Density Functional Theory Calculations of Magnesium Hydride: A

Article pubs.acs.org/JPCC

Density Functional Theory Calculations of Magnesium Hydride: A Comparison of Bulk and Nanoparticle Thermodynamics A. C. Buckley,† D. J. Carter,‡ D. A. Sheppard,† and C. E. Buckley*,† †

Department of Imaging and Applied Physics, Fuels and Energy Technology Institute, Curtin University, GPO Box U1987, Perth WA 6845, Australia ‡ Nanochemistry Research Institute, Curtin University, GPO Box U1987, Perth WA 6845, Australia ABSTRACT: Density functional theory calculations have been performed on a range of magnesium and magnesium hydride nanoclusters to examine the 1 bar desorption temperatures. The vibrational entropies and enthalpies are calculated for each cluster, within the harmonic approximation, which permits calculation of the desorption Gibbs free energy of reaction. For the bulk system, good agreement is found with experiment for the desorption temperature and a range of structural and electronic properties. For the nanoparticulate systems, we report binding energies, along with desorption reaction entropies and enthalpies. The finite-temperature effects on the vibrational energies of all system sizes are examined, and the findings suggest that the harmonic approximation is too restricted to account for the experimentally observed reductions in the nanoparticulate reaction enthalpies.

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temperatures as low as 358 K. However, recent studies5,7 have suggested that the large destabilization could also be attributed to the interaction of the surfactant with the surface of the nanoparticles and are not purely a result of nanoparticle size. Paskevicius et al.2 reported a much smaller destabilization for MgH2 nanoparticles, with a decrease in the 1 bar desorption temperature of approximately 6 K for mechanochemically synthesized nanoparticles (with a characteristic particle diameter of 7 nm) embedded within a salt matrix. The reduction in reaction enthalpy (δΔrH), relative to the bulk, that they observed is in agreement with the predictions of Kim et al.,4 for nanoparticles with a diameter of 4 nm. Paskevicius et al.2 note that the volume-weighted average, used to calculate the characteristic size, overestimates the nanoparticle size, which in turn corresponds to the measured desorption temperature. This helps justify their direct comparison to the theoretical results of Kim et al.4 Despite the good agreement between predicted and observed values for δΔrH, there is disagreement over the change in desorption temperature (ΔTd). Paskevicius et al.2 found that bulk-sized particles/crystallites embedded within the salt matrix reproduced the bulk desorption temperature, indicating that the salt matrix is unlikely to have significantly altered the chemical environment of the nanoparticle. Their experimental result is in disagreement with the theoretical predictions of Kim et al.4 who report a ΔTd of −15 K for nanoparticles with a 7 nm diameter. Paskevicius et al. suggest that the discrepancies are because the theoretical calculations assume that the nanoparticulate desorption reaction entropy change is equivalent to that in the bulk (ΔrSnano = ΔrSbulk). To date, no theoretical study in the

INTRODUCTION Magnesium hydride (MgH2) is an attractive hydrogen storage material due to its high gravimetric hydrogen storage capacity of 7.7 wt %.1 The use of bulk MgH2 as a commercial hydrogen storage material is, however, restricted by the undesirably high 1 bar desorption temperature (Td) of approximately 553 K (280 °C).2 In the interest of improving the desorption reaction thermodynamics, recent experimental2,4,6 and theoretical studies3,5 have examined the thermodynamic stability of nanoparticulate MgH2 systems. The theoretical focus has been primarily on the changes in the 0 K desorption energies (ΔEd), which are used to predict desorption temperatures for the nanoparticulate systems. These quantities are defined with respect to their bulk values. ΔEd = Ed(nano, T = 0 K) − Ed(bulk, T = 0 K)

(1)

ΔTd = Td(nano) − Td(bulk)

(2)

Though experiments have shown that lower desorption temperatures can be achieved for MgH2 nanoparticles,2 density functional theory (DFT) calculations may be restricted in their ability to predict reaction thermodynamics over a range of temperatures and particle sizes. Reactions that become spontaneous at high temperatures are difficult to model since DFT is strictly appropriate only at 0 K, and hence, one must extrapolate 0 K material properties to finite temperatures. Furthermore, DFT only minimizes energies (maximizing entropy is not relevant at 0 K); thus, at finite temperatures, real material configurations could vary from their 0 K DFT optimum structures. An experimental study of MgH2 nanoparticles by Aguey et al.6 showed that significant destabilization can be achieved with surfactant-stabilized MgH2 nanoparticles (approximately 5 nm in diameter), which can desorb hydrogen (with fast kinetics) at © 2012 American Chemical Society

Received: July 4, 2012 Published: July 23, 2012 17985

dx.doi.org/10.1021/jp306613t | J. Phys. Chem. C 2012, 116, 17985−17990

The Journal of Physical Chemistry C

Article

Starting geometries for the nanoparticles were obtained by extracting the cluster from supercells of the corresponding bulk structure. The geometries of the Mgn and MgnH2n clusters were optimized using simulation cells that were large enough to ensure complete isolation of the clusters from their periodic images. Each optimized cluster was tested for stability via calculation of the vibrational frequencies using the PHON10 program. If any of the (3N − 6) vibrational frequencies were imaginary, then the configuration was considered unstable. For most nanoparticle sizes of n > 5, multiple stable local minima were found. In these cases, the configuration with the lowest energy was used for the subsequent thermodynamic calculations. In practice, without exploring the entire configurational space, there exists no systematic method for determining whether a given configuration is the global minimum. However, we have calculated the vibrational frequencies for each cluster, to ensure their dynamical stability. Methods such as simulated annealing, ab initio molecular dynamics, or genetic algorithm based structure searching could be used to explore the configuration space but are well beyond the scope of this current investigation.

literature has attempted to calculate the nanoparticulate MgH2 desorption entropy. Theoretical studies3,5 investigating the 0 K desorption energies for MgnH2n clusters (where n < 65) appear to reach conclusions in direct contradiction with each other. Wagemans et al.3 used DFT calculations to show significant destabilization of MgnH2n clusters (where n < 20) with respect to the bulk and predicted that MgH2 nanoparticles with a diameter of 0.9 nm will have a desorption temperature of approximately 473 K. In contrast, Wu et al.5 using diffusion Monte Carlo (DMC) coupled with DFT, calculated that MgnH2n clusters (where n > 5) were stabilized with respect to the bulk. They estimated the error in DFT-calculated desorption energies, by treating their DMC calculations as having zero error, and concluded that the errors in DFT values were strongly dependent on cluster size. Despite the chemical accuracy observed using DMC for the Mg1H2 molecular system,5 there is no conclusive evidence to suggest it accurately calculates desorption energies (Ed) for all nanoparticle systems, and it relies on vibrational spectra and zero-point energies calculated using standard DFT. The DMC results suggest that changes in the 0 K desorption energy of MgH2 nanoparticles might not be sufficient (or even significant) to explain the experimentally observed reductions in reaction enthalpy since they indicate that, for the size range examined, an increase in reaction enthalpy should be observed (although the size range is much smaller than can be produced experimentally). The literature currently lacks a comprehensive theoretical treatment of the desorption reaction entropy for a nanoparticulate Mg−H system, and experimental results clearly demonstrate that its effect on the desorption temperature is significant. Furthermore, the vibrational properties of the nanoparticulate Mg−H systems have not been investigated either, and hence, the experimentally observed reductions in desorption temperature2 remain unexplained. The primary goal of this study is to use DFT to characterize the effect that 0 K ground state thermodynamic properties of bulk and nanoparticulate Mg−H systems have on the desorption temperature.

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THERMODYNAMICS The complete MgH2 desorption reaction (r) for a cluster with x Mg atoms is given by MgxH 2x(s) → x Mg(s) + x H 2(g)

(3)

11

The Gibbs−pressure relationship for an ideal gas serves to define the desorption temperature ⎛ p⎞ ΔG ln⎜⎜ ⎟⎟ = − r RT ⎝ p0 ⎠

(4)

where T is the temperature, R is the universal gas constant, G is the Gibbs free energy, p0 is 1 (bar), and p is the hydrogen equilibrium pressure (in bar). The desorption temperature is the temperature at which the hydrogen equilibrium pressure is 1 bar (when ΔrG = 0).

■

Δr G = Δr H − T Δr S

METHODOLOGY All geometry optimizations and cluster electronic energies were calculated using the SIESTA8 DFT code. Calculations were performed using the Wu-Cohen (WC) exchange-correlation functional9 and triple-ζ polarized atomic orbital basis sets (with split norms of 0.15 and 0.20 for Mg and H, respectively). The basis sets were radially confined to an extent that induces an energy shift in each orbital of 0.001 Ry. Hartree and exchangecorrelation energies are evaluated on a real space grid with a maximum kinetic energy of 300 Ry. Structural optimizations were considered converged when all atomic forces were less than 10−3 eV/Å. For bulk Mg metal and MgH2, the reciprocal cell was sampled using 26 × 26 × 16 and 18 × 18 × 27 k-point grids, respectively. The hydrogen molecule was optimized in a rigid box with dimensions of 14 × 14 × 14 Å. Atomic vibrational frequencies were calculated using the PHON10 program, which diagonalizes a dynamical matrix constructed from finite displacement force constant data. The displacement chosen for the calculation of force constants was 0.002 Å. For the bulk metals (Mg and MgH2), supercells of 3 × 3 × 3 primitive cells were constructed for the force constant calculations (5 × 5 × 5 supercells were also used to confirm that convergence was sufficiently achieved).

(5)

where ΔrG is the change in Gibbs free energy of the reaction, ΔrH is the change in enthalpy of the reaction, and ΔrS is the change in entropy of the reaction. For the desorption of MgH2, the reaction differential of any of the state functions (f) above can be expressed as Δr f = f (Mg) + f (H 2) − f (MgH2)

(6)

In the following equations, we adopt the notation described by Hector et al.12 and define Eel as the negative of the cohesive energy (with the zero-point energy, ZPE, excluded), Evib as the total vibrational energy (including the ZPE), Erot as the rotational energy (of H2), and Strans as the translational entropy (of H2). All quantities are normalized per Mg atom (or equivalently, per H2 molecule). Herein, the 0 K desorption energy (Ed) is defined as the reaction enthalpy at 0 K Ed = Eel(Mg) + ZPE(Mg) + Eel(H 2) + ZPE(H 2) − [Eel(MgH2) + ZPE(MgH2)]

(7)

For the solids, the enthalpy (H) is approximated as the sum of electronic and vibrational contributions to the energy (with pV considered negligible) 17986



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H(Mg) = Eel(Mg) + Evib(Mg)

(8)

H(MgH2) = Eel(MgH2) + Evib(MgH2)

(9)

where a 3:1 ratio of ortho to para hydrogen has been assumed for all temperatures, I is the moment of inertia of the hydrogen molecule calculated using the reduced mass (μ), j is the rotational quantum number, and R is the calculated equilibrium bond length such that

For hydrogen gas, the rotational and translational contributions to the energy are accounted for with pV substituted for kBT (by an ideal gas approximation, where kB is the Boltzmann constant) and with 3kBT/2 substituted for the translational kinetic energy H(H 2) = Eel(H 2) + Evib(H 2) + Erot(H 2) +

■

I = μR2

(20)

RESULTS AND DISCUSSION Bulk Systems. In Table 1, we compare the 0 K properties of the bulk Mg−H system with other theoretical and

3kBT + kBT 2 (10)

Table 1. Summary of the 0 K Properties of the Bulk Mg−H System Compared to Other Theoretical and Experimental Valuesa

For the solids, the vibrational entropy is considered the most significant term, and all other contributions to the entropy (S) are neglected S(Mg) = Svib(Mg)

(11)

S(MgH2) = Svib(MgH2)

(12)

For hydrogen gas, only the electronic entropy is neglected S(H 2) = Svib(H 2) + Srot(H 2) + Strans(H 2)

(13)

Vibrational energies and entropies were calculated within the harmonic approximation. That is, atomic vibrations are considered harmonic at all temperatures, and the configuration of each bulk solid, cluster, or molecule is assumed unchanged from its configuration at 0 K. As a consequence, the harmonic approximation does not account for any cluster/lattice expansion. The thermal equilibrium occupancy of each vibrational mode is calculated using the Planck distribution at each temperature17 1

⟨n⟩ =

( )−1

exp

ℏω kBT

(14)

where ℏ is the reduced Plancks constant and ω is the angular frequency of vibration. The translational partition function (Ztrans) is calculated using the mass of the hydrogen molecule (m) and V = kBT/p (by the ideal gas law), where V is the volume per H2 molecule and Λ is the thermal de Broglie wavelength

Ztrans =

V Λ3

Zpara =

Zrot =

(17)

⎡ ℏ2j(j + 1) ⎤ ⎥ (2j + 1) exp⎢ − 2IkBT ⎦ ⎣ even j

(18)

∑

3Zortho + Zpara 4

−13.410 −4.326 0.784 0.257 4153 4.069

−13.652 −4.560 0.749 0.273 4399 4.287

−13.60614 −4.74715 0.74116 0.27321 440521 4.47821

Mg (hcp) Eel (eV) ZPE (eV) Ecoh (eV) a = b (Å) c (Å)

−1.589 0.031 1.557 3.19 5.22

−1.478 0.028 1.450 3.19 5.17

1.51217 3.2118 5.2118

MgH2 (Tetragonal) Eel (eV) ZPE (eV) Ecoh (eV) a = b (Å) c (Å) Ed (eV)

−6.650 0.397 6.253 4.50 3.03 0.628

−6.682 0.405 6.277 4.50 3.01 0.540

6.71 4.5019 3.0119 0.70520

experimental values. The desorption energy (Ed) of bulk MgH2 shows good agreement to the experimental value of 0.705 eV/Mg atom.20 The desorption energy of MgH2 is sometimes quoted as being 0.79 eV/Mg atom, which is in fact the enthalpy of formation of MgH2 at 298.15 K.20 Furthermore, the uncertainty in the experimental desorption energy is almost 0.1 eV/Mg atom, suggesting that not even experiment has achieved chemical accuracy for the desorption energy value. Good agreement is seen between the experimental and calculated lattice parameters in Table 1, although it should be noted that these calculations do not account for zero-point lattice expansion (while experiment does). The bond length of H2 is overestimated compared to experiment, which results in an underestimation of the zero-point frequency. Similar SIESTA calculations of H2 by Pupysheva et al.,22 using the PBE functional of Perdew, Burke, and Ernzerhof,23 report a bond length of 0.775 Å, close to our value of 0.784 Å. The localized basis of SIESTA has advantages of speed and allows larger system sizes to be more accessible than normal, but this is weighted up against the transferability and accuracy of the

(16)

⎡ ℏ2j(j + 1) ⎤ ⎥ ⎢− (2 j + 1) exp ∑ 2IkBT ⎦ ⎣ odd j

experiment

H2 Eatom (eV) Eel (eV) bond length (Å) ZPE (eV) ZPF (cm−1) Ecoh (eV)

All energies are given in eV/Mg atom. ZPE and ZPF are the zeropoint energies and frequencies, respectively, and a, b, and c refer to the primitive cell lattice vectors of bulk hcp Mg and tetragonal MgH2.

The rotational partition function is evaluated using the rotational partition function for a hydrogen molecule in the ortho-state and in the para-state within the rigid-rotor approximation13 Zortho =

PW9112

a

(15)

h 2πmkBT

Λ=

Wu−Cohen

(19) 17987



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pseudopotentials and basis set descriptions. For example, the Mg pseudopotential and basis set must be able to describe Mg in an isolated Mg atom, and in the bulk and nanoparticle forms/phases of Mg and MgH2; all very different local environments for the Mg atom. A similar story occurs for the local environments of hydrogen atoms. Our pseudopotentials and basis sets were chosen to provide the best overall description of the bulk MgH2 system. The enthalpies and entropies of the bulk Mg−H system at finite temperature are compared to experimental data in Table 2. The results in Table 2 show that the temperature

Table 3. Calculated and Experimental Bulk MgH2 Desorption Reaction Thermodynamics ΔrH (eV/Mg atom) ΔrS (meV/K/Mg atom) Td (K)

experiment20

H2 H(298.15) − H(0) H(500) − H(0) H(600) − H(0) S(298.15) S(500) S(600)

0.0877 0.1485 0.1787 1.274 1.430 1.485

0.0878 0.1487 0.1791 1.354 1.510 1.566

Mg (hcp) H(298.15) − H(0) H(500) − H(0) H(600) − H(0) S(298.15) S(500) S(600)

0.0506 0.1011 0.1265 0.336 0.465 0.511

0.0518 0.1064 0.1351 0.339 0.478 0.530

MgH2 (Tetragonal) H(298.15) − H(0) H(500) − H(0) H(600) − H(0) S(298.15) S(500) S(600)

0.0577 0.1513 0.2077 0.339 0.575 0.678

0.0551 0.1452 0.1998 0.322 0.550 0.649

experiment2

0.726 1.32 549

0.768 ± 0.004 1.382 ± 0.007 555.0 ± 2.2

experimental values. Overall, the results provides good evidence that our methodology is capable of accurately predicting the bulk thermodynamics of the MgH2 desorption reaction. Somewhat fortuitous error cancellation perhaps plays a small part in this, and it cannot be assumed that modeling of the nanoparticulate desorption reactions will benefit from the same error cancellation. Nanoparticulate Systems. The analysis of the nanoparticulate Mg and MgH2 systems begins with an examination of the binding energies as a function of nanoparticle size (n). In order to compare the Mg nanoparticle binding energies to those of the earlier DFT study of Kohn et al.,25 we define the relative binding energy as the binding energy of the nanoparticle expressed as a fraction of the binding energy of the corresponding bulk material. In Figure 1, the relative binding

Table 2. Calculated Enthalpy Changes and Entropy for the Bulk Mg−H System at Finite Temperatures Compared to Experimental Dataa this work

this work

a

Enthalpies (H) are in units of eV/Mg atom, entropies (S) are in units of meV/K/Mg atom, and temperatures (Kelvin) are given in parentheses.

Figure 1. Relative binding energy, expressed as a fraction of the binding energy of the corresponding bulk material, for Mg and MgH2 nanoparticles. For Mg nanoparticles, the DFT results from Kohn et al.25 are also shown for comparison.

dependence of the calculated enthalpy for the hydrogen molecule is well described, whereas the calculated entropy is consistently smaller than experimental by approximately 0.08 meV/K. Performing the same calculations on the hydrogen molecule using the experimental bond length increases the rotational entropy by approximately 0.07 meV/K, thus accounting for the underestimated values. Mg has an experimental melting temperature of 923 K20 and so the harmonic approximation is not expected to yield reliable results above approximately 450 K. The results in Table 2 show that the harmonic approximation underestimates the enthalpy and entropy of bulk Mg at near-desorption temperatures, while for bulk MgH2, the enthalpy and entropy are overestimated. Kelkar et al.24 show closer agreement with experiment for bulk MgH2 using a quasi-harmonic approximation; however, we could not use this method because its application to clusters is prohibitively expensive. Table 3 displays the overall desorption reaction thermodynamics for the MgH2 bulk system and includes a comparison to

energies of the Mgn and MgnH2n nanoparticles are plotted as a function of n. As n increases for both Mgn and MgnH2n nanoparticles, the relative binding energy increases, approaching the bulk binding energy value. The trend for Mg nanoparticles matches well with the earlier DFT results of Kohn et al.,25 which are also shown in Figure 1. The calculated thermodynamics of the MgnH2n desorption reaction are plotted in Figure 2, showing the enthalpy (ΔrH), entropy (ΔrS), and desorption temperature (Td) as a function of n. The thermodynamic data from Figure 2 is also reported in Table 4. The reaction enthalpies (Figure 2) are all greater than the bulk value but, in general, decrease as the nanoparticle size increases, approaching the bulk value. The desorption temperatures Td are all larger than the bulk value and, in general, decrease with increasing size. The nanoparticulate entropies of reaction are lower than the bulk value, for all values of n (except n = 13). These results are in qualitative agreement with those of 17988



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Δ[Evib(Mg) − Evib(MgH2)]T = [Evib(Mg, T ) − Evib(MgH2 , T )]nano − [Evib(Mg, T ) − Evib(MgH2 , T )]bulk

(21)

A temperature of 600 K was chosen as a representative value to illustrate the behavior at a temperature above the bulk desorption temperature. The calculated difference (Δ[Evib(Mg) − Evib(MgH2)]T) at 600 K is illustrated, for all n, in Figure 3.

Figure 2. (a) Enthalpy (ΔrH), (b) entropy (ΔrS), and (c) desorption temperature (Td) of the desorption reaction of MgnH2n versus nanoparticle size.

Figure 3. Difference between the nano and bulk finite-temperature contributions (Δ[Evib(Mg)−Evib(MgH2)]T) to the vibrational reaction enthalpy, at 600 K.

Table 4. Thermodynamic Quantities for the Desorption Reaction of MgnH2na n

Ed (eV)

ΔrH(Td) (eV)

ΔrS(Td) (meV/K)

Td (K)

5 6 7 8 9 10 11 12 13 19 20 25 31 64 Bulk

0.728 0.761 0.748 0.729 0.715 0.707 0.726 0.767 0.820 0.721 0.754 0.742 0.721 0.672 0.628

0.812 0.836 n/a 0.809 0.774 0.795 0.803 0.848 0.910 0.808 0.842 0.830 0.805 0.766 0.726

1.28 1.26 n/a 1.28 1.26 1.28 1.24 1.26 1.34 1.27 1.29 1.28 1.26 1.30 1.32

636 666 n/a 631 615 623 647 674 679 638 651 646 640 591 549

The positive values in Figure 3 show that, at 600 K, the vibrational reaction enthalpy for the nanoparticulate systems is larger than for the bulk system, across the range of sizes examined. The difference (Δ[Evib(Mg) − Evib(MgH2)]T) follows the general trend of decreasing as the n increases, and for sufficiently large n, the difference should tend to zero. The results in Table 4 and Figure 3 show a general downward trend in the desorption energy as n increases, which is in qualitative agreement with the results of Wu et al.5 Deviations from the downward trend indicate that the electronic properties of Mgn and/or MgnH2n clusters are very sensitive to the value of n. This may also be a reflection of the difficulty involved in finding the true global minima for large clusters. The general trend is of interest for hydrogen storage, and clearly shows that, within the harmonic approximation, all system sizes have a desorption temperature higher than the bulk. The finding that the nanoparticulate enthalpies of reaction, shown in Table 4 and Figure 3, are in excess of the bulk value, is qualitatively contrary to the experimental results of Paskevicius et al.,2 who report a reduction relative to the bulk value. This suggests that the harmonic approximation is overly simplified in describing the thermodynamics of nanoparticulate systems. Geometry changes including different configurations and cluster expansion along with shifts in the vibrational spectra, in reality, all occur at finite temperatures. The harmonic approximation does not take these effects into account. A quasiharmonic approximation includes these effects, but still assumes that all vibrational modes are harmonic. A more sophisticated model that includes effects neglected by the harmonic approximation will likely be computationally prohibitive but

a

All energies and entropies are given per Mg atom, and an entry marked n/a indicates that the reaction, as calculated, did not become spontaneous for temperatures less than 1000 K.

Paskevicius et al.,2 who also found that the nanoparticulate reaction entropies are reduced relative to the bulk. To examine the dependence of the Gibbs free reaction energy on temperature, in eq 21, we calculate the difference between the nano and bulk finite-temperature contributions (Δ[Evib(Mg) − Evib(MgH2)]T) to the vibrational reaction enthalpy, at a given temperature T. 17989



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(12) Hector, L. G.; Herbst, J. F.; Wolf, W.; Saxe, P.; Kresse, G. Phys. Rev. B 2007, 76, 014121. (13) Kittel, C. Elementary Statistical Physics; John Wiley & Sons: New York, 1958. (14) Serway, R. A.; Jewett, J. W. Principles of Physics: A Calculus Based Text, 4th ed.; Thomson Learning: Belmont, CA, 2006. (15) Kolos, W.; Roothaan, C. C. J.; Sack, R. A. Rev. Mod. Phys. 1960, 32, 178. (16) Herzberg, G.; Howe, L. L. Can. J. Phys. 1959, 37, 636. (17) Kittel, C. Introduction to Solid State Physics, 8th ed.; John Wiley & Sons: New York, 2005. (18) Raynor, G. V. Proc. R. Soc. London, Ser. A 1940, 174, 457. (19) Zachariasen, W. H.; Holley, C. E., Jr.; Stamper, J. F., Jr. Acta Crystallogr. 1963, 16, 352. (20) Chase, M. W., Jr. NIST-JANAF Themochemical Tables. J. Phys. Chem. Ref. Data, 4th ed.; American Institute of Physics: New York, 1998. (21) Huber, K. P.; Herzberg, G. Molecular Spectra and Molecular Structure. IV. Constants of Diatomic Molecules; Van Nostrand Reinhold Company: New York, 1979. (22) Pupysheva, O. V.; Farajian, A. A.; Yakobson, B. I. Nano Lett. 2007, 8, 767. (23) Perdew, J. P.; Burke, K.; Ernzerhof, M. Phys. Rev. Lett. 1996, 77, 3865. (24) Kelkar, T.; Kanhere, D. G.; Pal, S. Comput. Mater. Sci. 2008, 42, 510. (25) Köhn, A.; Weigend, F.; Ahlrichs, R. Phys. Chem. Chem. Phys. 2001, 3, 711.

should be the focus of future theoretical studies on these systems.

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CONCLUSIONS This study has used DFT to calculate the 0 K desorption energies and the vibrational free energies of the bulk and nanoparticulate MgH2 desorption reaction (up to 64 Mg atoms) within the harmonic approximation. The bulk desorption temperature was calculated to be 276 °C, which is in excellent agreement with the experiment value of approximately 280 °C.2 We report binding energies as well as harmonic desorption reaction entropies and enthalpies for all the Mg and MgH2 cluster sizes. A general downward trend is observed in the desorption energy with increasing particle size, in qualitative agreement with the results of recent diffusion Monte Carlo calculations.5 It is found that, in all but one case, the desorption reaction entropy change (ΔrS) is larger for the nanoparticulate systems than the bulk, in agreement with experimental results for large nanoparticles.2 However, the experimental values also show a corresponding decrease in reaction enthalpy for the nanoparticle systems, which is contrary to our computational results. Although, this work clearly demonstrates the finite-temperature effects on the vibrational energies of the nanoparticulate systems, when considered within the framework of the harmonic approximation, it appears too simple to account for the experimentally observed reductions in reaction enthalpy.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Financial support from the Australian Research Council under grant DP0877155 is gratefully acknowledged. A.C.B. also acknowledges the financial support provided by the Australian Institute of Nuclear Science and Engineering through Research Award ALNGRA11160. We thank iVEC for the provision of computational resources.

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REFERENCES

(1) Das, L. M. Int. J. Hydrogen Energy 1996, 21, 789. (2) Paskevicius, M.; Sheppard, D. A.; Buckley, C. E. J. Am. Chem. Soc. 2010, 132, 5077. (3) Wagemans, W. P.; van Lenthe, J. H.; de Jongh, P. E.; van Dillen, A. J.; de Jong, K. P. J. Am. Chem. Soc. 2005, 127, 16675. (4) Kim, K. C.; Dai, B.; Johnson, J. K.; Sholl, D. S. Nanotechnology 2009, 20, 204001. (5) Wu, Z.; Allendorf, M. D.; Grossman, J. C. J. Am. Chem. Soc. 2009, 131, 13918. (6) Aguey-Zinsou, K.-F.; Ares-Fernández, J.-R. Chem. Mater. 2008, 20, 376. (7) Paskevicius, M.; Tian, H.-Y.; Sheppard, D. A.; Webb, C. J.; Pitt, M. P.; Gray, E. M.; Kirby, N. M.; Buckley, C. E. J. Phys. Chem. C 2011, 115, 1757. (8) Soler, J. M.; Artacho, E.; Gale, J. D.; García, A.; Junquera, J.; Ordejón, P.; Sánchez-Portal, D. J. Phys.: Condens. Matter 2002, 14, 2745. (9) Wu, Z.; Cohen, R. E. Phys. Rev. B 2006, 73, 235116. (10) Alfé, D. Comput. Phys. Commun. 2009, 180, 2622. (11) Atkins, P.; de Paula, J. Physical Chemistry, 8th ed.; W.H. Freeman and Company: New York, 2006. 17990


Density Functional Theory Calculations of Magnesium Hydride: A

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