Lifshitz theory

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... 

Flough D B and White L R 1980 The calculation of Flamaker constants from Lifshitz theory with applications to wetting phenomena Adv. Colloid Interface Sc/. 14 3-41... 

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... 

Lagrange Multiplier Method for programming problems, 289 for weapon allocation, 291 Lamb and Rutherford, 641 Lamb shift, 486,641 Lanczos form, 73 Landau, L. D., 726,759, 768 Landau-Lifshitz theory applied to magnetic structure, 762 Large numbers, weak law of, 199 Law of large numbers, weak, 199 Lawson, J. L., 170,176 Le Cone, Y., 726... 

Refinements in the theory of interparticle long-range van der Waals forces (the Landau-Lifshitz theory) are within reach. New techniques are now available for measuring the complex dielectric constants of various media required for the implementation of that theory. 

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 ... 

L. D. Landau and E. M. Lifshitz, Theory of Elasticity, Pergamon Press, New York, 1986. 

These eqnations are simple and become powerfnl if we know the valne of A for particular materials. Tables of A values which have been calculated using the Lifshitz theory are available and some examples are given in Table 7.2. 

Mahanty, J., and Ninham, B. W., Dispersion Forces, Academic Press, New York, 1976. (An advanced monograph on dispersion forces. Discusses topics such as London and Lifshitz theories.)... 

Both Hamaker and Lifshitz theories of van der Waals interaction between particles are continuum theories in which the dispersion medium is considered to have uniform properties. At short distances (i.e. up to a few molecular diameters) the discrete molecular nature of the dispersion medium cannot be ignored. In the vicinity of a solid surface, the constraining effect of the solid and the attractive forces between the solid and the molecules of the dispersion medium will cause these molecules to pack, as depicted schematically in Figure 8.5. Moving away from the solid surface, the molecular density will show a damped oscillation about the bulk value. In the presence of a nearby second solid surface, this effect will be even more pronounced. The van der Waals interaction will, consequently, differ from that expected for a continuous dispersion medium. This effect will not be significant at liquid-liquid interfaces where the surface molecules can overlap, and its significance will be difficult to estimate for a rough solid surface. 

Prieve, D.C. and Russel, W.B., (1988), Simplified predictions of Hamaker constants from Lifshitz theory , J. Colloid and Interface Science, 125 (1), 1-13. 

Hough, D. B. and White, L. R. (1980). The Calculation of Hamaker Constants from Lifshitz Theory with Applications to Wetting Phenomena. Adv. Colloid Interface Sci., 14, 3. 

Both formulations stumble when the materials are real conductors such as salt solutions or metals. In these cases important fluctuations can occur in the limit of low frequency where we must think of long-lasting, far-reaching electric currents. Unlike brief dipolar fluctuations that can be considered to occur local to a point in a material, walls or discontinuities in conductivity at material interfaces interrupt the electrical currents set up by these longer-lasting "zero-frequency" fields. It is not enough to know finite bulk material conductivities in order to compute forces. Nevertheless, it is possible to extend the Lifshitz theory to include events such as the fluctuations of ions in salt solutions or of electrons in metals. 

These quartz measurements, together with several other less-successful attempts by others, had been fiercely contested.20 Theories had been fitted to faulty measurements there had been no adequate theory yet available for good measurements. "Measurement" drove theory. Hamaker constants (coefficients of interaction energy) were so uncertain that they were allowed to vary by factors of 100 or 1000 in order to fit the data. The Lifshitz theory put an end to all that. Disagreement meant that either there was a bad measurement or there was something acting besides a charge-fluctuation force. 

It is curious how people working in different disciplines seek and see validation in different kinds of experiments. Only the liquid helium up-the-walls measurement, described in the preceding subsection, seems to satisfy most people that the Lifshitz theory quantitatively accounts for measured forces (see note 37 in the preceding subsection). As with the historical comments, this review of measurements is not intended to be exhaustive.39... 

Other continuous profiles in e produce similarly intriguing behaviors. The nondivergence of free energy and of pressure, qualitatively different from the power-law divergences in Lifshitz theory, occurs here when there is no discontinuity in s itself or its z derivative. Deeper consideration of such behaviors would require going beyond macroscopic-continuum language. 

How does Hamaker pairwise summation emerge from reduction of the Lifshitz theory ... 

It is remarkable how many people think it difficult to compute van der Waals forces directly by using the Lifshitz theory. Invariably, after a few minutes instruction, there is the reaction "I didn t know how easy it was." Essentially it is a matter of introducing tabulated experimental information for e s and numerically summing or integrating for the interaction energy. 

The Lifshitz theory uses only the so-called "local" dielectric and magnetic responses. That is to say, the electric field at a place polarizes that place and that place only. What if the field is from a wave sinusoidally oscillating in space Then the material polarization must oscillate in space to follow the field. What if that oscillation in space is of such a short wavelength that the structure of the material cannot accommodate the spatial variation of the wave We are confronted with what is referred to as a "nonlocal" response a polarization at a particular place is constrained by polarizations and electric fields at other places. 

NB Rough numbers Values vary from sample to sample and from handbook to handbook. Source Table modified from D. Gingell and V. A. Parsegian, "Prediction of van der Waals interactions between plastics in water using the Lifshitz theory," J. Colloid Interface Sci., 44, 456-463 (1973). 

The interaction of real metal plates is in fact far more complicated than what is derived assuming ideal infinite conductance. See B. W. Ninham and J. Daicic, "Lifshitz theory of Casimir forces at finite temperature," Phys. Rev. A, 57, 1870-80 (1998), for an instructive essay that includes the effects of finite temperature, finite conductance, and electron-plasma properties. The nub of the matter is that the Casimir result is strictly correct only at zero temperature. 

E. S. Sabisky and C. H. Anderson, "Verification of the Lifshitz theory of the van der Waals potential using liquid-helium films," Phys. Rev. A, 7, 790-806 (1973). 

P. Richmond, B. W. Ninham, and R. H. Ottewill, "A theoretical study of hydrocarbon adsorption on water surfaces using Lifshitz theory," J. Colloid Interface Sci., 45, 69-80 (1973). See also I. M. Tidswell, T. A. Rabedeau, P. S. Pershan, and S. D. Kosowsky, "Complete wetting of... 

See the seminal paper by B. W. Ninham and V. Yaminsky, "Ion binding and ion specificity The Hofmeister effect and Onsager and Lifshitz theories," Langmuir, 13, 2097-108 (1997), for the connection between solute interaction and van der Waals forces from the perspective of macroscopic continuum theory. 

C. M. Roth and A. M. Lenhoff, "Improved parametric representation of water dielectric data for Lifshitz theory calculations,"). Colloid Interface Sci., 179, 637-9 (1996), present another set of parameters for water. 

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