Confidence Limits in the Oxygen Transport Parameters of the (La0.8Sr0.2)(Cr0.2Fe0.8)O3-δ Determined By the Isotopic Exchange, Depth Profiling Method
The isotopic exchange, depth profiling method (IEDP) method allows the bulk oxygen transport parameters, the diffusion coefficient, D(T) and surface reaction coefficient, k(T) both functions of temperature, T, to be determined once an appropriate solution of the diffusion equation  has been identified and assuming compliance with the boundary conditions of that solution. The IEDP results dataset, normalised isotopic fraction measured at N depth intervals, (xi, Cxi ) for i=1 to N is matched by the ‘least squares’ process that determines the most probable values of the two parameters, DFit and kFit, that minimises the ‘sum-of-squares’ of the residual differences, rxi between the measured profile data Cxi and the values calculated from the solution, CxiFit , where rxi = Cxi-CxiEst . Ideally the conditions, ∂/∂D(∑(rxi)2)=0 and ∂/∂k(∑(rxi)2)=0 are satisfied. This process assumes that the variance associated with the depth measurement set, Vx is small compared to the variance of the measured isotopic fraction set, VCxFit . VDFit , the variance in DFit is calculated as a product, ( ∂D/∂rx)2VCxFit. where the partial differential, (∂rx/∂D) , is the gradient of the sum of the residuals squared along the D axis for a fixed value of kFit in D and k space. The variance in kFitis calculated from a similar product, VkFit=( ∂k/∂rx)2VCxFit .
This statistical quantification of the variance of D and k for each IEDP measurement of the LSCrF oxide is reported in this contribution for three different exchange anneal ambients with an oxygen activity range between 1 and 1x10-14 . All the calculations are done by numerical evaluation so the effect of the finite element size, (ΔD,Δk) and termination conditions for the ‘best-fit’ is also reported. . A comparison is made with the ‘goodness-of-fit’ tests using the often quoted “R-squared” and “chi” values from their probability distributions.
The variances in DFit and kFit are known to be sensitive to the surface isotopic fraction, Cx=0 , which for the semi-infinite solution becomes, Cx=0=1-exp(h'2).erfc(h') where h'=k(tanneal/D)1/2 and tanneal is the time for the oxygen-18 exchange anneal. Values of h’ around 0.2 are preferred for similar variances in DFit and kFit whilst values of 0.02<h’<2 lead to higher errors with dissimilar upper and lower limits. This effect is shown from the results set for LSCrF with two cases where h’ is 7.3 in one dry oxygen exchange and h’is 0.073 in a steam and forming gas exchange reaction.
The variance of D and k for each IEDP measurement is used to weight the plotted points in the Arrhenius plot and in the ‘best straight-line fit’ for an estimate of the activation energy for both diffusion and the surface exchange reaction. In this situation with often just a few points it is essential to apply the F-test or t-test significance tests for the null-hypothesis. The results for LSCrF are plotted with the standard 2-theta confidence limits derived from the ‘best-fit’ so that a reasonable assessment can be made with data in the literature that has been measured under similar conditions .
1. Mizusaki, J., Solid State Ionics, 1992. 52(1-3): p. 79-91.
2. Crank, J., The mathematics of diffusion. 2nd ed. 1975, Oxford: Clarendon Press.
3. Chater, R.J. Ph.D. Thesis, Imperial College London, 2014
4. Ramos, T., Solid State Ionics, 2004 170(3-4): p. 275-286