Phenomenological mechanical theory

The first is the phenomenological mechanical theory. One form of the equation reads r6-9L... 

The existing phenomenological theories of catalysis bear approximately the same relation to the electron theory as the theory of the chemical bond, which was prevalent in the last century and which made use of valence signs (and dealt only with these signs), bears to the modern quantum-mechanical theory of the chemical bond which has given the old valence signs physical content, thereby disclosing the physical nature of the chemical forces. 

Because of the inadequacies of QED a fundamental theory of electrode processes is still lacking. The working theories are exclusively phenomenological and formulated entirely in terms of ionic distributions in the vicinity of electrode interfaces. An early, incomplete attempt [54] to develop a quantum mechanical theory of electrolysis based on electron tunnelling, is still invoked and extensively misunderstood as the basis of charge-transfer. It is clear from too many superficial statements about the nature of electrons that the symbol e is considered sufficient to summarize their important function. The size, spin and mass of the electron never feature in the dynamics of electrochemistry. 

Abstract Contribution of the Jahn-Teller system to the elastic moduli and ultrasonic wave attenuation of the diluted crystals is discussed in the frames of phenomenological approach and on the basis of quantum-mechanical theory. Both, resonant and relaxation processes are considered. The procedure of distinguishing the nature of the anomalies (either resonant or relaxation) in the elastic moduli and attenuation of ultrasound as well as generalized method for reconstruction of the relaxation time temperature dependence are described in detail. Particular attention is paid to the physical parameters of the Jahn-Teller complex that could be determined using the ultrasonic technique, namely, the potential barrier, the type of the vibronic modes and their frequency, the tunnelling splitting, the deformation potential and the energy of inevitable strain. The experimental results obtained in some zinc-blende crystals doped with 3d ions are presented. 

The parameters measured in an ultrasonic experiment are the amplimde and phase of the signal. They are determined by attenuation and phase velocity of a wave. In turn, the attenuation and phase velocity are associated with material constants. In our case they are elastic coefficients (or elastic moduli). These constants can be calculated using quanmm-mechanical approach. Finally, we will obtain the expressions for the measured (phenomenological) parameters in terms of the microscopic ones. In the present section we will discuss the basics of the phenomenological elasticity theory and the microscopic description of the Jahn-Teller contribution to the elastic moduli will be discussed later. 

A consistently quantum mechanical theory describing the coherent and stochastic dynamics of tetrahedral rotors has not been reported yet. Nevertheless, in the extreme situations where the facile re-orientations involve only one axis, the DQR theory will be rigorously valid also for such rotors. Specifically, if the unique axis is a three-fold axis, the stochastic term will have the same form as in Eq. (8) [or the equivalent form in Eq. (10)]. In the case of a two-fold axis, the AB term such as that in Eq. (5) will be obtained, but with the pair-permutation operator P replaced by the operator R defined above. One can thus reasonably expect that even in the cases where there are more than one facile re-orientation axes, applicability of the phenomenological AB approach will suffer similar restrictions as those specified in Subsection 4.4 for methyl-like rotors. 

Another route to find AG is what is usually called the theimodynamic or phenomenological approach. In this, we assume an appropriate expression for AG on the basis of experiments or some theoretical considerations and evaluate the parameters involved from a set of relevant equilibrium data. We then apply the expression of AG so determined to calculate other equilibrium properties and compare the results with the corresponding experimental data. The comparison will show us how adequate the chosen AG is and how it should be modified for better agreement with experiment. Thus, in principle, it is possible to approach step by step a more correct AG of the system under study. Being purely empirical, this method may not be appealing for those who are primcirily interested in events at the molecular level. However, it often allows us to know what thermodynamic factors play a role in controlling the phenomena concerned. The availablity of such information is indeed essential for theoreticians who want to make a more accurate formulation of AG by statistical mechanical theory. 

From the middle of the nineteenth century on, understanding of the phenomenology of interfaces became better at the molecular level, although the nature of the forces involved remained uncertain until the advent of quantum mechanical theory in the 1930s. The study of colloidal phenomena followed a similar track in that certain characteristics of colloidal systems were recognized and studied in the last century (and before), but a good quantitative understanding of the principles and processes involved remained elusive. 

The transmission coefficient k which is also included in the expression for the preexponential factor, cannot be calculated using a purely phenomenological approach, but involves the quantum mechanical theory of an elementary act. For most simple electron transfer reactions, k = 1. 

Strictly speaking, the values of a = 1 or 0 are not necessarily associated with complete disappearance of the energy barrier between the two states. This barrier suffices to be potential independent. Such processes may be called quasibarrierless and quasiactivationless. An explanation of the reason why, even with a significant potential shift, there remains a permanent barrier, is beyond the scope of the phenomenological theory. This phenomenon was predicted on the basis of the quantum mechanical theory of an elementary act, and it was shown that certain reactions, particularly those of chlorine evolution, are, in fact, quasibarrierless. A detailed discussion of this problem can be found in Ref. 70. Here we shall only point out that the reason for this... 

Phenomenological models describe the phenomena of craze and crack formation and crack growth in notched tensile bars in die long-term tensile test. The failure behaviour in the tensile tests is quantitatively parameterised and analysed using fracture-mechanics theories. 

The first notion on the deviation of elementary catalytic acts of enzyme reaction, from that prescribed by classical thermodynamic and kinetic approaches, was, probably, formulated in 1971 [19]. It had been shown that the application of basic postulates of activated state theory to the majority of enzyme processes can lead to physically meaningless values of the activation parameters (energy and entropy of activation). It was emphasized that enzyme functioning is more similar to the work of a mechanical construction than to the catalytic homogeneous chemical reaction. The selfconsistent phenomenological relaxation theory of enzyme catalysis was proposed in 1972 [20, 21]. 

Formulating a criterion to discriminate between brittle and ductile response based on atomistic features of the solid has been a long sought after goal in the mechanics of solids. At the phenomenological level, theories have been developed that characterize the two types of behavior in terms of cleavage of the crystal or the nucleation and motion of dislocations[138, 139], We review here the basic elements of these notions. 

A positive feamre about MPTA is that it has been generalized for several complex cases like liquid solutions, supercritical/high pressure and non-Langmuir adsorption behaviour. In addition, MPTA may probably be considered as a partial step from macroscopic , phenomenological adsorption theories towards the theories based on statistical mechanics and molecular dynamics, like, for example, density functional theory. 

On the phenomenological level, it is clear that there are many rough surfaces known to adhesion science and technology where the surface roughness plays an essential role in adhesion the mechanical theory of adhesion applies. Rather than simply ascribing adhesion to mechanical effects, it is useful to explore why roughness can lead to good adhesion. 

The simplified failure envelopes are not derived from physical theories of failure in which the actual physical processes that cause failure on a microscopic level are integrated to obtain a failure theory. We, instead, deal with phenomenological theories in which we ignore the actual failure mechanisms and concentrate on the gross macroscopic events of failure. Phenomenological theories are based on curve-fitting, so they are failure criteria and not theories of any kind (the term theory implies a formal derivation process). 

If it cannot be guaranteed that the adsorbate remains in local equilibrium during its time evolution, then a set of macroscopic variables is not sufficient and an approach based on nonequihbrium statistical mechanics involving time-dependent distribution functions must be invoked. The kinetic lattice gas model is an example of such a theory [56]. It is derived from a Markovian master equation, but is not totally microscopic in that it is based on a phenomenological Hamiltonian. We demonstrate this approach... 

As the density of a gas increases, free rotation of the molecules is gradually transformed into rotational diffusion of the molecular orientation. After unfreezing , rotational motion in molecular crystals also transforms into rotational diffusion. Although a phenomenological description of rotational diffusion with the Debye theory [1] is universal, the gas-like and solid-like mechanisms are different in essence. In a dense gas the change of molecular orientation results from a sequence of short free rotations interrupted by collisions [2], In contrast, reorientation in solids results from jumps between various directions defined by a crystal structure, and in these orientational sites libration occurs during intervals between jumps. We consider these mechanisms to be competing models of molecular rotation in liquids. The only way to discriminate between them is to compare the theory with experiment, which is mainly spectroscopic. 

The present theory can be placed in some sort of perspective by dividing the nonequilibrium field into thermodynamics and statistical mechanics. As will become clearer later, the division between the two is fuzzy, but for the present purposes nonequilibrium thermodynamics will be considered that phenomenological theory that takes the existence of the transport coefficients and laws as axiomatic. Nonequilibrium statistical mechanics will be taken to be that field that deals with molecular-level (i.e., phase space) quantities such as probabilities and time correlation functions. The probability, fluctuations, and evolution of macrostates belong to the overlap of the two fields. 

The elastic free energy given by the elementary and the more advanced theories are symmetric functions of the three extension ratios Xx, Xy, and Xz. One may also express the dependence of the elastic free energy on strain in terms of three other variables, which are in turn functions of Xx, Xy, and Xz. In phenomenological theories of continuum mechanics, where only the observed behavior of the material is of concern rather than the associated molecular deformation mechanisms, these three functions are chosen as... 

No attempt will be made here to extend our results beyond the simple lowest-order limiting laws the often ad hoc modifications of these laws to higher concentrations are discussed in many excellent books,8 11 14 but we shall not try to justify them here. As a matter of fact, for equilibrium as well as for nonequilibrium properties, the rigorous extension of the microscopic calculation beyond the first term seems outside the present power of statistical mechanics, because of the rather formidable mathematical difficulties which arise. The main interests of a microscopic theory lie both in the justification qf the assumptions which are involved in the phenomenological approach and in the possibility of extending the mathematical techniques to other problems where a microscopic approach seems necessary in the particular case of the limiting laws, obvious extensions are in the direction of other transport coefficients of electrolytes (viscosity, thermal conductivity, questions involving polyelectrolytes) and of plasma physics, as well as of quantum phenomena where similar effects may be expected (conductivity of metals and semi-... 

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