Lifshitz theory model

However, it must be recalled that the Lifshitz theory was originally formulated23 25 for the model of beads (see Fig. 7 a). In this model, each monomer is represented as a material point thus, this model cannot be used for the description of the intramolecular liquid-crystalline phase. The description of the orientational ordering, requires the generalization of the Lifshitz consideration for the models, in which the state of an elementary monomer is defined not only by its spatial position but also by its orientation (see, for example, the models of Fig. 7 b-db Such a generalization will be our first aim in this section. 

Using the conventional Lifshitz-Slyozov arguments [74,75] based on the Kelvin equation, one might expect [73,76] that the 3D crystallite growth, occurring via 2D diffusion, should follow Eq. (18) with n = 4. This value, however, is much lower than observed in experiments. The appreciable difference between the theory and experiment is actually not surprising, because the applicability of the Lifshitz-Slyozov model to nm crystallites is far from obvious (e.g., the curvature of such crystallites is an ill-defined quantity). 

Values of e, n and ve and Hamaker constants for two identical types of a material in a vacuum, which are calculated from Equation (567) by taking e3 = 1 and 3 = 1, are given in Table 7.1. Unfortunately, the lack of material constants, such as the dielectric constant, as a function of frequency for most of the substances, and also the complexity of the derived formulae have hampered the general use of the Lifshitz model. However, Lifshitz theory made possible the advent of the first theories on the stability of hydrophobic colloids as a balance between London attraction and electrical double-layer repulsion. Later, these theories were further elaborated by Derjaguin and Landau, and independently by Verwey and Overbeek. The general theory of colloidal stability (which is beyond the scope of this book) is based on Lifshitz theory and has become known as the DLVO theory, by combining the initials of these four authors. 

Adsorption of enteric viruses on mineral surfaces in soil and aquatic environments is well recognized as an important mechanism controlling virus dissemination in natural systems. The adsorption of poliovirus type 1, strain LSc2ab, on oxide surfaces was studied from the standpoint of equilibrium thermodynamics. Mass-action free energies are found to agree with potentials evaluated from the DLVO-Lifshitz theory of colloid stability, the sum of electrodynamic van der Waals potentials and electrostatic double-layer interactions. The effects of pH and ionic strength as well as electrokinetic and dielectric properties of system components are developed from the model in the context of virus adsorption in extra-host systems. 

Although originally designed for astrophysical problems [86], as shown in [120], SPH can also be used for modeling polymers in the macroscale. However, smoothed particle hydrodynamics does not include thermal fluctuations in the form of a random stress tensor and heat flux as in the Landau and Lifshitz theory of hydrodynamic fluctuations. Therefore, the validity of SPH to the study of complex fluids is problematic at scales where thermal fluctuations are important. 

Elastic models lead to p=3 if the stress is uniformly distributed through the thickness of a bent sheet e.g. L. D. Landau and E. M. Lifshitz, "Theory of Elasticity," Pergamon, New York (1970) S. T. Milner and T. A. Witten, M. E. Cates, Europhysics, Lett. 5, 413... 

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. 

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]. 

Besides the kinetic energy release associated with cluster evaporation, it is also possible in a mass spectrometer (either the double focusing M/E type or in a reflectron TOP apparatus) to measure the ratio of the daughter to parent signal, that is, M/AM. A model that expresses this ratio as well as the kinetic energy release is one based on the Klots theory of cluster evaporation. Because this approach is very different from the microcanonical theory so far presented, some basic ideas of theijnal kinetics must be discussed. Two excellent reviews of the basic theory (Klots, 1994) and their application to cluster evaporation (Lifshitz, 1993) provide most of the information needed to understand this field. 

It would be highly desirable to derive the conditions for the appearance of a prepeak or a Lifshitz state directly from STL methods, such as the integral equation techniques, for example. Unfortunately, these methods are still very much approximate and in particular cannot handle complex liquids such as water. Much progress in improving these methods is required before we can think of using Equation 7.53 in more details. In the absence of better theories, we can explore simple models that can show us how domains emerge from fluctuating mixtures. 

In a large part of the (current) literature the Lifshitz-van der Waals component (o, is simply termed dispersion component and the Lewis acid-base interactions (o ) are interpreted as polar interactions even though the material s dipole moments may be zero or the interactions originating from permanent dipoles are very small and can be easily associated with the dispersion part [6]. The misleading denominations go back to a historical misidentification of the acid-base interactions as polar interactions in the Owens-Wendt-Rabel-Kaelble [7-9] approach to calculate the IFT [6] (OWRK model). However, as an impact on the SFE calculation by this misinterpretation of this old theory occurs only when a monopolar base interacts with a monopolar acid, this nomenclature is still widely used. And here in this work we will also use the terms dispersion and po/ar interactions to differentiate the two major contributions to SFE, ST, and IFT. For a detailed discussion of the use of contact angles in determining SFE of solids and other methods of determining SFE, see Etzler [10]. 

Finally, it is worth mentioning another approach used to describe nonlinear viscoelastic solids nonlinear differential viscoelasticity [49, 178, 179]. This theory has been successfully applied to model finite amplitude waves propagation [180-182]. It is the generalization to the three-dimensional nonlinear case of the rheological element composed by a dashpot in series with a spring. Thus in the simplest case, the stress depends upon the current values of strain and strain rate rally. In this sense, it can account for the nonlinear short-term response and the creep behavior, but it fails to reproduce the long-term material response (e.g., relaxation tests). The so-called Mooney-Rivlin viscoelastic material [183] and the incompressible version of the model proposed by Landau and Lifshitz [184] belraig to this class. 

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