Modelling of surface exchange reactions and diffusion in composites and polycrystalline materials

Abstract

Abstract

Surface exchange reactions and chemical diffusion in composites, consisting of a dilute distribution of inclusions in a matrix, and polycrystalline materials have been modelled by application of both a square grain and a spherical grain model. The diffusion equations have been solved numerically by employing a finite element approach in the case of the square grain model and the Laplace transform method involving numerical Laplace inversion with respect to the spherical grain model. The boundary conditions refer to oxygen exchange reactions between a gas phase and a mixed ionically–electronically conducting ceramic sample within the linear response regime, i.e. small variations of the oxygen partial pressure. Diffusion profiles as well as the time dependence of the total amount of exchanged oxygen (relaxation curves) have been calculated. A necessary requirement for effective medium diffusion is proposed, and appropriate relations for the effective chemical surface exchange coefficient and the effective chemical diffusion coefficient are derived. On the contrary, when the time constant for diffusion from the matrix into the inclusions of a composite exceeds considerably the relaxation time for effective medium diffusion, relaxation curves with two separate time constants are observed. Analogously, in the case of polycrystalline materials the overall transport process is determined by slow (rate-limiting) bulk diffusion from the grain boundaries into the grains. Adequate formulae for the relaxation times are given based on analytical approximations of the solution functions to the diffusion equations. In addition, the spherical grain model is applied to interpret the re-oxidation kinetics of the positive temperature coefficient of resistivity (PTC) ceramics based on conductivity relaxation experiments.

Graphical abstract