Faster grain-boundary diffusion with a higher activation enthalpy than bulk diffusion in ionic space-charge layers

TL;DR

This work investigates whether cation diffusion along grain-boundary space-charge layers can be faster than bulk diffusion even when the grain-boundary activation enthalpy $\Delta H^\mathrm{gb}$ is higher, by modeling two diffusion mechanisms in acceptor-doped SrTiO$_3$ via continuum FEM in a 2D bicrystal. By solving Poisson's equation to obtain space-charge profiles and then diffusion with isolated Sr vacancies and defect associates $(v_{Sr}v_O)^x$, the authors extract the grain-boundary diffusion product $a^{gb}D^{gb}$ and the corresponding activation enthalpies, $\Delta H^\mathrm{gb}$ and $\Delta H^\mathrm{b}$. The results show that $r = \Delta H^\mathrm{gb}/\Delta H^\mathrm{b}$ can exceed unity (up to about $1.3$) at certain temperatures and $\Delta S_a$, especially when associates dominate diffusion in the bulk but isolated vacancies dominate in the space-charge layers; this yields a temperature-dependent, nontrivial diffusion enhancement along grain boundaries. The findings highlight a mechanism by which space-charge effects can invert conventional diffusion expectations and outline experimental conditions needed to observe $r>1$ in titanate perovskites, with potential applicability to other $ABO_3$ oxides.

Abstract

Faster diffusion of cations along grain boundaries is reported in the literature for a variety of acceptor-doped $AB\mathrm{O}_{3}$ perovskite-type oxides. The ratio $r$ of the activation enthalpy of grain-boundary diffusion ($ΔH^\mathrm{gb}$) to the activation enthalpy of bulk diffusion ($ΔH^\mathrm{b}$) is seen experimentally to lie in the range $0.7 < r = ΔH^\mathrm{gb} / ΔH^\mathrm{b} < 1.3$, albeit with substantial errors. In a previous publication [Parras and De Souza, Acta Mater. 195 (2020) 383] it was shown through a set of continuum simulations that cation-vacancy accumulation within negative space-charge layers at grain boundaries in acceptor-doped perovskites will give rise to faster grain-boundary diffusion of cations, with the associated values of $r$ approaching but not exceeding unity. In the present study, we demonstrate by means of continuum simulations that under certain conditions $r > 1$ is achievable for faster cation diffusion along grain boundaries in an acceptor-doped perovskite ceramic. Diffusion profiles for a two-dimensional bicrystal geometry are obtained by solving, first, Poisson's equation, and subsequently, the diffusion equation. The specific case we consider is cation migration occurring by two related mechanisms, by isolated cation vacancies and by defect associates of cation and anion vacancies; the electric potential within the space-charge layers shifts the association equilibrium so that associate diffusion dominates in the bulk whereas isolated vacancy diffusion dominates within the space-charge layers. The conditions under which $r > 1$ is observed are described, and issues with experimental confirmation are discussed.

Figures (8)

• Figure 1: Cation diffusion in selected $AB\mathrm{O}_3$ perovskite oxides: (a) Activation enthalpies for bulk diffusion $\Delta H^\mathrm{b}$ and for the grain-boundary diffusion product $\Delta H^\mathrm{gb}$. (b) The ratio of the activation enthalpies, $r=\Delta H^\mathrm{gb}/\Delta H^\mathrm{b}$. A: Ce in BaZrO3 Hasle.2021 B: ^52Cr in La_0.8Sr_0.2Ga_0.8Mg_0.2O_2.8 O.Schulz.2003, C: ^56Fe in LaGaO3 O.Schulz.2003, D: Fe-Co interdiffusion in La_0.6Sr_0.4CoO_3-$\delta$Kubicek.2014 E: ^86Sr in La_0.6Sr_0.4CoO_3-$\delta$Kubicek.2014 F: ^56Fe in La_0.8Sr_0.2Ga_0.8Mg_0.2O_2.8O.Schulz.2003
• Figure 2: Bulk concentrations of point defects in acceptor-doped SrTiO3 as a function of inverse temperature. Gray dashed line indicates [Acc']. Entropy of associate formation is varied: (a) $\Delta S_{\text{a}} = \qty{0}{\kb}$, (b) $\Delta S_{\text{a}} = \qty{-2}{\kb}$, (c) $\Delta S_{\text{a}} = \qty{-4}{\kb}$. Parameters are $[\ch{Acc'}] = \qty{8.43e24}{\per\cubic\meter}$, $\Delta H_\mathrm{a} = \qty{-1}{\electronvolt}$, and $a_{\ch{O2}} = 0.2$.
• Figure 3: Bulk diffusion coefficients of strontium in SrTiO3 calculated from defect concentrations and defect diffusivities for diffusion by means of isolated vacancies and defect associates. (a) Bulk diffusion coefficients as a function of inverse temperature. (b) Activation enthalpy of bulk diffusion over a rolling interval of 40, plotted as a function of temperature. Different datasets refer to differing values of the entropy of defect association $\Delta S_\mathrm{a}$.
• Figure 4: Calculated defect concentrations normal to a grain boundary. Entropy of associate formation is varied: (a) $\Delta S_{\text{a}} = \qty{0}{\kb}$, (b) $\Delta S_{\text{a}} = \qty{-2}{\kb}$, (c) $\Delta S_{\text{a}} = \qty{-4}{\kb}$. Simulation parameters: $T = \qty{1100}{\kelvin}$, $\Delta S_\mathrm{a} = \qty{-2}{\kb}$$[\ch{Acc'}] = \qty{8.43e24}{\per\cubic\meter}$. GB-core ($\qty{0}{\nano\meter} < y < \qty{1}{\nano\meter}$) calculated with $\Delta \mu_\mathrm{v}^\standardstate = \qty{-2}{\electronvolt}$.
• Figure 5: (a) Two-dimensional simulation cell of two grains separated by a grain boundary. The grain boundary consist of an interface core and adjacent space charge layer. Ions come from an undepletable source on the left side and diffuse via the red arrows, into the bulk, along the SCL and from the SCL into the bulk. (b) Concentration profile of strontium ions after finished simulation, with enhanced diffusion along the grain boundary.