Applicability of the Classical Theory

Since the development of the classical model and theory, almost all of the phenomena and kinetics of liquid phase sintering have been explained and analysed in terms of the three-stage model and theory. In particular, the theory of the second stage, contact flattening theory, has been the standard theory of liquid phase sintering for decades, despite doubts raised by some research-ergi3,i5,i6,i05,i06 validity of the basic assumptions. Modified contact 

The contact flattening theory contains some inherent problems. One is the continuous reduction in pore size with increasing sintering time. Therefore, in the pore size distribution, the maximum pore size and the frequency of large pores must decrease continuously as the sintering proceeds. Such a change 

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Equation 9.6 and Equation 9.9 through Equation 9.12 are the basis of the classic theory of capillarity [9], The moderate surface curvature that was assumed for these equations follows the fundamental Gibbs Equation 9.1 and Equation 9. lb. However, there was a problem of application of the classic theory of capillarity to the region of high surface curvatures that corresponds to the nanoparticles (down to 2 nm). 

Unraveling the concept of applicability of the classical theory, manifested in Appendix III, where the case of room temperature was considered, we present in Fig. 13 the temperature dependence of the quantum-break factor (A8a). As we see, this -factor is commensurable with unity, decreases monotonically with the increase in T and, just as in Appendix III, remains noticeably greater than the possible minimal value equal to 1. 

We give in conclusion a brief formulation of the ideas which have led to Bohr s atomic theory. There are two observations which are fundamental firstly the stability of atoms, secondly the validity of the classical mechanics and electrodynamics for macroscopic processes. The application of the classical theory to atomic processes... 

Gibbs found the solution of the fundamental Equation 9.1 only for the case of moderate surfaces, for which application of the classic capillary laws was not a problem. But, the importance of the world of nanoscale objects was not as pronounced during that period as now. The problem of surface curvature has become very important for the theory of capillary phenomena after Gibbs. R.C. Tolman, F.P. Buff, J.G. Kirkwood, S. Kondo, A.I. Rusanov, RA. Kralchevski, A.W. Neimann, and many other outstanding researchers devoted their work to this field. This problem is directly related to the development of the general theory of condensed state and molecular interactions in the systems of numerous particles. The methods of statistical mechanics, thermodynamics, and other approaches of modem molecular physics were applied [11,22,23],... 

The Cu-Co system is a particularly simple precipitation system in which a Corich /3 phase precipitates in a Cu-rich terminal a phase. The f.c.c. lattices of both phases are well matched in three dimensions, so that the precipitate interfaces are coherent with respect to either lattice as a reference structure and the interfacial energy is sufficiently isotropic so that they are almost spherical, as in Fig. 19.2. Both the interfacial energy and strain energy are therefore relatively low and the nucleation of the f3 phase is therefore relatively easy and occurs homogeneously. This system has been used to test the applicability of the classical nucleation theory (Section 19.1.1) [11, 12]. In this work, the experimental conditions under which... 

For combustion of simple hydrocarbons, the oxidation reactions appear to follow classical first-order reaction kinetics sufficiendy closely that practical designs can be established by application of the empirical theory (8). For example, the general reaction for a hydrocarbon ... 

While catalytic HDM results in a desirable, nearly metal-free product, the catalyst in the reactor is laden with metal sulfide deposits that eventually result in deactivation. Loss of catalyst activity is attributed to both the physical obstruction of the catalyst pellets pores by deposits and to the chemical contamination of the active catalytic sites by deposits. The radial metal deposit distribution in catalyst pellets is easily observed and understood in terms of the classic theory of diffusion and reaction in porous media. Application of the theory for the design and development of HDM and HDS catalysts has proved useful. Novel concepts and approaches to upgrading metal-laden heavy residua will require more information. However, detailed examination of the chemical and physical structure of the metal deposits is not possible because of current analytical limitations for microscopically complex and heterogeneous materials. Similarly, experimental methods that reveal the complexities of the fine structure of porous materials and theoretical methods to describe them are not yet... 

Although the escape of He from the atmosphere must be accepted, a formal application of the classical Jeans thermal escape theory to 4He shows that the temperature... 

Application of the Thermal Theory to Multiple Reactions. For a flame driven by a single exothermic reaction (of the type nA B + C. ..), the laminar burning velocity SL according to the classical thermal theory of Zeldovich, Frank-Kamentskii and Semenov (ZFKS) is given by (4)... 

The methods utilized to measure the viscoelastic functions are often close to the stress patterns occurring in certain conditions of use of polymeric materials. Consequently, information of technological importance can be obtained from knowledge of these functions. Even the so-called ultimate properties imply molecular mechanisms that are closely related to those involved in viscoelastic behavior. Chapters 16 and 17 deal with the stress-strain multiaxial problems in viscoelasticity. Application of the boundary problems for engineering apphcations is made on the basis of the integral and differential constitutive stress-strain relationships. Several problems of the classical theory of elasticity are revisited as viscoelastic problems. Two special cases that are of special interest from the experimental point of view are studied viscoelastic beams in flexion and viscoelastic rods in torsion. 

The development of the theory of charge-transfer elementary acts on superconductors on the basis of traditional theories of electrode kinetics [161,162] is limited by the state-of-the-art of the superconductivity theory as a whole, and particularly the HTSC theory [163]. So far theoreticians preferentially use the concepts of the classical Bardeen-Cooper-Schrieffe (BCS) theory [6] developed in the 1960s for common superconductors. Because the applicability of the BSC theory to HTSC materials is not without dispute, the theory of charge-transfer elementary acts will become transformed as adequate HTSC theory develops. 

Throughout the preceding section we have tabulated the measure of sorption expressed in terms of that obtained by the application of the BET theory. In general, the theory and data are compatible over the classical range limited to about 0.05 Pq to 0.35 Pq. Such a correlation should not be construed to serve as proof of the BET mechanism or even to measure a true specific surface area. The BET theory is, by its very nature and derivation, a theory for multilayer formation. Such a process is probably in play for nitrogen sorption but is questionable for water and... 

One of the most active areas of research in the statistical mechanics of interfacial systems in recent years has been the problem of freezing. The principal source of progress in this field has been the application of the classical density-functional theories (for a review of the fundamentals in these methods, see, for example, Evans ). For atomic fluids, such apphcations were pioneered by Ramakrishnan and Yussouff and subsequently by Haymet and Oxtoby and others (see, for example, Baret et al. ). Of course, such theories can also be applied to the vapor-liquid interface as well as to problems such as phase transitions in liquid crystals. Density-functional theories for these latter systems have not so far involved use of interaction site models for the intermolecular forces. 

Chaotic systems. Here the mere notion of synchrony is non-trivial, and several concepts have been developed. The effect of phase synchronization is a direct extension of the classical theory to the case of a subclass of self-sustained continuous time chaotic oscillators which admit a description in terms of phase. Synchronization of these systems can be described as a phase and frequency locking, in analogy to the theory of synchronization of noisy systems. An alternative approach considers a synchronization of arbitrary chaotic systems as a coincidence of their state variables (complete synchronization) or as an onset of a functional relationship between state variables of two unidirection-ally coupled systems (generalized synchronization). Although physical mechanisms behind the two latter phenomena essentially differ from the mechanisms of phase and frequency locking, all these effects constitute the field of application of the modern synchronization theory. 

This value cannot be properly derived from the non-relativistic theory because the application of the classical expression for the magnetic moment of a charged particle associated with some angular momentum /... 

In this paper the Weibull theory is applied to very small specimens. The analysis follows the ideas presented in [13]. The relationships between flaw population, size of the fracture initiating flaw and strength are discussed. It is shown that a limit for the applicability of the classical fracture statistics (i.e. Weibull statistics based on the weakest link hypothesis) exists for very small specimens (components). 

There exists now an enormous literature on the applications of the classical or semiclassical collision and transition state theories to different types of chemical reactions in gas phase and in solution (see,for instance, /1,3,19a,35f49/). For our purposes it is sufficient to show the applicability of the general formulations presented in Chapter III to some simple gas phase and dense phase reactions. In this way we would like to demonstrate, first, the computational possibility of these formulations, and, second, their utility for an understanding of the influence of various factors, such as nonseparability effects, quantum effects, isotope effects a.o. on the kinetic parameters. 

Recent dramatic advances in computational techniques and computer power have enabled us to simulate crystalline structures from first-principles by means of the electronic structure calculation of the whole system within the density functional theory. Even liquid and vitreous silica have come to be studied by the ab initio MD method or so-called Car-Parrinello method [59]. Thus the application of the classical MD method is to be shifted to study of dynamics with a larger system size and longer simulation time. For example, the simulation of the oxygen diffusivity mentioned in the previous section needs accumulation of positions of five hundred atoms over 120 ps at each pressure, for which the ab initio MD is too inefficient. On the other hand, a local structural deformation relevant for the diffusion could be simulated with a smaller cell and a shorter time scale. It is obviously fruitful to make proper use ofthese approaches, i.e. the classical MD supported by first-principles cluster calculations and the ab initio MD, in each problem of materials science. 

The forces of interaction (i.e., prior to contact) which a single, gas-borne particle can be subject to are treated from the perspective of its chemical and physical structure. To provide the requisite perspective for understanding the importance of these compositionally dependent factors, the role of the gas is discussed. Classical electrostatic and multipolar forces and the thermodynamic setting for any interaction involving a particle are described briefly. Principle emphasis in the chapter is given to the van der Waals forces. The modern (Lifshitz) theory is introduced and its relation to the classical Hamaker theory is described. A qualitative discussion of the computational approaches commonly used and experimental evidence for the theory are given. Inclusion of the chemical and physical factors necessary for treatment of cases that arise in actual application of the general theory is discussed. 

In this chapter, the application of the KB theory to the solubility in various systems is examined. The KB theory of solutions is a suitable tool to examine the solubility of a solute in a multicomponent solvent. The original Kirkwood and Buff equation for the composition derivative of the chemical potential (and respectively activity coefficient) of component i in an mixture (Kirkwood and Buff 1951) provides the theoretical basis for such an analysis. Such an expression has advantages when compared to classical thermodynamics, which cannot provide information about the activity coefficients of the components of multicomponent (even binary) mixtures. In addition, the KB theory of solutions is valid for all systems under any conditions. For example, it can be applied to supercritical mixtures. 

The Theory of Kuhn and Grun. The theory of birefringence of deformed elastomeric networks was developed by Kuhn and Griin and by Treloar on the basis of the same procedure as that used for the development of the classical theories of rubber-like elasticity (48,49). The pioneering theory of Kuhn and Griin is based on the affine network model that is, upon the application of a macroscopic deformation the components of the end-to-end vector for each network chain are assumed to change in the same ratio as that of the corresponding dimensions of the macroscopic sample. 

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