Native Defect Concentrations in NaAlH4 and Na3AlH6 - The Journal

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Native Defect Concentrations in NaAlH4 and Na3AlH6 Kyle Jay Michel and Vidvuds Ozolin-s* Department of Materials Science and Engineering, University of California, Los Angeles, P.O. Box 951595, Los Angeles, California 90095-1595, United States

bS Supporting Information ABSTRACT: Titanium-doped sodium alanate has been shown to have potentially useful properties for storing hydrogen in fuel cell vehicles. To quantitatively explain the kinetic rates and activation energies for hydrogen release and absorption, it is necessary to calculate the rate of metal diffusion that is required for forming the dehydrogenation products, Na3AlH6 and Al. The bulk defects existing in large concentrations will likely play a dominant role in the transport of metal species. In the first of a series of papers, we use first-principles density functional theory (DFT) calculations to determine the formation free energies and concentrations of native defects in NaAlH4 and Na3AlH6 as functions of temperature, including vibrational and H2 gasphase entropy contributions. We find that at low temperatures the largest concentrations of native defects in NaAlH4 are positively charged AlH4 vacancies and negatively charged Na vacancies, which can be thought of as the primary defect types for NaAlH4. At high temperatures (near 100 °C), neutral AlH3 vacancies, positively charged AlH4 vacancies, and negatively charged interstitial hydrogen ions and hydrogen vacancies have the largest concentrations at an interface between NaAlH4 and Al. At all temperatures, an interface between NaAlH4 and Na3AlH6 remains predominantly populated by positively charged AlH4 vacancies and negatively charged Na vacancies. In Na3AlH6, the highest defect concentrations belong to negatively charged Na vacancies and positively charged H vacancies throughout the entire temperature range considered. Inclusion of the gas-phase free energy of hydrogen is shown to be crucial for obtaining quantitative estimates of defect free energies and concentrations.

’ INTRODUCTION Onboard storage of hydrogen remains a significant obstacle for large-scale adoption of fuel-cell passenger vehicles due to the challenges posed by the complex set of requirements for an economically viable storage system.1 Significant effort has been expended to find new materials that can reversibly store hydrogen at near-ambient conditions. Complex hydrides have attracted particular attention due to their high volumetric and gravimetric hydrogen densities and favorable thermodynamics.24 Unfortunately, most complex hydrides are severely limited by very slow rates of hydrogen release and practical irreversibility, often requiring temperatures that are several hundred degrees Celsius above the target temperature set by the operating conditions of proton exchange membrane (PEM) fuel cells (approximately 80 °C). Therefore, considerable effort has gone into understanding the (de)hydrogenation properties of NaAlH4, which has been found to exhibit greatly improved reaction kinetics when mixed with small amounts of finely dispersed Ti.5 On heating, sodium alanate undergoes a two-step dehydrogenation reaction 1 2 NaAlH4 f Na3 AlH6 þ Al þ H2 3 3 3 Na3 AlH6 f 3NaH þ Al þ H2 2 r 2011 American Chemical Society

ð1Þ

ð2Þ

Carried to completion of Reaction 2, NaAlH4 has a theoretical gravimetric capacity of 5.6 wt % H2. The thermodynamic properties of Reactions 1 and 2 are well-understood. Experimental studies have found that the equilibrium temperature for Reaction 1 is equal to 33 °C at P = 1 bar hydrogen pressure,6 easily within the range of temperatures required for onboard hydrogenstorage reactions.7 However, as with many hydrogen-storage materials that are characterized by good thermodynamics,821 pure NaAlH4 is limited by unacceptably slow reaction kinetics. It has been shown that doping with low concentrations of Ti (a few mol %) greatly increases the reaction kinetics and lowers the activation energy to 73 kJ/mol from 118 kJ/mol in the undoped sample for Reaction 1 and to 97 kJ/mol from 121 kJ/mol for Reaction 2.22 In addition, high levels of reversibility, even after repeated cycles, have been observed in Ti-doped samples.5 Although the sodium alanate system does not meet all the targets that have been set for onboard hydrogen storage in fuel cell vehicles,7 an atomic-level understanding of the kinetics of Reactions 1 and 2 is important for achieving further improvement in other hydrogen storage materials. Both experimental and computational studies have suggested that a defect-related process may be the rate-limiting step in Reaction 1.23,24 For instance, Gunaydin et al.23 proposed that Received: April 20, 2011 Revised: August 15, 2011 Published: September 15, 2011 21443

dx.doi.org/10.1021/jp203672u | J. Phys. Chem. C 2011, 115, 21443–21453

The Journal of Physical Chemistry C diffusion of neutral AlH3 vacancies is the rate-limiting step when NaAlH4 is doped with Ti. However, it was noted in Wilson-Short et al.24 that the results of those calculations did not in fact lead to this conclusion as an incorrect multiplicative factor was used during comparison with experimental activation energies. Wilson-Short et al. proposed that the creation of charged hydrogen vacancy/interstitial pairs in the bulk NaAlH4 phase is the ratelimiting step in undoped samples. However, this is contradicted by experimental results that suggest the rate-limiting step is likely the diffusion of metal species.2527 Therefore, more than a decade after the original discovery,5 there is still no general agreement on which process limits the (de)hydrogenation rate of sodium alanate. This is the first in a series of three papers examining the free energies of formation, concentrations, and diffusion rates of point defects in the sodium alanate system. The second paper in this series (DOI: 10.1021/jp203673s) studies the effects of Ti doping on the concentrations of bulk defects,28 while the third paper (DOI: 10.1021/jp203675e) gives the calculated migration energies and diffusivities of bulk defects in NaAlH4 and Na3AlH6.29 Here we present results of first-principles densityfunctional theory (DFT) calculations of defect concentrations in undoped NaAlH4 and Na3AlH6. This is an important first step in understanding if metal diffusion represents the rate-limiting process of Reaction 1 since the defects that exist in the highest concentrations are likely to be involved in the fastest bulk diffusion pathways. Previous calculations have addressed the energetics of defects in NaAlH4 using static total energies as obtained from density-functional theory (DFT) calculations.24,30 We go beyond these studies by incorporating finite-temperature vibrational free energies and the free energy of gaseous hydrogen. We also present DFT calculations of defects in Na3AlH6, a product phase of Reaction 1, which to the best of our knowledge have not been previously published. Recent experimental work has found that large defect concentrations exist in at least one of the product phases of Reaction 1 and that they are even larger than defect concentrations in the initial NaAlH4 phase.31 Therefore, the calculation of these concentrations is important for understanding the kinetics of decomposition. Defect concentrations are computed using three levels of physical accuracy. First, only the static DFT energies are included, enabling direct comparison with the results of previous studies.24,30 Second, the temperature-dependent free energy of gaseous H2 is added to the static energies. Finally, temperaturedependent vibrational free energies are added to the static and H2 gas-phase free energies. Comparison of these results shows that the inclusion of the temperature-dependent energies, especially the H2 gas-phase free energy, is significant and necessary for obtaining accurate concentrations of defects at finite temperatures. Our calculations show that the highest defect concentrations in NaAlH4 are those of negatively charged Na and H vacancies, neutral AlH3 vacancies, positively charged AlH4 vacancies, and negatively charged H interstitials. In Na3AlH6, negatively charged Na vacancies and positively charged H vacancies have the highest concentrations, with all other defect concentrations being several orders of magnitude lower.

’ METHODS Model and Assumptions. One of the first in situ X-ray diffraction studies of thermal decomposition kinetics in sodium alanate was performed by Gross et al.,32 who showed that pure

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and Ti-catalyzed systems undergo very different decomposition pathways. In pure systems, the product phases of Reaction 1 are not observed in appreciable amounts until the onset of melting of NaAlH4 at 180 °C. At this point, Reaction 2 is thermodynamically favorable, and there is a coexistence of molten NaAlH4 with solid Na3AlH6, Al, and NaH. In addition, Gross et al. observed a transient intermediate phase, which they called X1 and which disappeared on continued heating. Other studies have also observed the formation of intermediate phases in superheated uncatalyzed samples,27,33 and the existence of a new γ-NaAlH4 phase has been proposed based on theoretical calculations.34 In contrast, Gross et al.32 clearly showed that Ti-doped systems decompose according to Reaction 1 when the temperature is only 100 °C. Since this is below the temperature where Reaction 2 becomes thermodynamically possible, during decomposition there is a coexistence of three crystalline phases (NaAlH4, Na3AlH6, Al) and hydrogen gas. When the temperature is raised above the onset of Reaction 2, solid NaH is observed as a decomposition product of Na3AlH6. These findings have been confirmed by subsequent studies on catalyzed samples.35,36 In this study, we aim to establish whether the intrinsic defect concentrations and mass transport rates in NaAlH4 and Na3AlH6 can explain the observed rates of hydrogen release and absorption in catalyzed samples, which decompose via Reaction 1 between crystalline bulk phases. This implicitly assumes that the Ti catalyst has negligible effects on the rates of bulk mass transport in NaAlH4, Na3AlH6, and Al and is largely incorporated in the form of TiAl alloys.3739 The validity of the latter assumption is demonstrated in the second paper of this series where we calculate the effect of substitutional Ti doping on the concentration of native defects and find that it cannot explain the observed decrease in the activation energy.28 Accordingly, the catalytic effect of Ti is attributed to accelerating process(es) other than bulk diffusion, such as nucleation of the product phases, formation/dissociation of H2 molecules, or interfacial reactions involved in breaking and reforming AlH bonds. Temperatures above the melting point of NaAlH4 may be required to get acceptable rates for these steps without the Ti catalyst. Some of these steps have been studied recently using first-principles methods (e.g., hydrogen dissociation and diffusion on TiAl,40), while others still await theoretical examination (e.g., nucleation of product phases and interfacial reactions). In this paper, we have examined native defects in NaAlH4, Na3AlH6, and Al under the assumption that they exist as crystalline bulk phases and that no other phases (besides H2) are present. Physically, this represents conditions in catalyzed samples below the onset temperature of Reaction 2. Incoherent solidsolid interfaces are assumed to form between these phases, and local equilibrium conditions are used to fix the chemical potentials and determine defect concentrations. The real interface morphology may be much more complex than assumed here, but we are concerned only with the concentrations of defects at these interfaces and the relative gradients of defect concentrations (therefore, mass fluxes) that exist between them. As long as the local equilibrium is maintained, the actual structure of the interface is not important for determining defect concentrations. The kinetics of interfacial reactions leading to the formation of defects are not examined here since realistic studies of chemical processes at incoherent interfaces are currently beyond the computational capabilities of accurate first-principles methods. We make use of an assumption that these kinetics are faster than typical diffusion times in bulk samples, and therefore 21444

dx.doi.org/10.1021/jp203672u |J. Phys. Chem. C 2011, 115, 21443–21453

The Journal of Physical Chemistry C

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local equilibrium is maintained at the interfaces. The latter assumption may break down in very small particles with short diffusion paths, where experimental evidence indicates that the measured activation energies depend on average particle size.41 The present study could be extended to describe the reaction kinetics in undoped samples by including NaH as a fourth solid phase and examining mass transport in molten NaAlH4, but this has not been attempted here. Defect Concentrations. Periodic bulk supercells were used to calculate the DFT formation energies of various defects. The formation energy of defect i is defined as42 ΔEi ¼ Ei ðdefectÞ EðsupercellÞ

Nspecies

∑

s¼1

nis μs þ qi ðEVBM þ εF þ ΔV Þ

ð3Þ

where Ei(defect) is the total energy of a supercell with the defect; E(supercell) is the energy of the perfect bulk supercell; nis is the number of atoms of type s that were added (nis > 0) or removed (nis < 0) to create the defect; μs is the chemical potential of atomic specie s; and the sum is taken over all elemental species, Nspecies. The electron chemical potential (i.e., the Fermi level), εF, is referenced to the valence-band maximum (VBM), EVBM, of the bulk material, and the ΔV term is an extra shift in the average electrostatic potential which aligns the VBM of defect supercells with the bulk VBM. This alignment term originates from the change in the electrostatic potential due to the finite size of the defect supercell.42 The concentration of defect i is given by Ci ¼

Di ≈ Di eΔEi =kB T Di þ eΔEi =kB T

ð4Þ

where Di is a multiplicative factor accounting for the number of equivalent ways that defect i can be introduced per site. ΔEi was defined in eq 3; kB is the Boltzmann constant; and T is the temperature. The Fermi level was fixed by enforcing charge neutrality according to Ndefects

∑ Ci q i ¼ 0 i¼1

Figure 1. Reaction morphologies when studying defect diffusion through NaAlH4 (a), Na3AlH6 (b), and Al (c).

reactants are out of equilibrium, except at the critical temperature for hydrogen release where the free energies of the left- and righthand sides of Reaction 1 are exactly equal. The mass fluxes will depend on the concentration gradients and diffusion rates of bulk defects, BJ i = Di3Ci. This paper is concerned with the local concentrations entering the gradient 3Ci, while ref 29 presents the calculated diffusivities Di. In the first morphology [Figure 1(a)], mass transport occurs by defect migration through the NaAlH4 phase between the right interface, where NaAlH4, Na3AlH6, and H2 coexist, and the left interface, where NaAlH4, Al, and H2 are present. These are consistent with interfaces that have been proposed in both experimental and computational work.23,24,26,43 Chemical potentials [μs in eq 3] were calculated separately at each interface assuming local thermodynamic equilibrium. They were fixed by the condition that the free energies of coexisting phases are equal to the sums of the elemental chemical potentials. At the interface between Na3AlH6 and NaAlH4, the chemical potentials satisfy 3μNa þ μAl þ 6μH μNa þ μAl þ 4μH 2μH

ð5Þ

where the sum runs over all defects and qi is the charge of defect i. This ensures that positively and negatively charged defects exist in compensating concentrations while not posing any restrictions on the types and absolute concentrations of these defects. Equation 5 was solved with eq 3 and eq 4 so that the Fermi level, formation energies, and concentrations of all defects were determined uniquely as functions of temperature and chemical potentials. Chemical Potentials. Chemical potentials μs in eq 3 were determined from local equilibrium conditions for coexisting phases at solidsolidgas interfaces. These conditions depend on the assumptions about the reaction morphologies and the associated solid-state diffusion mechanisms. We illustrate this concept in Figure 1, showing three possible diffusion morphologies for Reaction 1. It is assumed that the reactant (NaAlH4) remains in physical contact with the reaction products (Al and Na3AlH6 solids) and that long-range mass transport through a given phase (NaAlH4, Na3AlH6, or Al) is driven by chemical potential gradients between the coexisting interfaces. These gradients exist because the free energies of the products and

¼ GðNa3 AlH6 Þ, ¼ GðNaAlH4 Þ, ¼ GðH2 Þ

ð6Þ

Here, G(NaAlH4) represents the free energy of a single formula unit of NaAlH4 that, in the most detailed calculations, included the total electronic energy as calculated in DFT and the temperature-dependent vibrational free energy (and in the case of H2 also its gas-phase free energy). At the interface between Al and NaAlH4, these constraints become μNa þ μAl þ 4μH μAl 2μH

¼ GðNaAlH4 Þ, ¼ GðAlÞ, ¼ GðH2 Þ

ð7Þ

Solving for the chemical potential difference between the left (L) and right (R) interfaces from the above equations, we obtain μLAl μRAl μLNa μRNa 21445

3 ¼ ΔGr , 2 3 ¼ ΔGr 2

ð8Þ



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where ΔGr is the free energy change for Reaction 1. This relation shows that above the critical temperature for hydrogen release (where ΔGr < 0) the chemical potential for Al is higher at the NaAlH4/Na3AlH6 interface. Therefore, Al ions will diffuse toward the Al/NaAlH4 interface, or equivalently, Al vacancies will diffuse away from the Al/NaAlH4 interface toward the NaAlH4/Na3AlH6 interface, as suggested in ref 23. Similarly, under hydrogen release conditions, the Na chemical potential will be higher at the Al/NaAlH4 interface and drive the mass transport of Na toward the NaAlH4/Na3AlH6 interface. It is easy to see that in the case of hydrogen release the mass fluxes are consistent with the direction of the reaction and lead to the consumption of the reactant (NaAlH4) and growth of the products (Al and Na3AlH6). For the rehydrogenation reaction (corresponding to ΔGr > 0), these mass fluxes are reversed. The second morphology shown in Figure 1(b) is characterized by diffusive mass transport through the Na3AlH6 phase where at one of the relevant interfaces the coexisting phases are Na3AlH6, NaAlH4, and H2 and at the other they are Na3AlH6, Al, and H2. Just like above, these local environments were used to fix the chemical potentials of the elemental species, μs in eq 3. Equilibrium conditions at the right interface in Figure 1(b) are given by eq 6. At the left interface where Na3AlH6, Al, and H2 are present, the chemical potentials satisfy 3μNa þ μAl þ 6μH μAl 2μH

¼ GðNa3 AlH6 Þ, ¼ GðAlÞ, ¼ GðH2 Þ

ð9Þ

Following the preceding discussion and solving for the chemical potential difference between the left (L) and right (R) interfaces, we obtain μLAl μRAl μLNa μRNa

3 ¼ ΔGr , 2 1 ¼ ΔGr 2

ð10Þ

showing that above the critical temperature for hydrogen release (ΔGr < 0) Al ions will diffuse through Na3AlH6 toward the left Al/Na3AlH6 interface, while Na ions will diffuse toward the right Na3AlH6/NaAlH4 interface. Similarly, Al vacancies will be driven to the right interface, and Na vacancies will migrate toward the left interface in Figure 1(b). The third possible reaction morphology is shown in Figure 1(c). In this case, the Al product phase is the medium through which diffusion proceeds. The chemical potential of Al is equal at both interfaces [μAl = G(Al)], and the only relevant diffusion flux is that of Na through metallic Al. It is well-known that Na is practically insoluble in Al. Under the local equilibrium conditions relevant for Figure 1(c) and using only static DFT total energies, we find that at T = 0 K the energy cost to substitute a Na ion for an Al ion in an Al supercell is 1.77 eV at the interface with NaAlH4 and 1.50 eV at the interface with Na3AlH6. As expected, these energies result in negligible concentrations of Na in Al. The total activation energy for vacancy-mediated diffusion of substitutional Na in Al will also include the energy required to create an Al vacancy (approximately 0.6 eV4446) and the migration energy for swapping the sites of the vacancy and Na atom, minus the binding energy between the vacancy and Na. The resulting mass fluxes for the morphology shown in Figure 1(c) will be much smaller than those in Figure 1(a)

Figure 2. Unit cell of NaAlH4. Yellow spheres show Na ions; pink, H ions; and blue, Al ions. Tetrahedra are added to show the orientation of AlH4 units.

or (b), and therefore this morphology will not contribute significantly to the reaction rates and will not be considered here. Ab Initio Calculations. We used the generalized gradient approximation (GGA) to the electronic exchange-correlation functional (Perdew and Wang47) of density-functional theory (DFT) as implemented in the Vienna Ab-initio Simulation Package48 (VASP). The experimentally determined lattice parameters of NaAlH4 (space group I41/a, Figure 2) are a = 5.010 and c = 11.323 Å.49 From static GGA calculations, these were found to be equal to a = 4.894 and c = 10.775 Å. Part of the unusually large difference between the experimental and static GGA lattice parameters can be attributed to the neglect of lattice expansion due to anharmonic zero-point energy effects. It has been shown that a quasiharmonic treatment of vibrational effects gives an approximately 1.2% expansion in the a parameter and a 1.5% expansion in the c parameter at T = 0 K, increasing to 1.5% and 2.5% for a and c at 300 K, respectively.50,51 Given the fact that we were attempting to explain experimental results, we chose to fix the lattice parameters to their experimentally determined values in all calculations involving NaAlH4. The Na3AlH6 structure (space group P21/n, Figure 3) has a = 5.402, b = 5.507, and c = 7.725 Å with β = 89.491° from experiments.49 From DFT calculations, these were predicted to be a = 5.344, b = 5.517, and c = 7.663 Å. Since experimentally and computationally derived lattice parameters agree well for this structure, the latter values were used in all calculations involving Na3AlH6. 21446



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Formation energies were found to be sufficiently converged with these settings. For example, the formation energy of an AlH3 vacancy in NaAlH4 was converged to better than 1 meV/defect with respect to results using a 2 2 2 Γ-centered k-point mesh. Charged defect states were achieved in the usual way by manually adjusting the total number of electrons in the system and adding a compensating uniform background charge. In the special case of Na substitutions in Al, a 256-atom supercell of Al was used with a 4 4 4 Γ-centered k-point mesh and a cutoff energy of 500 eV. The supercell frozen phonon method56 was used to find the vibrational free energy of each structure as a function of temperature. Defects were introduced into a 2a 2a c {2a 2b c} supercell of NaAlH4 {Na3AlH6} that contains 96 {80} atoms and subsequently relaxed using the methods described above, with the exception that the Brillouin zone was sampled on a 2 2 2 Γ-centered k-point mesh. The forces were calculated for all symmetry-inequivalent atomic displacements in each relaxed structure for two 0.03 Å steps on each side of the equilibrium configuration. The force constants were obtained by fitting a third-order Chebyshev polynomial to the calculated forces and extracting the linear term. The resulting dynamical matrix (evaluated at the Γ point) was diagonalized to obtain the phonon mode frequencies within the harmonic approximation. These frequencies were used to calculate the vibrational enthalpy and entropy as functions of temperature for each defect type. The vibrational free energies for the 96- {80-} atom supercells were scaled to the 768- {960-} atom supercell using large

Gdef ðTÞ ¼ Gsmall def ðTÞ þ Figure 3. Unit cell of Na3AlH6. Yellow spheres show Na ions; pink, H ions; and blue, Al ions. Octahedra are added to show the orientation of AlH6 units.

Bulk NaAlH4 {Na3AlH6} supercells that contain 768 {960} atoms were used for all ground-state total-energy calculations; this corresponds to a supercell of dimension 4a 4a 2c {4a 4b 3c} where a, {b}, and c are the conventional unit cell lattice parameters. This large cell reduces both electrostatic and strain interactions between periodic images of defects due to finite supercell size. We neglected the shift in the VBM at the cell boundary, ΔV in eq 3, which is a reasonable simplification since ΔV for a positively charged AlH4 vacancy in NaAlH4 was calculated to be less than 20 meV. Similar small corrections were found for other defects, likely below the numerical accuracy of the methods that were used to determine total energies and vibrational free energies. Dipole corrections due to electrostatic interactions between periodic images52 were not used here since they are known to be often unreliable.53,54 Furthermore, since the calculated low-frequency dielectric constants of NaAlH4 are reasonably high (ε0xx = 10.1 and ε0zz = 9.250), and the magnitude of the dipole corrections decreases as the third power of the defect separation, we expect that these corrections are small for the very large supercells employed here. For each defect supercell, the atomic positions were fully relaxed until the forces were below 0.01 eV/Å, while the lattice parameters and cell shape were fixed. We have checked that allowing for changes in the cell shape and lattice parameters leads to very small corrections in the calculated defect energetics (20 meV for an AlH3 vacancy in Na3AlH6). Ultrasoft pseudopotentials55 (USPP) were used with a cutoff energy of 250 eV, and the electronic k-point mesh was set to sample only the Brillouin zone center, or Γ point.

Nlarge Nsmall small Gpure ðTÞ Nsmall

ð11Þ

where Nlarge is the number of atoms in the large supercell, 768 {960} for NaAlH4 {Na3AlH6}, and Nsmall is the number of atoms in the small supercell, 96 {80} for NaAlH4 {Na3AlH6}. Glarge def (T) is the vibrational free energy of the 768- {960-} atom supercell with defect; Gsmall def (T) is the vibrational free energy of the 96{80-} atom supercell with defect; and Gsmall pure (T) is the vibrational free energy of the 96- {80-} atom pure NaAlH4 {Na3AlH6} supercell. This approximate method of scaling the vibrational free energies was used to avoid the prohibitive computational expense of calculating vibrational free energies in large supercells for which the symmetry has been broken by the introduction of defects. The following expression was adopted for the free energy of gaseous H257 GðH2 Þ ¼ Ecalc ðH2 Þ þ

7 kB T TSemp ðH2 Þ 2

ð12Þ

where kB is the Boltzmann constant and Ecalc(H2) is the calculated energy (electronic and zero-point) of an isolated H2 molecule in a periodic cubic cell of side length equal to 10 Å. Semp gives the empirical H2 gas-phase entropy 7 lnðTÞ lnðpÞ þ C Semp ðH2 Þ ¼ kB ð13Þ 0 2 where p is the pressure (here p = 1 bar) and C0 = 4.22 is a dimensionless constant. We refer to the factor of [(7/2)kB Semp(H2)]T in eq 12 as the gas-phase free energy of H2.

’ BULK DEFECTS IN NaAlH4 In what follows, we label vacancies of type X as [X] and interstitial defects of type X as iX. Charged vacancies are labeled 21447



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Table 1. Defect Formation Enthalpies and Concentrations in NaAlH4 at T = 400 K Using Only the DFT Energiesa Na3AlH6/NaAlH4/H2 defect [Al]3

ΔH400K form 1.98

2

[AlH]

1.77

400K

C

4.74 1025 22

1.85 10

C400K

3.54

1.04 1044

2.95

3.01 1037

[AlH2] [AlH3]

1.29 0.87

2.51 10 4.61 1011

2.07 1.27

3.01 1026 4.08 1016

[AlH4]+

0.49

6.24 107

0.51

4.08 107

0.48

9.54 107

0.98

1.98 1012

0.55

5.13 107

1.72

7.50 1022

[Na]

[NaH] +

[H]

[H]

0.49 1.38 0.93 1.34

16

Al/NaAlH4/H2 ΔH400K form

7

6.24 10

17

1.75 10

12

6.95 10

17

5.53 10

H

1.03

4.50 10

0.64

3.32 108

H

0.77

8.46 1010

1.15

1.15 1014

i i

+

13

a

Values are given at the interface at which Na3AlH6, NaAlH4, and H2 are in contact as well as the interface at which Al, NaAlH4, and H2 are in contact. Free energies of formation are given in units of eV/defect, while concentrations are given as the number of defects per f.u. of NaAlH4.

so that [Al]3 represents an Al vacancy with a 3 charge, i.e., the removal of an Al3+ ion. Charged interstitial defects are simply labeled according to their charge state and superscript “i” so that iH+ represents an interstitial hydrogen ion with a positive charge. The NaAlH4 phase, shown in Figure 2, can be viewed as an ordered system of Na+, Al3+, and H ions. We consider the native defects for this structure that are listed in Table 1. Although we are primarily concerned with studying the concentration of metal-type defects, we include charged hydrogen defects, as these may exist in large concentrations and will therefore have a significant effect on the Fermi level. Interstitial Na and Al ions have been found previously to be high in energy relative to the other defects;24 therefore, we neglect these defect types as their concentrations will be too low to have a significant effect on the Fermi level or diffusion kinetics. Static Energetics. Here we report the results of total energy calculations for the formation of various defects in NaAlH4 without the inclusion of vibrational or H2 gas-phase contributions to the free energy. These results are used as a reference for later results in which these free energy contributions are included (see the next subsection). We find that the vibrational and H2 gas-phase contributions to the free energy significantly change the general qualitative conclusions about the relative abundance of defect types and therefore are essential for obtaining accurate defect concentrations at typical reaction temperatures. With the set of chemical potentials given by eqs 6 and 7, defect formation energies and concentrations are shown in Table 1. The defect types with the lowest formation energies at T = 400 K at the Na3AlH6/NaAlH4 interface are negatively charged Na vacancies and positively charged AlH4 vacancies. At the interface between Al and NaAlH4, negatively charged Na vacancies and positively charged AlH4 vacancies again have the lowest formation energies with positively charged H vacancies and positively charged hydrogen interstitials being only slightly higher. The given energies at each interface are consistent with previously reported results24 where any significant quantitative differences are attributed to the choice of the Fermi level. Here, εF is determined from eq 5, resulting in the values of εF = 3.08 eV

at the Na3AlH6/NaAlH4 interface and εF = 2.69 eV at the Al/NaAlH4 interface at T = 400 K. When comparing these results to those obtained in previous studies, it is important to note that here the Fermi levels are self-consistently calculated as functions of temperature, which differentiates this analysis from that in Wilson-Short et al.24 Temperature Effects. We consider two approaches to incorporating temperature-dependent free energies. First, both the vibrational and H2 gas-phase free energies are added to electronic energies. These results are expected to be the most accurate ones and are used in all subsequent calculations of defect concentrations. Second, only the gas-phase H2 free energy is added to the electronic energies, ignoring the vibrational free energies. These results are included to show the effect of the vibrational free energy contributions and to estimate the accuracy of simplified approaches that do not require CPU-intensive DFT calculations of the full phonon spectra. Full Free Energies. This section discusses the case when vibrational free energies of all phases as well as the gas-phase free energy of H2 are added to the static DFT energies introduced in the previous subsection. We find that the critical temperature for hydrogen release at p = 1 bar H2 pressure is predicted to be 68 °C, whereas experimental results show that this temperature is at Tc = 33 °C.6 Given the strong dependence of the defect concentrations on temperature in eqs 3 and 4, this difference in critical temperatures between calculations and experiments is significant. This error is likely due to inaccuracies in the exchange-correlation functional, but other sources of error could be important such as anharmonic phonon modes at high temperatures; its effects on the presented results are discussed in the Supporting Information available for this paper. Chemical potentials at the two interfaces are again fixed by eqs 6 and 7. Since the free energies of all phases are temperaturedependent, the chemical potentials and Fermi levels also depend on temperature. The Fermi level is plotted as a function of temperature in Figure 4 for calculations at various degrees of physical accuracy: (i) including static total energies only, (ii) with the addition of the H2 gas-phase free energy, and (iii) including the vibrational free energies and H2 gas-phase free energies. It is seen that including the gas-phase free energy of H2 (red lines in Figure 4) has a very large effect on the Fermi level in comparison with the value calculated from static energetics only (black lines). This effect is much more pronounced at the interface with Al (0.250.95 eV over the temperature range of 150450 K) than at the interface with Na3AlH6 (0.060.26 eV). Adding vibrational free energies provides a further correction to the calculated Fermi level by roughly 0.10.2 eV depending on the temperature and interface. In this section, we use the Fermi level that is calculated from the full free energies and given by the blue lines in Figure 4. The defect formation energies at each interface as functions of temperature can be found from eqs 3 and 5 and are given for T = 400 K in Table 2. We find that defects with the lowest formation energies at the interface between Na3AlH6 and NaAlH4 are positively charged AlH4 vacancies and negatively charged Na vacancies. At the interface between Al and NaAlH4, the lowest formation free energies belong to positively charged AlH4 vacancies, negatively charged Na vacancies, negatively charged hydrogen interstitials, neutral AlH3 vacancies, and negatively charged hydrogen and AlH2 vacancies. In addition to the defect free energies of formation, the enthalpies of formation at T = 400 K are also shown in Table 2. 21448



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interface with Al. The formation enthalpy of this defect from electronic-energy-only calculations is equal to 0.51 eV, while calculations that include temperature effects give a formation enthalpy of 1.45 eV at T = 400 K. This large difference is primarily a result of the dependence of the Fermi level on temperature that comes from the addition of vibrational and H2 gas-phase contributions to the free energy. At the Al/NaAlH4 interface at T = 400 K, electronic-energy-only calculations give εF = 2.69 eV, while including vibrational and gas-phase free energies changes this value to 3.63 eV. This results in substantially different results for the formation enthalpies of charged defects at elevated temperatures. The Fermi level at the Na3AlH6/NaAlH4 interface at T = 400 K has values of εF = 3.08 eV in electronic-energy-only calculations and εF = 3.41 eV when including vibrational and gasphase free energies. Consequently, the effect on the defect formation enthalpies is smaller here than at the Al/NaAlH4 interface, but still substantial. Another important contribution to the free energies of defects comes from the vibrational and H2 gas-phase entropies, which are manifested in substantial differences between the defect formation enthalpies, ΔH, and free energies, ΔG, in Table 2. Again, the major contributing factor is the entropy of H2 molecules in the gas phase that are either absorbed or released when forming a given defect. To understand these effects, we consider the formation of AlHx vacancies at the NaAlH4/Al interface x x ðNaAlH4 Þsc f ðNaAlH4 Þ½AlH þ Al þ H2 ð14Þ sc 2 Figure 4. Fermi level as a function of temperature (a) at the NaAlH4/Al interface and (b) at the NaAlH4/Na3AlH6 interface. Various levels of physical accuracy are achieved by adding the indicated temperaturedependent free energy terms to the static DFT energies.

Table 2. Defect Formation Energies and Concentrations in NaAlH4 at T = 400 K, Including Vibrational and Gas-Phase H2 Enthalpies and Entropiesa Na3AlH6/NaAlH4/H2 400K defect ΔH400K form ΔGform

[Al]3

0.99

1.60

Al/NaAlH4/H2 400K ΔH400K form ΔGform

C400K 2.50 1020 18

2½NaHx þ Na3 AlH6 ðNaAlH4 Þsc þ NaAlH4 f ðNaAlH4 Þsc

C400K

0.70

0.58

2.12 107

[AlH]

1.11

1.41

6.11 10

1.06

0.62

6.67 108

[AlH2]

0.96

1.01

7.55 1013

1.13

0.44

1.06 105

2

9

5

[AlH3]

0.87

0.71

4.74 10

1.27

0.37

8.56 10

[AlH4]+

0.82

0.43

4.29 106

1.45

0.32

9.96 105

[Na] [NaH]

0.16 1.38

0.43 1.42

4.18 10 0.47 5.13 1018 0.98

0.54 1.76

1.80 107 2.84 1022

[H]+

1.26

1.07

1.23 1013

1.46

1.30

1.58 1016

0.79

0.38

5.79 105

1.59

1.93

1.98 1024

0.21

0.41

3.02 105

[H] i

+

i

H H

1.01 1.36 0.44

0.61 1.70 0.64

6

8

7.44 10

21

1.54 10

8

3.88 10

where (NaAlH4)Xsc denotes a NaAlH4 supercell with a defect X. Equation 14 describes a process where an AlHx group is removed from sodium alanate, an Al atom is added to the Al phase, and hydrogen is released into the gas phase, possibly through the intermediate step of dissolving H atoms as interstitial impurities in Al. With each 1/2 H2 molecule released, the defect formation entropy increases by roughly 8kB at T = 400 K, resulting in an increase in the concentration by a factor of e8 ≈ 3000. The formation of NaHx vacancies at the NaAlH4/Na3AlH6 interface follows a slightly more complicated formula

a

Values are given at the interfaces between Na3AlH6, NaAlH4, and H2 as well as Al, NaAlH4, and H2. Enthalpies and free energies of formation are given in units of eV/defect, while concentrations are given as the number of defects per f.u. of NaAlH4.

Comparing ΔH400K form in Table 1 (electronic energy only) to Table 2 (full free energy), the effects of vibrational and gas-phase contributions to the formation enthalpies are seen to be significant. As an example, consider the AlH4 vacancy at the

þ ðx 1ÞH2

ð15Þ

where a formula unit of NaAlH4 is converted into Na3AlH6 by taking two Na ions from the bulk sodium alanate and leaving behind two NaHx vacancies. Equation 15 shows that the difference in ΔS when forming a Na vacancy vs a NaH vacancy is given by one-half the entropy term of gaseous H2 (8kB at T = 400 K), again resulting in an entropic prefactor of e8 in the concentrations. It will be shown in the following subsection that the vibrational free energies have smaller, but significant, effects on the calculated defect concentrations. The free energies of formation of the lowest-energy defects at both interfaces are shown as functions of temperature in Figure 5. They vary strongly in the temperature range between T = 150 and 450 K due to both temperature dependence of the Fermi level and H2 gas-phase free energy contributions. At the Al/ NaAlH4 interface, the energies of negatively charged Na vacancies ([Na]) and positively charged AlH4 vacancies ([AlH4]+) are nearly identical, and these defects charge-balance each other until temperatures near 300 K. At higher temperatures the free energies of [AlH4]+ and [Na] start to diverge, which leads to 21449



Figure 5. Free energies of formation, including the vibrational enthalpy and entropy components, as a function of temperature for defects in NaAlH4 at the Al interface (a) and at the Na3AlH6 interface (b). Only the lowest-energy defects are shown.

qualitative changes in the charge-compensating defect types as the free energies of negatively charged hydrogen vacancies ([H]) and hydrogen interstitials (iH) become lower than those of [Na]. In other words, we observe a crossover from a low-T regime where pairs of [AlH4]+ and [Na] are most prevalent to a high-T regime where two [AlH4]+ are chargebalanced by [H] and iH. This crossover can be explained by noting that the formation of a Na vacancy in NaAlH4 at the interface with Al requires the incorporation of Al and two H2 molecules from the gas phase (a process which becomes increasingly unfavorable with rising temperature), while the formation of hydrogen interstitial-vacancy pairs can be accomplished without involving gaseous H2. The formation energy of neutral AlH3 vacancies is primarily dependent on the free energy of gaseous H2 and as a result decreases roughly linearly with increasing temperature. At the interface with Na3AlH6, the energies of negatively charged Na vacancies and positively charged AlH4 vacancies are nearly degenerate over the entire temperature range. H2 Gas-Phase Free Energy Only. Here we discuss the results obtained when the gas-phase free energies of H2 according to eqs 12 and 13 are added to the electronic energies, but the vibrational free energies are not included. These results are compared with those that include these vibrational free energies to quantify the contribution of vibrations to the defect thermodynamics. Of particular importance to the formation energies of defects is the temperature dependence of the Fermi level. Comparison of the Fermi levels in Figure 4 shows that there is as much as a

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Figure 6. Defect concentrations in NaAlH4 at the Al/NaAlH4 (a) and Na3AlH6/NaAlH4 (b) interfaces as a function of temperature including the vibrational and H2 gas-phase free energy. Defects are shown that have the highest concentrations at each interface.

200 meV difference in these values when vibrational free energies are included in the defect energetics as compared to calculations in which they are not included. This difference is significantly larger than other corrections that have been neglected, such as the shift in the VBM far from defects which is only about 1020 meV in the large supercells employed. This correction to the Fermi level will be of increasing importance with the absolute charge of a defect. Therefore, we conclude that including only the H2 gasphase free energy and electronic DFT energies can give reasonable estimates of defect thermodynamics and Fermi levels, but the corrections from vibrational free energies may also be significant. It should be noted here that qualitatively accurate results are achieved even neglecting vibrational contributions in this particular system, but quantitative results are of course significantly different. Furthermore, without prior knowledge of the contributions of vibrations to a particular system, it is difficult to exclude their necessity. Defect Concentrations. Concentrations of defects in NaAlH4 at each interface at T = 400 K are given in Tables 1 and 2 for calculations using electronic energies only and including vibrational and gas-phase free energies, respectively. Compared to the exponential dependence of the defect concentrations on the formation energy, the effect of the factor Di is small. Therefore, the defects with the highest concentrations are those with the lowest free energies of formation. Figure 6 shows the calculated 21450


The Journal of Physical Chemistry C defect concentrations obtained by including the vibrational and gas-phase free energies. Large changes in these concentrations with temperature are due to both the change in the formation free energies of defects with temperature and the Boltzmann factor in the exponent of eq 4. Over the entire temperature range shown in Figure 5, the defects with the largest concentrations at the Na3AlH6/NaAlH4 interface are [Na] and [AlH4]+. These concentrations are nearly identical at all temperatures and therefore constitute the two primary charge-compensating defect types. As discussed above, the charge compensation of defects is more complicated at the interface between Al and NaAlH4. Here, [Na] and [AlH4]+ again have by far the highest concentrations below roughly 300 K. However, at high temperatures the concentration of [Na] is significantly lower than that of [AlH4]+, with [H] and iH balancing the charge of [AlH4]+. Furthermore, AlH3 vacancies also have significant concentrations at the interface with Al at high temperatures. Compared to Table 1 (electronic energies only), the calculated concentrations of defects in Table 2 (full free energies) are significantly different. Most importantly, the metal defect with the largest concentration (positively charged AlH4 vacancies) at T = 400 K is predicted to be 2 orders of magnitude higher in concentration when vibrational and gas-phase free energies are included as opposed to when only the electronic energy is used. Other defects, such as AlH3 vacancies and negatively charged hydrogen interstitials that exist in low concentrations when only accounting for the electronic energy, are actually found to have large concentrations when the vibrational and gas-phase free energies are included; these concentrations are even comparable to those of positively charged AlH4 vacancies at the Al/NaAlH4 interface. Therefore, to accurately describe defect concentrations at the temperatures of interest for Reaction 1, vibrational and H2 gas-phase free energies should be included with the static DFT energies.

’ BULK DEFECTS IN Na3AlH6 We consider the native defects in Na3AlH6 (Figure 3) listed in Table 3. These are identical to the defects discussed for NaAlH4 with the addition of [AlH5]2+ and [AlH6]3+ that account for partial or full removal of AlH6 groups. Two interfaces are considered when defects are created in Na3AlH6 [see Figure 1(b)], and the chemical potentials are accordingly fixed using eqs 6 and 9. Having shown the significance of vibrational and H2 gasphase free energies on the concentrations of defects in NaAlH4, we only show the results obtained by including all free energy contributions. At the interface between Na3AlH6 and Al, the Fermi level varies near-linearly between T = 150 and 450 K from εF = 1.69 to 1.99 eV. At the interface between Na3AlH6 and NaAlH4, this increase is from εF = 1.70 to 1.92 eV over the same temperature range. This dependence of the Fermi level on temperature is much larger than the potential alignment and bulk relaxation effects that were neglected in our calculations and therefore both physically and numerically significant. Defect free energies of formation and concentrations at T = 400 K are given at both interfaces in Table 3 and shown in Figures 7 and 8 as functions of temperature. There are two symmetry-inequivalent Na sites in the Na3AlH6 structure. We label Na-type defects with a superscript 1 or 2 to distinguish between these two sites. Physically, site 1 is located in the octahedral void formed by AlH6 units

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Table 3. Defect Formation Energies and Concentrations in Na3AlH6 at T = 400 K, Including Vibrational and Gas-Phase Enthalpies and Entropiesa Na3AlH6/NaAlH4/H2 defect [Al]3

ΔG400K form 4.02

2

[AlH]

[AlH2] [AlH3]

3.13 2.64 1.67

400K

C

1.33 1050 39

1.98 10

33

3.07 10

21

6.05 10

C400K

3.51

3.21 1044

2.68

9.39 1034

2.25

2.84 1028

1.33

1.09 1016

[AlH4] [AlH5]2+

1.09 1.35

1.02 10 6.26 1017

0.81 1.12

3.59 1010 4.31 1014

[AlH6]3+

1.75

8.03 1023

1.58

1.08 1020

0.30

1.71 104

0.55

2.54 107

0.58

5.28 108

0.73

1.43 109

0.36

1.71 104

+

1

[Na ]

2

[Na ]

1

[NaH ] 2

[NaH ] +

[H]

0.24 0.49 0.47 0.61 0.30

13

Al/Na3AlH6/H2 ΔG400K form

4

8.74 10

6

1.30 10

6

1.38 10

8

3.74 10

4

8.76 10

[H] i + H

1.59 1.10

5.99 10 7.76 1014

1.53 1.16

3.07 1019 1.52 1014

H

1.65

9.13 1021

1.60

4.68 1020

i

20

a

Values are given at the interfaces between Na3AlH6, NaAlH4, and H2 as well as Al, Na3AlH6, and H2. Enthalpies and free energies of formation are given in units of eV/defect, while concentrations are given as the number of defects per f.u. of Na3AlH6.

Figure 7. Free energies of formation of defects in Na3AlH6 at the Al/ Na3AlH6 (a) and NaAlH4/Na3AlH6 (b) interfaces as a function of temperature. Defects are shown that have the lowest free energies of formation at each interface. [Na1] and [Na2] differentiate the two inequivalent Na sites. 21451



Figure 8. Defect concentrations in Na3AlH6 at the Al/Na3AlH6 (a) and NaAlH4/Na3AlH6 (b) interfaces as a function of temperature. Defects are shown that have the highest concentrations at each interface.

(commonly referred to as the large site), and site 2 is located in a similar tetrahedral void (small site) with the static T = 0 K DFT formation energy of Na vacancy being 0.24 eV lower at the large site. At both interfaces the primary charge compensating defects are negatively charged Na vacancies and positively charged H vacancies over the entire temperature range between 150 and 450 K. No other defect concentrations are within even 2 orders of magnitude of these concentrations. Therefore, these defects are expected to be the dominant type involved in Reaction 1 if the mass transport takes place via the Na3AlH6 phase, as required in the morphology pictured in Figure 1(b).

’ DISCUSSION AND CONCLUSIONS Free energies of formation and concentrations of native defects in NaAlH4 and Na3AlH6 have been calculated as functions of temperature. For each phase, these values have been determined at possible two-phase interfaces that may exist during Reaction 1. It has been shown that results can be made much more accurate by including the gas-phase free energy of H2 in addition to the static DFT total energy. Additional improvements in accuracy are obtained by including the vibrational free energies of all phases as functions of temperature. In NaAlH4, the metal-type defects that exist in the highest concentrations at the interfaces with both Na3AlH6 and Al are negatively charged Na

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vacancies and positively charged AlH4 vacancies. At the Al/ NaAlH4 interface, there are also large concentrations of AlH3 and H vacancies, as well as interstitial H, but these are only comparable to the highest defect concentrations at temperatures above T = 300 K. In Na3AlH6, the largest concentration of metal-type defects belongs to negatively charged Na vacancies at both interfaces with Al and NaAlH4, over the entire temperature range relevant for mobile hydrogen storage. Some insights into the reaction kinetics can be obtained by comparing concentrations of defects in NaAlH4 to those in Na3AlH6. Large concentrations of defects are needed to rapidly drive Reaction 1 to completion. Since experimental results strongly suggest that diffusion of metal defects is likely the rate-limiting process,2527 we focus only on those defects that contain Al or Na with hydrogen vacancies and interstitials balancing the charge carried by these metal vacancies. Comparison of the defect concentrations in NaAlH4 (Figure 6) and Na3AlH6 (Figure 8) shows that the highest concentrations belong to negatively charged Na vacancies in Na3AlH6; this defect also has the lowest free energy of formation. This result is consistent with recent experimental work that suggests that high defect concentrations are produced in at least one of the product phases of Reaction 1.31 From these results, it is tempting to conclude that diffusion of negatively charged Na vacancies dominates the metal transport that is necessary for Reaction 1. However, defect concentrations presented here only include information about the local equilibrium thermodynamics at all possible solid/solid interfaces. There is no information on purely kinetic effects such as the defect migration barriers and diffusion rates that, together with concentration gradients, determine the mass fluxes. Therefore, the results of this paper by themselves are not sufficient when attempting to determine which defects and atomic species are the rate-limiting ones for bulk diffusion, but they do give essential pieces of the puzzle. The diffusion rates of defects are addressed in ref 29, and the effect of Ti dopants on the concentrations of native defect is discussed in ref 28.

’ ASSOCIATED CONTENT

bS

Supporting Information. The error in the calculated free energy change of Reaction 1 compared to experiments leads to an underestimation of the equilibrium temperature of the reaction by roughly 100 K. The effect of such an error on the concentrations of native defects and on the activation energy of a flux of defects is discussed. This material is available free of charge via the Internet at http://pubs.acs.org.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected].

’ ACKNOWLEDGMENT The authors gratefully acknowledge financial support from the U.S. Department of Energy, Office of Science, Basic Energy Sciences under Grant No. DE-FG02-05ER46253. This research used resources of the National Energy Research Scientific Computing Center, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DEAC02-05CH11231. 21452



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Native Defect Concentrations in NaAlH4 and Na3AlH6 - The Journal

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