Hydrogen storage in Mg transition metal alloys : a DFT study

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Hydrogen storage in Mg transition metal alloys : a DFT study

Citation for published version (APA):

Tao, S. (2011). Hydrogen storage in Mg transition metal alloys : a DFT study. Technische Universiteit Eindhoven. https://doi.org/10.6100/IR711018

DOI:

10.6100/IR711018

Document status and date: Published: 01/01/2011

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Hydrogen storage in Mg transition metal

alloys - A DFT study

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan de

Technische Universiteit Eindhoven, op gezag van de

rector magnificus, prof.dr.ir. C.J. van Duijn, voor een

commissie aangewezen door het College voor

Promoties in het openbaar te verdedigen

op maandag 23 mei 2011 om 16.00 uur

door

Shuxia Tao

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prof.dr. R.A. van Santen en prof.dr. P.H.L. Notten Copromotor: dr. A.P.J. Jansen Shuxia Tao

Hydrogen storage in Mg transition metal alloys - A DFT study

A catalogue record is available from the Eindhoven University of Technology Library ISBN:978-90-386-2472-3

Copyright c⃝2011 by Shuxia Tao

The research described in this thesis has been carried out at the Schuit Institute of Catal-ysis within the Laboratory of Inorganic Chemistry and CatalCatal-ysis, Eindhoven University of Technology, the Netherlands.

Cover design: Shuxia Tao and Bob Kielstra (Ipskamp Drukkers) Printed by Ipskamp Drukkers.

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Chapter 1 Introduction

1.1 The hydrogen economy and hydrogen storage

A hydrogen economy is proposed to solve the negative eﬀects of using hydrocarbon fuels with carbon dioxide emissions. Modern interest in the hydrogen economy can generally be traced to a 1970 technical report by Lawrence W. Jones of the University of Michigan.1

In the current hydrocarbon economy, burning of hydrocarbon fuels emits carbon dioxide and other pollutants. The supply of economically usable hydrocarbon resources in the world is limited, and the demand for hydrocarbon fuels is increasing. We should therefore seek new ways to ensure the future energy needs. These resources should preferable be sustainable and renewable in nature, e.g., solar, biomass, wind, geothermal. These resources can typically be used for stationary applications. For mobile applications, like a fuel-cell driven vehicle or portable electronics, hydrogen is expected to play an important role as a energy carrier. The attraction of using hydrogen as an energy currency is that, if hydrogen is prepared without using fossil fuel inputs, vehicle propulsion would not contribute to carbon dioxide emissions. The challenges facing the use of hydrogen in vehicles include production, storage, transportation and distribution. The eﬃciency for the whole system, because of all these challenges will not exceed 25%.2–4

Hydrogen storage is one of the most important issues to be solved before a hydrogen vehicle can be used an alternative fuel vehicle that uses hydrogen as its onboard fuel for motive power. Although molecular hydrogen has very high energy density on a mass basis, partly because of its low molecular weight, as a gas at ambient conditions it has very low energy density by volume. If it is to be used as fuel stored on board the vehicle, pure hydrogen gas must be pressurized or compacted into smaller volume to provide suﬃcient driving range. The current goal, according to the technical requirements for an on board

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hydrogen storage system by the U. S. Department of Energy, is to ﬁnd a metal hydride with a gravimetric capacity that exceeds 6 wt.% hydrogen, absorb and desorb hydrogen at atmospheric pressures around room temperature.5

Fig. 1.1 illustrates various ways in which hydrogen can be compacted into a smaller volume with respected to the size of a car. The first way is that the pure hydrogen gas is pressurized under high presser (200 bars). Increasing gas pressure improves the energy density by volume. However, the heavy container tanks increase the total weight dramatically. Alternatively, higher volumetric energy density liquid hydrogen or slush hydrogen may be used. However, liquid hydrogen is cryogenic and boils at 20.268 K (-252.882 oC). Cryogenic storage cuts weight but requires large liquification energies. The liquefaction process, involving pressurizing and cooling steps, is energy intensive. Beside the high energy cost, the liquefied hydrogen still has lower energy density by volume than gasoline by approximately a factor of four, because of the low density of liquid hydrogen.

Figure 1.1: Volume of 4 kg of hydrogen compacted in diﬀerent ways, with size relative to the

size of a car. This ﬁgure is rearanged from Ref.6

Distinct from storing molecular hydrogen, a signiﬁcant increase of the energy density and strong reduction of the volume can be achieved by other hydrogen-containing com-pounds, such as, LaNi5H6 and MgNiH4. Hydrogen gas is reacted with other elements to

produce the hydrogen storage material, which can be transported relatively easily and safely. At the point of use the hydrogen storage material can be made to decompose, yielding hydrogen gas. As well as the mass and volume density problems associated with molecular hydrogen storage, current barriers to practical storage are high pressure and

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1.1 The hydrogen economy and hydrogen storage 3

Figure 1.2: Stored hydrogen per mass and per volume. Comparison of metal hydrides, carbon

nanotubes, petrol and other hydrocarbons. This ﬁgure is rearanged from Ref.6

temperature conditions needed for hydride formation and hydrogen release. For many po-tential systems hydriding and dehydriding kinetics and heat management are also issues that need to be overcome.

Fig. 1.2 shows a comparison of metal hydrides, petrol and hydrocarbons with respect to the stored hydrogen per mass, per volume, the decomposition temperature and pressure. Based on the weight constraints, most of the materials are not suitable for on board hydrogen system. Despite some light weight elements, such as, CH4 and C4H10 having a

very high gravimetric capacity, they are gas or liquid at ambient temperatures, which will lower the volumetric capacity signiﬁcantly compared to the solid metal hydride. During the last few decades, many metal hydrides were developed that readily absorb and desorb hydrogen at room temperature and atmospheric pressure. A notable example is the LaNi5H6, which shows very promising properties, including fast and reversible sorption

and pressure of a few bar at room temperature (see Fig. 1.2) and good cycling life. This is successful for hydrogen storage in metal hydride for rechargeable electrodes for portable electronics. However, their application is still limited, as they are capable of storing only about 2 weight percent of hydrogen due to the large weight proportion of the LaNi5. This

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temperature.

Currently the only hydrides which are capable of achieving the 6 wt% gravimetric goal for 20105 are limited to lithium, boron and aluminum, magnesium based compounds. Proposed hydrides for use in a hydrogen economy include simple hydrides of magnesium or transition metals and complex metal hydrides, typically containing sodium, lithium, or calcium and aluminium or boron.

1.2 Recent progress on hydrogen storage in Mg based

hydrides

As mentioned in the last section, Mg is one of the most promising elements for hydrogen storage due to its attractive gravimetric capacity of 7.7 wt.%. But a major disadvantage is that its formation from bulk Mg and gaseous hydrogen is extremely slow. The slow kinetics arises both from diﬀusion limitations and high oxidation sensitivity of the Mg surface hindering the dissociation of hydrogen molecule. Another disadvantage is that MgH2 is

too stable, reaching an equilibrium pressure of 1 bar at 300o_{C. Various processes have been}

tried to improve the hydrogen storage properties by addition of catalytic additives such as Ni,7 _LaNi

58 or La-Mg-Ni alloys,9 reducing the particle and grain size by mechanical

milling,10–17 _{alloying the Mg with transition metals to alter the crystal structure of the}

hydride.18–23

Among the improvements is Mg2Ni that enables 1 bar H2 pressure at 255 oC, which is

only 24oC lower compared to MgH2.28,29The fairly rapid kinetics is due to the presence of

Ni as catalyst for the dissociation of molecular hydrogen. The alloys Mg2Cu, Mg17La2and

MgAl, and some other known alloys or intermetallic compounds of Mg, react readily with hydrogen and decompose into MgH2 and another compound or hydride. One example is

Mg2Cu and H2 which decomposes to MgH2 and MgCu2. The reactions are only reversible

at high temperature. It comes as no surprise that the capacities are reduced by the addition of the heavy metal elements.

Theoretical predictions made by Wagemans et al.24 _{indicate that the hydrogen storage}

properties of Mg can be enhanced by decreasing the particle size of the crystallites. They calculated that decreasing the grain size of magnesium hydride below proximately 1.3 nm results in a substantial decrease of the hydrogen desorption enthalpy. Similar destabiliza-tion upon reducdestabiliza-tion in the Mg particle size has been predicted theoretically by several other researchers.25–27 _{For these small Mg particles the hydrogen diﬀusion rate is of}

mi-nor importance as the diﬀusion lengths are short. In fact, signiﬁcant hydrogen sorption enhancement have been observed by reducing the particle and grain size by mechanical

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1.2 Recent progress on hydrogen storage in Mg based hydrides 5

milling.10–17 However, no profound destabilization of the MgH2 was reported

experimen-tally. This is probably due to the relatively large size of the milled materials caused by the limitation of the mechanical milling.

Fluorite transition metals, such as Sc, Ti were alloyed with Mg by Notten and his co-workers and it was shown that the binary alloys reveal excellent sorption kinetics.18–21

It was argued that fast kinetics was due to a face centered cubic (fcc) structure of the hydride, whereas the alloys with slow kinetics revealed the body centered tetragonal (bct) MgH2 structure. The Mg0.80Sc0.20 composition revealed excellent kinetic properties and

the highest reversible hydrogen capacity of 6.7 wt% of hydrogen. The high costs of Sc, however, prevent Mg-Sc alloys from being employed commercially as hydrogen storage medium. To limit the gravimetric capacity loss by substituting a heavier element for Sc, the ﬁrst row transition metals are most promising, therefore Ti, V and Cr would be excep-tionally suitable, as these elements not only have a reasonably high gravimetric capacity, but also form a fcc-structure hydride. Unfortunately, the absorbed hydrogen atoms in the Mg-Cr and Mg-V can be released only at a low rate.19 _Mg

0.80Ti0.20 behaves similar to

Mg0.80Sc0.20. This attractive kinetic property makes Mg-Ti alloys a very promising system

and therefore Mg-Ti-H will be the main focus of this thesis.

Ti is immiscible with Mg. Mg-Ti alloys can therefore not be prepared by conventional methods.30 _{However, based on X-ray diﬀraction results (XRD) it has been reported that}

alloying of Mg and Ti does take place in mechanically alloyed bulk samples31 _{and in thin}

ﬁlms by using physical vapor deposition,32 e-beam deposition,33,34 and sputtering.35,36 Thin ﬁlms of Mg-based alloys are a subject of extensive research, as they are becoming increasingly more utilized for optical hydrogen sensing,37 _{switchable mirrors and solar}

absorbers,22 and as model systems for designing and understanding bulk hydrogen stor-age materials. Vermeulen et al.38 _{investigated the structural transformations}

through-out the (de)hydrogenation process in Mg-Ti thin ﬁlm alloys. XRD and electrochemical (de)hydrogenation were performed in situ to monitor the symmetry of the unit cells of MgxTi1−x along the pressure-composition isotherms at room temperature. The lattice

spacings found for Mg0.7Ti0.3H2y revealed that the Mg:Ti ratio changed continuously in

the two-phase coexistence region. Experimental measurements implied a nano-structured alloy with Ti-poor and Ti-rich regions in which Mg and Ti atoms were not randomly distributed. In addition, Srinivasan et al.39 _{investigated Mg}

0.65Ti0.35Dx prepared by ball

milling and gas-phase deuterium absorption. By combined use of XRD, neutron diﬀrac-tion, and magic-angle-spinning H nuclear magnetic resonance(NMR), partial segregation structures with rutile MgD2 domain and inter-dispersed TiDy and a separated fcc TiDz

phases were observed. By continuously monitoring the structure during hydrogen uptake, Borsa et al.32 _{have obtained data that are compatible with a coherent structure. They}

proposed that the average structure resembled rutile MgH2 at high Mg content and was

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systems by combining XRD and Extended X-ray Absorption Fine Structure (EXAFS) spectroscopy. Despite the positive enthalpy of mixing of Mg and Ti, they found that the degree of ordering did not vary upon loading and unloading with hydrogen. The robustness of this system and the fast and reversible kinetics of (de)hydrogenation were attributed to the formation of nanoscale compositional modulations in the intermetallic alloy.

Beside the binary Mg-based alloys, ternary Mg-based alloys with Fe, Al, Ti as alloy-ing elements were also studied by Kaliswaart and his coworkers.41 _{The binary Mg}

70Al30

and Mg75Ti25 alloys showed degradation of the kinetics over time because of alloying of

Pd catalyst layers or decomposition of a metastable ternary hydride phase. The ternary alloys showed superior performance over the binary ones. Kinetically, the Mg70Al15Ti15

and Mg70Fe15Ti15 showed the best performance, fully desorbing their total capacity of

approximately 4 wt% within 15 min, which remained stable for over 100 cycles. This is very likely due to their multi-phase structure, combining Mg as the main storage phase with Ti-Al or Ti-Fe intermetallic as pathways for enhanced hydrogen diﬀusion and surface catalysts. Similar enhanced hydride sorption kinetics were found in Mg-Cr-Ti system.42 The role of the stable secondary alloy phase as a heterogeneous nucleation site was pro-posed. Because of the signiﬁcant volume and chemical interfacial energy mismatch be-tween hexagonal close packed metallic magnesium and rutile type MgH2, there is a large

nucleation barrier for bulk formation of either one in the other. With many more nano-sized Ti-Cr nucleation sites available, the diﬀusion distances are subsequently reduced and so are the times for full absorption/desorption.

Besides the improved kinetics, Baldi et al.43,44 _{managed to tune the thermodynamics}

of hydrogen absorption in Mg-Ti-Pd system by means of elastic clamping. Their re-sults show that the thermodynamics of hydrogen absorption in Pd-capped Mg ﬁlms are strongly dependent on the magnesium thickness.43 _{This dependency can be suppressed}

by inserting a thin Ti layer between Mg and Pd. Furthermore, Mg/Ti multilayers with various monolayer thicknesses between 0.5 and 20 nm were prepared. The layer thickness dependence of hydrogenation properties was again reported.44 _{Beside the elastic}

clamp-ing effect of the Pd layer, the interface effect was also proposed to be responsible for the different thermodynamics. The fcc structures of the MgH2 layer was identified in the 0.5

nm multilayers, which indicates the structural eﬀect may contribute to the tuned ther-modynamics. Instead of preparing Mg-Ti alloys, nano-structured Mg-Ti-H from mixing MgH2 and TiH2 by ultrahigh-energy-high-pressure mechanical milling have been prepared

by Lu et al.45 _{They proposed that TiH}

2 is uniformly distributed in MgH2, and not only

the kinetics but also the thermodynamics are changing with diﬀerent MgH2/TiH2 ratios

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1.3 This thesis 7

1.3 This thesis

As summarized above, a systematic investigation of Mg-transition metal (TM) based hydrogen storage materials with regard to the thermodynamic, kinetics, geometric as well as electronic structures is desired. By using density functional theory (DFT), this thesis describes an extensive theoretical study on tuning the thermodynamics and kinetics of the MgH2 by means of alloying Mg with a secondary TM. The studied TMs include Sc,

Ti, V, Cr, Y, Zr, Nb, La and Hf with a main focus on Ti. Besides changing the transition metals, diﬀerent dimensions and geometries of the materials, i.e., pure bulk Mg and Ti, bulk Mg-TM alloys, 2D Mg/TM multilayers, and 3D Ti(core)/Mg(shell) structures are studied, respectively.

Chapter 2 gives a brief physical background of computational tools (DFT) and

tech-niques (density of states, charge analysis, nudged elastic band) used in this work.

Chapter 3 deals with hydrogen diﬀusion barriers in MgH2 with three diﬀerent

struc-tures: equilibrium rutile, rutile with deformation twins and fluorite polymorph. The hy-drogen diffusion barriers are calculated and compared. The mechanisms of the improved kinetics in the deformation twins and fluorite are analyzed. The results demonstrate how the hydrogen kinetics is altered by the structural control of the hydrides.

After the kinetic study in Chapter 3, the following chapters mainly focus on thermo-dynamic properties. Before the Mg-TM alloys are studied, the hydrogen storage properties in pure MgH2 and TiH2 are studied and compared in Chapter 4. Absorption energies as

a function of the hydrogen concentration in metal hosts with diﬀerent structures fcc, hcp, and bct for Mg, hcp and fcc for Ti were calculated. The diﬀerences in the hydrogenation behavior of Mg and Ti are compared and analyzed using the electron density.

Chapters 5 and 6 focus on the bulk Mg-TM-H systems. Chapter 5 deals with

structural transformation from rutile to ﬂuorite of MgxTM1−xH2 (TM=Sc, Ti, Zr, Hf).

The stability of two crystallographic modiﬁcations of MgxTM1−xH2, i.e., rutile and ﬂuorite

is studied. Beyond a certain transition metal content, the rutile structure of pure MgH2

is no longer stable, and the hydride transforms into a ﬂuorite-type structure, similar to that of the transition metal dihydride. The transition point for both Mg-Sc and Mg-Ti hydrides is estimated and compared with experimental studies. For Mg-Zr and Mg-Hf, no experimental data are available for comparison, so the calculations have a predictive value for these systems.

In Chapter 6 hydrogen storage properties in ﬂuorite Mg0.75Ti0.25H2 is further

stud-ied since Mg-Ti system is particular interesting as the excellent kinetics, high hydrogen capacity and the low cost compared to Sc. Four types of structures of Mg0.75Ti0.25 alloys

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The formation energies of the alloys and hydrides show that segregation and dispersion are thermodynamically favored for alloys and hydrides, respectively. Therefore hydrogen induced structural rearrangement of the intermetallic structures of the Mg0.75Ti0.25might

occur upon hydrogen cycling. For the non-homogenous Mg-Ti-H system, further phase segregation of Ti in Mg might occur. This prediction is in agreement with experimental ﬁnding that Mg0.75Ti0.25 alloys showed degradation of the kinetics over time. Partial

de-hydrogenation with some hydrogen remaining in the Ti-rich region is proposed to improve the reversibility.

Chapters 7 and 8 deal with the hydrogen storage properties in nano-sized MgH2/TMH2

(TM=Sc, Ti, V, Cr, Y, Zr, Nb, La, Hf) multilayers. In Chapter 7 MgH2/TiH2

multilay-ers with various Mg:Ti ratios and Mg thicknesses are studied. In MgH2/TiH2 multilayers,

the ﬂuorite TiH2develops an epitaxial contact with MgH2. Three types of hydrogen atoms

can be distinguished, hydrogen in the Mg phase, in the Mg/Ti interface and in the Ti phase. The structural transformation from rutile to ﬂuorite in the MgH2 phase observed

in Chapter 5 is again found in the multilayered structures. The hydrogen desorption energies in the MgH2 phase are calculated to be much less than that of rutile MgH2.

To futher conﬁrm the origins of the tuned thermodynamics, nine TMs instead of only Ti, i.e., MgH2/TMH2(TM=Sc, Ti, V, Cr, Y, Zr, Nb, La, Hf) multilayers are studied in

Chapter 8. Models of MgH2(rutile)/TiH2(fluorite) and MgH2(fluorite)/TiH2(fluorite)

multilayers with different Mg:TM ratios have been designed and studied. With a fixed thickness of the TMH2 layer, structure transformation of MgH2 from rutile to fluorite

occurs with decreasing thickness of the MgH2 layer. The structural transformation points

and the tuned hydrogen desorption energies are calculated and compared for the nine transition metals. The origins of the destabilization of the Mg-H bonding is conﬁrmed. The results provide an important insight for the development of new hydrogen storage materials with desirable thermodynamic properties. However, the disadvantage of the MgH2/TMH2 multilayers is the largely reduced hydrogen capacity not only due to the

introduction of the heavy transition metals but also the trapped hydrogen in the transition metals and the Mg/TM interfaces.

In Chapter 9, by extending 2D MgH2/TMH2 multilayers system described in

Chap-ters 7 and 8 to a 3D TMH2(core)/MgH2(shell), we hope to obtain larger hydrogen

ca-pacity from the Mg hydride phase. Nano-sized structures: Mg/Ti/Mg sandwich and Ti(core)/Mg(shell) with three diﬀerent Mg:Ti ratios are studied. Besides the three types of hydrogen species in the Mg/Ti multilayers (hydrogen in the Mg phase, hydrogen in the Ti phase, hydrogen at the Mg/Ti interface), hydrogen on the Mg surface can be found in the Ti(core)/Mg(shell) structures. The (de)hydrogenation reactions, thermodynam-ics, structures as well as the reversible hydrogen capacities are studied and compared to 2D MgH2/TiH2 multilayers. Similar to the multilayers, the destabilization of the Mg-H

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BIBLIOGRAPHY 9

the hydrogen on the Mg surface are observed. Based on the diﬀerent thermodynamics of the four types of hydrogen atoms, solutions are proposed to make use of the potential useful Ti(core)/Mg(shell) materials. The reversible hydrogen capacity is calculated to be reasonably high.

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Chapter 2 Methodology

2.1 Introduction

Density functional theory (DFT) is based on the principle that all the properties of a system of interacting particles can be described by a unique functional of the ground state single particle density (r), which provides all information embedded in the many body wave function of the ground state of the system. DFT is among the most popular and versatile methods available in condensed-matter physics, computational physics, and computational chemistry. In the ﬁrst part of the chapter, before presenting details of DFT, a brief discussion on quantum chemistry is presented. It starts with a short introduction of the Schr¨odinger equation, which is followed by the Born-Oppenheimer Approximation. After that DFT, Kohn-Sham equations, exchange and correlation functionals as well as the VASP code used to perform the research in the thesis are discussed, respectively. The second part of the chapter includes the details of approaches used in the VASP to analyze the electronic (density of states and charge analysis) and kinetic (Nudged Elastic Band) properties of the studied systems.

2.2 Theory

2.2.1 Schr¨

odinger equation

In physics, specifically quantum mechanics, the Schrödinger equation, formulated in 1926 by Austrian physicist Erwin Schrödinger, is an equation that describes how the quantum state of a physical system changes in time. It is as central to quantum mechanics as

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Newton’s laws are to classical mechanics. In the standard interpretation of quantum mechanics, the quantum state, also called a wavefunction or state vector, is the most complete description that can be given to a physical system. Solutions to Schr¨odinger equation describe not only molecular, atomic and subatomic systems, but also macroscopic systems, possibly even the whole universe.1

The most general form is the time-dependent Schrödinger equation, which gives a description of a system evolving with time. For systems in a stationary state, the time-independent Schrödinger equation is sufficient. Approximate solutions to the time-independent Schrödinger equation are commonly used to calculate the energy levels and other properties of atoms and molecules.

The many body form of the time independent Schr¨odinger equation consisting of M nuclei and N electrons is written as:

ˆ

HΨi( ⃗r1, ..., ⃗rN, ⃗R1, ..., ⃗RM) = EΨi( ⃗r1, ..., ⃗rN, ⃗R1, ..., ⃗RM), (2.1)

where ˆH denotes the Hamiltonian, and Ψi denotes the wave function in terms of all the

electronic (r) and nuclear coordinates (R). The Hamiltonian ˆH in Eq. 2.1 which represents

total energy consisting of diﬀerent kinetic and potential energy terms can be written as:

ˆ H =−1 2 N ∑ i=1 ∇2 i − 1 2 M ∑ A=1 1 MA ∇2 A− N ∑ i=1 M ∑ A=1 ZA riA + N ∑ i=1 N ∑ j>i 1 rij + M ∑ A=1 M ∑ B>A ZAZB RAB . (2.2)

A and B represent M nuclei, i, j denote the N electrons in the system and ∇2 is the Laplacian operator. The first two terms in Eq. 2.2 are the kinetic energies for the electrons and the nuclei, respectively. The remaining three terms constitute the potential energy part of the total energy and represent coulombic interactions in order: electron-nuclei attractions, electron-electron and the nucleus-nucleus repulsions. In principle, the above equation is applicable to any physical system regardless of size (i.e., from atoms to a whole crystal). In practice, however, a complete analytical solution is impossible for most real life systems, since the number of variables to deal with is determined by the 3(M +N ) degrees of freedom. If we consider the benzene molecule, then there are in total 12 atomic nuclei and 42 electrons. The time independent Schrödinger equation for a single benzene molecule consequently becomes a partial differential eigenvalue problem in 162 variables. Since the exact solution of the Schrödinger equation is formidable to solve for almost every real system, various approximations have to be made.

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2.2 Theory 15

2.2.2 The Born-Oppenheimer approximation

As shown in last section, the computation of the energy and wavefunction of an average-size molecule is a formidable task which has to be alleviated by approximations. The Born-Oppenheimer (BO) approximation2_{permits us to separate the electronic and nuclear}

motions. The justiﬁcation for this is the diﬀerence between masses of nuclei and electrons. Because nuclei are much heavier than electrons, their coordinates are assumed to evolve classically. In basic terms, it allows the wavefunction of a molecule to be broken into its electronic and nuclear (vibrational, rotational) components. After exclusion of nuclear terms, Eq. 2.2 takes the form:

ˆ Helec =− 1 2 N ∑ i=1 ∇2 i − N ∑ i=1 M ∑ A=1 ZA riA + N ∑ i=1 N ∑ j>i 1 rij . (2.3)

The solution of the Eq. 2.1 with the electronic Hamiltonian of Eq. 2.3 above gives the electronic wave function Ψelecand the electronic energy Eelecfor a particular conﬁguration

of the M nuclei. Therefore, the electronic Schr¨odinger equation is reduced to

ˆ

HelecΨelec = EelecΨelec. (2.4)

The total energy of the system is obtained by including the constant nuclear repulsion term of the complete Hamiltonian as Eq. 2.2:

Etotal = Eelec+ Enuc, (2.5)

where Enuc= M ∑ A=1 M ∑ B>A ZAZB RAB . (2.6)

Even after this simpliﬁcation, the task of solving the electronic Schr¨odinger wave equa-tion remains challenging.

2.2.3 Density functional theory

Decoupling the Schr¨odinger equation into electronic and nuclear components by the Born-Oppenheimer Approximation simplify the problem. However, the electronic problem is

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still too complicated to be solved exactly due to the interactions between the electrons. Instead of using the 3N -dimensional Schr¨odinger equation for the many electron wave function, Ψi(r1; r2; ...; rN), Density Functional Theory (DFT) reduces the problem into a

series of coupled one-body problems. This is accomplished by using the electronic density distribution, n(r), and a universal functional of the density Exc[n(r)].

2.2.4 Kohn-Sham equations

Although DFT has its conceptual roots in the Thomas-Fermi model, DFT was put on a ﬁrm theoretical footing by the two Hohenberg-Kohn theorems (H-K).3 _{The original H-K}

theorems held only for non-degenerate ground states in the absence of a magnetic ﬁeld, although they have since been generalized to encompass these.4,5 _{The Hohenberg-Kohn}

theorem consist of two main parts:

Theorem I: In a system of N interacting particles under the inﬂuence of an external po-tential Vex(r), the ground state electron density, n(r), uniquely determines the potential,

Vex(r). Since n(r) determines Vex(r), then the full ground state Hamiltonian is known.

Consequently, n(r) completely determines all the properties of a physical system, such as the eigenfunctions, Ψi(r1; r2; ...; rN) and the eigenvalues, Ei. In other words, the ﬁrst

theorem states that the energy is a unique functional of the electron density:

E[n(r)] = F [n(r)] +

∫

Vex(r)n(r)d3r; (2.7)

Theorem II: There exists a functional F [n(r)] for the ground state energy, for any given

Vex(r). The global minimum of this functional deﬁnes the exact ground state energy of

the physical system, and the density that minimizes the total energy is the exact ground state density, n0(r). The second theorem states that the ground state electron density,

n0(r), minimizes the universal functional, F , such that the electronic energy reaches its

minimum value of E[n0(r)]:

E[n0(r)] = F [n0(r)] +

∫

Vex(r)n0(r)d3r. (2.8)

It should be noted that F [n(r)] depends purely on the electron density. It does not depend on the external potential, Vex(r).

2.2.5 Exchange and correlation functional

Density functional theory reduces the quantum mechanical groundstate many-electron problem to self-consistent one-electron form, through the Kohn-Sham equations. The only

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2.2 Theory 17

undetermined component of the equation is the exchange-correlation energy functional,

Exc = Ex + Ec. This exchange-correlation energy as a functional of the density must be

approximated. In doing so, the local (spin) density approximation (LDA) has long been the standard choice6 and it is written as:

E_xcLDA = ∫

n(r)εxc[n(r)]d3r. (2.9)

Although simple, the LDA results in a realistic description of the atomic structure, elastic, and vibrational properties for a wide range of systems. Yet the LDA is generally not accurate enough to describe the energetics of chemical reactions (heats of reaction and activation energy barriers), leading to an overestimate of the binding energies of molecules and solids in particular.

In the late 90’s, Generalized gradient approximation (GGA) have overcome such de-ﬁciencies to a considerable extent,7–9 giving for instance a more realistic description of energy barriers in the dissociative adsorption of hydrogen on metal and semiconductor surfaces.10,11 _{Gradient corrected or GGA functionals depend on the local density as well}

as on the spatial variation of the density:

E_xcGGA = ∫

n(r)εxc[n(r)]Fxc[n(r),∇n(r)]d3r. (2.10)

Fxc is dimensionless and numerous forms of it exist in literature. The most widely used

forms in solid state physics are proposed by Perdew and Wang (PW91)12 _{and Perdew,}

Burke, and Ernzerhof (PBE)7_{. The PW91 is always used in the calculations in the thesis.}

Compared to LDA, GGA improves the binding energies, especially when there is chem-ical bonding between the atoms. The description of the geometries, however, is not uni-versally better. Similar to LDA, screening of exchange hole is not fully taken into account in GGA. Consequently, GGA cannot account for noticeable improvements to the band gap problem or to the calculation of the dielectric constants. The chemical accuracy of the calculated energies is still too limited to achieve the desired chemical accuracy better than 1 kcal/mol or 50 meV/atom in general.

2.2.6 The VASP program

All the DFT calculations presented in the thesis were performed using the Vienna Ab initio Simulation Package (VASP). This code was developed by Georg Kresse, Jurgen Furthmuller and their collaborators in the University of Vienna.13,14 _{This DFT based}

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an approximate solution to the many-body Schrödinger equation, either within density functional theory (DFT), solving the Kohn-Sham equations, or within the Hartree-Fock (HF) approximation, solving the Roothaan equations. Hybrid functionals that mix the Hartree-Fock approach with density functional theory are implemented as well. Fur-thermore, Green’s functions methods (GW quasiparticles, and ACFDT-RPA) and many-body perturbation theory (2nd-order Møller-Plesset) are available in VASP. In VASP, central quantities, like the one-electron orbitals, the electronic charge density, and the local potential are expressed in plane wave basis sets. The interactions between the electrons and ions are described using norm-conserving or ultrasoft pseudopotentials, or the projector-augmented-wave method. To determine the electronic groundstate, VASP makes use of efficient iterative matrix diagonalisation techniques, like the residual min-imisation method with direct inversion of the iterative subspace (RMM-DIIS) or blocked Davidson algorithms. These are coupled to highly efficient Broyden and Pulay den-sity mixing schemes to speed up the self-consistency cycle. A detailed description of all the methods used in the VASP program can be found at the VASP manual homepage: http://cms.mpi.univie.ac.at/vasp/vasp/vasp.html.

2.3 Approaches used to analyze calculated results

2.3.1 Density of states

In solid-state and condensed matter physics, the density of states (DOS) of a system describes the number of states per interval of energy. Unlike isolated systems, like atoms or molecules in gas phase, the density distributions are not discrete like a spectral density but continuous. DOS delivers invaluable electronic structural information about the bonding within solids and in classiﬁcation of materials as metallic, semiconductor or insulators. A high DOS at a speciﬁc energy level means that there are many states available for occupation. A DOS of zero means that no states can be occupied at that energy level. Metals or semi metals have non-localized electrons and no separation between valence and conduction band. Materials with a large gap larger than 4 eV are called insulators, whereas systems with a smaller gap are categorized as semiconductors. A large majority of the electronic structures and band plots are calculated using DFT. Although the band gap in insulators and semiconductors are underestimated by about 30-40%, the shape of the plots of DOS can be well reproduced compared with the experimental measurements.

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2.3 Approaches used to analyze calculated results 19

2.3.2 Charge analysis

The charge density describes the distribution of negative charge in real space, it is a physically measurable quantity. The linear, surface, or volume charge density is the amount of electric charge in a line, surface, or volume respectively. When used as a basis for the discussion of chemistry, the charge density allows for a direct physical picture and interpretation. In particular, the forces exerted on a nucleus in a molecule by the other nuclei and by the electronic charge density may be rigorously calculated and interpreted in terms of classical electrostatics. Thus, given the molecular charge distribution, the stability of a chemical bond may be discussed in terms of the electrostatic requirement of achieving a zero force on the nuclei in the molecule. In covalent bonding the valence charge density is distributed over the whole molecule and the attractive forces responsible for binding the nuclei are exerted by the charge density equally shared between them in the internuclear region. In ionic bonding, the valence charge density is localized in the region of a single nucleus and in this extreme of binding the charge density localized on a single nucleus exerts the attractive force which binds both nuclei.

2.3.3 Nudged Elastic Band

The nudged elastic band (NEB)15–17 _{is a method for ﬁnding saddle points and minimum}

energy paths between known reactants and products. The NEB method is used to find reaction pathways when both the initial and final states are known. Using this code, the Minimum Energy Path (MEP) for any given chemical process may be calculated, however both the initial and final states must be known. The code works by linearly interpolating a set of images between the known initial and final states (as a ”guess” at the MEP), and then minimizes the energy of this string of images. Each ”image” corresponds to a specific geometry of the atoms on their way from the initial to the final state, a snapshot along the reaction path. Thus, once the enery of this string of images has been minimized, the true MEP is revealed.

A new tangent definition and a climbing image method18combine to allow for the more accurate saddle points using the NEB with fewer images than the original method. With this code, it is possible to calculate not only the MEP for the reaction but also the tran-sition state configuration at the saddle point. The climbing image NEB method has one modification which drives the image with the highest energy up to the saddle point. This image does not see the spring forces along the band. Instead, the true force at this image along the tangent is inverted. In this way, the image tries to maximize it’s energy along the band, and minimize in all other directions. When this image converges, it will be at the ex-act saddle point. More information about the implement of the climbing NEB in the VASP code can be found at the following website: http://theory.cm.utexas.edu/vtsttools/neb/.

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Bibliography

[1] E. Schr¨odinger, Phys. Rev. 28, 1049(1926).

[2] M. Born and J. R. Oppenheimer: Zur Quantentheorie der Molekeln. Ann. Phys. 84, 457(1927) .

[3] P. Hohenberg and W. Kohn, Phys. Rev. 136, B864(1964). [4] L. H. Thomas, Proc. Cambridge Philos. Soc. 23, 542(1927).

[5] E. Fermi, Z. Phys. 48, 37(1928).

[6] R.O. Jone and O. Gunnarsson, Rev. Mod. Phys. 61, 689(1989).

[7] J. P. Perdew, K. Burke and M. Ernzerhof, Phys. Rev. Lett. 77, 3865(1996); 78, 1396(1997).

[8] J. P. Perdew, K. Burke and Y. Wang, Phys. Rev. B 54, 16533(1996). [9] A. D. Becke, J. Chem. Phys. 102, 8554 (1997).

[10] B. Hammer, M. Scheﬄer, K. W. Jacobsen and J. K. Nørskov, Phys. Rev. B 73, 1400(1994).

[11] E. Penev, P. Kratzer and M. Scheﬄer, J. Chem. Phys. 110, 3986 (1999).

[12] J. P. Perdew, J. A. Chevary, S. H. Vosko, K. A. Jackson , M. R. Pederson, D. J. Singh and C. Fiolhais, Phys. Rev. B 46, 6671(1992).

[13] G. Kresse and J. Furthm¨uller, Phys. Rev. B 54, 11169(1996).

[14] G. Kresse and J. Furthm¨uller, J. Comp. Mat. Sci. 6, 15(1996). [15] G. Henkelman and H. J´onsson, J. Chem. Phys. 113, 9978(2000).

[16] G. Henkelman and H. J´onsson, J. Chem. Phys. 113, 9901(2000). [17] H. Jonsson, Ann. Rev. Phys. Chem. 51, 623(2000).

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Chapter 3 Hydrogen diﬀusion in Mg hydride

ABSTRACT

The transport properties of hydrogen are crucial to the kinetics of hydrogen storage in MgH2. We use ﬁrst principle calculations to identify the hydrogen

diﬀusion paths and barriers in three diﬀerent MgH2 structures: equilibrium

rutile, rutile with ball-milling-induced deformation twins and fluorite poly-morph. We observed that both hydrogen diffusion barriers in deformation twins and fluorite structure are lower compared to that in the equilibrium rutile. This is because the hydrogen diffusion is facilitated by new interstitial sites in the Mg lattice: a new hexahedral site formed by the reconstruction of Mg lattice at the twinning interface in the deformation twins and the oc-tahedral sites in the fluorite structure. Furthermore, the hydrogen vacancy density effect on the diffusion barrier was estimated. For all three types of Mg hydrides the general trend is the higher the density of hydrogen vacancies, the lower the hydrogen diffusion barrier. Our results demonstrate how the hydrogen kinetics is altered by the structural control of the hydrides.

3.1 Introduction

MgH2 is a promising hydrogen storage material due to its low cost, availability and

non-toxicity. Furthermore, MgH2 has a high gravimetric capacity, up to 7.6 wt%. 1–3However,

MgH2 is too stable4 and the kinetics of the hydrogen uptake and release in Mg are poor.

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tetragonal TiO2-rutile-type structure, i.e. P42/mmm. It is well known that the activation

energy for hydrogen desorption from pure rutile-MgH2 is relatively high. It is generally

accepted that the diﬀusion limitations and high oxidation sensitivity of the Mg surface leads to poor hydrogen sorption kinetics. The experimental values reported in literature are in the range of 120-200 kJ/mol.5–9

High-energy ball milling of MgH2 is known to improve the hydride desorption kinetics

signiﬁcantly. This eﬀect has been attributed to various characteristics of the mechan-ically processed powder. Among the proposed explanations for this phenomenon are reduction in particle and grain size,10–13 _{the formation of a sub-stoichiometric hydride}

phase,14,15 milling-induced dislocations16,17 and deformation twins.8 For instance, using transmission electron microscopy Danaie et al.8 _{observed deformation twins with}

twin-ning planes (101) of the hydride in the high energy mechanical milled samples. Significant milling-induced kinetic enhancement, i.e. a reduction in activation energy from 196 to 127 kJ/mol was measured. It is proposed that the deformation twins contribute significantly to the observed milling-induced kinetic enhancement by acting as high diffusivity paths for hydrogen.

Another focus of the attempts to improve the kinetics is alloying Mg with a secondary transition metal (TM). It has been proven that the role of the TM at facilitating the hydro-gen ab/desorption kinetics, is either acting as a catalyst17–22_{or changing the crystal}

struc-tures of the hydrides.23–27 _{For the Mg}

xTM1−xHy system, electrochemical measurements

revealed that inserting and extracting hydrogen is greatly facilitated when containing more than 20 at.% of TM=Sc, Ti, V, Cr.23–27_{The ﬂuorite structure of the Mg}

xTM1−xHy system

has been proposed to be responsible for the improved kinetics, whereas the rutile structure of unalloyed Mg hydride strongly inhibited hydrogen transport.25 This structural trans-formation from rutile to ﬂuorite with the increasing proportion of TM in MgxTM1−xHy

was also conﬁrmed by our earlier theoretical calculations.28,29

It should be stressed that the experimental measurements of the hydrogen desorption activation barrier reﬂect an overall value of many microscopic processes involved in the desorption process (H2 dissociation, surface desorption, hydrogen diﬀusion in the bulk,

phase nucleation and growth, etc.). Some of the single processes, such as H2

dissocia-tion and reassociadissocia-tion on the Mg surface, hydrogen ab/desorpdissocia-tion from the surface have been studied theoretically. For instance, the activation energy for desorption of hydrogen molecule on the MgH2(110) surface was calculated to be 172 kJ/mol, which is proposed to

be likely the rate-limiting step in the whole hydrogenation process.30 _{However, activation}

energies of 14831 and 160 kJ/mol32 were also recently reported to correspond to hydro-gen diﬀusion through the rutile MgH2 phase, being the rate-limiting step in desorption.

Moreover, recent modeling work also indicates that, because of the interdependence of hydrogen diﬀusion and dissociation mechanisms, a single rate-limiting step may not be valid.33

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3.2 Computational methods 23

Despite numerous published experimental studies on the subject, the mechanism of the improved kinetics, and particularly the structural eﬀect of the hydride on the kinetics is not yet fully understood. Inspired by the observed deformation twins formed by high-energy milling8 and the ﬂuorite structure of the MgxTM1−xHy,23–25 we propose there is

a close relation between the structures of the hydride and the hydrogen ab/desorption kinetics. Therefore, the goal of this chapter is to elucidate the structural effect by calcu-lating the hydrogen diffusion barriers in different structures of the MgH2 by first principles

density functional theory (DFT). Three different structures including equilibrium rutile, ball-milling-induced deformation twins in rutile and fluorite polymorph are studied. By changing the size of the unit cell of the model, the effect of the density of vacancies on the activation barrier is also estimated.

3.2 Computational methods

All calculations were performed using DFT as implemented in the Vienna Ab-Initio Sim-ulation Package (VASP).34,35 _{The Kohn-Sham equations were solved using a basis of}

Projector Augmented Wave-functions with a plane-wave energy cut-oﬀ of 300 eV,36 _and

using pseudo potentials37 to describe the core electrons. Energy cut-off of 300 eV and 400 eV show a difference in formation energies only about 0.02%. Therefore 300 eV is sufficient. The Perdew-Wang 1991 generalized gradient approximation was used for the electron-exchange correlation potential.38 A total of 13×13×13 k points were used to model the Brillouin zone for the four metal atom cell. For larger cells k points were scaled down proportionally, e.g. for a lattice parameter of double length, only half number of

k points were required. Therefore for sixteen metal atom cells, 13×7×7, 13×13×3, and

7×13×7 k points were used for rutile, deformation twins and ﬂuorite, respectively. For all structures the lattice parameters, the volume and the atom positions were allowed to relax prior to the calculation of formation energies and subsequent diﬀusion pathway calculations. The calculation of atomic hydrogen and molecule H2 has been done using a

cubic supercell with size 10×10×10 ˚A3. The bond length is predicted as 0.746 ˚A and the binding energy as 461 kJ/mol H2. The agreement with the experimental data (0.741 ˚A

and 456 kJ/mol H2) is satisfactory. Additionally, we emphasize that the contribution of

zero point energy corrections (ZPE) has been estimated to be 14 kJ/mol for both rutile and deformation twin MgH2.

Minimum energy paths for diffusion were calculated using the nudged elastic band (NEB) method,39–41 with 5 to 9 images forming a discrete representation of the path, including the two fixed endpoints (see Fig. 3.1(c) and (d)). The transition states have been confirmed by the saddle points obtained from the frequency calculations. In order to estimate the hydrogen diffusion barrier in the magnesium hydride, we considered the

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vacancy mediated mechanism. This means to create one vacancy one hydrogen atom has to be taken out of the hydride. The diﬀusion pathways in this study are the adjacent hydrogen atoms hopping to the manually created vacancy, i.e. the diﬀusion path ac in Fig. 3.1(a), where the hydrogen at site c was taken out and the hydrogen atom hops from the site a to the vacancy at site c.

3.3 Results

Figure 3.1: (a) Optimized rutile Mg4H8; (b) Mg16H32 (c) Selected three conﬁgurations of one

hydrogen atom along the NEB path from site a to site c; (d) The minimum energy path for a diﬀusion hop of a hydrogen atom from site a to site c.

3.3.1 Equilibrium rutile

The optimized rutile structure and structural data with a four metal atom cell are shown in Fig. 3.1(a) and Table 3.1. In rutile MgH2 the hydrogen atoms are arranged approximately

octahedrally around the magnesium atoms, which in turn are arranged trigonally around the hydrogen atoms. The calculated formation energy of rutile MgH2 is -77 kJ/mol (with

ZPE correction), which is very close to the decomposition enthalpy (78 kJ/mol) of the unmilled MgH2.8 Three diﬀusion paths, ab, ac, and be (see Fig. 3.1(a)) were calculated.

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3.3 Results 25

The diﬀusion barriers are 86, 58 and 66 kJ/mol, respectively. Structural information of transition state is summarized in Table 3.1. The transition states of the paths ab and

be are located in two diﬀerent octahedral sites. The minimum energy path ac and a

graphical representation of the lowest energy diﬀusion path, i.e., the hydrogen hopping between a and c sites, are shown in Fig. 3.1(c) and (d), respectively. The hydrogen atom migrates at a minimum energy cost through a curve along the free space between a and c sites and the transition state is located in a distorted trigonal site. The diﬀusion barrier of the minimum energy path (ac) in rutile MgH2 is 58 kJ/mol (shown Fig. 3.1(a)).

Figure 3.2: Multiple deformation twins. In order to highlight the twin structures (the black

lines), supercells are shown. (a) Mg4H8 with 100% deformation twins (2×2×2

supercell); (b) Mg16H32 with 25% deformation twins (2×2×1 supercell); (c)

Schematic of the twin geometry looking down [0 1 0]; the numbers show the depth of the atoms with reference to (0 1 0) plane; (d) Hydrogen diﬀusion paths.

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3.3.2 Rutile with deformation twins

For the deformation twins, two diﬀerent unit cells are considered to model diﬀerent den-sities of the multiple deformation twins. Fig. 3.2(a) and (b) are with 100% and 25% deformation twins in the cells, respectively. The detailed twin geometry and the positions of the hydrogen atoms are shown in Fig. 3.2(c). According to the calculated hydride for-mation energies (-77 kJ/mol) both of the two structures are from a thermodynamic point view as stable as the rutile MgH2. This indicates that the formation of the deformation

twins is not likely responsible for the observed minor destabilization (3 kJ/mol) in the milled sample described in Ref. 8. The destabilization probably comes from a mechanism proposed by Berube et al.,42consisting of high energy milling inducing localized regions in the hydride with excess volume. In the four metal atom unit cell with 100% deformation twins, three hydrogen hopping paths were considered: ab, ac, cd shown in Fig. 3.2(a). The hydrogen diﬀusion barriers are 29 (path ab), 37 (path ad ), and 87 kJ/mol (path cd ). The structural data of transition states are summarized in Table 3.1. For the path ad, the transition state is located in a distorted tetrahedral site in a tetrahedron which is formed by the rearrangement of the Mg atoms at the twinning interface. For the path

cd, the transition state is located at the trigonal site. For path ab, the transition state is

located in a distorted hexahedral site in a trigonal bipyramid structure which is formed by two tetrahedrons sharing the twinning interface (see Fig. 3.2(b)). The diﬀusion barrier of minimum energy path (ab) in the deformation twinning structure is 29 kJ/mol and it is 29 kJ/mol less than that of the rutile MgH2. Therefore, the lower energy path way at the

twinning interfaces is conﬁrmed to be an important factor contributing to the enhanced hydrogen kinetics in the milled powder sample.

Figure 3.3: Structures of the ﬂuorite (a) Mg4H8, where ab, ac, and ad are the hydrogen

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3.3 Results 27

3.3.3 Fluorite polymorph

Fluorite polymorph MgH2 is a meta-stable phase which can only be observed under

ex-treme high pressure43 _{or when alloyed with TMs.}24–27 _{In the ﬂuorite hydride, the Mg}

atoms have a fcc symmetry and the hydrogen atoms occupy tetrahedral sites. Three hop-ping paths in the four metal unit cell were considered: ab, ac, and ad (see Fig. 3.3(a)). The transition states of path ab and ad are both located in octahedral sites with hydrogen hopping barriers of 20 and 72 kJ/mol, respectively. No local minimum has been found for the path ac. It is probably because at the halfway of the pathway, the structure is very unstable due to the very small Mg-H bond distance of around 1.2 ˚A. It is very likely that the hydrogen hopping follows a curved path from a to the octahedral site (where the tran-sition state of the path ad is located) to c. Therefore, the same diffusion barrier as that of path ad is expected for ac. The diffusion barrier of the minimum energy path (ab) in fluorite MgH2 is 20 kJ/mol, which is 38 kJ/mol lower than that of the rutile MgH2. This

probably explains the signiﬁcant enhancement of the hydrogen sorption kinetics observed experimentally in ﬂuorite MgxTM1−xHy systems.24–26

3.3.4 The eﬀect of the vacancy density on the diﬀusion barrier

To test the eﬀect of the vacancy density on the diﬀusion barriers, cells with sixteen metal atoms were also calculated and compared (see Table 3.1) with the four metal unit cell for the three structures. In this case, when one hydrogen vacancy is created, the density of the vacancy will be 1/16 instead of 1/4 in the four metal atom cell. The minimum energy pathways remain the same as those in the four metal atom cells: ac, ab,

ab for rutile, deformation twins and ﬂuorite, respectively. Moreover, the diﬀusion barriers

in deformation twins and fluorite are still lower than that in the rutile. However, the hydrogen diffusion barriers are much larger than in the four metal unit cells: 124 vs 58, 68 vs 29, 65 vs 20. The diffusion barriers in the hydrides with high vacancy densities are low mainly because the atoms can find a position with the optimum structure, where the Mg metal lattice also has more space to adjust themselves to form more stable Mg-H bonds. This indicates that the vacancy density has a significant effect on the hydrogen diffusion barrier. The general trend is that the higher the vacancy density, the lower the diffusion barrier. In fact, very low activation energy of 13 kJ/mol for complete deuterium exchange among NMR-distinct sites were found in Mg0.65Ti0.35D1.1,44where the hydrogen

vacancy concentration is extremely high.

It should be pointed out that the above diﬀusion calculations were all done with one existing hydrogen vacancy. On one hand, it is conceivable that vacancies and more com-plex aggregates of vacancies should occur in the milled and alloyed materials. These defects should contribute to the hydrogen mobility. On the other hand, for the unmilled

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Table 3.1: The structures of the three types of hydrides, hydrogen diﬀusion barrier (in kJ/mol)

and the corresponding transition states (TS in ˚A) in the hydrides. For the TS locations, the trigon in rutile and hexahedron and tetrahedron in deformation twins are not regular but non-equilateral structures

Hydride Lattice Mg-H Path Barrier Location of TS Structure (Mg-H)

Rutile a=4.45 1.94 ac 58 Trigonal 1.84,2.13,2.58

Mg4H8 c=2.99 1.94 ab 86 Octahedral 1.54,1.54,3.09,3.15,3.15,3.17

1.92 be 66 Octahedral 2.12,2.16,2.61,2.61,2.54,2.54

Rutile a=8.95 1.90 ac 124 Trigonal 2.13,2.13,2.34

Mg16H32 c =6.00 1.93 ab 139 Octahedral 1.42,1.81,3.15,3.15,3.19,3.20

1.98 be 129 Octahedral 1.96,2.33,2.65,2.62,2.55,2.55

Twins a=5.37 1.90 ab 29 Hexahedral 1.90,2.30,2.31,2.42,2.45

Mg4H8 b=4.90 1.95 ad 37 Tetrahedral 1.93,1.82,2.56,2.58

c=4.44 1.98 cd 87 Trigonal 1.98,2.20,2.25

Twins a=5.41 1.94 ab 68 Hexahedral 2.03,2.32,2.40,2.42,2.49

Mg16H32 b=20.14 1.94 ad 93 Tetrahedral 1.69,2.09,2.43,2.80

c=4.46 1.93 cd 141 Trigonal 2.17,2.25,2.53

Fluorite a=4.71 2.04 ab 20 Octahedral 1.77,1.77,2.91,2.91,2.91,2.91

Mg4H8 ac - -

-ad 72 Octahedral 2.36,2.36,2.36,2.36,2.55,2.15

Fluorite a=9.46 2.05 ab 65 Octahedral 1.87,1.87,2.83,2.83,2.99,2.99

Mg16H32 c=4.73 ac - -

-ad 103 Octahedral 2.37,2.37,2.29,2.29,2.28,2.28

sample, hydrogen vacancies may not exist or the density of vacancies is extremely low. Therefore the activation barrier for hydrogen desorption may be a summation of the hy-drogen diffusion barrier and the formation energy of hyhy-drogen vacancies. As simple tests of these ideas, we calculated the formation energies of one hydrogen vacancy and a sec-ond hydrogen vacancy which is the closest to the first one, respectively. As shown in Table 3.2, the formation energies of the first vacancy are both high for both the rutile and the deformation twins. For instance, the formation energy of the first vacancy in the rutile MgH2 within four metal unit cell is as high as 128 kJ/mol. If adding the

forma-tion energy of the vacancy to the hydrogen diﬀusion barrier, the activaforma-tion barrier of the hydrogen desorption can be expected to be 186 kJ/mol (128kJ/mol+58 kJ/mol). It is very close to that of the experimental activation barrier of 196kJ/mol measured for the unmilled MgH2.8 Furthermore, one can notice that except for the ﬂuorite Mg16H32, the

formation energy of the second vacancy is always smaller than the ﬁrst one. Assuming the ﬁrst vacancy already present in the milled sample, the activation barrier of the hydrogen desorption will be the summation of the formation energy of the second hydrogen vacancy

Hydrogen storage in Mg transition metal alloys : a DFT study

Hydrogen storage in Mg transition metal alloys : a DFT study

Hydrogen storage in Mg transition metal

alloys - A DFT study

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan de

Technische Universiteit Eindhoven, op gezag van de

rector magnificus, prof.dr.ir. C.J. van Duijn, voor een

commissie aangewezen door het College voor

Promoties in het openbaar te verdedigen

op maandag 23 mei 2011 om 16.00 uur

door

Shuxia Tao

Contents

Chapter 1

Introduction

1.1

The hydrogen economy and hydrogen storage

1.2

Recent progress on hydrogen storage in Mg based

hydrides

1.3

This thesis

Bibliography

Chapter 2

Methodology

2.1

Introduction

2.2

Theory

2.2.1

Schr¨

odinger equation

2.2.2

The Born-Oppenheimer approximation

2.2.3

Density functional theory

2.2.4

Kohn-Sham equations

2.2.5

Exchange and correlation functional

2.2.6

The VASP program

2.3

Approaches used to analyze calculated results

2.3.1

Density of states

2.3.2

Charge analysis

2.3.3

Nudged Elastic Band

Bibliography

Chapter 3

Hydrogen diﬀusion in Mg hydride

3.1

Introduction

3.2

Computational methods

3.3

Results

3.3.1

Equilibrium rutile

3.3.2

Rutile with deformation twins

3.3.3

Fluorite polymorph

3.3.4

The eﬀect of the vacancy density on the diﬀusion barrier