The Oxygen Tracer Diffusion Coefficient

The self-diffusion coefficient is usually obtained from measurement of the tracer diffusion coefficient, in this case the oxygen tracer diffusion coefficient D = f D here, / is called the correlation factor and represents the deviation from randomness of the jumps ( 1). We will follow an analysis given in [2, 3] for a material that has an oxygen deficiency accommodated by oxygen vacancies which are mobile, D can be derived in terms of atomistic parameters from random walk theory. 

Kroger-Vink notation [4]), the mobile oxygen vacancy concentration, expressed as a site fraction, ao, is the distance between equivalent sites, Vq is a characteristic lattice frequency, and vq exp y is the jump frequency for the migrating ion. Hy and are the enthalpy and entropy of the migrating ion. 

the oxygen self-diffusion coefficient becomes 

At this point, it is instructive to examine both components of this equation and the likely contribution to the tracer diffusion coefficient for these perovskite materials. Here the term [F ] refers to the mobile vacancy concentration, which may be different from the stoichiometric vacancy concentration (i.e., that determined by the oxidation states of the constituent cations) because of vacancy trapping, as observed in the fluorite oxides [2], or due to vaeaney ordering [5]. contains the terms relevant to mobility of the vacancies, i.e., the ease with which the oxygen atoms can jump from an adjacent lattice site into a vacancy. Mizusaki et al. [6] have previously shown that data for D, the vacancy diffusivity, show remarkably little variation for a number of perovskite materials. This is a very interesting observation and one to which we return later. It is thus important to understand the changes that occur in the vacancy concentration in these materials and how this affects the oxygen self-diffusion coefficient. 

One point to emphasize at this stage concerns the measurement of the self-diffusion coefficients, particularly in the mixed conducting perovskite materials. An extended review and description of diffusion measurement and techniques have been published by Philibert [1]. There are many electrochemical methods whereby various diffusion coefficients can be extracted from ceramic samples. In the main, data obtained using these methods must be treated with some circumspection because there are many sources of possible error, particularly with porous samples when gas diffusion down pores can 

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Fig. 3. Temperature dependence of the oxygen tracer diffusion coefficient in the crystals Lai- SryMnOs-s (y = 0.025-circle). Also shown are this work data for Lax- Sr MnO s (y = 0.075-square) and SIMS data for La0.65Sro.35Mn03- (diamond) [21] and extrapolated Arrhenius diffusivity plot for Lao.9Sr0.xMn03 (lower line) [22].

The electrical conductivity of sapphire in a particular crystallographic direction was found to be 1.25mSm 1 at 1773 K. An independent experiment on the same material at the same temperature determined the oxygen tracer diffusion coefficient to be 0.4nm2s 1, the diffusion occurring by a vacancy mechanism. Do these data favour oxygen ion movement as the dominant charge transport mechanism (Relative atomic masses, A1 = 27 and 0=16 density of sapphire, 3980 kgm-3.)... 

The thermodynamic factor d ioJdco in Eq. (10.12) can be determined directly from experiment by measuring the oxygen stoichiometry as a function of oxygen partial pressure, either by gravimetric or coulometric measurements. In view of Eqs. (10.6) and (10.7), it comprises contributions from both ionic and electronic defects, which reflect their non-ideal behaviour. For materials with prevailing electronic conductivity Eq. (10.12) may be simplified to yield an exact relation between the chemical diffusion coefficient D and the oxygen tracer diffusion coefficient D ... 

The chemical diffusion coefficient Dchem and the oxygen tracer diffusion coefficient D are correlated by the thermodynamic factor y ... 

A more complex example is La2Cu04. The doping dependence of oxygen tracer diffusion coefficients in La2Cu04 and their anisotropy are illustrated in Fig. 6.19. Whilst the behaviour at low x-values is in accordance with simple defect chemistry, the interactions and structmral changes at high doping concentrations lead to deviations from the ideal mass action laws (see Fig. 5.55). 

We remember that minority point defect concentrations in compounds depend on the activity of their components. This may be illustrated by the solubility of hydrogen in olivine since it depends on the oxygen potential in a way explained by the association of the dissolved protons with O" and O- as minority defects [Q. Bai, D. L. Kohlstedt (1993)]. Similarly, tracer diffusion coefficients and mobilities of Si and O are expected to depend on the activity of Si02. The value (0 lnDf/0 In aSio2)> = Si and O, should give information on the disorder type as discussed in Section 2.3. 

The tracer diffusion coefficient for oxygen ions in a particular cubic stabilized zirconia is measured and found to fit the relationship... 

The obtained value is smaller by about four orders of magnitude from experimental values15,16 of the chemical diffusion coefficient for oxygen in the ab-plane of the 123 matrix (D0(i23)ab), and higher by five orders from the experimental value15 of the tracer diffusion coefficient for copper in the ab-plane of the 123 matrix (D Cu(i23)ab), Table 1 the latter can be of the same order or by one order lower than the chemical diffusion coefficient15,16. 

As already mentioned the tracer experiment (which can also be conceived as a counter motion of the isotopes) delivers the tracer diffusion coefficient. In the case of oxides ideally the natural oxygen gas phase is instantaneously replaced by a gas phase with the same oxygen partial pressure but exhibiting a different isotope ratio (or an oxide is contacted with the same oxide in which a cation isotope is... 

In the first place our understanding of factors that control and limit the interfacial kinetics is still rudimentary, and therefore should be a fruitful area for further investigation. The apparent correlation between the surface oxygen exchange coefficient and the tracer diffusion coefficient D for different classes of oxides, the fluorite-related and the perovskite-related oxides, as noted by Kilner et al. [73], clearly indicate the potential of isotopic exchange. 

Fig. 83. Arrhenius plot of the oxygen tracer self-diffusion coefficient Z>. Comparison of the data of Conder et al. (1994b) with those of Rothman et al. (1989, 1991). After Conder et al. (1994b).

The tracer diffusion coefficient of oxide ions in single crystals was measured by using the gas-solid isotopic exchange technique with 1 0 tracer. Least-squares analysis of tracer diffusion coefficients under an oxygen pressure of 4.5 x lO Pa gave ... 

A vacancy diffusion coefficient D is also commonly introduced in the literature as the transport of oxygen ions can also be viewed as the transport of oxygen vacancies in the opposite direction. The relation between this vacancy diffusion coefficient and the self-diffusion coefficient or the tracer diffusion coefficient of oxygen ions is... 

Until recently, wehave assumed that diffusion is a random process. However, in real experiments diffusion coefficients are often determined by the use of isotopic tracers. If the tracer (in this case an oxygen isotope) has jumped into a vacancy it is clearly in the right environment to jump immediately back to the unoccupied vacancy that it leaves behind. Such correlated jumps serve to reduce the diffusion coefficient to a value lower than that which would be expected for a random walk. The resulting diffusion coefficient in the presence of such correlated jumps is known as the tracer diffusion coefficient (as its value can be determined from tracer experiments) or the self-diffusion coefficient. 

Fig. 5.9 Relationships between the activation energy of oxygen tracer diffusion and the logarithm of pre-exponential coefficient in ferrites (a), chromates (a), cobaltites (a), manganites (b), titanates (b). The lines are a guide to the eye...

Here D is the tracer diffusion coefficient Dv is the diffusion coefficient of oxygen vacancy / is the correlation factor N is the total volimie concentration of oxygen ions in the substance. Eqn. (9) includes only free oxygen vacancies, i.e. not associated with the dopant cation. The diffusion coefficient is defined by ionic mobility mechanism and may be expressed in terms of ion hopping parameters ... 

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