Fast Mass Transport Kinetics in B20H16: A High-Capacity Hydrogen

Article pubs.acs.org/JPCC

Fast Mass Transport Kinetics in B20H16: A High-Capacity Hydrogen Storage Material Kyle Jay Michel,* Yongsheng Zhang, and C. Wolverton Department of Materials Science and Engineering, Northwestern University, Evanston, Illinois 60208, United States ABSTRACT: First-principles calculations are used to study the role of point defects in the rehydrogenation and dehydrogenation of B20H16; in particular, we focus on the energetics of long-range mass transport through bulk phases. We find that interstitial H2 in B20H16 exists in the largest concentrations of all native point defects during both the formation and decomposition reactions. Using kinetic Monte Carlo simulations, we show that the diffusivity is high for this particular defect and that the overall activation energies for mass transport are 6 and 70 kJ/mol for the rehydrogenation and dehydrogenation reactions, respectively.

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INTRODUCTION The storage of hydrogen in an economical and practical way remains a significant challenge in the development of hydrogen fuel cell vehicles.1 Reversible storage in the solid state (such as in complex metal hydrides) has gained much interest because of the favorable storage densities that can be achieved.2 A candidate material must release hydrogen at a few bar and operate using the waste heat from the proton exchange membrane fuel cell.3,4 Well-established methods such as the grand canonical linear programming (GCLP)5 method have been used to screen hydrogen storage reactions using energetics obtained from first-principles calculations to determine those that are thermodynamically allowed.6 Besides releasing and absorbing hydrogen in the target range of temperatures and pressures, these reactions should occur rapidly; the targeted rehydrogenation rate proposed by the Department of Energy (DOE) is 2.0 kg H2/min.3 All of the known hydrogen storage reactions that have thermodynamic properties and storage capacities that meet the DOE targets suffer from unacceptably slow reaction rates. There are many processes that may limit the dehydrogenation or rehydrogenation rates of metal hydrides, such as nucleation, growth, and H2 dissociation and recombination at surfaces. In particular, many studies have focused on the kinetics of long-range mass transport during the growth stage. There is both experimental and theoretical evidence that this mass transport step occurs via the diffusion of point defects through bulk phases and that it may be rate-limiting in many reactions.7,8 This provides motivation to determine the mass transport barrier for proposed reactions. Those reactions with high mass transport barriers can be disregarded for practical applications because their kinetics will be unacceptably slow unless a suitable catalyst can be found. Recently, the decomposition of B20H16 has been predicted to occur via a single-step reaction:9 B20H16 → 20B + 8H 2

The enthalpy change of reaction 1 is equal to 33 kJ/mol of H2 with a calculated equilibrium temperature of 20 °C (at a pressure of 1 bar H2), releasing 6.9 wt % H2 with a volumetric density of 64.1 g of H2/L. These thermodynamic properties and the large storage capacity make this reaction attractive for applications in passenger vehicles. In this paper, we focus on the kinetics of mass transport in reaction 1. We study the formation of point defects in B20H16 and B and identify those that exist in the largest concentrations. We then calculate the flux of these defects using diffusivities obtained from kinetic Monte Carlo simulations to determine if reaction 1 may be kinetically limited by mass transport. We find that this process is dominated by the flux of interstitial H2 in B20H16 and that the associated activation energy is 6 kJ/mol for rehydrogenation and 70 kJ/mol for dehydrogenation. With both fast mass transport kinetics and ideal thermodynamic properties, we argue that reaction 1 may be well-suited for hydrogen storage in passenger vehicles. Similar methods have been applied to the sodium alanate system, and the results of that study support the accuracy of these calculations in determining activation energies; the calculated activation energy for mass transport during dehydrogenation is 70 kJ/mol,10 while the measured activation energy for the reaction is 80 kJ/mol.11

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METHODS Reaction Morphology and Chemical Potential. At a morphological scale, reaction 1 can occur in two ways (Figure 1). In Figure 1a, a layer of B20H16 is maintained at the surface while the product B phase grows in the interior (or is consumed during rehydrogenation). In this case, mass transport occurs through B20H16 as hydrogen must be transported between the surface and the B/B20H16 interface. In the second morphology, Figure 1b, the B product phase is maintained at the surface and grows inward, consuming B20H16 in the interior Received: March 17, 2013 Revised: August 28, 2013

(1) © XXXX American Chemical Society

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dx.doi.org/10.1021/jp402669u | J. Phys. Chem. C XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry C

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the following analysis, we study this kinetic process at the atomic scale using ab initio calculations of point defects in bulk phases. The flux of defects (which is directly related to the mass transport rate) is given by

J = −D∇C

(7)

where D is the defect diffusivity and ∇C = ΔC/d is its gradient in concentration between interfaces (where ΔC is the difference in concentration between two interfaces separated by a distance d). Using the Arrhenius equation, the flux is related to the activation energy, Eact, by Figure 1. Model morphologies used to describe reaction 1.

J = J0 exp( −Eact /kBT )

In practice, we calculate the flux of defects (described in the following) and fit eq 8 to the associated Arrhenius plot in order to calculate the activation energy for mass transport. The concentrations needed to evaluate eq 8 for a defect i are obtained using12,13

(or B20H16 grows into the B region during rehydrogenation). Here, the relevant mass flux is of hydrogen through boron. We study both morphologies to determine which provides the fastest mass transport medium. We note that the morphologies shown in Figure 1 are not meant to imply that the system must be cylindrical or spherical; instead, they simply show the interfaces that exist in the system and the regions through which mass transport may occur. The driving force for mass transport can be easily viewed in terms of the elemental chemical potentials. Local equilibria at each interface fix these chemical potentials such that at the interface between B20H16 and H2 they are set by eqs 2 and 3: 20μB + 16μH = G(B20H16)

Ci = NisitesNieqv exp( −ΔGiform /kBT )

where is the number of possible defect sites per unit volume and Neqv is the number of equivalent ways that the i defect can be introduced at a particular site. ΔGform is the free i energy of formation of the defect, which, for an insulating material (the predicted band gap of B20H16 is 3.7 eV), is equal to13

(2)

ΔGiform = Gidef − G pure −

nisμ s + qi(εF + E VBM) (10)

Here, is the energy of a supercell with this defect and Gpure is the free energy of the pure supercell. The third term of eq 10 represents the total chemical potential of the atoms in the defect; μs is the chemical potential of an atom of type s (either B or H) and nsi is the number of atoms of this type in defect i, which is negative for atoms that are removed (vacancies) and positive for atoms that are added (interstitials). The final term represents the energy to introduce or remove electrons in the creation of a charged defect with charge state qi and Fermi level εF, which is referenced to the valence band maximum, EVBM. The formation energy of point defects in boron can also be described by eq 10 by excluding the final term because the bulk phase is metallic. The Fermi level was fixed by enforcing a local charge neutrality at each interface. Formally, this was accomplished by solving Gdef i

(4)

Finally, at the interface between B and H2, the chemical potentials are set by eqs 3 and 4. Referring to Figure 1a, let μint H and μext H represent the chemical potentials of H at the interior (B/B20H16) and exterior (B20H16/H2) interfaces, respectively. Using eqs 2, 3, and 4 to solve for the difference between them yields 1 μHint − μHext = − ΔGrxn (5) 16 where ΔGrxn = 20G(B) + 8G(H 2) − G(B20H16)

∑ s ∈ {B,H}

(3)

where G(B20H16) and G(H2) are the free energies of B20H16 and H2, respectively. At the interface between B20H16 and B, the chemical potentials satisfy eqs 2 and 4:

μB = G(B)

(9)

Nsites i

and

2μH = G(H 2)

(8)

(6)

Ndef

∑ qiCi = 0

is the change in free energy of reaction 1. At high temperatures (above the onset of dehydrogenation), ΔGrxn < 0 and a ext chemical potential gradient (μint H > μH ) is formed such that there is a driving force for hydrogen diffusion to the surface, as is expected for dehydrogenation. At low temperatures, corresponding to rehydrogenation conditions, ΔGrxn > 0 and ext μint H < μH so that the chemical potential gradient reverses direction and hydrogen is driven toward the interior where it can react with boron to form B20H16. Using a similar argument for the morphology in Figure 1b, it can be shown that the chemical potential gradients under dehydrogenation conditions drive hydrogen through boron to the surface and in the opposite direction under rehydrogenation conditions. Defect Flux. While the previous discussion involved mass transport at the morphological scale and provides the basis for

(11)

i

for the Fermi level as a function of temperature and pressure where Ndef is the total number of defects considered, including distinct sites and configurations. Generation of Interstitial Defect Sites. Locating the stable interstitial sites in B20H16 is difficult largely because of the size of the structure; the primitive cell contains 144 atoms and has a volume of 1671 Å3. We have devised a geometric method to predict the location of possible interstitial sites by selecting voids in the structure. In doing so, we defined a radially decaying exponential function centered at each atomic site, fij (r) = exp( −|r − r ij0| /a) B

(12)



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where r0ij is the vector to the Cartesian position of atom j in unit cell i and a controls the rate of decay (a = 0.25 in our calculations). We then defined an overall function for the crystal Ncells Natom

F(r) =

∑ ∑ fij (r) i

j

(13)

where Natoms is the number of atoms in a single unit cell and Ncells is the number of cells in the crystal. The resulting function has peaks at all atomic sites and decays toward local minima in voids. Using periodic boundary conditions, the sum over cells is infinite; however, this sum can be truncated by taking only those terms within some absolute distance from r such that the excluded terms are negligible to a fixed tolerance. A list was made of stationary points (all three Cartesian derivatives are equal to zero and are well-defined) of F(r) in a single unit cell. We identified candidate interstitial sites as those stationary points that are local minima, thus excluding single and double saddle points. Using this method on the conventional cell of B20H16, we generated more than 500 candidate sites. However, this was reduced to 22 distinct sites using the underlying crystal symmetry operations. These candidate sites are shown in Figure 2. It should be emphasized that these are only starting guesses for interstitial defects; once atoms had been placed at these sites, atomic coordinates were fully relaxed using ab initio calculations (described in Ab Initio Methods). This same procedure was used to predict interstitial sites in boron. Kinetic Monte Carlo Simulations of Point Defects. Where necessary, defect diffusivities were calculated from transition-state theory14,15 using kinetic Monte Carlo (KMC) simulations. The rate of jumps from site i to site j was calculated with R ij = νij exp( −ΔHijmig /kBT )

(14)

ΔHmig ij

where νij is the attempt frequency for the jump and is the difference in enthalpies of the system in the ground and transition states at T = 0 K. In principle, the attempt frequency can be calculated from the harmonic vibrational modes of the defected system in the ground and transition states (see ref 8 and references therein). However, we have not attempted such calculations in the current work and instead choose νij = 1013 s−1 in all cases. Because νij is independent of temperature in the classical limit,15 this choice will have a negligible effect on the calculated activation energies. In each simulation, the displacement vector, d, was taken as the difference between the initial and final positions of the defect. The diagonal elements of the diffusivity tensor were then calculated according to ⟨dγ2⟩ (15)

Figure 2. Locations of candidate interstitial sites in B20H16 (wire frame) represented by gold spheres. Symmetry distinct sites are shown as larger red spheres.

where dγ is the component of the displacement vector in the γ direction (equal to x, y, or z) and Δt is the length of a single simulation. Averages of dγ and Δt were taken over all simulations at a single temperature; these simulations were performed until the standard error was within 1% of the mean diffusivity at that temperature (typically about 20,000 simulations) with 10,000 jumps made in each simulation. The diffusivity was calculated at 10 K intervals using isolated defects in an infinite crystal to simulate diffusion in the bulk. As discussed previously, H2 dimers were initially placed at 22 candidate interstitial sites. Following relaxations using ab initio

calculations, only four distinct stable sites were found. The full list of interstitial sites in the conventional cell was then populated by applying the 32 crystal symmetry operations of the host structure to each of the four distinct sites. From this list, all transitions between stable sites were enumerated to give more than 150 allowed jumps in the conventional cell of the B20H16 structure. Finally, using the crystal symmetry, this list was reduced to eight distinct jumps between stable interstitial sites. Nudged elastic band (NEB) calculations were performed to obtain each diffusion path, and climbing image NEB

Dγγ =

2⟨Δt ⟩

C



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calculations were performed to find the barriers needed for the KMC simulations.16 Ab Initio Methods. All total energies were obtained from density functional theory (DFT) calculations using the Vienna ab initio simulation package (VASP).17 We used the PW9118 generalized gradient approximation (GGA) to the exchangecorrelation energy with the projector augmented wave (PAW)19 method and a plane wave cutoff energy of 875 eV. The 2s and 2p electrons were treated as valence for B. Defects were introduced in cells of B20H16, P43 boron, and R3̅m boron that contained 288, 196, and 324 atoms, respectively. For calculations involving B20H16, we used a k-point mesh of 3 × 3 × 1, giving a linear density of approximately one point per 0.03 Å−1 along each reciprocal lattice direction. The atomic coordinates were relaxed for all defected structures until the total energy converged to 10−5 eV and the interatomic forces were below 10−2 eV/Å. When determining the chemical potentials according to eqs 2−4, we added to the DFT energy, EDFT, of the H2 molecule a temperature-dependent correction to the enthalpy and a temperature- and pressure-dependent correction to the entropy so that20 G = EDFT +

⎤ ⎡7 7 kBT − kBT ⎢ ln T − ln p + C0 ⎥ ⎦ ⎣2 2

(16)

Here, C0 = −4.22 is an empirical parameter obtained from a fit of the entropy to experimental data. In all calculations we have taken p = 1 bar H2 and left the temperature as variable.

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RESULTS When studying point defects in B20H16 (structure shown in Figure 3 and obtained from ref 21), we considered the following vacancies in the specified charge states: VB (5 symmetry-distinct sites) with q = +3, +2, +1, 0, −1, −2, −3; VH (4 distinct sites) with q = +1, 0, −1; and VBH (4 distinct configurations) with q = +1, 0, −1. We also examined interstitial B with q = +1, −1, H with q = +1, 0, −1, and H2 with q = 0. Because the chemical potential of boron is the same at both interfaces in Figure 1b, the relevant mass flux in this second morphology is of neutral interstitial H, which we have studied in both the P43 and R3̅m structures of boron. In total, more than 100 unique point defects (including distinct sites or configurations) were considered in this study. Free energies of formation for these defects are plotted versus temperature in Figure 4 at both of the interfaces shown in Figure 1a. Because each defect has at least four distinct sites or configurations, we show only the lowest formation energy for each. We also omit those defects that have formation energies greater than 3 eV/ defect because the relative concentrations will be negligible compared to those that are shown. Of the defects in B20H16, we find that interstitial H2 has the lowest free energy of formation under all assumed conditions; it is equal to 0.55 eV/defect at the B/B20H16 interface and increases from 0.28 to 0.63 eV/defect at the B20H16/H2 interface as the temperature increases from 250 to 500 K (the distributions of sites and energies for this defect are shown in Table 1). This temperature dependence of the free energy of formation is due to the chemical potential of hydrogen gas, which decreases with increasing temperature (see eq 16). Of the remaining defects, the formation energy of neutral BH vacancies is several hundred meV above that of interstitial H2, and all other defect formation energies are still higher by at

Figure 3. Structure of B20H16 with B atoms represented by purple spheres, H atoms by pink spheres, and B20H16 clusters by polyhedra.

least 1 eV. From steric arguments used in most crystals, it is somewhat surprising that an H2 dimer, instead of some other monatomic type, is the lowest-energy interstitial defect. However, because B20H16 is a molecular crystal, there exist large voids in the structure in which it is possible that both atoms in an H2 dimer can each be more than 1.5 Å from any other atom in the crystal. This large distance between the H2 dimer and the host B20H16 clusters allows the formation energy of this interstitial defect to remain relatively low. Having established that interstitial H2 forms with the lowest energy of all defects, we now turn to the diffusion of interstitial H2. From KMC simulations, we find that the energy landscape D



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Figure 5. Calculated flux of interstitial H2 across a 1 μm region of B20H16. The vertical dashed line shows the equilibrium temperature for the reaction. Separate linear fits to high- and low-temperature regions are shown as dotted black lines with the associated activation energy.

Taking the slopes above and below the critical temperature in Figure 5, we find that the activation energies for dehydrogenation and rehydrogenation are 70 and 6 kJ/mol respectively. We note that the activation energy, especially for dehydrogenation, is lower than the free energies of formation shown in Figure 4. However, these formation energies are free energies whereas the activation energy is a function of the defect formation enthalpy, which is not shown in Figure 4. Considering interstitial hydrogen in boron, we find that the lowest formation energy occurs in the P43 phase (as opposed to the R3̅m phase). At the interface with B20H16, its free energy of formation is 1.31 eV/defect, while at the interface with H2 it increases from 1.13 to 1.31 eV/defect as the temperature increases from 250 to 500 K. Because these energies are much larger than those calculated for interstitial H2 in B20H16, the concentrations of hydrogen-containing defects in boron will be relatively negligible and the reaction will proceed most rapidly via the diffusion of defects in B20H16.

Figure 4. Free energies of formation for defects in B20H16. Only those defects with formation energies of less than 3 eV are shown. The bold red lines on the inset morphologies show the interfaces at which each set of formation energies is calculated.

Table 1. Unique Interstitial H2 Positions in Fractional Coordinates with Multiplicities in the Conventional B20H16 Unit Cell, Formation Free Energies at T = 300 K (eV/H2) at the B20H16−H2 Interface, and Distances to Nearest Atom in This Host Cell (Å) site 1 2 3 4

position 0.0000 0.0000 0.0322 0.3027

0.0000 0.0000 0.2500 0.2500

0.1886 0.2500 0.3750 0.3750

multiplicity

ΔGform (T = 300K)

dnear

16 8 16 16

0.369 0.346 0.481 0.475

2.42 2.63 2.05 1.89

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SUMMARY AND CONCLUSIONS We have shown that interstitial H2 in B20H16 forms in the largest concentration of all defects relevant for mass transport in the bulk phases involved in reaction 1. The calculated activation energy for H2 transport is 6 kJ/mol under rehydrogenation conditions and 70 kJ/mol under dehydrogenation conditions. While the activation energy for reaction 1 has not been determined experimentally, insight into the reaction kinetics can still be gained by comparison to other systems. For example, Ti-doped NaAlH4 is generally considered to have fast reaction kinetics compared to other complex metal hydrides. The measured activation energy for the dehydrogenation of this system is 80 kJ/mol,11 while it is 62 kJ/mol for rehydrogenation;22 both of these are even higher than the energies calculated in this study. Therefore, the mass transport rates in reaction 1 should be sufficiently fast for practical applications. It must be emphasized that rapid mass transport is a necessary, but not sufficient, condition for reaction kinetics that are acceptably fast. Other processes, such as nucleation and H2 dissociation, may also limit the reaction rate. However, methods for catalyzing these processes in particular have

between interstitial H2 sites in B20H16 is relatively flat and that the largest barrier for diffusion is only 116 meV. Using the concentration gradients obtained from the energies shown in Figure 4 and the diffusivities obtained from KMC simulations, the overall flux of interstitial H2 is shown on an Arrhenius plot in Figure 5. Because the flux depends on the length of the region over which the concentration gradient is calculated, we have made the arbitrary choice to show the results when the length of the B20H16 region is 1 μm; however, such a choice will have no effect on the calculated activation energy because this distance does not depend on temperature. The vertical, dashed line in this figure shows the location of the equilibrium temperature for reaction 1 where the concentration gradient and thus flux decrease to zero because the chemical potential of hydrogen is exactly equal at both interfaces at this temperature. E



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already been demonstrated in other hydrogen storage systems (using product seeding23 and transition metal doping,24 respectively). With this, the combination of fast mass transport kinetics and large hydrogen content in B20H16 makes this material promising for the vehicular storage of hydrogen.

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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 Funding for K.J.M. was provided by the U.S. Department of Energy, Office of Science, Basic Energy Sciences under Grant DE-FG02-07ER46433. Funding for Y.Z. was provided by the U.S. Department of Energy under Grant DE-FC3608GO18136.

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Fast Mass Transport Kinetics in B20H16: A High-Capacity Hydrogen

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