Slow, Steady-State Transport with "Loading" and Bulk Reactions: the Mixed Ionic Conductor La${}_2$CuO${}_{4+\delta }$

We consider slow, steady transport for the normal state of the superconductor La${}_2$CuO${}_{4+\delta }$ in a one-dimensional geometry, with surface fluxes sufficiently general to permit oxygen to be driven into the sample (loaded'') either by electrochemical means or by high oxygen partial pressure. We include the bulk reaction O$\to$O$^{2-}+2h$, where neutral atoms ($a$) go into ions ($i$) and holes ($h$). For slow, steady transport, the transport equations simplify because the bulk reaction rate density $r$ and the bulk loading rates $\partial_t n$ then are uniform in space and time. All three fluxes $j$ must be specified at each surface, which for a uniform current density $J$ corresponds to five independent fluxes. These fluxes generate two types of static modes at each surface and a bulk response with a voltage profile that varies quadratically in space, characterized by $J$ and the total oxygen flux $j_O$ (neutral plus ion) at each surface. One type of surface mode is associated with electrical screening; the other type is associated both with diffusion and drift, and with chemical reaction (the {\it diffusion-reaction mode}). The diffusion-reaction mode is accompanied by changes in the chemical potentials $\mu$, and by reactions and fluxes, but it neither carries current (J=0) nor loads the system chemically ($j_O=0$). Generation of the diffusion-reaction mode may explain the phenomenon of turbulence in the voltage’’ often observed near the electrodes of other mixed ionic electronic conductors (MIECs).