Atomic order, electronic structure and thermodynamic stability of nickel aluminate

Ishfaque Eliasa, Aloysius Soonb, Jun Huanga, Brian S. Haynesa and Alejandro Montoya*a
aSchool of Chemical and Biomolecular Engineering, The University of Sydney, Sydney, NSW 2006, Australia. E-mail:
bDepartment of Materials Science and Engineering, Yonsei University, Seoul 03722, Korea

Received 4th August 2019 , Accepted 27th September 2019

First published on 28th September 2019

The atomic order, electronic structure and thermodynamic stability of nickel aluminate, NiAl2O4, have been analyzed using periodic density functional theory and cluster expansion. NiAl2O4 forms a tetragonal structure with P4122 space group. At temperatures below 800 K, it is an inverse spinel, with Ni occupying the octahedral sites and Al occupying both the octahedral and the tetrahedral sites. Some Niocta + Altetra ⇌ Nitetra + Alocta exchange occurs above 800 K, but the structure remains largely inverse at high temperatures, with about 95% Niocta at 1500 K. Various functionals, with and without van der Waals corrections, were used to predict the experimental formation energy, lattice parameters and electronic structure. In all cases, the NiAl2O4 is found to be ferromagnetic and a semiconductor with an indirect band gap along the ΓM symmetry points. NiAl2O4 is found to be thermodynamically stable at operating conditions of 900–1000 K and 1 atm relative to its competing oxide phases, NiO and Al2O3.


The ability to control the structure, composition, morphology and valence of spinel oxides has made them suitable as catalysts in various processes1–8 ranging from chemical looping combustion9 and direct methane combustion10 to hydrogen and oxygen evolution reactions.11,12 Nickel aluminate, NiAl2O4, for instance, has been doped with various other atoms, such as Cu, Zn and Nb, to tune its properties as a pigment,13 as a photocatalyst to remove toxic organic pollutants from industrial wastewater,14,15 and as a catalyst in the oxidative dehydrogenation of n-octane.16 NiAl2O4 has applications as a refractory material,17 volatile organic sensor18 and high temperature fuel cell anode.19,20 It is used as a ceramic skeleton for metal-ceramic composites as it has a porous structure, superior strength and good wettability with metals at high temperature.21 It has recently gained widespread interest as a catalyst,22–27 especially in high temperature catalytic processes, because of its ability to reduce carbon deposition.28–36 Dry reforming of methane34,37 is one such process which provides an unique opportunity to convert the greenhouse gases, CO2 and CH4, into syngas, CO and H2, which are the building blocks for valuable chemicals such as methanol. Ni is the most preferred catalyst for this reaction due to its high catalytic activity, however, catalyst deactivation due to carbon deposition and sintering are the major drawbacks of this process.38 Under the required high temperature reaction conditions (900–1000 K and 1 atm), NiAl2O4 provides 32% conversion of CH4 and has a carbon deposition of 7.5 wt% compared to rapid carbon formation of 79 wt% with a commercial Ni/α-Al2O3 catalyst which results in immediate catalyst deactivation and eventually leads to a reactor shutdown.37 An understanding of this resistance of NiAl2O4 to carbon deposition is yet to be achieved. However, this catalytic activity may depend on the atomic order of NiAl2O4 as Farahani et al. reported that Niocta was responsible for the activity of NiAl2O4 in oxidative dehydrogenation of n-octane.16 Thus, a better understanding of its atomic structure is necessary which may allow the use of tunable properties of NiAl2O4 effectively in heterogeneous catalysis.16

Experimental methods using conventional diffraction techniques, such as X-ray diffraction, indicate that the NiAl2O4 adopts the general face-centered cubic spinel structure with the Fd3m space group39–43 and the tetragonal R3m44 structure in NiAl2O4 nanoparticles. However, studies using total neutron diffraction have reported that NiAl2O4 adopts a lower symmetry group, the tetragonal P4122 configuration.45,46 Moreover, Ni and Al may distribute across the tetrahedral and octahedral sites and form a range of structures, [Ni1−i2+Ali3+]tetra[Nii2+Al2−i3+]octaO4. The degree of inversion, i, is defined as the fraction of Niocta present, which can take a range of values from zero, a normal spinel, to one, an inverse spinel. Although NiAl2O4 is generally reported to adopt the inverse spinel structure, evidence of normal spinel structure is also reported and thus provides additional points of disagreement regarding its structure.46–51 This indicates that Niocta + Altetra [left over right harpoons] Nitetra + Alocta exchange is possible, which is generally seen to occur at higher temperatures.

Several theoretical approaches52–55 predicted inverse cubic spinel structures, but either magnetic order53,54 or atomic order52 were not taken into consideration in these studies. Stevanović et al., however, reported a possibility of NiAl2O4 adopting an ordered tetragonal P4122 structure using an electrostatic model, but their results were not in agreement when magnetic order, using density functional theory (DFT) calculations, was taken into account.55 Magnetic order may be important in determining the atomic order in NiAl2O4 as Gouda et al. argued that the difference in magnetic order of NiAl2O4 compared to other spinel oxides, such as ZnAl2O4 and CuAl2O4, is due to the different occupation of the cations in the octahedral and tetrahedral sites.56 A difference in dielectric constant was observed as well for different particle sizes of NiAl2O4 nanoparticles and Kurien et al. argued that this was due to different occupation of Ni and Al in the tetrahedral and octahedral sites, as well.57 This relationship of physical properties with the atomic order of NiAl2O4 is consistent with other spinel oxides where experimental observations of physical properties such as dielectric behavior, magnetism, heat capacity, infrared spectra, electron spin resonance and optical fluorescence spectra have been difficult to reconcile with the cubic Fd3m structure.58 A better understanding of the atomic scale structure of NiAl2O4 would lead to better control and exploitation of their physiochemical properties to engineer materials with novel functionalities.59


First-principles calculation

The self-consistent periodic DFT calculations for the electronic properties of NiAl2O4 were performed using the Vienna Ab-initio Simulation Package (VASP)60–62 within the MedeA® software environment.63 Projector augmented wave (PAW) potentials64 were used to model the interaction between ionic cores and electrons. The electron exchange–correlation effects were described using various generalized gradient approximation (GGA) functionals such as the PBE,65 PBE+U,66 the self-consistent van der Waals corrected functional optB88-vdW67 and the hybrid functional HSE06.68,69 Two distinct U values have been used for the PBE+U exchange–correlation functional. Stevanović et al. suggested a value of U = 3.0 eV, obtained using a fitted elemental-phase reference energies (FERE) method70 while Jain et al. suggested a value of U = 6.2 eV based on a mixed PBE/PBE+U method.71 The formation energies of NiAl2O4 image file: c9cp04325j-t1.tif for different exchange–correlation functionals were calculated from its bulk elemental species as defined in eqn (1).
image file: c9cp04325j-t2.tif(1)
where ENiAl2O4, ENi, EAl and EO2 are the total energies of the bulk NiAl2O4, Ni, Al and gas phase O2 respectively. Almost all exchange–correlation functionals have large error in the O2 binding energy72 and thus, the O2 energy was corrected using the reaction 2H2O(g) → 2H2(g) + O2(g).73 The O2 correction energies were 0.46, 0.38, and 0.09 eV per O2 for PBE, optB88-vdW and HSE06 respectively. However, for the PBE+U methods, the empirical oxygen correction used by Stevanović et al.70 for the FERE method (0.46 eV per O2) and by Jain et al.71 for the mixed PBE/PBE+U method (1.36 eV per O2)74 was employed. The Kohn–Sham DFT equations were solved using a plane-wave basis set with convergence of total energies (less than 1 kJ mol−1 per formula unit) and forces (less than 0.02 eV Å−1) with respect to kinetic energy cutoffs (620 eV) and Monkhorst–Pack Γ-centered k-point mesh spacing (0.2 Å−1). The atomic positions were fully relaxed by allowing the cell volume and cell shape to change. A linear-tetrahedron smearing with Bloechl corrections was used. The magnetism of NiAl2O4 was obtained by carrying out spin polarized calculations and by initializing the calculation using different initial magnetic moments of Ni.

Cluster expansion

The iterative search to find the most stable configuration of NiAl2O4 was carried out by cluster expansion (CE)75–77 using the Universal Cluster Expansion (UNCLE)78 program package in the MedeA® software environment.63 Possible configurations were generated with varying distribution of Ni and Al in the tetrahedral and octahedral sites of a cubic Fd3m spinel structure using one to four unit cells. The use of more than four unit cells was not possible due to computational limitation. However, configurations in the order of 105 were generated using four unit cells which resulted in an extensive exploration of the NiAl2O4 structure. The ratio of the number of Ni to Al was fixed to 1[thin space (1/6-em)]:[thin space (1/6-em)]2 as the stoichiometric NiAl2O4 structure was of interest in this study. A genetic algorithm was applied for the selection of the clusters. Enthalpy of mixing (ΔHmixing) per active site (two Al and one Ni) of the generated structures were calculated using eqn (2).
image file: c9cp04325j-t3.tif(2)
where ENi2Al4O8, ENi2Ni4O8 and EAl2Al4O8 are the total energies of the distinct structures generated with 1[thin space (1/6-em)]:[thin space (1/6-em)]2 ratio of Ni and Al, spinel structure with only Ni and spinel structure with only Al respectively. The CE fitting was done by least-square minimization to check its quality in terms of cross-validation score (CVS).

Monte Carlo simulation

Canonical Monte Carlo (MC) simulation with the Metropolis algorithm79 was used to evaluate the cation disorder of NiAl2O4 with respect to temperature in the range of 0 K to 2000 K. MC simulation was performed on 10 × 10 × 10 supercells of the primitive rhombohedral cell of the cubic Fd3m unit cell. The MC simulation was equilibrated over and sampled for 4.374 × 107 steps per cation for calculating the thermodynamic averages of energy and cluster functions. The temperature interval of the MC simulation was set to 100 K while cooling from 2000 K to 0 K and to 200 K while heating from 0 K to 2000 K.

Chemical potential limits analysis

The thermodynamic stability of NiAl2O4 with respect to its competing phases was analyzed using the Chemical Potential Limits Analysis Program (CPLAP).80 Eqn (3)–(6) describe the constraints necessary to avoid the precipitation of the metals (Ni or Al) or the release of the lattice O atoms into the atmosphere.
image file: c9cp04325j-t4.tif(3)
ΔμNi ≤ 0 (4)
ΔμAl ≤ 0 (5)
ΔμO ≤ 0 (6)
where image file: c9cp04325j-t5.tif is the enthalpy of formation of NiAl2O4 and ΔμNi, ΔμAl and ΔμO are changes in chemical potentials of Ni, Al and O respectively. The vibrational entropy and the pressure effects on the solid crystal were considered negligible, and the Gibbs free energy was approximated with the total energy obtained from the DFT functionals. Eqn (7) and (8) describe the constraints necessary for the stability of NiAl2O4 with respect to the precipitation of its competing phases, NiO and Al2O3.
ΔμNi + ΔμO ≤ ΔHNiOf (7)
image file: c9cp04325j-t6.tif(8)
where ΔHNiOf and image file: c9cp04325j-t7.tif are the enthalpies of formation of NiO and Al2O3 respectively. The chemical potentials of Ni and Al were replaced by the deviations from the chemical potentials per metal atom in the bulk. The change in the oxygen chemical potential (ΔμO(T,PO2)) can be expressed as a function of temperature (T) and oxygen partial pressure (PO2) as shown in eqn (9).
image file: c9cp04325j-t8.tif(9)
where image file: c9cp04325j-t9.tif is the change in oxygen Gibbs free energy at a pressure of P0 (1 atm) and temperature (T) with respect to Gibbs free energy at T0 (298.15 K) and kB is the Boltzmann constant. The image file: c9cp04325j-t10.tif was calculated by using experimental data from standard thermodynamic tables.81 Eqn (3)–(9) were used to draw the phase diagram of NiAl2O4 with respect to ΔμNi, ΔμAl and temperature at atmospheric pressure (derived from μO) using formation energies from experimental data82 and from calculations done using different functionals (the formation energies are shown in Table S1 in the ESI).

Results and discussion

Atomic order of NiAl2O4

Fig. 1 (top left) shows the enthalpy of mixing of all the structures generated using one to four unit cells with the functionals PBE, PBE+U (U = 3.0 eV and 6.2 eV) and optB88-vdW. The performance of the CE was evaluated using the CVS where an optimal set of clusters up to six-body interactions using four unit cells resulted in a small CVS between 0.15 to 7.6 meV/cation for all the functionals (the effective cluster interactions are shown in Fig. S1 in the ESI). The enthalpy of mixing for structures generated using two to four unit cells reveals that a tetragonal structure with a space group P4122 is the most stable NiAl2O4 structure. It is crucial to consider possible configurations beyond one unit cell because in this case, the orthorhombic Imma, is incorrectly preferred when the cluster expansion is carried out with only one unit cell. The P4122 is ∼4 kJ mol−1 more stable than the Imma structure. The PBE and optB88-vdW functionals predict similar enthalpy of mixing within 5 kJ mol−1 for the configurations generated, but PBE+U predicts lower (absolute values) enthalpy of mixing by 25 to 45 kJ mol−1. However, all the functionals predict similar order of stability for the structures generated and thus, identify the tetragonal P4122 as the most stable structure.
image file: c9cp04325j-f1.tif
Fig. 1 (top left) Enthalpy of mixing for all the different structures generated for 1, 2, 3 and 4 unit cells using cluster expansion and DFT for PBE, PBE+U and optB88-vdW. Empty circles show the cluster expansion energy for all the structures generated. Filled-in red circles show the structures in the training set for which the DFT energy has been calculated. (top right) Enthalpy of mixing and the degree of inversion for the structures in the training set using 4 unit cells and different functionals. (bottom) Polyhedral representation (viewed through the [001] plane) of the tetrahedral and octahedral sites of four structures which have been generated from the cluster expansion. Blue, and green and red circles represent Al, Ni and O respectively. Al and Ni are also shown as blue and green polyhedral. The solid red outline shows the size of the unit cell.

Fig. 1 (top right) shows the enthalpy of mixing of the structures generated in the training set using four unit cells with respect to their degrees of inversion for the different functionals. The training set consists of 14, 24, 41 and 36 structures out of 1.5 × 105 possible configurations for PBE, PBE+U (U = 3.0 eV and 6.2 eV) and optB88-vdW respectively. The structures become more stable as the degree of inversion increases for all the functionals. The completely inverted tetragonal P4122 structure is the most stable for all the functionals while the normal cubic Fd3m structure with no inversion is the least stable. A previously suggested structure, tetragonal R3m,44 is found here to exist with a degree of inversion of 0.5 but is less stable than the fully inverted tetragonal P4122 structure by about 4.5 kJ mol−1 using PBE (about 7.7 kJ mol−1 using PBE+U).

Fig. 1 (bottom) shows the polyhedral representation of the atomic position in the Fd3m, P4122, Imma and R3m NiAl2O4 structures. A clockwise 45° rotation of the cubic Fd3m structure results in the tetragonal P4122 structure which is a image file: c9cp04325j-t11.tif superstructure of the cubic Fd3m structure. In the cubic Fd3m structure, the octahedral site is occupied by Al only. However, in the P4122 and Imma structure, all the Ni occupies the octahedral site as well and thus they form inverse spinel structures with degrees of inversion of one. R3m, on the other hand, has Ni on both octahedral and tetrahedral sites in equal numbers and thus forms a spinel structure with a degree of inversion of 0.5.

The methods of Stevanović et al.70 and Jain et al.71 were used to further fit the U values to reproduce the experimental formation energy using the P4122 structure (the formation energy for different U values is shown in Fig. S2 in the ESI). The U values, 2.8 eV and 6.4 eV, slightly different from those reported by the original authors’ values, 3.0 eV and 6.2 eV respectively, were obtained. These are small deviations, corresponding to differences less than 6 kJ mol−1 in the computed formation energies. It is possible that Stevanović et al.70 and Jain et al.71 based their calculations on structures of NiAl2O4 that differed from the most stable configuration identified in this study.

Table 1 shows the experimental46,82 as well as the calculated lattice parameters, volume and formation energy of the tetragonal P4122 NiAl2O4 for different exchange–correlation functionals. The hybrid functional, HSE06, cancels the self-interaction energy and accurately reproduces the formation energies of transition metal spinel structured compounds, such as LiMn2O4 and LiTi2O4.83 Thus, it was used to compare the energy and geometry of NiAl2O4 with other functionals. However, it was not used in cluster expansion as it is computationally very expensive, about 40 times higher than PBE+U.83

Table 1 Experimental and calculated (using different functionals) lattice parameters (a and c), volume and formation energy image file: c9cp04325j-t12.tif of tetragonal P4122 NiAl2O4 bulk structure
  a (Å) c (Å) V3)

image file: c9cp04325j-t13.tif

PBE 5.745 8.060 266.1 −1654.7
PBE+U (U = 2.8 eV) 5.736 8.058 265.1 −1920.2
PBE+U (U = 6.4 eV) 5.724 8.054 263.9 −1920.6
OptB88-vdW 5.728 8.037 263.7 −1923.3
HSE06 5.688 7.986 258.3 −2021.9
Experimental 5.674 8.164 262.8 −1920.5

PBE underestimates the formation energy significantly by 266 kJ mol−1 per formula unit because it is known that PBE cannot cancel the large self-interaction energy due to localized electrons in d orbitals of Ni. The hybrid functional, HSE06, gives a better agreement but the error is still 101 kJ mol−1 per formula unit. PBE+U and the van der Waals corrected functional, optB88-vdW, show the best agreement with reported experimental formation energy with error that are less than 3 kJ mol−1 per formula unit.

The ability of the optB88-vdW to predict image file: c9cp04325j-t14.tif within 3 kJ mol−1 prompted further analysis to evaluate the effect of van der Waals corrections using the PBE functional. The PBE-D2, zero damping PBE-D3 and BJ damping PBE-D3BJ were used to predict the image file: c9cp04325j-t15.tif as shown in Table S2 (ESI), resulting in the reduction of the error in the image file: c9cp04325j-t16.tif. However, the image file: c9cp04325j-t17.tif are still underestimated by 188, 222 and 227 kJ mol−1 using PBE-D2, PBE-D3 and PBE-D3BJ respectively. Clearly, the optB88 functional appears to contribute substantially to decrease the errors in image file: c9cp04325j-t18.tif.

The PBE, PBE+U and optB88-vdW functionals predict the lattice parameters to within 0.1 Å relative to the experimental data. HSE06 predicts one of the lattice parameters, a, within 0.01 Å, but it gives a larger deviation of 0.18 Å in the other lattice parameter, c. Comparing the computed volumes of the cells with experimental data, PBE, PBE+U (U = 2.8 eV) and HSE06 produce deviations of 3 Å3, 2 Å3 and 5 Å3 respectively, while optB88-vdW and PBE+U (U = 6.4 eV) are within 1 Å3.

The average bond lengths of Niocta–O, Alocta–O and Altetra–O are predicted to be of similar lengths by all the functionals, with Altetra–O bonds being the shortest bonds, averaging ∼1.8 Å. Niocta–O are the longest bonds, ∼2.1 Å and Alocta–O bond lengths are close to 1.93 Å.

Temperature dependence of nickel and aluminium distribution

Fig. 2 shows the changes in energies per cation and the corresponding degrees of inversion of NiAl2O4 structures at a range of temperatures from 0 K to 2000 K calculated using Monte Carlo simulation.
image file: c9cp04325j-f2.tif
Fig. 2 (top) Relative energies of different structures generated at various temperatures starting from 0 K up to 2000 K using MC simulation. The empty circles (connected by a blue line) represent the energies calculated during cooling down to 0 K. The empty squares (connected by a red line) represent the energies calculated during heating from 0 K. (middle) Degrees of inversion of the structures generated at various temperatures. The ×'s (connected by a black line) shows the degrees of inversion of structures generated in this work using MC simulation. The crosses, solid circles and solid triangles show experimental degrees of inversion at various temperature.47,50,51 (bottom) Snapshot of 10 × 10 × 10 NiAl2O4 structure generated from MC simulation at 800 K, 900 K and 2000 K.

Snapshot of the structures generated at 800 K, 900 K and 2000 K are also shown in Fig. 2. The structure is first cooled from 2000 K to 0 K and then heated up to 2000 K to find any hysteresis. The energy per active site increases continuously with increasing temperature but there is a sudden reduction in the rate of increase at around 800 K, which appears to correspond with the onset of reverse inversion, when Niocta + Altetra ⇌ Nitetra + Alocta exchange begins to take place. This Niocta + Altetra ⇌ Nitetra + Alocta exchange continues to take place with further increase in temperature which leads to a decrease in the degree of inversion to about 0.95 at 2000 K.

The reported experimental results for temperatures above 900 K shown in Fig. 2 confirm that there is a high degree of inversion, albeit with more scrambling (i.e. more Niocta + Altetra ⇌ Nitetra + Alocta exchange), especially at higher temperatures. The only experimental study (not shown in Fig. 2) that reports the tetragonal P4122 structure states a degree of inversion of 0.86.46 Comparison between different experiments and theory is difficult because experimental observation of the cation disorder may be affected by the rate of cation exchange which may be too slow to reach equilibrium at temperatures below 1000 K and too fast for the quenching method to be reliable at temperatures above 1500 K.47 However, the difference in the degree of inversion between experimental and theoretical results are about 10% with both indicating that NiAl2O4 remains largely inverse.

Electronic structure

The electronic structure of NiAl2O4 has not been reported in the literature, possibly due to the uncertainties in its atomic structure. The band structure of the tetragonal P4122 NiAl2O4 using four functionals is shown in Fig. 3. As can be seen in the figure, the valence-band maximum is located at the Γ point, while the conduction-band minimum is positioned at the M point which results in an indirect band gap in the ΓM symmetry points for all the functionals used. The optB88-vdW functional underestimates and HSE06 overestimates the band gap by 2.2 eV and 1.8 eV respectively compared to the experimental band gap of 3.0 eV.14 The band gaps estimated using the PBE+U methods are comparatively closer to the experimental band gap where U = 2.8 eV underestimates by about 0.7 eV and U = 6.4 eV overestimates by 0.9 eV. The opening of the band gap of the PBE+U approach observed here is consistent with the Hubbard correction for electron correlated systems,66 in which larger U values moves the localized states away from the Fermi level.
image file: c9cp04325j-f3.tif
Fig. 3 Band structure of the tetragonal P4122 NiAl2O4 along the high-symmetry directions of the Brillouin zone for different functionals. Red lines show the spin down states and the blue lines show the spin up states. The black arrows show the indirect band gap between the conduction band minimum and valence band maximum (adjusted to 0.0 eV). The band gaps estimated using the PBE+U methods are comparatively closer to the experimental band gap where U = 2.8 eV underestimates by about 0.7 eV and U = 6.4 eV overestimates by 0.9 eV.

Fig. 4 presents the density of states showing the Ni 3d, O 2p and Al p states. Ni 3d states occupy most of the conduction band with spin down states for all the four functionals. The band energies that the Ni 3d, O 2p and Al p states occupy are different for each functional and this gives rise to the differences in the band gaps. The shapes of the O 2p and Al p states do not vary with the use of different functionals where the density of Al p states is almost negligible compared to Ni 3d and O 2p states. The density of the Ni 3d states in the conduction band is estimated to be highest using PBE+U (U = 6.4 eV) and lowest using HSE06. PBE+U (U = 2.8 eV) and optB88-vdW show similar density of Ni 3d states in the conduction band. In the valence band, PBE+U (U = 2.8 eV), optB88-vdW and HSE06 show similar band structures with sharp peaks of Ni 3d of both spin up and down states between 0.0 eV and −1.8 eV and sharp peaks of spin down states between −1.8 eV and −3.0 eV. On the other hand, PBE+U (U = 6.4 eV) shows a peak of Ni 3d states in the spin up state between −2.0 eV and −3.0 eV and another spin up state between −6.0 eV and −7.0 eV.

image file: c9cp04325j-f4.tif
Fig. 4 Density of states with respect to band energies of tetragonal P4122 NiAl2O4 for different functionals. The shaded grey area represents the total density of states. Density of Ni 3d, O 2p and Al p states are shown by blue, red and green lines respectively. The area above the horizontal line show the spin up state and the area below show the spin down state. The vertical dotted lines represent the valence band maximum (adjusted to 0.0 eV) and the conduction band minimum.

This difference in the electronic structure using a U value of 6.4 eV can be seen beyond U values of 3.0 eV. Fig. 5 shows the density of Ni 3d states using U values from 0.0 eV to 8.0 eV for the functional PBE+U. As can be seen in the figure, the increment of the U values results in the enlargement of the band gap, but it comes at the expense of dramatic modification of the Ni 3d states. Beyond U values of about 3.0 eV, the sharp peaks of Ni 3d states near the Fermi level continue to move further away and thus results in artificial modification of the dispersion of the Ni 3d states.

image file: c9cp04325j-f5.tif
Fig. 5 Density of states with respect to band energies of tetragonal P4122 NiAl2O4 for PBE+U functional using U values from 0.0 eV to 8.0 eV. The shaded grey area represents the total density of states. Ni 3d states are shown by blue lines. The area above the horizontal line shows the spin up state and the area below show the spin down state of the electrons. The vertical dotted lines represent the valence band maximum which has been adjusted to 0.0 eV.

The magnetic moment of NiAl2O4 is found to be in the range of 1.7–1.8 μB with almost all the contribution coming from Ni whose partial magnetic moment lies in the range of 1.6–1.7 μB for all the functionals. It forms a ferromagnetic structure, in agreement with experimental observation.84

Phase diagram

The thermodynamic stability of NiAl2O4 at varying temperatures and changes in chemical potentials of Ni (ΔμNi) and Al (ΔμAl) using experimental data and different functionals is presented in Fig. 6. The functionals PBE+U and optB88-vdW are used as they provide better approximation to the experimental formation energy of NiAl2O4 compared to PBE and HSE06. The colored area is the region of stability for NiAl2O4 that is spanned by ΔμNi and ΔμAl and the lines show the limits set by NiAl2O4 and its competing phases, NiO and Al2O3. Experimental data and all the functionals predict that Al-poor and Ni-rich conditions are necessary for the NiAl2O4 bulk to be stable. The PBE+U functional show very similar region of stability compared to the phase diagram constructed using experimental data where NiAl2O4 is stable for similar range of ΔμNi (about −2.5 to 0.0 eV) and ΔμAl (about −8.8 to −5.0 eV). The optB88-vdW, on the other hand, show a stability region between −2.0 to 0.0 eV for ΔμNi and −9.0 to −6.0 eV for ΔμAl. However, optB88-vdW performs better than the PBE+U functionals in estimating the area of the NiAl2O4 stability region. The PBE+U functional show a larger area of stability, but the region of stability found with optB88-vdW is more similar to the experimental data. This means that for a specific ΔμNi, NiAl2O4 would be stable for a larger range of the corresponding ΔμAl and vice versa when PBE+U functionals are used. The phase diagram shows the temperature at atmospheric pressure corresponding to ΔμNi and ΔμAl. The optB88-vdW shows different ΔμNi and ΔμAl for a given temperature compared to experimental data and the PBE+U functionals. For example, at 800 K, experimental data and PBE+U functional predict that NiAl2O4 is stable when ΔμNi is −2.1 eV and ΔμAl is −8.3 eV. However, optB88-vdW estimate a slightly lower value of −8.6 eV for ΔμAl and −1.6 eV for ΔμNi in the same condition.
image file: c9cp04325j-f6.tif
Fig. 6 Thermodynamic phase stability diagram of tetragonal P4122 NiAl2O4 using different functionals and experimental data with respect to changes in Al, Ni and O chemical potential. The solid black, solid red and dashed green lines are the lines of stability limits of NiAl2O4, NiO and Al2O3 set by the changes in Al and Ni chemical potential. The colored area shows the region of stability of NiAl2O4 compared to its competing phases, NiO and Al2O3. The different shades of color are defined by the temperature of the system at atmospheric pressure.

Nevertheless, all the functionals used correctly reproduce the experimental phase diagram that NiAl2O4 is thermodynamically stable at operating conditions, 900–1000 K and 1 atm, of catalytic processes, such as dry reforming of methane, given that Al poor and Ni rich conditions are present. This is consistent with the dry reforming of methane experiments performed by Rogers et al.37 at 973 K using untreated NiAl2O4 and pre-treated NiAl2O4 under reducing conditions. In both the cases, NiAl2O4 did not get reduced to metallic Ni or NiO and in addition, provided conversion of 32% CH4 with 7.5 wt% and 12.6 wt% carbon deposition using untreated and pre-treated NiAl2O4 catalyst respectively. This clearly shows that the active sites for dry reforming of methane are ionic sites in the NiAl2O4 spinel phase and not metallic Ni.


The local atomic order in the NiAl2O4 spinel structure has been identified using periodic DFT combined with cluster expansion. NiAl2O4 adopts the tetragonal P4122 structure with Niocta, Alocta and Altetra, thus forming an inverse spinel structure. The Niocta + Altetra ⇌ Nitetra + Alocta exchange starts to take place at a temperature above 800 K, and thus the degree of inversion starts to deviate from 1 beyond 800 K. However, the structure remains largely inverse with further increase in temperature, as is also found experimentally.

The optB88-vdW and PBE+U functionals provide accurate results for the experimental formation energy and the lattice parameters compared to PBE and HSE06, but only PBE+U gives better agreement in terms of the experimental band gap of NiAl2O4. However, using U values above 3.0 eV results in the artificial modification of the shape of Ni 3d states. Thus, PBE+U with U = 2.8 eV gives the best agreement overall with experimental data in terms of the energetics, geometry and electronic structure compared to PBE, optB88-vdW, HSE06 and PBE+U with U = 6.4 eV.

NiAl2O4 is found to be thermodynamically stable at operating conditions, 900–1000 K and 1 atm, of catalytic processes, such as dry reforming methane with the constraint that Al poor and Ni rich conditions are present to prevent the decomposition of NiAl2O4 to its competing phases, NiO and Al2O3.

Conflicts of interest

There are no conflicts to declare.


I. E., A. S. and A. M. gratefully acknowledge the fund provided by the University of Sydney – Yonsei Joint Research Funding Scheme, International Program Development Fund. I. E. and A. M. also gratefully acknowledge the computational resources provided by Artemis HPC system at the University of Sydney and by Intersect with the assistance of the Australian National Computational Infrastructure (NCI).


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Electronic supplementary information (ESI) available: Formation energies of NiAl2O4, NiO and Al2O3 from experimental data and different functionals; effective cluster interactions from cluster expansion; fitting of Hubbard U value with the experimental formation energy of NiAl2O4; formation energies of NiAl2O4 using van der Waals corrected PBE functionals. See DOI: 10.1039/c9cp04325j

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