Structure Element Fluxes

The total flux JK of SE s in sublattice x and the total flux J of SE s in the whole crystal are given by 

If all fluxes vanish and the number of lattice sites is conserved, only two types of homogeneous reactions between SE s are possible 

In view of the assumed site conservation in each sublattice we then have 

The structural conditions of a crystal lattice are, in accordance with Eqn. (2.12) 

Electroneutrality imposes a further condition on the fluxes, namely 

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Knowing the heat flux from a fire and temperatures, the time to structural failure can be estimated. A somewhat more detailed approach is to evaluate the heat transfer to the structural element and compare the resulting temperature to critical failure temperatures. Failure of a structural metal element occurs... 

In order to describe interfaces kinetically, we choose the equilibrium state of the interface as the reference state. In (dynamic) equilibrium, the net fluxes of components k vanish across an interface. Since the mobilities of the components in the interface are finite, there can be no driving forces acting upon component k at equilibrium. For isothermal and isobaric crystals with electrically charged structure elements, this means that Ari, = 0 (/ denoting the (charged) reversible carrier of type /). The explicit form of this equilibrium condition is... 

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. 

We will introduce basic kinetic concepts that are frequently used and illustrate them with pertinent examples. One of those concepts is the idea of dynamic equilibrium, as opposed to static (mechanical) equilibrium. Dynamic equilibrium at a phase boundary, for example, means that equal fluxes of particles are continuously crossing the boundary in both directions so that the (macroscopic) net flux is always zero. This concept enables us to understand the non-equilibrium state of a system as a monotonic deviation from the equilibrium state. Driven by the deviations from equilibrium of certain functions of state, a change in time for such a system can then be understood as the return to equilibrium. We can select these functions of state according to the imposed constraints. If the deviations from equilibrium are sufficiently small, the result falls within a linear theory of process rates. As long as the kinetic coefficients can be explained in terms of the dynamic equilibrium properties, the reaction rates are directly proportional to the deviations. The thermodynamic equilibrium state is chosen as the reference state in which the driving forces X, vanish, but not the random thermal motions of structure elements i. Therefore, systems which we wish to study kinetically must first be understood at equilibrium, where the SE fluxes vanish individually both in the interior of all phases and across phase boundaries. This concept will be worked out in Section 4.2.1 after fluxes of matter, charge, etc. have been introduced through the formalism of irreversible thermodynamics. 

Since doped zirconia allows one to extend the oxide electrochemistry up to very high temperatures and since it can serve as a fuel cell electrolyte and even as a heating element in high temperature furnaces, we will briefly formalize the structure element transport in zirconia, which is the basis for all of this. Let us introduce the SE fluxes in their usual form. We know that only oxygen ions and electronic defects contribute to the electrical transport (/ = 02, e, h )... 

The extinction of the luminous flux passing through a foam layer occurs as a result of light scattering (in the processes of reflection, refraction, interference and diffraction from the foam elements) and light absorption by the solution. In a polyhedral foam there are three structural elements, clearly distinct by optical properties films, Plateau borders and vertexes. The optical properties of single foam films have been widely studied (see Section 2.1.3) but these of the foam as a disperse systems are poorly considered. 

A conclusion has been drawn in [67] that the extinction of the luminous flux (7//o, where, 7o is the intensity of the incident light and 7 is the intensity the light passed through the foam) is a linear function of the specific foam surface area. A similar dependence has been used also for the determination of the specific surface area of emulsions [68]. Later, however, it has been shown [69,70] that the quantity 7//o depends not only on the specific surface area (or dispersity) but also on the liquid content in the foams, i.e. on the foam expansion ratio, that during drainage can increase without changing the dispersity. Since foam expansion ratio and dispersity are determined by the radii of border curvatures and film thicknesses, all the structural elements of the foam will contribute to the optical density of foams. This means that... 

As noted earlier, quantitative element fluxes in subduction zones are tricky. They depend on magma production rates, which are often poorly known. They also depend on specific fluid-fluxing or sediment melting models, which implicitly or explicitly invoke a specific thermal structure and mineralogy for the slab, a subject of much debate. 

Nonuniform Surface Temperature. The previous section was devoted to uniform-temperature plates. In practice, however, this ideal condition seldom occurs, and it is necessary to account for the effects of surface temperature variations along the plate on the local and average convective heat transfer rates. TTiis is required especially in the regions directly downstream of surface temperature discontinuities, e.g., at seams between dissimilar structural elements in poor thermal contact. These effects cannot be accounted for by merely utilizing heat transfer coefficients corresponding to a uniform surface temperature coupled with the local enthalpy or temperature potentials. Such an approach not only leads to serious errors in magnitude of the local heat flux, but can yield the wrong direction, i.e., whether the heat flow is into or out of the surface. 

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