The Van der Waals Interaction

When an atom or molecule approaches a surface, the electrons in the particle - due to quantum fluctuations - set up a dipole, which induces an image dipole in the polarizable solid. Since this image dipole has the opposite sign and is correlated with fluctuations in the particle, the resulting force is attractive. In the following we construct a simple model to elucidate the phenomenon. 

Consider a hydrogen atom with its nucleus at the origin located above the surface of a conducting metal at the position d = (0,0,d) and an electron at r = (x,y,z) (Fig. 6.1).The nucleus and the electron both induce image charges in the metal equal to 

Thus the net result of all interactions is an attractive potential behaving as 

---

Microscopic analyses of the van der Waals interaction have been made for many geometries, including, a spherical colloid in a cylindrical pore [14] and in a spherical cavity [15] and for flat plates with conical or spherical asperities [16,17]. 

The interest in vesicles as models for cell biomembranes has led to much work on the interactions within and between lipid layers. The primary contributions to vesicle stability and curvature include those familiar to us already, the electrostatic interactions between charged head groups (Chapter V) and the van der Waals interaction between layers (Chapter VI). An additional force due to thermal fluctuations in membranes produces a steric repulsion between membranes known as the Helfrich or undulation interaction. This force has been quantified by Sackmann and co-workers using reflection interference contrast microscopy to monitor vesicles weakly adhering to a solid substrate [78]. Membrane fluctuation forces may influence the interactions between proteins embedded in them [79]. Finally, in balance with these forces, bending elasticity helps determine shape transitions [80], interactions between inclusions [81], aggregation of membrane junctions [82], and unbinding of pinched membranes [83]. Specific interactions between membrane embedded receptors add an additional complication to biomembrane behavior. These have been stud-... 

Rare-gas clusters can be produced easily using supersonic expansion. They are attractive to study theoretically because the interaction potentials are relatively simple and dominated by the van der Waals interactions. The Lennard-Jones pair potential describes the stmctures of the rare-gas clusters well and predicts magic clusters with icosahedral stmctures [139, 140]. The first five icosahedral clusters occur at 13, 55, 147, 309 and 561 atoms and are observed in experiments of Ar, Kr and Xe clusters [1411. Small helium clusters are difficult to produce because of the extremely weak interactions between helium atoms. Due to the large zero-point energy, bulk helium is a quantum fluid and does not solidify under standard pressure. Large helium clusters, which are liquid-like, have been produced and studied by Toennies and coworkers [142]. Recent experiments have provided evidence of... 

At large separation r, equation (C2.6.3) decays as oc r just as the van der Waals interactions between... 

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.

In some force fields the interaction sites are not all situated on the atomic nuclei. For example, in the MM2, MM3 and MM4 programs, the van der Waals centres of hydrogen atoms bonded to carbon are placed not at the nuclei but are approximately 10% along the bond towards the attached atom. The rationale for this is that the electron distribution about small atoms such as oxygen, fluorine and particularly hydrogen is distinctly non-spherical. The single electron from the hydrogen is involved in the bond to the adjacent atom and there are no other electrons that can contribute to the van der Waals interactions. Some force fields also require lone pairs to be defined on particular atoms these have their own van der Waals and electrostatic parameters. 

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. 

In the second type of interaction contributing to van der Waals forces, a molecule with a permanent dipole moment polarizes a neighboring non-polar molecule. The two molecules then align with each other. To calculate the van der Waals interaction between the two molecules, let us first assume that the first molecule has a permanent dipole with a moment u and is separated from a polarizable molecule (dielectric constant ) by a distance r and oriented at some angle 0 to the axis of separation. The dipole is also oriented at some angle from the axis defining the separation between the two molecules. Overall, the picture would be very similar to Fig. 6 used for dipole-dipole interaction except that the interaction is induced as opposed to permanent. 

It is clear from Table 1 that, for a few highly polar molecules such as water, the Keesom effect (i.e. freely rotating permanent dipoles) dominates over either the Debye or London effects. However, even for ammonia, dispersion forces account for almost 57% of the van der Waals interactions, compared to approximately 34% arising from dipole-dipole interactions. The contribution arising from dispersion forces increases to over 86% for hydrogen chloride and rapidly goes to over 90% as the polarity of the molecules decrease. Debye forces generally make up less than about 10% of the total van der Waals interaction. 

The remarkable stability of onion-like particles[15] suggests that single-shell graphitic molecules (giant fullerenes) containing thousands of atoms are unstable and would collapse to form multi-layer particles in this way the system is stabilized by the energy gain from the van der Waals interaction between shells [15,26,27],... 

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... 

Matsui441 has computed energies (Evdw) due to the van der Waals interaction between a-cyclodextrin and some guest molecules by the use of Hill s potential equation 451 ... 

The properties of glassy polymers such as density, thermal expansion, and small-strain deformation are mainly determined by the van der Waals interaction of adjacent molecular segments. On the other hand, crack growth depends on the length of the molecular strands in the network as is deduced from the fracture experiments. 

The fracture behavior can be attributed to strain softening [91] in the deformation zone [92, 93] or to stress-activated devitrification [89, 96]. The strands are comparatively free to move in the strain softened regions of the deformation zone. The van der Waals interaction between adjacent strands is greatly reduced and the clearence between molecular segments is enlarged. 

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... 

In Eq. (6) Ecav represents the energy necessary to create a cavity in the solvent continuum. Eel and Eydw depict the electrostatic and van-der-Waals interactions between solute and the solvent after the solute is brought into the cavity, respectively. The van-der-Waals interactions divide themselves into dispersion and repulsion interactions (Ed sp, Erep). Specific interactions between solute and solvent such as H-bridges and association can only be considered by additional assumptions because the solvent is characterized as a structureless and polarizable medium by macroscopic constants such as dielectric constant, surface tension and volume extension coefficient. The use of macroscopic physical constants in microscopic processes in progress is an approximation. Additional approximations are inherent to the continuum models since the choice of shape and size of the cavity is arbitrary. Entropic effects are considered neither in the continuum models nor in the supermolecule approximation. Despite these numerous approximations, continuum models were developed which produce suitabel estimations of solvation energies and effects (see Refs. 10-30 in 68)). 

---