1. Introduction

2. Oxygen and hydrogen diffusion in MIEC materials

2.1. 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 is satisfied:

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

(1)

where D_i is a diffusion coefficient of i-th species, C_i is their concentration. Strictly speaking, the Equation (1) 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 [68,69],[70]. In this case, in a linear non-equilibrium thermodynamics region, the Equation (1) can be written as follows:

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

(2)

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 $\overrightarrow{\nabla }{C}_l=0$.

The Fick’s second law follows from the Fick’s first law and the mass conservation:

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

(3)

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 [71,72,73,74]:

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

(4)

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:

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

(5)

where f_i is a correlation factor which is related to influence of counterflows of ions of different isotopes [71,73,74,75]. The ratio

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

(6)

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. [75,76,77,78,79,80,81]. 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:

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

(7)

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 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 (1)), the Nernst – Plank equation [82,83,84]

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

(8)

or the modified Fick’s first law [74,82,85]

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

(9)

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 [71,74,75]:

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

(10)

For oxide-ionic and mixed oxide-ionic – electronic conductors the following equations for the thermodynamic factor are known:

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

(11)

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

(12)

where t_h is the hole transport number, $P_{{\mathrm{O}}_2}$ is the partial pressure of oxygen in the gas phase, $C_{{\mathrm{V}}_{\mathrm{O}}^{••}}$ and C_h are the concentrations of oxygen vacancies and holes in the oxide, respectively [74,75,85,86,87,88].

For protonic conductors, more complex relationship of chemical diffusion coefficient and self-diffusion coefficients of charge carriers is given in the work [68]:

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

(13)

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,

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

(14)

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

(15)

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 [89].

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 is given in the work [90]:

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

(16)

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.

2.2. Oxygen and hydrogen diffusion mechanisms

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

• Vacancy mechanism (Figure 2 (a)): transport of regular oxide anions into neighboring vacancies; this mechanism is typical for perovskites, fluorites and many other types of oxides [53,91,92,93,94];
• Interstitial mechanism (Figure 2 (b)): transport of interstitial oxide anions into neighboring interstitial sites; this mechanism is typical for some pyrochlores, mayenites and some other oxides [91,93,94,95,96,97];
• Cooperative mechanism (Figure 2 (c)): 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 [75,91,93,94,95,96,98].

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 4.

The main mechanisms of hydrogen diffusion are:

• Diffusion of protons through interstitial defects (Figure 3 (a)); this mechanism is typical for metals and alloys [90,99,100,101,102];
• Grotthuss mechanism (Figure 2 (b)): 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 [68,102,103];
• Vehicle mechanism (Figure 3 (c)): transport of protons together with the neighboring oxide anion as a hydroxyl; this mechanism is also typical for proton-conducting oxides [68,102,103];
• Diffusion of structurally bound water (Figure 3 (d)): transport of water species embedded into the lattice; this mechanism is proposed for some oxides [104,105].

Figure 2. General types of oxygen diffusion mechanisms in oxides: (a) vacancy mechanism; (b) interstitial mechanism; (c) cooperative mechanism [93,94,95,96].

Figure 2. General types of oxygen diffusion mechanisms in oxides: (a) vacancy mechanism; (b) interstitial mechanism; (c) cooperative mechanism [93,94,95,96].

3. Isotope exchange of oxygen and hydrogen

4. Oxygen and hydrogen mobility of materials for membranes and SOFC

5. Importance of oxygen and hydrogen transport properties for the performance of membranes and SOFCs

5.1. 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’ pathway 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 27). This allows to use such membranes for pure oxygen production as well as a part of catalytic membrane reactors for fuels transformation reactions [52,54,62,75,94,310,311,312,313,314,315].

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

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

(44)

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 the different sites of the membrane, σ_O and σ_el are oxide-ionic and electronic conductivity, respectively [94]. In MIEC materials, σ_O << σ_el. If the ionic conductivity is constant across the entire membrane thickness, the Equation (44) can be simplified as follows:

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

(45)

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 (Equation (4)) can be re-written as follows:

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

(46)

where δ₀ is the nonstoichiometric oxygen at the reference oxygen pressure (1 atm), V_m is the molar volume of the oxide. Combining this with the Equation (44) and assuming σ_O << σ_el one can obtain Sievert’s law:

(47)

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

Considering the effect of the surface exchange of oxygen, the Wagner equation (45) is transformed into the modified Wagner equation introduced by Bouwmeester et al. [316]:

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

(48)

where L_C is the characteristic thickness (Equation (33)).

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 [94], Xu and Thomson’s model [94,317], Zhu’s model [94,314,315]. E.g., Zhu’s model (Figure 28) is based on the Wagner equation and takes into account 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 taken into account along with the properties of gas-phase diffusion in a porous support [54,62,318,319,320]. However, gas-phase phenomena are out of scope of this review.

5.2. 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 to reach high hydrogen permeation fluxes for obtaining pure hydrogen including its production in catalytic membrane reactors for fuel transformation reactions [41,54,62,170,236,304,305,321,322,323,324]. There are advantages in using triple (H⁺/O^2–/e^–) conducting materials for hydrogen separation membranes since the presence of oxide-ionic component of the conductivity can enable the following features:

• Some proton transport mechanisms being mediated by oxygen transport as mentioned in the Section 2.2 [101,102,104];
• Oxide ions counterpermeation across the membrane allows to increase the hydrogen yield due to the water splitting reaction [324,325,326];
• 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 its chemical potential gradient [327,328,329].

The processes in the triple-conductive hydrogen separation membrane are illustrated in Figure 29.

For dense metallic membranes, hydrogen concentration in metal is proportional to $P_{{\mathrm{H}}_2}^{0.5}$ [323]. Similar to MIEC oxides with the variation of oxygen nonstoichiometry on the oxygen partial pressure (Equation (47)), the Sievert’s law can be obtained:

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

(49)

where

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

(50)

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 ideal case is equal to 0.5 (for real membranes it lies in the range of ~0.5 – 1), K_s is the hydrogen solubility constant (Sievert’s constant) [322,323,330].

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

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

(51)

where σ_H is the proton conductivity [99,322,331]. Since proton and electronic conductivity may depend on $P_{{\mathrm{H}}_2}$, the result of integrating in the Equation (51) 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 $j_{{\mathrm{H}}_2}=-\frac{RT}{4F^{2}L}{\sigma }_{H,0}\ln \left(\frac{P_{{\mathrm{H}}_2}^{II}}{P_{{\mathrm{H}}_2}^I}\right)$ [99,322,332,333,334].

In the case of a cermet membrane, the equation for its hydrogen permeation flux combines those for the ceramic (Equation (51)) and metallic (Equation (49)) components:

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

(52)

where x_ceram is the volume fraction of the ceramic component, Pe_metal is the permeability of the metallic component [321].

For triple-conductive membranes, the oxide-ionic component of the conductivity should be accounted [324,334]:

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

(53)

(54)

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

(55)

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 [54], [99], [335], [336], [337]. However, gas-phase phenomena are out of scope of this review.

6. Conclusions and perspectives

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