Thermodynamics, phenomenological theory

D. Blankschtein, G. Thurston, G. Benedek. Phenomenological theory of equilibrium thermodynamic properties and phase separation of micellar solutions. J Chem Phys 25 7268-7288, 1986. 

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 form of W as a function of the set of must be derived from molecular theory or from experimental measurement. It cannot be deduced from the phenomenologic theory of continua, just as the free energy cannot be deduced from thermodynamics. However, the phenomenologic theory imposes the following restrictions on the form of W if the material is isotropic 18 First, W must be an even power function of X,-(restriction A). Second, W must be invariant for permutations of Xt (restriction B). Third, W must be invariant for the transformation of coordinate axes (restriction C). 

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. 

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,... 

The study of ferroelectrics has been greatly assisted by so-called phenomenological theories which use thermodynamic principles to describe observed behaviour in terms of changes in free-energy functions with temperature. Such theories have nothing to say about mechanisms but they provide an invaluable framework around which mechanistic theories can be constructed. A.F. Devonshire was responsible for much of this development between 1949 and 1954 at Bristol University. 

The Kauzmann temperature plays an important role in the most widely applied phenomenological theories, namely the configurational entropy [100] and the free-volume theories [101,102]. In the entropy theory, the excess entropy ASex obtained from thermodynamic studies is related to the temperature dependence of the structural relaxation time xa. A similar relation is derived in the free-volume theory, connecting xa with the excess free volume AVex. In both cases, the excess quantity becomes zero at a distinguished temperature where, as a consequence, xa(T) diverges. Although consistent data analyses are sometimes possible, the predictive power of these phenomenological theories is limited. In particular, no predictions about the evolution of relaxation spectra are made. Essentially, they are theories for the temperature dependence of x.-jT) and r (T). 

The formulation of linear nonequilibrium thermodynamics is based on the combination of the first and second laws of thermodynamics with the balance equations including the entropy balance. These equations allow additional effects and processes to be taken into account. The linear nonequilibrium thermodynamics approach is widely recognized as a useful phenomenological theory that describes the coupled transport without the need for the examination of the detailed coupling mechanisms of complex processes. 

In addition to a general knowledge of laboratory techniques, creative research work requires the ability to apply two different kinds of theory. Many an experimental method is based on a special phenomenological theory of its own this must be well rmderstood in order to design the experiment properly and in order to calculate the desired physical property from the observed raw data. Once the desired result has been obtained, it is necessary to understand its significance and its interrelationship with other known facts. This requires a sound knowledge of the fundamental theories of physical chemistry (e g., thermodynamics, statistical mechanics, qrrantrrm mechanics, and kinetics). Cortsider-able emphasis has been placed on both kirrds of theory in this book. 

The foregoing appears to be as satisfactory as can be expected from a phenomenological theory, and seems to indicate that more etching experiments with ionic crystals in aqueous solutions (or other media for which the thermodynamic data are known) should be made in order to see whether or not the correlation between critical undersaturation for etching and surface energy is confirmed. In spite of its success, this simple approach has been criticized to a surprising extent (see for... 

Recently, a quantitative model for the size-dependent Curie temperature has been established based on mechanic and thermodynamic considerations using the Landau-Ginsburg-Devonshire (LGD) phenomenological theory by equalizing free energies of the ferroelectrics and paraelectrics phases [1-4],... 

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. 

As for polymer blends, the phenomenological theory of block copolymers attributes the thermodynamic properties to an interplay of configurational... 

A further phenomenological theory, which uses the concept of strain-energy functions, deals with more general kinds of stress than uniaxial stress. When a rubber is strained work is done on it. The strain-energy function, U, is defined as the work done on unit volume of material. It is unfortunate that the symbol U is conventionally used for the strain-energy function and it will be important in a later section to distinguish it from the thermodynamic internal-energy function, for which the same symbol is also conventionally used, but which is not the same quantity. 

Considerable amount of work has been done on both charged and uncharged membranes [46-53]. We shall discuss some of the recent data in the context of the conclusion of the thermodynamic theory developed in an earlier section. It may be noted that Eqs. (4.14) and (4.15) are used as axioms in the development of the phenomenological theory. The only justification for these is that these are consistent with the theory developed in the subsequent section. A direct test of the validity of the linear phenomenological equation has been performed recently. For pyrex sinter with water as permeant, (7)totd. (J) 4,=o have been measured individually. It is found... 

One approach has been suggested by WALDMANN [2.89] and WALDMANN and VESTNER [2.90]. This invoives the use of a truncated version of Maxwell s moment equations. Boundary conditions for Waldmann s "generalized hydrodynamics are established by the methods of nonequilibrium thermodynamics with restrictions arising from the second law and Onsager-Casimir symmetries. This gives, of course, phenomenological coefficients for the boundary laws. These coefficients are implicitly dependent on the accommodation coefficients, but the nature of this dependence cannot be determined solely from the phenomenological theory. 

A phenomenological theory of ordinary diffusion in multicomponent systems based on irrevernble thermodynamics suggests that the ordinary diffusive flux of any species is a linear function of all ffie independent composition gradients. In one dimension. 

In Landau s phenomenological theory of the mean field the thermodynamic potential G (or F) is considered as a series in terms of the order parameter Q. I he series coefficients, generally, depend on temperature and pressure. Depending on the presence of the series term with a certain power n Q ), on the coefficients values, and on the relationships between them, the thermodynamic potential assumes a functional dependence being characteristic for the first-order transitions, transitions of the continuous type or for the first-order ones of the near-continuous type. 

The accurate description of the thermodynamic behaviors of polymeric systems provides benchmarks to phenomenological theories. Scaling analysis... 

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