Deviation from local equilibrium

In this section we address formation of concentration shocks in reactive ion-exchange as an asymptotic phenomenon. The prototypical case of local reaction equilibrium of Langmuir type will be treated in detail, following [1], [51], For a treatment of the effects of deviation from local equilibrium the reader is referred to [51]. The methodological point of this section consists of presentation of a somewhat unconventional asymptotic procedure well suited for singular perturbation problems with a nonlinear degeneration at higher-order derivatives. The essence of the method proposed is the use of Newton iterates for the construction of an asymptotic sequence. 

We deal in this section with quasi-binary systems in which more than one product phase A, B forms between the reactants A(=AX) and B(=BX) (Fig. 6-9). The more interfaces separating the different product phases, the more likely it is that deviations from local equilibrium occur (the interfaces become polarized during transport as indicated in Fig. 6-9, curve b). Polarization of interfaces is the theme of Chapter 10. If, however, we assume that local equilibrium is established during reaction, the driving force of each individual phase (p) in the product is inversely... 

The superscripts in [] are not to be confused with the prior superscripts in (). When A is smaller, the frequency of collisions is larger and the mean free path is smaller, whereby there is less deviation from local equilibrium. Thence, coefficients of equal powers of X may be equated in the usual fashion to continue with the solution, but it should be mentioned that the convergence of the series in Eq. (229) is open to some doubt. 

We now introduce the stable state assumption [4,14]. The projector onto the fast variables Q = 1 — P has two contributions a contribution describing the deviations from local equilibrium in each stable state well and a contribution from the intermediate region I. We will assume that the characteristic time for equilibration of each stable state Tint is much smaller that the characteristic time Trei for the transition A — B to occur. Under these conditions we have approximately ... 

Next we consider how to evaluate the factor 6p. We recognize that there is a local variation in the Gibbs free energy associated with a fluctuation in density, and examine how this value of G can be related to the value at equilibrium, Gq. We shall use the subscript 0 to indicate the equilibrium value of free energy and other thermodynamic quantities. For small deviations from the equilibrium value, G can be expanded about Gq in terms of a Taylor series ... 

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. 

We consider the reactive solute system with coordinate x and its associated mass p, in the neighborhood of the barrier top, located at x=xi=0, and in the presence of the solvent. We characterize the latter by the single coordinate. v, with an associated mass ps. If the solvent were equilibrated to x in the barrier passage, so that there is equilibrium solvation and s = seq(x), the potential for x is just -1/2 pcc X2, where (, , is the equilibrium barrier frequency [cf. (2.2)]. To this potential we add a locally harmonic restoring potential for the solvent coordinate to account for deviations from this equilibrium state of affairs ... 

In addition, deviations from known equilibrium structures denoted by their symmetry can be avoided. Sometimes, definition of local symmetry for specific subunits is recommended. On the other hand, symmetry constraints outside potential energy minima should be handled with great care,... 

In other cases, however, and in particular when sublattices are occupied by rather immobile components, the point defect concentrations may not be in local equilibrium during transport and reaction. For example, in ternary oxide solutions, component transport (at high temperatures) occurs almost exclusively in the cation sublattices. It is mediated by the predominant point defects, which are cation vacancies. The nearly perfect oxygen sublattice, by contrast, serves as a rigid matrix. These oxides can thus be regarded as models for closed or partially closed systems. These characteristic features make an AO-BO (or rather A, O-B, a 0) interdiffusion experiment a critical test for possible deviations from local point defect equilibrium. We therefore develop the concept and quantitative analysis using this inhomogeneous model solid solution. 

Here, X is the. lA -dimensional vector consisting of K 3-dimensional vectors = AM s(Rs — Rs), where s = 1,2, 3,. .. K numbers the nuclei in the system of the local vibrations, Ms is the mass of 5th nucleus, (Rs — Rs) is the vector of deviation from the equilibrium position Rs, Ol f are the frequency tensors of second rank in the initial and final states. Let us introduce such unitary operators Sy that transformations 2 — Sy Y reduce the quadratic... 

F. Observed Deviations from Local Thermodynamical Equilibrium... 

All these flow types appear more or less in a series one after the other during the evaporation of a liquid in a vertical tube, as Fig. 4.30 illustrates. The structure of a non-adiabatic vapour-liquid flow normally differs from that of an adiabatic two-phase flow, even when the local flow parameters, like the mass flux, quality, etc. agree with each other. The cause of this are the deviations from thermodynamic equilibrium created by the radial temperature differences, as well as the deviations from hydrodynamic equilibrium. Processes that lead to a change in the flow pattern, such as bubbles coalescing, the dragging of liquid drops in fast flowing vapour, the collapse of drops, and the like, all take time. Therefore, the quicker the evaporation takes place, the further the flow is away from hydrodynamic equilibrium. This means that certain flow patterns are more pronounced in heated than in unheated tubes, and in contrast to this some may possibly not appear at all. 

The interfacial energy is contributed both by deviations from the equilibrium density levels in the transitional region and by the distortion energy localized there. The ID interaction kernel Q z) lumps intermolecular interaction between the layers 2 = const. It is computed by lateral integration using as an... 

The most common deviation from local thermal equilibrium is caused by the relative oversaturation of the ground state. The higher the energies of these levels, the more precisely the Boltzmann law gives the population ratio between two energy levels NjNj). If the deviations from the local equilibrium are great, the contribution of the electron density must be added to the temperature dependent Boltzmann law. Several plasma models have been proposed for these cases. 

At the present time, of all EXAFS-like methods of analysis of local atomic structure, the SEES method is the least used. The reason is that the theory of the SEES process is not sufficiently developed. However the standard EXAES procedure of the Fourier transformation has been applied also to SEES spectra. The Fourier transforms of MW SEES spectra of a number of pure 3d metals have been compared with the corresponding Fourier transforms of EELFS and EX-AFS spectra. Besides the EXAFS-like nature of SEES oscillations shown by this comparison, parameters of the local atomic structure of studied surfaces (the interatomic distances and the mean squared atomic deviations from the equilibrium positions [12, 13, 15-17, 21, 23, 24]) have been obtained from an analysis of Fourier transforms of SEES spectra. The results obtained have, at best, a semi-quantitative character, since the Fourier transforms of SEES spectra differ qualitatively from both the bulk crystallographic atomic pair correlation functions and the relevant Fourier transforms of EXAFS and EELFS spectra. 

Since the intensity of spontaneous Raman lines is proportional to the density N(vi, Ji) of molecules in the initial state (u/, 7/), Raman spectroscopy can provide information on the population distribution A(u/, 7/), its local variation, and on concentrations of molecular constituents in samples. This allows one, for instance, to probe the temperature in flames or hot gases from the rotational Raman spectra [372, 373, 377, 378] and to detect deviations from thermal equilibrium. 

Absolute values of potentials in metal and electrolyte phases do not matter besides, they cannot be measured. For the determination of catalyst layer local overpotentials, it only matters by how much the local values of and deviate from their equilibrium values. 

It is known [5], that at elastoplastic behavior a system crack-local deformation zone deviates from thermodynamical equilibrium and for its analysis a principles, correct for close to equilibrium systems, for example, Griffith theory, are inapplicable. Besides, prefailure zone structure is differed from elastically deformed material structure (Fig. 5.3) that complicates additionally process analysis. As it was noted above, for polymers this effect is displayed as the formation of local deformation zones near crack tip, containing microvoids and oriented material (crazes) or oriented material only (ZD) [20]. Therefore, for fracture analysis in such cases fracture fractal theory is applied, using fractal analysis and general principles of synergetics [28]. 

Equation (2-38) is valid for every region of the surface. In this case only weight loss corrosion is possible and not localized corrosion. Figure 2-5 shows total and partial current densities of a mixed electrode. In free corrosion 7 = 0. The free corrosion potential lies between the equilibrium potentials of the partial reactions and U Q, and corresponds in this case to the rest potential. Deviations from the rest potential are called polarization voltage or polarization. At the rest potential = ly l, which is the corrosion rate in free corrosion. With anodic polarization resulting from positive total current densities, the potential becomes more positive and the corrosion rate greater. This effect is known as anodic enhancement of corrosion. For a quantitative view, it is unfortunately often overlooked that neither the corrosion rate nor its increase corresponds to anodic total current density unless the cathodic partial current is negligibly small. Quantitative forecasts are possible only if the Jq U) curve is known. 

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