Diffusion in Materials with Ionic and Electronic Disorder
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DIFFUSION IN MATERIALS WITH IONIC AND ELECTRONIC DISORDER JOACHIM MAIER Max-Planck-Institut fuir Metallforschung, Pulvermetallurgisches Laboratorium, Heisenbergstr. 5, 7000 Stuttgart 80, Federal Republic of Germany ABSTRACT Besides some necessary reviewing of conventional diffusion theory, this paper deals with the modification of the mass and charge transport equations originating from the occurence of internal defect-chemical reactions (especially valence changes of the defects) which can be considered to be in local equilibrium. It is shown how the phenomenological transport coefficients for chemical diffusion, tracer diffusion and ionic conduction depend on the individual defect diffusivities under such general conditions. Moreover, the evaluation formulae of well-known electrochemical techniques such as Wagner-Hebb polarization and concentration cell experiments have to be modified. Application is made to the influence of trapping effects in doped SrTi0 3 , to the valence changes in YBa 2 Cu306+x as well as to the mixed conduction in orthorhombic PbO. INTRODUCTION Most simple examples of internal defect-chemical reactions are Frenkel reactions or electron-hole formation. As will be shown these reactions do not lead to qualitatively interesting effects in regard to the transport behavior in local equilibrium. More important in this context are association reactions which occur in many materials especially at lower temperatures. Examples are valence changes of point defects in pure materials e.g. high temperature superconductors, donor-acceptor reactions involving dopant impurities in semiconductors or ion-ion interactions in heavily disordered or heavily doped ionic conductors. In all these cases source terms appear in the continuity equations for the ions or electrons. This is - in contrast to statements in many textbooks - also true for the important special case of local equilibrium, which will be exclusively adopted in the following. Let us consider e.g. an elementary chemical reaction
A#B
(1)
with the forward and backward reaction ratesF'= k[A] and'F= k[B]. The condition of local equilibrium means that the net reaction rate 7 - F'is small with respect tor, 7"but not small with respect to the divergence of the flux densities JA, jB, e.g. (e/8t)[B] = -VjB + k[A] -k[B] = -VjB + k(K[A] - [B]),
(2)
where K = [B]eq/[A]eq = k/k is the mass action constant. In local equilibrium the bracketed term is close to zero, but this does not hold for the complete source term because of the comparably large values of the rate constants. In other words any change of [B] due to a net in- or efflux of B into or out of a given volume represents a perturbation of the local equilibrium situation and is accompanied by a shift of the chemical reaction. Consider e.g. the association of 02 interstitials in a pure oxide with a comparably high O-excess according to (3a) 0' + he i- Of. Mat. Res. Soc. Symp. Proc. Vol. 210. @1991 Materials Research Society
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A net effiux of 0' and/or h" out of a given volume will immediately lead
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