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Theoretical calculation of the energy of formation of LiBH4
Chemical Physics Letters 405 (2005) 73–78 www.elsevier.com/locate/cplett
Theoretical calculation of the energy of formation of LiBH4 Terry J. Frankcombe a
a,*
, Geert-Jan Kroes a, Andreas Zu¨ttel
b
Leiden Institute of Chemistry, Gorlaeus Laboratories, Leiden University, P.O. Box 9502, NL-2300 RA Leiden, The Netherlands b Physics Institute, University of Fribourg, Perolles 1700, Switzerland Received 29 September 2004; in final form 4 February 2005
Abstract We report density functional theory calculations on the energy of LiBH4, relative to solid B and LiH and gaseous H2. The calculated energy is 71.3 (76.1) kJ/mol H2 which can be approximately corrected for zero-point energy to give an enthalpy of 52 (57) kJ/mol H2 at the PW91 (LDA) level, smaller than but similar to the experimental value of 68.9 kJ/mol H2. Calculations on four different LiBH4 phases indicate that alternative phases are not accessible at low temperatures without applying high pressures. These results indicate that complete decomposition to H2, B and LiH is not an attractive means of obtaining a reversible hydrogen storage system based on LiBH4. Ó 2005 Elsevier B.V. All rights reserved.
1. Introduction Using hydrogen as an energy carrier is an attractive alternative to the current global dependence on hydrocarbon-based fuels, particularly for vehicular transport. The energy density of molecular hydrogen is very high and this energy can be liberated readily and rapidly using current internal combustion technology with nothing more polluting than water vapour as a by-product [1]. More efficient fuel-cell technology is on the horizon [2], increasing the attractiveness of hydrogen. However, effective storage of hydrogen presents a significant challenge that must be overcome for the hydrogen economy to replace current energy technologies [3,4]. Storing pure hydrogen on board passenger cars is unattractive from an engineering point of view, primarily due to the low density of hydrogen gas and the effort required to compress it to a reasonable volume. Adsorp-
*
Corresponding author. Fax: +31 71 527 43 97. E-mail addresses: [email protected] (T.J. Frankcombe), [email protected] (G.-J. Kroes), andreas.zuettel@ unifr.ch (A. Zu¨ttel). 0009-2614/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2005.02.017
tion into a host medium presents many advantages, potentially including reasonable operating temperatures and pressures and being loss-less. While nano-structured carbon is being heavily investigated as an adsorbent, complex metal hydride systems are proving to have many advantages, such as near-constant temperature operation and high hydrogen density [3,5,6]. An important criterion for such metal hydrides to be useful is that the enthalpy change for the release of hydrogen lies in the range 30–48 kJ/mol H2 [3]. It is to be expected that the highest gravimetric density of hydrogen can be obtained in hydrides of low-mass metals, making the complex hydride lithium borohydride (LiBH4) attractive with a total hydrogen content of 18.5% by mass. In this Letter, we describe our use of computational methods based on density functional theory to calculate the reaction energy for the formation of LiBH4 from its decomposition products. The release of 75% of the available hydrogen is achieved by the decomposition LiBH4 ! LiH þ B þ 32H2
ð1Þ
Decomposition of LiH to release the last 25% of the LiBH4 hydrogen content was not considered as the
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T.J. Frankcombe et al. / Chemical Physics Letters 405 (2005) 73–78
temperatures required for this final decomposition are too high for on board hydrogen storage applications. LiBH4 is known to undergo a structural transition on heating. The structure of the low-temperature phase is well established [7,8], exhibiting Pnma symmetry. While a structure of P63mc symmetry has been proposed for the high-temperature phase based on X-ray diffraction data [8], recent calculations have called this structure into question [9,10]. We investigated several of the proposed structures for the high-temperature phase of LiBH4 to determine their relative stability. The energy of all phases as a function of the cell size was investigated to determine whether low-temperature, pressureinduced phase changes present more candidates for hydrogen storage systems, as recently proposed for NaAlH4 [11].
2. Method To calculate the reaction energy of the formation of LiBH4 from the basic decomposition products [the reverse of Eq. (1)] the total potential energy for solid LiH, boron and LiBH4 and for gaseous H2 are required. The energies were calculated using density functional theory [12,13]. The Dacapo package [14,15] was used for all calculations, using Vanderbilt ultra-soft pseudopotentials [16]. The Perdew–Zunger local density approximation (LDA) [17] and PW91 generalised gradient approximation (GGA) [18] exchange-correlation functionals were used. All calculations on solid phases used Monkhorst– Pack k-point sampling [19] and a plane wave basis set, with the k-point grid and plane wave cutoff energy varied to achieve convergence of the total energy to within 50 meV for each cell. This convergence criterion for the total energy translates to 0.8 kJ/mol H2 for LiBH4, where the energy unit is per mole of hydrogen released via the reaction in Eq. (1), or 1.2 kJ/mol per LiBH4 formula unit. The error in calculated energy differences, such as the reaction energy for the formation of LiBH4, would be expected to be smaller than this total energy convergence level due to cancellation of errors. Atomic positions and unit cell parameters were relaxed according to the Hellmann–Feynman forces to achieve the minimum total potential energy. For the calculation on H2, the molecule was isolated in a repeating cube
˚ and only the H–H bond length alwith edge length 10 A lowed to relax.
3. Results The LiH and H2 calculations produced structures in reasonable agreement with experiment. Using the PW91 exchange-correlation functional resulted in the LiH lattice constant [20] and the H2 bond length [21] being overestimated by 1.1% and 1.7%, respectively. Using the LDA functional also lead to an overestimation of these parameters, by 1.3% and 3.5%, respectively. The structure of solid boron is significantly more complicated. Three structures were investigated with the PW91 functional, being the tetrahedral structure with 50 atoms in the unit cell [22] and the a- and b-rhombohedral phases with 12 [23] and 105 atoms [24] in the unit cell, respectively. No attempt was made to model the defective 106.7 atom structure of Slack et al. [25] due to the excessively large super cell that would be required to model the partially occupied sites. The a-rhombohedral boron structure was found to be the most stable structure, with the b-rhombohedral and tetrahedral structures 1.8 kJ/mol per atom and 9.3 kJ/mol per atom higher in energy, respectively. Based on these results, only the energy of a-rhombohedral boron was calculated with the LDA functional. The structure of the low-temperature Pnma phase of LiBH4 has been determined experimentally [7,8] and calculated with DFT methods [9,10]. Our PW91 and LDA relaxed structures were similar to the previously reported structures, as shown in Tables 1 and 2. The lattice constants calculated in this work were slightly closer to the experimental values than those of the previously reported calculations. In this regard the c lattice constant stands out, with the PW91 and LDA calculations converging to the same value 1% larger than the experimental values while the previous calculations underestimated this value by 3%. The relative atomic coordinates within the unit cell varied from the experimental values by similar amounts to the previously reported calculations, with the atoms displaced from the experimentally determined positions by distances of around 0.01–0.03 units in coordinate space. Using the total energies calculated for LiH, a-rhombohedral boron, H2 and LiBH4 the reaction energy for
Table 1 ˚) Lattice constants for the Pnma phase of LiBH4 (A Parameter
a b c
Present work
DFT [9]
DFT [10]
PW91
LDA
PW91
PBE
7.214 4.468 6.877
7.149 4.419 6.877
7.226 4.355 6.576
7.343 4.399 6.588
Expt. [7]
Expt. [8]
7.173 4.434 6.798
7.179 4.437 6.803
T.J. Frankcombe et al. / Chemical Physics Letters 405 (2005) 73–78 Table 2 Relative internal coordinates of atoms in the Pnma phase of LiBH4 Atom
Site
Atomic coordinates
Li
4c
PW91: (0.1572,1/4,0.1000)a LDA: (0.1576,1/4,0.0988)a PW91: (0.1560,1/4,0.1123)b PBE: (0.1552,1/4,0.1137)c X-ray: (0.1568,1/4,0.1015)d
B
4c
PW91: (0.2957,1/4,0.4209)a LDA: (0.2967,1/4,0.4200)a PW91: (0.3100,1/4,0.4232)b PBE: (0.3141,1/4,0.4229)c X-ray: (0.3040,1/4,0.4305)d
H
4c
PW91: (0.8944,1/4,0.9268)a LDA: (0.8963,1/4,0.9273)a PW91: (0.9089,1/4,0.9251)b PBE: (0.9131,1/4,0.9263)c X-ray: (0.900,1/4,0.956)d
H
4c
PW91: (0.3882,1/4,0.2705)a LDA: (0.4001,1/4,0.2688)a PW91: (0.4046,1/4,0.2673)b PBE: (0.4061,1/4,0.2656)c X-ray: (0.404,1/4,0.280)d
H
8d
PW91: (0.1949,0.0278,0.4235)a LDA: (0.1945,0.0268,0.4226)a PW91: (0.2086,0.0242,0.4221)b PBE: (0.2145,0.0246,0.4224)c X-ray: (0.172,0.054,0.428)d
a b c d
DFT, this work. DFT, Łodziana and Vegge [9]. DFT, Miwa et al. [10]. Experiment, Soulie´ et al. [8].
the formation of LiBH4 was calculated with the PW91 functional to be 71.3 kJ/mol H2 and with the LDA functional to be 76.1 kJ/mol H2. For comparison, the experimental value of the enthalpy is 68.9 kJ/mol H2 [26] and Miwa et al. [10] calculated 75 kJ/mol H2 for the reaction energy (using ultra-soft pseudopotentials and the PBE exchange-correlation functional [27,28]), or 56 kJ/mol H2 for the enthalpy including zero-point energy (ZPE) corrections. If we assume that the ZPE difference calculated by Miwa et al. using the PBE functional is reasonably transferable to our PW91 calculations, an estimate of around 52 kJ/mol H2 is obtained for the ZPE-corrected enthalpy of formation from these calculations. The transferability of the ZPE difference is supported in general by the observation that calculated vibrational frequencies are insensitive to the exchange-correlation functional [29], and specifically in this case by the similarity of the PW91 ZPE for LiBH4 of 108 kJ/mol calculated by Łodziana and Vegge [9] to the PBE value of 103 kJ/mol of Miwa et al. The corresponding estimate from the LDA functional, again using the PBE ZPE correction of Miwa et al., is around 57 kJ/mol H2, slightly closer to the experimental value. The experimentally observed structure of the lowtemperature phase of LiBH4 exhibits a highly distorted
75
BH4 tetrahedron, with B–H bond lengths varying by up to 23% [8,7]. Like the previously reported calculations [9,10], the BH4 unit in the relaxed Pnma structures reported here did not exhibit this variation yielding B–H ˚ . The H–B–H bond lengths in the range 1.224–1.231 A angles within the BH4 tetrahedron varied from 106.9° to 110.5°. Łodziana and Vegge [9] determined several stable structures as candidates for the high-temperature phase observed above 384 K, using the projected augmented plane-wave approach with the VASP code [30]. Two new candidates were identified of space group P21/c and Cc, in addition to the previously suggested P63mc structure [8]. We have performed structural relaxation on these phases using the PW91 functional at zero pressure, confirming the zero-pressure stability ordering found by Łodziana and Vegge. The P21/c, Cc and P63mc phases were found to be 2.1, 7.2 and 26 kJ/mol higher in energy per formula unit than the ground-state Pnma phase without ZPE corrections. The equivalent energies reported by Łodziana and Vegge are 2.2, 3.3 and 23 kJ/mol, respectively. Clearly the energy differences between the Pnma and the Cc and P63mc phases were larger in our calculation, but this has no impact on the stability ordering. Differences of this magnitude (<10 kJ/mol) are to be expected when comparing results from calculations with differing treatments of core electrons (ultra-soft pseudopotentials in the work reported here versus projected augmented plane-waves in the previous work). The fully relaxed, zero-pressure lattice parameters and atomic coordinates for the P21/c, Cc and P63mc structures are shown in Tables 3–6. Like that for the Pnma phase (Table 1), the relaxed units cells determined in this work for the P21/c, Cc and P63mc phases were consistently larger than the corresponding unit cells determined by Łodziana and Vegge. In general, the relaxed structures calculated in this work with the PW91 functional agree very well with the PW91-based results of Łodziana and Vegge, with fractional atomic coordinates differing by more than 0.001 in only a few cases Table 3 Lattice parameters for the P21/c, Cc and P63mc phases of LiBH4 ˚ and °) (A Phase
a
b
c
b
P21/c
PW91a PW91b
7.363 7.267
7.275 7.174
7.822 7.683
148.53 148.52
Cc
PW91a PW91b
4.202 4.131
7.396 7.270
6.719 6.601
97.29 97.26
P63mc
PW91a PW91b Expt.c
4.306 4.164 4.276
a b c
DFT, this work. DFT, Łodziana and Vegge [9]. Experiment, Soulie´ et al. [8].
6.995 6.766 6.948
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T.J. Frankcombe et al. / Chemical Physics Letters 405 (2005) 73–78
Table 4 Relative internal coordinates of atoms in the P21/c phase of LiBH4 Atom
Site
Table 6 Relative internal coordinates of atoms in the P63mc phase of LiBH4
Atomic coordinates a
Li
4e
PW91: (0.8659,0.8498,0.0433) LDA: (0.8628,0.8418,0.0381)a PW91: (0.8673,0.8486,0.0439)b
B
4e
PW91: (0.1666,0.6949,0.5028)a LDA: (0.1572,0.6859,0.4900)a PW91: (0.1664,0.6946,0.5026)b
4e
PW91: (0.1017,0.6004,0.3211)a LDA: (0.0922,0.5919,0.3103)a PW91: (0.1013,0.5998,0.3204)b
H
4e
a
PW91: (0.8913,0.7674,0.3305) LDA: (0.8835,0.7575,0.3198)a PW91: (0.8902,0.7674,0.3295)b
H
4e
PW91: (0.3011,0.6008,0.7253)a LDA: (0.2938,0.5933,0.7125)a PW91: (0.3013,0.6003,0.7258)b
H
4e
PW91: (0.3615,0.8217,0.6164)a LDA: (0.3509,0.8118,0.6011)a PW91: (0.3615,0.8221,0.6160)b
H
a b
DFT, this work. DFT, Łodziana and Vegge [9].
Table 5 Relative internal coordinates of atoms in the Cc phase of LiBH4 Atom
Site
Atomic coordinates
Li
4a
PW91: (0.9538,0.6405,0.1560)a LDA: (0.9463,0.6323,0.1489)a PW91: (0.9541,0.6418,0.1562)b
B
4a
PW91: (0.5127,0.1765,0.5349)a LDA: (0.5023,0.1710,0.5257)a PW91: (0.5125,0.1762,0.5349)b
H
4a
PW91: (0.2080,0.7124,0.4155)a LDA: (0.2023,0.7017,0.4075)a PW91: (0.2088,0.7126,0.4157)b
H
4a
PW91: (0.6669,0.1048,0.6850)a LDA: (0.6539,0.0992,0.6743)a PW91: (0.6667,0.1046,0.6850)b
H
4a
PW91: (0.4003,0.3219,0.5864)a LDA: (0.3920,0.3155,0.5768)a PW91: (0.4000,0.3217,0.5866)b
H
4a
PW91: (0.2975,0.0758,0.4547)a LDA: (0.2873,0.0724,0.4460)a PW91: (0.2970,0.0755,0.4541)b
a b
DFT, this work. DFT, Łodziana and Vegge [9].
for the P21/c and Cc phases. Differences of this magnitude are easily accounted for as arising from differences in optimisation method and convergence criteria. For the P63mc phase the current and previous results did not agree quite as precisely. The agreement was still very good with differences of the order of 0.01 in fractional coordinates. For all considered LiBH4 phases the geometry optimised with the LDA functional differed from
Atom
Site
Atomic coordinates
Li
2b
PW91: (1/3,2/3,0.0040)a LDA: (1/3,2/3,0.0084)a PW91: (1/3,2/3,0.0906)b X-ray: (1/3,2/3,0)c
B
2b
PW91: (1/3,2/3,0.5570)a LDA: (1/3,2/3,0.5525)a PW91: (1/3,2/3,0.5402)b X-ray: (1/3,2/3,0.553)c
H
2b
PW91: (1/3,2/3,0.3809)a LDA: (1/3,2/3,0.3767)a PW91: (1/3,2/3,0.3616)b X-ray: (1/3,2/3,0.370)c
H
6c
PW91: (0.1783,0.3567,0.6178)a LDA: (0.1758,0.3517,0.6131)a PW91: (0.1742,0.3484,0.6009)b X-ray: (0.172,0.344,0.624)c
a b c
DFT, this work. DFT, Łodziana and Vegge [9]. Experiment, Soulie´ et al. [8].
that optimised with the PW91 functional by 0.01– 0.001 in fractional coordinates. Miwa et al. [10] also performed calculations on the hexagonal P63mc phase of LiBH4, finding that when the geometry was optimised using the PBE exchangecorrelation functional the minimum-energy configuration did not reproduce the experimental structure of the high-temperature phase of Soulie´ et al. [8]. In the present work, the P63mc structure optimised using the PW91 exchange-correlation functional [18] agrees well with the experimental structure, overestimating the lattice constants by 0.7% and giving fractional positions of all atoms to within 1%, except for the 6c hydrogen which is significantly shifted. The B–H distance is reduced from the experimental geometry by around 6%. Our calculated structure is very similar to that reported by Łodziana and Vegge [9], who also performed calculations on the P63mc phase with the PW91 functional. Our calculations on LiBH4 in the same geometry as the ground state of NaAlH4 (space group I41/a) reveal that this structure is significantly less stable than the Pnma ground state, resulting in endothermic formation with a reaction energy of 27 kJ/mol H2. This is in qualitative agreement with the calculations of Łodziana and Vegge [9]. A previous study [31] calculated a formation reaction energy of LiBH4 in the NaAlH4 structure of around 60 kJ/mol H2, in good agreement with the experimental value. This is apparently at odds with the present calculations and those of Łodziana and Vegge, though it is not clear whether the difference is due to the calculation of the minimum potential energy of the LiBH4 phase or one of the other phases that go into the reaction energy calculation.
T.J. Frankcombe et al. / Chemical Physics Letters 405 (2005) 73–78
Fig. 1. Enthalpy as a function of external pressure for the P21/c and Cc structures of LiBH4, relative to the low-temperature and pressure ground state Pnma structure.
Pressure-induced phase transitions can be predicted by calculating the free energy of the competing phases as a function of applied pressure. At zero temperature, the free energy is just the enthalpy, H = U + pV. Scaling the relaxed unit cell for each of the four considered phases (without allowing the cell shape to change), relaxing the atomic positions within that cell and calculating the stress on the cell predicted that the P21/c phase becomes more stable than the Pnma phase at pressures greater than 1.0 GPa (10 kbar), as indicated in Fig. 1. The Cc phase, predicted by Łodziana and Vegge to take over from the Pnma phase as the most stable low-temperature structure at pressures greater than 3.0 GPa, becomes more stable than the Pnma phase above 1.8 GPa and then the most stable beyond 2.2 GPa of applied pressure. Not shown in Fig. 1 is the relative enthalpy of the P63mc phase which starts 26 kJ/mol higher in energy than the Pnma phase at zero pressure and, having a larger molar volume than the Pnma phase, the enthalpy difference increases with increasing pressure. Note that the enthalpy difference, shown in Fig. 1, is DH = DE + pDV, where DE is the difference in the calculated internal energy without ZPE corrections. Excluding ZPE is valid, provided the ZPE does not differ significantly between the phases. That this is the case for LiBH4 is supported by similarities in the high frequency components of the phonon spectra calculated by Łodziana and Vegge [9].
4. Discussion The calculations presented here are the second theoretical determination of the reaction energy of the formation of LiBH4 from its decomposition products. The value calculated using the PW91 exchange-correla-
77
tion functional of 71.3 kJ/mol H2, without ZPE corrections, is closer to the experimental value for the enthalpy of formation than the value of 75 kJ/mol H2 determined by Miwa et al. [10] using the PBE exchange-correlation functional and than the value of 76.1 kJ/mol H2 calculated using the LDA exchangecorrelation functional. However, including ZPE differences would significantly reduce the level of agreement if the PW91 ZPE differences are similar to those calculated at the PBE level, as seems likely. The same correction applied to the LDA result yields an estimate for the enthalpy slightly closer to the experimental value than the GGA results. This is not completely unexpected as experience has shown that the bond energy of H2 is calculated more accurately with the LDA than with the PW91 functional [32,33]. None of the available calculations reproduce the distorted BH4 tetrahedra observed experimentally [8,7]. Distorted BH4 tetrahedra may result in softer phonon modes, reducing the ZPE for the LiBH4 solid. Were this to be the case, it would indicate that the calculated ZPE difference was overestimated and help to explain the difference between the calculated and observed formation enthalpies. This and related work raises interesting questions about the P63mc phase. It is clearly a local minimum on the LiBH4 potential energy surface (PES) calculated with the PW91 exchange-correlation functional. It appears not to be a local minimum on the PBE PES, with the description of the interactions between the Li + and (BH4) units yielding a completely different P63mc configuration [10]. PW91 calculations suggest that this phase is not stable at non-zero temperatures as the phonon spectrum exhibits modes with imaginary frequencies [9]. At the same time, the atomic positions at the minimum on the PW91 PES agree well with the P63mc fit to the X-ray powder diffraction pattern for the high-temperature LiBH4 phase [8]. While this inconsistency could well be simple coincidence, it seems further experimental characterisation of the high-temperature phase of LiBH4 is needed. If the stable high-temperature phase of LiBH4 turns out to be the P63mc structure proposed by Soulie´ et al. [8] then further computational exploration of the system with the PW91 functional is required to determine the origin of the reported unphysical phonon modes and the destabilisation of the P63mc phase relative to the P21/c and Cc phases. It is worth pointing out that there remains significant uncertainty surrounding the proposal of the Cc phase as the high temperature structure, with the predicted transition temperature of 520 K significantly higher than the experimentally observed transition temperature of 384 K [9]. There are a clear differences between the low-temperature relative stabilities upon compression reported here and those reported by Łodziana and Vegge [9]. We found the Pnma structure to be the most stable of all
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T.J. Frankcombe et al. / Chemical Physics Letters 405 (2005) 73–78
considered phases for pressures below 1.0 GPa, above which the P21/c structure becomes more stable. Conversely, Łodziana and Vegge predicted that the Cc structure becomes the most stable at around 3.0 GPa of applied pressure, with the P21/c phase remaining significantly less stable than the Pnma phase for all pressures presented by them (up to 4.0 GPa). The present calculations indicate that the Cc structure does not become more stable than the Pnma structure until pressures greater than 1.8 GPa are applied and not more stable than the P21/c structure below a pressure of 2.2 GPa. In neither calculation are ZPE effects taken into account. Whereas no shape relaxation on compression was allowed in our calculations, Łodziana and Vegge do mention cell shape changes. While these are interesting results and comparisons from a computational point of view and suggest that further calculations or experimental determination of the structure of LiBH4 at high pressures and low temperatures could yield some insight, it is unlikely that a system operating at these pressures would yield a practical hydrogen storage system. Like that suggested recently for NaAlH4 [11], any such pressure-induced modification to LiBH4 would need to be somehow stabilised at ambient conditions. The experimental enthalpy of 68.9 kJ/mol H2, supported by these calculations, suggests that the 75% hydrogen release described by the reaction in Eq. (1) is unlikely to be favourable for hydrogen storage applications, requiring high-temperature conditions for hydrogen release. Lower available hydrogen densities released through catalysed partial dehydrogenation [7] remain an attractive option, though this significantly reduces the effective gravimetric density of available hydrogen for LiBH4-based hydrogen storage systems.
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Acknowledgements We are grateful to Zbigniew Łodziana and Tejs Vegge for making their results available before publication. This work was supported financially by a CW/ NWO programme grant.
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Theoretical calculation of the energy of formation of LiBH4 Terry J. Frankcombe a
a,*
, Geert-Jan Kroes a, Andreas Zu¨ttel
b
Leiden Institute of Chemistry, Gorlaeus Laboratories, Leiden University, P.O. Box 9502, NL-2300 RA Leiden, The Netherlands b Physics Institute, University of Fribourg, Perolles 1700, Switzerland Received 29 September 2004; in final form 4 February 2005
Abstract We report density functional theory calculations on the energy of LiBH4, relative to solid B and LiH and gaseous H2. The calculated energy is 71.3 (76.1) kJ/mol H2 which can be approximately corrected for zero-point energy to give an enthalpy of 52 (57) kJ/mol H2 at the PW91 (LDA) level, smaller than but similar to the experimental value of 68.9 kJ/mol H2. Calculations on four different LiBH4 phases indicate that alternative phases are not accessible at low temperatures without applying high pressures. These results indicate that complete decomposition to H2, B and LiH is not an attractive means of obtaining a reversible hydrogen storage system based on LiBH4. Ó 2005 Elsevier B.V. All rights reserved.
1. Introduction Using hydrogen as an energy carrier is an attractive alternative to the current global dependence on hydrocarbon-based fuels, particularly for vehicular transport. The energy density of molecular hydrogen is very high and this energy can be liberated readily and rapidly using current internal combustion technology with nothing more polluting than water vapour as a by-product [1]. More efficient fuel-cell technology is on the horizon [2], increasing the attractiveness of hydrogen. However, effective storage of hydrogen presents a significant challenge that must be overcome for the hydrogen economy to replace current energy technologies [3,4]. Storing pure hydrogen on board passenger cars is unattractive from an engineering point of view, primarily due to the low density of hydrogen gas and the effort required to compress it to a reasonable volume. Adsorp-
*
Corresponding author. Fax: +31 71 527 43 97. E-mail addresses: [email protected] (T.J. Frankcombe), [email protected] (G.-J. Kroes), andreas.zuettel@ unifr.ch (A. Zu¨ttel). 0009-2614/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2005.02.017
tion into a host medium presents many advantages, potentially including reasonable operating temperatures and pressures and being loss-less. While nano-structured carbon is being heavily investigated as an adsorbent, complex metal hydride systems are proving to have many advantages, such as near-constant temperature operation and high hydrogen density [3,5,6]. An important criterion for such metal hydrides to be useful is that the enthalpy change for the release of hydrogen lies in the range 30–48 kJ/mol H2 [3]. It is to be expected that the highest gravimetric density of hydrogen can be obtained in hydrides of low-mass metals, making the complex hydride lithium borohydride (LiBH4) attractive with a total hydrogen content of 18.5% by mass. In this Letter, we describe our use of computational methods based on density functional theory to calculate the reaction energy for the formation of LiBH4 from its decomposition products. The release of 75% of the available hydrogen is achieved by the decomposition LiBH4 ! LiH þ B þ 32H2
ð1Þ
Decomposition of LiH to release the last 25% of the LiBH4 hydrogen content was not considered as the
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T.J. Frankcombe et al. / Chemical Physics Letters 405 (2005) 73–78
temperatures required for this final decomposition are too high for on board hydrogen storage applications. LiBH4 is known to undergo a structural transition on heating. The structure of the low-temperature phase is well established [7,8], exhibiting Pnma symmetry. While a structure of P63mc symmetry has been proposed for the high-temperature phase based on X-ray diffraction data [8], recent calculations have called this structure into question [9,10]. We investigated several of the proposed structures for the high-temperature phase of LiBH4 to determine their relative stability. The energy of all phases as a function of the cell size was investigated to determine whether low-temperature, pressureinduced phase changes present more candidates for hydrogen storage systems, as recently proposed for NaAlH4 [11].
2. Method To calculate the reaction energy of the formation of LiBH4 from the basic decomposition products [the reverse of Eq. (1)] the total potential energy for solid LiH, boron and LiBH4 and for gaseous H2 are required. The energies were calculated using density functional theory [12,13]. The Dacapo package [14,15] was used for all calculations, using Vanderbilt ultra-soft pseudopotentials [16]. The Perdew–Zunger local density approximation (LDA) [17] and PW91 generalised gradient approximation (GGA) [18] exchange-correlation functionals were used. All calculations on solid phases used Monkhorst– Pack k-point sampling [19] and a plane wave basis set, with the k-point grid and plane wave cutoff energy varied to achieve convergence of the total energy to within 50 meV for each cell. This convergence criterion for the total energy translates to 0.8 kJ/mol H2 for LiBH4, where the energy unit is per mole of hydrogen released via the reaction in Eq. (1), or 1.2 kJ/mol per LiBH4 formula unit. The error in calculated energy differences, such as the reaction energy for the formation of LiBH4, would be expected to be smaller than this total energy convergence level due to cancellation of errors. Atomic positions and unit cell parameters were relaxed according to the Hellmann–Feynman forces to achieve the minimum total potential energy. For the calculation on H2, the molecule was isolated in a repeating cube
˚ and only the H–H bond length alwith edge length 10 A lowed to relax.
3. Results The LiH and H2 calculations produced structures in reasonable agreement with experiment. Using the PW91 exchange-correlation functional resulted in the LiH lattice constant [20] and the H2 bond length [21] being overestimated by 1.1% and 1.7%, respectively. Using the LDA functional also lead to an overestimation of these parameters, by 1.3% and 3.5%, respectively. The structure of solid boron is significantly more complicated. Three structures were investigated with the PW91 functional, being the tetrahedral structure with 50 atoms in the unit cell [22] and the a- and b-rhombohedral phases with 12 [23] and 105 atoms [24] in the unit cell, respectively. No attempt was made to model the defective 106.7 atom structure of Slack et al. [25] due to the excessively large super cell that would be required to model the partially occupied sites. The a-rhombohedral boron structure was found to be the most stable structure, with the b-rhombohedral and tetrahedral structures 1.8 kJ/mol per atom and 9.3 kJ/mol per atom higher in energy, respectively. Based on these results, only the energy of a-rhombohedral boron was calculated with the LDA functional. The structure of the low-temperature Pnma phase of LiBH4 has been determined experimentally [7,8] and calculated with DFT methods [9,10]. Our PW91 and LDA relaxed structures were similar to the previously reported structures, as shown in Tables 1 and 2. The lattice constants calculated in this work were slightly closer to the experimental values than those of the previously reported calculations. In this regard the c lattice constant stands out, with the PW91 and LDA calculations converging to the same value 1% larger than the experimental values while the previous calculations underestimated this value by 3%. The relative atomic coordinates within the unit cell varied from the experimental values by similar amounts to the previously reported calculations, with the atoms displaced from the experimentally determined positions by distances of around 0.01–0.03 units in coordinate space. Using the total energies calculated for LiH, a-rhombohedral boron, H2 and LiBH4 the reaction energy for
Table 1 ˚) Lattice constants for the Pnma phase of LiBH4 (A Parameter
a b c
Present work
DFT [9]
DFT [10]
PW91
LDA
PW91
PBE
7.214 4.468 6.877
7.149 4.419 6.877
7.226 4.355 6.576
7.343 4.399 6.588
Expt. [7]
Expt. [8]
7.173 4.434 6.798
7.179 4.437 6.803
T.J. Frankcombe et al. / Chemical Physics Letters 405 (2005) 73–78 Table 2 Relative internal coordinates of atoms in the Pnma phase of LiBH4 Atom
Site
Atomic coordinates
Li
4c
PW91: (0.1572,1/4,0.1000)a LDA: (0.1576,1/4,0.0988)a PW91: (0.1560,1/4,0.1123)b PBE: (0.1552,1/4,0.1137)c X-ray: (0.1568,1/4,0.1015)d
B
4c
PW91: (0.2957,1/4,0.4209)a LDA: (0.2967,1/4,0.4200)a PW91: (0.3100,1/4,0.4232)b PBE: (0.3141,1/4,0.4229)c X-ray: (0.3040,1/4,0.4305)d
H
4c
PW91: (0.8944,1/4,0.9268)a LDA: (0.8963,1/4,0.9273)a PW91: (0.9089,1/4,0.9251)b PBE: (0.9131,1/4,0.9263)c X-ray: (0.900,1/4,0.956)d
H
4c
PW91: (0.3882,1/4,0.2705)a LDA: (0.4001,1/4,0.2688)a PW91: (0.4046,1/4,0.2673)b PBE: (0.4061,1/4,0.2656)c X-ray: (0.404,1/4,0.280)d
H
8d
PW91: (0.1949,0.0278,0.4235)a LDA: (0.1945,0.0268,0.4226)a PW91: (0.2086,0.0242,0.4221)b PBE: (0.2145,0.0246,0.4224)c X-ray: (0.172,0.054,0.428)d
a b c d
DFT, this work. DFT, Łodziana and Vegge [9]. DFT, Miwa et al. [10]. Experiment, Soulie´ et al. [8].
the formation of LiBH4 was calculated with the PW91 functional to be 71.3 kJ/mol H2 and with the LDA functional to be 76.1 kJ/mol H2. For comparison, the experimental value of the enthalpy is 68.9 kJ/mol H2 [26] and Miwa et al. [10] calculated 75 kJ/mol H2 for the reaction energy (using ultra-soft pseudopotentials and the PBE exchange-correlation functional [27,28]), or 56 kJ/mol H2 for the enthalpy including zero-point energy (ZPE) corrections. If we assume that the ZPE difference calculated by Miwa et al. using the PBE functional is reasonably transferable to our PW91 calculations, an estimate of around 52 kJ/mol H2 is obtained for the ZPE-corrected enthalpy of formation from these calculations. The transferability of the ZPE difference is supported in general by the observation that calculated vibrational frequencies are insensitive to the exchange-correlation functional [29], and specifically in this case by the similarity of the PW91 ZPE for LiBH4 of 108 kJ/mol calculated by Łodziana and Vegge [9] to the PBE value of 103 kJ/mol of Miwa et al. The corresponding estimate from the LDA functional, again using the PBE ZPE correction of Miwa et al., is around 57 kJ/mol H2, slightly closer to the experimental value. The experimentally observed structure of the lowtemperature phase of LiBH4 exhibits a highly distorted
75
BH4 tetrahedron, with B–H bond lengths varying by up to 23% [8,7]. Like the previously reported calculations [9,10], the BH4 unit in the relaxed Pnma structures reported here did not exhibit this variation yielding B–H ˚ . The H–B–H bond lengths in the range 1.224–1.231 A angles within the BH4 tetrahedron varied from 106.9° to 110.5°. Łodziana and Vegge [9] determined several stable structures as candidates for the high-temperature phase observed above 384 K, using the projected augmented plane-wave approach with the VASP code [30]. Two new candidates were identified of space group P21/c and Cc, in addition to the previously suggested P63mc structure [8]. We have performed structural relaxation on these phases using the PW91 functional at zero pressure, confirming the zero-pressure stability ordering found by Łodziana and Vegge. The P21/c, Cc and P63mc phases were found to be 2.1, 7.2 and 26 kJ/mol higher in energy per formula unit than the ground-state Pnma phase without ZPE corrections. The equivalent energies reported by Łodziana and Vegge are 2.2, 3.3 and 23 kJ/mol, respectively. Clearly the energy differences between the Pnma and the Cc and P63mc phases were larger in our calculation, but this has no impact on the stability ordering. Differences of this magnitude (<10 kJ/mol) are to be expected when comparing results from calculations with differing treatments of core electrons (ultra-soft pseudopotentials in the work reported here versus projected augmented plane-waves in the previous work). The fully relaxed, zero-pressure lattice parameters and atomic coordinates for the P21/c, Cc and P63mc structures are shown in Tables 3–6. Like that for the Pnma phase (Table 1), the relaxed units cells determined in this work for the P21/c, Cc and P63mc phases were consistently larger than the corresponding unit cells determined by Łodziana and Vegge. In general, the relaxed structures calculated in this work with the PW91 functional agree very well with the PW91-based results of Łodziana and Vegge, with fractional atomic coordinates differing by more than 0.001 in only a few cases Table 3 Lattice parameters for the P21/c, Cc and P63mc phases of LiBH4 ˚ and °) (A Phase
a
b
c
b
P21/c
PW91a PW91b
7.363 7.267
7.275 7.174
7.822 7.683
148.53 148.52
Cc
PW91a PW91b
4.202 4.131
7.396 7.270
6.719 6.601
97.29 97.26
P63mc
PW91a PW91b Expt.c
4.306 4.164 4.276
a b c
DFT, this work. DFT, Łodziana and Vegge [9]. Experiment, Soulie´ et al. [8].
6.995 6.766 6.948
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T.J. Frankcombe et al. / Chemical Physics Letters 405 (2005) 73–78
Table 4 Relative internal coordinates of atoms in the P21/c phase of LiBH4 Atom
Site
Table 6 Relative internal coordinates of atoms in the P63mc phase of LiBH4
Atomic coordinates a
Li
4e
PW91: (0.8659,0.8498,0.0433) LDA: (0.8628,0.8418,0.0381)a PW91: (0.8673,0.8486,0.0439)b
B
4e
PW91: (0.1666,0.6949,0.5028)a LDA: (0.1572,0.6859,0.4900)a PW91: (0.1664,0.6946,0.5026)b
4e
PW91: (0.1017,0.6004,0.3211)a LDA: (0.0922,0.5919,0.3103)a PW91: (0.1013,0.5998,0.3204)b
H
4e
a
PW91: (0.8913,0.7674,0.3305) LDA: (0.8835,0.7575,0.3198)a PW91: (0.8902,0.7674,0.3295)b
H
4e
PW91: (0.3011,0.6008,0.7253)a LDA: (0.2938,0.5933,0.7125)a PW91: (0.3013,0.6003,0.7258)b
H
4e
PW91: (0.3615,0.8217,0.6164)a LDA: (0.3509,0.8118,0.6011)a PW91: (0.3615,0.8221,0.6160)b
H
a b
DFT, this work. DFT, Łodziana and Vegge [9].
Table 5 Relative internal coordinates of atoms in the Cc phase of LiBH4 Atom
Site
Atomic coordinates
Li
4a
PW91: (0.9538,0.6405,0.1560)a LDA: (0.9463,0.6323,0.1489)a PW91: (0.9541,0.6418,0.1562)b
B
4a
PW91: (0.5127,0.1765,0.5349)a LDA: (0.5023,0.1710,0.5257)a PW91: (0.5125,0.1762,0.5349)b
H
4a
PW91: (0.2080,0.7124,0.4155)a LDA: (0.2023,0.7017,0.4075)a PW91: (0.2088,0.7126,0.4157)b
H
4a
PW91: (0.6669,0.1048,0.6850)a LDA: (0.6539,0.0992,0.6743)a PW91: (0.6667,0.1046,0.6850)b
H
4a
PW91: (0.4003,0.3219,0.5864)a LDA: (0.3920,0.3155,0.5768)a PW91: (0.4000,0.3217,0.5866)b
H
4a
PW91: (0.2975,0.0758,0.4547)a LDA: (0.2873,0.0724,0.4460)a PW91: (0.2970,0.0755,0.4541)b
a b
DFT, this work. DFT, Łodziana and Vegge [9].
for the P21/c and Cc phases. Differences of this magnitude are easily accounted for as arising from differences in optimisation method and convergence criteria. For the P63mc phase the current and previous results did not agree quite as precisely. The agreement was still very good with differences of the order of 0.01 in fractional coordinates. For all considered LiBH4 phases the geometry optimised with the LDA functional differed from
Atom
Site
Atomic coordinates
Li
2b
PW91: (1/3,2/3,0.0040)a LDA: (1/3,2/3,0.0084)a PW91: (1/3,2/3,0.0906)b X-ray: (1/3,2/3,0)c
B
2b
PW91: (1/3,2/3,0.5570)a LDA: (1/3,2/3,0.5525)a PW91: (1/3,2/3,0.5402)b X-ray: (1/3,2/3,0.553)c
H
2b
PW91: (1/3,2/3,0.3809)a LDA: (1/3,2/3,0.3767)a PW91: (1/3,2/3,0.3616)b X-ray: (1/3,2/3,0.370)c
H
6c
PW91: (0.1783,0.3567,0.6178)a LDA: (0.1758,0.3517,0.6131)a PW91: (0.1742,0.3484,0.6009)b X-ray: (0.172,0.344,0.624)c
a b c
DFT, this work. DFT, Łodziana and Vegge [9]. Experiment, Soulie´ et al. [8].
that optimised with the PW91 functional by 0.01– 0.001 in fractional coordinates. Miwa et al. [10] also performed calculations on the hexagonal P63mc phase of LiBH4, finding that when the geometry was optimised using the PBE exchangecorrelation functional the minimum-energy configuration did not reproduce the experimental structure of the high-temperature phase of Soulie´ et al. [8]. In the present work, the P63mc structure optimised using the PW91 exchange-correlation functional [18] agrees well with the experimental structure, overestimating the lattice constants by 0.7% and giving fractional positions of all atoms to within 1%, except for the 6c hydrogen which is significantly shifted. The B–H distance is reduced from the experimental geometry by around 6%. Our calculated structure is very similar to that reported by Łodziana and Vegge [9], who also performed calculations on the P63mc phase with the PW91 functional. Our calculations on LiBH4 in the same geometry as the ground state of NaAlH4 (space group I41/a) reveal that this structure is significantly less stable than the Pnma ground state, resulting in endothermic formation with a reaction energy of 27 kJ/mol H2. This is in qualitative agreement with the calculations of Łodziana and Vegge [9]. A previous study [31] calculated a formation reaction energy of LiBH4 in the NaAlH4 structure of around 60 kJ/mol H2, in good agreement with the experimental value. This is apparently at odds with the present calculations and those of Łodziana and Vegge, though it is not clear whether the difference is due to the calculation of the minimum potential energy of the LiBH4 phase or one of the other phases that go into the reaction energy calculation.
T.J. Frankcombe et al. / Chemical Physics Letters 405 (2005) 73–78
Fig. 1. Enthalpy as a function of external pressure for the P21/c and Cc structures of LiBH4, relative to the low-temperature and pressure ground state Pnma structure.
Pressure-induced phase transitions can be predicted by calculating the free energy of the competing phases as a function of applied pressure. At zero temperature, the free energy is just the enthalpy, H = U + pV. Scaling the relaxed unit cell for each of the four considered phases (without allowing the cell shape to change), relaxing the atomic positions within that cell and calculating the stress on the cell predicted that the P21/c phase becomes more stable than the Pnma phase at pressures greater than 1.0 GPa (10 kbar), as indicated in Fig. 1. The Cc phase, predicted by Łodziana and Vegge to take over from the Pnma phase as the most stable low-temperature structure at pressures greater than 3.0 GPa, becomes more stable than the Pnma phase above 1.8 GPa and then the most stable beyond 2.2 GPa of applied pressure. Not shown in Fig. 1 is the relative enthalpy of the P63mc phase which starts 26 kJ/mol higher in energy than the Pnma phase at zero pressure and, having a larger molar volume than the Pnma phase, the enthalpy difference increases with increasing pressure. Note that the enthalpy difference, shown in Fig. 1, is DH = DE + pDV, where DE is the difference in the calculated internal energy without ZPE corrections. Excluding ZPE is valid, provided the ZPE does not differ significantly between the phases. That this is the case for LiBH4 is supported by similarities in the high frequency components of the phonon spectra calculated by Łodziana and Vegge [9].
4. Discussion The calculations presented here are the second theoretical determination of the reaction energy of the formation of LiBH4 from its decomposition products. The value calculated using the PW91 exchange-correla-
77
tion functional of 71.3 kJ/mol H2, without ZPE corrections, is closer to the experimental value for the enthalpy of formation than the value of 75 kJ/mol H2 determined by Miwa et al. [10] using the PBE exchange-correlation functional and than the value of 76.1 kJ/mol H2 calculated using the LDA exchangecorrelation functional. However, including ZPE differences would significantly reduce the level of agreement if the PW91 ZPE differences are similar to those calculated at the PBE level, as seems likely. The same correction applied to the LDA result yields an estimate for the enthalpy slightly closer to the experimental value than the GGA results. This is not completely unexpected as experience has shown that the bond energy of H2 is calculated more accurately with the LDA than with the PW91 functional [32,33]. None of the available calculations reproduce the distorted BH4 tetrahedra observed experimentally [8,7]. Distorted BH4 tetrahedra may result in softer phonon modes, reducing the ZPE for the LiBH4 solid. Were this to be the case, it would indicate that the calculated ZPE difference was overestimated and help to explain the difference between the calculated and observed formation enthalpies. This and related work raises interesting questions about the P63mc phase. It is clearly a local minimum on the LiBH4 potential energy surface (PES) calculated with the PW91 exchange-correlation functional. It appears not to be a local minimum on the PBE PES, with the description of the interactions between the Li + and (BH4) units yielding a completely different P63mc configuration [10]. PW91 calculations suggest that this phase is not stable at non-zero temperatures as the phonon spectrum exhibits modes with imaginary frequencies [9]. At the same time, the atomic positions at the minimum on the PW91 PES agree well with the P63mc fit to the X-ray powder diffraction pattern for the high-temperature LiBH4 phase [8]. While this inconsistency could well be simple coincidence, it seems further experimental characterisation of the high-temperature phase of LiBH4 is needed. If the stable high-temperature phase of LiBH4 turns out to be the P63mc structure proposed by Soulie´ et al. [8] then further computational exploration of the system with the PW91 functional is required to determine the origin of the reported unphysical phonon modes and the destabilisation of the P63mc phase relative to the P21/c and Cc phases. It is worth pointing out that there remains significant uncertainty surrounding the proposal of the Cc phase as the high temperature structure, with the predicted transition temperature of 520 K significantly higher than the experimentally observed transition temperature of 384 K [9]. There are a clear differences between the low-temperature relative stabilities upon compression reported here and those reported by Łodziana and Vegge [9]. We found the Pnma structure to be the most stable of all
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T.J. Frankcombe et al. / Chemical Physics Letters 405 (2005) 73–78
considered phases for pressures below 1.0 GPa, above which the P21/c structure becomes more stable. Conversely, Łodziana and Vegge predicted that the Cc structure becomes the most stable at around 3.0 GPa of applied pressure, with the P21/c phase remaining significantly less stable than the Pnma phase for all pressures presented by them (up to 4.0 GPa). The present calculations indicate that the Cc structure does not become more stable than the Pnma structure until pressures greater than 1.8 GPa are applied and not more stable than the P21/c structure below a pressure of 2.2 GPa. In neither calculation are ZPE effects taken into account. Whereas no shape relaxation on compression was allowed in our calculations, Łodziana and Vegge do mention cell shape changes. While these are interesting results and comparisons from a computational point of view and suggest that further calculations or experimental determination of the structure of LiBH4 at high pressures and low temperatures could yield some insight, it is unlikely that a system operating at these pressures would yield a practical hydrogen storage system. Like that suggested recently for NaAlH4 [11], any such pressure-induced modification to LiBH4 would need to be somehow stabilised at ambient conditions. The experimental enthalpy of 68.9 kJ/mol H2, supported by these calculations, suggests that the 75% hydrogen release described by the reaction in Eq. (1) is unlikely to be favourable for hydrogen storage applications, requiring high-temperature conditions for hydrogen release. Lower available hydrogen densities released through catalysed partial dehydrogenation [7] remain an attractive option, though this significantly reduces the effective gravimetric density of available hydrogen for LiBH4-based hydrogen storage systems.
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Acknowledgements We are grateful to Zbigniew Łodziana and Tejs Vegge for making their results available before publication. This work was supported financially by a CW/ NWO programme grant.
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