Spatially inhomogeneous systems

Currently the authors are developing three classes of models of extreme intermediate states (MEIS) (1) with variable parameters (2) with variable flows, and (3) those describing spatially inhomogeneous systems. All these classes of the models are formulated and analyzed in terms of MP, which, in the authors opinion, can be defined as a mathematical theory of equilibrium states. It is natural to start the analysis of the created modifications with the MEIS with variable parameters, which is the closest in character to the traditional equilibrium thermodynamics models. 

Let us write the MEIS of spatially inhomogeneous system with intensive parameters changing only along the vertical axe, and with fixed y and, in each k-th zone, T and P has the form find... 

Classification of MEISs. Models with variable parameters with variable flows and spatially inhomogeneous systems with constraints on the macroscopic kinetics and without them. Specific features of modifications and their comparative capabilities. 

Devising the methods for analysis of spatially inhomogeneous systems, applied first of all to nonisothermal natural systems and installations for fuel combustion and processing. 

Description of nonstationary kinetics and transfer in spatially inhomogeneous systems. 

The thermodynamic force (affinity) X is a pivotal concept in thermo dynamics of nonequilibrium processes because of its relationship to the concept of driving force of a particular irreversible process. Evidently, thermodynamic forces arise in spatially inhomogeneous systems with, for example, temperature, concentration, or pressure inhomogeneity. In spatially uniform homogeneous systems, such forces arise either in the presence of chemically reactive components that have not reached thermodynamic equiHbrium via respective chemical transformations or at the thermodynamic possibility of some phase transformations. 

Calculating the Thermodynamic Forces in Spatially Inhomogeneous Systems... 

In the general case of a spatially inhomogeneous system, the total rate of energy dissipation is described by the space integral ... 

In such cases a local-equilibrium structure may be obtained theoretically by minimization of the free energy of the system under the constraint of a fixed alloy composition in the surface region [8,17-24]. Although this approach is very similar to the one used for bulk systems, it should be modified due to the specific features introduced by the surface. First of all, since the structure of the underlying bulk system is fixed, it acts as the source of an external field for the surface alloy, creating, for instance, epitaxial strain. Secondly, since the surface is an open system, it allows the formation of a great variety of different structures, which may not have any connection at all to the crystal structure of the substrate. Finally, the surface is a spatially inhomogeneous system, and thus different alloy components have their own... 

In a macroscopic system a supersaturated vapor, Ap = p — Pcoex > 0, is metastable and the excess number of particles will condense into a drop of radius R that consists of the thermodynamically stable liquid. In the framework of classical nucleation theory, the excess free energy of such a spatially inhomogeneous system can be decomposed into a surface and a volume contribution ... 

Besides issues related to the accuracy of force fields in spatially inhomogeneous systems comprising many chemically distinct components, the basic restriction related to the chemically detailed models is the rather small length and time scales that they can access. This limitation imposes severe restrictions for considering collective phenomena in amphiphilic vesicles, i.e., processes that involve large particle numbers. Typical examples include vesicle assembly, vesicle fusion, phase separation and shape transformations of multicomponent amphiphilic vesicles. For many of these processes, it is expected that the underlying atomistic details of the molecular constituents can be captured by a small number of relevant characteristics and universality classes, comprised of systems with a rather different atomistic structure, can be identified. These phenomena can be successfully investigated via minimal... 

In two classic papers [18. 46]. Cahn and Hilliard developed a field theoretic extension of early theories of nucleation by considering a spatially inhomogeneous system. Their free energy functional, equations tA3.3.52k has already been discussed at length in section A3.3.3. They considered a two-component incompressible fluid. The square gradient approximation implied a slow variation of the concentration on the... 

Furthermore, we measured the time-dependent halfwidth of the emission spectra, which provides useful information on the extent of the studied process. It has been shown that the halfwidth should be constant in continuously relaxing homogeneous systems. In spatially inhomogeneous systems, the relaxation behavior proceeds differently. Because the properties of the system vary in space, individual fluorophores distributed in the system are nonequivalent and their solvent shells respond with different rates to the local electric field. This inhomogeneity gives rise to a new phenomenon that reflects the time distribution of phases of... 

There are also other reasons that truncate the order parameter divergence such as spatial inhomogeneities or external fields. For example, to describe a spatial inhomogeneous system, a term quadratic in the gradient of the order parameter G(Vri) must be added to the density of free energy and all the Landau expansion should be integrated over the system volume ... 

A basic approach for the description of polymer chains in the continuum is the Gaussian thread model [26, 31]. Treating interactions among monomers in a mean-field-like fashion, one obtains the self-consistent field theory (SCFT) [11, 32-36] which can also be viewed as an extension of the Hory-Huggins theory to spatially inhomogeneous systems (like polymer interfaces in blends, nucrophase separation in block copolymer systems [11, 13], polymer bmshes [37, 38], etc.). However, with respect to the description of the equation of state of polymer solutions and blends in the bulk, it is stiU on a simple mean-field level, and going beyond mean field to include fluctuations is very difficult [11, 39-42] and outside the scope of this article. 

Upon involving two or even more variables in connection with spatially inhomogeneous systems, the higher-order non-linearity gives rise to more complex phenomena. The best examples are systems controlled by simultaneous chemical reactions and mass diffusion. From the mathematical point of view, the system becomes localized at the thermodynamic branch and the initially stable solution of the appropriate balance equation bifurcates and new stable solutions suddenly appear often overlapping. One such a possibility is time-symmetry breaking, associated with the merging of time-periodic solutions known as limit... 

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