Equilibrium non-local

X can be regarded as the local non-equilibrium distribution of the species a at a position rj. As the system approaches equilibrium, x becomes -small. In a like manner, the doublet density fjf3 can be decomposed into an equilibrium term, two contributions from the local non-equilibrium departures of the particle a and (3 separately and then their mutual effect... 

Peak broadening is a result of ordinary diffusion, eddy diffusion (due to flow along longer or shorter paths in packed columns), and local non-equilibrium. The eddy diffusion is absent in capillary columns. The zone spreading due to the ordinary diffusion od can be expressed by the formula ... 

When a laser pulse passes through a material it produces a local non equilibrium state that induces a modification of the optical properties. This is a transient effect that relaxes back to the equilihrium state through a variety of processes. In a typical pump-prohe experiment, a second laser pulse is sent on the material probing the optical modifications induced by the pump pulse. Since the second laser pulse arrives with a controlled delay, it monitors and measures the transient optical excitation and hence the relaxation of the nonequilibrium state. In the time-resolved OKE both the pulses, pump and probe, are linearly... 

If, within the diffusion zone, there is no active vacancy source or sink, then no drift of lattice planes could occur and the difference in the diffusion fluxes of substitutional chemical species would result in vacancy supersaturation and build-up of local stress states within the diffusion zone. Return to local equilibrium in a stress-free state could be achieved by the nucleation of pores leading to the well-known Kirkendall porosity (Fig. 2.2d). All intermediate situations are possible depending on local stress states and the density, distribution and efficiency of vacancy sources or sinks. However, it should be emphasized that complete Kirkendall shift would occur only in stress-free systems in local equihbrium. Therefore, all obstacles to the free relative displacement of lattice planes would lead to local non-equilibrium. Such a situation corresponds to the build-up of stress states that modify the conditions of local equilibrium and the action of vacancy sources or sinks these stress states must therefore be taken into account to define and analyse these local conditions and their spatial and temporal evolutions. 

The third approach is called the thermodynamic theory of passive systems. It is based on the following postulates (1) The introduction of the notion of entropy is avoided for nonequilibrium states and the principle of local state is not assumed, (2) The inequality is replaced by an inequality expressing the fundamental property of passivity. This inequality follows from the second law of thermodynamics and the condition of thermodynamic stability. Further the inequality is known to have sense only for states of equilibrium, (3) The temperature is assumed to exist for non-equilibrium states, (4) As a consequence of the fundamental inequality the class of processes under consideration is limited to processes in which deviations from the equilibrium conditions are small. This enables full linearization of the constitutive equations. An important feature of this approach is the clear physical interpretation of all the quantities introduced. 

Within non-equilibrium thermodynamics, the driving force for relaxation is provided by deviations in the local chemical potential from it s equilibrium value. The rate at which such deviations relax is determined by the dominant kinetics in the physical system of interest. In addition, the thermal noise in the system randomly generates fluctuations. We thus describe the dynamics of a step edge by the equation. 

The concept of transport resistances localized in the outermost regions of NS crystals was introduced in order to explain the differences between intracrystalline self-diffusion coefficients obtained by n.m.r methods and diffusion coefficients derived from non-equilibrium experiments based on the assumption that Intracrystalline transport is rate-limiting. This concept has been discussed during the past decade, cf. the pioneering work [79-81] and the reviews [2,7,8,23,32,82]. Nowadays, one can state that surface barriers do not occur necessarily in sorption uptake by NS crystals, but they may occur if the cross-sections of the sorbing molecular species and the micropore openings become comparable. For indication of their significance, careful analysis of... 

This situation with thermal equilibrium, where the population of the excited states and hence emission intensity is determined by collisions, is known as local thermodynamic equilibrium (LTE) and holds in the atmosphere up to altitudes of 50-60 km (Lenoble, 1993). Above this altitude, non-LTE models must be used (e.g., see Lopez-Puertas et a.l., 1998a, 1998b). 

The last few years have seen a minor revolution in determining solar and stellar abundances (Asplund, 2005). Much of the previous work assumed that the spectral lines originate in local thermodynamic equilibrium (LTE), and the stellar atmosphere has been modeled in a single dimension. Since 2000, improved computing power has permitted three-dimensional modeling of the Sun s atmosphere and non-LTE treatment of line formation. The result has been significant shifts in inferred solar abundances. 

The work that was performed in this set of experiments was an extension of work performed by Inoue and Kaufman (7). In the previous work, the migration of strontium in glauconite was modeled using conditions of local equilibrium for flows up to 6.3 kilometers per year (72 cm/hr). The differences between the predicted and experimental results in the experiments performed by Inoue and Kaufman may be due to the existence of non-equilibrium behavior. 

The coefficient C, related to the resistance to mass transfer between the two phases, becomes important when the flow rate is too high for equilibrium to be obtained. Local turbulence within the mobile phase and concentration gradients slow down the equilibrium process (Cs <=> Cm). The diffusion of solute between the phases is not instantaneous, hence the solute will be in a non-equilibrium process. 

Solid state reactions occur mainly by diffusional transport. This transport and other kinetic processes in crystals are always regulated by crystal imperfections. Reaction partners in the crystal are its structure elements (SE) as defined in the list of symbols (see also [W. Schottky (1958)]). Structure elements do not exist outside the crystal lattice and are therefore not independent components of the crystal in a thermodynamic sense. In the framework of linear irreversible thermodynamics, the chemical (electrochemical) potential gradients of the independent components of a non-equilibrium (reacting) system are the driving forces for fluxes and reactions. However, the flux of one independent chemical component always consists of the fluxes of more than one SE in the crystal. In addition, local reactions between SE s may occur. 

In conclusion, we observe that the crossing of crystal phase boundaries by matter means the transfer of SE s from the sublattices of one phase (a) into the sublattices of another phase (/ ). Since this process disturbs the equilibrium distribution of the SE s, at least near the interface, it therefore triggers local SE relaxation processes. In more elaborated kinetic models of non-equilibrium interfaces, these relaxations have to be analyzed in order to obtain the pertinent kinetic equations and transfer rates. This will be done in Chapter 10. 

The aim of this chapter is to clarify the conditions for which chemical kinetics can be correctly applied to the description of solid state processes. Kinetics describes the evolution in time of a non-equilibrium many-particle system towards equilibrium (or steady state) in terms of macroscopic parameters. Dynamics, on the other hand, describes the local motion of the individual particles of this ensemble. This motion can be uncorrelated (single particle vibration, jump) or it can be correlated (e.g., through non-localized phonons). Local motions, as described by dynamics, are necessary prerequisites for the thermally activated jumps responsible for the movements over macroscopic distances which we ultimately categorize as transport and solid state reaction.. 

The time evolution of a system may also be characterized according to the degree of perturbation from its equilibrium state. Linear theories hold if local equilibrium prevails, that is, each volume element of the non-equilibrium system can still be unambiguously defined by the usual set of (local) thermodynamic state variables. Often, a crystal is in (partial) equilibrium with respect to externally predetermined P and 7j but not with external component chemical potentials pik. Although P, T, and nk are all intensive functions of state, AP relaxes with sound velocity, A7 by heat conduction, and A/ik by matter transport. In solids, matter transport is normally much slower than the other modes of relaxation. 

As an illustration, consider the isothermal, isobaric diffusional mixing of two elemental crystals, A and B, by a vacancy mechanism. Initially, A and B possess different vacancy concentrations Cy(A) and Cy(B). During interdiffusion, these concentrations have to change locally towards the new equilibrium values Cy(A,B), which depend on the local (A, B) composition. Vacancy relaxation will be slow if the external surfaces of the crystal, which act as the only sinks and sources, are far away. This is true for large samples. Although linear transport theory may apply for all structure elements, the (local) vacancy equilibrium is not fully established during the interdiffusion process. Consequently, the (local) transport coefficients (DA,DB), which are proportional to the vacancy concentration, are no longer functions of state (Le., dependent on composition only) but explicitly dependent on the diffusion time and the space coordinate. Non-linear transport equations are the result. 

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