Macroscopic classical theory

In this section we examine the solutions of Maxwell s equations for a system with a broad and dispersionless electronic resonance. We show that these conditions result in the appearance of the end-points of the lower and upper polariton branches. These end-points restrict the intervals in which the polariton states have well-defined wavevectors. This consideration is applicable, in particular, to the disordered system of J-aggregates since each J-aggregate chain possesses rather narrow electronic transitions instead of broad dispersion (Fig. 10.3). The disorder present in the system does not influence the following arguments, since for small-cavity photon wavevectors, the system can be treated as effectively homogeneous. 

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The next step is to connect the macroscopic susceptibility to individual molecular moments and finally to the number of unpaired electrons. From classical theory, the corrected or paramagnetic molar susceptibility is related to the permanent paramagnetic moment of a molecule, y. by ... 

In the optical case the transition amounts to taking wavelength into account - in the mechanical case Planck s constant becomes a factor, also in the short-wavelength region. There is no implication that the classical equations describe fundamentally different situations. They are simply less detailed than their non-classical analogues and more convenient to use in the macroscopic world. The two sets of equations deal with the same concepts at different levels of refinement. Apart from Planck s constant quantum theory does not introduce any additional concepts, unknown to classical theory, but it has the ability to explain some experimental results that baffled classical science. 

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... 

Affine deformation This model assumes that the deformation of each configuration of the chains is affine in the macroscopic deformation. It is not compatible with known classical theories of rubber elasticity. 

The BBB model is a means of macroscopic approximation to the system on an exclusively electrostatic basis it describes the solvation effects with the aid of classical field theories, primarily electrostatics and hydrodynamics. The nomenclature BBB, an abbreviation of the expression Brass Balls in a Bathtub , originates from Frank [Fr 65]. The more important classical theories included in this group have led to a result only for dilute solutions nevertheless, their refinement has continued up to the present [Ab 79, Be 78, Kr 79, Li 79]. 

Classical nucleation theory uses macroscopic properties characteristic of bulk phases, like free energies and surface tensions, for the description of small clusters These macroscopic concepts may lack physical significance for typical nucleus sizes of often a few atoms as found from experimental studies of heterogeneous nucleation. This has prompted the development of microscopic models of the kinetics of nucleation in terms of atomic interactions, attachment and detachment frequencies to clusters composed of a few atoms and with different structural configurations, as part of a general nucleation theory based on the steady state nucleation model [6]. The size of the critical nucleus follows straightforwardly in the atomistic description from the logarithmic relation between the steady state nucleation rate and the overpotential. It has been shown that at small supersaturations, the atomistic description corresponds to that of the classical theory of nucleation [7]. 

Such decoupling in the liquid may be strictly justified only in the long-wave approximation.In this sense, such a procedure is justified for the macroscopic description. However, one should remember that this is the correct method in a number of cases also for short wavelengths. For example, this is the case for phonons in solids. In other cases, such as the electron gas in metals (plasmons), acoustic phonons in quantum liquids and so on, this decoupling may be considered as the self-consistent field method or the random phase approximation (the analog of the superposition approximation in the classical theory of liquids). 

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. 

The classical models of adsorption processes like Langmuir, BET, DR or Kelvin treatments and their numerous variations and extensions, contain several uncontrolled approximations. However, the classical theories are convenient and their usage is very widespread. On the other hand, the aforementioned classical theories do not start from a well - defined molecular model, and the result is that the link between the molecular behaviour and the macroscopic properties of the systems studied are blurred. The more developed and notable descriptions of the condensed systems include lattice models [408] which are solved by means of the mean - field or other non-classical techniques [409]. The virial formalism of low -pressure adsorption discussed above, integral equation method and perturbation theory are also useful approaches. However, the state of the art technique is the density functional theory (DFT) introduced by Evans [410] and Tarazona [411]. The DFT method enables calculating the equilibrium density profile, p (r), of the fluid which is in contact with the solid phase. The main idea of the DFT approach is that the free energy of inhomogeneous fluid which is a function of p (r), can be... 

The classical theories meet difficulties, as well, when it comes to dealing with k=0 macroscopic fields, an the usual procedure In that case is to evaluate limits for k - 0. This approach is not viable in the se -ggnslstent procedures - although attempts have been made recently to find out how to circumvent the problem of non-periodicity. 

We conjecture that the actual adsorption mechanism results in a source term rather than a storage that provides a coefficient to the term dc/dt. This perspective, however, is relevant only if the theory is developed through a formulation that couples the microscale phenomena with the macroscale behavior. This is because in the classical theory, it is simple to evaluate an experimental result macroscopically due to adsorption as Kd. Here we will develop an alternative adsorption/desorption/diffusion theory, which is based on MD (molecular dynamics) and HA (homogenization analysis). 

We show a few selected experimental results in Fig. 6.10. The measured nucleation mercury-drop flux (in drops per cm per second) is plotted as a function of the vapor supersaturation in the nucleation zone of maximal supersaturation. Fig. 6.10 also shows two estimates of the nucleation rate. One estimate is based on the classical theory employing the capillarity approximation and macroscopic values of the surface... 

High moduli, memory effects, and SANS results which are inconsistent with classical theories of rubber elasticity provoke the need for a new theory. The ideas of junction rearrangement, if correct, require that none of the models of affine deformation should be expected to apply. A statistical mechanical partition function, properly formulated for a polymeric elastomer, should yield predictions of chain deformation, and additional assinnptions relating macroscopic and molecular geometry are superfluous. 

The macroscopic development of crystallinity in polymers, is generally described by the following equation obtained from the classical theory of Avarmi for phase-transformation kinetics [1] ... 

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