On the relationship between chemical expansion and hydration thermodynamics of proton conducting perovskites

In this contribution, we determine the compositional dependence of the chemical expansion and entropy of hydration of the proton conducting perovskites BaZrO 3 , BaSnO 3 , BaCeO 3 , and SrZrO 3 by ﬁ rst principles phonon calculations. The calculations reveal that the cubic BaZrO 3 and BaSnO 3 , which display the least favourable hydration enthalpies, (cid:1) 72 and (cid:1) 65 kJ mol (cid:1) 1 , respectively, exhibit the most favourable entropies, (cid:1) 108 and (cid:1) 132 J mol (cid:1) 1 K (cid:1) 1 , respectively. The strong compositional dependency of the hydration entropy primarily originates from the entropy gain upon ﬁ lling the oxygen vacancy, which is closely related to the chemical expansion coe ﬃ cient of oxygen vacancies, and thus the chemical expansion upon hydration. The chemical expansion coe ﬃ cient of oxygen vacancies is more negative for the cubic than the orthorhombic perovskites, leading to a considerably larger chemical expansion upon hydration of the former. The calculations therefore suggest that challenges associated with chemical expansion upon hydration of BaZrO 3 proton conducting electrolytes to some extent can be avoided, or reduced, by partial substitution of Zr by Ce.


Introduction
Protons dissolve in many oxides 1 through hydration of oxygen vacancies, resulting in high temperature proton conduction with potentials for important applications such as in fuel cells and gas separation membranes. The relative dominance of protons and oxygen vacancies is determined by the enthalpy and entropy of the hydration reaction with the corresponding equilibrium constant: where c i are the molar concentrations of protons, oxygen vacancies, and lattice oxygen species. The enthalpy of this reaction can be evaluated by both indirect 2 and direct measurements, [3][4][5] or computationally, and correlates with various materials' parameters. [5][6][7][8][9] The entropy on the other hand, has never been measured directly, and its nature and compositional dependence is in general poorly understood. We recently determined D Hydr S of BaZrO 3 computationally, 10 and revealed that its major contribution is the vibrational entropy of oxygen vacancies ðv O Þ, in addition to loss of water vapour.
Although the vibrational contribution from protons was found to be small, there is a signicant congurational contribution due to their site degeneracy around the oxide ions in the perovskite structure. In this contribution, we continue our efforts to understand the entropy of hydration, by for the rst time assessing its compositional dependence and relationship with the chemical expansion, computationally. D Hydr H has been shown to correlate with material specic parameters such as the difference in the electronegativity of the B-and A-site cations 8 and the tolerance factor, 9 which has been argued to reect its dependence on the basicity, or partial charge density of the oxide ions. 7,11 D Hydr S , however, seemingly does not display any similar correlations, 8 which partly may stem from the large scatter in the experimental range of values. Further, the compositional dependence of D Hydr S is less intuitive than that of D Hydr H as it is related to both the chemical expansion of the material upon v O and OH O formation, the bonding strengths of H + and O 2À , and the local structural relaxations induced by the two defects.
To assess the compositional and structural dependency of D Hydr S , we evaluate the vibrational formation entropies of v O and OH O from rst principles phonon calculations, and the congurational contribution originating from the site degeneracy of protons (and oxygen vacancies). We have chosen to focus on four structurally different proton conducting perovskites; BaZrO 3 , BaSnO 3 , BaCeO 3 and SrZrO 3 . The two former take on a cubic structure (Pm 3m), while the two latter adopt an orthorhombic structure (Pnma), thus allowing us to judge the effect of both symmetry and composition on D Hydr S . As shown in Table 1, the experimental D Hydr H and D Hydr S for these perovskites cover a wide range of values, with the cubic BaZrO 3 and BaSnO 3 exhibiting the least exothermic D Hydr H , but also the least negative (most favourable) D Hydr S .
First principles phonon calculations allow us to determine both D Hydr H and D Hydr S as a function of temperature, and explore the individual contributions that constitute both thermodynamic parameters. The calculations are performed under constant volume and/or constant/zero pressure conditions, in order to separate contributions from local relaxations and chemical expansion to the vibrational entropies.

Defect formation thermodynamics
The free energy of defect formation, D F F defect (or D F G defect ) is given by where DE el defect is the total energy difference between the defective and perfect supercells, while D F F vib defect is the vibrational (phonon) contribution to the formation energy. Further, Dn i is the change in the number of atoms i with chemical potential m i , q is the effective charge of the defect, 3 f is the Fermi level and D3 aligns the core potentials of the perfect and the defective supercells to remedy shis in the band edges due to the jellium background charge. The chemical potential of H 2 O is given by The standard chemical potential, m H2O , is set to the calculated total electronic energy of the isolated H 2 O molecule.
The contribution from phonons to the free energy, F vib (T) or G vib (T), and entropy, S vib (T), is in the harmonic approximation given by where n(q,s) are the phonon frequencies throughout the q-space. Although the vibrational frequencies in principle should be evaluated at all q-points, the phonon spectra for the defective cells were only evaluated at the G-point of the defective supercells due to their large sizes. From eqn (6), the vibrational formation entropy of a defect is simply and similar for D F H vib defect , and D F F vib defect or D F G vib defect . Note that we are assuming a negligible change in the thermal expansion coefficient upon defect formation, such that D F S vib defect and D F G vib defect calculated by full volume relaxations represent zero/ constant pressure properties. We are currently also exploring anharmonic contributions to the defect formation entropies and thermal expansion of BaCeO 3 in more detail, which indicate that these contributions are of minor importance for the defect thermodynamics of BaCeO 3 .

Hydration thermodynamics
The enthalpy and entropy of hydration according to eqn (1) can readily be determined from eqn (3)-(7): Upon experimental determination of D Hydr S by curve tting of eqn (2) to e.g. water uptake data from thermogravimetric measurements, it is usually assumed that the number of regular positions for OH O and v O , and thus K Hydr . Not accounting for these congurations when evaluating defect thermodynamics by curve tting results in an apparent hydration entropy, D Hydr S app , which, in addition to the vibrational contributions, also includes congurational contributions: N O is the number of oxide ions per formula unit, while p i,H and p i,v are the conditional probabilities for occupation of each conguration at an oxide ion by OH O and v O , respectively. The general expression for the probability of occupation of a given conguration is for instance for the proton given by where E i are the relative energies of the different N H congurations around the N O oxide ions (3 for perovskites). It is important to note that D Hydr S conf app in eqn (10) only accounts for the congurational contributions stemming from distributing a proton or an oxygen vacancy over N H and N v congurations at one specic oxide ion, as the remaining contributions are accounted for through the mass action law (eqn (2)). At higher temperatures or in perovskites where the different N H and N v are degenerate, eqn (10) converges to

Computational details
The rst principles calculations are performed with the planewave Density Functional Theory (DFT) approach within the VASP (v.5.3.5) code, 16,17 at the GGA-PBE 18 and/or LDA 19 level with a constant plane-wave cut-off energy of 500 eV. For the cubic BaZrO 3 and BaSnO 3 , all defect calculations were performed with 3 Â 3 Â 3 (135 atoms) supercell expansions, while 2 Â 2 Â 2 supercells (160 atoms) were used for orthorhombic BaCeO 3 and SrZrO 3 . Electronic integration was performed using a 2 Â 2 Â 2 Monkhorst-Pack k-mesh over the supercell. Further, all calculations were performed using a reciprocal projection scheme, with ionic and electronic convergence criteria of 10 À4 eVÅ À1 and 10 À8 eV, respectively. The number of electrons in the supercell was adjusted to simulate the desired charge state of the defects, which was compensated by a homogeneous, opposite background charge. Finally, the defect calculations were performed both by xing the volume to that of the defectfree bulk, or by relaxing both the supercell shape and its volume, in order to simulate constant volume, or zero/constant pressure conditions, respectively. The phonon frequencies were calculated within the harmonic approximation using nite ionic displacements of AE0.01Å. Fourier transform and diagonalisation of the dynamical matrix were performed using the phonopy code. 20,21 While BaZrO 3 , BaCeO 3 and SrZrO 3 displayed no imaginary modes, BaSnO 3 exhibited imaginary modes at the R-point of the Brillouin zone with the GGA-PBE functional. With the LDA functional however, all modes of BaSnO 3 were found to be positive and this functional was therefore chosen to avoid erroneous contributions to the vibrational thermodynamics. All electronic contributions to the defect thermodynamics were, however, evaluated with the GGA-PBE functionalfor consistency. Table 2 shows the calculated lattice parameters and unit cell volumes of the four included perovskites. As is typically observed, GGA consistently overestimates the lattice parameters slightly, while the LDA parameters for BaSnO 3 are somewhat underestimated.

Site degeneracy and congurational entropy
The orientation of the protonic defect around each oxide ion has previously been shown to depend on the symmetry of the structure, but also on the lattice constant/volume of the unit cell. 26 In BaZrO 3 , BaSnO 3 , BaCeO 3 and SrZrO 3 , the proton takes on a position along the bisector of two O connecting lines and there are thus 4 proton positions per oxide ion. [26][27][28][29][30] While these positions are degenerate in BaZrO 3 and BaSnO 3 , they are nondegenerate in the orthorhombic BaCeO 3 and SrZrO 3 . In the denser SrTiO 3 , on the other hand, protons have been argued to take on a position along the edge of the TiO 6 polyhedra, resulting in 8 equivalent positions around each oxide ion. 31,32 Our calculations are in general agreement with those of refs [26][27][28][29][30] showing that all included compositions exhibit proton positions at, or close to, the bisector of two oxygen connecting lines (see Fig. 1). The site is 4-fold degenerate in BaZrO 3 and BaSnO 3 , while BaCeO 3 and SrZrO 3 display 4 unique minima around each of the O1 and O2 ions within 0.20 and 0.18 eV of each other, respectively (cf. Fig. 1 for SrZrO 3 ). In BaCeO 3 and SrZrO 3 , we also nd that v O is favoured by merely 70 and 60 meV, for BaCeO 3 and SrZrO 3 , respectively, at the O1 compared to the O2 site. As shown in Fig. 2, the non-degenerate proton and oxygen vacancy congurations in SrZrO 3 and BaCeO 3 render the congurational contribution to the hydration entropy temperature dependent. At the lowest temperatures, the contribution is actually negative due to the preference for occupation of the O1 site by oxygen vacancies. However, due to the small energy difference between the various proton and oxygen vacancy congurations, the congurational contribution approaches that for BaZrO 3 even at $600 K. Both SrZrO 3 and BaCeO 3 also undergo several phase transitions upon heating, which may affect the relative stability of the different proton and vacancy congurations. Hence, the congurational contribution to D Hydr S is approximately 2R ln(4) (23 J K À1 mol À1 ) within the majority of the experimental temperature window (600-1100 K) for all included perovskites. Any compositional dependence of D Hydr S , in contrast to the conclusions of Tauer et al., 33 is therefore of a purely vibrational nature, and will be the focus in the remaining parts of this contribution.

Chemical expansion
In our previous contribution on BaZrO 3 , 10 the formation volume of a defect was shown to have major implications for its formation entropy under zero/constant pressure conditions. In addition, the formation volume determines the chemical expansion coefficient of defects, and thus also the chemical expansion of the material upon e.g. hydration (eqn (1)).
The normalised volumetric chemical expansion coefficient, 3 i , of a defect i is here taken as: 34 where V i and V 0 are the volumes of the defective and perfect supercells, respectively, while d is the defect concentration in mole fractions (1/27 for BaZrO 3 and BaSnO 3 , and 1/32 for BaCeO 3 and SrZrO 3 ). Correspondingly, the chemical expansion upon hydration (eqn (1)) may be quantied through where 3 OH O and 3 v O are the chemical expansion coefficients of OH O and v O , respectively. 3 Hydr is thus given as the relative expansion upon hydration of 1 mole of v O , and with respect to the volume of the defect-free lattice. Experimentally, however, chemical expansion coefficients are taken as the relative volume of the hydrated and dry materialsi.e. relative to the defective lattice. Experimental 3 Hydr will therefore appear somewhat larger than those determined computationally in this work and in e.g. ref. 34.      Table 3 Linear (3 a , 3 b , 3 c ) and volumetric (3 i ) chemical expansion coefficients, formation volumes (D F V i ), of OH O and v O , and chemical expansion coefficient upon hydration (3 Hydr ) for the included perovskites. The v O in BaZrO 3 and BaSnO 3 is aligned with the Zr(Sn)-v O -Zr(Sn) axis parallel to the a-axis, while in BaCeO 3 and SrZrO 3 it is parallel to the b-axis. Note that the calculations on BaSnO 3 are performed with the LDA functional   D Hydr H results in a slight temperature dependence, and D Hydr H therefore becomes less negative with increasing temperature for all compositions. Experimentally, D Hydr H is determined from eqn (2) over a wide temperature interval such that the temperature dependence of D Hydr H (and D Hydr S ) is not observed. In a recent work, 3 we indicated, from combined thermogravimetric measurements of water uptake and differential scanning calorimetry (TG-DSC), that D Hydr H of Sc and In doped BaTiO 3 actually displays weak temperature dependencies in accordance with Fig. 5. The experimentally determined D Hydr H and D Hydr S listed in Table 1 should therefore be compared with the temperature averaged D Hydr H and D Hydr S from linearisation of the computational D Hydr G vs. T (see Table  4). For comparative reasons, D Hydr S app values in Table 4 represent the temperature averaged apparent entropy of hydration,
The D Hydr H values are, as expected, very close to the low temperature D Hydr H in Fig. 5, and are in reasonable agreement with the experimental ranges of enthalpies for all compositions in Table 1. The calculated values also reect the experimentally observed compositional dependence, with the two cubic perovskites displaying signicantly less exothermic enthalpies than the two orthorhombic members. Also the calculated D Hydr S values are in good agreement with the experimental values in Table 1. The entropies display a distinct compositional dependence, and are in general less negative for the cubic BaZrO 3 and BaSnO 3 than for the orthorhombic SrZrO 3 and BaCeO 3 . The less negative D Hydr S of BaZrO 3 and BaSnO 3 stabilises protons compared to in the orthorhombic BaCeO 3 and SrZrO 3 , in line with Kreuer's remark that the favourable proton transport properties of BaZrO 3 stems from entropic stabilisation of protons. 6 As noted in Fig. 3, D F S vib v O is the dominant vibrational contribution to D Hydr S , and also shows a stronger compositional dependency than D F S vib OH O . Hence, one may argue that the entropic stabilisation of protons in BaZrO 3 /BaSnO 3 is, more precisely, a result of entropic destabilisation of v O . Turning to the hydration enthalpy, the strong compositional dependency (cf. Table 4) is generally in line with experimental observations; orthorhombic perovskites exhibit more exothermic values than the cubic, i.e. it becomes more exothermic with a decreasing electronegativity difference between the B-and A-site cations, but also with a decreasing Goldschmidt tolerance factor. 8,9 These correlations are usually attributed to the basicity/ionicity of the oxide, which affects the bonding strength of v O and OH O ; the oxide with the most exothermic hydration enthalpies exhibits the most stable protonic defects, and the most unstable oxygen vacancies (i.e. most stable oxide ions). There is further a striking correlation between both the calculated and experimental D Hydr    hydration. These properties can again be related to the size of the cations in the perovskite structure (which determines the unit cell volume and symmetry), and their electronegativity (which determines the interatomic bonding strengths), whichsomewhat speculativelyexplains the observed correlation between D Hydr H and D Hydr S . Conclusively, the oxide exhibiting the most stable protons and least stable vacancies enthalpywise, also exhibits weaker entropic destabilisation of vacancies, and thus the least favourable entropies of hydration.

Conclusions
We have in this work assessed the thermodynamics of hydration, with emphasis on vibrational contributions, and the chemical expansion of the four proton conducting perovskites BaZrO 3 , BaSnO 3 , BaCeO 3 and SrZrO 3 from rst principles phonon calculations. The calculations show that both OH O and v O are smaller than the native oxide ions in all materials. Further, v O is signicantly smaller than the proton and is therefore the main contributor to the chemical expansion of the materials upon hydration. The chemical expansion shows a distinct compositional dependence, and decreases in the order BaZrO 3 / BaSnO 3 / BaCeO 3 / SrZrO 3 . As such, proton conducting electrolytes composed of BaZrO 3 -BaCeO 3 or BaZrO 3 -SrZrO 3 solid solutions should be less prone to issues stemming from chemical expansion upon hydration than pure BaZrO 3 electrolytes. There is further a signicant contribution from phonons to the formation entropy of both OH O and v O . The contribution is most pronounced for v O , and negative for all compositions, and becomes less negative in the order BaZrO 3 / BaSnO 3 / BaCeO 3 / SrZrO 3 , i.e. with increasing (less negative) chemical expansion of v O . Both the calculated D Hydr H and D Hydr S are in good agreement with the experimental ranges of values, and display a distinct compositional dependence. D Hydr S is in general less negative (more favourable) for the cubic BaZrO 3 and BaSnO 3 than the orthorhombic members, reecting its dependence on the vibrational formation entropy, and again the chemical expansion, of v O . Interestingly, there is a general correlation between D Hydr H and D Hydr S ; the two cubic members display the least exothermic D Hydr H , but also the least negative (and thus most favourable) D Hydr San effect which may stem from both parameters being related to the bonding strengths of O 2À and H + .