# Probing oxygen ion transport in solid state oxides: A technical review

Zonghao Shen, Jacinthe Gamon, Clement Nicollet, Ji Wu, Jean-Marc Bassat,

Monica Burriel, Stephen J Skinner

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Zonghao Shen, Jacinthe Gamon, Clement Nicollet, Ji Wu, Jean-Marc Bassat, et al. Probing oxygen ion transport in solid state oxides: A technical review. Journal of physics, 2026, 8 (1), pp.012001. 10.1088/2515-7655/ae1255. hal-05373548

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TOPICAL REVIEW

# Probing oxygen ion transport in solid state oxides: a technical review

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Zonghao Shen<sup>1,2,*</sup> <sup> ID </sup>, Jacinthe Gamon<sup>3</sup> <sup> ID </sup>, Clément Nicollet<sup>4</sup> <sup> ID </sup>, Ji Wu<sup>5</sup> <sup> ID </sup>, Jean-Marc Bassat<sup>3</sup> <sup> ID </sup>, Mónica Burriel<sup>1</sup> <sup> ID </sup> and Stephen J Skinner<sup>2,6,7</sup> <sup> ID </sup>

E-mail: zonghao.shen12@imperial.ac.uk

**Keywords:** oxygen mass transport, solid state ionics, oxygen isotope exchange, electrical conductivity relaxation

Abstract

The study of ion transport in solid state ionic materials underpins the fast development of many electrochemical technologies such as solid state batteries, solid oxide cells, proton conducting solid oxide fuel cells. In order to obtain the ion transport dynamics, different *in situ* and *ex situ* techniques have been developed to probe ion transport while each method has its inherent merits and weaknesses. The choice of the method for measuring ionic mass transport primarily depends upon the appropriacy and accessibility of the technique based on sample properties, targeted information of interest and technical limitations. This review introduces aspects of the underlying principles, discusses and compares different techniques available to date in the literature to obtain oxygen mass transport kinetics, primarily focussing on experimental techniques, in particular the isotope exchange depth profiling and electrical conductivity relaxation methods, together with a brief introduction on computational tools.

<footnote n="1">1 Université Grenoble Alpes (UGA), CNRS, Grenoble-INP, LMGP, Grenoble 38000, France</footnote>

<footnote n="2">2 Imperial Global Singapore (IGS), Imperial College London, 1 CREATE Way, #11-01, CREATE Tower, Singapore 138602, Singapore</footnote>

<footnote n="3">3 Université de Bordeaux, CNRS, Bordeaux-INP, ICMCB, UMR 5026, Pessac F-33600, France</footnote>

<footnote n="4">4 Nantes Université, CNRS, IMN, Nantes F-44000, France</footnote>

<footnote n="5">5 School of Physical and Chemical Sciences, Queen Mary University of London (QMUL), London, United Kingdom</footnote>

<footnote n="6">6 Department of Materials, Imperial College London, Exhibition Road, London SW7 2AZ, United Kingdom</footnote>

<footnote n="7">7 International Institute for Carbon Neutral Energy Research (I2CNER), Kyushu University, 744 Motooka, Nishi-ku, Fukuoka
819-0395, Japan</footnote>

<footnote>* Author to whom any correspondence should be addressed.</footnote>
## Contents

<table><tr><td>1. Introduction</td><td>2</td></tr><tr><td>2. Theoretical background</td><td>3</td></tr><tr><td>2.1. Oxygen bulk diffusion</td><td>3</td></tr><tr><td>2.1.1. Tracer diffusion coefficient D*</td><td>4</td></tr><tr><td>2.1.2. Chemical diffusion coefficient D<sub>chem</sub> (or written as D<sup>δ</sup> or D̃)</td><td>4</td></tr><tr><td>2.1.3. Conductivity diffusion coefficient D<sup>σ</sup> (or D<sup>q</sup>)</td><td>4</td></tr><tr><td>2.2. Oxygen surface exchange</td><td>5</td></tr><tr><td>3. Isotope-coupled techniques to probe oxygen mass transport dynamics</td><td>5</td></tr><tr><td>3.1. Isotope exchange depth profiling coupled with secondary ion mass spectrometry (IEDP-SIMS)</td><td>6</td></tr><tr><td>3.1.1. Importance of applying relevant initial and boundary conditions</td><td>8</td></tr><tr><td>3.1.2. Back exchange methodology</td><td>9</td></tr><tr><td>3.1.3. h-plot</td><td>9</td></tr><tr><td>3.1.4. Fast diffusion tails</td><td>10</td></tr><tr><td>3.1.5. Versatile atmospheres</td><td>12</td></tr><tr><td>3.1.6. Versatile samples and anisotropic diffusion</td><td>13</td></tr><tr><td>3.1.7. Composite phases (macroscopic/effective kinetics)</td><td>14</td></tr><tr><td>3.1.8. Summary and perspective on combining other techniques</td><td>14</td></tr><tr><td>3.2. Isotope exchange-atom probe tomography (IE-APT)</td><td>14</td></tr><tr><td>3.3. Isotope exchange Raman spectroscopy (IERS)</td><td>15</td></tr><tr><td>3.4. Gas phase analysis</td><td>17</td></tr><tr><td>3.5. Pulsed isotopic exchange (PIE)</td><td>17</td></tr><tr><td>4. Probing chemical and/or electrochemical-driven mass transport</td><td>21</td></tr><tr><td>4.1. Electrical conductivity relaxation (ECR)</td><td>21</td></tr><tr><td>4.1.1. Theory</td><td>21</td></tr><tr><td>4.1.2. Experiment in practice</td><td>23</td></tr><tr><td>4.1.3. Identifiability problems: Surface reaction vs. diffusion control</td><td>25</td></tr><tr><td>4.1.4. Link between coefficients obtained with other measurement techniques</td><td>29</td></tr><tr><td>4.1.5. Conclusion and perspective to proton conductors</td><td>31</td></tr><tr><td>4.2. Other 'property-relaxation'-based methods</td><td>32</td></tr><tr><td>4.3. Oxygen permeation measurements</td><td>32</td></tr><tr><td>4.4. Electrochemical impedance spectroscopy-based methods</td><td>34</td></tr><tr><td>4.5. Electrochemical polarised measurements</td><td>34</td></tr><tr><td>5. Computational related techniques</td><td>35</td></tr><tr><td>5.1. Molecular dynamics</td><td>35</td></tr><tr><td>5.2. Density functional theory simulation</td><td>36</td></tr><tr><td>5.3. Emerging trends: machine learning and high-throughput screening</td><td>37</td></tr><tr><td>6. Other directions</td><td>37</td></tr><tr><td>7. Conclusions</td><td>37</td></tr><tr><td>Data availability statement</td><td>37</td></tr><tr><td>Acknowledgment</td><td>37</td></tr><tr><td>References</td><td>38</td></tr></table>

## 1. Introduction

The unambiguous determination of ion mass transport kinetics and understanding of the mechanisms of solid state materials are of fundamental importance and are a great challenge. One important example is oxygen mass transport, which is fundamental to the functioning of a range of electrochemical devices such as solid oxide cells (SOC) [1–3], oxygen transport membranes (OTMs) [4–7] oxygen ion batteries [8, 9], yttria-stabilized zirconia (YSZ)-based sensors [10] etc. The oxygen mass transport here refers to two ‘phenomenologically’ different processes: (1) the transport of oxygen across the gas-solid interface and (2) the diffusion of oxygen in the solid state. These are characterised by two kinetic parameters: the surface exchange coefficient (k) and the diffusion coefficient (D).

Over the past decades, a wide range of macroscopic and microscopic, experimental and computational approaches have been developed to probe oxygen ion transport. Experimentally, the kinetics of the transport of oxygen are determined under typical driving forces such as isotopic tracer concentration gradient, chemical potential gradient or electrical potential gradient, which measures tracer, chemical or
electrical species respectively [11, 12]. These kinetics are conceptually distinct but interrelated [13], and it is essential to clearly define the experimental conditions and what is measured in each case.

In this article we review the techniques developed to date for determining oxygen ionic mass transport. Their working principles, metrics and drawbacks, ongoing or potential improvements, particularly on technical aspects, will be discussed. While this review is not exhaustive, it focuses on isotope exchange depth profiling (IEDP), isotope exchange Raman spectroscopy (IERS), electrical conductivity relaxation (ECR), and gas phase analysis methods particularly pulsed isotope exchange (PIE), alongside a brief introduction to other relevant techniques. Electrochemical impedance spectroscopy (EIS) and simulation tools are also of significant importance, though only a general introduction of these methods is provided here.

It should be noted that this article discusses only oxygen transport, although many of the discussed techniques can be extended to study the transport of other elements, such as lithium, hydrogen/deuterium, carbon, and nitrogen.

## 2. Theoretical background

This section only serves as a brief introduction to the underlying theories related to oxygen mass transport in solid state ionic materials as well as the different kinetic parameters [11, 13, 14]. More detailed fundamental equations governing transport processes and how the parameters contribute to the apparent performance of the electrochemical devices, are extensively covered in the literature [12, 15] and will not be discussed further here.

### 2.1. Oxygen bulk diffusion

In a featured article, two complementary ways to describe diffusion were provided by De Souza [14]: on a microscopic scale upon atomic random walk theory and on a macroscopic scale upon Fick's laws of diffusion. The latter one describes the migration of species under a concentration gradient and provides the experimental basis for methods such as tracer exchange and ECR to determine the tracer diffusion coefficients.

On an atomic scale, diffusion in a solid is mediated by point defects—interstitials (i) and/or vacancies (v). Here the defects are noted with lower-case symbols as first suggested by Norby [16] and further emphasised by De Souza and Harrington [17] to avoid ambiguity with iodine (I) and vanadium (V).

In a dilute solution, where the defect (here taking oxygen vacancies as an example) concentration is small, on the assumption that the oxygen vacancies do not interact with each other, the oxygen vacancy diffusion coefficient $D_v$ shows independence from the oxygen vacancy concentration and is given by the expression in equation (2.1)) [18, 19].

$$D_v = \frac{z}{6} (1 - n_v) \lambda^2 \nu \exp \left( \frac{\Delta S}{k_B} \right) \exp \left( -\frac{\Delta H}{k_B T} \right) \quad (2.1)$$

where a random walker moves to one of the adjacent neighbouring positions: the number 6 relates to the six different directions in a 3-dimensional structure, z is the number of equivalent neighbouring sites and $(1-n_v)$ presents the probability that the adjacent sites are occupied by an ion, approximately equal to 1 in a dilute system. $\lambda$ is the jump distance (distance between the equivalent sites) and $\Gamma = \nu \exp(-\frac{E}{k_B T})$ is an average jump frequency, where $\nu$ is the frequency of ion vibration and $E$ is the activation energy for migration equal to $\Delta H - T\Delta S$. $\Delta S$ is the entropy, $\Delta H$ is the enthalpy associated with the migration, $k_B$ is the Boltzmann constant and $T$ is the absolute temperature.

The self-diffusion coefficient ($D_{self}$), the diffusion of ions in the lattice, relies on an adjacent vacancy site for the ion to jump to [18]. Thus, $D_{self}$ shows dependence upon the oxygen vacancy concentration [$v_{\ddot{O}}$] and can be further written as:

$$D_{self} = [v_{\ddot{O}}] D_v = \frac{z}{6} [v_{\ddot{O}}] \lambda^2 \nu \exp \left( \frac{\Delta S}{k_B} \right) \exp \left( -\frac{\Delta H}{k_B T} \right). \quad (2.2)$$

Of particular note is that in equations (2.1) and (2.2) the activation energy is associated with migration while for some materials oxygen loss upon heating exists and the vacancy formation can influence the term: oxygen vacancy concentration [$v_{\ddot{O}}$] as a function of temperature and oxygen partial pressure can be described by an enthalpy and entropy of formation. As highlighted by Kilner and Steele [20], the experimentally obtained activation energy for ionic diffusion comprises both the migration term and the vacancy formation term.
<figure><img src="image_4.png" /><figcaption>Figure 1. Schematic presents the movement of a tracer ion in a 2-dimension hexagonal lattice where the blue ion is the tracer ion and the dash square is the vacancy. After the exchange between the tracer ion 6 and vacancy position 0 (a), for the subsequent jump, the tracer ion has the finite probability to jump back to its original position 6 (b) and the probability here is 1/6 [22].</figcaption></figure>

In the literature, various methods have been used to determine the diffusion coefficients of metal oxides, with the three most common being the tracer diffusion coefficient ($D^*$), chemical coefficient ($D_{chem}$), and conductivity diffusion coefficient ($D^\sigma$), each of which has an associated surface exchange coefficient ($k$) [12, 15].

### 2.1.1. Tracer diffusion coefficient $D^*$

Given that it is difficult to directly measure $D_{self}$, a common approach using a chemically similar element or identical isotope is normally adopted, known as tracer diffusion. Although the tracer diffusion provides a good estimation, $D^*$ is not completely identical to the self-diffusivity $D_{self}$ but they are related through a correlation factor $f$, which is related to the migration mechanisms and the crystal structure, for example $f=0.69$ for perovskite-structured materials [14, 21].

The reason is ascribed to the fact that according to the presence of an oxygen vacancy, the motion of the tracer ion is not random in direction and after the jump of the tracer, it provides the possibility that the tracer will jump back to its original position due to the vacancy being left behind (figure 1). Thus, the measured $D^* = f D_{self}$ ($f < 1$).

$$D^* = fD_{self} \qquad (2.3)$$

### 2.1.2. Chemical diffusion coefficient $D_{chem}$ (or written as $D^\delta$ or $\tilde{D}$)

Different from tracer diffusion under an isotopic tracer gradient where it is at chemical equilibrium with the surrounding gas phase, diffusion can also be driven by a change in the chemical potential of oxygen, which defines the chemical diffusion coefficient. It is related to the tracer diffusion coefficient by $D_{chem} = \gamma D^*$ with the thermodynamic factor $\gamma$ (or the enhancement factor $W$ in section 4.1.4 Link between coefficients obtained with other measurement techniques) given in equation (2.4) [23, 24]:

$$\gamma = \frac{\delta\ln(pO_2)}{2\delta\ln(C_0)} \qquad (2.4)$$

where $C_0$ is the oxygen concentration in the solid. Experimentally, $\gamma$ can be measured by thermogravimetric analysis (TGA) on oxygen non-stoichiometry as a function of $pO_2$.

### 2.1.3. Conductivity diffusion coefficient $D^\sigma$ (or $D^q$)

Under an electrical potential gradient, the obtained total electrical conductivity, $\sigma_t$, from Ohm's law consists of contributions from all mobile charged, electronic and ionic species:

$$\sigma_t = \sum [i] q_i u_i \qquad (2.5)$$

where $q_i = z_i e$ is the charge, $u_i$ is the mobility, $[i]$ is the concentration, and the subscript $i$ denotes the charge type. For pure ionic conducting materials, determining ionic migration from the ionic conductivity is straightforward. For mixed ionic and electronic conducting (MIEC) materials, the key is to isolate the ionic conductivity from the electronic contributions, which generally predominate the total conductivity. Further details can be found in [14]. From the Nernst-Einstein relation, assuming that the charge carrier is dilute, the conductivity diffusion coefficient can be calculated from equation (2.6):
$$D_i^\sigma = \sigma_i \frac{k_B T}{[i] q_i^2} \qquad (2.6)$$

where $k_B$ is the Boltzmann constant and $T$ is the temperature. $D_i^*$ and $D_i^\sigma$ are related with the Haven ratio $H_R = D_i^*/D_i^\sigma$ and for a system with dilute and non-interacting defects, the Haven ratio equals to the correlation factor $f$, $H_R = f$ [15, 25].

## 2.2. Oxygen surface exchange

The overall and reversible oxygen surface exchange reaction, referring to oxygen transfer across the gas-solid interface, is shown in equation (2.7). The kinetic parameter, surface exchange coefficient $k$ presents the rate of a multistep exchange process between gaseous oxygen and oxygen in the solid. Hence it is a lumped parameter, consisting of adsorption, dissociation, charge transfer and incorporation [26]. However, there is no consensus on whether the subsets of reactions are all included for different materials such as electron-rich or electron-poor compounds, and which one is the rate determining step (RDS).

$$\frac{1}{2}O_2(g) + v_{\ddot{O}} + 2e' \leftrightarrow O_{\ddot{O}}^{\times}. \qquad (2.7)$$

Under the dynamic equilibrium, two equal and opposite fluxes in and out of sample surface make the net oxygen flux null.

In oxygen isotope exchange experiments on a macroscopic scale, the tracer surface exchange coefficient ($k^*$) obtained from the experiments is defined as [26]:

$$j = k^* C_o (n_g - n_s) \qquad (2.8)$$

where $j$ is the tracer flux, $C_o$ is the concentration of oxygen ions and $n_g$, $n_s$ are the fraction of the isotope in the annealing gas and at the surface of the solid respectively. Similar to the diffusion coefficients, the chemical surface exchange coefficient is related to the tracer coefficient with a thermodynamic factor, $\gamma$. More details on the electrical oxygen exchange coefficient are provided in section 4.4.

Microscopically, Maier [13] outlined the derivation of atomistic expressions for $k$ through the application of chemical kinetics to detailed reaction mechanisms. An atomic expression for the surface exchange coefficient was derived by Kilner et al [27] upon several assumptions, some of which are not necessarily true such as a perfect surface termination. A universal expression for $k$ was further provided by De Souza [26], empirically. More surface exchange theories will be presented in the following section 3.5 Pulsed isotopic exchange (PIE).

In general, compared to the bulk diffusion, oxygen surface exchange is much less understood on atomistic levels. Meanwhile, by plotting the diffusion coefficient against the surface exchange coefficient, a clear correlation between the two parameters has been observed [27–29], which suggests that oxygen defect concentration, here vacancies for fluorite- and perovskite-structured materials, play a similar or correlated role in these two processes.

## 3. Isotope-coupled techniques to probe oxygen mass transport dynamics

The movement of oxygen across the gas-solid interface (surface) and through the bulk of oxide compounds is briefly presented in the previous section. The oxygen mass transport process is thermally activated at the atomic level while experimentally it is normally described on a macroscopic scale.

Of the techniques developed to date for measuring oxygen mass transport kinetics ($D$ and $k$), isotope-coupled techniques are commonly applied by taking advantage of the use of chemically identical isotopes<fnref n="8" />. However, this also gives rise to challenges in distinguishing between them, which can be performed in several ways. Among all the oxygen stable isotopes, ¹⁸O is often chosen as the ideal tracer, mainly because of its higher abundance (0.204%), commercial availability in highly enriched form (>95%) and relatively lower cost compared to ¹⁷O (0.0347% abundance) though the price of ¹⁸O is still high. An exception is nuclear magnetic resonance (NMR) spectroscopy where ¹⁷O is used due to its odd number of nuclides [30, 31].

<footnote n="8">⁸ This does not consistently hold for hydrogen and deuterium.</footnote>
All the approaches involve annealing a sample in a gaseous medium enriched with the isotope. The original method of measurement relies on monitoring changes in the isotopic concentration of the gas phase during annealing, using a mass spectrometer [32]. The main drawback of this technique is that information about the diffusion profile within the solid, such as fast diffusion pathways along grain boundaries or dislocations cannot be straightforwardly obtained. To overcome this drawback, ion beam techniques (depth profiling techniques) were developed for the post-annealing determination of isotope penetration profile in solids. Nuclear Reaction Analysis or, more commonly, Secondary Ion Mass Spectrometry (SIMS) are used for this purpose [33]. In addition, whilst vibrational spectroscopy is known to be sensitive to isotopes, its application in determining oxygen mass transport has been scarcely reported until recently when it has been developed as a novel methodology [34, 35].

In general, the measurements can be categorised as measuring the post-annealed sample and measuring the labelled gas phase. In the following sections, five techniques will be introduced where the samples themselves are analysed in the first three examples and the gas phases are analysed in the last two techniques. To measure oxygen mass transport kinetics ex situ in the sample, it is necessary to introduce a distribution of oxygen tracer concentration into the bulk of the material, on the lattice oxygen sites, which can be measured before and after isotopic exchange treatment. For *in situ* measurements, a prerequisite is to continuously monitor changes in oxygen tracer concentration within the solid or gas phase during thermal annealing.

### 3.1. Isotope exchange depth profiling coupled with secondary ion mass spectrometry (IEDP-SIMS)

The IEDP-SIMS technique is conceptually straightforward and was one of the earliest techniques developed to determine $D^*$ and $k^*$ [33, 36–38]. Secondary ion mass spectrometry (SIMS) with trace level sensitivity to all elements and their isotopes over the periodic table, has been widely applied to measure oxygen isotopic fractions in samples.

Kilner, Skinner and Brongersma [39] reviewed the challenges associated with IEDP-SIMS, including technical aspects and sample preparation. Indeed, IEDP-SIMS has been successfully applied to various sample formats including but not limited to polycrystals [40–44], single crystals [45, 46] and thin films [47, 48], to extract diffusion and surface exchange kinetics, even for samples with inhomogeneous defect distributions [49, 50]. Regardless of sample geometry, a critical requirement is that the sample must be highly dense and free of open porosity, as open pores can serve as short circuit pathways for oxygen diffusion, resulting in an overestimated diffusivity. Ideally, samples should achieve densities over 95% of their theoretical values to ensure that only closed porosity remains [51]. Furthermore, another essential step in preparing bulk samples after sintering (densifying) is grinding and polishing. This serves several purposes: removing surface impurities and segregated species formed during heat treatment, minimizing the effective isotope exchange surface area, and preserving SIMS depth resolution by reducing surface roughness, which is particularly important for depth profiling analysis mode.

Technically, $^{18}$O can be introduced into the oxygen sublattice of a sample by various experimental methods [52] such as $^{18}$O ion implantation, deposition of an $^{18}$O enriched surface layer and gas-solid exchange annealing. Among these, gas-solid exchange annealing is the most commonly applied: the sample is thermally annealed in an $^{18}$O-enriched atmosphere (figures 2(a) and (b)). In practice, isotopic exchange annealing experiments are typically performed in two-steps: oxygen-16 pre-annealing followed by oxygen-18 exchange-annealing. The main purposes of the pre-anneal step, as summarised by De Souza and Chater [53], are to eliminate the oxygen concentration gradient for both oxygen isotopes ($\nabla ^{16}\text{O} = 0, \nabla ^{18}\text{O} = 0$) at the chosen temperature and oxygen partial pressure, and to remove any mechanical damage caused by polishing during sample preparation, where applicable. The pre-anneal ensures that the measured oxygen diffusivity is upon 'random walk' behaviour free from additional chemical driving force and consequently yields the tracer diffusion coefficient. It is recommended that the pre-annealing duration be 4–10 times longer than the $^{18}$O exchange annealing duration ($t_{\text{pre}} > 4–10 t_{\text{ex}}$). The selection of $t_{\text{ex}}$ to obtain accurate $D$ and $k$ values has been discussed by Fielitz and Borchardt [54], who define a characteristic time constant $\tau \equiv D/k^2$. This time constant determines the annealing duration required to reach equilibrium between the $^{18}$O concentration in the annealing gas and that of the sample surface.

To freeze ion transport, a rolling-on/off furnace is commonly used; that is, the sample is heated and cooled as rapidly as possible at each step. After the exchange anneal, an isotope gradient is formed in the solid (figure 2(c)). Depending on the oxygen diffusion length, estimated by $l = 2\sqrt{Dt_{\text{ex}}}$, SIMS measurements can be performed either by depth profiling (practically $l < 5 \mu\text{m}$) [33] or line scanning (practically $l > 10 \mu\text{m}$). The latter was first reported by Chater *et al* in 1992 [56] and is commonly performed in SIMS mapping mode (figure 2(d)).
<figure><img src="image_5.png" /></figure>

SIMS measurement of oxygen isotopic fraction $C(^{18}O)$ is normally conducted by measuring the intensities of the negative secondary ions of oxygen isotopes<fnref n="9" />: $^{16}O^-$ and $^{18}O^-$ (equation (3.1))

$$C(^{18}O) = \frac{I(^{18}O^-)}{I(^{18}O^-) + I(^{16}O^-)} \quad (3.1)$$

It is well established, taking advantage of the high yields of these species, that oxygen isotopes are considered as chemically identical and are not affected by *matrix effects*. In addition to the aforementioned relatively low cost compared to $^{17}O$, another benefit of using $^{18}O$ as a tracer for IEDP-SIMS measurements is that there is no potential influence from $^{17}OH^-$ species compared to the case of $^{17}O^-$ and $^{16}OH^-$, particularly for mass spectrometers with lower mass resolving power such as a quadrupole mass spectrometer [57]. Furthermore, using other species to determine the isotopic fraction, such as $O_2^-$ and $MO^-$, is also possible and indeed often recommended. This is especially relevant when $O^-$ signals are excessively high, leading to detector saturation ('dead time'). From a SIMS technical perspective, there are several options to mitigate detector saturation effects; however this is beyond the scope of this review and can be found elsewhere [57]. When $O_2^-$ negative ions are used, a more precise calculation of the oxygen isotopic fraction $C(^{18}O)$ is provided in equation (3.2), by considering the probability of forming

<footnote n="9">⁹ Here $^{17}O$ intensity is neglected due to its low natural abundance.</footnote>
different diatomic O₂ species on the surface during SIMS analysis [58]

$$C(^{18}\text{O}) = \frac{I(^{16}\text{O}^{18}\text{O}^-)}{2\sqrt{I(^{16}\text{O}_2^-)}} \cdot \frac{1}{\sqrt{I(^{16}\text{O}_2^-) + I(^{16}\text{O}^{18}\text{O}^-) + I(^{18}\text{O}_2^-)}} \quad (3.2)$$

### 3.1.1. Importance of applying relevant initial and boundary conditions

As fully discussed by De Souza and Chater [52, 53], prior to obtaining the oxygen mass transport parameters $D^*$ and $k^*$ by non-linear least square fitting of the measured isotopic fraction data using one of the solutions to Fick's laws developed by Crank [59], choosing the appropriate boundary conditions for the exchange experiments is vital. Here, an overview of the two most common solutions, namely the semi-infinite and plane sheet solutions, is provided. A first order surface exchange reaction is presented in equation (3.3)), where the flux of the exchanging oxygen across the surface is proportional to the change between the isotopic fraction of the gas and the isotopic fraction on the surface and the proportionality constant is the surface exchange coefficient $k^*$

$$-D^* \left(\frac{\partial C}{\partial x}\right)_{x=0} = k^* [C_g - C(x=0, t)] \quad (3.3)$$

#### (1) Semi-infinite solution:

For thick bulk samples where the isotope diffusion length is much shorter than the sample thickness or where the two diffusion fronts from opposite sides do not meet with each other, the diffusion process can be considered as occurring in a semi-infinite medium. The corresponding solution to the diffusion equation, as given by Crank, is presented in equation (3.4)

$$C'(x,t) = \frac{C_0 - C_{bg}}{C_g - C_{bg}} = \text{erfc}\left[\frac{x}{2\sqrt{D^*t}}\right] - \left[\exp\left(\frac{kx}{D^*} + \frac{k^2t}{D^*}\right) \times \text{erfc}\left(\frac{x}{2\sqrt{D^*t}} + k\sqrt{\frac{t}{D^*}}\right)\right] \quad (3.4)$$

where $C'(x,t)$ is the normalised isotopic fraction corrected by the background isotopic fraction concentration (referred to as $C_{bg}$, natural $^{18}$O isotope abundance-0.2% is normally used$^{10}$) and the isotope fraction of the exchange annealing gas (referred to as $C_g$); $t$ is the time of the exchange anneal and erfc is the error function complement.

The important boundary conditions for the semi-infinite solution include: $C_g$ remains constant during the exchange anneal, $C_{bg}$ is the natural abundance of $^{18}$O which corresponds to the value throughout the sample before the exchange anneal, $C(x \ge 0, t = 0) = C_{bg}$, and at large values of $x$, $C(x = \infty, t > 0) = C_{bg}$, during the anneal.

#### (2) Plane-sheet solution:

For samples in which the solid is only semi-infinite in two dimensions, such as solid slabs where the isotope diffusion length is long enough that the diffusion fronts from opposite sides meet, or thin films where there is a significant isotopic concentration at the film-substrate interface, the semi-infinite medium becomes inapplicable. In such cases, a different solution, known as plane-sheet medium, is typically used. The two parallel surfaces of a sheet (slab) are defined as equidistant from the centre of the sheet at positions $x = l$ and $x = -l$, so the actual thickness of the sample is $2l$. It should be noted that in the case of thin films the total thickness of the films corresponds to $x = l$. The corresponding solution, also provided by Crank, is given in equation (3.5)

$$C'(x,t) = 1 - \sum_{n=1}^{\infty} \frac{2L \cdot \cos\left(\beta_n \frac{l-x}{l}\right) \cdot \exp\left(-\frac{\beta_n^2 \cdot D \cdot t}{l^2}\right)}{(\beta_n^2 + L^2 + L) \cdot \cos\beta_n} \quad (3.5)$$

where $L = l \cdot \frac{k}{D^*}$, the gas-film interface is at $x = 0$, $\beta_n$ are positive roots of $\beta_n \cdot \tan \beta_n = L$, and $x$ is the diffusion depth.

The important boundary conditions for the plane sheet solution include: $C_g$ remains constant during the exchange anneal, and $C_{bg}$ is the natural abundance of $^{18}$O which is the value throughout the sample before the exchange anneal, $C(x \ge 0, t = 0) = C_{bg}$.

<footnote>$^{10}$ High purity oxygen gas can raise the values of $C_{bg}$ after the pre-anneal [53]. This can be measured using an unexchanged sample or Si after pre-annealing to verify $C_{bg}$.</footnote>
Additionally, solutions for diffusion in samples with other geometries such as cylinders, spheres etc. have also been provided by Crank [59]. These, however, are scarcely reported for oxygen mass transport studies in the open literature, with only a few studies available by treating powder samples as spheres using the gas phase analysis technique (more in section 3.4 Gas phase analysis). In addition to the numerical solutions to Fick's law, numerical methods such as finite difference methods [53, 59] can also be applied, which are often faster and easier to implement.

To analyse thin film samples, two configurations are commonly used: the transverse configuration (i.e. out-of-plane diffusion) and the longitudinal configuration (i.e. in-plane diffusion). Thin films with epitaxial growth, specific surface orientations etc, are therefore ideal for studying anisotropic mass transport behaviour [47, 48, 60] (figure 2(e)) as well as for investigating the effects of strain [61], dislocations [62] and other factors on mass transport behaviour. In the transverse configuration, diffusion typically follows the plane-sheet boundary conditions; however, if the film thickness is much larger than the isotope diffusion length, equation (3.4) can be used, as it meets the boundary conditions of the semi-infinite medium. In the longitudinal configuration, diffusion takes place in-plane, parallel to the substrate surface, and it generally is considered to meet the boundary conditions of the semi-infinite medium.

### 3.1.2. Back exchange methodology

With the growing interest in and importance of studying oxygen mass transport under more realistic conditions, the still common practice of recovering the oxygen isotope-enriched gas, due to its relatively high cost compared to regular oxygen gas, largely precludes the use of enriched multicomponent gas mixtures. This limitation confines isotopic exchange experiments mostly to dry oxygen conditions, with limited reports involving water vapour or other gases [63]. To this end, in the last decade, a back exchange method has been reported [64], in which the sample is first enriched with $^{18}$O via a forward exchange process (figure 2(f)) and is then back exchanged under any chosen unenriched gas environment (natural isotopic fraction). A one-dimensional Crank-Nicolson type [65] simulation was firstly applied to the semi-infinite boundary conditions to obtain the kinetics for forward exchange ($D_1^*, k_1^*$) and back exchange ($D_2^*, k_2^*$) [64]. An example of an oxygen diffusion profile after back exchange is shown in figure 2(g) [55]. Additionally, an explicit solution has been provided by Akbay et al [66] A key aspect for these analytical solutions is whether the four parameters ($D_1^*, k_1^*, D_2^*, k_2^*$) are fitted independently; within the experimental uncertainty, $D_1^*$ and $D_2^*$ are generally found to be nearly identical while a large discrepancy can be found between $k_1^*$ and $k_2^*$. Using the back-exchange method, a 250% improvement in the oxygen surface exchange rate under ambient atmosphere has been observed compared to pure oxygen atmosphere at the same $pO_2$ [64].

This oxygen isotope back exchange approach provides opportunities to investigate oxygen mass transport performance in situ at relatively low cost. An example will be discussed in the following section 3.3 Isotope exchange Raman Spectroscopy (IERS).

### 3.1.3. h-plot

Although the IEDP method has demonstrated a strong capability for measuring $D^*$ and $k^*$ through nonlinear least square fitting of diffusion data, the unambiguous determination of oxygen transport kinetics within certain ranges remains challenging due to fundamental limitations. These include low isotopic fraction, resulting in low statistics for pure ionic conductors, ion beam mixing and developed surface topography during primary ion bombardment etc [67, 68].

From a practical perspective, for a measurable diffusion length by SIMS (10 nm- 1 mm), a wide range of $D^*$ values ($10^{-19}-10^{-4}$ cm$^2$ s$^{-1}$) can be determined by the IEDP-SIMS method for exchange durations ranging from 60 s—8 × $10^6$ s (3 months), assuming no significant structural or surface chemistry changes occur during annealing at elevated temperatures [33, 68].

Here using the h-plot ($h = k/D$, in units of length$^{-1}$), the best range for measuring $D^*$ and $k^*$ is reviewed under the semi-infinite condition which is most common for bulk pellets [67]. Of particular interest is the plane defined at $x = 0$, i.e. at the surface of the solid, where equation (3.4) can be further written as:

$$C'(0) = 1 - \exp(h^2 D^* t) \text{erfc}(h\sqrt{D^* t}). \qquad (3.6)$$

Figure 3(a) presents this expression plotted as a function of a dimensionless parameter $h' = h\sqrt{D^* t}$, which can be considered in two ways: (1) as the evolution of the theoretical isotope surface concentration as a function of isotope exchange time (for a fixed value of $h$); or (2) as the theoretical isotope
<figure><img src="image_6.png" /><figcaption>**Figure 3.** (a) The calculated normalised oxygen fraction at the surface C'(0) as a function of the reduced variable *h*', and (b)–(d) representative diffusion profiles of regions (I–III), respectively. Adapted with permission from [55]. CC BY-NC-ND 4.0.</figcaption></figure>

surface concentration as a function of *h* (for a fixed isotope exchange time *t*). For accurately measuring both *D*\* and *k*\*, Kilner and De Souza [67] have divided the *h*-plot into three characteristic regions.

In region I, where *h*' has low values, i.e. for short isotope exchange time or for small *h* values where *D*\* ≫ *k*\*, the surface oxygen isotope concentration is very low. This situation is common in fast oxygen ionic conductors such as YSZ, Gd-doped ceria (CGO) etc, which have limited surface exchange rates and thus exhibit relatively flat diffusion profiles with low isotope concentration. Under these conditions, the resulting diffusion profiles are often noisy and have poor statistical reliability, making it difficult to accurately determine both *D*\* and *k*\*. (figure 3(b)) In order to improve measurement accuracy for *D*\*, it is common practice to enhance the surface exchange rate by coating the surface with catalysts such as Pt or Ag; however, this invalidates the true value of *k*\*.

In region II ($h' = 0.1–4.0$ [67]), well defined diffusion profiles can be obtained, yielding reliable values for both oxygen transport parameters, i.e. the best region for measuring $D^*$ and $k^*$ (figure 3(c)).

In region III, where $h'$ is large, (i.e. for long isotope exchange time or for large $h$ values where $k^* \gg D^*$), the surface oxygen isotope concentration is very high, approaching that of the gas phase. Materials with low bulk diffusivity but high surface exchange rates such as (La,Sr)MnO$_{3-\delta}$ (LSM), (La,Sr)(Cr,Fe)O$_{3-\delta}$ (LSCrF) typically fall into this region. Under these conditions, the obtained value is the lower bound of $k^*$, i.e. obtainable minimum value, and it is impossible to obtain reliable surface exchange coefficients. In such cases, alternative techniques such as pulsed isotope exchange (PIE) are able to provide more accurate results (figure 3(d)).

In addition to bulk samples, Fisher and Hertz [68] described the measurable ranges of $D^*$ and $k^*$ for both bulk and thin film samples using the conventional IEDP-SIMS method. They suggest that $5 \times 10^{-6} \le k^2t/D \le 20$ should be met for accurate determination.

### 3.1.4. Fast diffusion tails

In some cases, especially at low temperatures and/or for poor ionic conductors where the lattice (intra-granular) diffusion coefficient is below $10^{-11}–10^{-12} \text{ cm}^2 \text{ s}^{-1}$ [69–72], a mismatch between the experimental data and fitted profile can be observed where the $^{18}$O depth profiles show extended tails, such as early reports on LSM [70, 73]. An example of a diffusion profile with an observable ‘grain boundary fast diffusion tail’ for LSCrF bulk sample is shown in figure 4(a). It is commonly called a ‘grain boundary fast diffusion tail’, but it can result from any fast diffusion pathways such as dislocations, twin boundaries etc. In addition, a different physical process—a surface space-charge layer [74, 75], can result in a similar oxygen isotope diffusion profile to the fast diffusion case as pointed out by Schraknepper
<figure><img src="image_7.png" /><figcaption>Figure 4. (a) $^{18}$O diffusion profile of (La$_{0.8}$Sr$_{0.2}$)$_{0.95}$Cr$_{0.7}$Fe$_{0.3}$O$_{3-\delta}$ (LSCrF73) isotopically exchanged at 800°C for 40 min with $D^* = 3.1 \times 10^{-14}$ cm$^2$ s$^{-1}$ k = 3.1 $\times 10^{-9}$ cm s$^{-1}$, presents a mismatch between the experimental data and the fitted profile. (b) Diffusion profile of figure (a) plotted as $ln[C^*]$ vs $\eta^{6/5}$ with grain boundary products fitting. (a), (b) Adapted from [69] with permission from the Royal Society of Chemistry. (c) $^{18}$O- ion image on the four sputtered craters with different sputtering durations (1–30 s, 2–60 s, 3–120 s 4–240 s) shows the mismatch in figure (a) origins from the grain boundary fast diffusion. (d) FIB image of the intragranular $^{18}$O depth profile (inset image) on a single LSM grain after isotopic exchange at 1000 °C for 35 min under 5 mbar. Reprinted from [77], Copyright (2008), with permission from Elsevier.</figcaption></figure>

and De Souza where a means to unambiguously assign these two features has also been provided [75]. Notably the term ‘fast’ here is usually a relative term compared to the intragranular diffusivity, instead of an absolute one, highlighted by Kilner et al [76].

To take into account this intergranular diffusion via grain boundaries, an additional term to the solution to the Fick’s second law of diffusion in a semi-infinite medium is used, as provided by De Souza et al [70] (equation (3.7)):

$$C'(x,t) = \text{erfc}\left[\frac{x}{2\sqrt{D^*t}}\right] - \left[\exp\left(\frac{kx}{D^*} + \frac{k^2t}{D^*}\right) \times \text{erfc}\left(\frac{x}{2\sqrt{D^*t}} + k\sqrt{\frac{t}{D^*}}\right)\right] + A_{gb}\exp\left(-Z_{gb}x^{6/5}\right) \quad (3.7)$$

where $A_{gb}$ and $Z_{gb}$ are fitted tailing function parameters. For a reliable estimation, Harrison type B kinetics [78] are required where the bulk diffusion length is shorter than the grain size, which has been formulated by Kaur et al [79], as shown in equation (3.8):

$$20(D_b t)^{1/2} < d_g \qquad (3.8)$$

where $D_b$ is the bulk diffusion coefficient, $t$ is the isotopic exchange duration and $d_g$ is the grain size. Such profiles can be fitted using either Le Claire’s approximation [80] or the expression proposed by Chung and Wuensch [81]. The latter is derived from the exact Whipple solution [82], with the fitted parameter being the grain boundary diffusion product $D_{gb}\delta$. A common practice for estimating $D_{gb}\delta$ is to plot $ln[C^*]$ vs $\eta^{6/5}$ where $\eta = \frac{x}{\sqrt{D_b t}}$. The grain boundary product $D_{gb}\delta$ is then obtained using a linear approximation over a specific range of $\eta$, applying either the Le Claire or the Chung and Wuensch method (figure 4(b)). In the literature, Le Claire’s approximation has been more widely used, although the Chung and Wuensch method is recommended for its higher accuracy [70, 83]. Shen et al [69] compared these two approximations and confirmed that within the range of $\beta = 10–100$ where $\beta = \frac{D_{gb}\delta}{2D_b^{3/2}t^{1/2}}$ and $\delta$ is the thickness of grain boundaries, both methods yield similar results, while outside this range
the Chung and Wuensch approach is more accurate. Conventionally, the grain boundary diffusion coefficient $D_{gb}$ is estimated assuming a grain boundary thickness of approximately 1–2 nm. By contrast a chemical thickness of 11 nm for grain boundaries in (La,Sr)(Co,Fe)O$_{3-δ}$ (LSCF) has been experimentally measured by Sha *et al* [84] using atom probe tomography (APT).

As mentioned above, grain boundaries are not the only possible fast diffusion pathways. To confirm their contribution, $^{18}$O enrichment along grain boundaries can be visualised using advanced techniques such as focused ion beam (FIB)/nano-SIMS [69] and APT [85]. For example, four craters at different depths (1 <2 <3 <4) were sputtered at the fitting mismatch (figure 4(a)), using a FIB-SIMS (2000 workstation, Hiden Ltd, UK). The $^{18}$O⁻ ion image over the four craters (figure 4(c)) confirms that the mismatch is associated with fast diffusion along the grain boundaries, and the $^{18}$O⁻ signal diminishes once beyond the mismatch region [55, 69]. To exclude grain boundary contributions, intragranular depth profiling is an effective strategy; an example is provided by Fearn *et al* [77] for an LSM pellet (figure 4(d)). By measuring within grains of different orientation, this strategy can also be used to measure orientation-dependent oxygen ion transport properties in polycrystalline anisotropic materials [86]. The origin of fast oxygen diffusion along LSM grain boundaries, whether due to a space-charge phenomenon, has been investigated with simulation tools [62, 87, 88] but consensus has not yet been reached.

### 3.1.5. Versatile atmospheres

In the literature, oxygen isotope exchange experiments are conventionally performed in a dry $^{18}$O-enriched oxygen atmosphere (<1 vppm H$_2$O) usually at around 200 mbar with an extendable range from a few mbar up to 1 bar. However, considering the actual operating conditions of solid oxide fuel cells (SOFC)/ solid oxide electrolyser cells (SOEC), the effects of humidity and highly pressurised oxygen on oxygen mass transport have been relatively underreported, and many early studies were conducted without precise control of the water and oxygen partial pressures [89].

Sakai *et al* [90, 91] reported that the presence of water vapour enhances the oxygen surface exchange rate of YSZ compared to dry molecular oxygen, while oxygen bulk diffusion remains unaffected. They further showed that the oxygen surface exchange rate increases with the partial pressure of water vapour, later verified by Pietrowski *et al* [92] (figure 5(a)). The enhancement of the surface exchange kinetics in the presence of water vapour, particularly for SOFC electrolyte materials, has been attributed to a mechanism that does not involve electronic species [92]. Instead, the surface exchange primarily involves water molecules, while direct oxygen molecule exchange is negligible [93]. However, for the ionic conductor La$_2$Mo$_2$O$_9$ [94], though a similar improvement in $k^*$ was observed in wet atmosphere, a decrease in $D^*$ was reported which is attributed to the reduced oxygen vacancy sites resulting from the incorporated hydroxyl from water molecules.

Similar effects have also been observed for MIEC materials, where even small amounts of water vapour can significantly influence oxygen transport across the surface [97–100]. To elucidate the underlying mass transport mechanisms under such conditions, experiments using isotopically labelled oxygen species in different molecules have provided valuable insights. For example, Thoréton *et al* [63] performed two sets of isotope exchange experiments: A ($^{18}$O$_2$ + H$_2$ $^{16}$O) and B ($^{16}$O$_2$ + H$_2$ $^{18}$O). They observed a significant increase in $k^*$ for LSCF at 350 °C for case B, while the diffusion coefficients in both cases were relatively similar (figure 5(b)). This suggests that water molecules monopolise the active oxygen gas-solid exchange sites on the LSCF surface. Conversely, using labelled H$_2$$^{18}$O, Sha *et al* [101] reported suppressed oxygen surface exchange rates of LSCRF at 900 °C under humid atmosphere (30 mbar H$_2$$^{18}$O + 200 mbar $^{16}$O$_2$) compared to dry conditions (200 mbar $^{16}$O$_2$, <1 vppm H$_2$O), supported by detailed surface chemistry studies [102]. This suppression of $k^*$ in the presence of water vapour has been attributed to a competition between water and oxygen molecules for oxygen vacancy sites on the surface.

In addition, a dedicated high-pressure oxygen exchange set-up has been developed to perform oxygen isotope exchange at pressures up to 40 bar [95, 103] considering the real operation conditions of SOCs. This system employs a long (1 m) stainless steel rod as a cryo-pump, immersed into liquid helium and a Mg-doped zirconia ceramic exchange tube designed to withstand stationary high pressure and temperature. High pressure experiments have been performed on Ln$_2$NiO$_{4+δ}$ (Ln = La, Pr, Nd) Ruddlesden-Popper (RP) phase materials. For these over-stoichiometric lanthanide nickelates containing oxygen interstitials, improved oxygen surface exchange and diffusion coefficients were observed at 500–700 °C under 6.5 bar pO$_2$ compared to atmospheric pressure (0.2 atm). An example for Pr$_2$NiO$_{4+δ}$ is shown in figure 5(c). These findings suggest that RP phases with oxygen over-stoichiometry could be promising candidates for oxygen electrodes under pressurised conditions, outperforming conventional materials such as LSCF, whose oxygen vacancy concentration becomes limited at high oxygen pressures. However, their long-term stability and performance under polarisation still need to be thoroughly explored.
<figure><img src="image_8.png" /></figure>

### 3.1.6. Versatile samples and anisotropic diffusion

Most of the studies cited above focus on dense ceramics while the IEDP-SIMS technique has been widely applied to various sample formats<fnref n="11" /> including bulk pellets, thin films, and single crystals. Particularly, the crystal structure of the studied materials can induce a high degree of anisotropy in both the oxygen diffusion coefficient and the surface exchange coefficient. To characterise such anisotropy, films with different crystallographic orientations, as mentioned above, can be deposited on single crystals substrates, such as SrTiO₃ or NdGaO₃ [47, 48]. In reference [47] GdBaCo₂O₅+δ, a SOFC cathode material, was investigated and found to exhibit a *D*\* value one order of magnitude lower along the *c*-axis compared to the *a*-axis, while no anisotropy was observed for the *k*\* coefficient; this is obviously only an example and not necessarily a general trend.

Another approach to evidence such anisotropy effects is to use single crystals with precisely defined crystallographic orientation (prepared by accurate cutting, polishing, and Laüe diffraction) and directly measure the diffusion profiles in the corresponding pieces for each desired direction. A drawback of this method is that two separate samples (one for each orientation) must be prepared for each diffusion temperature, which can become very labour-intensive if the material is to be studied over a wide temperature range. As a case study, Ln₂NiO₄+δ (Ln = La, Pr, Nd) rare-earth nickelates are considered as promising intermediate temperature oxygen electrode materials. Among them, Nd₂NiO₄+δ and especially Pr₂NiO₄+δ exhibit the most interesting electrochemical properties. Understanding the relevant oxygen transport parameters in these MIEC oxides is of high importance, especially for clarifying the underlying oxygen transport mechanisms which are highly anisotropic. Corresponding measurements have been reported in [104]. A large anisotropy of the *D*\* (figure 5(d)) and *k*\* coefficients was previously determined on La₂NiO₄+δ thin films [48] and single crystals [105]. Furthermore, significant differences in the

<footnote n="11">¹¹ In some cases, it is also possible to perform the experiments on powders [285].</footnote>
activation energy for diffusion were observed, depending on the composition studied, and the corresponding diffusion mechanisms were discussed in detail [48, 104]. Neutron diffraction studies can also be performed to study the delocalization of different oxygen involved, and further refine the understanding of these mechanisms.

### 3.1.7. Composite phases (macroscopic/effective kinetics)

On top of the predominant single phase samples in the open literature, there are relatively modest attempts on studying oxygen mass transport via IEDP-SIMS in the area of dual-phase composite materials. The majority of the materials studied are related to SOC oxygen electrode components such as LSM-YSZ [106, 107], LSCF-CGO [108] and oxygen transport membranes such as LSCrF-(Sc,Ce)-stabilised zirconia (ScCeSZ) [109], where one key component is an oxygen ionic conductor dominating the oxygen bulk diffusion and the other one is either a MIEC phase or a purely electronic conducting material.

For composite materials the obtained oxygen mass transport parameters $D^*$ and $k^*$ are known as the ‘macroscopic’ or ‘effective’ kinetics by treating the composites as homogenous single phase materials. As the oxygen isotope diffusion profiles are still obtained from $^{16}$O and $^{18}$O signals, the matrix effect does not present a limitation for composite phases. However, in order to obtain reliable data, a homogenous distribution of the two phases, normally above the percolation limit particularly for the ionic conducting phase, is a prerequisite [109]. With IEDP-SIMS, one key finding for composite materials, for example LSM-YSZ [107], was the apparent existence of a synergistic effect on oxygen surface exchange. Three mechanisms: (1) spillover type mechanism, (2) surface activation via transition metal migration and (3) self-cleaning mechanism were further proposed for the LSCF-CGO system via a combination of SIMS and low energy ion scattering (LEIS) analysis on the surface composition [108]. A bespoke sample configuration was then designed for the LSCrF-ScCeSZ dual-phase composite, attributing a combination of both spillover and self-cleaning mechanisms to the observed synergistic effect [110].

### 3.1.8. Summary and perspective on combining other techniques

IEDP-SIMS is a powerful, versatile and widely used technique to investigate oxygen mass transport dynamics, particularly for obtaining the tracer diffusion coefficient ($D^*$) and surface exchange coefficients ($k^*$). It can be applied to different materials in various sample formats for direct comparison (figure 6) and to all elements with isotopes over the periodic table. However, to obtain reliable results careful experimental control is needed.

This method can further be complemented by other surface analysis techniques to link transport behaviour to surface chemistry and microstructure. Compared to oxygen diffusion in the bulk materials, the oxygen surface exchange processes are not yet fully understood, and the literature reports considerable variation in $k^*$ values for many materials. To better understand the origin of the differences in $k^*$ for nominally similar samples the IEDP technique can be combined with LEIS measurements which provides first atomic layer resolution giving their exact outermost surface chemical termination. As an example, the benefits of this combined approach were demonstrated by two sets of $La_2NiO_{4+\delta}$ single crystals with precisely cut crystal faces [105]. The oxygen surface exchange coefficient and diffusion coefficient were measured along two different orientations at 450–600 °C for each of the two sets. The different $k^*$ values were ascribed to different surface chemistry such as extrinsic impurities, near-surface arrangement and different surface termination, where a predominantly La-terminated outermost surface was evidenced using LEIS.

To conduct IEDP-SIMS, requires a SIMS instrument, which involves unavoidable high cost, ex situ measurements and destructive processes. Despite these challenges, IEDP-SIMS remains an indispensable and powerful tool while alternative and complementary approaches are also in high demand and under development.

## 3.2. Isotope exchange-atom probe tomography (IE-APT)

Until now, most measurements of isotope gradients in solids have been conducted using SIMS. However, limited by its lateral resolution, obtaining information on grain boundaries remains challenging. To this end, atom probe tomography (APT) with sub-nanometre lateral resolution for 3D quantitative compositional mapping, has demonstrated as a powerful complementary technique to SIMS. Oxygen isotope exchange experiments for IE-APT are generally performed similarly to IEDP-SIMS but the requirement for a needle-shaped specimen significantly complicates sample preparation. For ceramic materials in solid state ionics, specimen preparation is typically carried out using the focused ion beam (FIB) lift-out method [114]. In APT, ionic 3D mapping is achieved by mass spectrometry based upon the time-of-flight of ions, allowing the determination of the mass to charge ratio, which is conceptually similar
<figure><img src="image_9.png" /><figcaption>Figure 6. Oxygen tracer diffusion coefficient $D^*$ for a range of oxides. Reproduced from [76], © The Author(s). Published by IOP Publishing Ltd CC BY 4.0. The original data are taken from La$_{0.8}$Sr$_{0.2}$MnO$_{3-δ}$ (LSM82) [41], La$_{0.5}$Sr$_{0.5}$MnO$_{3-δ}$ (LSM55) [41], (La$_{0.8}$Sr$_{0.2}$)$_{0.95}$Cr$_{0.7}$Fe$_{0.3}$O$_{3-δ}$ (LSCrF73) [69], LaCoO$_{3-δ}$ (LCO) [38], LaCo$_{0.6}$Ni$_{0.4}$O$_{3-δ}$ (LCN64) [111], La$_{0.6}$Sr$_{0.4}$CoO$_{3-δ}$ (LSCF6428) [112], La$_{0.6}$Sr$_{0.4}$CoO$_{3-δ}$ (LSC64) [44], La$_{2}$NiO$_{4+δ}$ (LNO214) [40] and Sm$_{0.5}$Sr$_{0.5}$CoO$_{3-δ}$ (SSO55) [113].</figcaption></figure>

to the detection principle of time-of-flight (ToF)-SIMS. However, the ionisation process differs fundamentally for SIMS and APT: in APT field evaporation of ions is induced due to the high bias applied to the sample, either by varying the electrical field or by applying a laser-pulse, the latter being particularly important for ceramic samples. The difference between voltage pulsing and laser pulsing lies in the specific mechanism by which each imparts an additional energy impulse sufficient to temporarily raise the energy of the specimen above the field-evaporation threshold.

Due to the limited accessibility and complexity of the novel APT technique, despite the significant insights it provides into grain boundaries and interfacial phenomena [115, 116], relatively few studies have explored mass transport using this technique, especially in combination with isotope exchange experiments [117–119]. For instance, Sha et al [84] measured a 11.10 nm chemical thickness of grain boundary of LSCF using APT, showing that diffusivity along grain boundaries can be three orders of magnitude higher than the bulk diffusion coefficients for LSCF at 350 °C, although the diffusion coefficients were obtained by SIMS. Furthermore, Baiutti and Chiabrera et al [85, 120] extracted local oxygen mass transport kinetics along grain boundaries $D_{gb}^*$, and $k_{gb}^*$, using an accurate 3D finite element model (FEM) reconstructed from APT measurements. These studies confirmed enhanced oxygen surface exchange and bulk diffusion along grain boundaries in both (La,Sr)CrO$_{3-δ}$ (LSCr) and LSM thin films. Additionally, APT’s ability to map both cationic and anionic species has revealed highly inhomogeneous distributions of cations such as Sr within these materials.

### 3.3. Isotope exchange Raman spectroscopy (IERS)

In vibrational spectroscopy, based upon the harmonic approximation, the oscillation frequency of each vibration mode, $\omega$ is proportional to its reduced mass (effective mass), $\mu$:

$$ \omega \propto \left( \frac{k}{\mu} \right)^{1/2} \qquad (3.9) $$

where $\mu = \frac{m_1 m_2}{m_1 + m_2}$. $m_1$, $m_2$ are the masses of the two bodies, and $k$ is the spring constant. Hence, each wavenumber corresponds to a reduced mass which is determined by the isotope fraction of the involved atoms.
<figure><img src="image_10.png" /></figure>

This phenomenon was initially explored using isotopic labelling to assign the vibrational modes of materials such as the Raman bands in monoclinic zirconia [121]. Early attempts were also made to study oxygen mass transport in yttria-stabilised zirconia (YSZ) pellets [122, 123]. However, due to limitations in instrumentation, only a few data points were collected along the diffusion length, leading to significant uncertainty in the obtained diffusion kinetics. Furthermore, micro-Raman spectroscopy was performed on a bevelled surface of a YSZ single crystal and compared to SIMS depth profile for the same sample; the depth profile obtained from the Raman spectra was still less satisfying [34].

With advances in instrumentation, Raman mapping over a chosen area now enables efficient line scan measurements on cross sections, as demonstrated for Gd-doped ceria (CGO) in figure 7(a) [124]. In this example, the oxygen diffusion profile obtained by Raman spectroscopy agrees very well with that by SIMS (cleaned surface). Beyond providing isotopic fraction information, each Raman spectrum can provide additional insights into the chemistry and structure of the solid, benefits that cannot be achieved with SIMS alone. However, on the other hand, limited by the laser spot size (submicron) it is challenging to measure short diffusion profiles using Raman. A minimum of 25 datapoints along the diffusion profile has been suggested by Kilner et al to achieve reliable kinetics [33].

Additionally, when combined with an external heating stage, IERS can also be performed in situ. Stangl et al [35, 125], were the first to report the oxygen surface exchange kinetics of CGO thin films studied in situ by IERS with unprecedented time resolution using the aforementioned back-exchange approach. Furthermore, by using a capping layer both $D^*$ and $k^*$ of in-plane oxygen mass transport in CGO thin films were determined (figure 7(b)) for a co-limited exchange process. The oxygen mass transport kinetics obtained from both studies are in good agreement with values obtained by SIMS and other techniques reported in the literature.

From a technical point of view, sample preparation such as pre-/exchange-annealing, polishing where applicable etc, for IERS is similar to that for IEDP-SIMS, with the difference being how the oxygen isotopic fraction is determined. Based on its principle, a prerequisite for characterising oxygen mass transport by IERS is that the material must have at least one active oxygen vibrational Raman mode.

Consequently, the applicability of IERS is limited to a certain range of materials on a case-by-case basis. Since IERS does not require an ultra-high vacuum system, it is simpler and more practical for conducting in situ and in operando measurements. Practically, various atmospheres and electrochemical
bias can be applied at elevated temperatures, providing significant insights into the oxygen mass transport under realistic operational conditions.

Although most work using Raman spectroscopy for ionic mass transport has focused on oxygen isotopes, there have been a few investigations involving other elements such as hydrogen/deuterium [126, 127]. Clearly, the novel IERS method, based upon the frequency shift of specific Raman modes due to changes in isotope concentration, has proven to be a powerful and standardised *in situ* and ex situ technique to study ion transport dynamics, particularly due to its accessibility and easy set-up.

### 3.4. Gas phase analysis

As reported in the 1960s [32, 128, 129], gas phase analysis is one of the earliest techniques to study oxygen mass transport. In general, the gas phase analysis has been more widely applied for materials with catalyst applications, so the focus was more on the oxygen exchange reactions on the oxide surfaces.

This method is based on monitoring the time-dependent equilibration of the oxygen composition *in situ* using a mass spectrometer, by following three oxygen isotopologues ($^{16}$O$_2$, $^{16}$O$^{18}$O and $^{18}$O$_2$) upon exchange with the solid oxide powders in a closed vessel. The typical procedures for catalyst applications include a pretreatment under pure oxygen at elevated temperature followed by an anneal under a reducing atmosphere or vacuum and then oxygen isotopic gas is introduced [130, 131]. To be noted, from an oxygen mass transport study point of view, these powder samples are not at thermodynamic equilibrium. Therefore, though an oxygen isotope was introduced in these experiments, the driving force for mass transport is not only the concentration gradient of the tracer oxygen.

Several variations of the gas phase analysis techniques have been reported thereafter, and the pulsed isotopic exchange (PIE) method will be introduced in detail in the following section 3.5.

Kan et al [132], have reported an isothermal isotopic exchange (IIE) approach where the temperature and oxygen partial pressure were maintained before and after the rapid switching to introduce the isotopically labelled gases. By treating the powder sample as spheres, both $D^*$ and $k^*$ can be obtained from this experiment using Crank's solution to Fick's diffusion law in a sphere medium [129, 133]. With this method, oxygen transport in various fluorite- and perovskite-structured materials such as YSZ, CGO, LSM, LSCF etc, with applications for SOFCs has been studied [133, 134]. Overall the obtained activation energies for the perovskites are significantly lower compared to the values determined by other techniques such as IEDP, ECR and an apparent negative activation energy was observed for $k^*$, contradicting the conventional understanding that the oxygen surface exchange reaction presents the Arrhenius behaviour. The authors have attributed it to a precursor-mediated mechanism with variable oxidation states of transition metals for the dissociative adsorption process of oxygen while it does not explain the incorporation process as $k^*$ is a lumped parameter.

Additionally, Nau et al [135], used labelled $C^{18}$O$_2$ to study the bulk oxygen transport in YSZ for catalytic methane combustion. The polarised oxygen isotopic exchange approach [136, 137] was further reported to study oxygen transport in operando under applied potential in SOFC type cells, where more discussion will be provided in section 4.5 Electrochemical polarised measurements. This approach is of particular interest as the gas phase analyses were performed firstly on symmetric cells and secondly under polarisation.

To summarise, one advantage of gas phase analysis, or IIE is that it does not require dense pellets. This allows the determination of oxygen transport kinetics and mechanisms in powders or porous samples. However, it does require the gas phase to reach a steady state, which means experiments often last for long durations. Moreover, recent studies have demonstrated that in operando gas phase analysis can be performed, which is of great importance for understanding oxygen mass transport mechanisms under real conditions.

### 3.5. Pulsed isotopic exchange (PIE)

The pulsed isotopic exchange (PIE) technique was first developed by Bouwmeester et al in 2009 [138], with the aim of rapidly determining the oxygen surface exchange rate of oxide powders, along with the extraction of important mechanistic information. Compared to the gas phase analysis method presented above, the PIE technique also relies on *in situ* gas analysis using a spectrometer but is a more rapid approach.

A powder bed is placed in a tubular reactor and heated to the desired temperature in a furnace (figure 8(a)). A carrier gas flow of Ar/$^{16}$O$_2$ at a given $pO_2$ passes through the reactor, and the powder is initially equilibrated with $^{16}$O at the chosen $pO_2$. A pulse of $^{18}$O$_2$ at the same $pO_2$ is then sent through the reactor, and the fractions of the different isotopic species are analysed at the reactor outlet by a mass spectrometer (figure 8(b)). The ratio of these fractions is used to calculate the global exchange rate ($R_0$) at a given temperature, as well as to determine the contributions from oxygen dissociative adsorption and incorporation into the structure of the materials.
<figure><img src="image_11.png" /><figcaption>Figure 8. Schematic of the Pulsed Isotopic Exchange (PIE) technique. (a) Equilibrating step: a carrier gas $^{16}$O$_2$/Ar passes through an oxide powder bed installed in a reactor in a tubular furnace at chosen temperature and pO$_2$, and a $^{18}$O$_2$/N$_2$ gas mixture with the same pO$_2$ is prepared in an external side loop. (b) Pulsing step: valves are switched so that the carrier gas pushes the labelled gas in the loop towards the reactor, thereby imposing a pulse of $^{18}$O$_2$ gas through the oxide powder bed sample. Gases are analysed at the outlet by a mass spectrometer.</figcaption></figure>

After each isotope pulse, the sample can be equilibrated again with $^{16}$O under different conditions (pO$_2$, temperature), before performing another pulse. This method has the advantage of relatively rapid screening of material exchange properties without going through a densification step and under multiple conditions on the same batch of powders.

Because the carrier gas and the $^{18}$O gas pulse have the same oxygen partial pressure, chemical equilibrium is maintained between the gas phase and the oxide. The surface exchange reaction is defined by:

$$O_2^* + 4e' + 2v_O^{\bullet\bullet} \leftrightarrow 2O_O^{*\times} \qquad (3.10)$$

where $\mathfrak{R}_0$ is the balanced surface exchange rate under these equilibrium conditions, $O^*$ is the tracer oxygen [139]. $\mathfrak{R}_0$ is ruled by the first order rate equation from the $^{18}$O fraction in the gas phase, $f_g^{18}$, and oxide, $f_o^{18}$, the oxide surface area, S, and the total number of oxygen atom in the gas phase, n:

$$n \frac{\partial f_g^{18}}{\partial t} = -\mathfrak{R}_0 S (f_g^{18} - f_o^{18}). \qquad (3.11)$$

The following conditions are required for the simple evaluation of $\mathfrak{R}_0$:

(i) In comparison to the number of exchangeable atoms in the oxide phase, the number of $^{18}$O atoms is extremely small, so that it can be considered negligible, i.e. $f_o^{18} \approx 0$;

(ii) The oxide diffusion length in the bulk oxide phase is large;

(iii) The reactor is under ideal plug flow behaviour (channelling, interphase and interparticle diffusion are neglected) [138, 140].

Then, through integration over time, the following expression for $\mathfrak{R}_0$ is obtained:

$$\mathfrak{R}_0 = \frac{n}{\tau_r S} \ln \frac{f_{g,i}^{18}}{f_{g,e}^{18}} \qquad (3.12)$$

where $f_{g,i}^{18}$ and $f_{g,e}^{18}$ are the $^{18}$O isotope fractions in the pulse volume at the inlet and outlet of the reactor, respectively and $\tau_r$ is the mean residence time of the reactor.

The fraction of $^{18}$O isotope at the outlet of the reactor is determined from the molar fractions of $^{18}$O$^{16}$O and $^{18}$O$_2$, with mass-to-charge ratios of 34 and 36 respectively: $f_g^{18} = f_g^{36} + 0.5f_g^{34}$. These are determined from signal integration of the mass spectrometer peaks after normalization with the mass 28 response (N$_2$) at each temperature.

A typical temperature dependent isotope fraction curve is presented in figure 9(a) for Pr$_6$O$_{11}$. Generally, at low temperature, the mass 36 peak is at its maximum, meaning that oxygen exchange
<figure><img src="image_12.png" /><figcaption>Figure 9. Isotope fraction obtained from PIE measurement on (a) Pr$_{6}$O$_{11}$ and (b) PrNiO$_{3}$ powders, and the corresponding oxygen surface exchange rates $\mathfrak{R}_{0}$, $\mathfrak{R}_{a}$ and $\mathfrak{R}_{i}$ for (c) Pr$_{6}$O$_{11}$ and (d) PrNiO$_{3}$ powders. Reproduced from [141] with permission from the Royal Society of Chemistry.</figcaption></figure>

between the oxide and the pulse does not occur. As temperature increases, its intensity decreases, while the intensity of the mass 34 peak increases until reaching a maximum before decreasing again. Less $^{18}$O$^{16}$O formation may be related to high rate of incorporation (high oxygen diffusivity) and a comparison to PrNiO$_{3-\delta}$ is shown in figure 2(b). At very high temperature, the rate of oxygen exchange is so fast that the probability to exchange two oxygen atoms is high, therefore favouring the release of $^{16}$O$_{2}$ instead of $^{18}$O$^{16}$O, and leading to an increase of the intensity of mass 32 peak.

The surface exchange coefficient, $k^{*}$, can be obtained from the surface exchange rate using:

$$k^* = \mathfrak{R}_0/c_0 \qquad (3.13)$$

where $c_0$ is the concentration of oxygen atoms in the oxide phase [140]. This enables direct comparison with $k^*$ obtained with other techniques such as ECR or IEDP methods. Discrepancies are often attributed to different surface chemistry—due to different thermal history of the sample as well as post annealing treatments (grinding, polishing). The PIE method enables access to the surface exchange rates with high accuracy (providing careful material preparation and reactor design), as its determination is not limited by oxygen bulk diffusivity of the materials, particularly for diffusion-limited materials such as LSM (cf IEDP methods), and is free from errors coming from the fitting of profiles.

The surface exchange rate $\mathfrak{R}_0$ is an overall parameter that regroups many different steps, each of which can be rate limiting (adsorption, dissociation, charge transfer, incorporation). Boukamp et al [139, 142] proposed the two-step oxygen exchange mechanism model involving adsorption at the surface, defined by the rate $\mathfrak{R}_a$ and incorporation into the lattice, defined by the rate $\mathfrak{R}_i$:
$$ \text{Step 1: } O_2^* \leftarrow \mathfrak{R}_a \leftrightarrow 2O_{ad}^* $$

$$ \text{Step 2: } O_{ad}^* \leftarrow \mathfrak{R}_i \leftrightarrow O_{lattice} $$

From these reactions, a probability of incorporation can be defined:

$$ p = \frac{\mathfrak{R}_i}{\mathfrak{R}_a + \mathfrak{R}_i} \qquad (3.14) $$

As shown by den Otter et al [143], the transient for mass 36 can be derived from the measurement of the different isotopic fractions in the gas phase:

$$ n \frac{\partial f_g^{6}}{\partial t} = S \mathfrak{R}_a \left( -f_g^{6} + \left( f_g^{8} (1-p) \right)^2 \right) \qquad (3.15) $$

After integration, a model equation is derived for the molar fraction $f_g^{6}$ in the case of the two-step oxygen exchange model. The solving of this equation enables to determine $p$. Moreover, in the steady state approximation, Boukamp et al showed that the inverse of the overall exchange rate $1/\mathfrak{R}_0$ acts as a resistance consisting of two series resistances $1/\mathfrak{R}_a$ and $1/\mathfrak{R}_i$, thereby giving [139]:

$$ 1/\mathfrak{R}_0 = 1/\mathfrak{R}_a + 1/\mathfrak{R}_i \qquad (3.16) $$

$\mathfrak{R}_a$ and $\mathfrak{R}_i$ can then be calculated from equations (3.14) and (3.16) at each temperature. Figures 9(c) and (d) presents the calculated surface exchange rates $\mathfrak{R}_0$, $\mathfrak{R}_a$ and $\mathfrak{R}_i$ for $Pr_6O_{11}$ and $PrNiO_3$, where the rate-determining step (RDS) can be determined. For example, for $Pr_6O_{11}$ the RDS is absorption while a competing behaviour between absorption and incorporation is seen for $PrNiO_3$.

This two-step model is a particular case of the earlier developed statistical ‘parallel path’ model by Klier et al [144], as has been demonstrated by Den Otter et al [143]. In the latter, the surface exchange reaction can be divided into three ‘parallel’ routes:

(1) Homoexchange of the two oxygen atoms with oxygen atoms from the gas phase only:

$$ ^{18}O_2 + ^{16}O_2 \leftrightarrow 2^{18}O^{16}O. $$

(2) Heteroexchange of one oxygen atom with one oxygen atom from the oxide phase:

$$ ^{18}O_{2,g} + ^{16}O_{\text{lattice}} \leftrightarrow ^{18}O_{\text{lattice}} + ^{18}O^{16}O. $$

(3) Heteroexchange of two oxygens atoms with two oxygens atoms from the oxide phase:

$$ ^{18}O_{2,g} + ^{16}O_{\text{lattice}} \leftrightarrow ^{18}O_{\text{lattice}} + ^{16}O_2. $$

These three reactions, shown in figure 10, are ruled by constant rates $r_0$, $r_1$, and $r_2$, as well as by the independent probabilities $p_1$ and $p_2$ for each of the two oxygen atoms of the molecule. In the two-step model, $p_1 = p_2 = p$, implying that breaking the oxygen bond in the $O_2$ molecule leads to the loss of information about the isotopic distribution within the molecule. In this case, it can be shown that: $\mathfrak{R}_a = r_0 + r_1 + r_2$, $\mathfrak{R}_0 = 1/2r_1 + r_2$ and $r_1 = 2\sqrt{r_0 r_2}$.

Therefore, the PIE method provides significant insights into oxygen surface exchange mechanisms and has been applied to a wider range of environments and samples. The importance of both oxygen vacancy and redox-active centre has been demonstrated using an extended set of doped ceria samples under varying $pO_2$ [145]. The influence of humidity was reported on Pr-substituted $La_2Ce_2O_7$, where a decreased exchange rate was observed, particularly at the temperature below 750 °C, due to the formation of hydroxide defects [146]. For double-perovskite $BaGd_{0.3}La_{0.7}Co_2O_{6-\delta}$ with surface exsolved $Co_3O_4$ nanoparticles, the effect of water was further suggested to depend on oxygen partial pressure with an enhanced surface exchange rate reported at low $pO_2$ [147]. This also demonstrated the versatility of PIE in studying various samples with surface decoration, reflecting more realistic applications.

Furthermore, enhanced surface exchange rates, including both dissociative adsorption and incorporation, were observed for $(Bi_{0.8}Tm_{0.2})_2O_{3-\delta}$-LSM composites compared to their parent single phase materials, suggesting synergistic effects on oxygen surface exchange [148]. In addition, as in general gas phase analyses, powder samples are typically used, same as in PIE. For materials that undergo structural and/or chemical change during densification at high temperatures, PIE, as well as general gas phase analyses,
<figure><img src="image_13.png" /><figcaption>Figure 10. Schematic of a stochastic model for oxygen isotope exchange.</figcaption></figure>

enables the determination of more reliable kinetics for the material of interest. For example, PIE allows direct comparison of F-doped and undoped materials [149], which may suffer F-loss during high temperature treatment, providing more refined understanding.

In summary, though only surface exchange kinetics can be obtained and there are very few studies available in the open literature using the PIE approach, it has established itself as a very powerful *in situ* technique to understand oxygen surface exchange processes under more realistic conditions.

# 4. Probing chemical and/or electrochemical-driven mass transport

## 4.1. Electrical conductivity relaxation (ECR)

### 4.1.1. Theory

Chemical relaxation experiments are based on the equilibration of the oxygen chemical potential of mixed conducting oxides with their surroundings following equation (4.1):

$$ \mu_{O, int} = \mu_{O^{2-}} - 2\mu_{e^{-}} = -\mu_{v_{O}^{-}} - 2\mu_{e^{-}} = \mu_{O, ext} \quad (4.1) $$

in which $\mu_{O^{2-}}$, $\mu_{e^{-}}$, and $\mu_{v_{O}^{-}}$ are the chemical potentials of oxide ions, electrons, and oxygen vacancies, respectively. Here is the example for oxides with oxygen vacancies and similar equation can be established for oxides with oxygen interstitials. Since the chemical potential of oxygen vacancies in the mixed conducting oxide is directly dependent on the vacancy concentration according to:

$$ \mu_{v_{O}^{-}} = \mu_{v_{O}^{\circ}} + RT \cdot \ln[v_{O}^{\ddot{O}}] \quad (4.2) $$

and since the oxygen potential of the gas phase is defined by its oxygen partial pressure, a solid oxide material subjected to an instantaneous change in oxygen partial pressure ($pO_2$) will accommodate its vacancy concentration to balance equation (4.2).

Then, the kinetics of the uptake and release of oxygen are dependent on chemical diffusion and surface exchange coefficients, $D_{chem}$ and $k_{chem}$, and any method to monitor oxygen vacancy concentration as a function of time after a $pO_2$ step change can enable the determination of those coefficients. There are many examples of such experiments in the literature (more in the following section 4.2), among which the most widespread technique is to use electrical conductivity. The ECR method was first introduced by Carl Wagner in 1950 to measure the self-diffusion coefficient of lead in silver chloride [150]. It is now widely applied for the determination of oxygen diffusion and surface exchange coefficients in oxides [151–155], since the conductivity is directly proportional to oxygen vacancy concentration. This method has the benefits of being cheap, sensitive, and easy to implement as compared to the aforementioned alternatives. Indeed, in these materials, the electronic conductivity is particularly sensitive to variation in oxygen stoichiometry, as electronic and ionic defect concentrations are correlated through the charge neutrality requirement [151].

For instance, in a p-type doped perovskite oxides such as (La,Sr)(Co,Fe)O$_{3-\delta}$, one can write the redox reaction:

$$ v_{O}^{\bullet\bullet} + \frac{1}{2}O_{2} \leftrightarrow O_{O}^{\times} + 2h^{\bullet} \quad (4.3) $$

showing that an increase in the oxygen partial pressure leads to an oxygen uptake and increase in the hole concentration, and therefore of the conductivity.
On the contrary, in n-type oxide semiconductors such as acceptor doped CeO₂, the electron concentration, and hence the electronic conductivity decreases when the oxygen partial pressure increases, following the reaction, equation (2.7).

The relaxation of the electronic conductivity to the new equilibrium value is controlled by the kinetic rates of O₂ surface exchange (embodying many elementary steps such as adsorption, dissociation and incorporation into the lattice) and by O²⁻ diffusion within the crystal structure [143]. Then, by fitting the relaxation data with an appropriate model, one can extract the chemical diffusion and surface exchange coefficients, Dₚₚ and kₚₚ, respectively [151, 153, 154].

### 4.1.1.1. Models

Models describing the variation of oxygen concentration to the diffusion and surface exchange coefficient are solutions to Fick's second laws of diffusion associated to specific boundary conditions. In the majority of derived models, the linear kinetic boundary conditions are assumed (small step change) as well as a constant chemical diffusion coefficient (linear ECR model), leading to solving a linear partial differential equation. These simplifications have the advantage of yielding analytical solutions when simple sample geometries are considered (thin sheets, 3D bar...).

In this case, considering a plane sheet geometry, Fick's second law can be written as [156]:

$$ \frac{\partial c(x, t)}{\partial t} = D_{\text{chem}} \frac{\partial c(x, t)}{\partial x} \qquad (4.4) $$

where c is the oxygen concentration. The surface exchange process can be modelled at the boundary of the sample (x = ±l) by [156]:

$$ -D_{\text{chem}} \frac{\partial c(x, t)}{\partial t} \Bigg|_{x=\pm l} = \pm k_{\text{chem}} (c(x = \pm l, t) - c_{\infty}) \qquad (4.5) $$

where c_∞ is the equilibrium concentration of oxygen after infinite time following the step change in pO₂. Equations (4.4) and (4.5) can be solved analytically using the eigenfunction expansion technique to obtain the concentration profile as a function of time [156]. Then, the normalized conductivity σ̄(t) is derived by integrating the concentration over the sample volume, by considering a linear proportionality between the changes in electronic conductivity and oxygen concentration [59]:

$$ \bar{\sigma}(t) = \frac{\sigma(t) - \sigma_0}{\sigma_\infty - \sigma_0} \equiv \frac{c(t) - c_0}{c_\infty - c_0} = 1 - \sum_{n=1}^{\infty} \frac{2L\alpha^2}{\beta_n^2(\beta_n^2 + L_{\alpha}^2 + L_{\infty})} \exp\left(-\left(\frac{\beta_n^2 D_{\text{Chem}} t}{l^2}\right)\right) \qquad (4.6) $$

with σ₀ and σ_∞ the initial and equilibrated electronic conductivity, respectively, and c₀ and c_∞ the corresponding oxygen concentrations. The parameter L_α is expressed by: L_α = Ikₚₚ/Dₚₚ, and β_n are the eigenvalues, which are the non-zero roots of equation: β_n tan β_n = L_α, same as equation (3.5).

The general equation for 2D or 3D bar sample can be found in other reference publications [157–159]. Therefore, using the correct sample geometry, fitting of the conductivity relaxation data as a function of time enables the derivation of both Dₚₚ and kₚₚ. ERCTools [160], NETL ECR analysis tools [161] and ECReX [162] are examples of programs developed by researchers and made freely accessible for that matter. An example of normalized conductivity transient and curve fitting is shown in figure 11 for the compound of La₀.₈Ca₀.₂FeO₃⁻δ.

It has to be remembered that the linear ECR model is only valid when particular assumptions are made:

(1) A linear response assumption: a 'vanishingly small perturbation of the materials' characteristics' [154] is obtained, i.e. the pO₂ step change and the bias imposed to measure conductivity are sufficiently small.

(2) After a step change in oxygen partial pressure, the new desired pO₂ is constant at the surface during the relaxation experiment and hence independent from the oxygen surface exchange rate, meaning that the mass transport in the gas phase must be sufficiently effective [158].

Many papers pointed out the limit for the validity of the linear model, in particular when materials with high surface exchange rate as well as low pO₂ conditions are studied [153, 158]. Typically, discrepancies between Dₚₚ and kₚₚ values in oxidation and reduction runs are obtained, which can be explained by limiting mass transport phenomena in the gas phase (figure 12) [158]. In order to be exempt from these assumptions, Ciucci et al developed a nonlinear model to study kinetic parameters of LSCF enabling considerations of chamber dynamics [154].
<figure><img src="image_14.png" /></figure>

**Figure 11.** Typical normalized conductivity curves following a step change in $pO_2$ from 0.100 to 0.215 atm at different temperatures. Data shown were obtained for $La_{0.8}Ca_{0.2}FeO_{3-δ}$. The solid lines show the least square fitting to the model equations. Corresponding values of $D_{chem}$ and $k_{chem}$ obtained from fitting are listed. Reproduced with permission from [163].
CC BY-NC 4.0.

<figure><img src="image_15.png" /></figure>

**Figure 12.** Normalized conductivity data for $La_{0.6}Sr_{0.4}Co_{1-y}Fe_yO_{3-d}$ at 750 °C following oxidation and reduction step changes to the same final oxygen partial pressure: (a) $pO_2$=0.35 bar and (b) $pO_2$=0.28 · 10⁻³ bar. Reproduced from [164], with permission from Springer Nature.

### 4.1.1.2. 'Flush-time' consideration

In the derivation of the above model, the flush-time, i.e. the time needed to switch between two different $pO_2$ values, is considered negligible in comparison to the time needed to equilibrate the oxygen concentration in the sample. The 'instantaneous switching' must be validated before data analysis, through measurement of the chamber volume and considering the gas flow rate. Also, the flush-time can be determined by a conductivity measurement on a porous and thin layer of a typical MIEC material deposited on an insulating support, for which diffusion and surface exchange will be extremely fast (the sample is said to be in the sensor regime: the regimes are defined for flush-time correction) [156, 161].

Den Otter *et al* studied the influence of the flush-time on proper analysis of ECR data [156]. The general rule that was concluded in their study is that in order for the flush time to have a negligible influence on the determination of kinetic parameters, it must be at least 500 times lower than the sample equilibration time, $\tau_{eq}$, in the case of diffusion limitation, and 5000 times lower in the case of surface control (cf section 4.1.3, identifiability of $D_{chem}$ and $k_{chem}$). For higher values of the flush-time (while keeping $\tau_f < \tau_{eq}/50$), a correction to the above linear ECR model can be added in order to take the flush-time into account [156]. Non negligible flush times will lead to the addition of an extra surface limitation phenomenon, and, if not taken into account in the model, will result in the underestimation of $k_{chem}$ [157].

### 4.1.2. Experiment in practice

A typical conductivity relaxation experiment requires at least three components: a furnace to control the temperature, mass flow controllers to control the $pO_2$ precisely, and electrical measurement equipment
<figure><img src="image_16.png" /><figcaption>Figure 13. Example of ECR set-up, showing the two preparation gas lines, with the use of four mass flow controllers (MFCs) and four-way valves to switch rapidly between lines. The sample chamber volume is limited due to the use of alumina stoppers.</figcaption></figure>

to monitor the conductivity of the sample (digital multimeter or source-measure unit). From such initial setup, additional equipment and design considerations should be implemented for optimal measurements (figure 13). Indeed, the accurate determination of $D_{chem}$ and $k_{chem}$ is prone to identifiability problems, and those problems will be even greater if the quality of the experimental data is poor, e.g. noisy, or contains measurement artefacts. Moreover, since the reliability of the technique can be impacted by the flushtime, it should be carefully addressed when designing the experimental setup. Last, temperature monitoring should be as accurate as possible since the temperature dependence of $D_{chem}$ and $k_{chem}$ are the primary information that can be interpreted.

Step changes of $pO_2$ in the range ($10^{-2}$–1 atm) are usually achieved by mixing $O_2$ with $N_2$ or Ar without further precaution. To reach lower $O_2$ partial pressure (down to ~$10^{-7}$ atm), oxygen pumps must be used for the purification of $N_2$ gas, whereas other gas mixtures must be considered for highly reductive atmospheres: $CO/CO_2$ ($pO_2$ between $10^{-13}$–$10^{-9}$ atm) and $H_2/H_2O$ ($pO_2$ < $10^{-22}$–$10^{-13}$ atm) [165].

For gas mixing, two approaches can be considered depending on the targeted pressure range. At high $pO_2$ (0.1–1 atm), gas mixing is fast and a simple change of setpoint on two mass flow controllers is enough to ensure a fast $pO_2$ change in the gas stream. For lower $pO_2$, since gas mixing is slower, it is necessary to pre-mix the gas in an additional line before flowing it into the reactor. In this case, two pairs of mass flow controllers are used to mix $O_2$ and $N_2$ in two distinct $pO_2$s, and a 4-way crossover valve allows fast switching of the line sent into the reactor (figure 13). In any case, an oxygen sensor should be included in the reactor to monitor the $pO_2$ and flushtime precisely. Moreover, Haugsrud et al have employed changes in total pressure to achieve instantaneous gas switching [166]. One possible design is illustrated in figure 13.

### 4.1.2.1. Flushtime reduction

The flushtime is primarily a function of gas flow rate and reactor volume according to the equation defining its time constant in the case of a continuously stirred tank reactor model:

$$ \tau_{\text{flush}} = \frac{V_{\text{reactor}}}{F_{\text{tot}}} \cdot \frac{T_{\text{STP}}}{T_r} \qquad (4.7) $$

with $\tau_{flush}$ the flushtime constant, $V_{reactor}$ the reactor volume, $F_{tot}$ the gas flow rate entering the reactor, $T_{STP}$ and $T_r$ the temperature at standard conditions (25 °C) and the temperature of the reactor.

Then, a high gas flow rate such as 300 sccm and a small reactor volume will enable small flushtimes while keeping the flow laminar. The reactor volume is harder to manage since it requires custom-made quartz or alumina parts that can fit into a tube furnace (inset figure 13). However, the volume of an
existing tube reactor can be reduced by filling the empty space with plain alumina rods, thereby reducing the flushtime without re-designing a reactor from scratch.

### 4.1.2.2. Temperature monitoring (thermocouple placement, joule heating mitigation)

The temperature is usually controlled by placing the reactor into a laboratory tube furnace that can be controlled with a PID controller. The sample temperature has to be measured precisely, usually via an additional thermocouple placed as close as possible to the sample. Since the conductivity is temperature dependent, and the baseline conductivity has to be as flat as possible for the transient not to be skewed by baseline variations, the furnace should have enough inertia and the PID controller should be fine-tuned to limit temperature variation below 0.1 °C.

The actual temperature of the sample can also be affected by local Joule heating while current is applied to measure the electrical resistance of the sample. To avoid such issues, the current level should be kept to a minimum and should be optimized beforehand to avoid those local heating effects. A simple protocol to ensure that Joule heating is negligible is to measure the sample at various current levels to determine the current range at which the resistance is constant.

### 4.1.2.3. Noise reduction

Lowering the current applied to the sample to avoid Joule heating can yield noisy measurements, especially for highly conductive materials such as (La,Sr)CoO$_{3-δ}$. Then, it is important to reduce the level of electrical noise to ensure clean experimental data. First, the reactor can be surrounded by a Faraday cage that will protect the sample from electromagnetic interference and inductive noise coming from the heating elements of the furnace. Additionally, the measuring equipment (digital multimeter or source measure unit) can be programmed to trigger batches of measurements for all data points. Those batches can be averaged programmatically to improve the signal to noise ratio.

### 4.1.2.4. Stabilization time

Before each $pO_2$ step, it is mandatory that the electrical conductivity is stable, which means that the sample is fully equilibrated with respect to both the $pO_2$ and the temperature. To achieve such stability, all parameters should be kept constant for several hours before the $pO_2$ step change. It is noteworthy that at lower temperature, i.e. when the equilibration kinetics are getting more sluggish, stabilization times should be increased accordingly to avoid baseline drift.

### 4.1.3. Identifiability problems: Surface reaction vs. diffusion control

It is well known that the results coming from the fit of conductivity relaxation data suffer from identifiability problems, meaning that multiple sets of $k_{chem}$ and $D_{chem}$ values yield good fits, rendering it difficult to distinguish the true properties of the sample [160, 161, 167, 168]. This results in large errors of the obtained parameters and explains the value discrepancies of both parameters for the same compositions in the literature.

Model sensitivity on $D_{chem}$ and $k_{chem}$ heavily depends on the rate limiting step, surface reaction vs. diffusion, occurring in a particular experimental condition. In the most preferable case, the fit heavily depends on both parameters, meaning that neither diffusion nor surface exchange dominate the mass transport process, in which case it becomes possible to determine accurately both parameters at the same time from fitting. In other cases, the experiment contains relatively much more information on the parameter associated with the rate limiting step, and much less on the faster step, making the associated parameter non-identifiable.

The determination of the ratio between the time constants of the diffusion ($\tau_D \sim l/D_{chem}$) and surface exchange ($\tau_k \sim l/k_{chem}$) processes, referred to as the Biot number ($L_\alpha$ or $Bi$) [161], enables the type of regime which the sample is subjected to, to be distinguished:

$$L_\alpha \text{ (or } Bi\text{)} = \frac{l \times k_{chem}}{D_{chem}} = \frac{\tau_D}{\tau_k} \qquad (4.8)$$

where $l$ is the shortest dimension of the sample. Den Otter et al showed that for $L_\alpha$ values larger than ~30, diffusion will be the main limiting mechanism, and the model will be insensitive to $k_{chem}$. On the contrary, if $L_\alpha$ falls below ~0.03, mass transport will be mostly defined by the surface exchange rate, and the fit will be relatively independent from the value of $D_{chem}$ [156]. This is further visualized when plotting the sensitivity of the electrical conductivity with respect to $D_{chem}$ or $k_{chem}$ as a function of the Biot number (figure 14). The sensitivity of a signal, $\sigma$, with respect to parameter $q$ is defined by:
<figure><img src="image_17.png" /><figcaption>Figure 14. Sensitivity of the ECR signal with respect to $k_{chem}$ and $D_{chem}$ in function of the Bi number (and therefore sample dimension) for a given set of ($k_{chem}$, $D_{chem}$) showing (a) the influence of the thickness on the sensitivity values and (b) the influence of the relaxation time for the surface exchange reaction—the Bi number is controlled by increasing surface area. Reprinted from [161], Copyright (2021), with permission from Elsevier.</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>Figure 15. (a), (b) Estimated errors on $k_{chem}$ (a) and $D_{chem}$ (b) in function of their true values ($\bar{D}$ and $\bar{k}$) for a sample with cubic dimension $l_x = l_y = l_z = 1$ mm. (c, d) Sensitivity of the ECR signal with respect to $D_{chem}$ and $k_{chem}$ in the worst case scenario corresponding to $\bar{D} = 10^{-5}$ cm$^2$/s and $\bar{k} = 10^{-2}$ cm/s and for a sample dimension $l_x = l_y = l_z = 1$ mm (c) or $l_x = l_y = l_z = 5.7 \times 10^{-2}$ mm (d). Reprinted from [160], Copyright (2013), with permission from Elsevier.</figcaption></figure>

$$S = q \frac{\partial \sigma}{\partial q}, \quad \text{with } q = D_{chem} \text{ or } k_{chem} \qquad (4.9)$$

#### 4.1.3.1 Solving identifiability issues computationally

Ciucci et al showed that lack of sensitivity leads to large errors on the parameter values, making the concept of sensitivity and accuracy deeply related [160]. Based on sensitivity analysis, the Optimal Experimental Design approach was developed to test which combination of time span and sample geometry is the most favourable for reducing uncertainty on $D_{chem}$ and $k_{chem}$ values [160, 161]. Other methods, such as the Bayesian approach [167] or Euclidian Norm [161] can be used for uncertainty determination and correlation to the experimental conditions. In figures 15(a) and (b), the estimated errors of $D_{chem}$ and $k_{chem}$ were calculated based on simulated experiments with ‘true’ values ranging from $10^{-5}$ to $10^{-4}$ cm$^2$ s$^{-1}$ for $D_{chem}$ and from $10^{-3}$ to $10^{-2}$ cm s$^{-1}$ for $k_{chem}$. In the case of $D_{chem}$ ($\bar{D}$) = $10^{-5}$ cm$^2$ s$^{-1}$ and $k_{chem}$ ($\bar{k}$) = $10^{-2}$ cm s$^{-1}$, the error on $k_{chem}$ is very large, whereas that on $D_{chem}$ is small. Concomitantly, one observes that the sensitivity to the ECR data is much smaller for $k_{chem}$ than for $D_{chem}$ for the same experimental conditions (figure 15(c)). Decreasing sample size enables a better compromise between the sensitivity with respect to both parameters (figure 15(d)).
### 4.1.3.2. Solving identifiability issues experimentally

From these analyses, two main strategies stand out in the literature to improve parameter identifiability and reduce fitting errors experimentally. The first one consists of sample geometry optimization to obtain most favourable sensitivity on $D_{chem}$ and $k_{chem}$ at the same time. The second strategy consists of a two-step determination of both parameters using conditions where diffusion or surface exchange is the rate limiting process in order to yield an accurate value of each value independently [161].

#### – Forced $D_{chem}$ limited regime

In order to force the $D_{chem}$ limiting regime, the sample thickness can be increased, but such a strategy is limited to the realistic dimensions that should fit in the reactors. Moreover, a thicker sample also yields longer relaxation times, which increase the overall duration of the experiment. Another strategy to force the $D_{chem}$ limiting regime is to modify the surface of the sample, leading to high surface area and increased surface reaction rate [161, 166, 169]. In the $D_{chem}$ limiting regime, the Biot number increases such that:

$$L_{\alpha} (\text{or } Bi) = \frac{l \times k_{\text{chem}}}{D_{\text{chem}}} \rightarrow \infty. \qquad (4.10)$$

Then, a simple solution becomes: $\beta_n = \pi/2$. Thus, the normalized conductivity can be simplified from equation (4.6), giving:

$$\bar{\partial}(t) = \frac{\partial(t) - \partial_0}{\partial_{\infty} - \partial_0} = 1 - \sum_{n=1}^{\infty} \frac{8}{(2n+1)^2 \pi^2} \exp\left(-\frac{(2n+1)^2 \pi^2 D_{\text{chem}} t}{4l^2}\right). \qquad (4.11)$$

For instance, porous coatings were applied on dense ceramics of $La_{0.6}Sr_{0.4}CoO_{3-\delta}$ [169], $La_2NiO_4$ [166], and $La_{0.6}Sr_{0.4}Co_{0.2}Fe_{0.8}O_{3-\delta}$ [161] through deposition of a slurry of the same material followed by moderate temperature annealing, only to ensure correct mechanical strength and adhesion of the porous layer. The porous layer thickness is typically of a few micrometres and does not contribute to the diffusion rate, the diffusion pathway of the porous layer being much shorter than that of the dense material. A significant difference in the $D_{chem}$ values was obtained from fitting the data coming from the dense sample with and without the porous layer. Following the determination of $D_{chem}$, measurements using regular dense samples enabled the derivation of $k_{chem}$ values by fixing the $D_{chem}$ values during the fitting procedure. In all cases, the author showed an increased accuracy on the determination of both coefficients.

#### – Forced $k_{chem}$ limited regimes

In the case of studies focusing on surface exchange rates rather than diffusion coefficients, it is possible to reduce the dimensions of the samples to experimentally shift the relaxation problem to the strictly surface limited regime. With such a strategy, since the flux of oxygen in the sample is controlled by the surface, there is no concentration gradient build up inside the bulk of the sample and the relaxation is no longer a diffusion problem. Rather, it can be modeled by a simple first order reaction equation such as:

$$j = k_{\text{chem}} \cdot (c(t) - c_0). \qquad (4.12)$$

in which $j$ is the oxygen flux, $k_{chem}$ is the surface exchange coefficient, $c(t)$ is the concentration of oxygen in the sample at time $t$ and $c_0$ is the concentration of oxygen at the beginning of the relaxation experiment. Then, the time dependent equation for the conductivity transient is:

$$G(t) = \frac{\sigma(t) - \sigma_0}{\sigma_{\infty} - \sigma_0} \equiv \frac{c(t) - c_0}{c_{\infty} - c_0} = 1 - \exp\left(-\frac{t}{\tau}\right) = 1 - \exp\left(-\frac{A}{V} \cdot k_{\text{chem}} \cdot t\right) \qquad (4.13)$$

where $G(t)$ is the normalized conductivity, $\sigma$ is the conductivity, $\tau$ is the time constant associated with the reaction, and $A$ and $V$ are the area and volume of the sample, respectively.

In this case, identifiability issues are completely removed since the fitting model only has one fitting parameter, namely $k_{chem}$. Experimentally, decreasing the dimensions of the samples to the surface limited regime is not simple, since sintered ceramics become impossible to handle for thicknesses below hundreds of microns. Then, two main approaches to achieve a surface limited relaxation experiment can be considered: working with thin films, and working with porous ceramics, both of which have their advantages and drawbacks that are developed in the next section.
<figure><img src="image_19.png" /><figcaption>Figure 16. Surface exchange coefficients measured on thin films and dense ceramics by electrical conductivity relaxation. Data taken from refs Mosleh et al [170], Sogaard et al [175], Burriel et al [171], Bucher et al [176], Mérieau et al [177] and Egger et al [178].</figcaption></figure>

#### 4.1.3.3 Thin film

Thin films are a logical path to decreasing the dimensions of the sample since it allows a well defined geometry to be maintained. In the case of a thin film of a mixed conducting oxide on a substrate, the geometry can be considered as a slab with only the top surface participating to the reaction, since the edges are negligibly small and the bottom side is blocked by the substrate. Then, the relationship between the relaxation time and the oxygen exchange coefficient $k_{chem}$ reduces to:

$$k_{chem} = \frac{L}{\tau} \qquad (4.14)$$

with L the thickness of the film. Several examples of conductivity relaxation measurements on thin films are available in the literature. Mosleh et al measured the chemical surface exchange coefficient $k_{chem}$ of $La_{0.6}Sr_{0.4}FeO_{3-\delta}$ thin films prepared by pulsed laser deposition [170]. The main advantage of thin films over bulk dense ceramics is the possibility of obtaining single crystal surfaces (epitaxial growth) that eliminate the potential influence of grain boundaries on the reaction rate. For example, Burriel et al used a frequency resolved conductivity relaxation technique to measure the $k_{chem}$ coefficients of $Ba_{0.5}Sr_{0.5}Co_{0.2}Fe_{0.8}O_{3-\delta}$ epitaxial films [171]. Kim et al used conductivity relaxation to measure $k_{chem}$ on epitaxial $PrBaCo_2O_{5+\delta}$ [172] and $La_2NiO_{4+\delta}$ [173]. Although interesting, the thin film approach to conductivity relaxation has a significant drawback that has limited its widespread use. In the two studies by Kim and coworkers [172, 173], the simple reaction model with one time constant proves insufficient to correctly describe the relaxation transients. The authors used a combination of two relaxation time constants and attributed such behaviour to heterogeneity of the surface. Skiba et al proposed a thorough analysis of conductivity relaxation measurements on $SrTi_{0.65}Fe_{0.35}O_{3-\delta}$ [174], and suggested that the precious metal contacts could locally change the surface exchange rate in the vicinity of the contacts. The same authors also discuss the limits of the technique for resistive samples: since the sheet resistance is much higher than a bulk ceramic sample, the relaxation data becomes highly noisy at lower temperature, which increases the error in determining $k_{chem}$. Moreover, it appears that $k_{chem}$ determined on thin films tend to show lower surface exchange rate coefficients than their dense ceramic counterparts. Figure 16 shows a comparison of $k_{chem}$ determined on thin films as compared to bulk ceramics for three materials, namely $La_{0.6}Sr_{0.4}FeO_{3-\delta}$, $Ba_{0.5}Sr_{0.5}Co_{0.2}Fe_{0.8}O_{3-\delta}$, and $La_{0.6}Sr_{0.4}CoO_{3-\delta}$. For all three compositions, the surface exchange coefficient of thin films is lower than that of the dense ceramics, and the difference can be as high as one order of magnitude, e.g. for $La_{0.6}Sr_{0.4}FeO_{3-\delta}$.

#### 4.1.3.4. Porous materials

As thin film preparation requires specific and costly equipment, alternative ways of preparing samples with small diffusion lengths and to force the $k_{chem}$ limited regime have been proposed. The simplest way
to reduce diffusion distances experimentally is to prepare porous ceramics, for which the gas phase penetrates inside the pores and the oxygen exchange reaction takes place on the whole specific surface area of the ceramics. Concomitantly, the bulk diffusion distance is reduced to the size of each grain, which ensures that the sample remains in the surface limited regime.

Measuring porous samples by conductivity relaxation has been considered by several groups in the past. In the mid-2000s, Ganeshanathan and Virkar proposed the first thorough analysis of conductivity relaxation on porous ceramics, which laid the groundwork for future studies [179]. In such work, the microstructure of the porous body is described macroscopically using the porosity and the specific surface area of the sample. However, such an approach failed to interpret the relaxation transients correctly, since it assumed a perfectly homogeneous microstructure. More recently, two main approaches have been considered to better account for the sample microstructure: fitting the data with a distribution of relaxation times [180–182], or determining the distribution of grain size of the ceramics prior to the relaxation measurements to include such distribution into the relaxation-fitting model [183]. In the latter, the relaxation model includes the grain size distribution and a flushtime correction such as:

$$G(t) = 1 - \exp\left(\frac{t}{\tau_{\text{flush}}}\right) - \int_{0}^{\infty} f(R_n) \cdot \frac{\tau_n}{\tau_n - \tau_{\text{flush}}} \cdot \left[\exp\left(-\frac{2 \cdot k_{\text{chem}}}{R_n} \cdot t\right) - \exp\left(-\frac{t}{\tau_{\text{flush}}}\right)\right] dR_n \quad (4.15)$$

where $τ_{flush}$ is the flushtime constant, $f(R_n)$ is a grain size distribution function, and $τ_n$ is the time constant associated with the relaxation of a grain of radius $R_n$. In such an experiment, the grain size distribution can be determined experimentally by electron microscopy following a procedure called ‘local thickness’ thoroughly described in ref [183].

Since the dimensions of the grains for porous ceramics is well under the critical length, such samples enable the comparative study of the surface exchange coefficients for various surface modification strategies. Indeed, the influence of such surface modifications were more questionable in the case of dense samples since the modifications systematically change the Biot number and therefore the sensitivity of the transient to $k_{chem}$. Then, a number of recent studies have taken advantage of porous samples to study surface modification with various oxides. As compared to thin films, the surface modifications on porous ceramics can be achieved with simple infiltrations of precursor solutions that are calcined to form surface oxides in situ.

Such strategies have enabled the demonstration of the central role of surface acidity in determining the surface exchange coefficient of mixed conducting oxides, e.g. for Pr doped ceria [184, 185], Fe doped SrTiO$_3$ [186], or La(Sr,Ca)CoO$_3$ [187]. However, studies on porous ceramics have also yielded different values of $k_{chem}$ for the same compositions. Figure 17 shows discrepancies between $k_{chem}$ measured on porous samples by various authors for Pr$_{0.1}$Ce$_{0.9}$O$_{2-δ}$ [183], SrTi$_{0.65}$Fe$_{0.35}$O$_{3-δ}$ [186, 188], La$_{0.6}$Sr$_{0.4}$Co$_{0.2}$Fe$_{0.8}$O$_{3-δ}$ [181, 189], and La$_{0.6}$Sr$_{0.4}$CoO$_{3-δ}$ [183], exemplifying this issue. For La$_{0.6}$Sr$_{0.4}$CoO$_{3-δ}$ and La$_{0.6}$Sr$_{0.4}$Co$_{0.2}$Fe$_{0.8}$O$_{3-δ}$, a reference on dense ceramics is given for comparison, also exemplifying the discrepancies between dense and porous samples. From such observations, it becomes unclear which sample (dense or porous) yields the more reliable values of $k_{chem}$. More importantly, it suggests a strong influence of the surface morphology on oxygen exchange kinetics. This point is partially clarified by the study of Pr$_{0.1}$Ce$_{0.9}$O$_{2-δ}$ porous samples [183], in which a wide variety of microstructures were measured. All samples showed $k_{chem}$ values falling on either one of two lines (reproduced in figure 17), suggesting that surface morphologies can significantly affect $k_{chem}$. Then, the discrepancies in $k_{chem}$ between dense and porous samples simply demonstrate that the surface exchange coefficient is highly extrinsic in nature, contrary to the bulk diffusion coefficient.

### 4.1.4. Link between coefficients obtained with other measurement techniques

#### 4.1.4.1. Thermodynamic factor

Diffusion and reaction coefficients determined by conductivity relaxation are different from coefficients of self-diffusion or those determined by impedance spectroscopy. This difference arises from the various driving forces used in each case. In the case of chemical driving forces, the diffusion of both electronic and ionic defects is impacted by the chemical gradient, and their mutual interaction leads to an enhancement of rate constants that should be accounted for. The factor binding coefficients determined by chemical gradient or by tracer experiment is called the enhancement factor $W$ [191], but is sometimes referred to as the thermodynamic factor $γ$ by other authors [151] (equation (2.4)). The factor takes the general form:

$$W = t_e \frac{\delta \ln \mu_O}{\delta \ln C_O} \quad (4.16)$$
<figure><img src="image_20.png" /><figcaption>Figure 17. Chemical oxygen surface exchange coefficient $k_{chem}$ measured on porous samples by various authors for $Pr_{0.1}Ce_{0.9}O_{2.δ}$ [183], $SrTi_{0.65}Fe_{0.35}O_{3.δ}$ [186, 188], $La_{0.6}Sr_{0.4}Co_{0.2}Fe_{0.8}O_{3.δ}$ [181, 189], and $La_{0.6}Sr_{0.4}CoO_{3.δ}$ [183]. The references measured on dense samples are from ten Elshof et al [190] and Tripkovic et al [162].</figcaption></figure>

where $t_e$ is the electronic transference number, $μ_O$ is the chemical potential of oxygen, and $C_O$ is the oxygen concentration of the oxide. It is important to reinforce that this is not the most common expression of the enhancement (or thermodynamic) factor. In most work in the field of solid state ionics, $W$ is written in the form:

$$W (or \gamma) = \frac{\delta \ln \mu_O}{\delta \ln C_O} \qquad (4.17)$$

This difference is simply a consequence that most state-of-the-art mixed conducting oxides are overwhelmingly electronic conductors, in which case $t_e = 1$ and $W$ is reduced to equation (4.17). Weppner and Huggins [191] have described in detail a series of other cases and the related expression of the enhancement factor, and one has to carefully choose the right expression for $W$ depending on the mobile defects and their concentrations. Maier laid the theoretical groundwork to better understand how diffusion and reaction rate constants relate to each other [11, 13]. Since the chemical potential $μ_O$ is defined as:

$$\mu_O = \mu_O^0 + RT \ln \sqrt{pO_2} \qquad (4.18)$$

The enhancement factor can be expressed as a function of the oxygen partial pressure as:

$$W = -t_e \frac{1}{2} \frac{\delta \ln pO_2}{\delta \ln C_O} \qquad (4.19)$$

and it can be readily determined experimentally with thermogravimetric measurements or coulometric titrations. The link between diffusion coefficients and surface exchange coefficients are:

$$D_{chem} = W \cdot D^* \qquad (4.20a)$$

$$k_{chem} = W \cdot k^* \qquad (4.20b)$$

Equations (4.20a) and (4.20b) hold true only if oxygen defects are considered non-interacting and randomly distributed [192]. In addition, the enhancement factor for oxygen vacancies ($W_v$) is introduced in the literature [170, 193] to relate $D_{chem}$ to the vacancy diffusion coefficient $D_v$ as $D_{chem} = W_v \cdot D_v$ with $W_v$ defined by:

$$W_v = -t_e \frac{1}{2} \frac{\delta \ln pO_2}{\delta \ln \delta} \qquad (4.21)$$
<figure><img src="image_21.png" /><figcaption>Figure 18. Comparison between values of k (a) and D (b) obtained from ECR and IEDP measurements in the literature for different class of materials: kchem, Dchem values from ECR were obtained from refs. [194] for BSCF (Ba$_{0.5}$Sr$_{0.5}$Co$_{0.8}$Fe$_{0.4}$O$_{3}$$_{−}$$_{δ}$) [195], for LSCF (La$_{0.6}$Sr$_{0.4}$Co$_{0.2}$Fe$_{0.8}$O$_{3}$$_{−}$$_{δ}$) [192], for LSC82 (La$_{0.8}$Sr$_{0.2}$CoO$_{3}$$_{−}$$_{δ}$) [162, 196], for LSC64 (La$_{0.5}$Sr$_{0.2}$CoO$_{3}$$_{−}$$_{δ}$) and [197, 198] for LNO (La$_{2}$NiO$_{4}$$_{+}$$_{δ}$) and NNO (Nd$_{2}$NiO$_{4}$$_{+}$$_{δ}$); D* values were obtained from IEDP from refs. [199] for BSCF [112], for LSCF [44], for LSC64, and [40, 200] for LNO and NNO. Thermodynamic factors were obtained for refs. [176] for BSCF [195], for LSCF [192], for LSC82 [196], for LSC64 and [198, 201] for LNO and NNO.</figcaption></figure>

where δ is the oxygen non-stoichiometry. The two factors are then related by equation (4.22):

$$ \frac{W}{3 - \delta} = \frac{W_v}{\delta}. \qquad (4.22) $$

### 4.1.4.2. Comparison with isotopic exchange measurements

A comparison between k* and D* values obtained from IEDP measurements and from ECR measurements through the use of equations (4.20a) and (4.20b) was made for different reference materials, as shown on figure 18. Although good agreement between measurement methods is found for D* values, stronger discrepancies are observed concerning k* values. As explained above, the surface exchange coefficient is very sensitive to sample preparation (temperature used for thermal treatment, potential surface absorbed impurities, polishing treatment etc...), which can vary from study to study, and are sometimes difficult to control precisely.

In the case of La$_{1-x}$Sr$_x$CoO$_{3-δ}$ materials, W is found to be nearly independent of the oxygen partial pressure. Because D* is proportional to the oxygen stoichiometry, which is itself nearly independent of the pO$_{2}$ (3-δ ~ 3), it follows that Dchem should be poorly dependent on the oxygen partial pressure, in the range considered by the authors. Although this is observed for pO$_{2}$ > 10$^{-2}$, at lower pO$_{2}$, experimentally determined Dchem decreases, which was attributed by the authors to some degree of oxygen vacancy ordering, previously evidenced by x-ray diffraction (XRD) and high-resolution transmission electron microscopy (HRTEM) analysis [202, 203], thereby invalidating the use of equations (4.20a). Similar observations were found in lanthanum strontium ferrate (LSF) materials, with a stronger dependency of the surface exchange coefficient on the pO$_{2}$ compared to the diffusion coefficient in the 0.01-0.15 pO$_{2}$ range [193]. Additionally, in the case of oxygen over-stoichiometry, La$_{2}$Cu$_{0.5}$Ni$_{0.5}$O$_{4+δ}$, no clear link between W and pO$_{2}$ was observed [204].

### 4.1.5. Conclusion and perspective to proton conductors

To summarise, the electrical relaxation conductivity technique provides a means to determine the oxygen mass transport coefficient through the development of a physical model that relates electronic and ionic defects, which in turn allows for the deconvolution of electronic and ionic contributions to the total electronic conductivity in MIEC oxide materials. Using the correct experimental conditions and model, one can get three transport parameters (kchem, Dchem and σe) easily scanning different pO$_{2}$ and temperature ranges on the same sample without requiring the need for postmortem analysis, contrary to the IEDP method which is highly time consuming, and for which one sample is needed per temperature and pO$_{2}$. However, getting reliable and reproducible results is not trivial as identifiability problems are often encountered for the determination of kchem and Dchem. It is crucial to exercise great caution in
selecting suitable instrumentation and experimental conditions such as utilizing a small chamber volume, ensuring fast gas switching, and having small pO₂ step changes. Sample preparation also requires meticulous attention and should be tailored to the specific properties of the sample.

In addition, some authors have started adapting the technique to proton conducting ceramics and triple conducting oxides, starting from the work of Yoo et al [205-207]. Lim et al applied the technique to derive H⁺ and vO₂ transport coefficients in BaZr₀.₂Ce₀.₆₅Y₀.₁₅O₃-δ [208], BaZr₀.₆Ce₀.₂₅Y₀.₁₅O₃-δ [209], and BaZr₀.₄Ce₀.₄₅Y₀.₁₅O₃-δ [210]. Building on these studies, Falkenstein et al [211] expanded the mathematical model to define amplitude factors for both proton and oxide transport to diffusion controlled and surface exchange controlled regimes, and applied it to La₅.₄WO₁₁.₁. Lee et al [212] used the same technique to study BaPr₀.₉Y₀.₁O₃-δ and Ba₁-xCo₀.₄Fe₀.₄Zr₀.₁Y₀.₁O₃-δ.

Conductivity relaxation is a powerful technique to determine oxygen transport coefficients in oxide ceramics and films, but is not limited to the study of oxide ion transport which are expected to be with much broader possibilities.

## 4.2. Other ‘property-relaxation’-based methods

In addition to electrical conductivity, many measurable property changes such as lattice parameters by diffraction [213-217], volume changes by dilatometry [218], mass monitoring with a quartz microbalance [219], transmitted light intensity [220, 221], sample curvature [222] etc associated with uptake and release oxygen after changing the pO₂ or external stimuli such as an electric field [216, 217] (more details in section 4.5) can be applied to obtain oxygen mass transport kinetics.

For example, a change in lattice parameters as small as 2 × 10⁻⁴ Å with a time resolution of 10 s for La₂NiO₄+δ epitaxial films has been reported using lab x-ray by Moreno et al [213]. The obtained surface exchange kinetics kₚₚ of La₂NiO₄+δ is comparable to the results measured by ECR and the literature value using IEDP-SIMS [48]. Further work using lab time-resolved-XRD on an epitaxial GdBaCo₂O₅.₅+δ thin films verified the reliability of this method [214]. A key factor to perform this measurement is to have good time resolution aligned with the relaxation time, which requires good statistics of the peak intensity. This does not mean epitaxial films are necessarily required, though they do provide satisfying intensities of the diffraction peaks. Also to be noted, the pO₂ changes in these experiments are significant while a small pO₂ step is generally preferred in relaxation experiments.

Simultaneous optical absorption and EIS measurements were first reported on Pr₀.₁Ce₀.₉O₂-δ thin films, validating that optical absorption can be used to determine defect concentration [223]. Optical transmission relaxation (OTR) [220, 221] was then applied to obtain oxygen surface exchange kinetics, and compared to the EIS method, the continuous and contact free OTR method is also free from degradation effects from current collectors.

In general, all these novel approaches complement the well-developed conventional methods, providing significant insights into the transport studies, while require a measurable property during relaxation—chemo-mechanically active or optical active, thus only have been applied for limited samples on individual bases and remain to be exploited.

## 4.3. Oxygen permeation measurements

The oxygen permeation method is a common technique in the literature to determine oxygen mass transport kinetics, particularly for MIEC materials with oxygen permeation membrane applications [7, 224]. In such experiment, oxygen transport is also driven by oxygen partial pressure differences. Different from relaxation-based experiments, there is no switching of surrounding atmosphere with different pO₂ in permeation experiments while it relies on the different pO₂ across the membrane, as shown in figure 19(a).

It measures the oxygen flux JO₂ through an oxygen permeation ceramic membrane under a chemical gradient, given by equation (4.23) based on Wagner's theory:

$$J_{O_2} = \frac{RT}{16F^2L} \int_{P'_{O_2}}^{P'_{O_2}} \sigma_{amb} d(\ln P^*_{O_2}) = \frac{RT\sigma_{amb}}{16F^2L} \ln \left( \frac{P'_{O_2}}{P''_{O_2}} \right) \quad (4.23)$$

where σₚₚ is the ambipolar conductivity of the membrane related to the electronic conductivity (σₑ) and ionic conductivity (σᵢ) shown in equation (4.24), R and F are the gas constant and Faraday constant, respectively, L is the membrane thickness, P'₀ and P''₀ are oxygen partial pressure on the rich and lean side, respectively

$$\sigma_{amb} = \frac{\sigma_i \sigma_e}{\sigma_i + \sigma_e} \quad (4.24)$$
<figure><img src="image_22.png" /><figcaption>Figure 19. (a) Oxygen chemical potential gradient across an oxygen separation ceramic membrane. Adapted with permission from [55]. CC BY-NC-ND 4.0. (b) Schematic of a 3-chamber set-up for the oxygen semi-permeation measurement. Reproduced from [225]. © IOP Publishing Ltd. All rights reserved.</figcaption></figure>

It is clear the measured oxygen flux is determined by the ambipolar conductivity, membrane thickness and the partial pressure difference. Given that electronic conductivity is much higher than the ionic conductivity of MIEC materials ($\sigma_e \gg \sigma_i$), the ambipolar conductivity tends to be ionic conductivity ($\sigma_{amb} \approx \sigma_i$). Thus, the ionic conductivity $\sigma_i$ is the rate determining factor considering the charge neutrality needs to be maintained. Combined with the Nernst-Einstein relation which connects conductivity and diffusivity (D) and on the assumption that $\sigma_i$ does not vary under different $pO_2$, equation (4.23) can be rewritten as equation (4.25):

$$J_{O_2} = \frac{DC_O}{4FL} \ln \left( \frac{P'_{O_2}}{P''_{O_2}} \right) \quad (4.25)$$

where $C_O$ is the oxygen concentration in the solid. For the application of OTMs, higher oxygen flux can be achieved by increasing the oxygen diffusivity of the materials, increasing the partial pressure difference across the membrane and decreasing the thickness of the ceramic layer. The latter is often true until reaching a point where the thickness is at the critical (or characteristic) length $L_c = D/k$ which is the reciprocal value of h as defined in section 3.1 IEDP-SIMS. Under this situation, the oxygen permeation process is controlled by both surface exchange and bulk diffusion and equation (4.23) will be further modified as equation (4.26) with a factor of $(1 + \frac{2L_c}{L})$ [226]:

$$J_{O_2} = \frac{DC_O}{4FL} \ln \left( \frac{P'_{O_2}}{P''_{O_2}} \right) \cdot \frac{1}{1 + \frac{2L_c}{L}} = \frac{Dk}{Lk + 2D} \frac{C_O}{4F} \ln \left( \frac{P'_{O_2}}{P''_{O_2}} \right). \quad (4.26)$$

More limiting cases have been discussed and provided by Sunarso et al [7]. For semi-permeation experiments, similar to the IEDP methods, dense pellets without percolated open porosity are required and a well-sealed system is also essential while technically challenging to achieve. Failure to meet either of these requirements can lead to a short-circuited oxygen pathway, resulting in an undesirable overestimation of the oxygen flux, and consequently, the ion diffusivity.

In the literature, various chamber configurations have been designed to measure oxygen flux such as two-chamber [227–229], three-chamber [225, 230] and internal-chamber set-ups [231, 232]. Among these, the three-chamber configuration is preferred due to its relatively simple design compared to the more complex internal-chamber set-up and because it generally results in less leakage through O-rings than the two-chamber arrangement [225]. A schematic illustrating semi-permeation measurements using the three-chamber configuration is shown in figure 19(b).

Reviews of the literature data [7, 224] reveal considerable discrepancies in semi-permeation measurements, which are closely related to the experimental conditions and sample preparation processes. Geffroy et al [225] summarised several experimental factors that significantly affect the accuracy of the results, including the diameter-to-thickness ratio, the sealing (O-ring) materials and the presence of the $pO_2$ gradient on the surface.

Furthermore, a method to determine the oxygen surface exchange coefficient (k) for both incorporation and desorption was proposed using the oxygen semi-permeation technique [233]. For LSCF materials, the k values measured by this method are lower than those obtained using the IEDP method.
This difference is attributed to the actual physicochemical properties of the surfaces and whether measurements are performed under chemical equilibrium conditions. Although this method has so far been applied to a limited range of materials, broader application could provide more valuable insights. A key advantage of this method is its ability to determine both *D* and *k* (incorporation and desorption) on samples under realistic conditions for oxygen permeation membrane applications.

### 4.4. Electrochemical impedance spectroscopy-based methods

Oxygen mass transport kinetics are often determined using electrochemical measurements such as DC measurements, impedance spectroscopy (EIS) etc, particularly for materials with SOFC applications. Common DC measurements can be performed using electronic (or ionic) blocking design [234], the Hebb-Wagner polarisation method [235], etc. Additionally, numerous related studies have been reported in the open literature using EIS [236–242]. The essential requirement for applying this technique is the accurate interpretation of impedance spectra to ensure the validity of the extracted parameters. Practical guidelines and guidelines for EIS measurements are also widely available [243–246] including those focusing on reliable measurements related to SOFCs [247]. Therefore, in this section, only a very brief discussion is provided to compare the electrical and tracer parameters obtained from EIS and IEDP.

For MIEC materials as electrodes of SOFCs, by applying a small AC voltage, an area specific resistance $R_s$ of the oxygen surface exchange reaction can be determined, thus yielding the electrical oxygen exchange coefficient ($k_q$) [11–13, 237], as shown in equation (4.27):

$$k^q = \frac{k_b T}{(ze)^2 C_o R_s} \qquad (4.27)$$

where $C_o$, $k_b$, T and (ze) are standard symbols as defined in equations (2.4) and (2.6) in this article. Similarly, upon the Nernst-Einstein relation, oxygen bulk diffusion coefficients can be estimated using equation (2.5).

For oxygen surface exchange, the electrical surface exchange coefficient ($k^q$) is often assumed to be approximately equal to the tracer exchange coefficient ($k^*$) [13]. This assumption is empirically valid in most cases after applying minor corrections. For example, in LSC/YSZ thin films at $pO_2 = 200$ mbar under oxygen atmospheres [248], a surface exchange correlation factor similar to the correlation factor of diffusion coefficients (0.69 for perovskites [21]) has been reported, with its value depending on the exact elementary mechanisms. The importance of applying an appropriate fitting model to achieve accurate interpretation of mass transport via impedance analysis has also been emphasised by Kubicek et al [248]. A good agreement in ionic conductivity between EIS and IEDP-SIMS measurements has been reported for $La_2(Mo/W)_2O_9$-based oxygen ionic conducting materials [94, 249] under both dry and wet conditions.

However, Nenning et al [99] compared $k^q$ and $k^*$ for MIEC $La_{0.6}Sr_{0.4}FeO_{3-\delta}$ and $SrTi_{0.3}Fe_{0.7}O_{3-\delta}$ thin films where EIS was performed during the isotopic exchange experiments on the same sample. Their study demonstrated that the assumption of a 1:1 correlation between $k^q$ and $k^*$ generally holds in a dry oxidising atmosphere but can be dramatically inaccurate under humid oxidation or humid reducing conditions.

Overall, only limited reports have directly compared the electrical and tracer parameters under identical conditions. Caution should be exercised when assuming $k^q / k^* \approx 1$ as even trace amounts of residual humidity can significantly increase $k^*$, as highlighted by Nenning et al [99]. In addition, a general concern regarding electrochemical measurements is the effect of the Pt current collector on oxygen surface exchange, as it has been observed that oxygen exchange coefficients can be significantly altered when Pt remobilised to the sample surface [221, 250]. Therefore, it is always beneficial to compare the obtained results with measurements conducted free from a current collector.

### 4.5. Electrochemical polarised measurements

Considering the applications of most oxygen ion conducting materials such as SOCs, there is a clear motivation to study oxygen mass transport in understanding of both the kinetics and the transport pathways under polarisation.

Early studies [251–254] using oxygen isotopic exchange and SIMS analysis at the $O_2/LSM$ mesh pattern (Au)/YSZ interface, revealed the significant impact of the electric field on oxygen exchange. These studies identified the triple phase boundaries (TPBs) as the active sites for oxygen reduction. Although isotopic exchange combined with SIMS imaging on LSM/YSZ samples has proven to be a useful technique for investigating the distribution of reaction sites, the resulting images were largely qualitative rather than providing unambiguously quantitative results, such as TPB parameters.
Subsequently, an expansion of the $^{18}$O incorporation zone with increasing polarisation on Pt/YSZ films was observed [255] and more pronounced bulk transport pathways in LSM/YSZ films under higher overpotential was evidenced [256]. Fleig et al [257] further elucidated the conditions under which a quantitative analysis of TPB width under polarisation becomes possible using finite element modelling.

In terms of transport kinetics, the effective surface exchange coefficient ($k$) can be dramatically enhanced by an applied potential. A linear relationship between the effective $k$ and current density was provided by Fleig [258]. This tunability of $k$ is particularly useful for extending the measurable range of oxygen transport parameters using the tracer exchange methods, especially in thin films of fast ion conductors. For example, a study on YSZ thin films demonstrated that its inherently slow oxygen surface exchange rate does not limit the accurate determination of the diffusion coefficients ($D^*$) [259].

In addition, the effect of cathodic bias on both oxygen surface exchange and bulk diffusion has been studied in materials of various types, one example being the perovskite-structured MIEC LSM [260, 261]. Under polarisation, tracer oxygen diffusion profiles in LSM thin films present an unusual shape—an uphill-like diffusion. This behaviour is potentially attributed to fast diffusion along grain boundaries and polarised stoichiometry under potential [260].

While all the aforementioned studies were performed on thin films using oxygen isotope exchange combined with SIMS analysis, investigations of oxygen transport are certainly not limited to this approach. A significant improvement of dry oxygen surface exchange rate for the bulk BIMEVOX family of compounds under current flow has been reported by Vannier et al [262]. As briefly discussed in section 3.4 Gas phase analysis, polarised oxygen isotope exchange has also been conducted with gas phase analysis [136, 137, 263]. Cohn and Wachsman [263], using isothermal isotope exchange, also observed a dramatic increase in $k$ under cathodic polarisation and different behaviour in $k$ for LSC and LSF powders with and without polarisation, emphasising the importance of studying oxygen transport behaviour under realistic conditions. Nau et al [136, 137] further explored polarisation effects on different symmetric cells, identifying the rate limiting steps of oxygen surface exchange. Their work demonstrated a clear enhancement in the surface transport pathway, while the bulk pathway remained largely unaffected.

As highlighted by many studies, investigating oxygen transport under polarisation is of great importance. This research can be performed versatilely, which is not able to be summarised into a standardised protocol. One can always perform the studies using one or more technique(s) under polarisation and different atmosphere conditions to gain realistic insights into oxygen mass transport behaviour in practical applications.

# 5. Computational related techniques

Computational simulations have long been used to probe oxygen transport processes and mechanisms since the early ages of oxygen transport studies in the 1980s [264, 265]. These methods, being atomistic by nature, can explore local defects and individual processes where the experimental techniques cannot easily determine or decouple. Techniques such as classical or ab initio molecular dynamics (MD) and density functional theory (DFT) simulations provide complementary insights into defect energetics, migration barriers, diffusion coefficients, and the influence of dopants, temperature, and interfaces.

## 5.1. Molecular dynamics

MD simulates the diffusion process by using Newtonian mechanics, given the starting positions, velocities and a set of interatomic potentials of the atoms studied. The interatomic potentials may be empirically determined via fitting against a set of model systems experimentally measured (Classical MD) or calculated on-the-fly using Ab Initio simulation method (AIMD). Typical MD simulation models may contain 10 000s to 100 000s of atoms and therefore can capture certain statistical features like the diffusion coefficient $D$ but are still relatively local compared to the physical properties measured with experimental techniques. These simulations can run over nano-second timescales and are conducted at a temperature similar to the operational temperature in experiments. Due to the short process timescale simulated, a temperature ~200–400 K higher than the operational temperature may be used to study an accelerated diffusion.

In practice, the mean squared displacements (MSD) of the oxygen atoms can be directly taken from the MD simulation results, and on the assumption of isotropic diffusion, the ionic diffusivity $D$ is linked to the MSD by:

$$MSD = 6Dt. \qquad (5.1)$$
By performing simulations at multiple temperatures and plotting the D–T graph, the diffusion activation energy $E_a$ at this temperature range can be calculated with the Arrhenius equation:

$$D = D_0 \exp \left( \frac{E_a}{k_B T} \right) \qquad (5.2)$$

where $D_0$ corresponds to the maximum diffusion coefficient at infinite temperature, $k_B$ is the Boltzmann constant and $T$ is the temperature. It is important to note that, these two calculated values are not directly comparable to experimentally determined values, due to the many differences between the ideal simulation model and the imperfect real-world sample [266]. What the simulation results provide is how the oxygen diffusion behaviour may change due to a varied factor, e.g. defect location, defect ordering, local strain and so on. For example, the impact of yttria content on defect clustering and oxygen ionic mobility in YSZ was determined by Devanathan et al [267], and the effect of strain on the oxygen diffusivity in CGO was studied via MD simulation by Rushton et al [268]. In both cases, the possibility to precisely adjust local atomic defects and orderings without affecting other factors are essential to decouple and study how one particular factor may change the oxygen diffusion.

The trajectories of the oxygen atoms are commonly plotted from the MD results, which provides a very intuitive visualisation of the diffusion pathway. Oxygen diffusion pathways related to specific structural features were determined in different materials like the Ruddlesden-Popper type $La_2NiO_{4+\delta}$ [269] and the Fergusonite type $CeNbO_{4+\delta}$ [270], which would be very challenging by experiments alone.

The success of MD simulations depends on the accuracy of the force fields or interatomic potentials employed. The various potentials used by Classical MD are fixed throughout the simulation and therefore cannot capture electronic change of the ions during the diffusion process. Ab initio molecular dynamics (AIMD), based on on-the-fly DFT calculations, offers higher accuracy but is computationally expensive, limiting simulations to small systems and short timescales. Due to the cost, the AIMD is more commonly used at the surface, where the system scale is small and the electronic interaction is important, e.g. the study of oxygen surface exchange on the surface of NiO by Sugiura et al [271] and the study of oxygen ion diffusion in oxide-based resistance switches by Xiao et al [272].

## 5.2. Density functional theory simulation

The DFT simulation is an ab initio simulation method based on the time independent Schrodinger equation. It is a ground-state or zero Kelvin method widely used to calculate the energetics associated with point defects—such as oxygen vacancies, interstitials, and dopant-vacancy complexes—and to identify oxygen migration barriers via transition state method like the nudged elastic band.

Compared to MD, the DFT simulations give more accurate energetics and reveal detailed electronic interactions during individual oxygen transport steps and, therefore, is a great tool for studying mechanistic details. However, the DFT method is static, at 0 K and limited to very small system size (100 ~ 200 atoms). It cannot capture features due to long-range order or elevated temperature [273] and is often used together with MD or experimental techniques to complement each other.

The quality of DFT simulation is highly dependent on the input structure and will excel when the target question is very specific. The effect of local strain on the oxygen ion hopping in YSZ is such a well-defined question. Starting with a very well-studied structure and a precisely designed atomic strain model, Kushima et al utilized DFT simulation to reveal the oxygen hopping barrier correlates with two competing processes associated with lattice strain, namely the migration space and the oxygen—cation bond strength [274].

For materials containing strongly correlated metals, e.g. Fe, Ni or Co, a careful selection of the DFT implementation is required to obtain a good accuracy of the transport energetics. These methods including the usage of the Hubbard U correction (commonly noted as DFT +U) [275] and the usage of the hybrid functionals (e.g. PBE0) [276] or range-separated hybrid functional (e.g. HSE06) [277]. The DFT +U method is semi-empirical and provides qualitatively accurate energetics but has a significantly lower computational cost compared to the hybrid functional approaches. For materials like LSF series, Ritzmann et al used the DFT +U method to identify that the holes introduced by the Sr dopants are mainly compensated by the oxygen vacancies but not by other potential defects and therefore, increasing oxygen vacancy formation and transport [278]. However, with the development of more powerful computing infrastructure, the usage of the hybrid functionals in oxygen transport simulation is becoming more common.
### 5.3. Emerging trends: machine learning and high-throughput screening

Recent advances in data science and machine learning (ML) have begun to transform the landscape of atomistic simulations. ML potentials, like the neural network-based force fields, may achieve near-DFT accuracy while enabling simulations of larger systems over longer timescales [279].

Simultaneously, high-throughput DFT workflows, such as those implemented by Meng *et al* allow for rapid screening of materials with high oxygen ion conductivity by evaluating migration barriers, defect formation energies, and electronic structure properties across thousands of compounds [280]. These data sets, coupled with ML-based descriptors, informatics tools and automated experiments, are facilitating the discovery of novel oxide materials with tailored oxygen transport properties for next-generation energy devices.

## 6. Other directions

The methods used to probe oxygen transport are certainly not limited to the aforementioned ones and are continuously being developed. Some isotope-sensitive techniques coupled with IEDP methods can be electron energy loss spectroscopy, and ion scattering spectroscopy, particularly LEIS. In this method, ions with known masses and energies are used as projectiles, enabling differentiating isotopes, though it mainly focuses on (sub)surface regions [281, 282].

Although most experimental approaches provide only macroscopic information on oxygen mass transport kinetics, ¹⁷O NMR can identify different crystallographic oxygen sites and their different roles in diffusion, as well as estimate localised oxygen exchange rates between sites, for example in Aurivillius-type 2D phases [30]. When combined with DFT calculations, this technique further demonstrates its capability of understanding the local structure and surface chemistry of nanosized ceria with $H_2^{17}O$ [283] and the enhanced oxygen mobility at the interface of CeO₂-SrTiO₃ vertically aligned nanocomposite thin films [31].

Additionally, different external stimuli for oxygen surface exchange are being explored such as UV light [174, 284]. For example, recently Skiba *et al* [174] reported a novel UV-driven oxygen surface exchange in $Sr(Ti,Fe)O_{3-δ}$ thin films, compared with the conventional $pO_2$-driven oxygen transport. The oxygen surface exchange kinetics obtained using OTR for both cases are in good agreement, and the study also shows that above-band-gap illumination results in oxidation of STF films.

## 7. Conclusions

Over the last half-century, there have been major advances in techniques for probing oxygen mass transport in solid state oxide materials. A wide variety of methods have been developed and continue to be developed, each with its inherent merits and limitations. Some techniques, such as IEDP-SIMS have proven to be indispensable for unambiguously determining ion transport ($D^*$ and $k^*/k$) in solid state systems, while others, such as ECR enable rapid screening of $D_{chem}$ and $k_{chem}$ under varying temperatures and oxygen partial pressures. It is also evidenced that the oxygen diffusion coefficients ($D$) obtained from these experiments generally show good agreement, whereas a significant scatter in the surface exchange coefficients ($k$) is still observed. This variability highlights the critical role of surface chemistry, the need for more systematic studies of its evolution and the associated exchange mechanisms under realistic operating conditions. Anticipatedly, the development of more accessible, reliable and practical techniques is still crucial to achieve a comprehensive understanding of oxygen mass transport. Meanwhile, the real challenges remain at conducting a meaningful measurement, understanding what to be measured and bridging between the measured kinetics and the actual performance of materials in real devices.

### Data availability statement

All data that support the findings of this study are included within the article (and any supplementary files).

### Acknowledgment

This work has received funding from the Horizon Europe programme under the Marie Skłodowska-Curie Grant Agreement No. 101064349 (NATFOX). This article is supported by the National Research Foundation, Prime Minister's Office, Singapore under its Campus for Research Excellence and Technological Enterprise (CREATE) programme.
## Conflict of interest

The authors declare that there is no conflict of interest.

## ORCID iDs

Zonghao Shen ID 0000-0002-0538-8409

Jacinthe Gamon ID 0000-0002-0888-4248

Clément Nicollet ID 0000-0002-8715-906X

Ji Wu ID 0000-0003-3938-8834

Jean-Marc Bassat ID 0000-0002-0489-6399

Mónica Burriel ID 0000-0002-7973-7421

Stephen J Skinner ID 0000-0001-5446-2647

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