Samuel J.
Cooper
*a,
Mathew
Niania
b,
Franca
Hoffmann
c and
John A.
Kilner
b
aDyson School of Design Engineering, Imperial College London, London, SW7 1NA, UK. E-mail: samuel.cooper@imperial.ac.uk
bDepartment of Materials, Imperial College London, London, SW7 2BP, UK
cDPMMS, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, UK
First published on 27th April 2017
A novel two-step Isotopic Exchange (IE) technique has been developed to investigate the influence of oxygen containing components of ambient air (such as H2O and CO2) on the effective surface exchange coefficient (k*) of a common mixed ionic electronic conductor material. The two step ‘back-exchange’ technique was used to introduce a tracer diffusion profile, which was subsequently measured using Time-of-Flight Secondary Ion Mass Spectrometry (ToF-SIMS). The isotopic fraction of oxygen in a dense sample as a function of distance from the surface, before and after the second exchange step, could then be used to determine the surface exchange coefficient in each atmosphere. A new analytical solution was found to the diffusion equation in a semi-infinite domain with a variable surface exchange boundary, for the special case where D* and k* are constant for all exchange steps. This solution validated the results of a numerical, Crank-Nicolson type finite-difference simulation, which was used to extract the parameters from the experimental data. When modelling electrodes, D* and k* are important input parameters, which significantly impact performance. In this study La0.6Sr0.4Co0.2Fe0.8O3−δ (LSCF6428) was investigated and it was found that the rate of exchange was increased by around 250% in ambient air compared to high purity oxygen at the same pO2. The three experiments performed in this study were used to validate the back-exchange approach and show its utility.
To extract D* and k* from experimental isotopic fraction data, the appropriate solution to the diffusion equation, eqn (1), is fitted by a non-linear least squares method. This expression was originally derived by Carslaw and Jaeger4 as the solution to sys. 3, but was more famously communicated by Crank in his book “The Mathematics of Diffusion” (hence “Crank's solution”).5
(1) |
This function is the solution to the 1-dimensional (1D) diffusion equation in a semi-infinite domain, with a surface exchange boundary and uniform initial conditions, where C′ is the normalised isotopic fraction found using eqn (2) (N.B. this normalisation notation is consistent throughout this paper and the dash should not be confused with a derivative); t > 0 is the duration of the exchange; and x ≥ 0 is the depth of the profile into the sample from the exchange surface.
(2) |
(3) |
The background isotopic fraction and that of the enriched exchange gas, Cbg and Cgas respectively, are required for the normalisation in eqn (2). System 3 uses the notation ∂t to represent a partial derivative with respect to t.
(4) |
System 4, has two key differences from the single step system. Firstly, by considering the system time to start at the beginning of the second exchange, the initial condition is the Crank solution resulting from the first exchange, rather than a uniform distribution at background level. Secondly, the boundary condition no longer has the Cgas stimulation and now simply sets the surface gradient proportional to the current surface concentration (Neumann boundary condition).
The existence and uniqueness of a weak solution, in the sense of distributions, to this system follows from the Lax–Milgram Theorem once it has been converted into Laplace space.7 The solution was found by considering the following two physical constraints. Firstly, that the diffusion length, which is the convenient measure of the degree of penetration of a profile, must be approximately ; secondly, that C′ must be positive for all values of t and ϕ1. The solution found is given by,
(5) |
C′(t,x) = CCrank′((ϕ1 + t),x) − CCrank(t,x) | (6) |
By simple differentiation and substitution, it can be shown that eqn (5) is the unique solution to sys. 4. Furthermore, this approach can be extended to n exchanges, eqn (7), where D* and k* are constant; ϕj is the duration of exchange j; and the normalised gas concentration at the boundary alternates between 1 and 0 at each step.
(7) |
Although useful for validation purposes, these analytical solutions are only suitable for fitting profiles where D* and k* are the same in each exchange step. The following section describes a numerical scheme constructed for fitting all other cases.
To ensure that the semi-infinite boundary was modelled accurately, a hybridised boundary condition was utilised, which involved running the simulation twice (once with a mirror boundary (Neumann, ∂xC′ = 0) and once with a fixed boundary (Dirichlet, C′ = 0)) and taking the mean (validated using the analytical form described above). However, as all the profiles measured in this study are very close to C′ = 0 at the edge of the analysis area, the choice of this boundary condition was of minimal significance.
The simulation was used to explore the space of possible profiles, Fig. 1, under the constraint that D* was constant between the two exchanges, which is a property we expect from the systems investigated here, i.e. constant temperature and pO2. Seven families of curves were generated at equally spaced values of the exchange time ratio, θ, defined as,
(8) |
(9) |
In order to allow Fig. 1 to generalise to all possible back-exchange scenarios (with D2* = D1*), the isotopic fraction required an additional normalisation step defined by,
(10) |
(11) |
All the required normalisation equations, described above, have been inset into Fig. 1 for convenience.
In order to efficiently fit the back-exchange data with the CN derived profiles, it is beneficial to select initial values of D* and k* with the correct order of magnitude. This is achieved by first fitting the data with the relevant analytical solution described earlier in this section and then using the results of this much faster method to initialise a second fitting using the CN profiles.
Before the first exchange, each pellet was annealed in research grade 99.999% pure dry oxygen (16O) at the desired experimental temperature and oxygen partial pressure. By doing so, the oxygen vacancy concentration of the sample is fixed and ensures chemical equilibrium with the following exchanges. The pre-anneal time is calculated to be ten times that of the first exchange to provide a sufficiently large region with a constant oxygen non-stoichiometry. In order to determine accurately the effect of the atmosphere in the second exchange step, strict precautions were taken to ensure that the pre-anneal step and first exchange were undertaken in a dry, very high purity oxygen atmospheres. All experiments were performed in a bakeable ultra high vacuum exchange apparatus fabricated from stainless components and sealed with “conflat” flanges. Before each experimental step, the exchange volume was evacuated to a pressure of <4 × 10−7 mbar using a vacuum system employing a turbomolecular pump backed by a dry membrane pump to reduce the chance of back-streamed oil vapour. The 18O enriched oxygen for the exchanges was derived from a 5 Å molecular sieve reservoir, which ensured that the gas was dry (c. 1 vppm water). The samples were not re-polished between exchanges.
Three experiments, illustrated in Fig. 2, were performed using an initial static volume partial pressure of 200 mbar and a nominal annealing temperature of 785 °C. Experiments A and B were used to validate the back-exchange approach; experiment C was used to demonstrate its utility by investigating the effect of ambient air on the exchange process:
A. Dry 18O exchange ⇒ ToF-SIMS
B. Dry 18O exchange ⇒ Dry 16O exchange ⇒ ToF-SIMS
C. Dry 18O exchange ⇒ Ambient air exchange ⇒ ToF-SIMS
Due to the high oxygen diffusivity of LSCF6428 it was necessary to prepare the sample for line-scan SIMS profiling2 after the final exchange. To do so, the same grinding and polishing regime was applied to a face perpendicular to the exchange surface, prior to imaging. The sample was polished to a depth greater than a predicted value of x′ = 4 to ensure analysis occurs beyond the region where the diffusion front from this face will affect the isotopic ratio.
The exposed cross-section was then imaged using a ToF-SIMS IV machine (ION-TOF GmbH, Münster, Germany), as illustrated in Fig. 3. The technique collects a full mass spectrum at each pixel and rasters over the surface to generate a 2D image. A burst analysis mode was utilised to measure the 16O/18O ratio due to high oxygen sputter yields.3 This mode prevents detector saturation by using a series of short primary ion pulses rather than a larger, single ion pulse.
Fig. 3 Schematic representation of the progress grinding of the pellet surface between exchanges in order to capture ToF-SIMS images from the cross-section. |
The value of Cbg was measured from an unexchanged, but similarly prepared sample using the same imaging method and Cgas was measured during the exchange using the silicon wafer technique described in ref. 12.
Bilinear interpolation was used to produce an image rotation series at 0.1° increments. After generating a profile at each step, it was possible to align the exchange surface by choosing the angle with the maximum gradient of total counts (16O + 18O), this same process was also used to find the location of the “surface”. Typically, however, the degree of misalignment was small (<±2°), which has only a modest effect on the fitted values of D* and k* as long as the surface location is specified as the mean surface, rather than the leading point. Fig. 4 shows the mean oxygen counts (16O + 18O) against distance from the surface, superimposed onto the 2D ToF-SIM image for sample C, which required 1° of clockwise rotational alignment.
Following alignment, the isotopic fraction was calculated in each pixel and then the mean of each column was used to construct a 1D depth profile. Fig. 5 shows the isotopic fraction (18O/(16O + 18O)) against distance from the surface, superimposed onto the normalised 2D ToF-SIM data.
For each exchange, the time-temperature profile inside the furnace was recorded using a shielded thermocouple. The nominal duration of the first and second exchange steps were 1.5 h and 0.75 h respectively. By analysing the thermocouple data, the effective temperature was set to 785 °C and, using the method described by Killoran,6 the effective exchange times were calculated to be, ϕ1 = 1.4 h and ϕ2 = 0.66 h.
In order to make this correction, the Killoran method requires an activation energy of diffusion for the LSCF6428. The value used here was Ea = 1.917 eV and although this value was itself derived from a similar experiment,14 a sensitivity study suggests the resulting effective anneal times were highly insensitive to its value (<1 min eV−1). A separate activation energy is associated with the surface exchange coefficient, which is more difficult to account for as the surface exchange boundary condition is a function of both D* and k*. However, the rapid heating/cooling rates (c. 170 °C min−1) combined with suitable anneal times (>40 min in all cases) should minimise any resulting discrepancies.
Fig. 6 Graph showing the measured isotopic profiles for the three samples (A, B and C), as a function of depth from the sample surface. |
The values of k1* extracted from the fits was essentially identical to within reasonable experimental accuracy for all three experiments, as can be seen in Table 1. All three fits had values of R2 above 0.999 measured for the region between the surface and x′ = 2. The value of D* was assumed to be constant for the two exchange steps.
Sample | D*/10−8 cm2 s−1 | k 1*/10−6 cm s−1 | k 2*/10−6 cm s−1 |
---|---|---|---|
A | 1.6 | 1.1 | — |
B | 1.7 | 1.4 | 1.6 |
C | 1.7 | 1.3 | 3.3 |
Table 1 also shows that k2* of sample B, was identical to within experimental accuracy to k1* of sample B; whereas k2* of sample C was c. 250% larger than k1* of sample C. Although it is possible to approximate the uncertainty associated with the fitting process, it is expected that this will be small in comparison to other sources, such as sample to sample variation. As such, no uncertainty values are reported. The surface position and the size of the raster step are determined through calibration and contribute both a systematic and a statistical error.
There are several possible mechanism for the augmentation of k* in the presence of air compared to dry oxygen. Ambient air contains a wide range of oxygen containing components and it is known that H2O and CO216–18 can both significantly influence the overall oxygen exchange rate. Literature data suggests that both the temperature and overall composition of the anneal environment effects the measured oxygen exchange rate. It was beyond the scope of this paper to quantify the direct effect of each gaseous component; however, it should be clear that the back-exchange technique is capable of making these measurements in a systematic study.
For a number of common perovskite-based MIEC materials, exsolution of A-site cations to form secondary phase surface particles is a common occurrence.19–22 These particles are typically non-conductive and thus reduce the catalytically active area for the reduction and incorporation of oxygen. It has been shown that significant ‘strontium segregation’ can occur in LSCF6428 at relatively low temperatures and short anneal times.20,22 However, high temperature environmental scanning electron microscopy was used to inspect the surface of the LSCF6428 before and after annealing and no surface particles were observed.
The k1* value of sample B and C show excellent accordance, however, sample A appears to be lower. As mentioned previously, the statistical dead-time corrections are an additional source of error. Using the line-scan method and ToF-SIMS analysis, a constant total ion count is necessary to ensure accurate isotopic fraction measurement; however, the geometry and surface roughness of the sample can alter secondary ion extraction rates. Polishing the surface to a 0.25 μm grade is sufficient to negate these errors across the bulk surface, but at the edge of the sample it is very difficult to prevent curvature during to the polishing process. The radius of curvature (and thus number of affected pixels in the 2D ion image) determines surface uncertainty of the profile. In this work samples were adhered together prior to the second polishing step using a commercially available epoxy resin. This minimised the curvature to approximately 1 μm to 2 μm, greatly reducing the uncertainty of the line-scan profile near the surface.
The red curve highlights the case where λ = 1 (i.e. k2* = k1*), which was close to the measured result of experiment B. The green curve is where k2* = 2.5 × k1*, similar to experiment C. For very small or very large value of λ, the profiles approach an asymptote, which would undermine the techniques ability to accurately measure the corresponding values of k2*.
When D1* and D2* were allowed to float independently during the fitting, a family of similar curves could be generated, which all closely fit the profile. However, all samples tested were sections cut from the same pellet and also exchanged for identical times and temperatures, thus making it reasonable to assume that the profiles generated in the first exchange was the same for experiments A–C. This suggests that, for the back-exchange fitting, profiles where D1* and k1* are close to that resulting from experiment A should be preferred.
All of the data analysis, curve fitting and graphs were generated using a custom software package, TraceX,23 which is a GUI developed using the MatLab guide platform. The latest version of TraceX is freely available from the supplementary materials section of this publication and those interested are encouraged to contact the author.
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