# Hydrogen Diffusion on, into and in Magnesium Probed by DFT: A Review

## Abstract

**:**

_{2}. Many attempts have been made to overcome these shortcomings. On a microscopic level, hydrogen absorption by metal is a complex multistep process that is impossible to survey experimentally. Theoretical studies help to elucidate this process and focus experimental efforts on the design of new effective Mg-based materials for hydrogen storage. This review reports on the results obtained within a density functional theory approach to studying hydrogen interactions with magnesium surfaces, diffusion on Mg surfaces, into and in bulk Mg, as well as hydrogen induced phase transformations in MgH

_{x}and hydrogen desorption from MgH

_{2}surfaces.

## 1. Introduction

_{2}, high natural abundance, low cost and non-toxicity, is considered one of the most attractive materials for the hydrogen storage application and is already used in practice [6,7]. However, its rather slow hydrogen absorption and desorption kinetics [8] as well as high dissociation temperature (above 673 K) [9,10] and the noticeable reactivity toward air (oxygen) impede the economic efficiency of Mg-based hydrogen storage devices.

_{2}with transition metals (TM) [13,14,15] or their oxides [16,17], nanostructuring of Mg using high energy ball milling [18,19,20] or severe plastic deformation [18,20,21] or ion irradiation [22,23].

_{2}molecule on the metallic surface and stepwise diffusion of atomic hydrogen into and further in bulk metal, hydride phase nucleation and growth, etc. Hydrogen desorption likewise comprises surface desorption, hydrogen atom or vacancy diffusion, phase nucleation and growth, etc. To follow this whole process experimentally is an impossible task. Experiments provide access either to a dynamic process upon (de)hydrogenation, or to the initial and final states of the substance in an equilibrium state. However, even in situ experiments are limited by a dramatic mismatch between hydrogen jump rates responsible for diffusion and the experimentally accessible timescale. Compared to many other atoms in solids, hydrogen in metals exhibits rather fast diffusion [24]. This entails applications of a large variety of experimental methods, including electrochemistry, mechanical relaxation, nuclear magnetic resonance (NMR), quasi-elastic neutron scattering (QENS), perturbed angular correlation, and others [24,25,26].

^{−1}to 10

^{11}s

^{−1}[28], which makes it a powerful tool for studying hydrogen self-diffusion in metals [28,29,30,31]. In addition, in an experiment we survey the statistical ensemble behavior. Being very effective at measuring both the hydrogen diffusivity and jump frequency, experimental methods cannot describe microscopic hydrogen migration processes and steps in the hydride formation, although this knowledge is essential for the development of a strategy for the development of new materials with specified parameters. From this perspective, theoretical calculations are very helpful, especially in combination with experimental studies.

_{x}and hydrogen desorption from MgH

_{2}surfaces. However, first it is necessary to give a short introduction, with a brief description of hydrogen’s interaction with metal and to introduce the basic concepts and approaches used to calculate hydrogen diffusion paths, activation barriers and jump rates within the framework of DFT.

## 2. Interaction of Hydrogen with Metals

_{4}]

^{–}, [AlH

_{4}]

^{–}, or polymeric structures. Magnesium hydride represents an intermediate case: a mixture of covalent and ionic bonding [32,33].

_{x}is a hydride of metal M with hydrogen concentration x = H/M, ΔH is the enthalpy of formation. If at normal conditions the reaction is exothermic, the corresponding hydride is stable, if endothermic, the hydride is unstable. For MgH

_{2}ΔH = −75.7 kJ/mol H

_{2}[34] that determines the high temperature of its hydrogen release.

_{2}molecules to the metal surface under an external pressure; (ii) attraction of H

_{2}molecules to the metal surface due to the van der Waals interaction (physisorption); (iii) dissociation of physisorbed H

_{2}molecules on the metal surface under external impact (temperature and/or pressure); (iv) penetration of H atoms into the host metal lattice (chemisorption) and occupation of interstitial sites in the sub-surface layer; (v) hydrogen diffusion into the metal lattice with formation, first, a solid solution M-H, α-phase (H/M < 0.1) and then a hydride MHx, β-phase. The energy diagram for this multistep process is schematically shown in Figure 2.

_{2}splitting (non-activated hydrogen absorption) [29]. To make H

_{2}dissociate on an Mg surface, an extra portion of energy—the so-called activation energy E

_{act}—has to be supplied. The origin of this activation barrier is mainly determined by electronic factors [30,31,32,33], but the hydrogen behavior is also influenced by an oxide film on the metal surface. In practice, to achieve better hydrogen capacity and kinetics before first hydrogenation, an activation is required. The activation normally involves internal cracking of metal particles to increase reaction surface area [34].

## 3. DFT Method to Study Hydrogen Diffusion

#### 3.1. Potential Energy Surface and Activation Energy

#### 3.2. Hydrogen Jump Rate and Diffusion Coefficient

_{2}are normally above 600 cm

^{−1}[45]. Within this low temperature approximation the hydrogen jump rate can be written as follows [44,46]:

#### 3.3. Hydrogen Solubility and Hydrogen Vacancy Energies

_{x}, the hydrogen diffusion coefficient decreases with hydrogen concentration x: H atoms penetrating into metal lattice occupy more and more interstitial sites making unavailable a part of diffusion paths. The simplest way to account for x is to weight the diffusion coefficient by the probability β that the target position is effectively accessible [49]. This parameter can be easily evaluated for regular lattice of pure metals, but for substituted lattices distribution of hydrogen over possible interstitial sites must be calculated in an accurate way. The distribution of H atoms over interstitials is given by a Fermi–Dirac distribution [50,51], derived in terms of general thermodynamic description:

_{2}the hydrogen solubility energy ${E}_{\mathrm{sol}}$ can be calculated as:

_{2}molecule $E\left({\mathrm{H}}_{2}\right)$ must be calculated using the same theory level, e.g., accounting for ZPE or not (see Section 3.4). Such an approach was successfully applied to study hydrogen distribution and interstitial diffusion in disordered Ti-V-Cr alloys [52,53]. Note that the hydrogen solubility energy in Equation (7) taken with the opposite sign is the energy for a single H-vacancy formation.

#### 3.4. Zero-Point Energy

## 4. DFT Modelling of Mg-H Systems

#### 4.1. Mg-H Bonding and Effect of TM Substitution

_{2}is a complex mixture of ionic and covalent contributions that determines the rather high thermodynamic stability of MgH

_{2}[33,69,70]. Partial substitution of TM for Mg weakens the bonding between magnesium and hydrogen [11,70,71,72,73,74], but the TM-H bonding appears rather strong: TM d-states are strongly hybridized with s-states of the hydrogen atoms [70,74]. Substitution effects are normally accompanied by local distortions that can be modeled within a cluster approach. In general, the main conclusions obtained to study effects of TM doping on MgH

_{2}within the cluster approach [71,73,75] are in line with supercell periodic boundary calculations [11,70,71,72,73,74].

_{7}MH

_{x}and Mg

_{6}MH

_{x}hydrides with M = Ti, V, Nb is lower compared to MgH

_{2}[76,77,78]. However, as soon as Mg does not form any alloys with these TMs, the structures are stabilized by hydrogen; no binary compounds exist after hydrogen release. As it was shown by Shelyapina et al. [79] co-substitution Mg by Ti and Zn or Al leads to stabilization of the alloys with simultaneous decreasing stability of the corresponding hydrides due to the TM–H bond weakening.

#### 4.2. Mg/TM Thin Films and bcc Mg Phase Stability

#### 4.3. Hydrogen Molecule Dissociation on Mg(0001) Surfaces and Migration into the Subsurface

_{2}, or by a highly stable surface hydride film blocking the diffusion of atomic hydrogen into magnesium.

_{2}dissociation with an activation energy of about 1.4 eV and barrier for H atomic diffusion from surface to bulk of 0.15–0.53 eV (depending on the method of calculation and initial and final hydrogen sites). These results suggest that the dissociation and recombination of H

_{2}are the rate-limiting processes in ab- and desorption of hydrogen at the Mg(0001) surface [87].

_{2}dissociation and atomic H diffusion on the Mg surface. On the Ti-incorporated surface, the H

_{2}molecule dissociates almost spontaneously, the activation barrier is rather low, but diffusion away from the Ti site is very unfavorable due to strong Ti-H bonding.

_{2}molecule dissociation is small: the barrier still exists and is high. However, such as Ni atom prefers to migrate inside Mg matrix rather than to occupy in or over the topmost Mg(0001) surface [95]. Being placed inside Mg bulk Ni not only does not improve H

_{2}dissociation but has a detrimental effect on atomic hydrogen diffusion [94,95], whereas Ni atom locating in/over the topmost Mg(0001) surface is an excellent catalyst for H

_{2}dissociative chemosorption. Theoretical studies of hydrogen dissociation and diffusion mechanism on the Mg(0001) surface co-doped with V and Ni have shown that the presence of V atom at the subsurface layer not only stabilizes Ni at the surface layer but also facilitates H

_{2}dissociation and hydrogen diffusion on and inside Mg [94].

_{2}and the diffusion of atomic H on the Mg(0001) surfaces. A carbon atom on the Mg(0001) surface can easily migrate into the subsurface layer and occupy an fcc interstitial site. This process is accompanied by charge transfer from nearby Mg atoms to carbon, that (i) facilitates the dissociative chemisorption of H

_{2}on the Mg(0001) surface [88], and (ii) enhances the surface migration and subsequent diffusion of H atoms into the sub-surface layers, reducing the activation barrier by 0.16 eV [102]. MEP for this process for pure and C-doped Mg(0001) surface is shown in Figure 8. However, this result was obtained without accounting for zero-point vibrations. ZPE correction makes the difference between the activation barriers for pure and C-doped Mg(0001) surface less pronounced: 0.011 eV only, but as soon as the reaction paths are very similar in each case (see Figure 8), the inclusion of the ZPE correction does not affect the main conclusion concerning the effect of the carbon dopant on hydrogen adsorption on the Mg(0001) surface.

#### 4.4. Hydrogen Induced Phase Transformations in MgH_{x}

_{x}exists in a very narrow hydrogen concentration range [103]. Hydrogen charging results in volume expansion of the metallic lattice with subsequent phase transformations and formation eventually MgH

_{2}. It is impossible to survey experimentally these transforms upon hydrogenation; only the initial and final states can be caught. From this point of view, theoretical calculations are a unique tool for studying this process.

_{2}were theoretically studied by Klyukin et al. [85]. Heat of formation for various phases of MgH

_{x}calculated without ZPE contribution, are shown in Figure 9. According to calculations at 0.1 < x < 1.5 the fcc structure is more energetically favorable. At x ≈ 1.5 an fcc-to-rutile phase transformation occurs that is accompanied by atomic bonding change from metallic to ionic-covalent. It should be noted that a metastable fcc-MgH

_{2}phase was found experimentally very recently [107]. Being obtained upon cold rolling of reacted ball milled MgH

_{2}it exhibits lower stability compared to rutile and better hydrogen desorption kinetics.

_{x}structure is unstable within the whole hydrogen concentration range; however, in absence of hydrogen [82,83,84] or at low hydrogen concentration [85] it can be stabilized by presence of an adjacent layers of a bcc metal, such as Nb, V or Ti-V-Cr disordered alloys [52,53].

_{x}structures reveal that for all the studied systems [85,106] except bcc MgH

_{x}[85] hydrogen atoms occupy adjacent interstitial sites forming a hydride layer that prevents hydrogen propagation into the Mg bulk. This so-called blocking layer effect is absent in bcc-Mg, in which hydrogen is spread over the lattice without forming clusters, that may explain acceleration of hydrogen sorption kinetics in Mg/Nb multilayers [81,108,109] and Mg@Nb (or Mg@V) composites [6,14,110,111].

#### 4.5. Modelling of Hydrogen Diffusion in Bulk MgH_{x}

_{x}are not very numerous. Hydrogen diffusion coefficient in hcp-Mg has been theoretically calculated in Refs. [30,48,87,104] exhibiting a wide variation in two orders of magnitude. Klyukin et al. [30,48] studied hydrogen diffusion in hcp-, bcc- and fcc-MgH

_{x}with low (x = 0.0625) and moderate (x = 0.5) hydrogen concentrations. The activation energies for all possible diffusion paths were calculated using CI-NEB approach accounting for ZPE. The results are summarized in Figure 10a. It is worth noting that ZPE does not contribute much for MgH

_{x}, less than 5% [48], but is of extreme importance for correct calculations of TM-H systems [30,52,53,112].

_{x}the minimal activation energy was found for a T1→T2 hydrogen jump; however, accounting that diffusion assumes translational motion, in hcp Mg the following multistep hydrogen diffusion path is realized: O2→T2→T1→O1→T1′→… (where T1′ site is situated in an adjacent unit cell). The activation energy for this pathway is in fair agreement with experimental 0.25 eV [113]. In both bcc- and fcc-MgH

_{x}the hydrogen atoms prefer to occupy T-sites. The hydrogen diffusion paths were found T1→T3 (or T1→O1→T2, where hydrogen atoms move between two adjacent T-sites via an intermediate O-site) for the bcc lattice and T1→O1→T4 for the fcc one. Among considered structures bcc-MgH

_{x}exhibits the lowest activation energy for translational hydrogen motion.

_{x}with low hydrogen concentration calculated using equations like Equations (4) and (5) was found equal to 1.11 × 10

^{–8}m

^{2}/s that agrees well with the experimental value of 2.07 × 10

^{–8}m

^{2}/s obtained by Nishimura et al. using gas permeation technics [113] and other experimental studies of hydrogen diffusion in Mg (see Ref. [114] and references therein). Expanding this method to bcc- and fcc-MgH

_{x}, it was found that bcc lattice exhibits the highest diffusion coefficient, whereas fcc shows the lowest one (see the calculated temperature dependence of hydrogen diffusion coefficient D for various MgH

_{x}lattices in Figure 10b).

_{x}does not share the blocking layer effect [85] it was supposed that this structure should promote fast hydrogen diffusion via the nucleation process [48]. The presence of TM layers or particles on the surface of Mg grains should help to locally stabilize the bcc-Mg arrangement, in which hydrogen diffusion occurs faster due to a lower activation energy and absence of the blocking layer effect.

#### 4.6. Hydrogen Desorption from (001) MgH_{2} Surfaces

_{2}rutile surfaces, MgH

_{2}(001) and MgH

_{2}(110) was theoretically studied by Du et al. [98]. They found that the MgH

_{2}(110) surface is more stable than MgH

_{2}(001), that can be perceived in term of bonds cut for Mg atom on the surface: two bonds for MgH

_{2}(001) and only one for MgH

_{2}(110). This conclusion is supported by experimental observations for thin films [115,116] and more resent calculation of a wider set of MgH

_{2}surfaces [117].

_{2}(110) surface [98]. This relatively high barrier reflects the slow kinetics of hydrogen release and is in good agreement with experiment [118,119].

_{2}also depends on the subsequent hydrogen diffusion from bulk to surface, that is mediated by H-vacancy interactions [120,121]. Surface hydrogen vacancy formation results in the breaking of atomic bonds that affects the surface stability and the dehydrogenation process [117]. As it was shown by Kurko et al. [121] the increased number of surface hydrogen vacancies reduces the potential barrier for further H-desorption.

_{2}.

_{2}. Impact of different catalysts on hydrogen desorption barrier in MgH

_{2}have been theoretically studied is several works [11,122,123,124,125,126]. Reich et al. [122] studied the role of Ti dopant. The activation barrier was found generally lower as compared to pure MgH

_{2}(up to 14%) but very sensible to the step and doping configuration. As it was shown by Giusepponi and Celino [123] due to Fe atom has a higher coordination than Mg iron doping leads to a local destabilization of the MgH

_{2}lattice that increases probability of hydrogen diffusion towards surface. According to Ri et al. [124] co-substitution by Ti and Fe (adjacent each other at a short distance were found the most energetically favorable) on the MgH

_{2}(110) surface results in a large distortion of the lattice and decreases ionicity of H atoms. Compared to single substituted cases such a co-substituted surface is more favorable for hydrogen vacancy formation and leads to further improvement of hydrogen desorption.

_{2}(110) surface was theoretically studied by Kurko et al. [126]. It was found that boron forms strong covalent bonds with hydrogen that perturbs its first and second coordinations and results in both decreasing hydrogen desorption energies and improvement of hydrogen diffusivity. However, boron substitution does not affect much the energy barriers for hydrogen desorption. It is interesting to note that the effect of boron on hydrogen absorption by magnesium is more pronounced: doping with boron leads to a significant decrease in the activation energy of absorption of hydrogen acting as an active center [126].

## 5. Molecular Dynamics Simulation of Mg-H Systems

^{–8}m

^{2}/s) is in a good agreement with both experimental studies [113] and DFT calculations [48] (see Section 4.5).

## 6. Conclusions

_{2}molecules’ dissociation of Mg surfaces with further atomic hydrogen absorption, diffusion into and in bulk Mg, hydrogen induced phase transformations in magnesium, hydrogen desorption from MgH

_{2}surfaces and the effect of dopants and vacancies, etc. Due to its accessibility and flexibility, as well as the validity of the results obtained, DFT takes a worthy place among the tools aimed at the design of new materials. In relation to Mg, it allowed us not only to explain the mechanisms responsible for the improvement of MgH

_{2}stability and hydrogen sorption kinetics in the presence of TM additives or other non-metallic dopants, but also to predict new metastable MgH

_{x}phases with better properties, which was further proved experimentally.

## Funding

## Conflicts of Interest

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**Figure 1.**Interaction of hydrogen with metal. Adapted with permission from Shelyapina, M.G. Metal hydrides for energy storage; Springer Nature Switzerland AG, 2019 [5].

**Figure 2.**Energy diagram for hydrogen interaction with metal. Reproduced with permission from Shelyapina, M.G. Metal hydrides for energy storage; Springer Nature Switzerland AG, 2019 [5].

**Figure 3.**Octahedral (O) and tetrahedral (T) interstitial sites in hcp, bcc and fcc lattices. Reproduced with permission from Shelyapina, M.G. Metal hydrides for energy storage; Springer Nature Switzerland AG, 2019 [5].

**Figure 4.**Schematic diagram to show the forces and their relation to MEP: the total force on each image ${F}_{i}^{\mathrm{NEB}}$ is the sum of the true force perpendicular to the local tangent ${F}_{i}^{\perp}$ and the spring force along the local tangent ${F}_{i}^{\mathrm{S}\left|\right|}$. Reproduced with permission from Sheppard, D. et al. J. Chem. Phys.; published by American Institute of Physics, 2008 [42].

**Figure 5.**Energy profile for an indirect hydrogen diffusion path with a local metastable state on the hydrogen diffusion path. Reproduced with permission from Klyukin, K. et al, J. Alloys Compd.; Elsevier B.V., 2015 [48].

**Figure 6.**Schematic representation (

**a**) and PES counter plot (

**b**) for the hcp-to-bcc phase transformation in magnesium (dashed line represents MEP). Reproduced with permission from Klyukin, K. et al, J. Alloys Compd.; Elsevier B.V., 2013 [85].

**Figure 7.**Schematic diagram of hydrogen atom diffusion in bulk Mg from the Mg(0001) surface. Different H interstitial sites with their local environment are shown. Reproduced with permission from Han, Z. et al. Appl. Surf. Sci.; Elsevier B.V., 2017 [91].

**Figure 8.**The energy profiles for H

_{2}molecule dissociation on a Mg(0001) surface: with (squares) and without (triangles) carbon substitution. The large gray, large black, and small gray balls represent Mg, C, and H atoms, respectively. Three special configurations, initial state, transition state, and final states along the MEPs are shown. Reproduced with permission from Du, A.J. et al. J. Phys. Chem. B; American Chemical Society, 2006 [102].

**Figure 9.**Energy diagram for hydrogen induced phase transformations in Mg; insert: heat of formation for various phases of MgH

_{x}. Reproduced with permission from Grbović Novaković, J. et al. ChemPhysChem; Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, 2019 [11].

**Figure 10.**(

**a**)—Possible H diffusion pathways in hcp, bcc and fcc MgH

_{x}and corresponding activation energy values; (

**b**)—temperature dependence of the H diffusion coefficient D calculated for the most favorable diffusion pathways in the hcp, bcc and fcc Mg lattices. Reproduced with permission from Klyukin, K. et al, J. Alloys Compd.; Elsevier B.V., 2015 [48].

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Shelyapina, M.G.
Hydrogen Diffusion on, into and in Magnesium Probed by DFT: A Review. *Hydrogen* **2022**, *3*, 285-302.
https://doi.org/10.3390/hydrogen3030017

**AMA Style**

Shelyapina MG.
Hydrogen Diffusion on, into and in Magnesium Probed by DFT: A Review. *Hydrogen*. 2022; 3(3):285-302.
https://doi.org/10.3390/hydrogen3030017

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Shelyapina, Marina G.
2022. "Hydrogen Diffusion on, into and in Magnesium Probed by DFT: A Review" *Hydrogen* 3, no. 3: 285-302.
https://doi.org/10.3390/hydrogen3030017