Solids in Thermodynamic Potential Gradients

The content of this chapter is closely related to permeation, which is the transport of a solute across a layer of solvent (or membrane) under the action of a difference in activity. For example, the permeation of hydrogen through a metal foil has been studied, particularly for palladium [F.A. Lewis (1967)] and iron [J. P. Hirth (1980) H. H. Johnson (1988)]. One reason for studying the permeation of hydrogen through iron is to understand the hydrogen embrittlement of steel. 

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Processes in which solids play a rate-determining role have as their principal kinetic factors the existence of chemical potential gradients, and diffusive mass and heat transfer in materials with rigid structures. The atomic structures of the phases involved in any process and their thermodynamic stabilities have important effects on drese properties, since they result from tire distribution of electrons and ions during tire process. In metallic phases it is the diffusive and thermal capacities of the ion cores which are prevalent, the electrons determining the thermal conduction, whereas it is the ionic charge and the valencies of tire species involved in iron-metallic systems which are important in the diffusive and the electronic behaviour of these solids, especially in the case of variable valency ions, while the ions determine the rate of heat conduction. 

After the formulation of defect thermodynamics, it is necessary to understand the nature of rate constants and transport coefficients in order to make practical use of irreversible thermodynamics in solid state kinetics. Even the individual jump of a vacancy is a complicated many-body problem involving, in principle, the lattice dynamics of the whole crystal and the coupling with the motion of all other atomic structure elements. Predictions can be made by simulations, but the relevant methods (e.g., molecular dynamics, MD, calculations) can still be applied only in very simple situations. What are the limits of linear transport theory and under what conditions do the (local) rate constants and transport coefficients cease to be functions of state When do they begin to depend not only on local thermodynamic parameters, but on driving forces (potential gradients) as well Various relaxation processes give the answer to these questions and are treated in depth later. 

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 heterogeneous solid state reactions, the phase boundaries move under the action of chemical (electrochemical) potential gradients. If the Gibbs energy of reaction is dissipated mainly at the interface, the reaction is named an interface controlled chemical reaction. Sometimes a thermodynamic pressure (AG/AK) is invoked to formalize the movement of the phase boundaries during heterogeneous reactions. This force, however, is a virtual thermodynamic force and must not be confused with mechanical (electrical) forces. 

These assumptions, however, oversimplify the problem. The parent (A,B)0 phase between the surface and the reaction front coexists with the precipitated (A, B)304 particles. These particles are thus located within the oxygen potential gradient. They vary in composition as a function of ( ) since they coexist with (A,B)0 (AT0<1 see Fig. 9-3). In the Af region, the point defect thermodynamics therefore become very complex [F. Schneider, H. Schmalzried (1990)]. Furthermore, Dv is not constant since it is the chemical diffusion coefficient and as such it contains the thermodynamic factor /v = (0/iV/01ncv). In most cases, one cannot quantify these considerations because the point defect thermodynamics are not available. A parabolic rate law for the internal oxidation processes of oxide solid solutions is expected, however, if the boundary conditions at the surface (reaction front ( F) become time-independent. This expectation is often verified by experimental observations [K. Ostyn, et al. (1984) H. Schmalzried, M. Backhaus-Ricoult (1993)]. 

The starting point for the mathematical description of diffusion in membranes is the proposition, solidly based in thermodynamics, that the driving forces of pressure, temperature, concentration, and electrical potential are interrelated and that the overall driving force producing movement of a permeant is the gradient in its chemical potential. Thus, the flux,. /,(g/cm2 s), of a component, i, is... 

We will see that in the steady state of the blocking cells, we can extract partial conductivities, and from the transients chemical diffusion coefficients (and/or interfacial rate constants). Cell 7 combines electronic with ionic electrodes here a steady state does not occur but the cell can be used to titrate the sample, i.e., to precisely tune stoichiometry. Cell 1 is an equilibrium cell which allows the determination of total conductivity, dielectric constant or boundary parameters as a function of state parameters. In contrast to cell 1, cell 2 exhibits a chemical gradient, and can be used to e.g., derive partial conductivities. If these oxygen potentials are made of phase mixtures212 (e.g., AO, A or AB03, B203, A) and if MO is a solid electrolyte, thermodynamic formation data can be extracted for the electrode phases. 

Solid state electrochemistry is concerned with all kinds of electrical phenomena associated with chemical changes, and vice versa, in solid state. These are induced by migration of charged mass particles or ions under the action of a variety of thermodynamic forces, gradients of component chemical potentials, electrical potential, temperature, stress, and the like. Naturally the solids composed of ions, viz., ionic solid compounds serve the main stages for solid state electrochemistry, hi solid state, electrons may also be mobile, which makes solid state electrochemistry even more versatile and interesting. 

It is commonly assumed in physical kinetics that flux is proportional to the gradient of the driving thermodynamic potential. Such a relation has the same form as Eq. (46), but the mobility coefficient would be usually independent of the transported variable (or order parameter ). Thus, the total flux through a thin layer of thickness d adjacent to the solid support can be expressed simply as... 

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