Wagner transport theory

Wagner Transport Theory for Dense Ceramic Hydrogen-Separation Membranes 1.4.1... 

In 1937, dost presented in his book on diffusion and chemical reactions in solids [W. lost (1937)] the first overview and quantitative discussion of solid state reaction kinetics based on the Frenkel-Wagner-Sehottky point defect thermodynamics and linear transport theory. Although metallic systems were included in the discussion, the main body of this monograph was concerned with ionic crystals. There was good reason for this preferential elaboration on kinetic concepts with ionic crystals. Firstly, one can exert, forces on the structure elements of ionic crystals by the application of an electrical field. Secondly, a current of 1 mA over a duration of 1 s (= 1 mC, easy to measure, at that time) corresponds to only 1(K8 moles of transported matter in the form of ions. Seen in retrospect, it is amazing how fast the understanding of diffusion and of chemical reactions in the solid state took place after the fundamental and appropriate concepts were established at about 1930, especially in metallurgy, ceramics, and related areas. 

Wagner s theory is valid only for the growth of relatively thick films. The Nemst-Einstein relationship and the equation for transport by gradients of chemical and electrical potential (66) are valid only for small electric fields, whereas according to Eqs. (77 and 78) the voltage between the two interfaces of a film... 

Wagner s oxidation theory assumes that volume diffusion of point defects limits the growth of oxide layers. However, other transport mechanisms are possible, notably grain boundary diffusion. At relatively low temperatures, Tmelting temperature of the oxide, this mechanism contributes considerably to the transport, and the rate of oxidation exceeds that calculated using Wagner s theory. The rate of grain boundary diffusion depends on the microstructure of the oxide films formed, which is difficult to control [8]. For this reason, measured oxidation rates are often not well reproducible. 

Electronic Transport Theory of Liquid Non-Simple and Amorphous Metals, in Proc. 5th Int. Conf. on Liquid and Amorphous Metals, eds C.N.J. Wagner and W.L. Johnson (North-Holland, Amsterdam) pp. 1267-1272. Johansson. B., and N. M4rtensson, 1987, Thermodynamic Aspects of 4f Levels in Metals and Compounds, in Handbook on the Physics and Chemistry of Rare Earths, eds K.A. Gschneidner Jr and L. Eyring (North-Holland, Amsterdam) ch. 69. 

Wagner-type transport theory case studies... 

Wagner, M.R., Three-dimensional Nodal Difiusion and Transport Theory Methods for Hexagonal - Z Geometry, Proc. Conf. Jackson Hole, Wyoming, USA 28-30 Sept., 1988. 

In many cases, the layer growth can be described by a parabolic rate law x = kpt, where x is the scale thickness at time t and kp is the parabolic rate constant. This law may be derived from Wagner s theory of metal oxidation. The parabolic rate corrstants contain diffusion coefficients which are related to the concentration of the defects responsible for material transport through the layer. In fact, the higher the deviation from stoichiometry, the larger the diffusion coefficient and, consequently, the faster the oxidation rate of a metal at a given temperature. 

Rapid evaporation introduces complications, for the heat and mass transfer processes are then coupled. The heat of vaporization must be supplied by conduction heat transfer from the gas and liquid phases, chiefly from the gas phase. Furthermore, convective flow associated with vapor transport from the surface, Stefan flow, occurs, and thermal diffusion and the thermal energy of the diffusing species must be taken into account. Wagner 1982) reviewed the theory and principles involved, and a higher-order quasisteady-state analysis leads to the following energy balance between the net heat transferred from the gas phase and the latent heat transferred by the diffusing species ... 

The main difficulty with the first mode of oxidation mentioned above is explaining how the cation vacancies that arrive at the metal/oxide interface are accommodated. This problem has already been addressed in Section 7.2. Distinct patterns of dislocations in the metal near the metal/oxide interface and dislocation climb have been invoked to support the continuous motion of the adherent metal/oxide interface in this case [B. Pieraggi, R. A. Rapp (1988)]. If experimental rate constants are moderately larger than those predicted by the Wagner theory, one may assume that internal surfaces such as dislocations (and possibly grain boundaries) in the oxide layer contribute to the cation transport. This can formally be taken into account by defining an effective diffusion coefficient Del( = (1 -/)-DL+/-DNL, where DL is the lattice diffusion coefficient, DNL is the diffusion coefficient of the internal surfaces, and / is the site fraction of cations located on these internal surfaces. 

M. Wagner, Nonadiabatic transport in finite systems. I. Formal theory, Phys. Rev. B 45 (1992) 11595. 

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