Lifshitz interaction

The first term is the Lifshitz interaction of A with Bi across m at separation Z the second is the interaction of B with A across m and B Because of the difference in the velocity of light in materials Bi and m, there is a difference in the pBl and pm that measures thicknesses bi and 1. The second term has almost, but not exactly, the simplest Lifshitz form. 

As in the L m R geometry of the Lifshitz interaction between planar half-spaces, fluctuations in potential have the form of waves in the x,y directions parallel to the surfaces and an exponential f(z) that dies away from the surfaces.30 The general form is like that used in the derivation of the Lifshitz result. For each radial wave vector iu + jv, the potential [Pg.315]

Good, van Oss, and Caudhury [208-210] generalized this approach to include three different surface tension components from Lifshitz-van der Waals (dispersion) and electron-donor/electron-acceptor polar interactions. They have tested this model on several materials to find these surface tension components [29, 138, 211, 212]. These approaches have recently been disputed on thermodynamic grounds [213] and based on experimental measurements [214, 215]. 

The Hamaker constant can be evaluated accurately using tire continuum tlieory, developed by Lifshitz and coworkers [40]. A key property in tliis tlieory is tire frequency dependence of tire dielectric pennittivity, (cij). If tills spectmm were tlie same for particles and solvent, then A = 0. Since tlie refractive index n is also related to f (to), tlie van der Waals forces tend to be very weak when tlie particles and solvent have similar refractive indices. A few examples of values for A for interactions across vacuum and across water, obtained using tlie continuum tlieory, are given in table C2.6.3. 

This model was later expanded upon by Lifshitz [33], who cast the problem of dispersive forces in terms of the generation of an electromagnetic wave by an instantaneous dipole in one material being absorbed by a neighboring material. In effect, Lifshitz gave the theory of van der Waals interactions an atomic basis. A detailed description of the Lifshitz model is given by Krupp [34]. 

With the reader bearing in mind this framework, the Lifshitz theory of van der Waals interactions can readily be understood. According to the Lifshitz theory, van der Waals forces arise from the absorption of photons of frequency tu by a material with a complex dielectric constant... 

The interaction among the clusters via the common diffusion held leads in general to a coarsening of the clusters with time t. One denotes this by Ostwald ripening [58,96] (see Sec. HID). According to the Lifshitz-Slyozov theory [58] on this process, the typical cluster radius R increases as... 

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. 

The first term is related to the van der Waals interaction, with A being the Hamaker constant. The second term includes other forces that decay exponentially with distance. As discussed, these may include double-layer, solvation, and hydration forces. In our data analysis, B and C were used as fitting variables the Hamaker constant A was calculated using Lifshitz theory [6]. 

For the aq.KOH-graphite system, the van der Waals interaction should be repulsive, because Lifshitz theory predicts a negative Hamaker constant A, which we calculated to be approximately -7.7 X 10 ° J. Using this value, the fit gives ... 

The term molecular crystal refers to crystals consisting of neutral atomic particles. Thus they include the rare gases He, Ne, Ar, Kr, Xe, and Rn. However, most of them consist of molecules with up to about 100 atoms bound internally by covalent bonds. The dipole interactions that bond them is discussed briefly in Chapter 3, and at length in books such as Parsegian (2006). This book also discusses the Lifshitz-Casimir effect which causes macroscopic solids to attract one another weakly as a result of fluctuating atomic dipoles. Since dipole-dipole forces are almost always positive (unlike monopole forces) they add up to create measurable attractions between macroscopic bodies. However, they decrease rapidly as any two molecules are separated. A detailed history of intermolecular forces is given by Rowlinson (2002). 

P.-G. de Gennes later also considered the multisegment attraction regime. He suggested the so-called p-cluster model [11] in order to explain certain anomalies in behavior observed in many polymer species such as polyethyle-neoxide (PEO) see also [12]. The scenario of coil-globule transition with dominating multisegment interaction first considered by I.M. Lifshitz has been recently studied in [13]. The authors used a computer simulation of chains in a cubic spatial lattice to show that collapse of the polymer can be due to crystallization within the random coil. 

A particular complex problem has been the modelling of Si/W(l 10) Amar et have included pairwise interactions up to the sixth nearest neighbor shell, as estimated experimentally from field-ion microscopic studies The predicted phase diagram (Fig. 30) exhibits (5 x 1), (6 x 1) and p(2 x 1) commensurate phases, as well as a broad regime of an incommensurate phase. In contrast to the ANNNI model the present model does seem to have a finite-temperature Lifshitz point, where the incommensurate, commensurate... 

Fig. 30. Phase diagram of a model for Si/W(110) in the temperature versus 9 plane. Experimentally determined interactions J Jj,are used. Full dots are from Monte Carlo calculations, while triangles are based on transfer matrix finite size scaling using strip widths of 8 and 12. The point labelled L indicates approximate location of Lifshitz point. The dotted line indicates the transition region between the (5 x l)and(6 x 1) phases. (From...

Lifshitz (1955-60) developed a complete quantum electrodynamic (continuum) theory for the van der Waals interaction between macroscopic bodies. 

Later, considering the problem from a macroscopic point of view, H. Casi-mir and D. Polder (Netherlands, 1948), and E. M. Lifshitz (1954), obtained a different, more rapid law of interaction decay. Only recently L. P. Pitaevskii showed that the contradiction does not indicate an error Ya.B. studied an extreme case of large Debye radius, and this case is realizable in principle. 

Ya.B. applied formal perturbation theory to the interaction of an atom with the electrons of a metal, where the latter are assumed to be free. Meanwhile, Casimir and Polder and Lifshitz neglected the spatial dispersion of the dielectric permittivity of the metal. Therefore, in the region of small distances, frequencies of order ui0 are important at small distances in the sense indicated above, as are arbitrarily small frequencies at large distances. In both limits the dielectric permittivity of the metal is not at all close to one. Meanwhile, the perturbation theory used by Ya.B. corresponds formally to an expansion in powers of e - 1. and is therefore not applicable in this case. Neglecting the spatial dispersion is valid, however, only at distances r > a (a is the Debye radius in the metal) of the atom from the surface. At the opposite extreme, r a, the wave vectors kj 1/r > a vF/u>0 Me of importance (vF is the electron speed at the Fermi boundary). In this region of strong spatial dispersion perturbation theory can be applied, and the (--dependence satisfies Zeldovich s law. 

The expressions and calculated values for VDW interactions are given in Refs. 13 and 33. The calculations are based on a pair-summation procedure which includes retardation corrections and many-body orientation effects, and as shown Ref. 34 gives approximately the same results as the macroscopic Lifshitz approach (35,36) or other macroscopic... 

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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