Phenomenological parameters thermodynamics

Here the nucleation barrier AO is the excess thermodynamic potential needed to form the critical embryo within the uniform metastable state, while the prefactor Jq is determined by the kinetic characteristics for the embryo diffusion in the space of its size a. Expressions for both AO and Jo given by Zeldovich include a number of phenomenological parameters. 

Thermodynamic, statistical This discipline tries to compute macroscopic properties of materials from more basic structures of matter. These properties are not necessarily static properties as in conventional mechanics. The problems in statistical thermodynamics fall into two categories. First it involves the study of the structure of phenomenological frameworks and the interrelations among observable macroscopic quantities. The secondary category involves the calculations of the actual values of phenomenology parameters such as viscosity or phase transition temperatures from more microscopic parameters. With this technique, understanding general relations requires only a model specified by fairly broad and abstract conditions. Realistically detailed models are not needed to un-... 

Explain the emergence of the Seebeck, Peltier and Thomson effects in nonuniform conductors using the tools of phenomenological hnear thermodynamics. What is the physical sense of parameters that deter mine the magnitude of these effects ... 

From a multiple scale modeling perspective, the presence of phenomenological parameters in various effective theories provides an opportunity for information passage in which one theory s phenomenological parameters are seen as derived quantities of another. We have already seen that although the linear theory of elasticity is silent on the particular values adopted by the elastic moduli (except for important thermodynamic inequalities), these parameters may be deduced on the basis of microscopic analysis. The advent of reliable models of material behavior makes it possible to directly calculate these parameters, complementing the more traditional approach which is to determine them experimentally. 

In Eq. (A.17), as before X(f) is the vector of thermodynamic forces while M is a symmetric matrix of phenomenological parameters introduced by Machlup and Onsager [4]. We adopt Eq. (A.17) inasmuch as it is the simplest equation of motion that is consistent with the Machlup-Onsager Eq. (A.24). Notice that Eq. (A.17) is similar in form to Newton s equation of motion for a particle system. Thus, we denote the matrix of phenomenological parameters by M in order to emphasize the analogy to particle masses. The analogy, however, is not perfect because M may be nondiagonal [4]. 

In the framework the phenomenological approach - thermodynamics - the temperature is measured by the monitoring other physical parameters (expansion coefficient, resistivity, voltage, capacity etc.). That is why there are so many different kinds of thermometers. The procedure of temperature measinements consists of the thermal contact (energy exchange) between the system imder consideration and a thermometric body the physical state of which is monitored. This thermometric body should be as small as possible in order to do not disturb the state of the system dining measurement. In the realm of very small systems such a procedure is rather questionable. What the size should be for the thermometer to measure, for example, the temperature of a nanosystem Should the thermometric body be an atom or elementary particle in this case But the states of atoms and elementary particles are essential quantum ones and can not be changed continuously. The excellent treatment of the more sophisticated measurements of temperature (spectral temperature and radiation temperature) the reader can find in the very recent book (Biro, 2011). 

Huggins Molecular Models.—In a series of, so far, 14 papers, Huggins has developed an approach to solution thermodynamics that essentially builds on lattice-graph models, removing phenomenological parameters by re-expressing them in terms of molecular quantities obtained, as far as is possible, by independent experimental means. Thus far the theory has been applied mainly to non-polymer systems. The most recent part considers benzene solutions of n-alkanes as model oligomers for polyethylene. 

In this section we use Landau-de Gennes theory to calculate the thermodynamic properties of liquid-crystal phase transitions in terms of the phenomenological parameters a, B, C, and Tc. In particular, we will derive expressions for Tc, the transition temperature c, the equilibrium order parameter value in the low temperature phase at the transition and As, the transition entropy per molecule. We will also calculate the temperature dependence of (the equilibrium value of S) for T close to Tc, and h, the height of the free-energy barrier between = 0 and = c at T = Tc. 

Here the first term suppresses local density fluctuations [73,74] and the rest is an analog of the Ginzburg-Landau free energy associated with the instantaneous tensorial fields Q and B. Phenomenological parameters V,]i and A entering this functional are normally chosen such that the thermodynamic state of interest, e.g., a biaxial-nematic mesophase, is reproduced. In this respect, mean-field estimates can help to limit the physically adequate parameter ranges. Positive isothermal compressibility, for example, requires k >V + Ji + A [64]. 

Blends of immiscible polymers form liquid phases if the temperature is higher than the glass transition points (and/or melting points) of the components. Such blends behave as emulsions, and the thermodynamic interaction between the components is often quantified by a phenomenological parameter, the interfacial tension T, given that the components are isotropic and the interface is sharp compared with the phases. (Blends of lightly crosslinked mbbers in liquid matrices can be also classified as emulsions.) The stress Oim due to the interface between different phases is directly related to T (cf. eqn [19]) and the relaxation of Oint is slower than the relaxation of individual component chains. Thus, extensive rheological studies have been made for such emulsion-type blends,... 

If we turn from phenomenological thermodynamics to statistical thermodynamics, then we can interpret the second virial coefficient in terms of molecular parameters via a model. We pursue this approach for two different models, namely, the excluded-volume model for solute molecules with rigid structures and the Flory-Huggins model for polymer chains, in Section 3.4. 

Since we are considering equilibrium boundaries and interfaces, let us introduce some phenomenological thermodynamics. If 6 symbolizes the orientation (location) of two crystal parts (phases) relative to each other, and s designates a structure parameter that symbolizes the atomic structure of the boundary (composition and structural details), then... 

First-principle calculations of the thermodynamic properties are more or less hopeless enterprise. One of the most famous phenomenological approaches was suggested by van der Waals [6, 8, 9]. Using the dimensionless pressure it = p/pc, the density v = n/nc and the temperature r = T/Tc, the equation of state for the ideal gas reads it = 8zzr/(3 -u) — 3zA Its r.h.s. as a function of the parameter v has no singularities near u = 1 v = it = t = is the critical point) and could be expanded into a series in the small parameter 77 = [n — nc)/nc with temperature-dependent coefficients. Solving this... 

It is well known that a flow-equilibrium must be treated by the methods of irreversible thermodynamics. In the case of the PDC-column, principally three flows have to be considered within the transport zone (1) the mass flow of the transported P-mer from the sol into the gel (2) the mass flow of this P-mer from the gel into the sol and (3) the flow of free energy from the column liquid into the gel layer required for the maintenance of the flow-equilibrium. If these flows and the corresponding potentials could be expressed analytically by means of molecular parameters, the flow-equilibrium 18) could be calculated by the usual methods 19). However, such a direct way would doubtless be very cumbersome because the system is very complicated (cf. above). These difficulties can be avoided in a purely phenomenological theory, based on perturbation calculus applied to the integrated transport Eq. (3 b) of the PDC-column in a reversible-thermodynamic equilibrium. 

The electrostatic part, Wg(ft), can be evaluated with the reaction field model. The short-range term, i/r(Tl), could in principle be derived from the pair interactions between molecules [21-23], This kind of approach, which can be very cumbersome, may be necessary in some cases, e.g. for a thorough analysis of the thermodynamic properties of liquid crystals. However, a lower level of detail can be sufficient to predict orientational order parameters. Very effective approaches have been developed, in the sense that they are capable of providing a good account of the anisotropy of short-range intermolecular interactions, at low computational cost [6,22], These are phenomenological models, essentially in the spirit of the popular Maier-Saupe theory [24], wherein the mean-field potential is parameterized in terms of the anisometry of the molecular surface. They rely on the physical insight that the anisotropy of steric and dispersion interactions reflects the molecular shape. 

In conclusion, field dependent single-crystal magnetization, specific-heat and neutron diffraction results are presented. They are compared with theoretical calculations based on the use of symmetry analysis and a phenomenological thermodynamic potential. For the description of the incommensurate magnetic structure of copper metaborate we introduced the modified Lifshits invariant for the case of two two-component order parameters. This invariant is the antisymmetric product of the different order parameters and their spatial derivatives. Our theory describes satisfactorily the main features of the behavior of the copper metaborate spin system under applied external magnetic field for the temperature range 2+20 K. The definition of the nature of the low-temperature magnetic state anomalies observed at temperatures near 1.8 K and 1 K requires further consideration. 

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,... 

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