Design of Mixed Ionic-Electronic Materials for Permselective Membranes and Solid Oxide Fuel Cells Based on Their Oxygen and Hydrogen Mobility

Abstract

Oxygen and hydrogen mobility are among the important characteristics for the operation of solid oxide fuel cells, permselective membranes and many other electrochemical devices. This, along with other characteristics, enables a high-power density in solid oxide fuel cells due to reducing the electrolyte resistance and enabling the electrode processes to not be limited by the electrode-electrolyte-gas phase triple-phase boundary, as well as providing high oxygen or hydrogen permeation fluxes for membranes due to a high ambipolar conductivity. This work focuses on the oxygen and hydrogen diffusion of mixed ionic (oxide ionic or/and protonic)–electronic conducting materials for these devices, and its role in their performance. The main laws of bulk diffusion and surface exchange are highlighted. Isotope exchange techniques allow us to study these processes in detail. Ionic transport properties of conventional and state-of-the-art materials including perovskites, Ruddlesden–Popper phases, fluorites, pyrochlores, composites, etc., are reviewed.

1. Introduction

One of the major important challenges facing modern society is the necessity to search for renewable and environmentally friendly energy sources [1,2,3,4,5,6]. The progressive trend in alternative energy research is directed towards the intensive development of hybrid configuration systems that combine multiple energy sources and power systems to maximize the efficiency of energy production, use and storage [7,8,9,10,11,12]. Current polygeneration technologies include, among others, the combination of the solid oxide fuel cell (SOFC) as a promising source of direct energy production [13,14,15,16] with batteries, gas turbines, vehicles, heat systems, desalination systems, and hydrogen production systems [17,18,19,20,21,22,23]. The hybrid systems of SOFCs with proton-exchange membrane fuel cells [18,20,24] and waste-to-energy systems based on biofuels [25,26] should be mentioned separately.

The environmentally and economically efficient production of hydrogen and syngas for the above-mentioned polygeneration systems [27] and the dominant strategy to reduce the operating temperature of SOFCs [28] require the development of high-performance construction materials. They can be used for the design of electrodes [29,30,31,32,33,34,35] and electrolytes [36,37,38,39,40,41] in SOFCs and for the design of gas separation membranes [42,43,44,45,46] as well as in catalytic membrane reactors [47,48,49]. Oxide materials with mixed ionic-electronic conductivity (or mixed ionic electron conductors, MIECs) are considered as a prospective class of multifunctional materials and are widely investigated for their use in both SOFCs and membrane reactors [50,51,52,53,54,55,56].

Adler et al. [57] and Sadykov et al. [58] showed that the performance of SOFCs and permselective membranes, both based on MIECs, correlates with the oxygen mobility and surface reactivity, which can be characterized by the values of the oxygen self-diffusion coefficient and surface exchange constant. According to the Adler–Lane–Steele model, the electrode activity is defined both by the values of the oxygen self-diffusion coefficient and the surface exchange constant of the electrode material and by the microstructure characteristics [57,59]. In contrast to cathodes made of materials with dominant electronic conductivity, whose performance is limited by the triple-phase boundary (TPB), the performance of MIEC cathodes is limited by the double-phase boundary (DPB). It stimulates the oxygen reduction reaction at the cathode and enhances ion transport to the electrolyte [58,60,61,62].

Materials with mixed ionic-electronic conductivity are successfully applied in the design of oxygen separation membranes in catalytic membrane reactors to separate oxygen from air, while the oxygen is used in the oxidation of biofuels (methane, ethanol, etc.) to produce syngas as fuel for SOFCs. An asymmetric configuration of the membrane design includes a porous substrate and a gas-tight permselective layer, thus ensuring increased mechanical strength combined with a high oxygen flux [58]. High oxygen fluxes across the membrane and superior performance in catalytic reactions could be achieved if the materials used for the oxygen separation membranes (or for the permselective layer in the case of the asymmetric configuration) have a high mixed ionic-electronic conductivity [56,59].

The unique review work by Manthiram et al. [63] provides a comprehensive overview of the correlations between the chemical composition, crystal structure, transport properties and electrochemical performance of the mixed ionic-electronic oxide conductors that could be used in both SOFCs and membrane reactors. Oxides with the perovskite structure ABO₃, layered perovskites LnBaCo₂O_5+δ, Ruddlesden–Popper phases (La,Sr)_n+1M_nO_3n+1 and hexagonal phases RBa(Co_1−yM_y)₄O₇ have been widely discussed in the review. Meanwhile, it is worth noting that review [63] was published in 2011.

A global search in the scientometric database Scopus with the query {mixed ionic electron conductor} yielded 535 documents of the type of Article and Review, indexed in the period from 1986 to June 2023. Figure 1, constructed with the software package VOSviewer version 1.6.19 [64] considering a minimum number of occurrences equal to 20 author keywords, visualizes the map with thematic clusters related to MIECs according to the query {mixed ionic electron conductor}.

The graphical data shown in Figure 1 illustrate that the author keywords in the documents referring to MIECs can be divided into three thematic clusters: the green cluster focusing on SOFCs, the red cluster focusing on oxygen membranes, and the blue cluster focusing on electrical conductivity. It can be said that the blue cluster highlights the fundamental property of MIECs—the presence of ionic and electronic conductivity—while the red and green clusters characterize the applications of MIECs. The green cluster summarizes the electrochemical activity of MIECs as the anode, cathode, including composite, and electrolyte materials, and the electrochemical performance of MIEC-based SOFCs in general. The red cluster generalizes the topics related to oxygen transport in MIEC-based membranes: oxygen mobility, oxygen permeability, oxygen diffusion, surface reactivity and ion exchange. Thus, published documents on the mixed ionic-electronic conductors can, therefore, be grouped under the three research themes mentioned above.

Among the highly cited documents in the MIEC search list in the Scopus database, the reviews highlighting the applications of MIECs as the anode [65,66] and cathode [67,68] materials for SOFCs and as the ceramic separation membranes [50,51,52] were found. However, it is worth noting that all the above reviews were published up to the year 2017. Therefore, the aim of the present review is to summarize recent studies on the oxide materials with mixed ionic-electronic conductivity, such as perovskites, fluorites, Ruddlesden–Popper phases, pyrochlores, and composites, focusing on their ion transport properties: oxygen and hydrogen diffusion, oxygen and hydrogen isotope exchange, and oxygen and hydrogen mobility. The applications of MIECs in SOFCs as cathodes and as the functional layers of the oxygen separation membranes are also highlighted.

2. Importance of Oxygen and Hydrogen Transport Properties for the Performance of Membranes and SOFCs

2.1. Oxygen Separation Membranes

High oxygen mobility and surface reactivity as well as a high electronic conductivity are the crucial characteristics of oxygen separation membrane materials required for achieving high oxygen permeation fluxes. The oxygen bulk diffusion enables oxide ions’ transport across the membrane, while the oxygen surface exchange enables oxygen adsorption/desorption. Since the oxide ions’ transport across the membrane is coupled with the electron transport, a high electronic conductivity is required as well (Figure 2). This allows to use such membranes for pure oxygen production as well as a part of catalytic membrane reactors for fuels transformation reactions [53,55,58,69,70,71,72,73,74,75,76].

The oxygen permeation flux across the membrane $\left(j_{{\mathrm{O}}_2}\right)$ obeys the Wagner equation (Equation (1)):

$$j_{{\mathrm{O}}_2}=-\frac{RT}{16F^{2}L}{\displaystyle \underset{\ln P_{{\mathrm{O}}_2}^I}{\overset{\ln P_{{\mathrm{O}}_2}^{II}}{\int }}\frac{{\sigma }_O{\sigma }_{el}}{{\sigma }_O+{\sigma }_{el}}}d\ln P_{{\mathrm{O}}_2},$$

(1)

where F is the Faraday constant, L is the membrane thickness, $P_{{\mathrm{O}}_2}^I$ and $P_{{\mathrm{O}}_2}^{II}$ are the oxygen partial pressures at different sites of the membrane, and σ_O and σ_el are oxide-ionic and electronic conductivity, respectively [70]. In MIEC materials, σ_O << σ_el. If the ionic conductivity is constant across the entire membrane thickness, Equation (1) can be simplified as follows (Equation (2)):

$$j_{{\mathrm{O}}_2}\cong -\frac{RT}{16F^{2}L}{\sigma }_O\ln \frac{P_{{\mathrm{O}}_2}^{II}}{P_{{\mathrm{O}}_2}^I}.$$

(2)

For MIEC membrane materials with oxygen nonstoichiometry depending on the oxygen partial pressure proportional to $P_{{\mathrm{O}}_2}^n$, the Nernst–Einstein equation of their ionic conductivity (Section 3) can be re-written as follows (Equation (3)):

$${\sigma }_O=\frac{4F^2}{RT{V}_m}{D}_V{\delta }_0{P}_{{\mathrm{O}}_2}^n,$$

(3)

where δ₀ is the oxygen nonstoichiometry at the reference oxygen pressure (1 atm), V_m is the molar volume of the oxide, D_V is the oxygen vacancy’s self-diffusion coefficient. Combining this with Equation (1) and assuming σ_O << σ_el one can obtain Sievert’s law (Equation (4)),

$$j_{{\mathrm{O}}_2}=-\frac{D_V{\delta }_0}{4V_{m}L}{\displaystyle \underset{\ln P_{{\mathrm{O}}_2}^I}{\overset{\ln P_{{\mathrm{O}}_2}^{II}}{\int }}{P}_{{\mathrm{O}}_2}^n}d\ln P_{{\mathrm{O}}_2}=\left(\frac{A}{L}\right)\left({\left(P_{{\mathrm{O}}_2}^I\right)}^n-{\left(P_{{\mathrm{O}}_2}^{II}\right)}^n\right),$$

(4)

where $A=\frac{D_V{\delta }_0}{4V_{m}n}.$

Considering the effect of the surface exchange of oxygen, the Wagner Equation (2) is transformed into the modified Wagner equation introduced by Bouwmeester et al. (Equation (5)) [77]:

$$j_{{\mathrm{O}}_2}=\frac{1}{1+\frac{2L_C}{L}}-\frac{RT}{16F^{2}L}{\sigma }_O\ln \frac{P_{{\mathrm{O}}_2}^{II}}{P_{{\mathrm{O}}_2}^I}.$$

(5)

where L_C is the characteristic thickness (will be defined in Section 3).

Several models are used to model the membrane performance based on the membrane material oxygen mobility and surface reactivity, electronic conductivity, and other characteristics, such as Jacobson’s model [70], Xu and Thomson’s model [70,78], Zhu’s model [70,75,76]. E.g., Zhu’s model (Figure 3) is based on the Wagner equation and considers the area-specific resistance of membrane surfaces at the air and purge sides (r′ and r″, respectively), which are proportional to the reciprocal oxygen surface exchange constant, and the membrane bulk (r^b), which is proportional to the reciprocal oxygen self-diffusion coefficient.

In the case of a multi-layer asymmetric supported membrane, the characteristics of each layer should be considered along with the properties of gas-phase diffusion in a porous support [55,58,79,80,81]. However, gas-phase phenomena are out of the scope of this review.

2.2. Hydrogen Separation Membranes

Similar to the oxygen separation membranes, a high hydrogen mobility and surface reactivity as well as a high electronic conductivity are required for hydrogen separation membrane materials. This allows it to reach high hydrogen permeation fluxes for obtaining pure hydrogen including its production in catalytic membrane reactors for fuel transformation reactions [42,55,58,82,83,84,85,86,87,88,89]. There are advantages in using triple (H⁺/O²⁻/e⁻)-conducting materials for hydrogen separation membranes, since the presence of the oxide-ionic component of the conductivity can enable the following features:

• Some proton transport mechanisms being mediated by the oxygen transport as will be mentioned in Section 3.2 [90,91,92];
• Oxide ion counterpermeation across the membrane allows us to increase the hydrogen yield due to the water splitting reaction [89,93,94];
• Triple conductivity allows to enhance the performance in various catalytic reactions and to improve gas separation properties due to the coupled transport of all types of mobile species, forcing them to be transported against their chemical potential gradient [95,96,97].

The processes in the triple-conductive hydrogen separation membrane are illustrated in Figure 4.

For dense metallic membranes, the hydrogen concentration in metal is proportional to $P_{{\mathrm{H}}_2}^{0.5}$ [88]. Similar to MIEC oxides with the variation of oxygen nonstoichiometry on the oxygen partial pressure (Equation (4)), Sievert’s law (Equations (6) and (7)) can be obtained:

$$j_{{\mathrm{H}}_2}=\left(\frac{Pe}{L}\right)\left({\left(P_{{\mathrm{H}}_2}^I\right)}^n-{\left(P_{{\mathrm{H}}_2}^{II}\right)}^n\right),$$

(6)

where

$$Pe=0.5D_H{K}_s$$

(7)

is the hydrogen permeability, $P_{{\mathrm{H}}_2}^I$ and $P_{{\mathrm{H}}_2}^{II}$ are hydrogen partial pressures in retentate and permeate gases, respectively, n is the exponent which in the ideal case is equal to 0.5 (for real membranes it lies in the range of ~0.5–1), D_H is the hydrogen self-diffusion coefficient, and K_s is the hydrogen solubility constant (Sievert’s constant) [87,88,98].

For ceramic membranes containing only protonic-electronic conductors, the Wagner equation can be written as follows (Equation (8)):

$$j_{{\mathrm{H}}_2}=-\frac{RT}{4F^{2}L}{\displaystyle \underset{\ln P_{{\mathrm{H}}_2}^I}{\overset{\ln P_{{\mathrm{H}}_2}^{II}}{\int }}\frac{{\sigma }_H{\sigma }_{el}}{{\sigma }_H+{\sigma }_{el}}}d\ln P_{{\mathrm{H}}_2},$$

(8)

where σ_H is the protonic conductivity [87,99,100]. Since protonic and electronic conductivity may depend on $P_{{\mathrm{H}}_2}$, the result of integrating it in Equation (8) can be different. Assuming σ_H << σ_el and σ_H is proportional to $P_{{\mathrm{H}}_2}^n$, there are limiting cases which can be considered:

• n = 0.5, when protons are minority defects, then $j_{{\mathrm{H}}_2}=-\frac{RT}{2F^{2}L}{\sigma }_{H,0}\left({\left(P_{{\mathrm{H}}_2}^I\right)}^{0.5}-{\left(P_{{\mathrm{H}}_2}^{II}\right)}^{0.5}\right)$;

• n = 0.25, when protons are majority defects compensated by electrons, then $j_{{\mathrm{H}}_2}=-\frac{RT}{F^{2}L}{\sigma }_{H,0}\left({\left(P_{{\mathrm{H}}_2}^I\right)}^{0.25}-{\left(P_{{\mathrm{H}}_2}^{II}\right)}^{0.25}\right);$

• n = 0, when protons are majority defects compensated by acceptor dopants, then $j_{{\mathrm{H}}_2}=-\frac{RT}{4F^{2}L}{\sigma }_{H,0}\ln \left(\frac{P_{{\mathrm{H}}_2}^{II}}{P_{{\mathrm{H}}_2}^I}\right)$ [87,99,101,102,103].

In the case of a cermet membrane, the equation for its hydrogen permeation flux combines those for the ceramic (Equation (8)) and metallic (Equation (6)) components (Equation (9)):

$$j_{{\mathrm{H}}_2}=-\left(x_{ceram}\frac{RT}{4F^{2}L}{\displaystyle \underset{\ln P_{{\mathrm{H}}_2}^I}{\overset{\ln P_{{\mathrm{H}}_2}^{II}}{\int }}\frac{{\sigma }_H{\sigma }_{el}}{{\sigma }_H+{\sigma }_{el}}}d\ln P_{{\mathrm{H}}_2}+\left(1-x_{ceram}\right)\left(\frac{P{e}_{metal}}{L}\right)\left({\left(P_{{\mathrm{H}}_2}^I\right)}^n-{\left(P_{{\mathrm{H}}_2}^{II}\right)}^n\right)\right),$$

(9)

where x_ceram is the volume fraction of the ceramic component, Pe_metal is the permeability of the metallic component [86].

For triple-conductive membranes, the oxide-ionic component of the conductivity should be accounted for (Equations (10)–(12)) [89,103]:

$$j_{{\mathrm{H}}_2}=-\frac{RT}{8F^{2}L}{\displaystyle \underset{I}{\overset{II}{\int }}{\sigma }_H\left(2\frac{{\sigma }_O+{\sigma }_{el}}{{\sigma }_H+{\sigma }_O+{\sigma }_{el}}d\ln P_{{\mathrm{H}}_2}+\frac{{\sigma }_O}{{\sigma }_H+{\sigma }_O+{\sigma }_{el}}d\ln P_{{\mathrm{O}}_2}\right)},$$

(10)

$$j_{{\mathrm{H}}_2}=-\frac{RT}{4F^{2}L}{\displaystyle \underset{I}{\overset{II}{\int }}{\sigma }_H\left(\frac{{\sigma }_{el}}{{\sigma }_H+{\sigma }_O+{\sigma }_{el}}d\ln P_{{\mathrm{H}}_2}+\frac{{\sigma }_O}{{\sigma }_H+{\sigma }_O+{\sigma }_{el}}d\ln P_{{\mathrm{H}}_2\mathrm{O}}\right)},$$

(11)

$$j_{{\mathrm{H}}_2}=\frac{RT}{4F^{2}L}\frac{{\sigma }_H+{\sigma }_{el}}{{\sigma }_H+{\sigma }_O+{\sigma }_{el}}\ln \frac{P_{{\mathrm{H}}_2}^I}{P_{{\mathrm{H}}_2}^{II}}+\frac{RT}{8F^{2}L}\frac{{\sigma }_O+{\sigma }_{el}}{{\sigma }_H+{\sigma }_O+{\sigma }_{el}}\ln \frac{P_{{\mathrm{O}}_2}^{II}}{P_{{\mathrm{O}}_2}^I}.$$

(12)

In the case of the asymmetric supported hydrogen separation membrane, more complex description is required since mass and heat transfer phenomena take place in the gas phase in the layers of the porous support. Gas-phase mass transport certainly affects the membrane performance or even can determine its characteristics [55,99,104,105,106]. However, gas-phase phenomena are out of the scope of this review.

2.3. Solid Oxide Fuel Cells

By selecting solid oxide fuel cell materials with a high oxygen and/or hydrogen mobility, the fuel cell operating temperature can be lowered while maintaining or even increasing the power output. A high oxide-ionic or/and protonic conductivity of the electrolyte reduces its resistance which is a predominant component of the ohmic losses of the cell [41,53,58,69,107,108,109]. Figure 5 demonstrates SOFCs with oxide-ionic, protonic, and dual (oxide-ionic + protonic) conductive electrolytes.

Using the electrode materials with pure electronic conductivity leads to limiting the electrode process by the electrode–electrolyte–gas phase triple-phase boundary (TPB) (Figure 6a). The ionic (oxide-ionic or/and protonic) component of the conductivity allows the electrode process to take place on the electrode–gas phase double-phase boundary (DPB) (Figure 6b). This results in the improvement in the electrode reaction kinetics [53,55,56,58,65,66,67,68,69,108,110].

3. Oxygen and Hydrogen Diffusion in MIEC Materials

3.1. Self-Diffusion, Tracer Diffusion and Chemical Diffusion

Let us consider an oxide-based material, in which the following species are mobile: electrons/holes, oxide anions/oxygen vacancies, protons/hydroxyls, etc. The flux of each of these species is ${\overrightarrow{j}}_i$. In the absence of the gradients of electric potential and temperature, the Fick’s first law (Equation (13)) is satisfied:

$${\left.\overrightarrow{j_i}\right|}_{\overrightarrow{\nabla }T=0}=-D_i\overrightarrow{\nabla }{C}_i,$$

(13)

where D_i is a diffusion coefficient of i-th species, C_i is their concentration. Strictly speaking, Equation (13) is correct if the diffusing species do not interact with each other. The interaction of the following species in triple-conductive oxides: holes, oxygen vacancies and protons, was noted in the number of studies [111,112,113]. In this case, in a linear non-equilibrium thermodynamics region, Equation (13) can be written as follows (Equation (14)):

$${\left.\overrightarrow{j_i}\right|}_{\overrightarrow{\nabla }T=0}=-{\displaystyle \sum _k{\displaystyle \sum _l{D}_{k(l)}^i\overrightarrow{\nabla }{C}_l}},$$

(14)

where the coefficients $D_{k(l)}^i$ correspond to the effect of the l-th species concentration gradient, $\overrightarrow{\nabla }{C}_l$, on the i-th species diffusive flux, ${\overrightarrow{j}}_i$, and such coefficients are the sums of the respective species’ diffusion coefficients multiplied by the transference numbers. Three types of the $D_{k(l)}^i$ coefficients can be distinguished:

• “Direct” coefficients $D_{i(l)}^i$ corresponding to the effect of the gradient $\overrightarrow{\nabla }{C}_i$ on the flux ${\overrightarrow{j}}_i$;

• “Indirect” coefficients $D_{l(i)}^i$ corresponding to the effective diffusion coefficients of i-th species under the effect of the driving force $\overrightarrow{\nabla }{C}_l$ when $\overrightarrow{\nabla }{C}_i=0$, i.e., they correspond to the effect of the gradient $\overrightarrow{\nabla }{C}_l$ on the flux ${\overrightarrow{j}}_i$ when $\overrightarrow{\nabla }{C}_i=0$;

• The coefficients with three different indices $D_{k(l)}^i$ correspond to the effective diffusion coefficients of i-th species when $\overrightarrow{\nabla }{C}_k$ is a driven force and $\overrightarrow{\nabla }{C}_l=0$.

Fick’s second law (Equation (15)) follows from Fick’s first law and the mass conservation:

$$\frac{\partial C_i}{\partial t}=D_i\widehat{\Delta }{C}_i.$$

(15)

In the absence of the chemical potential gradient, D_i is referred to as a self-diffusion coefficient. It is related to the ionic conductivity (σ_i) according to the Nernst–Einstein equation (Equation (16)) [114,115,116,117]:

$${\sigma }_i=\frac{f_{I,i}{D}_i{C}_i{q}_i}{k_{B}T},$$

(16)

where f_I,i is a correlation factor, f_I,i ≈ 1. The self-diffusion coefficient is related to the tracer diffusion coefficient $\left(D_i^{*}\right)$ determined by isotope exchange techniques as follows (Equation (17)):

$$D_i^{*}=f_i{D}_i,$$

(17)

where f_i is a correlation factor which is related to influence of counterflows of ions of different isotopes [69,114,116,117]. The ratio (Equation (18))

$$H_R=\frac{f}{f_I}$$

(18)

is referred to as a Haven ratio.

It is to be noted that i-th species’ mobility may be non-uniform in the material’s bulk: the fraction θ₁ of these species possesses a self-diffusion coefficient of D_i1, the fraction θ₂ possesses a self-diffusion coefficient of D_i2, etc. [69,118,119,120,121,122,123]. In this case, the transport of the i-th species’ can be described by a set of parameters {D_ij(T), θ_j} or by an effective (mean) self-diffusion coefficient (Equation (19)):

$$D_{over,i}={\displaystyle \sum _j{\theta }_j}{D}_{ij}.$$

(19)

If one of D_ij significantly exceeds other self-diffusion coefficients (it can be denoted as D_i,fast) and its fraction θ_j is high enough (it can be denoted as θ_fast), then D_over,i ≈ θ_fast D_i,fast.

However, in the real operating conditions of SOFCs/SOECs and permselective membranes, the chemical or electrochemical potential gradient occurs due to different gas phase composition in various device compartments and the flowing electric current. In this case, instead of the Fick’s first law (Equation (13)), the Nernst–Plank equation (Equation (20)) [124,125,126]

$${\left.\overrightarrow{j_i}\right|}_{\overrightarrow{\nabla }T=0}=-D_i\overrightarrow{\nabla }{C}_i-\frac{D_i{C}_i{q}_i}{k_{B}T}\overrightarrow{\nabla }\phi$$

(20)

or the modified Fick’s first law (Equation (21)) [117,124,127]

$${\left.\overrightarrow{j_i}\right|}_{\overrightarrow{\nabla }T=0}=-{\Gamma }_V{D}_i\overrightarrow{\nabla }{C}_i$$

(21)

can be used, where Γ_V is referred to as the thermodynamic factor, or in other words, the factor of enhancement. In this case, such a gradient as a driving force (as well as electroneutrality conservation) causes net transport of species characterized by a chemical diffusion coefficient (D_chem), which is related to the self- and tracer diffusion coefficients as follows (Equation (22)) [69,114,117]:

$$D_{chem}={\Gamma }_V{D}_i=\frac{{\Gamma }_V}{H_R}{D}_i^{*}.$$

(22)

For oxide-ionic and mixed oxide-ionic–electronic conductors, the following equations for the thermodynamic factor (Equations (23) and (24)) are known:

$${\Gamma }_V=-t_h\cdot \frac{1}{2}\frac{\partial \ln P_{{\mathrm{O}}_2}}{\partial \ln C_{{\mathrm{V}}_{\mathrm{O}}^{••}}},$$

(23)

$${\Gamma }_V=t_h\left(1+\frac{4C_{{\mathrm{V}}_{\mathrm{O}}^{••}}}{C_{\mathrm{h}}}\right),$$

(24)

where t_h is the hole transport number, $P_{{\mathrm{O}}_2}$ is the partial pressure of oxygen in the gas phase, and $C_{{\mathrm{V}}_{\mathrm{O}}^{••}}$ and C_h are the concentrations of oxygen vacancies and holes in the oxide, respectively [69,117,127,128,129,130].

For protonic conductors, a more complex relationship of chemical diffusion coefficient and self-diffusion coefficients of charge carriers (Equation (25)) is given in the work [111]:

$$D_{chem}=\frac{\left(2-\frac{C_{{\mathrm{OH}}_{\mathrm{O}}^{•}}}{2C_{{\left.{\mathrm{V}}_{\mathrm{O}}^{••}\right|}_{P_{{ \mathrm{H}}_2\mathrm{O}}=0}}}\right)D_H{D}_V}{\frac{C_{{\mathrm{OH}}_{\mathrm{O}}^{•}}}{2C_{{\left.{\mathrm{V}}_{\mathrm{O}}^{••}\right|}_{P_{{ \mathrm{H}}_2\mathrm{O}}=0}}}{D}_H+2\left(1-\frac{C_{{\mathrm{OH}}_{\mathrm{O}}^{•}}}{2C_{{\left.{\mathrm{V}}_{\mathrm{O}}^{••}\right|}_{P_{{ \mathrm{H}}_2\mathrm{O}}=0}}}\right)D_V},$$

(25)

where $C_{{\mathrm{OH}}_{\mathrm{O}}^{•}}$ is the concentration of hydroxyl ions in the oxide, $P_{{\mathrm{H}}_2\mathrm{O}}$ is the partial pressure of water in the gas phase, D_H is the self-diffusion coefficient of protons, D_V is the self-diffusion coefficient of oxygen vacancies (Equation (26)),

$$D_V=\frac{C_{\mathrm{O}}}{C_{{\mathrm{V}}_{\mathrm{O}}^{••}}}{D}_O=\frac{1}{f_O}\frac{C_{\mathrm{O}}}{C_{{\mathrm{V}}_{\mathrm{O}}^{••}}}{D}_O^{*}.$$

(26)

For oxide-ionic and mixed oxide-ionic–electronic conductors, the temperature dependence of the oxygen self-diffusion coefficient is given according to the random walk theory (Equation (27)):

$$D_O=\frac{\zeta }{6}{\epsilon }^2{C}_{{\mathrm{V}}_{\mathrm{O}}^{••}}\nu \times \exp \left(\frac{{\Delta }_{m}S}{R}\right)\times \exp \left(-\frac{{\Delta }_{m}H}{RT}\right),$$

(27)

where ζ is a number of equivalent positions, ε is the random walk step length, ν is a frequency of particle vibrations, Δ_mS and Δ_mH are migration entropy and enthalpy, respectively [131].

For the self-diffusion coefficient of protons in interstitial sites of a metal face-centered cubic lattice with the parameter a (e.g., nickel), the following equation (Equation (28)) is given in the work [132]:

$$D_H=a^2\frac{k_{B}T}{h}\exp \left(-\frac{{\Delta }_{TS-oxt}G}{k_{B}T}\right),$$

(28)

where h is the Planck constant, Δ_TS–octG is the Gibbs’ energy of the proton transition from the transition state to the ground octahedral state.

3.2. Oxygen and Hydrogen Diffusion Mechanisms

There are three general types of oxygen diffusion mechanisms in oxides and composites:

• Vacancy mechanism (Figure 7a): transport of regular oxide anions into neighboring vacancies; this mechanism is typical for perovskites, fluorites and many other types of oxides [54,70,107,133,134];
• Interstitial mechanism (Figure 7b): transport of interstitial oxide anions into neighboring interstitial sites; this mechanism is typical for some pyrochlores, mayenites and some other oxides [70,107,134,135,136,137];
• Cooperative mechanism (Figure 7c): cooperative movement of different types of oxide anions (regular, interstitial); this mechanism is typical for Ruddlesden–Popper phases, apatites, brownmillerites, orthorhombic oxides and is proposed for some other oxides [69,70,107,134,135,136,138].

In the case of non-uniformity of the oxygen mobility in the materials’ bulk due to structural and defect features, more complex features of oxygen transport can take place. They will be reviewed in details in Section 6.

The main mechanisms of hydrogen diffusion are:

• Diffusion of protons through interstitial defects (Figure 8a); this mechanism is typical for metals and alloys [90,91,99,132,139];
• Grotthuss mechanism (Figure 8b): jumps of protons between neighboring oxide anions with reorientation of M–O–H bonds; this mechanism is typical for the most oxides possessing a protonic component of conductivity [91,111,140];
• Vehicle mechanism (Figure 8c): transport of protons together with the neighboring oxide anion as a hydroxyl; this mechanism is also typical for proton-conducting oxides [91,111,140];
• Diffusion of structurally bound water (Figure 8d): transport of water species embedded into the lattice; this mechanism is proposed for some oxides [92,141].

3.3. Surface Exchange of Oxygen and Hydrogen

In 1970s, V.S. Muzykantov demonstrated that the interaction of oxide-based materials with molecular oxygen takes place via a dissociative adsorption–desorption mechanism (Muzykantov–Boreskov mechanism) [142,143], which includes the following stages:

• Physical adsorption;
• Dissociative chemisorption (Equations (29)–(31)):

$${ \mathrm{O}}_{2\left(\mathrm{g}\right)}+2{\left(\right)}_{\mathrm{a}}=2{\left(\mathrm{O}\right)}_{\mathrm{a}},$$

(29)

$${ \mathrm{O}}_{2\left(\mathrm{g}\right)}+{\left(\right)}_{\mathrm{a}}+{\left[\right]}_{\mathrm{S}}={\left(\mathrm{O}\right)}_{\mathrm{a}}+{\left[\mathrm{O}\right]}_{\mathrm{S}},$$

(30)

$${ \mathrm{O}}_{2\left(\mathrm{g}\right)}+2{\left[\right]}_{\mathrm{S}}=2{\left[\mathrm{O}\right]}_{\mathrm{S}};$$

(31)

3.

Embedding (the exchange itself) (Equation (32)):

$${\left(\mathrm{O}\right)}_{\mathrm{a}}+{\left[\right]}_{\mathrm{S}}={\left(\right)}_{\mathrm{a}}+{\left[\mathrm{O}\right]}_{\mathrm{S}}.$$

(32)

Here, ( )_a and [ ]_S denote the adsorption site and oxygen vacancy on the surface, respectively, (O)_a and [O]_S are the adsorbed (weakly bound, capable of surface diffusion) and surface (strongly bound with oxygen vacancy) oxygen species, respectively.

The exchange of oxygen with carbon dioxide proceeds in a different way compared to that with molecular oxygen. First, the entire surface of the material participates in the exchange, not only the active sites. Secondly, the mechanism of exchange is different. As a result, the exchange with CO₂ proceeds 2–5 orders of magnitude faster compared to the exchange with O₂. The CO₂ molecule is adsorbed on the surface of the oxide (Equation (33)):

$${\mathrm{CO}}_{2\left(\mathrm{g}\right)}+{\left(\right)}_{\mathrm{a}}={\left({\mathrm{CO}}_2\right)}_{\mathrm{a}},$$

(33)

then interacts with a neighboring oxide anion to form the carbonate complex (Equation (34)):

$${\left({\mathrm{CO}}_2\right)}_{\mathrm{a}}+\left[\mathrm{O}\right]={\left[{\mathrm{CO}}_3\right]}_{\mathrm{S}},$$

(34)

or with a neighboring oxygen vacancy to form the carboxylate complex (Equation (35)) [144,145,146,147,148,149,150]:

$${\left({\mathrm{CO}}_2\right)}_{\mathrm{a}}+\left[\right]={\left[{\mathrm{CO}}_2\right]}_{\mathrm{S}},$$

(35)

or, especially at high temperatures, can dissociate to the adsorbed carbon monoxide molecule and adsorbed oxygen (Equation (36)) [150,151,152,153]:

$${\left({\mathrm{CO}}_2\right)}_{\mathrm{a}}+{\left(\right)}_{\mathrm{a}}={\left(\mathrm{CO}\right)}_{\mathrm{a}}+{\left(\mathrm{O}\right)}_{\mathrm{a}}.$$

(36)

The carbonate complexes can be negatively charged complexes like ${\mathrm{CO}}_3^{2-}$, ${\mathrm{CO}}_3^{-}$, neutrally charged complexes like CO₃, or complexes being something between CO₃ and CO₂ as well. They can have various configurations depending on the metal cation they are coordinated to. The examples of such complexes are given in Figure 9 [144,145,149,154].

The carboxylate ion-radical, formed by the interaction of the adsorbed CO₂ molecule and the oxygen vacancy, is unstable, and the free valence of carbon would tend to be saturated (Figure 10) [149].

The interaction of molecular hydrogen with metals is described by two main mechanisms: the Bonhoeffer–Farkas mechanism (Equation (37)) [155] corresponding to the dissociative adsorption of hydrogen with the formation of hydrogen adatoms H_a:

$${\mathrm{H}}_2=2{\mathrm{H}}_{\mathrm{a}},$$

(37)

and the Eley–Rideal mechanism (Equation (38)) [156,157], for which the formation of a hydrogen adatom on the metal surface proceeds through the stage of the formation of a three-atomic activated complex ${\left(\mathrm{H}\cdots \mathrm{H}\cdots \mathrm{H}\right)}_{\mathrm{a}}$:

$${\mathrm{H}}_2+{\mathrm{H}}_{\mathrm{a}}={\left(\mathrm{H}\cdots \mathrm{H}\cdots \mathrm{H}\right)}_{\mathrm{a}}.$$

(38)

For metals which can intercalate hydrogen in their structure as interstitial defects (H_i; Figure 8a), the intercalation reaction (Equation (39)) is considered as well [90,91]:

$${\mathrm{H}}_{\mathrm{a}}={\mathrm{H}}_i.$$

(39)

Oxide materials interact with molecular hydrogen via its adsorption, dissociation, and interaction with the surface oxygen. They interact with water via hydration. In both cases, hydroxyls are formed on the oxide surface. These reactions are given in Equations (40)–(44) [99,111,112,113,139].

$${\mathrm{H}}_2=2{\mathrm{H}}^{•}+2 {\mathrm{e}}^{\prime },$$

(40)

$${\mathrm{H}}_2+2{\mathrm{O}}_{\mathrm{O}}^{\times }=2{\mathrm{OH}}_{\mathrm{O}}^{•}+2 {\mathrm{e}}^{\prime },$$

(41)

$${\mathrm{H}}_2\mathrm{O}+2{\mathrm{h}}^{•}=2{\mathrm{H}}^{•}+\frac{1}{2}{\mathrm{O}}_2,$$

(42)

$${\mathrm{H}}_2\mathrm{O}+{\mathrm{V}}_{\mathrm{O}}^{••}=2{\mathrm{H}}^{•}+{\mathrm{O}}_{\mathrm{O}}^{\times },$$

(43)

$${\mathrm{H}}_2\mathrm{O}+{\mathrm{O}}_{\mathrm{O}}^{\times }+{\mathrm{V}}_{\mathrm{O}}^{••}=2{\mathrm{OH}}_{\mathrm{O}}^{•}.$$

(44)

The rate of surface exchange is typically determined in terms of a surface exchange constant (k). In the presence of a chemical potential gradient, there is a chemical surface exchange constant (k_chem). In the absence of such a gradient, there is a surface exchange constant (k_ex or k^* in the case of isotope studies) which, like diffusion coefficients, is related to the chemical surface exchange constant via a thermodynamic factor. The ratio of diffusion coefficient and surface exchange constant is referred to as a characteristic thickness (Equation (45)) [129,158]:

$$L_C=\frac{D^{*}}{k^{*}},\dots \dots L_{C,chem}=\frac{D_{chem}}{k_{chem}}.$$

(45)

4. Isotope Exchange of Oxygen and Hydrogen

Isotope exchange techniques are based on the substitution of one isotope of the element in the sample (e.g., oxygen, hydrogen) with the other isotope (e.g., ¹⁶O/¹⁸O, H/D) while interacting with a gas-phase reagent in equilibrium or steady-state. These methods allow us to acquire the data on D^* and k^*. They can be divided into methods with the solid-state-phase and gas-phase analysis. The methods with solid-state-phase analysis such as SIMS, in some cases do not allow one to study diffusion processes in detail and allow one to acquire the data on the mean integral diffusion coefficient. Therefore, the methods with the gas phase analysis are more suitable for studying oxygen and hydrogen mobility features for SOFC and permselective membrane materials [32,159]. The methods with gas phase analysis can be implemented using static or flow reactors. In the case of using flow reactors, gas-phase diffusion limitations are avoided, and more complete isotopic substitution can be achieved during the experiment, allowing oxygen and hydrogen transport features to be studied in greater detail [32,55,58,69,152,159,160,161,162,163,164]. In the authors’ previous experimental works and reviews [55,58,69,161,162,163,164], it has been demonstrated that temperature-programmed oxygen isotope exchange with ¹⁸O₂ and C¹⁸O₂ allows one to describe the oxygen surface-exchange mechanisms and bulk oxygen diffusion features, including the non-uniformity of bulk oxygen mobility, for many types of ceramic materials, and for SOFCs, permselective membranes and other applications. The most interesting feature of using C¹⁸O₂ as a ¹⁸O-containing gas-phase reagent instead of ¹⁸O₂ is its faster surface exchange with C¹⁸O₂ compared to that with ¹⁸O₂ (as marked in Section 3.3) [58,145,146,147,148], which generally allows one to avoid the limitation of the process by the surface exchange (i.e., to carry out the experiments in the diffusion-controlled or mixed-controlled regime) [153] and to obtain more detailed data on the oxygen bulk diffusion [58,69,161]. The comparison of various types of isotope exchange experiments with gas phase analysis is given in

Table 1. Options of oxygen isotope exchange experiments with gas-phase analysis.

Reactor Type
	Static	Flow
Oxygen exchanged in the oxide	<10%	up to ≅100%
Sensitivity to the diffusion rate	No	Yes
Low isotope consumption	Yes	No
Simple reactor construction	No	Yes
Exchangeable gas-phase reagent
	18O2	C18O2
Sensitivity to the diffusion rate	No	Yes
Sensitivity to the kinetics of interaction with the oxide surface	Yes	No
Temperature mode
	Isothermal	Temperature-programmed
Sensitivity to the oxygen non-uniformity in the oxide bulk	No	Yes

.

The reaction of isotope exchange of the solid states with the gas-phase reagent containing two identical atoms (¹⁸O₂, C¹⁸O₂, D₂, D₂O, etc.) proceeds via routes which are classified as three types of exchange mechanisms according to Muzykantov’s classification (Equations (46)–(48)) [165]:

• Homoexchange:

  o

  R⁰-type (0-atomic type, I type):

X₂ + Y₂ = 2 XY;

(46)

• Heteroexchange:

  o

  R¹-type (1-atomic type, II type):

  X₂ + (Y)_S = XY + (X)_S,

  (47)

  o

  R²-type (2-atomic type, III type):

X₂ + 2 (Y)_S = Y₂ + 2 (X)_S.

(48)

In many cases, several types of exchange occur simultaneously.

If isotope exchange takes place with molecules containing more than two identical atoms, the mechanism can be more complex. For example, for hydrogen isotope exchange with methane, the theory of five types of exchange mechanisms is used [166,167].

Several models or combinations thereof are used to describe oxygen diffusion in the bulk [58,162,163,164]:

• Uniform 1D model (e.g., simple oxides) (Figure 11a);
• Non-uniform 1D model with a single diffusion channel involving the weakest bound oxygen form and an exchange with the neighboring strongly bound oxygen forms (complex oxides) (Figure 11b);
• Non-uniform 1D model with several parallel diffusion channels involving different oxygen forms (composites) (Figure 11c);
• Non-uniform 2D model with a single fast diffusion channel along grain boundaries followed by diffusion of the isotope tracer within the grain bulk (monocrystalline) (Figure 11d);
• Non-uniform 2D model with a single fast diffusion channel along grain boundaries with subsequent diffusion of the isotope tracer within the balk of different grains (polycrystalline) (Figure 11e).

The generalized model of oxygen isotope exchange is the following (Equations (49)–(54)):

$$N_g\frac{\partial {\alpha }_g}{\partial t}+O=N_S{R}^{\Sigma }({\alpha }_s-{\alpha }_g)+O(N_g,{\alpha }_g),$$

(49)

$$\frac{\partial {\alpha }_s}{\partial t}=R^{\Sigma }({\alpha }_g-{\alpha }_s)-\frac{N_{bulk}}{N_s}{\left.\frac{D}{h^2}\frac{\partial {\alpha }_{bulk}}{\partial \eta }\right|}_{\eta =0},$$

(50)

$$\frac{\partial {\alpha }_{bulk}}{\partial t}=\frac{D}{h^2}\frac{{\partial }^2{\alpha }_{bulk}}{\partial {\eta }^2},$$

(51)

$$N_g\frac{\partial f_{16-18}}{\partial t}=N_s{R}^{(i)}(P_{16-18}-f_{16-18})+O(N_g,f_{16-18}),$$

(52)

where N_g, N_S and N_bulk are numbers of oxygen atoms in the gas phase, on the sample surface and in the sample bulk, respectively; α_g, α_S and α_bulk are ¹⁸O atomic fractions in the gas phase, on the sample surface and in the sample bulk, respectively; O(N_g,α_g) is a component which depends on the reactor type,

$$O(N_g,{\alpha }_g)=\left\{\begin{array}{l}0 \hspace{1em}\hspace{1em}\hspace{1em}\text{-}\mathrm{static} \mathrm{reactor}\\ \frac{1}{\tau }{N}_g\frac{\partial {\alpha }_g}{\partial \xi } \hspace{1em}\text{-}\mathrm{flow} \mathrm{reactor}\end{array}\right. ;$$

(53)

R and D are the heteroexchange rate and oxygen tracer diffusion coefficient, respectively, which are assumed to be constant for isothermal isotope exchange (IIE) experiments and dependent on the temperature according to the Arrhenius law for temperature-programmed (TPIE) experiments:

$$R^{(i)}=\left\{\begin{array}{l}{R}^2=0.5R^{\Sigma }\\ R^1=R^{\Sigma }\end{array}\right.\hspace{1em}{P}_{16-18}=\left\{\begin{array}{l}2{\alpha }_s(1-{\alpha }_s){ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\text{-}\mathrm{exchange} \mathrm{with} \mathrm{O}}_{2 } \\ {\alpha }_g(1-{\alpha }_s)+{\alpha }_s(1-{\alpha }_g){ \text{-} \mathrm{exchange} \mathrm{with} \mathrm{CO}}_2\end{array}\right..$$

(54)

The mathematical models for specific cases of oxygen diffusivity features (e.g., 2D diffusion) can be found in the works [162,163,164].

5. Relaxation Techniques

Relaxation techniques such as the electrical conductivity relaxation (ECR) [168,169,170,171,172,173,174,175,176], mass relaxation (MR, also referred to as weight relaxation or thermogravimetric relaxation) [69,111,112,113,176,177,178] and unit cell volume relaxation (UCVR) [69,162,179] techniques are based on the changing some characteristics of a sample with time after rapid change of the gas-phase composition such as the partial pressure of oxygen, carbon dioxide, hydrogen or water vapors. After such a rapid change, the system solid-state sample–gas phase becomes non-steady-state, and oxygen or/and hydrogen desorbs from the sample or adsorbs on the sample to reach a new steady state. This leads to the relaxation of characteristics dependent on the oxygen and hydrogen content in the sample such as the electrical conductivity, the sample weight, and the unit cell volume to the new steady-state values. These methods allow one to acquire the data on the coupled transport of the mobile particles (oxide anions, protons, holes, etc.) in the materials for SOFC, permselective membranes and other devices (D_chem and k_chem).

The data acquired from the relaxation (ECR, MR or UCVR) experiment are normalized, like those shown in Equation (55) for ECR:

$$\overline{\sigma }\left(t\right)=\frac{\sigma \left(t\right)-{\sigma }_0}{{\sigma }_{\infty }-{\sigma }_0},$$

(55)

where σ₀, σ(t) и σ_∞ are the sample electrical conductivity before changing pressure, at the moment of time t and after the relaxation, respectively; $\overline{\sigma }\left(t\right)$ is the normalized electrical conductivity depending on time, $0\le \overline{\sigma }\left(t\right)\le 1$. The experimental data (Equation (55)) are fitted by theoretical curves, which can be found by solving Fick’s second law (Equation (15)). Such a solution was found for the following cases [173,174,180,181,182]:

• The infinite plane sheet (Equations (56)–(62)):

$$\overline{\sigma }\left(t\right)={\displaystyle \sum _{i=1}^{\infty }{A}_i\exp \left(-\frac{t}{t_i}\right)}+1,$$

(56)

where

$$A_i=-\frac{2{\Lambda }^2}{{\beta }_i^2\left({\beta }_i^2+{\Lambda }^2+\Lambda \right)},$$

(57)

$$t_i=\frac{l^2}{4{\beta }_i^2{D}_{chem}},$$

(58)

$$\Lambda =\frac{l{k}_{chem}}{D_{chem}}=\frac{l}{L_{C,chem}};$$

(59)

β_i are the eigenvalues of Equation (60):

$${\beta }_i\tan {\beta }_i=\Lambda ;$$

(60)

l is the sheet thickness;

• The infinite cylinder (Equations (61)–(64)):

$$\overline{\sigma }\left(t\right)={\displaystyle \sum _{i=1}^{\infty }{A}_i\exp \left(-\frac{t}{t_i}\right)}+1,$$

(61)

where

$$A_i=-\frac{2{\Lambda }_{\rho }^2}{{\rho }_i^2\left({\rho }_i^2+{\Lambda }_{\rho }^2\right)},$$

(62)

$$t_i=\frac{r^2}{{\rho }_i^2{D}_{chem}},$$

(63)

and the parameters Λ_ρ and the eigenvalues ρ_i can be found while solving Equation (64):

$${\rho }_j{J}_1\left({\rho }_j\right)={\Lambda }_{\rho }{J}_0\left({\rho }_j\right)=\frac{r{k}_{chem}}{D_{chem}}{J}_0\left({\rho }_j\right),$$

(64)

where J₀ and J₁ are the zero-order and the first-order Bessel functions, respectively; r is the cylinder radius;

• The short cylinder (Equations (65)–(67)):

$$\overline{\sigma }\left(t\right)={\displaystyle \sum _{i=1}^{\infty }{\displaystyle \sum _{j=1}^{\infty }{A}_{ij}\exp \left(-\frac{t}{t_{ij}}\right)}}+1,$$

(65)

where

$$A_{ij}=-\frac{2{\Lambda }^2}{{\beta }_i^2\left({\beta }_i^2+{\Lambda }^2+\Lambda \right)}\frac{2{\Lambda }_{\rho }^2}{{\rho }_i^2\left({\rho }_i^2+{\Lambda }_{\rho }^2\right)},$$

(66)

$$t_{ij}=\frac{1}{D_{chem}\left[{\left(\frac{{\beta }_i}{l/2}\right)}^2+{\left(\frac{{\rho }_j}{r}\right)}^2\right]};$$

(67)

• The sphere (Equations (68)–(71)):

$$\overline{\sigma }\left(t\right)={\displaystyle \sum _{i=1}^{\infty }{A}_i\exp \left(-\frac{t}{t_i}\right)}+1,$$

(68)

where

$$A_i=-\frac{6{\Lambda }_{\rho }^2}{{\rho }_i^2\left({\rho }_i^2+{\Lambda }_{\rho }\left({\Lambda }_{\rho }-1\right)\right)},$$

(69)

$$t_i=\frac{r^2}{{\rho }_i^2{D}_{chem}},$$

(70)

and the eigenvalues ρ_i can be found while solving Equation (71):

$${\rho }_i\cot {\rho }_i-{\Lambda }_{\rho }-1=0;$$

(71)

r is the sphere radius;

• The rectangular bar (Equations (72)–(75)):

$$\overline{\sigma }\left(t\right)={\displaystyle \sum _{i=1}^{\infty }{A}_{x,i}\exp \left(-\frac{t}{t_{x,i}}\right)}\times {\displaystyle \sum _{j=1}^{\infty }{A}_{y,j}\exp \left(-\frac{t}{t_{y,j}}\right)}+1,$$

(72)

where

$$A_{x,i}=-\frac{2{\Lambda }_x^2}{{\beta }_{x,i}^2\left({\beta }_{x,i}^2+{\Lambda }_x^2+{\Lambda }_x\right)},\dots A_{y,j}=-\frac{2{\Lambda }_y^2}{{\beta }_{y,i}^2\left({\beta }_{y,i}^2+{\Lambda }_y^2+{\Lambda }_y\right)},$$

(73)

$$t_{x,i}=\frac{l_x^2}{4{\beta }_i^2{D}_{chem}},\dots t_{x,i}=\frac{l_x^2}{4{\beta }_i^2{D}_{chem}},$$

(74)

$${\Lambda }_x=\frac{l_x{k}_{chem}}{D_{chem}},\dots {\Lambda }_y=\frac{l_y{k}_{chem}}{D_{chem}},$$

(75)

l_x and l_y are the bar dimensions along the x and y axes, respectively.

For a more complex sample shape or, all the more so, for any geometrical shape, a more complex approach is required such as mathematical modeling, e.g., the generalized model based on the inverse algorithm [172,175].

The reactor flush time (t_f) is taken into account by introducing the factor $\frac{t_i}{t_i+t_f}$ and the addendum $\exp \left(-\frac{t}{t_f}\right)$ into Equations (56), (61), (65), (68) and (72) [182]. E.g., for the short cylinder case, the best description of disc-shaped pelletized samples, Equation (65) transforms into Equation (76):

$$\overline{\sigma }\left(t\right)={\displaystyle \sum _{i=1}^{\infty }{\displaystyle \sum _{j=1}^{\infty }{A}_{ij}\frac{t_{ij}}{t_{ij}-t_f}\left[\exp \left(-\frac{t}{t_{ij}}\right)-\exp \left(-\frac{t}{t_f}\right)\right]}}-\exp \left(-\frac{t}{t_f}\right)+1.$$

(76)

The relaxation techniques are being developed up to now, and include new approaches for processing the experimental data, elucidating the contribution of each phase and interphase of composite materials, the analysis of different charge carrier behavior in triple-conductive materials, etc. [111,112,113,168,170,172,175,178].

6. Oxygen and Hydrogen Mobility of Materials for Membranes and SOFC

6.1. Fluorites, Bixbyites and Rhombohedral Phases

Fluorites and fluorite-related materials are the most of important types of materials for application in electrochemistry including SOFCs/SOECs, oxygen/hydrogen separation membranes, etc.

Recent research efforts [183,184,185,186,187] have made it possible to increase the grain-boundary conductivity of proton-conducting zirconates with a perovskite structure. At the same time, there is another class of proton-conducting materials, with a fluorite-like structure, which have comparable total and bulk conductivities, whereas the contribution of grain-boundary conductivity is extremely small or zero. This class of materials comprises the following disordered pyrochlores and fluorites based on La compounds:

• Ca-doped La₂Zr₂O₇ ((La_2−xCa_x)Zr₂O_7−δ) pyrochlore, a proton conductor in the range of 200–600 °C [188,189];
• La₂Ce₂O₇ (50% CeO₂ + 50% La₂O₃) fluorite, a proton conductor below 450 °C and an oxygen ion conductor at high temperatures [190]; and
• Fluorite-like La_6−xWO_12−δ (x = 0–0.8), a proton conductor with conductivity up to (3–7) × 10⁻³ S cm⁻¹ at 800 °C and 1 Pa, depending on x [82,191].

Ln tungstates were revealed to have mixed ionic–electronic conductivity with a potential ability of using in solid oxide fuel cells and proton conducting membranes [82,191]. La_6−xWO_12−δ (x = 0.2–1) solid solutions based on lanthanum tungstate La₆WO₁₂ were of particular interest since they were found to have the highest proton conductivity among the few non-perovskite proton-conducting materials [82,191,192,193,194,195]. La_6−xWO_12−δ (x = 0.2–1) tungstates can be used as potential solid electrolytes for solid-state fuel cells and proton-conducting membranes for hydrogen separation. An important advantage of lanthanum tungstates over perovskite-acceptor-doped barium and strontium cerates BaCeO₃, SrCeO₃–is the absence of interaction with CO₂ and SO_x with the formation of carbonates and compounds containing sulfur [192].

Among single-phase materials La_6−xWO_12−δ (x = 0–0.8), the highest proton conductivity was provided by La_6−xWO_12−δ (x = 0.4, 0.5) materials, but subsequent investigation showed that their proton conductivity dropped rather sharply during prolonged holding in wet H₂ at 1100 °C, and the most stable materials were La_6−xWO_12−δ with x = 0.6 and 0.7 [82]. According to Partin et al. [196], who prepared samples by standard solid-state reactions, the most stable solid solution was La_6−xWO_12−δ with x = 0.4. It seems likely that the problem of low grain-boundary conductivity arises as well in the case of proton-conducting lanthanum tungstates. For example, in studies of the conductivity of La_6−xWO_12−δ (x = 0.4, 0.6, 0.8, 1.0) [196], comparison of impedance plots before and after holding in a wet atmosphere showed a marked increase in grain-boundary resistance at 800–900 °C. By contrast, in the range 300–500 °C, the grain-boundary resistance decreased with increasing partial pressure in various atmospheres [82,196]. Since W⁶⁺ and Mo⁶⁺ are similar in ionic radius, Savvin et al. [197,198] expected to obtain proton-conducting materials based on the Ln₆MoO₁₂ (Ln = La–Lu) molybdates. Indeed, they succeed to extend the class of proton-conducting fluorite-like materials by synthesizing new mixed electron–proton-conducting molybdates: La_5.8Zr_0.2MoO_12.1 and Ln_5.4Zr_0.6MoO_12.3 (Ln = Nd, Sm, Dy) [197,198]. Doping with zirconium ensured a higher stability of molybdates to reduction, but as in the case of tungstates [191], Zr was found to be a donor dopant, reducing the proton conductivity of materials [197]. Among proton-conducting Ln_6−xZr_xMoO_12+δ (Ln = La, Nd, Sm, Gd, Dy, Ho; x = 0.2–0.6) molybdates, most of which have a fluorite structure (sp. gr. $Fm\overline{3}m$), the highest conductivity was found for the rhombohedral La_5.8Zr_0.2MoO_12.1 phase (sp. gr. $R\overline{3}$), which exhibited a total conductivity of 2.5 × 10⁻⁵ S cm⁻¹ at 500 °C (3 × 10⁻⁴ S cm⁻¹ at 800 °C) in wet air [198]. It should be noted that solid solutions based on rare-earth tungstates and molybdates are predominantly oxygen ion conductors in dry air at low temperatures, and predominantly proton conductors in wet air [82,198]. At high temperatures (above 600 °C) in an oxidizing atmosphere (air), the charge transport is dominated by p-type conduction, whereas under reducing conditions n-type conduction prevails [82,198]. Doping with Ti, Zr, and Nb on the Mo site and with fluorine on the oxygen site was studied using La_5.4MoO_11.1 as an example, but essentially all of the dopants reduced ionic conductivity of the material [197,199,200]. A similar situation was observed in La_6−xWO_12−δ (x = 0.4, 0.5) lanthanum tungstates [82,191,192,193,194,195]. Due to the fact that cation doping [197,199,200,201] decreased the proton conductivity of RE molybdates, the main attention was paid to the study of pure solid solutions based on Ln₆MoO₁₂: Ln_6−xMoO_12−δ (Ln = La, Nd, Sm, Gd–Lu) [163,202,203,204,205,206,207,208,209,210,211,212]. It is known that, to a large extent, the proton conductivity depends on the crystal structure type, and, in this regard, the rich polymorphism of solid solutions based on RE molybdates and tungstates Ln₆MO₁₂ (M = Mo, W) should be noted [199,201,203,205,206,209,210,212]. In the series Ln_6−xMoO_12−δ (Ln = La, Nd, Sm, Gd–Lu), depending on the temperature and lanthanide ionic radii, various structural types are realized: rhombohedral $\left(R\overline{3}\right)$, fluorite $\left(Fm\overline{3}m\right)$, and bixbyite $\left(Ia\overline{3}\right)$. Proton conductivity was found in various solid solutions based on RE molybdates, and it was shown that it reached maximal values for lanthanum molybdates La_6−xMoO_12−δ (x = 0.5, 0.6) with a complex rhombohedral structure R1 [202,203,211].

The stability of solid solutions based on REE molybdates, as well as of lanthanum tungstates La_6−xWO_12−δ (x = 0–0.8) solid solutions, known proton conductors [82,191,192,193,194,195], is an important issue in the perspective of their practical application. As a rule, it is the process of reduction in variable valence cations in solid solutions, which results in a grain-boundary contribution growth, limiting the conductivity of materials in wet atmospheres at high temperatures. The stability of the Ho_5.4Zr_0.6MoO_12.3 fluorite structure and the La_6−xMoO_12−δ (x = 0.5) fluorite-like rhombohedral structure R1 in extremely dry conditions under dynamic vacuum was investigated by in situ variable temperature neutron diffraction (NDD) between 800 and 1400 °C [205]. The NDD results unambiguously demonstrated the dimensional stability of the fluorite-like rhombohedral La_6−xMoO_12−δ (x = 0.5) as compared to the Ho_5.4Zr_0.6MoO_12.3 fluorite in the heating–cooling cycle. According to the NDD, heating to 1100 °C followed by vacuum cooling does not change the c cell parameter of rhombohedral La_6−xMoO_12−δ (x = 0.5), whereas its a parameter decreases by 0.13%. It was also found that the a cell parameter of cubic fluorite Ho_5.4Zr_0.6MoO_12.3 decreases by ~2.6%. It may be result of the partial reduction of Mo⁶⁺ to Mo⁺⁵ in RE molybdates. It seems likely that the same cause, i.e., the decrease in cubic cell parameter as a result of the partial reduction of W⁶⁺ to W⁺⁵, accompanied by disordering on the La/W sites, and subsequent formation of a denser atomic packing in the La_6-xWO_12−δ (x = 0.4, 0.6, 0.8) lanthanum tungstates, underlies their relatively low stability [196,198,213,214,215]. We believe that the loss of dimensional stability under reducing conditions in Ln₆MO₁₂ (M = Mo, W)-based solid solutions, which results in a grain-boundary contribution, limiting their conductivity in wet atmospheres, is due to the partial reduction of Mo⁶⁺ and W⁶⁺ in the rare-earth molybdates and tungstates, respectively [205].

A follow-up study of the structure of La-containing molybdates La_6−xMoO_12−δ (x = 0.5, 0.6) showed that they have a new structure type based on rhombohedral cells, which has been discussed in series of papers [199,203,205,206,209,210,212]. Along with main peaks of the $R\overline{3}$ [205] or $R\overline{3}m$ [209] structure, additional lines are present. These are superstructure lines typical of complex crystallographic cells whose parameters are increased by seven (R1) or five (R2)) times according to López-Vergara et al. [209]. López-Vergara et al. [203] reported that, depending on the cooling rate, the La_6−xMoO_12−δ (x = 0.6) solid solution can be obtained either in the form of a complex rhombohedral modification R1 (slow cooling) or in the form of fluorite (quenching), which agrees with the high-temperature experiment in vacuum for La_6−xMoO_12−δ (x = 0.5) [205]. It also turned out that R1 phase La_6−xMoO_12−δ (x = 0.6) has better oxygen-ion and proton conductivity than that of fluorite [203,209]. The decrease in the lanthanum concentration led to a decrease in the rhombohedral distortion degree and to the decrease in the contribution of proton conductivity in the series La_6−xMoO_12−δ (x = 0.5, 0.6, 0.7, 1) [211]. The proton conductivity for the optimal composition of La_6−xMoO_12−δ (x = 0.5) was ~5 × 10⁻⁵ S cm⁻¹ at 500 °C in wet air, while for La_6−xMoO_12−δ (x = 1) it was ~9 × 10⁻⁶ S cm⁻¹ (Figure 12a) [30].

A tendency towards a decrease in the proton conductivity contribution for the rare-earth (RE) molybdates Ln_6−xMoO_12−δ (Ln = La–Yb) series has been established. For heavy RE molybdates, the conditions for the synthesis of new proton conductors with a bixbyite structure (Figure 12b) were found for the first time [202,204,206,208,212], and the bixbyite structure type was first presented in the ICDD PDF crystallographic database (Er₆MoO_12−δ (No. I11624) and Tm₆MoO_12−δ (No. I11626)). It was found that with decreasing of the Ln₂O₃ content by 1.8 mol.%, fluorites Ln_5.5MoO_11.25−δ (Ln = Er, Tm) are formed under the same conditions (Figure 12c) [212].

Fluorites and bixbyites turned out to be mixed electron-oxygen conductors in dry air and electron-proton conductors in wet air, while the dominant ionic contribution maintains up to 550–600 °C [163,202,206]. In wet air, Er and Tm fluorites and bixbyites had a close total conductivity of ~2 × 10⁻⁶ S cm⁻¹ at 500 °C, but at 200 °C, bixbyites performed better than that of fluorites. The using of the isotope exchange with C¹⁸O₂ made it possible to confirm the high mobility of oxygen in these compounds in air, starting from 200 °C (Figure 13) [212]. A high or at least intermediate oxygen mobility was demonstrated for other fluorites and bixbyites (in some cases due to defect features such as the effect of grain boundaries resulting in a fast oxygen diffusion along grain boundaries (2D diffusion)), while rhombohedral phases possess lower oxygen mobility (Figure 13) [55,58,163,206,212,216].

It is of interest to note that the existence of compounds and solid solutions with close composition, differing by only a few mole percent, but having different structure, is typical for the Ln₂O₃–Mo(W)O₃ (Ln = La, Nd, Pr, Sm) systems [217,218,219]. For example, in the Pr₂O₃–MoO₃ and Nd₂O₃-MoO₃ systems at 1000 °C, the compounds with Ln₂O₃:MoO₃ (Ln = Pr, Nd) molar ratios of 5:6 and 7:8 differ in composition by just ~3 mol.% [217]. According to Chambrier et al. [218,219], cubic solid solutions based on La₁₀W₂O₂₁ free of La₂O₃ and La₆W₂O₁₅ impurities exist up to ~1700 °C in a narrow composition range, 26–30 mol.% WO₃, and La₁₀W₂O₂₁ exact composition is 28.6 mol.% WO₃ + 71.4 mol.% La₂O₃. La₆WO₁₂ contains 25 mol.% WO₃. Thus, in the Ln₂O₃–WO₃ system, La₆WO₁₂ and La₁₀W₂O₂₁ differ in composition by just 3.6 mol.% WO₃.

Doped ceria materials being typically pure ionic conductors in air and MIECs in reducing atmospheres are generally used as intermediate-temperature SOFC buffer layers between the electrolyte and the cathode in order to prevent their chemical interaction as well as electrolytes or components of composites for intermediate-temperature SOFC electrodes and oxygen separation membranes [33,36,39,53,54,58,69,70,107,220]. For using ceria as electrode or membrane material itself, the electronic component of conductivity should be increased. This can be achieved by doping with cations possessing redox activity such as Pr^4+/3+ and Tb^4+/3+ [70,220]. Doping with Pr leads to an increase in oxygen mobility and surface reactivity as well, due to the formation of ordered chains of Pr^4+/3+ cations [69,221,222]. For Tb-doped ceria, it was demonstrated that it possesses a high oxygen heteroexchange rate comparable with that for Gd-doped ceria [220,222]. On the other hand, it was demonstrated that the oxygen mobility of Ce_1−xTb_xO_2−δ (x = 0, 0.2 and 0.5) decreases with increasing Tb content, probably due to interaction between defects resulting in forming local associates [223,224]. Nevertheless, the oxygen permeability of membranes based on some Pr- and Tb-doped ceria was comparable to that for similar membranes based on perovskites such as LFN and LSFC [70,220]. Figure 14 demonstrates comparison of the oxygen tracer diffusion coefficient values of MIEC-doped ceria materials.

6.2. Pyrochlores

The pyrochlore structure A₂B₂O₇ is a derivative of the fluorite structure in which half of the cubes are replaced by octahedra (more precisely, it consists of the alternating AO₈ polyhedra and BO₆ trigonal antiprisms). Pyrochlores possessing a highly mixed ionic-electronic conductivity such as doped Pr₂Zr₂O₇, Gd₂Ti₂O₇, Er₂RuMnO₇, etc., are used in SOFC cathodes [164,225,226], oxygen [83,227,228] and hydrogen separation membranes [229,230]. They contain high amounts of oxygen vacancies providing fine oxygen transport characteristics. Some pyrochlores contain interstitial oxide anions formed due to Frenkel disordering (Equation (77))

$${\mathrm{O}}_{\mathrm{O}}^{\times }={\mathrm{V}}_{\mathrm{O} (48f)}^{••}+{ {\mathrm{O}}^{″}}_{\mathrm{i} (8a)}$$

(77)

involved in the oxygen diffusion as well [137]. There are two forms of oxygen in the pyrochlore structure (O, O′), of which the content ratio is 6:1. However, according to TPIE C¹⁸O₂ studies [56,58,164,228,231,232,233], the oxygen bulk mobility is uniform, or, in the case of its nonuniformity, the ratio of various oxygen forms differing in their mobility differs from 6:1. This is evidence that the oxygen migration mechanism is rather complex and includes the oxygen of both O- and O’-sublattices. It was proposed as well that the oxygen forms differing in their mobility can be associated with A–O–A, A–O–B and B–O–B migration pathways with their fraction depending on the partial disordering of the pyrochlore structure [164,228]. The other feature of some pyrochlores (Mg-doped Sm and Gd zirconates) is the fast oxygen transport along grain boundaries being characterized by a very high mobility (D* ~10⁻⁷ cm² s⁻¹ at 1000 K) [164]. The comparison of the oxygen mobility of some pyrochlores is given in Figure 15.

Shimura et al. [234] studied the proton conductivity of Ln₂Zr₂O₇-based (Ln = La, Nd, Sm, Gd и Er) pyrochlore oxides and found that the conductivity of the Ln₂Zr_1.8Y_0.2O_7−δ (Ln = La, Nd, Sm, Gd и Er) solid solutions in a hydrogen atmosphere at T > 600 °C was comparable to that of perovskites. The effect of alkaline earth cation (Mg, Ca, Sr, and Ba) and Y substitutions for both the La and Zr sites in pyrochlore La₂Zr₂O₇ on its proton conductivity was studied in detail in [188,189,234,235]. The highest proton conductivity was obtained by substituting La with Ca and Sr. The conductivity of (La_1.97Ca_0.03)Zr₂O_7−δ between 600 and 700 °C was determined to be 4 × 10⁻⁴ S cm⁻¹ [188]. It is important to note that the degree of Ca substitution in such solid solutions is low, and not higher than x = 0.05 in (La_2−xCa_x)Zr₂O_7−δ. Eurenius et al. [236,237] recently studied the proton conductivity of rare-earth stannates and titanates with the pyrochlore structure A_2−xCa_xSn₂O_7−x/2 (A = La, Sm, Yb) and Sm₂Ti_1.92Y_0.08O_7−δ, Sm_1.92Ca_0.08Ti₂O_7−δ. The conductivity of the A-site acceptor-substituted pyrochlores was about one order of magnitude higher than that of the B-site substituted materials. On the other hand, the conductivity clearly depended on the nature of the B-site cation: an increase in the ionic radius and electronegativity of the B-site cation was accompanied by the increase in conductivity. The proton conductivity of the samarium titanate-based solid solutions, and especially that of the rare-earth stannates, was found to be lower than that of the Ca-doped La₂Zr₂O₇.

Calcium- and strontium-doped lanthanum zirconates, La_2−xD_xZr₂O_7−δ (x = 0.05, 0.1; D = Ca, Sr), were extensively studied as electrolyte materials for proton-conducting solid oxide fuel cells (PC-SOFCs) [188,235,238,239,240]. Calcium appears to be the most promising dopant because strontium doping results in the formation of a second phase, SrZrO₃ with a perovskite structure, on the surface of strontium-containing zirconate ceramics [239] and, more importantly, because the overall conductivity of strontium-containing ceramics is an order of magnitude lower than that of calcium-containing ceramics. It was reported that pyrochlore solid solutions La_1.95Ca_0.05Zr₂O_6.95 and La_1.9Ca_0.1Zr₂O_6.9 were almost identical in proton conductivity [188,238]: 7.0 × 10⁻⁴ S cm⁻¹ at 600 °C. As was shown earlier [241], the proton conductivity of Sm_2−xCa_xZr₂O_7−δ (x = 0.05) at 600 °C is ~7.5 × 10⁻⁴ S cm⁻¹ [241].

Gas-tight proton-conducting Nd_2−xCa_xZr₂O_7−δ (x = 0, 0.05) ceramics were prepared for the first time via mechanical activation of the oxide mixture, followed by the single-step firing at 1600 °C for 3 or 10 h [242]. Like in the case of (Ln_1−xCa_x)₂Zr₂O_7−x (Ln = La, Sm; x = 0.05) pyrochlore solid solutions, the unit-cell parameter of the Ca-doped material Nd_2−xCa_xZr₂O_7−δ (x = 0.05) was smaller than that of the undoped Nd₂Zr₂O₇. The Rietveld-refined XRD data demonstrated that Ca substitutes on both cation sites of zirconate and that most of the Ca cations resides in the Zr sublattice. As a result, the total conductivity of Nd_2−xCa_xZr₂O_7−δ (x = 0.05) in wet air was lower than that of the (Ln_1−xCa_x)₂Zr₂O_7−x (Ln = La, Sm; x = 0.05) pyrochlores, where Ca substituted predominantly on the Ln site. The proton conductivity in wet air was 3 × 10⁻⁴ S cm⁻¹ at 500 °C (7 × 10⁻⁴ S cm⁻¹ at 600 °C) in (La_1−xCa_x)₂Zr₂O_7−x (x = 0.05), 7 × 10⁻⁵ S cm⁻¹ at 500 °C (~2 × 10⁻⁴ S cm⁻¹ at 600 °C) in (Nd_1−xCa_x)₂Zr₂O_7−x (x = 0.05), and 1 × 10⁻⁴ S cm⁻¹ at 500 °C (7.5 × 10⁻⁴ S cm⁻¹ at 600 °C) in (Sm_1−xCa_x)₂Zr₂O_7−x (x = 0.05). Even though the total conductivity of the Ca-doped zirconate Nd_2−xCa_xZr₂O_7−δ (x = 0.05) was an order of magnitude higher than that of Nd₂Zr₂O₇, predominant Ca substitution on the Zr site leads to a lower proton conductivity in comparison with that of (Ln_1−xCa_x)₂Zr₂O_7−x (Ln = La, Sm; x = 0.05), where all of the Ca cations resided on the Ln site. It is also possible that this result was due to the higher firing temperature: the (Ln_1−xCa_x)₂Zr₂O_7−x (Ln = La, Sm; x = 0.05) materials were prepared by firing at 1550 °C for 10–50 h [188,241], whereas a higher firing temperature of 1600 °C (3 and 10 h) was chosen for (Nd_1−xCa_x)₂Zr₂O_7−x (x = 0.05) in order to obtain gas-tight ceramics.

6.3. Perovskites

Perovskite-like oxides are widely used materials for SOFC and permselective membranes components due to their typically high electronic or mixed ionic-electronic conductivity [55,99,107,128,129,131,152,243,244,245]. The general oxygen transport mechanism in perovskites is a vacancy mechanism (Figure 7a and Figure 16). Hence, increasing the oxygen vacancy content can increase the oxygen mobility, which can be achieved by doping A- and B-sites with various aliovalent cations [131,246]. The creation of an A-site deficiency also allows for an increase in the oxygen vacancy content; however, it may result in a decrease in their mobility due to their binding to defect complexes such as $\left[{ {\mathrm{V}}^{‴}}_{\mathrm{La}}-{\mathrm{V}}_{\mathrm{O}}^{••}\right]$ [131]. For some oxides with distorted perovskite structure, it was demonstrated that significant deviation from oxygen stoichiometry in such materials is accompanied by nanostructuring; at the same time, grain boundaries become fast channel of oxygen transport, while oxygen transport within the grain bulk is slower (Figure 17) [118,119,120,121,122,123].

Conventional strontium-doped lanthanum manganite (LSM) materials have poor oxygen mobility (Figure 18), which limits their application as air electrodes in SOFCs with decreased operating temperatures that are being intensively developed [69,247,248]. However, they can be successfully used in the composite electrodes in combination with different ionic conductors [249,250,251]. Lanthanum ferrite-nickelates (LNF), being predominantly electronic conductors, demonstrate low oxygen diffusion and, as a result, oxygen permeation properties [252,253,254,255]. Nevertheless, LaNi_0.6Fe_0.4O₃, as the most stable in the series, found widespread application in SOCs due to its superior conductivity, low thermal expansion coefficient value, and tolerance to chromium poisoning [256]. It is also successfully used in different composite electrodes for intermediate-temperature SOFCs [257,258,259,260], and as cathode contact materials [261,262]. Materials with mixed oxygen ion and electron conductivity (MIECs), such as Sr-doped lanthanum ferrites-nickelates/cobaltites (LSFN, LSFC) possess much higher oxygen mobility (Figure 18) enabling the O reduction reaction (ORR) along both triple- and double-phase boundaries, thus improving cathode performance, as well as oxygen permeation fluxes across oxygen-separation membranes [69,128,129,247,248,263]. The other state-of-the-art MIEC materials based on Sr-doped La cobaltites (LSC), showing a high catalytic activity in the ORR reaction and a high performance as SOFCs/SOECs air electrodes, demonstrate a high oxygen mobility and surface reactivity as well; moreover, it was reported that LSCs have higher D^* values compared to those for LSFCs (D^* up to 2 × 10⁻⁷ cm² s⁻¹ at 700 °C, Figure 18) [264,265,266,267,268]. Pr-nickelate-cobaltites (PNC) are stable to carbonation and interaction with electrolytes, which is a well-known issue for Sr-doped perovskites with an La-occupying A-site, possess total conductivity and oxygen diffusivity properties comparable or even exceeding those for LSFN and LSFC [55,58,69,243,269].

Mixed protonic-electronic or triple (H⁺/O²⁻/e⁻)-conductive perovskites and their composites based on compositions, such as doped Sr/Ba cerates/zirconates, are the materials for proton-conducting SOFCs (H-SOFC), including high-performance electrodes with triple-conducting behavior [55,111,113,167,183,184,185,186,270,271,272,273], as well as hydrogen separation membranes [55,99,107,139]. Protons in such perovskites are formed due to the hydrogenation or hydration of oxygen vacancies (Equations (40)–(44)). Therefore, one of the factors providing fine protonic transport properties is a high content of oxygen vacancies. Typical values of the hydrogen tracer diffusion coefficient for doped Ba and Sr cerates are ~10⁻⁶–10⁻⁵ cm² s⁻¹ at 700 °C (Figure 19) [272,273,274,275,276].

The group of promising materials recently studied in applications as low- and intermediate-temperature SOFC cathodes (including H-SOFC), as well as oxygen separation membranes, are double perovskites A₂B₂O_6−δ or A₂B₂O_5+δ (A = La, Pr, Ca, Ba, etc., B = Mg, Mo, Sn, Fe, etc.) [55,60,69,243,244,272,277,278,279]. Double perovskites are attractive because they can accommodate a large amount of nonstoichiometric oxygen, as well as having a wide variation in the effective charge of the B-site cations, and having high redox stability and moderate values of the thermal expansion coefficient. Along with this, they possess very high oxygen (D^*~10⁻⁸–10⁻⁷ cm² s⁻¹ at 700 °C) and, in some cases, hydrogen mobility (D^*~10⁻⁶ cm² s⁻¹ at 500 °C) [63,272,280]. The other promising perovskite-based layered materials to be mentioned here are triple [281,282], quadruple [283] and even quintuple perovskites [284].

6.4. Ruddlesden–Popper Phases

The Ruddlesden–Popper (RP) phases with the general formula of (AO)(ABO₃)_n or A_n+1B_nO_3n+1 consist of the perovskite layers ABO_3−δ alternating with the rock salt layers A₂O_2+δ [58,60,63,69,129,159,285,286,287,288,289,290]. The important feature of RP phases, which makes them attractive SOFC cathodes and oxygen-separation-membrane materials, is a fine oxygen transport provided via the cooperative mechanism of oxygen migration. In this case, both lattice and interstitial oxide anions accumulating at a high level are involved in the process of oxygen transport (Figure 20) [55,58,60,63,69,159,285,288,291,292,293,294,295,296,297]. This allows them to reach superior oxygen mobility compared to other MIECs (Figure 21). On the other hand, doping with alkaline earth metals (Ca, Sr, Ba), which significantly improves total conductivity, leads to an apparent decrease in the oxygen tracer diffusion coefficient values due to a decrease in the interstitial oxygen content and a larger size of dopant cations resulting in steric hindrances for the oxygen transport [287,288,292,298,299,300]. In some cases, it leads to the formation of slow diffusion channels with complicated pathways (Figure 21). The fraction of oxygen involved in the oxygen slow diffusion channel increases with the increasing cation-dopant radius in a row of Ca, Sr, Ba. With a decreasing host Ln cation size in the row of Ln = La, Pr, Nd, this effect becomes less pronounced. Introducing A-site deficiency can slightly increase oxygen diffusivity [55,69,292,301,302,303]. Doping La₂NiO_4+δ with other lanthanides (Nd, Sm, Gd, Eu, etc.) can slightly increase or decrease the oxygen mobility as well [177,304]. The information on the effect of doping RP nickelates in the B-site with such cations as Cu on the oxygen transport properties is still lacking and controversial. The oxygen diffusivity can increase while doping with Cu due to the elongation of Ni/Cu–O bonds [305,306] and anomalous grain growth can occur due to Cu-rich liquid phase presence during sintering [307]; it can decrease due to decreasing the oxygen content [308,309], and a non-monotonous dependence can be observed as well [310].

The RP phases of higher orders, different from the first-order ones being overstoichiometric and accumulating large amount of highly mobile interstitial oxygen, tend to be hypostoichiometric. Hence, they contain less amounts of interstitial oxygen in the rock salt layers and more oxygen vacancies in the perovskite layers. As a result, the oxygen diffusivity of the higher-order RP phases is lower compared to that of the first-order RP phases (Figure 22). For these materials, the contribution of the oxygen vacancy migration in the perovskite layers into the diffusion mechanism becomes predominant [108,288,311,312,313,314].

It was also reported [108] that some RP phases possess proton mobility, which results in accelerating the cathodic reaction process in H-SOFCs. Proton migration is believed to be implemented via the Grotthuss mechanism (Figure 8b). It includes two main pathways, namely, the inner-layer migration within the perovskite structure and the inter-layer migration between neighboring perovskite layers across the rock salt layer [108].

6.5. Composites

A promising approach in the design of materials for SOFC and permselective membranes is the synthesis of composites comprising both an electronic-conductive or MIEC material and an ionic conductive material [55,58,91,216,248,269,315]. The ionic-conductive component applied can be fluorite [55,60,244,316,317,318,319,320,321], pyrochlore [55,60,69], spinel [55,322], etc. Such an approach allows one to combine the features of the components, such as a high electronic conductivity of one component and a high ionic conductivity of another component, and even acquire new characteristics due to the synergetic effect of both phases, such as a fast oxide ionic transport across the interfaces or fast diffusion channels (Figure 23) [55,69,269]. For example, for PrNi_1−xCo_xO₃–Ce_0.9Y_0.1O_1.95 (x = 0.4–0.6), such a fast channel corresponds to the interfaces and the fluorite-like phase due to the redistribution of cations, namely, the incorporation of Pr^3+/4+ cations into the doped ceria structure [55,60,69,269,321].

However, using composites not always results in improving all transport properties. Thus, for the composites based on Pr_1.9NiO_4+δ and solid electrolytes (Ce_0.9Gd_0.1O_1.95, Y₂(Ti_0.8Zr_0.2)_1.6Mn_0.4O_7−δ) a decrease in the oxygen tracer diffusion coefficient values compared to the individual Pr_1.9NiO_4+δ material was reported, which is probably explained by the incorporation of the cations from the electrolyte into the RP structure leading to hampering the cooperative mechanism of oxygen migration mentioned above [58]. For the composites (Nd,La)_5.5(W,M)O_11.25−δ–Ni_0.5Cu_0.5O, a decrease in the oxygen tracer diffusion coefficient values compared to the individual defective fluorites was probably due to blocking the fluorite phase surface by Ni (II)–Cu (II) oxide nanoparticles possessing a low oxygen mobility, as well as the formation of admixture phases [216].

6.6. Other Materials

Some MIEC spinels such as Mn_xCo_3−xO₄ [55,323], Fe_0.6Mn_0.6Co_0.6Ni_0.6Cr_0.6O₄ [324], and LaFe₂O₄ [225] can be utilized as cathode materials for SOFCs, including proton-conducting cells, due to a high activity in the oxygen reduction reaction (ORR). MnFe₂O₄ spinel and its composite with Gd-doped ceria are used for the fabrication of the oxygen-permeable protecting (buffer) layer of asymmetric supported oxygen separation membranes [53,55,56,69,321].

Various types of oxide materials, which possess ionic conductivity due to cooperative oxygen migration mechanisms involving the cooperative motion of some forms of oxygen, can be used as SOFC electrolytes or, as a composite with electronically conductive or MIEC materials, as SOFC electrodes and oxygen separation membranes (or their permselective layers). Amongst these materials, doped La silicates/germanates with the apatite structure [58,69,107,134,325] (Figure 24), alkaline-earth-metal-doped La gallates with β-K₂SO₄ structure (Figure 25) [107,134,326], alkaline-earth metal ferrites, cobaltites, aluminates, gallates and indates with a brownmillerite structure (Figure 26) [107,134,327,328], M_3−xM’_xTi₂NbO_10−δ (M = Na, Ca, Cs; M = Bi, Ln, Rb) with a Dion–Jacobson-type layered perovskite structure [329,330,331], etc. [55,58,60,69] are to be mentioned. Mayenites based on Ca₁₂Al₇O_33, possessing a high oxygen mobility due to the fast transport of weakly bound intracellular ‘free’ oxygen (Figure 27), are to be mentioned as well [107,134,135,136,332]. Doping with Si mayenite possessing in general oxide-ionic type of conductivity allows one to increase the electronic conductivity, which is necessary for the cathode application [55,333]. It is to be noted that these materials, including apatites, brownmillerites, mayenites, etc., possess a high protonic conductivity [107,327,334,335]; hence, they can be used in H-SOFCs and hydrogen separation membranes as well. E.g., mayenites possess a high hydrogen diffusivity which is implemented by vehicle and Grotthuss mechanisms, including OH⁻ migration and the reorientation of O–H bonds to jump between neighboring oxygen species in (O–H–O)³⁻ transition states (Figure 28), as well as hydrogen jumps in a form of hydride H⁻ (Figure 29) and non-charged H⁰ [335].

Swedenborgite-like RBaCo_4−xM_xO₇ (R = Y, Ca, In, Lu, Yb, etc., M = Co, Zn, Fe, Al, Ga) phases were demonstrated to be potential cathodes for low-temperature SOFCs due to their low thermal expansion and excellent electrochemical performance; however, their phase decomposition at elevated temperatures of 700–800 °C limited their application [60,244,336,337].

Other materials with low thermal expansion coefficient values, high total conductivity and fine oxygen transport properties to be mentioned as candidate SOFC cathodes are yttrium iron garnet Y₃Fe₅O₁₂ [55,244,338], misfit layered Ca₃Co₄O₉-based phases [339,340,341,342,343,344], and Aurivillius oxides (Bi₂O₂)(A_m−1B_mO_3m+1) (A = Na⁺, K⁺, Ca²⁺, Sr²⁺, Pb²⁺, Bi³⁺, etc.; B = Ti⁴⁺, Nb⁵⁺, Ta⁵⁺, etc.) [345,346]. The Aurivillius oxide Bi₂Sr₂Nb₂MnO_12−δ notably demonstrates an excellent chemical stability (including CO₂ tolerance) as well. Ca₃Co₄O₉ demonstrates fast surface-exchange kinetics (k* = 1.6 × 10⁻⁷ cm s⁻¹ at 700 °C to be compared to 1.3 × 10⁻⁷ cm s⁻¹ for the nickelate) [347], and is promising for air cathodes used in all type SOFCs, H-SOFCs and reversible cells, individually or in composites with protonics [348], ionics [349] and MIECs [350].

Alkaline-earth-metal-doped lanthanide niobates with sheelite, defective perovskite, monoclinic and tetragonal structures possess ionic (protonic and/or oxide-ionic), electronic or mixed ionic-electronic conductivity [58,351,352,353,354,355,356,357]. They can be used as a component of the composites for hydrogen separation membranes such as (La,Ca)NbO₄–La₃NbO₇, (La,Ca)NbO₄–LaNb₃O₉ and (La,Ca)NbO₄–NiCu [58,351,352,353,354,355,356,357].

Figure 30 demonstrates the oxygen mobility of some non-conventional materials for SOFCs and permselective membranes.

Metals and their alloys which can intercalate and transport protons as a defect (Figure 8a) are widely used for hydrogen separation membranes. Precious metals such as Pt, Pd, Ru, Ag and their alloys are conventionally used as hydrogen separation membrane materials. They possess absolute selectivity with respect to hydrogen; however, they are too expensive and have issues with stability under operating conditions [49,55,56,84,85,99,139,358]. As an alternative to precious metals, Ni and its alloys, being cheap but also possessing a high mixed protonic-electronic conductivity, can be used in hydrogen separation membranes in an individual form or as a component of cermet composites [55,56,132,216,359,360]. V and its alloys with Ni, Cu, V, Nb, Ta and other metals are promising materials showing high hydrogen permeation fluxes exceeding those for Pd-based membranes and having a lower cost [55,359,360,361]. The comparison of the hydrogen self-diffusion coefficient values of various metals and alloys is given in Figure 31.

7. Conclusions and Perspectives

In this review, the importance of oxygen and hydrogen mobility for the performance of solid oxide fuel cells, oxygen and hydrogen separation membranes was highlighted. Detailed studies of ionic transport characteristics using modern techniques such as temperature-programmed isotope exchange of oxygen with C¹⁸O₂, ECR technique, etc., were shown to widen the possibility of the design of advanced materials for these applications. Analysis of the modern literature of isotope-exchange methods demonstrated the necessity of the further development of isotope-exchange techniques, including the usage of new labelled oxygen or hydrogen containing molecules to study the bulk diffusion and the surface exchange processes. The development of new approaches for such data analysis or improving the existing ones in combination with the data acquired by relaxation techniques, structural and spectroscopic methods will help to elucidate atomic-scale factors controlling the mechanisms of diffusion and surface exchange properties.

Moreover, a lack of data should be noted regarding the oxygen and hydrogen transport properties of many functionally attractive and promising materials for electrochemical devices with decreased operating temperature, which requires their further study. One of the interesting approaches in creating the materials for these devices is related to triple-conductive (H⁺ + O²⁻ + e⁻) oxides and composites, which may demonstrate better characteristics compared to the conventional mixed O²⁻ + e⁻ or H⁺ + e⁻ conductive materials. Studying the oxygen-transport properties of the proton conductors and vice versa would allow us to expand the known number of triple-conductive materials and find new applications for these materials. Selecting solid oxide fuel cell materials with a high oxygen and/or hydrogen mobility and surface reactivity allows to decrease the fuel cell operating temperature and increase its power density due to reducing the electrolyte resistance and enabling the electrode processes to take place on the electrode–gas phase double-phase boundary not to be limited by the electrode–electrolyte–gas phase triple-phase boundary. This opens new perspectives in the solid oxide fuel cells design and manufacturing.

Mixed ionic-electronic conducting materials for permselective membranes with a high oxygen and hydrogen mobility and surface reactivity, as well as a high electronic conductivity, allows the obtaining of high permeation fluxes of oxygen and hydrogen, respectively. Along with this, the presence of oxygen component of the conductivity of hydrogen separation membrane materials allows us to increase the hydrogen yield. This opens the opportunity for creating new membrane materials followed by their selection for the prospective practical use based on their superior transport properties.