Total potential energy of interaction

Our discussion of solids and alloys is mainly confined to the Ising model and to systems that are isomorphic to it. This model considers a periodic lattice of N sites of any given symmetry in which a spin variable. S j = 1 is associated with each site and interactions between sites are confined only to those between nearest neighbours. The total potential energy of interaction... 

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... 

These are considered to be additives so that the total potential energy of interaction VT is given by ... 

For the case of purely attractive forces (such as Lon-don-van der Waals forces) the length Sjr over which they act is a useful characteristic. An attractive force which acts over a distance which is much less than Sc will not contribute substantially to the overall rate. When repulsive forces (such as the electrostatic double-layer forces) are also present, they may effectively prevent particles from arriving at the collector, even when they act only over a very short distance. For this reason the decay length alone cannot characterize the relative importance of the joint effect of attractive and repulsive farces. Useful characteristics of their combined effect may be obtained by considering the total potential energy of interaction between the particle and the collector. 

If the double-layer repulsion is sufficiently strong to cause the total potential energy of interaction to assume a maximum (larger than kT) in its dependence on the particle-collector separation Jt, Ruckenstem and Prieve (10) have shown that the apparent rate con-... 

The contribution from unretarded van der Waals forces to the total potential energy of interaction between a sphere of radius o, whose center is separated from the surface of a semiinfinite plate by a distance x -f- a, was calculated by Hamaker (14) ... 

For the calculations discussed below, the total potential energy of interaction v/as decomposed into contributions from van der Waals forces and double-layer forces VAW(h) + < >DL(fi). The van... 

Neglecting orientation effects, then, we deal only with the centers of gravity of the molecules. We must now ask, how docs the potential energy depend on these centers of gravity We have seen the general nature of Van der Waals s answer to this question. For the moment, let us simply write the total potential energy of interaction between two molecules i andj, at a distance rXi apart, as (rtJ). Then we may reasonably assume that the whole potential energy of the gas is... 

Here, Uiri ) is the total potential energy of interaction betweai molecule 1 and molecule 2 the functions... 

The total potential energy of interaction between the N molecules is, in some cases, separable into a central (radial) part U r dependent only on position, and a non-central (angular) part F(r, 12 ) d endent on molecular orientation As stated in Sections 2 and 3, the... 

From the various equations given, under the appropriate conditions of Kfl, h, etc. the total potential energy of interaction between the particles can be calculated from the basic DLVO assumption that for lyophobic dispersions. 

Fig. 4 The total potential energy of interaction Vt as a function of distance of surface separation H for two similar oh droplets in an oil-in-water emulsion. (A) Electrostatic stabilization by a monolayer of ionic surfactant. (B) Steric stabilization by a monolayer of non-ionic surfactant. V van der Waals attractive force Vr electrostatic repulsive force Vs steric repulsive force.

The total potential energy of interaction between two colloidal particles, C/fot, is given by the sum of the energy of interaction of the double layers, C/, and the energy of interaction of the particles themselves due to van der Waals forces, C/. Consequently,... 

Equations (2)-(4) show that the total potential energy of interaction between two colloidal spherical particles depends on the surface potential of the particles, the effective Hamaker constant, and the ionic strength of the suspending medium. It is known that the addition of an indifferent electrolyte can cause a colloid to undergo aggregation. Furthermore, for a particular salt, a fairly sharply defined concentration, called critical aggregation concentration (CAC), is needed to induce aggregation. 

In the DLVO theory the combination of the electrostatic repulsive energy 17 with the attractive potential energy Vj gives the total potential energy of interaction... 

Figure 7.6 The influence of the surface potential (t/>o) on the total potential energy of interaction of two spherical particles.

The forces acting on a colloidal system include gravitational, diffusion, viscous, inertial, attractive Van der Waals, and electrical repulsive forces. Because most of these forces are functions of the particle size, it is important to know both particles size and size distribution. The classical Derjaguin-Landau-Verwey-Overbeek (DLVO) theory describes colloid stability on the basis of pair interaction, considering only attractive van der Waals forces and repulsive electrostatic forces (Molina-Bolfvar and Ortega-Vinuesa, 1999). The total potential energy of interaction, Ujc, between two particles is defined as ... 

In writing (7.187), we have assumed that the solute molecules are completely dissociated into A and B in the liquid U( Nw, NA, NB) is the total potential energy of interaction among the Nw, NA and NB molecules at a specific configuration (with ND = NA = NB). 

The total potential energy of interaction V is the sum of the potential energy of attraction VA and that of repulsion Vr ... 

The form of the curve for the total potential energy of interaction V depends on the ratio of the particle size to the thickness of the electrical double layer... 

FIGURE 9-3 Influence of electrolyte concentration (as measured by k) on the total potential energy of interaction of two spherical particles. Reprinted with permission from J. Th. G. Overbeek in Colloid Science, Vol. 1, H. Kruyt (Ed.), Elsevier, Amsterdam, 1952, Chap. 6, p. 276. 

Essentially what is done is to choose a model for the electrolyte solution specifying what interactions are inherent in the model. For each type of interaction, the total potential energy of interaction calculated pair-wise, i.e. as a sum of contributions from all pairs of particles, can be formulated. 

Total Potential Energy of Interaction See Gibbs Energy of Interaction. 

Figure 6. The effect of different repulsive potential energy curves VR (1) and VR (2) on the total potential energy of interaction curves V (I) and V (2), for a given attractive potential energy curve. (Reproduced with permission from reference 4. Copyright 1981 Butterworth-Heinemann.)...

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