two colloidal particles in suspension. In the 1930s, predictions for these interactions were obtained from the pairwise addition of molecular interactions between two particles [38]. The interaction between two identical spheres is given by... 

The Hamaker constant A can, in principle, be determined from the C6 coefficient characterizing the strength of the van der Waals interaction between two molecules in vacuum. In practice, however, the value for A is also influenced by the dielectric properties of the interstitial medium, as well as the roughness of the surface of the spheres. Reliable estimates from theory are therefore difficult to make, and unfortunately it also proves difficult to directly determine A from experiment. So, establishing a value for A remains the main difficulty in the numerical studies of the effect of cohesive forces, where the value for glass particles is assumed to be somewhere in the range of 10 21 joule. 

The major problem in calculating the van der Waals interaction between colloidal particles is that of evaluating the Hamaker constant, A. Two methods are available. 

The presence of a liquid dispersion medium, rather than a vacuum (or air), between the particles (as considered so far) notably lowers the van der Waals interaction energy. The constant A in equations (8.8)-(8.10) must be replaced by an effective Hamaker constant. Consider the interaction between two particles, 1 and 2, in a dispersion medium, 3. When the particles are far apart (Figure 8.1a),... 

Void [11] has established Hqn(I) for the attractive van dcr Waals interactions between two spherical particles of radius R covered with a shell of a different material of thickness 5, by assuming pair-wise additivity of molecular interactions. 

The electrostatic stabilization theory was developed for dilute colloidal systems and involves attractive van dcr Waals interactions and repulsive double layer interactions between two particles. They may lead to a potential barrier, an overall repulsion and/or to a minimum similar to that generated by steric stabilization. Johnson and Morrison [1] suggest that the stability in non-aqueous dispersions when the stabilizers are surfactant molecules, which arc relatively small, is due to scmi-stcric stabilization, hence to a smaller ran dcr Waals attraction between two particles caused by the adsorbed shell of surfactant molecules. The fact that such systems are quite stable suggests, however, that some repulsion is also prescni. In fact, it was demonstrated on the basis of electrophoretic measurements that a surface charge originates on solid particles suspended in aprotic liquids even in the absence of traces of... 

From Eqs (8) and (9) it is clear that the van der Waals interaction between two identical particles or emulsion... 

As in part II, we shall assume also in this part that the attraction between coUoidal particles is entirely based upon the London-Van der Waals forces. Hamaker showed how the London-Van der Waals interaction between two spherical particles may be found from the interaction b etween the elements of these spheres. His expression for the energy of attraction Va) runs, using our symbols,... 

The stability of suspensions containing solid particles are treated in the framework of the Derjaguin-Landau-Verwey-Overbeek (DLVO) theory, which accounts for the electrostatic and van der Waals interactions between the particles (Verwey and Overbeek 1948, Derjaguin 1989). In the past decades it has been shown that other types of inter-particle forces may also play an important role in the stability of dispersions - hydrodynamic interactions, hydration and hydrophobic forces, steric and depletion forces, oscillatory structural forces, etc. The hydrodynamic and molecular interactions between surfaces of drops and bubbles in emulsion and foam systems (compared to that of suspensions of solid particles) are more complex due to the particles fluidity and deformability. These two features and the possible thin film formation between the colliding particles have a great impact on the hydrodynamic interactions, the magnitude of the disjoining pressure and on the dynamic and thermodynamic stability of such systems (Ivanov and Dimitrov 1988, Danov et al. 2001, Kralchevsky et al. 2002). 

The total interaction between any two surfaces must also include the van der Waals attraction, which is largely insensitive to variations in electrolyte concentration and pH, and so may be considered as fixed for any particular solute-solvent system. Further, the van der Waals attraction wins out over the double-layer repulsion at small distances, since it is a power-law interaction, whereas the double-layer interaction energy remains finite or rises much more slowly as 0. This is the theoretical prediction that forms the basis of the so-called Derjaguin-Landau-Verwey-Over-beek (DLVO) theory (illustrated in fig. 7.2) [15]. In the DLVO theory, the interaction between two particles is assumed to consist of two contributions the van der Waals attraction and the electrostatic double-layer repulsion. At low salt concentration, the... 

The three most important forces for the long range interaction between macroscopic particles and a surface are steric-polymer forces, electrostatic interactions and Van der Waals forces. If we assume than the Van der Waals interactions between two atoms in a vaccuum are non-retarded and additive, we saw in the previous chapter that the form of the Van der Waals pair potential is w = —CJD where C is the coefficient in the atom-atom pair potential and D is the distance between the two... 

HI of, HI ay, and so on, inside the bracket of Eq. (2.9) can be safely neglected because the ratio H a is much smaller than unity. A schematic representation of the van der Waals interaction between two approaching particles is shown in Figure 2.3. It is the exceedingly deep potential energy well observed at small distance of separation between two interactive particles that is responsible for the loss of colloidal stability. The attractive van der Waals force decreases rapidly with increasing distance of separation. 

In Fig. 14 we consider the interaction between two particles of substance 1 embedded in substance 2 at two different distances. One particle is sketched as a square one, the other as a round one. When the round particle is brought nearer to the square one, inevitably an equal volume of liquid is displaced and taken farther away from the square particle. In order to evaluate all the energies involved, it is necessary to consider a "square volume of liquid which is not displaced and which is near the round particle or the round volume of liquid in the first or the second situation. The Van Der Waals energy of the whole system remains unaltered except for the interactions between the two particles and the two volumes of liquid sketched in Fig. 14. The change in Van Der Waals ener between the two situations is given by... 

Some studies have been made of W/O emulsions the droplets are now aqueous and positively charged [40,41 ]. Albers and Overbeek [40] carried out calculations of the interaction potential not just between two particles or droplets but between one and all nearest neighbors, thus obtaining the variation with particle density or . In their third paper, these authors also estimated the magnitude of the van der Waals long-range attraction from the shear gradient sufficient to detach flocculated droplets (see also Ref. 42). 

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... 

Note that Eq. (36) exhibits an apparent numerical singularity in that the van der Waals interaction diverges if the surface distance between two particles approaches zero. In reality, such a situation will never occur because of the... 

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