Prediction of high temperature demixing in oxides in an electric field or an oxygen potential gradient
Introduction
When engineering ceramics are in-service, they frequently operate in driving forces such as temperature gradients, electric fields, stress gradients and oxygen potential gradients. For ceramic oxides at high temperatures, these conditions can lead to a gradual separation or kinetic demixing of the cation components of the ceramic. Since the properties of the ceramic are usually highly dependent on the composition, demixing results in a loss of performance and a reduced longevity [1], [2]. This paper is concerned with the theory of demixing in ceramic oxides placed in an oxygen potential gradient or an electric field.
Kinetic demixing in an oxygen potential gradient of the cations A and B in ternary ceramic oxides of the (A,B)O type was first studied experimentally by Schmalzried et al. [3] for the case of (Co,Mg)O. Since that time, the phenomenon of kinetic demixing has been analysed in various other mixed ceramic oxides including (Fe,Cr)O [4]and (Co,Ga)O [5]. Kinetic demixing in electric fields (high current conditions) has also been studied in various ceramic oxides including (Co,Mg)O and yttria-stabilized zirconia [6], (Co,Ni)O [7]and (Co,Ga)O [8].
Kinetic demixing of the cations in these rock-salt structure oxides is a result of the cations competing for the vacancies that continuously flow through the material from the higher oxygen partial pressure end (higher concentration of cation vacancies) to the lower oxygen partial pressure end (lower concentration of cation vacancies). The more mobile cation species tends to move ahead, thereby leading to a spatial separation or demixing of the cation species. In these oxides, the system is ‘open’ so that new oxide can form (at the higher oxygen partial pressure end) whilst oxide is lost at the other end. The entire ceramic oxide sample then moves, eventually with a steady state velocity, towards the higher oxygen partial pressure end. In the electric field case of demixing, the flow of vacancies is now driven by the electric field rather than the vacancy gradient, but the overall demixing process is similar to that for the oxygen potential gradient.
Schmalzried et al. [3] provided a sound formalism for the analysis of the demixing process and this has provided the foundation for all subsequent theoretical work, both phenomenological and atomistic, in the area [2], [3], [4], [5], [6], [7], [8]. The common feature of these treatments is the neglect of the off-diagonal phenomenological transport coefficients. These coefficients arise from the interaction of the fluxes of the different cation species, in effect, coming from the fact that the different species ‘compete’ for the same vacancies in order to diffuse. The neglect of the off-diagonal coefficients is, of course, an assumption not just confined to treatments of kinetic demixing: it is commonly made in numerous theoretical treatments of diffusion kinetics in solids [9]. The difficulty with this assumption is predicting its impact on the result. In other areas of diffusion, for example chemical diffusion in ternary alloys, it turns out that neglect of the off-diagonal coefficients can result in the incorrect prediction of the direction of one of the atomic fluxes [9]. But in other cases the result of neglecting the off-diagonal phenomenological transport coefficients is quite small, for example in disordered concentrated binary alloys neglect of the vacancy-wind factor changes the final result for the interdiffusivity by less than 30% [9].
Previous theoretical kinetic demixing analyses have been confined to ternary mixed oxides. Kinetic demixing in quaternary mixed oxides (A,B,C)O has not been previously investigated, either experimentally or theoretically. Although it is qualitatively clear, by analogy with (A,B)O oxides, that the most mobile cation species will be enriched at the high vacancy concentration end of the sample (for oxygen potential gradients) and down-field (for electric fields) whilst the least mobile cation species is likely to be depleted at the same end, it is unknown how the third cation species with an intermediate mobility would behave.
In the present paper, the theoretical problem of demixing in ternary and quaternary mixed oxides in an oxygen potential gradient and an electric field is addressed by assuming a random distribution of cations and making use of the exact sum-rule expression that relates the off-diagonal and diagonal phenomenological transport coefficients [10]. In this way, the off-diagonal transport coefficients can be retained in full in the analysis. The results are verified with Monte Carlo simulation of the demixing process. Contact is made with experimental results in several mixed oxide systems, in particular, (Co,Mg)O (for oxygen potential gradient demixing) and (Co,Ni)O (for electric field demixing).
Section snippets
Theory
For convenience, in the following the quaternary mixed oxide (A,B,C)O taking the rock-salt structure is analysed; results for the corresponding ternary mixed oxide (A,B)O can then be obtained simply by ignoring the component C. Furthermore, the oxygen potential gradient and the electric field are considered together in the analysis. In these oxides, the oxygen ions are essentially immobile at the temperatures of interest and so largely act as ‘spectators’ for the cation diffusion processes. One
Computational analysis
Eq. (10) is readily solved using standard numerical methods to provide the steady state atom (and vacancy) composition profiles across the sample. In order to verify the analysis of Section 2 Monte Carlo computer simulations of demixing were performed along the lines of those performed some time ago by Zhang and Murch [12]. Their simulation program was designed originally to describe the demixing of A and B in (A,B)O in an oxygen potential gradient. It was extended here to the (A,B,C)O case in
Results and discussion
In Fig. 1, typical results are presented for the steady state composition profiles for the cations A and B after demixing of the ternary mixed oxide (A,B)O in an oxygen potential gradient (cation vacancy gradient). The data points are gathered from direct Monte Carlo simulations and the solid lines are obtained by numerical integration by way of Eq. (10). The agreement is seen to be excellent, thereby verifying the analytical result. As expected, the species with the higher mobility (A) is
Conclusions
Steady state demixing in semi-conducting and initially homogeneous mixed oxides of the rock-salt type in an oxygen potential gradient or an electric field at high temperature was analysed. The usual assumption of neglecting the off-diagonal phenomenological coefficients was avoided here by assuming a random distribution of cations and making use of the exact sum-rule expression that relates the phenomenological coefficients. The demixing analysis was then greatly simplified and made exact. The
Acknowledgements
The authors wish to thank the Australian Research Council (Large Grants and Discovery Project Grants Scheme) for its support of this research. One of the authors wishes to thank the Australian research Council for the award of a Queen Elizabeth II Fellowship and a Professorial Fellowship (I.V.B).
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