Lifshits theory

Experimental verification of Eq. (3.69) done in [68] with NaDoS film indicates a good agreement with theory. The van der Waals-Hamaker constant Kvw was recalculated from the experimental data about hcr and its value was very close to the theoretically calculated according to the Lifshits theory (Kvw = 10 21 J). 

The dependence of the van der Waals-Hamaker constant on the film thickness, predicted by Lifshits theory and calculated by some authors using approximate methods is presented in Fig. 3.17 [164-165]. 

Lifshits theory see Van der Waals interaction between colloids light scattering see electromagnetic radiation lipase 1.1.3... 

Van der Waals forces between macrobodies. The Lifshits theory... 

In passing it is noted that we met complex frequencies before in the Lifshits theory for dispersion forces. There, an integration of complex permittivities over real frequencies was replaced by an integration of real permittivities over imaginary frequencies. See [14.7.7]. This is just a mathematical device. 

Figure 5.18 gives an impression of the agreement between theory and experiment. choosing representative parameter values. The London-Van der Waals component was computed from the Lifshits theory, as described in sec. 1.4.7. For water and silica the Hamaker constants are similar, as a result these forces do not contribute substcuitially, except at very low h, where 77... 

In a number of alternative approaches, equations similar to (5.7.1 and 2) are derived, ignoring the finite thickness of the interfacial region, and/or applying Lifshits theory to obtain (the equivalents of) Hamaker constants (eq. (I.4.7.7) or its variants). In such models our parameter is replaced by the distance of shortest approach between the adjoining phases. However, as is as esoteric as, if only because surfaces are rarely flat on a molecular scale, these alternatives offer little comfort. We recall that our wedge approach (fig. 2.7) obviates the introduction of... 

Currently, the more elaborate and precise Lifshits theory for the calculation of Hamaker constants is applied rather than the Hamaker-de Boer theory it yields values that agree well with experiment. Some results... 

Division by 4.05 gives A relative to kBT at room temperature.) For the most part calculated by Lifshits theory. 

Applications. Despite these complications, the Hamaker-de Boer Lifshits theory is very useful. It is commonly applied in virtually all theories on colloidal interaction. If no other colloidal interaction forces act, it can be used directly. A case in point in foods is triglyceride crystals in triglyceride oil (Hamaker constant of order 0.5 kTtT). Here no other substantial forces... 

Note It would take far too much space to discuss the Lifshits theory here. For the interested reader, we give one example of an approximate equation (error < 5%) for the unretarded Hamaker constant of particles (1) in a medium (3). The equation is often useful and it reads... 

Application. To apply the DLVO theory in practice, several pieces of information have to be collected. Particle size (distribution) and shape can generally be experimentally determined. Hamaker constants often are to be found in the literature or can be calculated from Lifshits theory. The surface potential can be approximated by the zeta potential obtained in electrophoretic experiments. The ionic strength is generally known (or can be calculated) from the composition of the salt solution. All the other variables needed are generally tabulated in handbooks. This then allows calculation of V(h). To arrive at an aggregation rate, more information is needed this is discussed in Section 13.2. 

The difference in the values obtained for interaction energy between two bodies by the Hamaker and Lifshits theories is particularly great when i and S2 are greatly different. Under the condition that i - 2 1 > the results ob-... 

In Table II.2 we have listed experimental and calculated values of the constant as obtained from calculations based on Eqs. (11.49) and (11.50). The same as for the constant we can note order-of-magnitude agreement between values of the constant B obtained by calculation and derived from experiments. Particularly close agreement is noted [57] for the interaction between quartz surfaces (see Table II.2). The experimental data deviate by 10-15% from the results calculated on the basis of the Lifshits theory. 

Lifshits [55] used quantum electrodynamics to develop a theory for the molecular interaction of condensed macroscopic bodies, allowing for electromagnetic lagging. The value of this theory lies in that the forces of interaction calculated from it agree closely with the earlier-obtained [53, 56] experimental data on the interaction of spherical glass bodies with a plane metallic surface. According to the Lifshits theory, for a small gap between the contiguous surfaces, i.e., in the case H < A, the force of interaction between two similar plane surfaces equals... 

Lifshits, I.M., M.Ya. Azbel and M.I. Kaganov, 1973, Electron Theory of Metals (Consultants Bureau, New Yoric), transl. of Russian original, 1971, Elek-tronnaya teoriya metallov (Nauka, Moskva). 

I.M. Lifshits, S.A. Gradeskul and L.A. Pastur, Introduction to the Theory of Disordered Systems, Nauka, Moscow, 1982 (in Russian). 

Although the shortest way to the tunneling gap 8 is the solution of Landau and Lifshits [27], here we consider the problem from a different perspective. Like in the theory of electric circuits, instead of a detailed consideration of each particle, one can apply some simple rules that provide enough equations to solve the problem. One is the junction rule. It is based upon the probability conservation law for a stationary state, PiQ, t). At any point Q in the domain of 77(2, t), the probability density, I PiQ, t) 2 remains constant, dl P(Q. f)P/df = 0. Consider the part of a vibronic state that is located in a potential well. In this region, the probability density, P(Q, t) 2, looks like an octopus with its tentacles extended into the restricted areas under the barriers.2 If we construct a closed surface S around the body of the octopus , then, due to conservation of probability density, the total flux of probability through the surface S must be equal to zero,... 

In conclusion, field dependent single-crystal magnetization, specific-heat and neutron diffraction results are presented. They are compared with theoretical calculations based on the use of symmetry analysis and a phenomenological thermodynamic potential. For the description of the incommensurate magnetic structure of copper metaborate we introduced the modified Lifshits invariant for the case of two two-component order parameters. This invariant is the antisymmetric product of the different order parameters and their spatial derivatives. Our theory describes satisfactorily the main features of the behavior of the copper metaborate spin system under applied external magnetic field for the temperature range 2+20 K. The definition of the nature of the low-temperature magnetic state anomalies observed at temperatures near 1.8 K and 1 K requires further consideration. 

Another approach [34-37] for describing the dispersity changes due to gas diffusion is based on the theory of Lifshits-Slezov [38,39]. If the rate of change in dispersity is determined by diffusion alone, the critical radius Rc changes according to the following equation [36]... 

L.D. Landau, E.M. Lifshits, Quantum Mechanics Non-Relativistic Theory. 3rd edn. (Elsevier Science, 2003)... 

Lifshits [41] used quantum electrodynamics to develop a theory for the... 

Adsorbed on the surface of dispersed particles, polymer chains lessen the attraction energy by steric reasons (the minimum distance to which particles can approach increases) and because they change the efficient Hamaker s constant value. The attraction energy in expressions for Ur dependence on A is the function of not only interaction constants of dispersed phase A, dispersion medium A2 and the phase with the medium A 2, but of Gamaker s constant for adsorption layer 3 too. The effect of polymer adsorption layers on molecular attraction of particles has been described theoret-ically. Below is an expression for Ur, based on the Lifshits macroscopic theory... 

However, it was possible to use the method of I.M. Lifshits for this case [137] the corresponding theory is elaborated in [36]. We will only report the results obtained for the persistence mechanism of flexibility, which is most frequently encountered for real rigid-chain polymers. 

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