Simulating Charge Transport in Solid Oxide Mixed Ionic and Electronic Conductors: Nernst-Planck Theory vs Modified Fick's Law

Abstract

With the assumption that the Fermi level (electrochemical potential of electrons) is uniform across the thickness of a mixed ionic and electronic conducting (MIEC) electrode, the charge-transport model in the electrode domain can be reduced to the modified Fick's first law, which includes a thermodynamic factor A. A transient numerical solution of the Nernst-Planck theory was obtained for a symmetric cell with MIEC electrodes to illustrate the validity of the assumption of a uniform Fermi level. Subsequently, an impedance numerical solution based on the modified Fick's first law is compared with that from the Nernst-Planck theory. The results show that Nernst-Planck charge-transport model is essentially the same as the modified Fick's first law model as long as the MIEC electrodes have a predominant electronic conductivity. However, because of the invalidity of the uniform Fermi level assumption for a MIEC electrolyte with a predominant ionic conductivity, Nernst-Planck theory is needed to describe the charge transport behaviors. The methods of Electrochemical Impedance Spectroscopy (EIS) 1,2 and Electrical Conduction Relaxation (ECR) [3][4][5] have been utilized widely in characterizing the bulk transport properties and the electrochemical/chemical reactions at the surface of the mixed ionic and electronic conducting (MIEC) electrodes and electrolytes. Both of these methods rely upon transient responses of the system subject to external perturbations (electrical stimulus or partial pressure of oxygen change). To interpret the experimental data obtained from these two techniques, two theoretical approaches have been commonly applied to describe the flux of species in the MIEC: modified Fick's first law and Nernst-Planck theory.Fick's first law with inclusion of the thermodynamic factor A is referred to as the modified Fick's first law throughout this paper. It describes the transport of species j as a simple diffusion process driven by a concentration gradient of species j under the assumption that the electrochemical potential of the electron hole/electron is uniform in the MIEC electrodes. For this modified Fick's first law, the flux of species is written as,where D j is the diffusion coefficient of species j (component diffusion coefficient);D j = AD j is the chemical diffusion coefficient of species j. Eq. 1 was originally proposed by Carl Wagner, 6,7 and then adopted by Weppner and Huggins 8 in 1977 to describe the transport of neutral species in the ionization equilibrium for mixed-conducting Li 3 Sb system. In their original paper, the following expression was used for the flux of species j:where,D j is the chemical diffusion coefficient for species j; D j is the self-diffusion coefficient; W is an enhancement factor introduced by Weppner and Huggins and defined as follows for a system with only one ionic species and one electronic species with the activity coefficient as constant: [3] where, t e is the transference number of electronic species; z j is the charge number of species j; c j and a j are th...

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