Waals Interaction Energy

The general result for the van der Waals energy reads (cf. also ref. 21) 

The change in the radius of the spherical part of the droplet with the deformation is 

The numerical comparison between the exact expression (10.2) and the approximate one (10.4) shows that the latter is rather accurate for A/a 0.3 and r/a 0.5. In many practically important cases one may use the simplest formula (10.4) without substantial loss of accuracy. 

Most of the droplet interactions are amenable to analytical calculation only in the case of relatively weak droplet deformations, r /a 1, and/or small separations, 

The electrostatic free energy of a thin liquid film can be approximated with different expressions, depending on the specific conditions. For arbitrarily charged droplets and high electrolyte concentrations i.e. weakly overlapped double layers) one may use the non-linear superposition approximation  

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Figure 7-12. Plot of the van der Waals interaction energy according to the Lennard-Jones potential given in Eq. (27) (Sj, = 2.0 kcal mol , / (, = 1.5 A). The calculated collision diameter tr is 1.34 A.

Adsorption on a nonpolar surface such as pure siUca or an unoxidized carbon is dominated by van der Waals forces. The affinity sequence on such a surface generally follows the sequence of molecular weights since the polarizabiUty, which is the main factor governing the magnitude of the van der Waals interaction energy, is itself roughly proportional to the molecular weight. 

FIGURE 1.13 The van der Waals interaction energy profile as a fnnction of the distance, r, between the centers of two atoms. The energy was calcnlated nsing the empirical equation U= B/r — A/r. (Values for the parameters B = 11.5 X 10 kJnm /mol and A = 5.96 X 10 kJnmVtiiol for the interaction between two carbon atoms are from Levitt, M., Journal of Molecular Biology... 

Matsui et al.82) have analyzed the same data of log l/Kd(X) for cyclodextrin-phenol systems from a somewhat different standpoint. They computed the minimal van der Waals interaction energies (Emin) for the systems by using the same method as described in a previous section (Table 4). The calculated Emin values were applied, in place of such steric parameters as Ibrnch and B1( to the correlation analysis. The correlations obtained are given in Eqs. 24 to 27. 

The van der Waals interaction energy of two hydrogen atoms at large intemuclear distances is discussed by the use of a linear variation function. By including in the variation function, in addition to the unperturbed wave function, 26 terms for the dipole-dipole interaction, 17 for the dipole-quadrupole interaction, and 26 for the quadrupole-quadrupole interaction, the... 

The computation of the density of states within the LCT is nontrivial since the LCT free energy F T) = —(31nZ(7 ) is derived as an expansion in the product of the van der Waals interaction energies e and (3 = l/k T. This free energy series for semiflexible polymer systems has been developed [50, 51] through second order in p,... 

The above derivation shows that one can determine the van der Waals parameters a and b if a (a measure of the size of the atom or molecules) and the van der Waals interaction energy parameter /3U are known. Alternatively, one can estimate /3n from known values of a and b. This is illustrated in Example 10.2. 

With the exception of highly polar materials, London dispersion forces account for nearly all of the van der Waals attraction which is operative. The London attractive energy between two molecules is very short-range, varying inversely with the sixth power of the intermolecular distance. For an assembly of molecules, dispersion forces are, to a first approximation, additive and the van der Waals interaction energy between two particles can be computed by summing the attractions between all interparticle molecule pairs. 

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),... 

Ai3i for the interaction of particles of the same material is always positive - i.e. the van der Waals interaction energy is always one of attraction. This interaction will be weakest when the particles and the dispersion medium are chemically similar, since An and A33 will be of similar magnitude and the value of Al3i will therefore, be low. 

Table 3.2 shows the additive total interaction energy, the Coulombic, and the van der Waals interaction energy values from the molecular mechanics study for each polymer fragment. 

Apart from the term in Eq. (7) there are other contributions toward the nonpolar van der Waals interaction energy arising from the interaction of continually changing quadrupoles with dipoles and quadrupoles with quadrupoles. The total expression should, therefore, be written as... 

We shall use this expression in estimating the nonpolar van der Waals interaction energy of physically adsorbed atoms or molecules on metals and on charcoal. We see that also in this expression the potential energy is inversely proportional to the third power of the distance. 

A positive pressure between A and B in m, a positive van der Waals interaction energy, reminds us that the intrusion of an interface disrupts the interactions of materials A, m, and B with themselves. A positive pressure simply indicates a force for thickening medium m, a force to draw more material m from a region of pure m into the space between A and B. This occurs when the interaction of substance m with itself is stronger than the interaction of substance m with substance A but weaker than its interaction... 

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... 

A complete expression for the van der Waals interaction energy can be obtained, in the hypothesis of pairwise additivity, by adding the self-energy of the planar layers... 

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