Oxygen Surface Exchange and Chemical Diffusion: Measuring k and D in Functional Oxides

A practical guide to oxygen surface exchange and chemical diffusion in functional oxides, including isotope exchange depth profiling, electrical conductivity relaxation, boundary conditions, data fitting, and experimental limitations.

OXYGEN TRANSPORT · ADVANCED GUIDE 21

Oxygen transport in a functional oxide involves at least two distinct processes: oxygen exchange between the gas phase and the oxide surface, and oxygen diffusion through the solid. These processes are described by different kinetic parameters. The surface exchange coefficient, commonly written \(k^*\) in tracer experiments, quantifies the rate of isotope exchange at the surface. The oxygen tracer diffusion coefficient, \(D^*\), describes the self-diffusion of oxygen isotopes in the solid. Chemical diffusion measurements instead yield chemical transport parameters, often denoted \(D_{\mathrm{chem}}\) and \(k_{\mathrm{chem}}\).

These parameters are important for mixed ionic–electronic conducting electrodes, oxygen separation membranes, solid oxide cells and related devices. This article explains what each coefficient means, how isotope exchange depth profiling (IEDP) and electrical conductivity relaxation (ECR) are used to determine them, and how to avoid common errors in fitting and interpretation.

1. Two steps in oxygen transport

At a gas/oxide interface, oxygen-containing gas species must participate in surface reactions before oxygen can enter or leave the oxide lattice. A simplified sequence may include:

1. Transport of gas molecules to the surface.
2. Adsorption and dissociation of oxygen-containing molecules.
3. Charge transfer and formation of adsorbed oxygen species.
4. Incorporation of oxygen into lattice sites, often involving oxygen vacancies.
5. Diffusion of oxygen or oxygen-related defects through the solid.

The actual sequence depends on the material, atmosphere, surface termination and temperature. Some steps may be coupled or occur through parallel pathways.

A schematic surface incorporation reaction for a vacancy-containing oxide can be written as

\[ \mathrm{\tfrac{1}{2}O_2(g)+V_O^{\bullet\bullet}+2e' \rightleftharpoons O_O^x} \]

This is a useful defect-chemical representation, not a complete elementary mechanism. It does not specify adsorbed intermediates or the rate-determining step.

The two kinetic questions are different:

• Surface exchange: How rapidly does oxygen from the gas exchange with oxygen at the solid surface?
• Bulk diffusion: How rapidly does oxygen redistribute through the solid after entering it?

A material may have fast bulk diffusion but slow surface exchange, or fast surface exchange but slow bulk diffusion. Both can limit the response of an electrode or membrane.

2. Meaning and units of the transport coefficients

Oxygen tracer diffusion coefficient

The tracer diffusion coefficient \(D^*\) describes the diffusion of an isotopically labelled oxygen species, such as \(^{18}\mathrm{O}\), through the oxide. It is commonly reported in

\[ \mathrm{cm^2\,s^{-1}}\quad\text{or}\quad\mathrm{m^2\,s^{-1}} \]

The asterisk indicates tracer or isotope transport. It should not be confused automatically with the chemical diffusion coefficient.

Oxygen surface exchange coefficient

The surface exchange coefficient \(k^*\) describes the rate at which the isotope fraction at the solid surface approaches the isotope fraction imposed by the gas under the specified conditions. It is commonly reported in

\[ \mathrm{cm\,s^{-1}}\quad\text{or}\quad\mathrm{m\,s^{-1}} \]

The coefficient is an effective kinetic parameter. It can depend on temperature, oxygen partial pressure, surface composition, gas flow, humidity, surface preparation and the chosen kinetic model.

Chemical diffusion coefficient

The chemical diffusion coefficient \(D_{\mathrm{chem}}\) describes the relaxation of a chemical-composition or defect-concentration gradient. It is measured under a chemical-potential driving force, for example after a step change in oxygen partial pressure during ECR.

Chemical diffusion includes thermodynamic factors and the coupled response of charged defects and electronic carriers. It is therefore not generally equal to the tracer diffusion coefficient.

Chemical surface exchange coefficient

The coefficient \(k_{\mathrm{chem}}\) is obtained from a chemical relaxation experiment using a surface boundary condition that relates the flux to the difference between the surface composition and its equilibrium value under the gas atmosphere. Its numerical value is tied to the definition of the concentration or nonstoichiometry variable used in the model.

Do not compare \(k^*\) and \(k_{\mathrm{chem}}\) as if they were interchangeable material constants. Their definitions, driving forces and thermodynamic factors differ.

3. Tracer diffusion and chemical diffusion are not the same

Tracer diffusion follows the movement of labelled atoms without requiring a macroscopic chemical-composition gradient. Chemical diffusion describes the collective relaxation of a composition gradient under a chemical-potential difference.

For a simple single-component system, a chemical diffusion coefficient can be related to a tracer diffusion coefficient through a thermodynamic factor and, where relevant, a correlation factor. In oxide conductors, the relationship can be more involved because oxygen vacancies, electronic carriers and cation valence changes are coupled.

A useful conceptual expression is

\[ D_{\mathrm{chem}}=D_{\mathrm{tr}}\,\Theta \]

where \(D_{\mathrm{tr}}\) represents an appropriate tracer-based mobility term and \(\Theta\) represents the thermodynamic and correlation contributions under the chosen convention. This is schematic, not a universal formula for every mixed conductor. The precise expression depends on the defect model and the definition of the diffusion coefficients.

Consequently:

• IEDP is primarily a tracer method and yields \(D^*\) and \(k^*\).
• ECR is a chemical relaxation method and yields \(D_{\mathrm{chem}}\) and \(k_{\mathrm{chem}}\) under a specified model.
• A comparison between IEDP and ECR can provide mechanistic information, but only after accounting for the thermodynamic relationship between tracer and chemical transport.

4. Isotope exchange depth profiling (IEDP)

IEDP uses isotopically enriched oxygen, commonly \(^{18}\mathrm{O}_2\), to label oxygen in a solid oxide. A typical experiment consists of:

1. Prepare a dense, chemically stable specimen with a well-characterized surface.
2. Establish the desired temperature and oxygen partial pressure.
3. Expose the specimen to an isotope-enriched oxygen atmosphere for a controlled time.
4. Quench or cool the specimen in a way that minimizes further isotope redistribution.
5. Measure the isotope concentration as a function of depth, commonly using SIMS or ToF-SIMS.
6. Fit the measured depth profile using a diffusion model with an appropriate surface boundary condition.

The resulting isotope profile contains information about both bulk diffusion and surface exchange.

Why the profile shape matters

If surface exchange is very fast compared with diffusion, the surface isotope fraction approaches the gas-imposed value rapidly. The profile is then strongly influenced by bulk diffusion.

If surface exchange is slow, the surface isotope fraction remains below its equilibrium value during the exchange time. The profile becomes sensitive to both \(D^*\) and \(k^*\).

A reliable fit must account for both processes unless one can be independently shown to be negligible.

Diffusion equation

For one-dimensional diffusion into a semi-infinite solid with constant tracer diffusivity, the isotope fraction \(c^*(x,t)\) follows Fick’s second law:

\[ \frac{\partial c^*}{\partial t} = D^*\frac{\partial^2 c^*}{\partial x^2} \]

where \(x\) is depth measured from the surface.

A commonly used linear surface boundary condition is

\[ -D^*\left.\frac{\partial c^*}{\partial x}\right|_{x=0} = k^*\left(c^*_g-c^*_s\right) \]

where \(c^*_g\) is the isotope fraction imposed by the gas under the model convention and \(c^*_s\) is the surface isotope fraction in the solid. The sign depends on the chosen coordinate and flux convention; the physical condition is that the inward flux is proportional to the surface-to-gas isotope driving difference.

The boundary condition is essential. Fitting a diffusion-only solution with an assumed fixed surface concentration can bias the extracted \(D^*\) when surface exchange is not sufficiently fast.

Choosing the exchange time

The exchange time should be selected so that the isotope profile is measurable over a useful depth range and is sensitive to the parameters being fitted.

• If the profile is too shallow, depth resolution and surface artefacts may dominate.
• If the profile extends beyond the useful SIMS depth range, the diffusion tail may not be measured accurately.
• If exchange is too long, the profile may approach a regime in which the surface or bulk parameter is poorly constrained.
• If exchange is too short, the surface isotope fraction may be difficult to distinguish from instrumental background and surface contamination.

Using multiple exchange times can help test the model and reduce parameter correlation.

5. SIMS and ToF-SIMS depth-profile considerations

The quality of an IEDP result depends strongly on the depth-profile measurement.

Important considerations include:

• Surface preparation: roughness, polishing damage and contamination can alter exchange kinetics.
• Sputtering conditions: ion beam energy and incidence angle influence depth resolution and sputter rate.
• Matrix effects: secondary-ion yields can depend on local chemistry, so isotope ratios require suitable calibration or validated normalization.
• Depth calibration: crater depth and sputtering time should be measured rather than assumed.
• Near-surface resolution: the first part of the profile is often critical for determining \(k^*\).
• Background isotope fraction: natural abundance and instrumental background must be included.
• Grain boundaries: in polycrystalline samples, fast paths may produce non-Fickian tails or multiple transport populations.
• Surface layers: polishing residues, segregated phases or reaction layers can introduce an additional barrier.

A visually smooth profile does not guarantee a unique or physically meaningful fit. Residuals, parameter confidence intervals and sensitivity to the fitting range should be examined.

6. Electrical conductivity relaxation (ECR)

ECR determines chemical transport coefficients by measuring the electrical response of a sample after a controlled change in oxygen partial pressure.

A typical sequence is:

1. Equilibrate the specimen at a fixed temperature and initial oxygen partial pressure.
2. Apply a step change to the gas oxygen partial pressure.
3. Record conductivity as a function of time.
4. Convert the conductivity response to a composition or nonstoichiometry response using a calibrated relationship.
5. Fit the transient using a diffusion equation and a surface exchange boundary condition.

The method is attractive because it does not require isotope-enriched gases or destructive depth profiling. However, it requires the electrical conductivity to be a reliable and sufficiently sensitive proxy for oxygen nonstoichiometry.

Converting conductivity to oxygen nonstoichiometry

For a material written as \(\mathrm{ABO_{3-\delta}}\), the oxygen deficiency is represented by \(\delta\). A change in oxygen content can alter the oxidation state and concentration of electronic carriers, changing the electrical conductivity.

The relationship between conductivity and \(\delta\) must be established for the composition and temperature range being studied. It may be obtained using independent thermogravimetric measurements, coulometric titration, calibrated equilibrium data or another suitable method.

Assuming a linear relationship between conductivity and \(\delta\) without validation can produce incorrect relaxation curves and biased kinetic coefficients.

Normalized relaxation function

A normalized conductivity response can be defined as

\[ F(t)=\frac{\sigma(t)-\sigma_i}{\sigma_f-\sigma_i} \]

where \(\sigma_i\) and \(\sigma_f\) are the initial and final equilibrium conductivities. With this definition, \(F(0)=0\) and \(F(\infty)=1\) for an ideal step response.

The normalized signal is useful for comparing transients, but it is not itself a diffusion coefficient. The geometry, surface boundary condition and conversion between conductivity and composition are needed for parameter extraction.

7. ECR boundary condition and characteristic times

For a plane sheet with half-thickness \(L\), a common model uses one-dimensional chemical diffusion with a surface exchange boundary condition. The characteristic diffusion time is of order

\[ \tau_D\sim\frac{L^2}{D_{\mathrm{chem}}} \]

and the characteristic surface-exchange time is of order

\[ \tau_s\sim\frac{L}{k_{\mathrm{chem}}} \]

These are scaling estimates, not exact relaxation times. The relative importance of bulk diffusion and surface exchange can be expressed through a dimensionless Biot-type number:

\[ \mathrm{Bi}=\frac{k_{\mathrm{chem}}L}{D_{\mathrm{chem}}} \]

Interpretation:

• \(\mathrm{Bi}\gg1\): surface exchange is fast relative to bulk diffusion; the transient is more strongly diffusion-controlled.
• \(\mathrm{Bi}\ll1\): surface exchange is slow; the transient is more strongly surface-controlled.
• \(\mathrm{Bi}\sim1\): both processes can contribute significantly.

For the ideal plane-sheet model, the relaxation solution is represented by a series of exponential terms. The eigenvalues depend on the Biot number and the chosen geometry. A fit should use the correct specimen shape and exposed surfaces rather than applying a plane-sheet model to an arbitrary geometry.

8. Geometry and experimental design for ECR

Specimen dimensions influence the relative sensitivity to surface exchange and bulk diffusion.

For a plane sheet exposed on both large faces, the half-thickness is the characteristic diffusion length in the ideal one-dimensional model. If the edges are also exposed and their area is not negligible, a one-dimensional approximation may be insufficient.

A good experimental design considers:

• Dense, single-phase specimens with reproducible geometry.
• A thickness chosen to produce measurable transients.
• Uniform gas flow and rapid gas switching.
• Accurate measurement of sample temperature.
• Stable electrical contacts and a suitable four-probe configuration where possible.
• A sufficiently large conductivity change after the oxygen-pressure step.
• Repeated steps in both oxidation and reduction directions to test reversibility.

Gas-switching time can be comparable to the sample relaxation time. If the gas composition at the specimen does not change as an ideal step, the measured transient contains both reactor dynamics and sample kinetics. Measure or model the gas-switching response.

9. Separating surface-limited and diffusion-limited regimes

A single transient may not uniquely determine both \(D_{\mathrm{chem}}\) and \(k_{\mathrm{chem}}\). The parameters can be correlated, especially when the experimental time window does not capture the relevant parts of the relaxation.

Useful strategies include:

1. Measure specimens with different thicknesses.
2. Repeat measurements at multiple temperatures.
3. Use both oxidation and reduction steps.
4. Verify the gas-switching time independently.
5. Fit the complete transient rather than selecting only a visually linear region.
6. Examine confidence intervals and parameter covariance.
7. Compare fitted parameters with independent isotope-exchange or diffusion data where possible.

A thickness series is particularly useful. Diffusion-controlled relaxation scales approximately with the square of the characteristic length, whereas surface-controlled relaxation scales approximately linearly with length. This provides a diagnostic test, provided the surface chemistry and microstructure remain comparable between specimens.

10. Temperature dependence and activation energies

The temperature dependence of a transport coefficient is often analysed using an Arrhenius expression:

\[ D=D_0\exp\left(-\frac{E_D}{k_{\mathrm B}T}\right) \]

or

\[ k=k_0\exp\left(-\frac{E_k}{k_{\mathrm B}T}\right) \]

Taking natural logarithms:

\[ \ln D=\ln D_0-\frac{E_D}{k_{\mathrm B}}\frac{1}{T} \]

\[ \ln k=\ln k_0-\frac{E_k}{k_{\mathrm B}}\frac{1}{T} \]

The activation energies extracted from tracer and chemical measurements need not be identical. They can reflect different thermodynamic factors and rate-controlling processes.

For surface exchange, the apparent activation energy can depend on the kinetic regime. If bulk diffusion and surface exchange are coupled in the measured response, a fitted apparent value may include contributions from both. It should not automatically be interpreted as the barrier of one elementary surface reaction.

Report the fitting interval, the units of the coefficient, the logarithm base and the uncertainty in the fitted slope.

11. Atmosphere effects: oxygen pressure and water vapour

Surface exchange and bulk diffusion can depend on oxygen partial pressure because the concentrations of oxygen vacancies, electronic carriers and adsorbed oxygen species change with oxygen chemical potential.

Water vapour can also alter the surface reaction pathway. In some oxides, water competes with molecular oxygen for surface sites or participates in oxygen exchange through hydroxyl-related intermediates. Humidity can therefore change the apparent surface exchange coefficient even when the bulk composition and temperature are unchanged.

For meaningful comparison, report:

• Oxygen partial pressure;
• Water-vapour partial pressure or dew point;
• Total gas flow and gas composition;
• Temperature and equilibration time;
• Whether the sample was exposed to dry or humid gas before measurement.

Do not attribute a change in \(k^*\) solely to intrinsic catalytic activity if the gas composition, surface state or water content also changed.

12. Common sources of error

• Confusing tracer and chemical coefficients: \(D^*\) is not automatically \(D_{\mathrm{chem}}\).
• Assuming instantaneous surface equilibrium in IEDP: this can bias \(D^*\) when surface exchange is finite.
• Using the wrong geometry: slab, cylinder and finite-thickness models have different solutions.
• Ignoring gas-switching time in ECR: the gas transient can be mistaken for sample relaxation.
• Assuming conductivity is linearly proportional to \(\delta\): this must be calibrated.
• Fitting both coefficients from an insensitive transient: parameter covariance can make the result non-unique.
• Ignoring grain-boundary diffusion: polycrystalline profiles can contain multiple transport pathways.
• Ignoring surface contamination or segregation: the measured \(k^*\) may reflect a surface layer rather than the intrinsic oxide.
• Reporting only the best-fit curve: residuals, uncertainty and model assumptions are essential.
• Comparing humid and dry results without atmosphere details: water can change both surface chemistry and carrier populations.

13. Choosing between IEDP and ECR

Criterion	IEDP	ECR
Main driving force	Isotope exchange	Oxygen chemical-potential step
Typical outputs	\(D^*\) and \(k^*\)	\(D_{\mathrm{chem}}\) and \(k_{\mathrm{chem}}\)
Main measurement	Isotope depth profile	Conductivity versus time
Common instrumentation	Isotope furnace, SIMS/ToF-SIMS	Controlled-atmosphere furnace, gas switching, conductivity measurement
Main strength	Direct tracer depth information	Time-resolved chemical relaxation
Main limitation	Isotope handling and depth-profile analysis	Requires calibrated conductivity–composition relationship
Typical model sensitivity	Surface boundary and depth resolution	Geometry, gas switching and conductivity calibration

The methods are complementary rather than interchangeable. A strong transport study may combine them to distinguish tracer mobility, chemical diffusion and surface kinetics.

14. Practical reporting checklist

A reproducible oxygen-transport study should report:

1. Exact sample composition, phase purity and synthesis history.
2. Density, grain size, surface preparation and exposed surface area.
3. Sample dimensions and the geometry used in the model.
4. Temperature, oxygen partial pressure, humidity and gas flow.
5. For IEDP: isotope enrichment, exchange time, quench protocol, SIMS settings, depth calibration and isotope normalization.
6. For ECR: gas-switching response, conductivity configuration, equilibration criteria and conductivity–nonstoichiometry calibration.
7. Governing diffusion equation and surface boundary condition.
8. Whether coefficients are tracer or chemical quantities.
9. Fitting method, parameter uncertainty, residuals and parameter covariance.
10. Arrhenius fitting range and activation-energy uncertainty.
11. Evidence for reversibility and absence of phase changes or surface degradation.

Further reading

• R. J. Chater, S. Carter, J. A. Kilner and B. C. H. Steele, “Development of a novel SIMS technique for oxygen self-diffusion and surface exchange coefficient measurements in oxides of high diffusivity,” Solid State Ionics 53–56 (1992), 859–867. https://doi.org/10.1016/0167-2738(92)90266-R.
• P. Fielitz and G. Borchardt, “On the accurate measurement of oxygen self-diffusivities and surface exchange coefficients in oxides via SIMS depth profiling,” Solid State Ionics 144 (2001), 71–80. https://doi.org/10.1016/S0167-2738(01)00893-1.
• P. Fielitz and G. Borchardt, “Oxygen exchange at gas/oxide interfaces: how the apparent activation energy of the surface exchange coefficient depends on the kinetic regime,” Physical Chemistry Chemical Physics 18 (2016), 22031–22038. https://doi.org/10.1039/C6CP02131J.
• S. J. Cooper, M. Niania, F. Hoffmann and J. A. Kilner, “Back-exchange: a novel approach to quantifying oxygen diffusion and surface exchange in ambient atmospheres,” Physical Chemistry Chemical Physics 19 (2017), 12199–12205. https://doi.org/10.1039/C7CP01317E.
• V. Thoréton, M. Niania and J. Kilner, “Kinetics of competing exchange of oxygen and water at the surface of functional oxides,” Physical Chemistry Chemical Physics 23 (2021), 2805–2811. https://doi.org/10.1039/D0CP04953K.
• “The oxygen reduction reaction in solid oxide fuel cells: from kinetic parameters measurements to electrode design,” Journal of Physics: Energy (2020). https://doi.org/10.1088/2515-7655/ABB4EC.

Related articles

• Ionic transport in solid oxides
• Defect equilibria in functional oxides
• Oxygen vacancies and defect chemistry in functional oxides
• Measuring oxygen nonstoichiometry in functional oxides
• Electrochemical impedance spectroscopy: a practical guide
• Protonic ceramic electrochemical cells

A reported transport coefficient is meaningful only when its definition, driving force, geometry, atmosphere and fitting model are stated. Keep tracer and chemical transport parameters distinct throughout data analysis and discussion.