Predicting Thermochemical Equilibria with Interacting Defects: ${\mathrm{Sr}}_{1\ensuremath{-}x}{\mathrm{Ce}}_{x}{\mathrm{Mn}\mathrm{O}}_{3\ensuremath{-}\ensuremath{\delta}}$ Alloys for Water Splitting

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Predicting Thermochemical Equilibria with Interacting Defects: ${Sr}_{1 - x} {Ce}_{x} {Mn O}_{3 - δ}$ Alloys for Water Splitting

Anuj Goyal, Michael D. Sanders, Ryan P. O’Hayre, and Stephan Lany

PRX Energy 3, 013008 – Published 16 February 2024

Abstract

Solar thermochemical hydrogen is one of the few potential routes towards direct fuel production from renewable energy sources, but the thermodynamic boundary conditions for efficient and economic energy conversion are challenging. Success or failure of a given oxide working material depends on the subtle balance between enthalpy and entropy contributions in the redox processes. Developing a mechanistic understanding of the behavior of materials on the basis of atomistic models and first-principles calculations is an important part of advancing the technology. One challenge is to quantitatively predict thermochemical equilibria at high concentrations when the redox-active defects start to interact with each other, thereby impeding the formation of additional defects. This problem is of more general importance to applications that rely on high levels of off-stoichiometry or doping, including, for example, batteries, thermoelectrics, and ceramic fuel cells. To account for such repulsive defect interactions, we introduce a statistical mechanics approach, defining an expression for the free energy of defect interaction based on limited sampling of defect configurations in density functional theory supercell calculations. The parameterization of this energy contribution as a function of defect concentration and temperature allows on-the-fly simulation of thermochemical equilibria. The approach consistently incorporates finite temperature effects by including the leading contributions to the temperature-dependent free energy for the case at hand, i.e., the ideal gas and configurational enthalpies and entropies. We demonstrate the capability and utility of the approach by simulating the water splitting redox processes for ${Sr}_{1 - x} {Ce}_{x} {Mn O}_{3 - δ}$ alloys.

Received 1 September 2023
Revised 2 December 2023
Accepted 19 December 2023

DOI:https://doi.org/10.1103/PRXEnergy.3.013008

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Open access publication funded by the National Renewable Energy Laboratory (NREL) Library, part of a national laboratory of the U.S. Department of Energy.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Defects Hydrogen production Solar energy

Oxides

First-principles calculations Statistical methods

Condensed Matter, Materials & Applied PhysicsEnergy Science & TechnologyStatistical Physics & Thermodynamics

Authors & Affiliations

Anuj Goyal ^1,2, Michael D. Sanders ³, Ryan P. O’Hayre ³, and Stephan Lany ^1,*

¹National Renewable Energy Laboratory, Golden, Colorado 80401, USA
²Indian Institute of Technology Hyderabad, Kandi, Sangareddy, Telangana, 502284, India
³Colorado School of Mines, Golden, Colorado 80401, USA

^*stephan.lany@nrel.gov

Popular Summary

Renewable energy growth comes mostly from electricity sources like solar photovoltaics and wind energy. However, most of the global final energy consumption occurs in the form of fuels. Thus, the direct conversion of sunlight energy into fuels could be a game changer in the energy transition. Solar thermochemical fuels hydrogen production and carbon dioxide reduction are among the few known potential routes towards direct renewable fuels that have the potential for deployment on an industrial scale. However, they currently lack suitable oxide working materials that optimally satisfy the challenging thermodynamic boundary conditions. Computational modeling can help advance the field through both the identification of new candidate materials and the theoretical assessment of their performance. The present work advances computational approaches for modeling the thermochemical equilibria for hydrogen production from first principles. Such modeling can provide a mechanistic understanding of materials behavior from the atomistic origins and help to assess the potential and limitations of materials. The methods introduced here are validated and demonstrated by application to strontium cerium manganate alloys.

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Vol. 3, Iss. 1 — February - April 2024

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Figure 1
(a) Hexagonal phase of ${Sr Mn O}_{3}$ in its ground state orthorhombic crystal structure (space group $C 222_{1}$ , no. 20) with antiferromagnetic order of $Mn$ spins. The preferred $O$ sites for vacancy formation are indicated. (b) Perovskite phase in its metastable cubic structure ( $P m \bar{3} m$ , no. 221). $Ce$ substitutes for $Sr$ to form SCM alloys.
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Figure 2
Thermodynamic defect equilibria as a function of temperature in air ( $p O_{2} = 0.2$ bar) for the hexagonal and perovskite phases of ${Sr Mn O}_{3 - δ}$ , obtained from the noninteracting (red), interacting (blue), and interacting-adjusted (green) defect models. Experimental data from the literature are shown by gray symbols. The phase transition occurs at $1400^{\circ} C$ in experiments. The dashed lines show the results for the respective phases beyond the phase transition. The interacting-adjusted model includes a reduction of the $Δ H_{D}^{ref} (V_{O})$ dilute-limit defect formation energies by 0.05 and 0.18 eV in the hexagonal and perovskite phases, respectively, compared to the SCAN+ $U$ -calculated values given in Table 1. The interaction parameters $a_{0}$ and $a_{1}$ in Table 3 are unchanged.
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Figure 3
(a) Interacting defect model using supercells with vacancy pairs and triplets in supercells of different size. (b) Interaction energy $Δ E^{int} / n$ per vacancy in hexagonal ${Sr Mn O}_{3}$ obtained from supercell calculations of vacancy pairs ( $n = 2$ ) as a function of the pair distance for the three different cell sizes.
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Figure 4
Temperature-dependent free energy of defect interaction $Δ G_{D}^{int} (T)$ for $O$ vacancy formation. The data points show $Δ G_{D}^{int} (T)$ as obtained from the supercell sampling [Eq. (3)] and the lines show the parameterization according to Eq. (4) and Table 3. (a) $V_{O}$ - $V_{O}$ interactions as a function of $δ$ in hexagonal ${Sr Mn O}_{3 - δ}$ . (b) $V_{O}$ - $V_{O}$ interactions as a function of $δ$ in perovskite ${Sr Mn O}_{3 - δ}$ . “p” and “t” indicate pair and triplet configurations. (c) $V_{O}$ - ${Ce}_{Sr}$ interactions as a function of $x$ in the perovskite phase of ${Sr}_{1 - x} {Ce}_{x} {Mn O}_{3}$ .
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Figure 5
(a) Enthalpy of mixing $Δ H_{mix}$ of SCM alloys in the hexagonal and perovskite phases, showing a phase transition around $x = 0.1$ . (b) Distribution of oxygen vacancy formation energies in the SCM alloys as a function of the $Ce$ concentration. The hexagonal phase has two groups of $O$ sites with similar formation energies, whereas the perovskite phase has only a single $O$ site. The data points with the larger symbol size show the average formation energies.
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Figure 6
(a) Off-stoichiometry $\log (δ_{red})$ in ${Sr}_{1 - x} {Ce}_{x} {Mn O}_{3 - δ}$ in the perovskite phase as a function of $Ce$ content $x$ and temperature $T_{red}$ during reduction at $p O_{2} = 10^{- 4}$ bar. Note that $\log (δ) = - 0.7$ and $\log (δ) = - 0.5$ correspond approximately to $δ = 0.2$ and $δ = 0.3$ , respectively. (b) Hydrogen production in number of $H_{2}$ molecules per SCM formula unit, shown as a function of $x$ and the $H_{2}$ partial pressure in $\log (p H_{2} / bar)$ . The underlying process conditions are $T_{red} = 1400^{\circ} C$ and $p O_{2} = 10^{- 4}$ bar for the reduction step and $T_{ox} = 850^{\circ} C$ and $p H_{2} O = 1 bar$ for the oxidation step. Note that $p H_{2} / bar$ corresponds to the $H_{2}$ : $H_{2} O$ ratio. The dashed line ( $H_{2} / f.u. = 0$ ) represents the demarcation line at which net water splitting ( $H_{2} / f . u . > 0$ ) occurs.
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PRX Energy

Predicting Thermochemical Equilibria with Interacting Defects: Sr1−xCexMnO3−δ Alloys for Water Splitting

Anuj Goyal, Michael D. Sanders, Ryan P. O’Hayre, and Stephan Lany

PRX Energy 3, 013008 – Published 16 February 2024

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Predicting Thermochemical Equilibria with Interacting Defects: ${Sr}_{1 - x} {Ce}_{x} {Mn O}_{3 - δ}$ Alloys for Water Splitting