Theory of the van der Waals Forces

Instead, the older theory of the van der Waals forces has shown that on the basis of classical mechanics there exists no dynamical interaction between neutral spherically symmetric atoms, if the mutual influence of their inner motion is neglected. Only consideration of the polarisation yields weak forces, based on the interaction between induced dipoles. Since for these forces no saturation mechanism is present, they are not related to the chemical interactions of neutral atoms, but to the van der Waals attraction forces. These forces decrease with 1/E7 with the distance R. 

The surface force apparatus (SFA) is a device that detects the variations of normal and tangential forces resulting from the molecule interactions, as a function of normal distance between two curved surfaces in relative motion. SFA has been successfully used over the past years for investigating various surface phenomena, such as adhesion, rheology of confined liquid and polymers, colloid stability, and boundary friction. The first SFA was invented in 1969 by Tabor and Winterton [23] and was further developed in 1972 by Israela-chivili and Tabor [24]. The device was employed for direct measurement of the van der Waals forces in the air or vacuum between molecularly smooth mica surfaces in the distance range of 1.5-130 nm. The results confirmed the prediction of the Lifshitz theory on van der Waals interactions down to the separations as small as 1.5 nm. 

So-called solvation/structural forces, or (in water) hydration forces, arise in the gap between a pair of particles or surfaces when solvent (water) molecules become ordered by the proximity of the surfaces. When such ordering occurs, there is a breakdown in the classical continuum theories of the van der Waals and electrostatic double-layer forces, with the consequence that the monotonic forces they conventionally predict are replaced (or accompanied) by exponentially decaying oscillatory forces with a periodicity roughly equal to the size of the confined species (Min et al, 2008). In practice, these confined species may be of widely variable structural and chemical types — ranging in size from small solvent molecules (like water) up to macromolecules and nanoparticles. 

The solid particles in a suspension attract because of the Van der Waals forces and charged particles also attract due to Coulomb forces. The charge of the particles and consequently also the Coulomb forces can be affected by adding electrolytes. The relevant theory was already... 

For a typical experimental hydrosol critical coagulation concentration at 25°C of 0.1 mol dm-3 for z = 1, and, again, taking if/d = 75 mV, the effective Hamaker constant, A, is calculated to be equal to 8 X 10 20 J. This is consistent with the order of magnitude of A which is predicted from the theory of London-van der Waals forces (see Table 8.3). 

As a preparation let us first summarize London s18 theory of the van der Waals interaction between two neutral atoms. Let ii and r2 be the positions of the two atoms, and let r2 rj = r12e18, r12 being the distance between them. The distance is assumed to be so large that the dipole-dipole interaction predominates as the origin of intermolecular forces. 

Column (4) of table 3.04 shows the total lattice energies of the alkali halides as computed by more recent and refined methods. These values include the effect of the van der Waals forces and also include a contribution representing the zero-point energy. It will be seen that they do not differ greatly from the values in column (3) derived on the elementary theory. 

The full lines in Fig. 3.8 are best fits to the presmectic interaction, based on the LdG theory [22,24] with an addition of the van der Waals force [2,14]. Just like in previous nematic cases, the mesoscopic LdG theory superbly describes the structural force between the surfaces. The fitting of the measured presmectic forces to the theory allows for the determination of many important surface parameters, that are difficult to obtain using other methods. For example, the amplitude of the smectic order at the surface and the smectic correlation length can be obtained directly. For 8CB on silanated glass, one obtains a typical surface smectic order parameter iPfi which is of the order of 0.1 and the smectic correlation length y, which is in perfect agreement with X-ray data of Davidov et al. [25]. In addition to that, it has been observed that the smectic order is coupled to the nematic order and this amplifies the presmectic interaction close to the isotropic-nematic phase transition [23]. 

Later, the relevance of the van der Waals forces in foam films was realized and it was suggested that the black films correspond to a secondary minimum. However, the DLVO theory could not adequately describe... 

The specific value includes both polar and hydrograi bonding effects. Noticeably, the dispersion value indicates that approximately 30% of the surface tension value of water is due to dispersion forces, which agrees well with theoretical considerations and the molecular theories mentioned in Chapter 2. (The contribution of dispersion to water s potential energy is around 24% at 298 K.) This illustrates the importance of the van der Waals forces even for strongly hydrogen bonding compounds such as water. The dispersion surface tension value of water from the Fowkes theory can be considCTed to be widely accepted and is therefore used in further calculations. 

From 1954 to 1956, Lifshitz derived the theoretical description for the forces betv een two parallel plates of dielectric materials across a vacuum [19]. This theory was extended together with Dzyaloshinskii and Pitaevskii between 1959 and 1961 to include the effect of a third dielectric filling the gap between the plates [20]. However, the complicated structure of their solution hindered its widespread acceptance and initially caused doubt of its practical use [21]. A simplified derivation of the van der Waals forces between parallel plates was introduced by van Kampen et al. [22] based on a model in which the fiuctuations were represented by a sum of harmonic oscillators. Since the bulk modes are independent of distance between the surfaces, only surface modes contribute to the van der Waals force. Based on the van Kampen calculations, Parsegian and Ninham showed in a series of papers in 1970 that the van der Waals forces could be calculated based on available dielectric data [23]. This paned the way for a general quantitative description of van der Waals forces. 

At this point we should darify that the Casimir force is not a really new type of force. It is simply another term for a special case of the van der Waals forces, namely, the retrarded van der Waals force between metallic surfaces. While the terms retarded van der Waals force or retarded London dispersion force are prevalent in the physical chemistry and colloid community, the term Casimir force or Casimir-Polder force has become popular in the physics community. This means that in principle the lifshitz theory is apphcable to describe the Casimir forces. The problem with using Lifshitz theory for ideal metals is the fact that for these the didectric constant diverges (e oo) and therefore the Lifshitz theory breaks down. However, for real metals, the use of the Lifshitz theory is possible with corresponding dielectric models of the metals. 

We should mention here one of the important limitations of the singlet level theory, regardless of the closure applied. This approach may not be used when the interaction potential between a pair of fluid molecules depends on their location with respect to the surface. Several experiments and theoretical studies have pointed out the importance of surface-mediated [1,87] three-body forces between fluid particles for fluid properties at a solid surface. It is known that the depth of the van der Waals potential is significantly lower for a pair of particles located in the first adsorbed layer. In... 

The Hamaker constant A can, in principle, be determined from the C6 coefficient characterizing the strength of the van der Waals interaction between two molecules in vacuum. In practice, however, the value for A is also influenced by the dielectric properties of the interstitial medium, as well as the roughness of the surface of the spheres. Reliable estimates from theory are therefore difficult to make, and unfortunately it also proves difficult to directly determine A from experiment. So, establishing a value for A remains the main difficulty in the numerical studies of the effect of cohesive forces, where the value for glass particles is assumed to be somewhere in the range of 10 21 joule. 

The macroscopic properties of the three states of matter can be modeled as ensembles of molecules, and their interactions are described by intermolecular potentials or force fields. These theories lead to the understanding of properties such as the thermodynamic and transport properties, vapor pressure, and critical constants. The ideal gas is characterized by a group of molecules that are hard spheres far apart, and they exert forces on each other only during brief periods of collisions. The real gases experience intermolecular forces, such as the van der Waals forces, so that molecules exert forces on each other even when they are not in collision. The liquids and solids are characterized by molecules that are constantly in contact and exerting forces on each other. 

The van der Waals forces scale up from atomic distances to colloidal distances undiminished. How the molecular forces scale up in the case of large objects, expressions for such forces, definition of the Hamaker constant, and theories based on bulk material properties follow in Sections 10.5-10.7. 

In the microscopic calculation pairwise additivity was assumed. We ignored the influence of neighboring molecules on the interaction between any pair of molecules. In reality the van der Waals force between two molecules is changed by the presence of a third molecule. For example, the polarizability can change. This problem of additivity is completely avoided in the macroscopic theory developed by Lifshitz [118,119]. Lifshitz neglects the discrete atomic structure and the solids are treated as continuous materials with bulk properties such as the... 

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