Distributions of charged defects in mixed ionic-electronic conductors .1. General equations for homogeneous mixed ionic-electronic conductors

General analytical solutions to transport equations are presented for a mixed ionic-electronic conductor (MIEC) subject to various electrical and chemical conditions at the surfaces. The derived general expressions can be used to predict the steady-state distributions of defects and electrical potential within the MIEC as a function of an external stimulus for transport: an electric field, a gradient in chemical potential, or a combination of the two. Also, variations in conductivities, transference numbers, current carried by each type of defect, and chemical potential of oxygen within the MIEC can be readily calculated under different conditions. Analyses indicate that the distribution of mobile defects is approximately linear when the amount of uniformly distributed immobile charges is sufficiently small while the electrical potential distributes nearly linearly when the amount of uniformly distributed Immobile charges is sufficiently large. In addition, the derived equations can be used to determine the transport properties of an MIEC from observed steady-state behavior of the MIEC under controlled conditions. The derived expressions are applicable to a variety of MIECs with vastly different transport properties, ranging from MIECs with predominant ionic disorders to the ones with overwhelming electronic disorders, from MIECs with Intrinsic disorders to the ones with significant extrinsic disorders, and from the MIECs with uniform properties to the ones switching transport character through their thicknesses. Further, and in particular, the derived equations can provide valuable guidance in optimizing performances of devices or systems based on MIECs and in improving or redesigning MIECs for various applications.