Interaction energy between spheres

Using the result given in (6.49), we can then obtain the corresponding interaction energy between spheres ... 

The interaction energy between spheres at constant surface potential involves only the function G (0 (which depends only on the sphere radius a,), while the interaction at constant surface charge density is characterized by the function H (i) (which depends on both sphere radius u, and relative permittivity Ep,). The interaction energy in the mixed case involves both G (i) and // (/) ... 

Equations (12.1) and (12.7) show that interaction energy between spheres is proportional to the radius at any value of h. As shown in Figure 12.4d, this may imply, for instance, that the larger particles in a dispersion would be stable against aggregation in the primary minimum, whereas the smaller particles are not. In practice, this is often not observed. In other words, the dependence of stability on particle size is smaller than predicted. 

In 1873 van der Waais pointed out that real gases do not obey the ideal gas equation PV = RT and suggested that two correction terms should be included to give a more accurate representation, of the form (P + ali/) V - b) = RT. The term a/v corrects for the fact that there will be an attractive force between all gas molecules (both polar and nonpolar) and hence the observed pressure must be increased to that of an ideal, non-interacting gas. The second term (b) corrects for the fact that the molecules are finite in size and act like hard spheres on collision the actual free volume must then be less than the total measured volume of the gas. These correction terms are clearly to do with the interaction energy between molecules in the gas phase. 

FIG. 13.5 Interaction energy between two spheres of diameters dx and d2. (Redrawn with permission from C. J. Brinker G. and Scherer, Sol-Gel Science The Physics and Chemistry of Sol-Gel Processing, Wiley, New York, 1990.)... 

The desired interaction energy between the electron probability density within the nuclear sphere and the nuclear magnetic field is then... 

Fig. 31 Overall interaction energy between two DNA-coated colloids, (a) Sketch of the interacting surfaces of two spheres of radius R0 separated by d. The maximum length of hybridized strands is 2L. (b) Total interaction energy as a function of d. It is the sum of the attractive I/DNA from the binding of accessible DNA strands, the repulsive I/rep from electrostatics and/or polymer steric effect, and the van der Waals attraction t/vdw. (c) For weak, short-range I/rep, particles which are unbound at high temperatures are irreversibly trapped in the van der Waals well after DNA hybridization at low temperatures, (d) For strong, medium-range I/rep, DNA binding produces a secondary minimum of reversible aggregation. Reproduced with permission from [138]...

The van der Waals interaction energy between a sphere and a seiniinfinite plate has been estimated, as a function of the sphere-plate separation distance, by Hamaker... 

We will try first to obtain some general information about the interaction energy between two identical spheres of radius a, separated by the distance of closest approach z, at high electrolyte concentrations, using some simple approximations. The following expressions will be used... 

Inner-sphere (electron transfer) — is, historically, an - electron transfer between two metal centers sharing a ligand or atom in their respective coordination shells. The term was then extended to any case in which the interaction energy between the donor and acceptor centers in the -> transition state is significant (>20 kj mol-1). See also -> Marcus theory. 

Another method to calculate the electrostatic interaction energy, Vg, is from the Deijaquin approximation. Deijaguin [9] developed a method to utilize the parallel plate interaction energy, V%ih), as an approximation for the interaction energy between two spheres, V (h). This approximation is given by... 

Calculate the steric interaction energy between the two spheres given in problem 1. Use Table 10.7 for the value of the relevant Flory-Huggins x parameter. 

In Chapter 11, we derived the double-layer interaction energy between two parallel plates with arbitrary surface potentials at large separations compared with the Debye length 1/k with the help of the linear superposition approximation. These results, which do not depend on the type of the double-layer interaction, can be applied both to the constant surface potential and to the constant surface charge density cases as well as their mixed case. In addition, the results obtained on the basis of the linear superposition approximation can be applied not only to hard particles but also to soft particles. We now apply Derjaguin s approximation to these results to obtain the sphere-sphere interaction energy, as shown below. 

That is, the interaction energy between two crossed identical cylinders equals twice the interaction energy between two identical spheres. 

Consider the validity of Derjaguin s approximation. In this approximation, the interaction energy between two spheres of radii oj and 02 at separation H between their surfaces is obtained by integrating the corresponding interaction energy between two parallel membranes at separation h via Eq. (13.28). We thus obtain... 

The result for the interaction energy between a plate 1 carrying a constant surface potential i/ oi and a sphere 2 of radius 02 carrying a constant surface potential 1/ 02, separated by a distance H between their surfaces, immersed in an electrolyte solution is (Eig. 14.9)... 

Equations (15.49) and (15.50), respectively, agrees with the expression for the electrostatic interaction energy between two parallel hard plates at constant surface charge density and that for two hard spheres at constant surface charge density [4] (Eqs. (10.54) and (10.55)). 

The interaction energy between the particles is then obtained by integrating Eq. (7) from infinity to the separation distance h. The Derjaguin approximation can be applied to van der Waals, double-layer, and many other interactions. For van der Waals interaction, either Eq. (1) or (3) for E h) can be substituted into Eq. (7) to determine the force between two spheres. 

The third group of the approximate models includes various improvements of the Derjaguin approximation, linearization, and approximate solutions of PB Eq. (13) for spherical particles. The first improvement on the Derjaguin approximation for the interaction energy between identical spheres was probably obtained by the Debye-Hiickel linearization and the superposition approximation,given by ... 

Fig. 5 Variations of the interaction energy between particles versus the center-to-center distance, (a) Hertzian potentials for particles with bulk elasticity the dashed line represents the usual Hertz potential (1), and the solid line the generalized Hertzian potential (2). (b) Potentials for emulsions with surface elasticity the dashed line denotes the approximate solution for small compression ratios (3), and the solid line the general solution (4). (c) Ultrasoft potentials for star polymers (where a 1.3Rg, with Rq being the radius of gyration of the star, following [123]) (5) dashed line f = 256 solid line f = 128 dashed and dotted line f = 64. (d) Hard-sphere potential...

A geometric configuration, which is important for disperse systems, is the case of two spheres of radii and R2 interacting across a medium (component 3). Hamaker has derived the following expression for the van der Waals interaction energy between two spheres ... 

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