# The Development of New Perovskite-Type Oxygen Transport Membranes Using Machine Learning

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

^{†}

## Abstract

**:**

_{1−x}Ba

_{x}(Ti

_{1−y−z}V

_{y}Fe

_{z})O

_{3−δ}(cubic perovskite-type phases). We have evaluated available experimental data, determined missing crystallographic information using bond-valence modeling and programmed a Python code to be able to generate training data sets for property predictions using machine learning. Indeed, suitable compositions of cubic perovskite-type phases can be predicted in this way, allowing for larger electronic conductivities of up to σ

_{e}= 1.6 S/cm and oxygen conductivities of up to σ

_{i}= 0.008 S/cm at T = 1173 K and an oxygen partial pressure p

_{O2}= 10

^{−15}bar, thus enabling practical applications.

## 1. Introduction

_{x}Fe

_{1−x}O

_{3−δ}[9] and Ba

_{0.5}Sr

_{0.5}Co

_{0.8}Fe

_{0.2}O

_{3−δ}[10,11]. Oxygen transport in dense ceramic membranes is driven by the partial pressure gradient across the membrane [9,12,13]. As mixed conducting materials single-phase perovskite can be used. Based on SrTiO

_{3}(STO), materials with the general structural formula ABO

_{3}offer a range of uses as functional materials in a variety of energy applications. For example, pure STO is used as a dielectric in electronic components. The crystal structure allows for a large number of dopants, making it possible to selectively introduce conductivities for electrons and/or oxygen ions into the material. Functionalized and doped STO materials are therefore used, e.g., as thermoelectrics, in the photovoltaic industry as well as in ceramic fuel cells or as gas separation membranes (high electronic and ionic conductivity) [12]. At the same time, the STO host lattice offers high intrinsic stability, which enables true long-term operation. As an example, dopants with Ba (A-position in the crystal structure) and V/Fe (B-position) are selected and a methodology combining available experimental data, chemical bond modeling, and machine learning is developed. Doping with toxic Co will be deliberately avoided, as well as the use of rare earths (La, Ce, Sm), in order to keep production costs as low as possible. The properties of the chemical bonds and the resulting electron density distributions are decisive for the functional properties. Therefore, these are modeled using the bond valence method, as this requires significantly less computational power compared to ab initio methods (e.g., DFT) with similar information and provides faster results. In this way, screening of possible candidate materials is attainable, and oxygen diffusion as well as electronic conductivity can be optimized. However, there are frequent cases where ionic oxygen transport is accompanied by phase changes depending on temperature and oxygen partial pressure operating conditions. De Souza [14] published a comprehensive review of oxygen diffusion in undoped SrTiO

_{3}and related perovskite oxides that illustrates the relationship between defect chemistry, diffusion, and conductivity. There are some references regarding B-site substituted SrTiO

_{3}with V [15,16,17], but they are mainly related to the application of SrTiO

_{3}-based ceramics such as thermoelectrics, solar cells, and sensors. However, such phases can also be used as stable gas separation membranes suitable for long-term application [6]. Doping of SrTiO

_{3}with Ba at the A position (ABO

_{3}) enhances O

_{2}diffusion [6,18]. Inclusion of V at the B position increases electronic conductivity [6]. V

^{5+}reduces an equivalent amount of Ti

^{4+}to Ti

^{3+}, which in turn increases the electronic conductivity. As long as the cation radius of the dopant on the B side is smaller than that of ${\mathrm{Ti}}_{\mathrm{VI}}^{4+}$ (r

_{k}= 0.605 Å), the oxygen conductivity is always increased [14]. In fact, this is the case for ${\mathrm{V}}_{\mathrm{VI}}^{4+}$ (r

_{k}= 0.580 Å), ${\mathrm{V}}_{\mathrm{VI}}^{5+}$ (r

_{k}= 0.540 Å), and ${\mathrm{Fe}}_{\mathrm{VI}}^{3+}$ (r

_{k}= 0.55 Å), but not for ${\mathrm{Fe}}_{\mathrm{VI}}^{2+}$ (r

_{k}= 0.610 Å) (ionic radii: Shannon and Prewitt [19]). Here, Fe

^{2+}and Fe

^{3+}are assumed to be low-spin on the octahedral sites in the perovskite crystal structure. Experimentally determined cell constants of SrTi

_{x}Fe

_{1−x}O

_{3−δ}phases deliver strong evidence for this assumption, because cell constants clearly decrease with increasing iron content [9]. With relatively low doping with V crystal structure stability is maintained. The Ti-O and V-O bonds are almost equally strong, i.e., the incorporation of V does not affect the BO

_{6}network, which largely determines the stability of the perovskite structure, and V+Ti generally increases temperature stability, that can be estimated by calculating a tolerance factor t [6]. This factor is defined as $t=({r}_{A}+{r}_{O})/\sqrt{({r}_{B}+{r}_{O})}$ and for a stable structure t should be equal to 1.0. Bond-valence modeling (BVM) can be used to calculate stable compositions in advance before synthesis, hence saving a lot of time. One scientific goal of this project is to develop a better understanding of the relationship between chemical composition, tolerance factor t, critical radius r

_{c}, the free volume FV in the crystal and in the micro structure, temperature T, and the binding energies of the metal–oxygen bonds. By combining available reference data and BVM for data supply, and subsequent machine learning (ML) for the prediction of promising chemical compositions, based on the supplied data, empirical trial-and-error methods will be avoided and a systematic way for the development of new ceramic ionic conductors will be established. Therefore, in this project structural parameters and conductivities of SrTiO

_{3}, SrVO

_{3}, Sr(Ti

_{1−y}V

_{y})O

_{3}, Sr(Ti

_{1−z}Fe

_{z})O

_{3−δ}, (Sr

_{0.5}Ba

_{0.5})(Ti

_{0.5}Fe

_{0.5})O

_{3−δ}, (Sr

_{1−x}Ba

_{x})(Ti

_{1−y}V

_{y})O

_{3}, and (Sr

_{1−x}Ba

_{x})(Ti

_{1−y−z}V

_{y}Fe

_{z})O

_{3−δ}solid solutions, as a function of composition, temperature, and oxygen partial pressure were determined. The results are largely based on experimental data, and to a small extent on BVM. Conductivities are calculated, as far as possible, only for the practically relevant temperature range between 950 and 1223 K and oxygen partial pressures between 1 and 10

^{−20}bar (depending on composition).

## 2. Materials and Methods

#### 2.1. Experimental Reference Data

_{3}, and just for electronic conductivities, calculations were related to SrTiO

_{3}, O-O bond lengths (1. order), and the free volume. For Sr(Ti

_{1−y}V

_{y})O

_{3}also no experimental values for oxygen conductivities are available, just for electronic conductivities, and only at T = 1173 K. Therefore, calculations are also related to SrTiO

_{3}, O-O bond lengths (1. order), and the free volume (FV). In case of Sr(Ti

_{1−z}Fe

_{z})O

_{3−δ}conductivity data are only available at T = 1123 K for the whole compositional range, and especially conductivity data are available at T = 973 to 1223 K for z = 0.4 to 0.8. Here, the space group is P

_{m}− 3m even up to z = 0.8. Only data for oxidizing conditions (pO

_{2}= 0.213 bar) are available. For compositions containing the cations Sr, Ba, Ti, and Fe only data for the specific composition (Sr

_{0.5}Ba

_{0.5})(Ti

_{0.5}Fe

_{0.5})O

_{3−δ}are available and only data for oxidizing conditions (pO

_{2}= 0.213 bar). For the two compositions (Sr

_{1−x}Ba

_{x})(Ti

_{1−y}V

_{y})O

_{3}and (Sr

_{1−x}Ba

_{x})(Ti

_{1−y−z}V

_{y}Fe

_{z})O

_{3−δ}no experimental data are available at all. Therefore, cell constants and tolerance factors were calculated using BVM (see following section), and only reducing conditions were considered. A first evaluation and analysis of the experimental data was performed with the statistical program R (https://www.r-project.org, accessed on 1 April 2022 ), in order to detect outlier data and to select the most precise and accurate reference data. The finally applied reference data are taken from the references [8,9,14,16,17,20,21,22,23,24,25,26,27,28,29,30,31,32,33].

#### 2.2. Bond-Valence Modeling

_{ij}to an atom j is equal to its valence V

_{j}. For an atom symmetrically coordinated by M similar atoms, the relationship is s

_{ij}= V

_{j}/M. If the bonds are not equal, a relationship between bond length and bond order is required, such as s

_{ij}= $exp[({d}_{0}-{d}_{ij})/b]$, where ${d}_{0}$ is the length of a single bond between atom j and atom i and ${d}_{ij}$ is the actual distance [34,35,36]. The constant b is assumed to be $0.37\phantom{\rule{0.166667em}{0ex}}\AA $ for most structures. The method is good at assigning oxidation states, and ${\mathrm{O}}_{2}$, OH, and ${\mathrm{H}}_{2}\mathrm{O}$ can be distinguished. Possible H and Li positions can be predicted, and also conduction paths in ionic conductors [37,38,39,40]. Yamada et al. [42] recently showed that the structural stability of ${\mathrm{SrTiO}}_{3}$ and ${\mathrm{CaTiO}}_{3}$ can be calculated equally well by BVM and DFT. Inoue et al. [40] discovered a completely new family of oxide ionic conductors ${\mathrm{Ca}}_{0.8}{\mathrm{Y}}_{2.4}{\mathrm{Sn}}_{0.8}{\mathrm{O}}_{6}$ by the combined application of synchrotron powder diffraction experiments and BVM modeling. The SPuDS software [43,44] enables the prediction of perovskite-type crystal structures with BVM. Modeling was performed using the following strategy: Depending on composition one or two cations (${\mathrm{Sr}}^{2+}$, ${\mathrm{Ba}}^{2+}$) were allowed on the A-site of the perovskite crystal structure, and additionally one, two, or three cations on the B-site (${\mathrm{Ti}}^{4+}$, ${\mathrm{V}}^{4+}$, ${\mathrm{Fe}}^{2+}/{\mathrm{Fe}}^{3+}$). For a given temperature the correct space group was chosen (e.g., $Pm-3m$ for ${\mathrm{SrTiO}}_{3}$ at $T=973$ K), the Glazer tilt system (e.g., a0a0a0 for ${\mathrm{SrTiO}}_{3}$), and the fractions of the different cations on the two possible sites, respectively. Because no oxidizing conditions were relevant in this work for solid solutions containing vanadium, no ${\mathrm{V}}^{5+}$ was considered. Cation ordering on the B-site was allowed, as well as a variation of the average B-site volume. Tilt angles were not pre-defined, but were refined during modeling. Calculated values (as a function of temperature) were for example the global instability index GII, the tolerance factor t, the tilt angle, the bond valence sum for each ion, and the lattice parameters. The lattice parameters and the tolerance factor were used as input for the program Pecon.py, which is described in the following section. For detailed definitions of the parameters mentioned above see references [43,44].

#### 2.3. Data Analysis and Python Programming

**Pe**rovskite

**con**ductiviy). With this program the structural parameters and conductivities of the pure perovskite-type phases and the solid solutions of interest ${\mathrm{SrTiO}}_{3}$, ${\mathrm{SrVO}}_{3}$, $\mathrm{Sr}\left({\mathrm{Ti}}_{1-\mathrm{y}}{\mathrm{V}}_{\mathrm{y}}\right){\mathrm{O}}_{3}$, $\mathrm{Sr}\left({\mathrm{Ti}}_{1-\mathrm{z}}{\mathrm{Fe}}_{\mathrm{z}}\right){\mathrm{O}}_{3-\delta}$, $\left({\mathrm{Sr}}_{0.5}{\mathrm{Ba}}_{0.5}\right)\left({\mathrm{Ti}}_{0.5}{\mathrm{Fe}}_{0.5}\right){\mathrm{O}}_{3-\delta}$, $\left({\mathrm{Sr}}_{1-\mathrm{x}}{\mathrm{Ba}}_{\mathrm{x}}\right)\left({\mathrm{Ti}}_{1-\mathrm{y}}{\mathrm{V}}_{\mathrm{y}}\right){\mathrm{O}}_{3}$, and $\left({\mathrm{Sr}}_{1-\mathrm{x}}{\mathrm{Ba}}_{\mathrm{x}}\right)\left({\mathrm{Ti}}_{1-\mathrm{y}-\mathrm{z}}{\mathrm{V}}_{\mathrm{y}}{\mathrm{Fe}}_{\mathrm{z}}\right){\mathrm{O}}_{3-\delta}$ can be calculated as a function of composition, temperature, and oxygen partial pressure. The results are largely based on fits and interpolation of analyzed, experimental data [8,9,14,16,17,20,21,22,23,24,25,26,27,28,29,30,31,32,33], and to a small extent on bond-valence modeling. Electronic and oxygen ion conductivities are calculated, as far as possible, only for the practically relevant temperature range between $T=950$ to 1223 K (depending on composition). Appendix A shows the input that has to be given by the user: Chemical composition (characterized by three dimensionless variables x, y and z), temperature (K), and the oxygen partial pressure (bar). If the input parameters are outside pre-defined limits, the user receives an error message. Based on the user input Pecon.py calculates the space group and the related crystal data (cell constants, volume of the unit cell, atomic number density, bond lengths, and inter atomic distances). Additionally, the tolerance factor is calculated for the chosen temperature, as well as three conductivities (total, electronic, and ionic). Furthermore, parameter values that are related to conductivity are given (critical radius, free volume, and oxygen diffusion saddle point [20]). For this purpose within the program Pecon.py the data are fitted with polynomials of 2nd to 4th degree, or with exponential functions. Only interpolations between known data boundaries take place, but no extrapolations. Only experimental data were considered and no results from quantum mechanics (e.g., DFT) or other sources. Gaps of structural data (especially cell constants and tolerance factors) were filled with results achieved using BVM, as described above. All results calculated by Pecon.py are written to separate text files, which can be further used. One application is the generation of training data sets, that can be used for subsequent machine learning simulations (see following section). The detailed use of Pecon.py is shown in Appendix A and possible results are given in Appendix B.

#### 2.4. Machine Learning

## 3. Results

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

Auto-WEKA | Automatic Model Selection and Hyperparameter Optimization in WEKA |

BVM | Bond Valence Model |

CLI | Command Line Interface |

DFT | Density Functional Theory |

FV | Free Volume |

lazy.IBK | K-nearest Neighbours (WEKA Classifier) |

MD | Molecular Dynamics |

ML | Machine Learning |

$p{O}_{2}$ | Oxygen Partial Pressure |

SG | Space Group |

SVM | Support Vector Machine (WEKA Classifier) |

STO | ${\mathrm{SrTiO}}_{3}$ |

SVO | ${\mathrm{SrVO}}_{3}$ |

STVO | $\mathrm{Sr}\left({\mathrm{Ti}}_{1-\mathrm{y}}{\mathrm{V}}_{\mathrm{y}}\right){\mathrm{O}}_{3}$ |

STFO | $\mathrm{Sr}\left({\mathrm{Ti}}_{1-\mathrm{z}}{\mathrm{Fe}}_{\mathrm{z}}\right){\mathrm{O}}_{3-\delta}$ |

SBTFO | $\left({\mathrm{Sr}}_{0.5}{\mathrm{Ba}}_{0.5}\right)\left({\mathrm{Ti}}_{0.5}{\mathrm{Fe}}_{0.5}\right){\mathrm{O}}_{3-\delta}$ |

SBTVO | $\left({\mathrm{Sr}}_{1-\mathrm{x}}{\mathrm{Ba}}_{\mathrm{x}}\right)\left({\mathrm{Ti}}_{1-\mathrm{y}}{\mathrm{V}}_{\mathrm{y}}\right){\mathrm{O}}_{3}$ |

SBTVFO | $\left({\mathrm{Sr}}_{1-\mathrm{x}}{\mathrm{Ba}}_{\mathrm{x}}\right)\left({\mathrm{Ti}}_{1-\mathrm{y}-\mathrm{z}}{\mathrm{V}}_{\mathrm{y}}{\mathrm{Fe}}_{\mathrm{z}}\right){\mathrm{O}}_{3-\delta}$ |

WEKA | Waikato Environment for Knowledge Analysis |

## Appendix A

SrTiO3 | -> | T = 0 to 2313 K |

SrVO3 | -> | T = 0 to 1956 K |

SrTi(1-y)V(y)O3 | -> | T = 973 to 1173 K |

SrTi(1-z)Fe(z)O3 | -> | T = 973 to 1173 K |

Sr(0.5)Ba(0.5)Ti(0.5)Fe(0.5)O3 | -> | T = 1073 to 1223 K |

Sr(1-x)Ba(x)Ti(1-y)V(y)O3 | -> | T = 973 to 1173 K |

Sr(1-x)Ba(x)Ti(1-y-z)V(y)Fe(z)O3 | -> | T = 973 to 1173 K |

SrTiO3 | -> | T = 950 to 1173 K |

SrVO3 | -> | T = 973 to 1173 K |

SrTi(1-y)V(y)O3 | -> | T = 1173 K |

SrTi(1-z)Fe(z)O3 | -> | T = 973 to 1223 K |

Sr(0.5)Ba(0.5)Ti(0.5)Fe(0.5)O3 | -> | T = 1073 to 1223 K |

Sr(1-x)Ba(x)Ti(1-y)V(y)O3 | -> | T = no data available |

Sr(1-x)Ba(x)Ti(1-y-z)V(y)Fe(z)O3 | -> | T = no data available |

SrTiO3 | -> | pO2 = $1.0\times {10}^{-20}$ to 1.0 bar |

SrVO3 | -> | pO2 = $1.0\times {10}^{-20}$ to $1.0\times {10}^{-15}$ bar |

SrTi(1-y)V(y)O3 | -> | pO2 = $1.0\times {10}^{-20}$ to $1.0\times {10}^{-11}$ bar |

SrTi(1-z)Fe(z)O3 | -> | pO2 = 0.213 bar |

Sr(0.5)Ba(0.5)Ti(0.5)Fe(0.5)O3 | -> | pO2 = 0.213 bar |

Sr(1-x)Ba(x)Ti(1-y)V(y)O3 | -> | pO2 = $1.0\times {10}^{-20}$ to $1.0\times {10}^{-15}$ bar |

Sr(1-x)Ba(x)Ti(1-y-z)V(y)Fe(z)O3 | -> | pO2 = $1.0\times {10}^{-15}$ bar |

Input x-value equal 0.0 (no Ba2+ on the A-site) or > 0.0 and <= 0.5: |

Input y-value equal 0.0 (SrTiO3) or 1.0 (SrVO3) or > 0.0 and <= 0.5 (SrTi(1-y)V(y)O3): |

Input z-value equal 0.0 (no Fe2+/3+ on the B-site) or > 0.0 and <= 0.5 or <= 0.8 |

(only SrTi(1-z)Fe(z)O3): |

Input temperature T (K): |

Input oxygen partial pressure pO2 (bar): |

## Appendix B

Phase | = | SrTiO3 | |

User input | x | = | 0.000000 |

User input | y | = | 0.000000 |

User input | z | = | 0.000000 |

User input | T | = | 973.000000 K |

User input | pO2 | = | $1.000000\times {10}^{-15}$ bar |

Cubic crystal structure | SG | = | $Pm-3m$ |

Cell constant | a | = | 3.918795 Å |

Volume of the unit cell | V | = | 60.180762 Å${}^{3}$ |

Atomic number density | N | = | 0.083083 atoms/Å${}^{3}$ |

Tolerance factor | t | = | 1.014169 |

O-O distance (1. order) | = | 2.771007 Å | |

O-O distance (2. order) | = | 3.918795 Å | |

Ti-O distance | = | 1.959398 Å | |

Sr-O distance | = | 2.771007 Å | |

Ti-Ti distance | = | 3.918795 Å | |

Sr-Sr distance | = | 3.918795 Å | |

Ti-Sr distance | = | 3.393776 Å |

Critical radius | r(c) | = | 0.895343 Å |

Free volume | FV | = | 15.827530 Å${}^{3}$ |

O2- diffusion saddle point | ODSP | = | 0.439857 |

Total conductivity | Sigma(t) | = | $1.1316857455\times {10}^{-4}$ S/cm |

Electronic conductivity | Sigma(e-) | = | $1.0884879227\times {10}^{-4}$ S/cm |

Oxygen conductivity | Sigma(O2-) | = | $4.3197822829\times {10}^{-6}$ S/cm |

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**Figure 1.**The workflow applied in this work, using the programs R, SPuDS, Pecon.py, WEKA, and Auto-WEKA, respectively.

**Figure 2.**Electronic conductivity of ${\mathrm{SrTiO}}_{3}$ at $p{O}_{2}={10}^{-15}$ bar (Line: calculated by Pecon.py; Points: predicted by ML using WEKA).

**Figure 3.**Electronic conductivity of ${\mathrm{SrVO}}_{3}$ at $p{O}_{2}={10}^{-15}$ bar (Line: calculated; Points: predicted).

**Figure 4.**Electronic conductivity of $\mathrm{Sr}\left({\mathrm{Ti}}_{1-\mathrm{y}}{\mathrm{V}}_{\mathrm{y}}\right){\mathrm{O}}_{3}$ as a function of composition, expressed by the variable average cation radius on the B-site, corresponding to a y-range from 0.48 down to 0 (from left to right), at $T=1173$ K and $p{O}_{2}={10}^{-15}$ bar (Line: calculated; Points: predicted).

**Figure 5.**Electronic conductivity of $\mathrm{Sr}\left({\mathrm{Ti}}_{1-\mathrm{z}}{\mathrm{Fe}}_{\mathrm{z}}\right){\mathrm{O}}_{3-\delta}$ as a function of composition, expressed by the variable average cation radius on the B-site, corresponding to a z-range from 0.78 down to 0 (from left to right), at $T=1173$ K and $p{O}_{2}=0.213$ bar (Line: calculated; Points: predicted).

**Figure 6.**Ionic conductivity of ${\mathrm{SrTiO}}_{3}$ at $p{O}_{2}={10}^{-15}$ bar (Line: calculated; Points: predicted).

**Figure 7.**Electronic conductivity of $\left({\mathrm{Sr}}_{0.5}{\mathrm{Ba}}_{0.5}\right)\left({\mathrm{Ti}}_{0.5}{\mathrm{Fe}}_{0.5}\right){\mathrm{O}}_{3-\delta}$ at $p{O}_{2}=0.213$ bar (Line: calculated; Points: predicted).

**Figure 8.**Ionic conductivity of $\left({\mathrm{Sr}}_{0.5}{\mathrm{Ba}}_{0.5}\right)\left({\mathrm{Ti}}_{0.5}{\mathrm{Fe}}_{0.5}\right){\mathrm{O}}_{3-\delta}$ at $p{O}_{2}=0.213$ bar (Line: calculated; Points: predicted).

**Figure 9.**Figure of merit of the ambivalent conductivity of $\left({\mathrm{Sr}}_{0.5}{\mathrm{Ba}}_{0.5}\right)\left({\mathrm{Ti}}_{1-\mathrm{y}-\mathrm{z}}{\mathrm{V}}_{\mathrm{y}}{\mathrm{Fe}}_{\mathrm{z}}\right){\mathrm{O}}_{3-\delta}$ at $p{O}_{2}={10}^{-15}$ bar and $T=1173$ K (see text and Table 2).

**Table 1.**Availability of experimental electronic conductivity data ${\sigma}_{e}$ and ionic conductivity data ${\sigma}_{i}$, respectively. For more details see Appendix A.

Composition | ${\mathit{\sigma}}_{\mathit{e}}$ | ${\mathit{\sigma}}_{\mathit{i}}$ |
---|---|---|

SrTiO${}_{3}$ | YES | YES |

SrVO${}_{3}$ | YES | NO |

$\mathrm{Sr}\left({\mathrm{Ti}}_{1-\mathrm{y}}{\mathrm{V}}_{\mathrm{y}}\right){\mathrm{O}}_{3}$ | YES | NO |

$\mathrm{Sr}\left({\mathrm{Ti}}_{1-\mathrm{z}}{\mathrm{Fe}}_{\mathrm{z}}\right)\mathrm{O}$${}_{3-\delta}$ | YES | YES |

$\left({\mathrm{Sr}}_{0.5}{\mathrm{Ba}}_{0.5}\right)\left({\mathrm{Ti}}_{0.5}{\mathrm{Fe}}_{0.5}\right)\mathrm{O}$${}_{3-\delta}$ | YES | YES |

$\left({\mathrm{Sr}}_{1-\mathrm{x}}{\mathrm{Ba}}_{\mathrm{x}}\right)\left({\mathrm{Ti}}_{1-\mathrm{y}}{\mathrm{V}}_{\mathrm{y}}\right){\mathrm{O}}_{3}$ | NO | NO |

$\left({\mathrm{Sr}}_{1-\mathrm{x}}{\mathrm{Ba}}_{\mathrm{x}}\right)\left({\mathrm{Ti}}_{1-\mathrm{y}-\mathrm{z}}{\mathrm{V}}_{\mathrm{y}}{\mathrm{Fe}}_{\mathrm{z}}\right)\mathrm{O}$${}_{3-\delta}$ | NO | NO |

**Table 2.**Predicted electronic and ionic (oxygen) conductivities ${\sigma}_{e}$ and ${\sigma}_{i}$ at $T=1173$ K, respectively.

Composition | ${\mathit{pO}}_{2}$ (bar) | FV (Å${}^{3}$) | ${\mathit{\sigma}}_{\mathit{e}}\phantom{\rule{0.166667em}{0ex}}(\mathbf{S}/\mathbf{cm})$ | ${\mathit{\sigma}}_{\mathit{i}}\phantom{\rule{0.166667em}{0ex}}(\mathbf{S}/\mathbf{cm})$ |
---|---|---|---|---|

${\mathrm{SrTiO}}_{3}$ | $1.0\times {10}^{-15}$ | 16.011 | $1.67\times {10}^{-2}$ | $2.51\times {10}^{-5}$ |

${\mathrm{SrVO}}_{3}$ | $1.0\times {10}^{-15}$ | 13.377 | 545.09 | $2.27\times {10}^{-5}$ |

$\mathrm{Sr}\left({\mathrm{Ti}}_{0.5}{\mathrm{V}}_{0.5}\right){\mathrm{O}}_{3}$ | $1.0\times {10}^{-15}$ | 15.320 | 20.01 | $1.22\times {10}^{-4}$ |

$\mathrm{Sr}\left({\mathrm{Ti}}_{0.5}{\mathrm{Fe}}_{0.5}\right){\mathrm{O}}_{3-\delta}$ | 0.213 | 13.360 | 1.67 | 0.05 |

$\left({\mathrm{Sr}}_{0.5}{\mathrm{Ba}}_{0.5}\right)\left({\mathrm{Ti}}_{0.5}{\mathrm{Fe}}_{0.5}\right){\mathrm{O}}_{3-\delta}$ | 0.213 | 17.904 | 1.56 | 0.13 |

$\left({\mathrm{Sr}}_{0.5}{\mathrm{Ba}}_{0.5}\right)\left({\mathrm{Ti}}_{0.8}{\mathrm{V}}_{0.1}{\mathrm{Fe}}_{0.1}\right){\mathrm{O}}_{3-\delta}$ | $1.0\times {10}^{-15}$ | 16.729 | 1.60 | $6.94\times {10}^{-3}$ |

$\left({\mathrm{Sr}}_{0.5}{\mathrm{Ba}}_{0.5}\right)\left({\mathrm{Ti}}_{0.7}{\mathrm{V}}_{0.1}{\mathrm{Fe}}_{0.2}\right){\mathrm{O}}_{3-\delta}$ | $1.0\times {10}^{-15}$ | 16.827 | 1.59 | $7.04\times {10}^{-3}$ |

$\left({\mathrm{Sr}}_{0.5}{\mathrm{Ba}}_{0.5}\right)\left({\mathrm{Ti}}_{0.7}{\mathrm{V}}_{0.2}{\mathrm{Fe}}_{0.1}\right){\mathrm{O}}_{3-\delta}$ | $1.0\times {10}^{-15}$ | 16.654 | 1.60 | $6.38\times {10}^{-3}$ |

$\left({\mathrm{Sr}}_{0.5}{\mathrm{Ba}}_{0.5}\right)\left({\mathrm{Ti}}_{0.6}{\mathrm{V}}_{0.3}{\mathrm{Fe}}_{0.1}\right){\mathrm{O}}_{3-\delta}$ | $1.0\times {10}^{-15}$ | 16.585 | 1.60 | $5.99\times {10}^{-3}$ |

$\left({\mathrm{Sr}}_{0.5}{\mathrm{Ba}}_{0.5}\right)\left({\mathrm{Ti}}_{0.6}{\mathrm{V}}_{0.2}{\mathrm{Fe}}_{0.2}\right){\mathrm{O}}_{3-\delta}$ | $1.0\times {10}^{-15}$ | 16.757 | 1.59 | $6.93\times {10}^{-3}$ |

$\left({\mathrm{Sr}}_{0.5}{\mathrm{Ba}}_{0.5}\right)\left({\mathrm{Ti}}_{0.6}{\mathrm{V}}_{0.1}{\mathrm{Fe}}_{0.3}\right){\mathrm{O}}_{3-\delta}$ | $1.0\times {10}^{-15}$ | 17.149 | 1.59 | $7.71\times {10}^{-3}$ |

$\left({\mathrm{Sr}}_{0.5}{\mathrm{Ba}}_{0.5}\right)\left({\mathrm{Ti}}_{0.5}{\mathrm{V}}_{0.3}{\mathrm{Fe}}_{0.2}\right){\mathrm{O}}_{3-\delta}$ | $1.0\times {10}^{-15}$ | 16.678 | 1.60 | $6.59\times {10}^{-3}$ |

$\left({\mathrm{Sr}}_{0.5}{\mathrm{Ba}}_{0.5}\right)\left({\mathrm{Ti}}_{0.5}{\mathrm{V}}_{0.2}{\mathrm{Fe}}_{0.3}\right){\mathrm{O}}_{3-\delta}$ | $1.0\times {10}^{-15}$ | 16.860 | 1.59 | $7.10\times {10}^{-3}$ |

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**MDPI and ACS Style**

Schlenz, H.; Baumann, S.; Meulenberg, W.A.; Guillon, O. The Development of New Perovskite-Type Oxygen Transport Membranes Using Machine Learning. *Crystals* **2022**, *12*, 947.
https://doi.org/10.3390/cryst12070947

**AMA Style**

Schlenz H, Baumann S, Meulenberg WA, Guillon O. The Development of New Perovskite-Type Oxygen Transport Membranes Using Machine Learning. *Crystals*. 2022; 12(7):947.
https://doi.org/10.3390/cryst12070947

**Chicago/Turabian Style**

Schlenz, Hartmut, Stefan Baumann, Wilhelm Albert Meulenberg, and Olivier Guillon. 2022. "The Development of New Perovskite-Type Oxygen Transport Membranes Using Machine Learning" *Crystals* 12, no. 7: 947.
https://doi.org/10.3390/cryst12070947