Modeling of surface exchange reactions and diffusion in composites including transport processes at grain and interphase boundaries

Abstract

Surface exchange reactions and diffusion of oxygen in ceramic composites consisting of a dilute and random distribution of inclusions in a polycrystalline matrix (host phase) are modeled phenomenologically by employing the finite element method. The microstructure of the mixed conducting composite is described by means of a square grain model, including grain boundaries of the matrix and interphase boundaries between the inclusions and grains of the host phase. An instantaneous change of the oxygen partial pressure in the surrounding atmosphere may give rise to an oxygen exchange process, i.e., oxidation or reduction of the ceramic composite. Relaxation curves for the total amount of exchanged oxygen are calculated, emphasizing the role played by fast diffusion along the interfaces. The relaxation curves are interpreted in terms of effective medium diffusion, introducing appropriate equations for the effective diffusion coefficient and the effective surface exchange coefficient. When extremely fast diffusion along the grain and interphase boundaries is assumed, the re-equilibration process shows two different time constants. Analytical approximations for the relaxation process and relations for the separate relaxation times are provided for this limiting case as well as for blocking interphase boundaries. Furthermore, conductivity relaxation curves are calculated by coupling diffusion and dc conduction. In the case of effective medium diffusion, the conductivity relaxation curves do not deviate from those for the total amount of exchanged oxygen. On the contrary, the conductivity relaxation curves differ remarkably from the time dependence of the total amount of exchanged oxygen, when the different phases of the composite re-equilibrate with separate time constants.

Appendix

The effective diffusion coefficient of the polycrystalline matrix (grains of phase 1 and grain boundaries) can be written as

$$ {D{\prime}_1} = \frac{{s{\prime}D{\prime}}}{{1 - g{\prime} + s{\prime}g{\prime}}}\left[ {1 + \frac{{2({D_1} - s{\prime}D{\prime})(1 - g{\prime})}}{{{D_1} + s{\prime}D{\prime} + (s{\prime}D{\prime} - {D_1})(1 - g{\prime})}}} \right] $$

(A1)

by employing the two-dimensional Maxwell–Garnett relation, where g′ is the volume fraction of grain boundaries in the matrix and s′ is defined by Eq. 5b. If fast grain boundary diffusion is assumed and the thickness of the grain boundaries is much smaller than the grain size, i.e., $D{\prime} > > {D_1}$ and $g{\prime} < < 1$, Eq. A1 can be approximated by

$$ {D{\prime}_1} \approx \frac{{s{\prime}D{\prime}}}{{1 - g{\prime} + s{\prime}g{\prime}}} \times \frac{{(2 - g{\prime}){D_1} + s{\prime}D{\prime}g{\prime}}}{{2s{\prime}D{\prime}}} = \frac{{(1 - g{\prime}/2){D_1} + s{\prime}D{\prime}g{\prime}/2}}{{1 - g{\prime} + s{\prime}g{\prime}}}. $$

(A2)

Introducing the area fraction of grain boundaries, $\varepsilon = \delta /d = g{\prime}/2$, one arrives at Eq. 17, which has been derived for the effective diffusivity with respect to fast grain boundary diffusion previously [20, 24, 25, 31].

When the heterointerfaces are arbitrarily combined with the inclusions (phase 2), the effective diffusion coefficient, ${D{\prime}_2}$ is given by

$$ {D{\prime}_2} = \frac{{{{s{\prime}{\prime}}_2}D{\prime}{\prime}}}{{1 - g{\prime}{\prime} + {{s{\prime}{\prime}}_2}g{\prime}{\prime}}}\left[ {1 + \frac{{2({D_2} - {{s{\prime}{\prime}}_2}D{\prime}{\prime})(1 - g{\prime}{\prime})}}{{{D_2} + {{s{\prime}{\prime}}_2}D{\prime}{\prime} + ({{s{\prime}{\prime}}_2}D{\prime}{\prime} - {D_2})(1 - g{\prime}{\prime})}}} \right] $$

(A3)

in analogy to relation (A1). In this case g″ is given by $g{\prime}{\prime} = 4\delta /d = 2\varepsilon$, such that Eq. 18 is obtained assuming D″>> D₂ and g″ <<1. For the sake of simplicity grain, boundaries are neglected in the inclusions. However, the extension of the relations for effective medium diffusion to polycrystalline inclusions is straightforward.