Lifshitz approach

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... 

Figure 2. Ion ion dispersion interactions (curve 1) calculated using the Lifshitz approach in ref 4 dominate at small separations the water-screened Coulomb interactions (curve 2, e = 80, A = < >) however, they are negligible compared to the nonscreened Coulomb interactions (e = 10, A = < >) (curve 3), in the presence of a hard wall repulsion at r = 3.6 A.

The description of the van der Waals interaction based on the Lifshitz approach is now sufficiently advanced to provide accurate predictions for the complete interaction energy. For the geometry of two half-spaces, the exact theory is available in a formulation suited for computational purposes. " In parallel with work on planar systems, there has been a focus on the interaction between spheres. " These developed theories have been used as the exact solutions in the validation of the approximate predictions using the Hamaker approach. The significant contribution of the continuum approach to our understanding of the van der Waals interaction lies in the reliable prediction of the Hamaker constant. The interaction energy for two half-spaces and two spheres is summarized below. 

A particularly lucid treatment and comparison of the surface-mode and Lifshitz approaches to the calculation of the van der Waals interaction forces was given by... 

Strictly speaking, Eq. (3.25) is valid for two half-spaces only. In the case of two spherical particles, there is a complex interrelation between geometiy parameter and material properties. The exact analytical solution of the Lifshitz approach was presented by Langbein (1974, as cited by Thennadil and Garcia-Rubio 2001). However, the necessary computations are rather laborious and converge slowly. For that reason, it seems appropriate to approximate the exact analytical solution by combining the Lifshitz-Hamaker function of two half-spaces (Eq. (3.25)) with the Hamaker geometry function for two spherical particles (Eq. (3.23)). 

According to the more accurate macroscopic (Lifshitz) approach, the Hamaker constant for particles 1 and 2 in a medium 3 is given as (Israelachvili, 2011) ... 

If we have relative permittivities and refractive indices, it is better to use the Lifshitz approach rather than the combining rules for the Hamaker constant. In many applications, the expressions for close distances (Table 2.2, Chapter 2) would also be sufficient. 

A related approach carries out lattice sums using a suitable interatomic potential, much as has been done for rare gas crystals [82]. One may also obtain the dispersion component to E by estimating the Hamaker constant A by means of the Lifshitz theory (Eq. VI-30), but again using lattice sums [83]. Thus for a FCC crystal the dispersion contributions are... 

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]. 

A proposal based on Onsager s theory was made by Landau and Lifshitz [27] for the fluctuations that should be added to the Navier-Stokes hydrodynamic equations. Fluctuating stress tensor and heat flux temis were postulated in analogy with the Onsager theory. Flowever, since this is a case where the variables are of mixed time reversal character, tlie derivation was not fiilly rigorous. This situation was remedied by tlie derivation by Fox and Ulilenbeck [13, H, 18] based on general stationary Gaussian-Markov processes [12]. The precise fomi of the Landau proposal is confimied by this approach [14]. 

G. Gompper, R. Holyst, M. Schick. Interfacial properties of amphiphilic systems the approach to Lifshitz points. Phys Rev A 45 3157-3160, 1991. 

The transitions of the coil-globule type were considered not only in the usual space, but also in the space of monomeric units orientation, where such transition is equivalent to the nematic liquid crystal ordering [60,61]. Such an approach using the formalism developed by I.M. Lifshitz has led to the creation of the theory of liquid crystal ordering in the solutions of semi-flexible macromolecules [62,63]. 

The invalidity of the random heteropolymer model for proteins can also be understood from a more general point of view, since we know that protein sequences result from long evolution and are therefore, strictly speaking, not random. This consideration was the origin of a very productive approach that considers heteropolymer physics in connection with their evolution. This idea was not mentioned in any published work by I.M. Lifshitz. However, the authors of the present paper can witness that I.M. Lifshitz had evolutionary ideas on this subject and often discussed them. 

The derivation of the transmission coefficients for a square barrier can be found in almost every textbook on elementary quantum mechanics (for example, Landau and Lifshitz 1977). However, the conventions and notations are not consistent. Figure 2.5 specifies the notations used in this book. To make it consistent with the perturbation approach later in this chapter, we take the reference point of energy at the vacuum level. 

Lifshitz et al. (166) proposed a chi-square (x2) test for detecting adulteration in lemon juice which, for their data, is more sensitive to dilution than the multiple regression approach. The same group (133) used a multivariate normal test on five parameters (Brix, formol number, chloramine-T number, total sugars and chlorides) of Israeli orange and grapefruit juices. 

This relaxation time—which, to be specific, we have written in the Landau-Lifshitz representation—has the anticipated behavior the smaller the precession damping constant (the higher the quality factor of the oscillations), the slower does the particle magnetic moment approach its equilibrium position. For ferromagnet or ferrite nanoparticles the typical values of the material parameters are Is Is < 103G, Vm 10 18cm3, and a 0.1. Substituting them in formula (4.28) aty 2 x 108 rad/Oe s and room temperature, one obtains xD 10-9 s. 

Until now, most studies of dissociation dynamics of metastable cluster ions have been made using a double-focusing mass spectrometry method (Lifshitz et al. 1990 Lifshitz and Louage 1989, 1990 Stace 1986). As discussed herein, the novel technique of reflectron time-of-flight mass spectrometry is a valuable alternative approach to more standard methods. With carefully designed experiments, it is possible to derive both kinetic energy releases and decay fractions for... 

Furthermore, these van der Waals interactions are important only near the interface, where it is unlikely that either Lifshitz or Hamaker approaches are accurate for spheres of molecular sizes. For example, the magnitude of the interaction for Na+ ions at. 5 A from the interface is only approximately 0.02kT (the values of B used in the calculation, Z Na = — 1X10 50 J m3 was obtained from fit by Bostrom et al. [17] and ZJNa= +0.8X10 511 J m3 was calculated by Karraker and Radke [18]). Eq. (8) might provide a convenient way to account for the interfacial interactions, if suitable values for Bt (not related to the macroscopic Hamaker constants) would be selected. 

Alternatively, the Hamaker constant can be calculated in the Lifshitz quantum electrodynamic continuum approach [7], which incorporates all three types of van der Waals interactions for condensed systems (Lifshitz-van der Waals interactions) through surface tension determinations with apolar liquids (e.g. diiodomethane, a-bromonaphthalene)... 

An attractive interaction arises due to the van der Waals forces between molecules of colloidal particles. Depending on the nature of dispersed particles, the Keesom forces (or the dipole-dipole interaction), the Debye forces (or dipole-induced dipole interaction), and the London forces (or induced dipole-induced dipole interaction) may contribute to the van der Waals interaction. First, the van der Waals interaction was theoretically computed using a method of the pairwise summation of interactions between different pairs of molecules of the two macroscopic particles. This method called the microscopic approximation neglects collective effects, and, as a consequence, misrepresents the Hamaker constant. For many problems of a practical use, however, specific features of the total interaction are determined by a repulsive part, and such an effective, gross description of the van der Waals interaction may often be accepted [3]. The collective effects in the van der Waals interaction have been taken into account in the calculations of Lifshitz et al. [4], and their method is known in the literature as the macroscopic approach. 

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