Onsager approach described

In the following we briefly discuss the continuum models of the Onsager type. The development of continuum models describing structural phase behavior in the context of reversible assembly, largely parallels the Onsager approach to ordinary lyotropics and its extensions. In the simplest case we may write Eq. (20) as... 

The key differences between the PCM and the Onsager s model are that the PCM makes use of molecular-shaped cavities (instead of spherical cavities) and that in the PCM the solvent-solute interaction is not simply reduced to the dipole term. In addition, the PCM is a quantum mechanical approach, i.e. the solute is described by means of its electronic wavefunction. Similarly to classical approaches, the basis of the PCM approach to the local field relies on the assumption that the effective field experienced by the molecule in the cavity can be seen as the sum of a reaction field term and a cavity field term. The reaction field is connected to the response (polarization) of the dielectric to the solute charge distribution, whereas the cavity field depends on the polarization of the dielectric induced by the applied field once the cavity has been created. In the PCM, cavity field effects are accounted for by introducing the concept of effective molecular response properties, which directly describe the response of the molecular solutes to the Maxwell field in the liquid, both static E and dynamic E, [8,47,48] (see also the contribution by Cammi and Mennucci). 

The Onsager irreversible thermodynamics approach in terms of the Fick s law methodology is, between the most frequently applied procedures to describe diffusion in mixtures. For a binary mixture (i.e., two types of diffusing molecules A and B) it is given by [124,125] ... 

We can describe irreversibility by using the kinetic theory relationships in maximum entropy formalism, and obtain kinetic equations for both dilute and dense fluids. A derivation of the second law, which states that the entropy production must be positive in any irreversible process, appears within the framework of the kinetic theory. This is known as Boltzmann s H-theorem. Both conservation laws and transport coefficient expressions can be obtained via the generalized maximum entropy approach. Thermodynamic and kinetic approaches can be used to determine the values of transport coefficients in mixtures and in the experimental validation of Onsager s reciprocal relations. 

Another approach to the conductance of electrolytes, which is less complex than that of Lee and Wheaton, is due to Blum and his co-workers. This theory goes back to the original Debye-Hiickel-Onsager concepts, for it does not embrace the ideas of Lee and Wheaton about the detailed structure around the ion. Instead, it uses the concept of mean spherical approximation of statistical mechanics. This is the rather portentous phrase used for a simple idea, which was fully described in Section 3.12. It is easy to see that this is an approximation because in reality an ionic collision with another ion will be softer than the brick-wall sort of idea used in an MSA approach. However, using MSA, the resulting mathematical treatment turns out to be relatively simple. The principal equation from the theory of Blumet al. is correspondingly simple and can be quoted. It runs... 

Several combinations of QM approaches with a continuum dielectric model [175, 325, 360-369] have been focused upon. Continuum solvation models have their origin in the work of Onsager [325] for describing ions in solution these models have shown flexibility and accuracy enough to become a popular tool nowadays. Generally in these methods the solute is placed inside a cavity with appropriate shape, made in a continuous medium characterised by a dielectric constant. The electronic distribution of the solute induces a charge density at the surface of the cavity which creates a field that modifies the energy... 

The structural approach will also contribute to the analysis of the thermodynamics of nonequilibrium systems. It is the aim and purpose of thermod5mamics to describe structural features of systems in terms of macroscopic variables. Unfortunately, classical thermodynamics is concerned almost entirely with the equilibrium state it makes only weak statements about nonequilibrium systems. The nonequihbrium thermodynamics of Onsager (f), Prigogine (2), and others introduces additional axioms into classical thermodynamics in an attempt to obtain stronger and more useful statements about nonequilibrium systems. These axioms lead, however, to an expression for the driving force of chemical reactions that does not agree with experience and that is only applicable, as an approximation, to small departures from equilibrium. A way in which this situation may be improved is outlined in Section VII. 

In order to describe the diffusive dynamics of composition fluctuations in binary mixtures one can extend the time-dependent Ginzburg-Landau methods to the free energy functional of the SCF theory. The approach relies on two ingredients a free energy functional that accurately describes the chemical potential of a spatially inhomogeneous composition distribution out of equilibrium and an Onsager coefficient that relates the variation of the chemical potential to the current of the composition. 

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