Mesoscope Non-Equilibrium

For a chemical reaction, for example, non-equilibrium thermodynamics formulates a linear relationship between the reaction rate and the affinity, which constitute only the first term in the development of the law of mass action. To obtain the full law, one has to take into account not only the initial and final states of the kinetics but all intermediate configurations, i.e. one has to introduce a mesoscopic degree of freedom accounting for the different molecular configurations. When this is done, in the framework of mesoscopic non-equilibrium thermodynamics, one arrives at the law of mass action governing the kinetics for arbitrary values of the thermodynamic... 

The Mesoscopic Non-Equilibrium Thermodynamics Approach to Polymer Crystallization... 

Nucleation and growth are the most important processes in determining the final morphology of the crystallized polymer and consequently the properties of the material. Both processes can be quite conveniently described within the framework of mesoscopic non-equilibrium thermodynamics. 

Mesoscopic non-equilibrium thermodynamics provides a description of activated processes. In the case considered here, when crystallization proceeds by the formation of spherical clusters, the process can be characterized by a coordinate y, which may represent for instance the number of monomers in a cluster, its radius or even a global-order parameter indicating the degree of crystallinity. Polymer crystallization can be viewed as a diffusion process through the free energy barrier that separates the melted phase from the crystalline phase. From mesoscopic non-equilibrium thermodynamics we can analyze the kinetic of the process. Before proceeding to discuss this point, we will illustrate how the theory applies to the study of general activated processes. 

We have shown that mesoscopic non-equilibrium thermodynamics satisfactorily describes the dynamics of activated processes in general and that of polymer crystallization in particular. Identification of the different mesoscopic configurations of the system, when it irreversibly proceeds from the initial to the final phases, through a set of internal coordinates, and application of the scheme of non-equilibrium thermodynamics enable us to derive the non-linear kinetic laws governing the behavior of the system. 

Bedeaux, D., Mazur, P. (2001). Mesoscopic non-equilibrium thermodynamics for quantum systems. Physica A. 298 81-100. 

Applying these methods to systems in the vicinity of the non-equilibrium critical points, the conclusion was drawn [72] that the mesoscopic approach contains excess information about spatial particle distribution the details of how the whole system s volume is divided into cells become unimportant as — oo. The possibility to employ expansion in inverse powers of vo -similarly to a complete mixing case - was also discussed. Asymptotically it leads to the Focker-Planck equation equivalent to the Langevin-like equation. 

Summing this Section up, we would like to note that in the approach discussed here the introduction of stochasticity on a mesoscopic level restricts the applicability of a method by such statements of a problem where subtle details of particle interaction become unimportant. First of all, we mean that kinetic processes with non-equilibrium critical points, when at long reaction time the correlation length exceeds all other spatial dimensions. This limitation makes us consider in the next Section 2.3 the microscopic level of the kinetic description. 

The formalism introduced in the previous subsections is able to incorporate the effect of these influences in the crystallization kinetics, thus providing a more realistic modeling of the process, which is mandatoiy for practical and industrial purposes. Due to the strong foundations of our mesoscopic formalism in the roots of standard non-equilibrium thermodynamics, it is easy to incorporate the influence of other transport processes (like heat conduction or diffusion) into the description of crystallization. In addition, our framework naturally accounts for the couplings between all these different influences. 

This description has to be compared with that proposed by non-equilibrium thermodynamics in terms of only two states, corresponding to the melted and crystallized phases in the example we are discussing, from which only one may account for the linear domain, when the chemical potentials at the wells are not very different. This feature imposes serious limitations in the application of NET to activation processes since that condition is rarely encountered in experimental situations and has therefore restricted its use to only transport processes. The mesoscopic version of non-equilibrium thermodynamics, on the contrary, circumvents the difficulty offering a promising general scenario useful in the characterization of the wide class of activated processes, which appear frequently in systems outside equilibrium of different nature. 

Though this entry has focused on equilibrium properties, mesoscopic and continuum models in chemical physics can also describe non-equilibrium phenomena, and we shall mention some techniques briefly. 

This new theory of the non-equilibrium thermodynamics of multiphase polymer systems offers a better explanation of the conductivity breakthrough in polymer blends than the percolation theory, and the mesoscopic metal concept explains conductivity on the molecular level better than the exciton model based on semiconductors. It can also be used to explain other complex phenomena, such as the improvement in the impact strength of polymers due to dispersion of rubber particles, the increase in the viscosity of filled systems, or the formation of gels in colloids or microemulsions. It is thus possible to draw valuable conclusions and make forecasts for the industrial application of such systems. 

The concepts of meso-thermodynamics can be extended to some non-equilibrium phenomena. In particular, like the thermodynamic properties, transport coefficients, such as the diffusion coefficient, become spatially dependent at meso-scales. Moreover, away from equilibrium, generic long-range correlations emerge even in simple molecular fluids, making the famous concept of local equilibrium, at least, questionable. In this section we focus only on one application of mesoscopic nonequilibrium thermodynamics in fluids fluid phase separation. 

Santamaria Holek, I., e. a. (2005). Mesoscopic thermodynamics of stationary non-equilibrium... 

Chapter VI illustrates what a mesoscopic level of description is meant to contain. It involves the variables of interest (usually assumed to be slow) plus a suitable set of auxiliary variables, whose role is to mimic the influence of the thermal bath on the variables of interest themselves. This level of description (reduced model theory) is less detailed than the truly microscopic one, because an overwhelming number of microscopic degrees of freedom are simulated with fluctuation-dissipation processes of standard type. The mesoscopic level, however, is still detailed enough to preserve the essential information without which the theoretical investigation becomes difficult and obscure. A new class of non-Gaussian equilibrium properties is proven to be responsible for the acceleration of the fall transient described in Chapter V. To obtain these results, use is made both of the theoretical tools already mentioned and of computer simulation (one-dimensional for translation and two-dimensional for rotation). 

When a selected portion of a fluid exhibits a fluctuation, it is no longer in equilibrium with its surroundings. Consequently, the net thermodynamic force acting on a fluctuation is non-zero, and tends to drive the system back to equilibrium via the transport of mass and energy. The relaxation of fluctuations, occurs according to the macroscopic laws of hydrodynamics, with the macroscopic transport coefficients. This equality of macroscopic and mesoscopic relaxation is known as Onsager s principle. For example, in the absence of chemical reactions, the hydrodynamic equation expressing the conservation of concentration c in a binary fluid is... 

---