Macroscopic phenomenological theories

Below a critical thickness of interfacially confined liquids, macroscopic phenomenological theories have to be adjusted. Simple nonpolar liquids such as... 

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). 

A. E. Yaroshchuk, S. S. Durkhin. Phenomenological theory of reverse osmosis in macroscopically homogeneous membranes and its specification for the capillary charged model. J Memb Sci 79 133, 1993. 

This idea is elegant for its simplicity and also for its usefulness. While often in phenomenological theories of materials, control of parameters with molecular structure would provide useful properties, but the parameters are not related in any obvious way to controllable molecular structural features. Meyer s idea, however, is just the opposite. Chemists have the ability to control enantiomeric purity and thus can easily create an LC phase lacking reflection symmetry. In the case of the SmC, the macroscopic polar symmetry of this fluid phase can lead to a macroscopic electric dipole, and such a dipole was indeed detected by Meyer and his collaborators in a SmC material, as reported in 1975.2... 

The theory described so far is based on the Master Equation, which is a sort of intermediate level between the macroscopic, phenomenological equations and the microscopic equations of motion of all particles in the system. In particular, the transition from reversible equations to an irreversible description has been taken for granted. Attempts have been made to derive the properties of fluctuations in nonlinear systems directly from the microscopic equations, either from the classical Liouville equation 18 or the quantum-mechanical equation for the density matrix.19 We shall discuss the quantum-mechanical treatment, because the formalism used in that case is more familiar. 

All that remains to be done for determining the fluctuation spectrum is to compute the conditional average, Eq. (31). However, this involves the full equations of motion of the many-body system and one can at best hope for a suitable approximate method. There are two such methods available. The first method is the Master Equation approach described above. Relying on the fact that the operator Q represents a macroscopic observable quantity, one assumes that on a coarse-grained level it constitutes a Markov process. The microscopic equations are then only required for computing the transition probabilities per unit time, W(q q ), for example by means of Dirac s time-dependent perturbation theory. Subsequently, one has to solve the Master Equation, as described in Section TV, to find both the spectral density of equilibrium fluctuations and the macroscopic phenomenological equation. 

In the phenomenological theory of phase transitions, it is customary to attribute order parameters to the relevant thermodynamic quantities associated with the macroscopic transition of the state of the material. In our case,... 

There are general relationships of transport phenomena based on phenomenological theory, i.e., on the correlations between macroscopically measurable quantities. The molecular theories explain the mechanism of transport processes taking into account the molecular structure of the given medium, applying the kinetic-statistical theory of matter. The hydrodynamic theories are also applied especially to describe - convection. 

The macroscopic phenomenological equation for heat flow is Fourier s law, by the mathematician Jean Baptiste Joseph Fourier (1768-1830). It appeared in his 1811 work, Theorie analytique de la chaleur (The analytic theory of heart). Fourier s theory of heat conduction entirely abandoned the caloric hypothesis, which had dominated eighteenth century ideas about heat. In Fourier s heat flow equation, the flow of heat (heat flux), q, is written as ... 

A + B v[/ =6, which results in an explicit relation between the homogeneous system superfluid density p and the coefficients A and B, so that p = A /B. At this stage the phenomenological theory of Ginzburg and Sobyanin can be adopted, representing the bulk order parameter and its superfluid density p in terms of a critical exponent of the macroscopic system... 

Though perhaps expressed in different terms, the contrast just cited has been the subject of intensive inquiry and controversy ever since the enunciation of the first and second laws of thermodynamics in the 1850 s. Invariably, a reconciliation is proposed based on regarding thermodynamics as a statistical macroscopic or phenomenological theory. 

Eq. (A.15) is a basic equation of Onsager s theory. Notice that so far it rests solely on macroscopic phenomenology, since both the flux-force relations and the Second Law derive from purely macroscopic experiments. 

We next turn to a discussion of the physical content of Onsager s theory. In this discussion we will develop Onsager s Eq. (A.15) from slow variable assumptions, thus showing the close link between slow variable models and macroscopic phenomenology. 

In the classical diffusion theory the adsorption term of bentonite is commonly treated by the distribution factor K. Instead of this macroscopic phenomenological treatment we propose a microscale HA procedure which exactly represents the edge adsorption characteristics at the edges of clay minerals. 

Kinetic theory and transition-state theory try to calculate the rates of chemical reactions starting from a model of molecular interactions. A less ambitious task is to correlate reaction rates with phenomenological laws of various macroscopic processes which have been established experimentally. This type of theory can be termed a phenomenological theory of reaction rates. For the purpose of calculating theoretical reaction rates, chemical reactions are divided into three categories bimolecular associations, uni-molecular dissociations, and intramolecular transformations. 

In this Section we merely attempt to make a contact with phenomenological theories, and concentrate our attention on effective charges in polar crystals at the same time we will pay attention to the electric fields - macroscopic or not - as they appear in our self-consistent calculations on polar crystals with displaced atoms. 

What remains to be established is a criterion that will differentiate between the tendency of the solid to respond to external loading by brittle fracture or ductile deformation. This is not an easy task, because it implies a connection between very complex processes at the atomistic level and the macroscopic response of the solid in fact, this issue remains one of active research at present. Nevertheless, phenomenological theories do exist which capture the essence of this issue to a remarkable extent. In an early work, Griffith developed a criterion for the conditions under which brittle fracture will occur [140]. He showed that the critical rate of energy per unit area Gi, required to open an existing crack by an infinitesimal amount in mode I loading is given by... 

If melting is difficult to characterize in molecular terms, nucleation and growth of crystalline particles from the melt is an even more elusive phenomenon. Given the extreme difficulty of obtaining molecular level information, phenomenological, macroscopic nucleation theories have been formulated [13] before and aside from numerical molecular simulation. These theories constitute an almost completely parallel approach to the matter and their description does not belong in this book, although points of contact with molecular level simulations have been explored [14]. 

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. 

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