Interaction Between Two Spheres

1 Interaction Between Two Spheres from the Derjaguin Approximation 

Using the Derjaguin approximation (2.27) for the force between two spheres the interaction potential between the spheres can be obtained from 

Comparing the expression (2.21) for penetrable hard spheres and (2.58) for ideal chains reveals that we match the contact potentials for 

The result a = 2.35Rg agrees closely (within 5%) with the value r = ARgf /n = 2.2 Rg for flat plates. Hence in the limit R Rg ideal polymers behave almost as penetrable hard spheres with a diameter c 2Rg. just as for ideal chains between flat plates. In the next section we will see that this picture changes when R Rg. 

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Figure 6.13 Diagram used to explain the Derjaguin approximation for the interaction between two spheres.

In the same way it is possible to calculate the van der Waals energy between solids having different geometries. One important case is the interaction between two spheres with radii Ri and i 2. The van der Waals energy is [115]... 

Figure 6.5 Calculating the interaction between two spheres with Der-jaguin s approximation.

Equation (6.36) refers to the interaction between two spheres. For a sphere which is at a distance D from a planar surface we get a similar relation ... 

Figure 1. Schematic illustrating the form of the curve of potential energy vs. distance for the interaction between two spheres. Total interaction involving electrostatic repulsion and van der Waals attraction.

Lee, K. C. (1979). Aerodynamic Interactions Between Two Spheres at Reynolds Number Around 104. Aerosp. Q., 30, 371. 

Tsuji, Y., Morikawa, Y. and Terashima, K. (1982). Fluid-Dynamic Interaction Between Two Spheres. Int. J. Multiphase Flow, 8,71. 

The theory has been developed for two special cases, the interaction between parallel plates of infinite area and thickness, and the interaction between two spheres. The original calculations of dispersion forces employed a model due to Hamaker although more precise treatments now exist [194],... 

In 1934, B. V. Derjaguin showed how the interaction between two spheres or between a sphere and a plane near contact could be derived from the interaction between facing plane-parallel surfaces.1 There were two conditions ... 

Solution This weakness of inverse-sixth-power van der Waals forces between small particles is discussed at length in the main text. Its thermal triviality is easily seen. Begin with [—(16/9)](R6/z6)AHam for the energy of interaction between two spheres of radius R and center-to-center separation z and ask what the Ah am would have to be for the magnitude of this energy to be comparable with kT, (16/9)(R6/z6)AHam = kT or Anam = (9/16)(z6/R6)kT. Even if the center-to-center separation z were equal to 4R, spheres separated by a distance equal to their diameter, R6/z6 would be 46 = 4096. Anam would have to be a ridiculous 4096 x (9/16) kT = 2304 kT for there to be thermally significant attraction. 

The potential energy V(R) of the double-layer interaction between two spheres at separation R is given by... 

In this chapter, we give exact expressions and various approximate expressions for the force and potential energy of the electrical double-layer interaction between two parallel similar plates. Expressions for the double-layer interaction between two parallel plates are important not only for the interaction between plate-like particles but also for the interaction between two spheres or two cylinders, because the double-interaction between two spheres or two cylinders can be approximately calculated from the corresponding interaction between two parallel plates via Deijaguin s approximation, as shown in Chapter 12. We will discuss the case of two parallel dissimilar plates in Chapter 10. 

FIGURE 12.2 Interaction between two spheres 1 and 2 at a closest separation H, each having radii ai and 02, respectively. 

By applying the above method to two interacting spheres on the basis of the linearized Poisson-Boltzmann equations, we can derive series expansion representations for the double-layer interaction between two spheres 1 and 2 (Fig. 14.3). 

In 1937 Hamaker had the idea of expanding the concept of the van der Waals forces from atoms and molecules to solid bodies. He assumed that each atom in body 1 interacts with all atoms in body 2, and with a method known as pairwise summation (Figure 11.3), found an expression for the interaction between two spheres of radius f , and R2. 

Figure 7.2 Diagram of itie interaction between two spheres of radius o at a distance of separation H with a centre to centre distance of (R = H + 2a) used in calculating energies of interaction.

In Eq. (15), the electrostatic potential, iJ/, is for the overlapping electric double layer of the interacting particles. Numerous models have been created to predict the overlapping field electrostatic potential between parallel plates. However, calculation of the EDL interaction for the common geometry of two spheres has not been satisfactorily resolved, due mainly to the nonlinear partial differential terms in Eq. (13) arising because of the three-dimensional geometry of the system. As a consequence, a number of approximate and numerical models have been developed for the calculation of the EDL interaction between two spheres. These models are briefly described below. 

Exact Numerical Solutions for EDL Interaction Between Two Spheres... 

When the EDL are thin compared with the particle radii (k- electrostatic interaction between two spheres reads... 

From the discussion given in Section 3.3.2 it is also clear that, except in very exceptional circumstances, there is attraction between particles and Figure 3.31 illustrates the case of interaction between two spheres this shows that the energy of attraction increases, at a constant distance of separation, with the size of the spheres. Therefore in ordCT to stabilize particles an alternative mechanism is needed to prevent entry of the particles into the attractive well. Since the thermal... 

We first consider the balance of interactions for flat plates. Once this interaction is understood, the interaction between two spheres can be obtained within the Derejaguin approximation the virial coefficient and hence the stability of the system to phase separation can then be determined. 

As already mentioned, it is hard to obtain a quantitative measure of PMF between two hydrophobic solutes, say for example, between two phenylalanine residues in a protein. Theoretical studies have often modeled this process by studying the interaction between two spheres as a fimction of the distance between them, as shown in Figure 15.4. 

The quantity p can be considered as a characteristic of the microgeometry of the rough surface. By the use of this concept, we can regard the adhesive interaction between a smooth spherical particle and a rough surface as an interaction between two spheres with radii r and p, where the quantity p takes into account both the ordinary roughness and the roughness at the atomic-molecular level. The force of adhesion in this case will be proportional to the parameter rp/(r + p). 

J.W. Krozel, D.A. Saville, Electrostatic interactions between two spheres solutions of the Debye-Hiickel equation with a charge regulation boundary condition. J. Colloid Interface Sci. 150(2), 365-373 (1992). doi 10.1016/0021-9797(92)90206-2 J. Lyklema, J.F.L. Duval, Hetero-interaction between Gouy-Stem double layers Charge and potential regulation. Adv. Colloid Interface Sci. 114—115, 27-45 (2005). doi 10.1016. cis. 2004.05.002... 

The first positive term on the right-hand side represents the osmotic repulsion between the brushes and the second negative term originates from the elastic energy gain upon retraction of chains (less stretching). The repulsion dominates the interaction for hpressure yields the interaction potential between two plates from which also the interaction between two spheres can be derived. 

This result of (2.19a), in which r is the variable, was first obtained by Vrij [2]. In (2.19b) the variable is h and was already given (without explicit derivation) in (1.22). Both (2.19a) and (2.19b) are frequently used in the literature. Note that ITs( ) in (2.18) is equal to pressitie times the overlap volume Vov The reason for this simple form will become clearer after consideration of the interaction between two spheres using the extended Gibbs equation. In the limit that [Pg.62]

Fig. 2.6). When the range of interaction is short it is suflhcient to consider only small values of h/R or z/R, see Fig. 2.7. For z

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