Multi-Scale Modeling of Doped Magnesium Hydride Nanomaterials for Hydrogen Storage Applications

Abstract

This work presents the development of a novel multi-scale modeling framework for investigating the beneficial impact of Ti-, Zr-, and V-doped magnesium hydride nanomaterials on hydrogen storage performance. The proposed model integrates atomistic-scale simulations based on density functional theory (DFT) with system-level dynamic heat and mass transfer modeling. At the nanoscale, DFT analysis provides key thermodynamic and kinetic parameters, including reaction enthalpy, entropy, and activation energy, which are incorporated into the macroscopic model to predict the hydrogenation behavior of MgH₂ nanostructures under realistic thermal boundary conditions. Model validation is performed through comparison with experimental data from the literature, showing excellent agreement. The DFT analysis reveals that doping MgH₂ nanomaterials with Ti, V, and Zr modifies their thermodynamic properties, including enthalpy of formation and desorption temperature. At the reactor scale, these modifications lead to enhanced hydrogenation kinetics and improved thermal management. Compared to pristine MgH₂, hydrogenation time is reduced by 21%, 40%, and 42% for Ti-, Zr-, and V-doped nanomaterials, respectively, while thermal energy consumption during hydrogenation decreases by ~17% for V doping. These results highlight the strong correlation between nanoscale modifications and macroscopic system performance. The proposed multi-scale model provides a powerful tool for guiding the design and optimization of advanced nanostructured hydrogen storage materials for sustainable energy applications.

1. Introduction

The global shift toward sustainable energy systems has intensified research efforts on hydrogen as a clean and effective medium for energy transport. Among the various hydrogen storage technologies, solid-state storage using metal hydrides is widely recognized for its high volumetric energy density, safety, and reversibility [1,2]. Magnesium hydride (MgH₂), in particular, has received extensive attention because of to its relatively high theoretical hydrogen capacity (~7.6 wt%), low cost and abundance [3,4]. Nonetheless, its practical use is constrained by sluggish hydrogen sorption kinetics and elevated desorption temperatures, typically exceeding 300 °C [5].

This scientific interest is reflected in the steadily increasing number of publications on MgH₂-based hydrogen storage systems over the past two decades (Figure 1). According to data retrieved from the Scopus database using the search terms “magnesium hydride” and “hydrogen storage”, research output has grown significantly, indicating the ongoing relevance of MgH₂ in hydrogen energy research and the persistent efforts to improve its performance.

Recent advancements in hydrogen storage have shown that transition metal doping of MgH₂ greatly improves its performance. For instance, Xiao et al. [6] demonstrated that using ball milling to add Ce_0.6 Zr_0.4O₂ nanocrystals to MgH₂ significantly reduced the beginning dehydrogenation temperature by 85 °C to 201 °C. This composite exhibited rapid hydrogen release (6.15 wt% at 270 °C) and absorption (6.3 wt% in 2 min at 50 bar and 150 °C), maintaining remarkable cycling stability exhibiting 98.9% capacity retention over the course of 20 cycles.

Additionally, MgH₂-30 wt% TiFe_0.92Mn_0.04Co_0.04 was synthesized by ball milling by Li et al. [7], demonstrating remarkable hydrogen storage capabilities. In contrast to conventional TiFe, the addition of Mn and Co lowered the activation barriers of TiFe_0.92Mn_0.04Co_0.04 and increased its catalytic activity by facilitating the in situ synthesis of α-Fe.

Furthermore, Han et al. [8] examined the impact of Ti, Zr, V, and Nb doping on MgH₂ using first-principles methods. According to these investigations, these dopants speed up the kinetics of hydrogen desorption by facilitating the transport of hydrogen atoms far from the dopant sites. Together, our results demonstrate the significant promise of Ti, Zr, V, and Nb doping to enhance MgH₂’s usefulness as a practical hydrogen storage substance.

Magnesium hydride (MgH₂) is an attractive material for hydrogen storage because of its high gravimetric and volumetric hydrogen capabilities (7.6 wt% and 110 g/L, respectively). It outperforms several other options, including carbon nanostructures like nanotubes and graphene, which typically have low gravimetric capacities (<1 wt%) at ambient temperature, and complicated hydrides (e.g., NaAlH₄) with complex reaction paths and poor reversibility. While Metal–Organic Frameworks (MOFs) have large surface areas for hydrogen adsorption, their practical application is typically limited to cryogenic temperatures due to poor interactions with hydrogen molecules. MgH₂ is an excellent choice because it is lightweight, inexpensive, and abundant. However, its primary constraints are slow kinetics and great thermodynamic stability, which necessitate high temperatures for hydrogen release and absorption [9,10,11]. The MgH₂ lattice’s hydrogen desorption behavior can be significantly modified via the addition of vacancies to it. According to Gaztanaga et al. [12], MgH₂ regularly shows negatively charged divacancies in addition to positively and negatively charged hydrogen vacancies. The material’s ability to store hydrogen can be substantially improved by fixing these flaws. Additionally, the material’s band gap is significantly reduced when negative and neutral hydrogen vacancies are present, improving its semiconductor qualities. German et al. [13] further demonstrated that Zr doping is more effective than Nb in reducing dehydrogenation energy. Additionally, the creation of vacancies within the Mg lattice leads to a significant decrease in energy of dehydrogenation, with vacancies in the Mg sublattice representing the arrangement that yields the lowest desorption energies. In comparison to the pure hydride, C.R. Luna et al. [14] found that adding trace amounts of Zr or Nb impurities (about 0.01 weight percent) to immaculate MgH₂ had no discernible impact on the electron momentum density. Conversely, a significant change in electron density was observed at the vacancy site in MgH₂ where a Mg vacancy exists. Moreover, the electron momentum density at the vacancy site is significantly altered when a substitutional impurity is present next to a Mg vacancy at the same time.

Finally, Reddad et al. [15] employed Density Functional Theory (DFT) to investigate the effects of Cu and Zn doping on the kinetic and thermodynamic properties of MgH₂. They methodically investigate how different Cu and Zn dopant concentrations and configurations affect MgH₂’s hydrogen desorption properties.

While numerical results have demonstrated improvements in hydrogen storage, a complete mechanistic understanding linking atomic-scale modifications to system-level behavior has remained incomplete. To bridge this gap, multi-scale modeling approaches that couple atomistic simulations with thermal and kinetic system-level models are essential. Previous literature has shown that DFT provides accurate predictions of material properties (e.g., reaction enthalpy and activation energy), while dynamic thermal modeling effectively captures the effects of heat and mass transfer in real operating conditions. In this study, we propose a novel coupled atomistic and dynamic thermal modeling framework to investigate the impact of Ti, Zr, and V doping on MgH₂. This approach integrates DFT-calculated parameters directly into a system-level dynamic model, enabling the quantification of hydrogenation time, energy consumption, and thermal behavior. The system-level model is fundamentally based on energy and mass balances, meticulously coupled with the hydrogenation/dehydrogenation kinetic equations. It is interesting to note that in the present work, a lumped parameter approach was adopted to model the dynamic behavior of the MgH₂-based hydrogen storage reactor. While spatially resolved finite-element or CFD models can capture local gradients in temperature, pressure, and concentration with high fidelity, they are computationally expensive and often impractical for extensive parametric studies or integration into system-level analyses. The lumped model used here provides a balance between accuracy and computational efficiency, enabling fast simulations while still reproducing the dominant thermal and kinetic trends observed in experimental studies. Such an approach is particularly valuable for early-stage design, performance optimization, and coupling with real-time control strategies, whereas distributed models remain more suitable for detailed reactor-scale optimization and scale-up considerations. The results from this integrated framework are expected to highlight the direct correlation between atomic-scale doping effects and macroscopic system performance, offering a robust tool for the design and optimization of advanced hydrogen storage systems.

The novelty of the proposed work can be summarized as:

• Novel coupled atomistic and dynamic thermal modeling framework: This study pioneers the direct, integrated application of atomistic (DFT) and system-level (dynamic thermal) models specifically for the analysis and optimization of hydrogen storage materials. Unlike previous studies that might use these methods in isolation or in a less integrated fashion, our framework establishes a seamless connection to bridge the disparate length and timescales.
• Direct correlation of atomic-scale doping with macroscopic system performance: A central innovative aspect is the explicit linkage established between the fundamental effects of specific dopants (Ti, Zr, V) on MgH₂ at the atomic scale (e.g., changes in reaction barriers and enthalpies from DFT) and their direct impact on observable, macroscopic system-level outcomes (e.g., overall hydrogenation time, energy consumption, and thermal management). This provides a mechanistic understanding that was previously lacking.
• Comprehensive quantification of system behavior under realistic conditions: Beyond just material properties, our framework allows for the comprehensive quantification of crucial system performance indicators such as hydrogenation/dehydrogenation time, total energy consumption, and detailed thermal behavior under dynamic operating conditions. This moves beyond theoretical predictions to practical system-level performance assessment.

2. Multi-Scale Modeling Method

The proposed modeling approach follows a multi-scale framework, combining atomic-scale DFT calculations with macroscopic system-level dynamic modeling to analyze the hydrogen storage performance of MgH₂ doped with Ti, V, and Zr. This methodology enables a direct link between fundamental material properties and large-scale system behavior, providing a comprehensive understanding of doping effects from the atomic level to practical applications.

At the microscopic scale, DFT calculations are employed to determine key material properties such as formation energy, hydrogen desorption enthalpy, diffusion barriers, and electronic structure. These parameters provide insight into how Ti, V, and Zr influence the thermodynamics and kinetics of hydrogen absorption and desorption in MgH₂. The results obtained from the DFT simulations are then used as input parameters IN the macroscopic system-level model, which describes the dynamic behavior of a hydrogen storage tank using a lumped modeling approach. This model accounts for mass and heat transfer, reaction kinetics, and thermal management by using cooling/heating fluid, hydrogen uptake/release rates, and the overall performance of the storage system under realistic operating conditions.

By coupling the atomic-scale and system-level models, this multi-scale approach enables a direct correlation between material modifications at the microscopic level and their impact on storage performance at the macroscopic scale. This integration is crucial for optimizing MgH₂-based hydrogen storage systems, ensuring both improved material properties and enhanced practical performance. A schematic representation of this coupling is shown in Figure 2, where the atomic-scale structure of MgH₂ and the system-level storage configuration are illustrated.

2.1. Microscopic Scale: DFT Modeling of Doped MgH₂

2.1.1. Computational Theory

The Cambridge Serial Total Energy Package (CASTEP) and density functional theory serve as the foundation for all of the computations in this work [16,17]. The option guarantees energy convergence to less than 1 meV per atom. The electron wave function was expanded in plane waves up to a cutoff energy of 350 eV. The exchange-correlation functional was created based on the generalized gradient approximation (GGA) as outlined by Perdew–Burke–Ernzerhof revised for solids (PBEsol) [17,18]. Prior research has demonstrated that PBEsol is superior to PBE for examining point defects [19] and can forecast lattice constants with more accuracy [20].

The MgH₂ lattice was represented by a 3 × 3 × 1 supercell with 54 atoms that could accommodate different Ti, V, and Zr concentrations. In the Brillouin zone, 3 × 3 × 1 Monkhost–Pack grids were used to sample the reciprocal spaces. In contrast, the surface of Fermi structure and the electronic density of charges map were examined using a larger k-point mesh, measuring 9 × 9 × 9. The convergence process criteria for force and energy were 2 × 10⁻⁵ eV/atom, the highest Hellmann–Feynman ionic strength was restricted to 0.05 eV/Å, the highest stress level was capped at 0.1 GPa, and the maximum ion displacement was confined to 0.002 Å. All atoms were fully relaxed using a conjugate gradient algorithm. An atom was added or removed from the designed MgH₂ supercell to create the defects, such as vacancies and interstitials. Structure relaxation was then carried out with specified lattice constants. Atomic locations inside the supercell were simultaneously given complete relaxation.

2.1.2. Formation Energy and Atomic Structure

To determine the thermodynamic characteristics of our systems, using a relaxation approach, we first determined the equilibrium lattice parameters. In particular, by decreasing the total energy with respect to the volume of a 3 × 3 × 1 MgH₂ supercell, we optimized the architectures (Figure 3). The energy associated with the formation of a charged defect (Xq) is contingent upon the atomic chemical potentials and the electronic chemical potential (Fermi level). It can be calculated using the following expression [21]:

$$Mg+H_2\to Mg{H}_2$$

(1)

$$\left(1-x\right)Mg+xM+H_2\to {Mg}_{1-x}{M}_x{H}_2$$

(2)

The heat of formation for the reaction (Equation (2)) was calculated by subtracting the total energies of elemental Mg, the M atom (where M represents Ti, V, or Zr), and the H₂ molecule from the total energy of the relaxed Mg_1−xM_xH₂ hydride:

$$∆H\left(Mg{H}_2\right)=E_{tot}\left(Mg{H}_2\right)-E_{tot}\left(Mg\right)-E_{tot}\left(H_2\right)$$

(3)

$$\begin{gathered} ∆H\left(M_{g0.9455}{X}_{0.0555}{H}_2\right) \\ =E_{tot}\left(M_{g0.9455}{X}_{0.0555}{H}_2\right)-E_{tot}\left(Mg\right)-E_{tot}\left(X=Ti, V and Zr\right)-E_{tot}\left(H_2\right) \end{gathered}$$

(4)

E_tot represents the crystal supercell’s total energies with a defect X. Detailed results on thermodynamic properties are presented in Section 3.1.1.

2.2. Macroscopic Scale: Lumped Modeling for Hydrogen Storage Performance

To simulate the dynamic behavior of the metal hydride storage with its thermal management system at the system level, lumped energy and mass balances are applied to each component of the reactor. These balances enable the determination of time-dependent variation in the reactor temperature and the progress of the hydrogenation process. More details about this approach can be found in our previous works [4,22]. In fact, this proposed lumped method considers the metal hydride reactor as a control volume with uniform temperature and a heat source. The latter is used to take into account the released heat during the exothermic reaction of the hydrogen absorption process. In addition to the reactor bed, the thermal management system, by using a heat transfer fluid, is also included in the system-level model. Figure 4 shows an analogous scheme for the mass and heat transport in the hydrogen storage reactor.

2.2.1. Energy and Mass Balances

In order to predict the time-wise variation in the hydrogen storage temperature, the thermal management system temperature and the hydrogenation process, energy and mass balance at the system level are used and the following assumptions are considered:

• The hydrogen reactor has a uniform temperature and pressure.
• The hydrogen is considered as an ideal gas.
• The hydrogen reactor properties are independent of the temperature.
• The hydrogen storage tank is well insulted and exchanges heat only with the thermal management system.

Based on the previous assumptions, the governing equations can be given as the following:

• Energy balance:

$${\left(m_{H_2}{C}_{p_{H_2}}+m_s{C}_{p_s}\right)}_1{\displaystyle \frac{d{T}_{MH}}{dt}}={\displaystyle \frac{T_{TMS}-T_{MH}}{R_{MH-TMS}}}+\dot{q}$$

(5)

In this expression, T_MH and T_TMS are the metal hydride temperature and used thermal management system temperature. $R_{MH-TMS}$ is thermal resistance between the thermal regulation system and the MH reactor. Its expression is given by:

$$R_{MH-TMS}={\displaystyle \frac{1}{UA}}$$

(6)

A is the heat transfer area between the heating/cooling system and the MH reactor, and U is the global heat transfer coefficient.

During the storage of the hydrogen in the reactor, the heat generated due to the exothermal reaction between the hydrogen (gas) and the metal (Mg) is given by as the following [4]:

$$\dot{q}={\displaystyle \frac{{∆H}_r}{{MW}_{MH}}}{\displaystyle \frac{d{m}_{MH}}{dt}}$$

(7)

${∆H}_r$ is the enthalpy reaction and its value depending on the used doped MgH₂.

${\displaystyle \frac{d{m}_{MH}}{dt}}$ is the metal hydride mass variation during the hydrogenation/dehydrogenation process and its expression is given in the following section.

• Mass balance

The mass balance of the MH reactor, which expresses the evolution of the hydrogenation/dehydrogenation process, is given as a function of the kinetic reaction rate [23]:

$${\displaystyle \frac{d{m}_{MH}}{dt}}={r\cdot m}_s$$

(8)

• Reaction kinetic

During the absorption/desorption of hydrogen in the metal, the process is affected by a number of physical parameters including hydrogen pressure, temperature, equilibrium pressure and activation energy. The general expression of the kinetic reaction can be expressed as a function of the previous physical parameters as [4,24,25]:

$$r=C_a e^{{\displaystyle \frac{-E_a}{R{T}_{MH}}}} \mathrm{l}\mathrm{n}({\displaystyle \frac{P_a}{P_{eq}}}) (1- {\displaystyle \frac{m_{MH}}{m_s}})$$

(9)

where $P_a$ is the used inlet pressure at the MH reactor and $P_{eq}$ is the equilibrium pressure. The latter is given based on the Van’t Hoff law as follow [26,27]

$$P_{eq}=P_0 e^{\left({\displaystyle \frac{{∆H}_r}{R{T}_{MH}}}-{\displaystyle \frac{∆S}{R}}\right)}$$

(10)

It is interesting to note that the reaction enthalpy (${∆H}_r$) and the entropy ($∆S$) the studied doped MgH₂ reactors are introduced in this lumped model based on output results from DFT calculations (See Section 2.1). A flow chart of the proposed coupling procedure between the micro-scale model and the system level model is given in Figure 5. In this multi-scale approach, for each doped MgH₂ case study, DFT calculations are first carried out to determine the atomic structure, energetics, and thermodynamic properties of the material. These DFT-derived properties, are then integrated as input parameters into the lumped dynamic model to accurately simulate the hydrogen absorption behavior at the system level. This coupling ensures that material-scale modifications induced by doping are consistently reflected in the performance predictions of the reactor.

2.2.2. Key Performance Indicators

Several key performance indicators (KPIs) are established in order to evaluate and compare the performance of MgH₂-based hydrogen storage units under various doping techniques (Ti, V, Zr). These indicators provide quantifiable measurements for assessing the thermal and storage performance of hydrogen and are calculated using the results of the lumped dynamic simulations:

• Hydrogenation Time (t_hyd)

Defined as the time required to reach a target hydrogen uptake fraction (e.g., 99% of the total capacity). It is strongly influenced by the reaction kinetics and thermal management efficiency. Its expression can be given as follow:

$$t_{hyd}={\int }_{X_0}^{X_{final}}{\displaystyle \frac{dX}{r(P_{eq},T,P)}}$$

(11)

where $r(P_{eq},T,P)$ is the kinetic reaction given as a function MH bed temperature, the equilibrium pressure and the used hydrogen pressure (See Equation (9)).

• Maximum Reactor Temperature (T_max)

As mentioned previously, the MH temperature plays a critical role during the hydrogenation process. This KPI indicates the peak temperature reached during the hydrogen absorption process and it reflects the effectiveness of heat dissipation during the exothermic reaction.

• Required heating/cooling thermal energy (Qth)

This KPI measures the consumed thermal energy by the thermal management system during the hydrogenation process. Its expression is given by integrating the thermal power exchanged between the reactor and the thermal management system as follow:

$$Q_{th}={\int }_0^{t_{hyd}}{\dot{Q}}_{th}\left(t\right)dt$$

(12)

2.2.3. Model Validation

In this section, the proposed system level model is validated through comparing our numerical results with experimental results from the work of Laurencelle et al. [28]. This experimental study considers a MH reactor with a capacity of 25 g. This cylindrical reactor is heated/cooled using a water loop as TMS and the inlet hydrogen pressure is fixed at 12 bar. More details about this experimental work and the validation approach can be found in our previous work [4]. Based on obtained results in Figure 6, the proposed dynamic model is able to predict the real behavior of the MH hydrogenation and dehydrogenation process with a good accuracy (Figure 6a). In addition to the hydrogenation kinetic, the proposed system level-scale model is also able to predict the MH temperature during both hydrogenation/dehydrogenation processes (Figure 6b).

3. Results and Discussion

3.1. Microscopic Investigation: DFT Results

In this section, we employed the DFT to investigate the atomic-scale effect of Ti, V, and Zr doping on the electronic and thermodynamic properties of magnesium hydride (MgH₂). The purpose of this study was to explore how dopant atoms disrupt the local atomic environment and change the fundamental properties of the material. The in-depth electronic and thermodynamic analyses that follow are based on the knowledge obtained through this atomic-level analysis.

3.1.1. Electronic and Structural Parameters

Comprehending the electronic properties of materials is vital for understanding the mechanics of carrier transport, since it facilitates the distinction between metals, semiconductors, and insulators through the examination of their band gaps. Under ambient conditions, our MgH₂ system has a rutile structure with a P4₂/mnm space group. Experimentally, The primary lattice parameters of bulk MgH₂ were found to be a = b = 4.501 Å and c = 3.010 Å. After optimization, the calculated equilibrium lattice parameters were a = b = 4.460 Å and c = 2.989 Å, closely matching experimental data and other computational results.

Figure 3 shows that magnesium is surrounded by six anionic hydrogen atoms. These hydrogen atoms can be separated into two groups:

• Two ‘apical’ hydrogen atoms are located above and below the magnesium, with an Mg-H distance of approximately 1.94 Å.
• Four ‘equatorial’ hydrogen atoms are located around the magnesium, forming a square, with a slightly larger Mg-H distance of about 2.03 Å.

The band gap width indicates the strength of the Mg-H bond: a smaller band gap suggests stronger electronic hybridization and less hydride stability, lowering the desorption temperature and reaction enthalpy. These modifications accelerate the kinetics of hydrogen sorption/desorption. Faster kinetics increase instantaneous thermal power during hydrogenation/dehydrogenation, whereas lower enthalpy requirements minimize overall thermal energy consumption.

This subsection examines the electronic structures and bonding properties of both pure MgH₂ surfaces and those doped with Ti, V, and Zr. We analyze the total and partial densities of states (DOS) to understand the band energy, hybridization between different states, and charge transfer characteristics. This analysis helps determine how the addition of these transition metal elements affects hydrogen stability and desorption in MgH₂.

This analysis provides a detailed picture of the electronic state distribution and reveals the bonding and electronic structure of the material (Figure 7). Pure MgH₂ reveals a non-metallic property, having a calculated band gap of 3.71 eV. While this is somewhat lower than the experimental value of 5.16 eV [29], it is consistent with the theoretical value of 3.6 eV referenced in [30]. The acknowledged limits of first-principles calculations utilizing PBE density functional theory are probably the cause of the discrepancy between our calculated and observed band gaps. As shown in Figure 7, doping with Ti, V, and Zr significantly reduces the band gap to 1.8 eV, 1.378 eV, and 1.199 eV, respectively, clearly demonstrating the impact of doping on the electronic structure.

3.1.2. Charge Density Distribution

An analysis of the electronic charge density for pure MgH₂ and MgH₂ doped with Ti, V, and Zr provides key insights into their bonding characteristics, particularly in cases of neutral charge distribution. The electron charge density was computed using the charge density difference approach from the CASTEP program, which is based on the GGA-PBE functional. This method was chosen because it emphasizes the redistribution of electronic density after Ti, V, and Zr doping, emphasizing the changes in Mg-H bonding properties and dopant-host interactions. Figure 8b shows a distinct localization of charge around the hydrogen atoms, indicating a strong covalent interaction with the Mg and V atoms. This is further evidenced by the directional charge localization observed between the H and V atoms. While the contribution of V atoms to the overall charge density is minimal, suggesting their ionic character, a mixed covalent-ionic bonding character exists between the hydrogen atoms and the dopants. This implies a weakening of the Mg-H bond compared to pure MgH₂. These mixed interactions play a crucial role in stabilizing the structure by strengthening covalent bonds and weakening short-range interactions, ultimately influencing the material’s overall electronic and structural properties. Mg_0.9455Ti_0.0555H₂ and Mg_0.9455Ti_0.0555H₂ exhibit charge distribution patterns similar to Mg_0.9455Ti_0.0555H₂, their patterns are not depicted.

Figure 9a clearly shows hybridization peaks in the valence and conduction bands of pure MgH₂, indicating strong Mg-H bond orbital interactions. The conduction band is primarily characterized by Mg-s and p states. These strong interactions likely contribute to the high desorption temperature of MgH₂. In contrast, the DOS plots for Ti, V, and Zr-doped MgH₂ (Figure 9b–d, respectively) reveal defect states within the band gap. These defect states arise from interactions between H and the Ti, V, or Zr atoms. Were the defect states’ refer to localized electronic energy levels introduced into the band gap of MgH₂ as a result of dopant atoms (Ti, V, or Zr) and the associated lattice distortions or vacancies. These states arise from the perturbation of the host crystal’s electronic structure and can influence charge transfer and, consequently, the kinetics of hydrogen absorption/desorption.

Figure 9b–d shows weak hybridization between the (V, Ti, and Zr) d orbitals and the neighboring H s orbital, as evidenced by the alignment of their respective peaks. In addition, the Mg-H hybridization within the high-energy segment of the valence band is lessened. The V d and H s orbitals show the highest degree of hybridization, reflecting the most robust chemical interaction between V and H. While pure MgH₂ displays strong Mg-H hybridization, this hybridization weakens significantly in the doped structures, with a near absence of MgH₂ hybrid peaks. These strong inter-state hybridizations confirm the high stability of the doped materials, consistent with the calculated ΔH values. Our findings are consistent with [8].

3.1.3. Thermodynamic Properties

(a)

Hydrogen desorption energy

To demonstrate the changes made to the thermodynamic properties, by doping MgH₂ with the transition elements Ti, V and Zr. We examined the hydrogen desorption activation energy (E_d) for the 3 × 3 × 1 supercell using the equation below [7]:

E_d = E(MgH₂ + x) + x/2 E(H₂) − E(MgH₂)

(13)

E(MgH₂+ xVH) represents the energy of the MgH₂ in which an H₂ atom is taken away from the relaxed MgH₂, E(H₂) refers to the energy associated with the H₂ molecule at 0 K, and E(MgH₂) indicates the energy of the MgH₂ surface before the process of dehydrogenation. Our calculated value of 109.21 kJ/mol agrees with previous theoretical calculations [31], although it is below the experimental values of approximately 143.0 and 160.6 kJ/mol [32]. This high desorption energy is attributed to the strong ionic bonds between the hydrogen atoms.

Table 1. Summary of formation enthalpy, the desorption temperature and the activation energy calculated as a function of Equations (2), (3), and (13) for pure and doped MgH₂ and for comparative studies.

Structures	ΔH (kJ/(mol H2))	Td (K)	Activation Energy (kJ/mol)
MgH2	−63.961	489.37	109.21
Mg0.9455Ti0.0555H2	−51.212	391.82	78.24
Mg0.9455V0.0555H2	−50.117	383.45	69.11
Mg0.9455Zr0.0555H2	−51.192	391.67	75.49

shows the evaluated hydrogen desorption energies for pure and Ti, V, and Zr-doped MgH₂ at various doping levels. The desorption energies for the doped systems range from 69 ➔ 78 kJ/mol, significantly lower than that of pure MgH₂. MgVH₂ exhibits the lowest desorption energy, likely due to the reduced number of H-Mg ionic bonds, as each hydrogen atom is coordinated by only two Mg atoms.

• (b) Formation enthalpy

The enthalpy of formation (ΔH) influences the heat of hydrogenation and dehydrogenation reactions. It is computed by subtracting the total energy of the reactants and products of the reaction.

ΔH = ∑E_tot(products) − ∑E_tot(reactants)

(14)

Equations (3), (4), and (14) have been used to determine the formation enthalpies for the systems. Our calculated enthalpy of formation for pure MgH₂ is −63.961 kJ/mol H₂, which closely agrees with previous theoretical and experimental values: −63.68 kJ/mol H₂ [33], ~−75 kJ/mol H₂ [34], ~−62.01 kJ/mol H₂ [8], 88.571 kJ/mol H₂ [35], and −82 kJ/mol H₂ [36]. The calculated enthalpies of formation for Mg_0.9455V_0.0555H₂, Mg_0.9455Zr_0.0555H₂, and Mg_0.9455Ti_0.0555H₂ are −50.117, −51.192, and −51.212 kJ/mol H₂, respectively. The enthalpies of formation for the doped structures are lower than those of pure MgH₂, suggesting a reduction in stability and a decrease in the dehydrogenation temperature. Doping MgH₂ with Ti, V, and Zr alters its thermodynamic properties, including the enthalpy of formation and the desorption temperature, which arise from the structural and electronic changes induced within the lattice. Due to their smaller atomic radii and distinct electronic properties, each dopant induces crystalline distortions that disrupt electron distribution and weaken Mg-H bonds by modifying the local electron density, thereby promoting hydrogen release. The Mg_0.9455V_0.0555H₂ system exhibits the highest average neutral hydrogen removal energy among the three doped materials, consistent with the Density of States (DOS) analysis presented in Section 3.1.1.

• (c) Desorption temperature of the pure and doped MgH₂

The temperature of the desorption process is a crucial factor in assessing the practical applicability of materials for hydrogen storage, directly impacting the conditions required for hydrogen release from MgH₂. This temperature can be evaluated using the Van’t Hoff equation [37]:

$${\displaystyle \frac{P_{H_2}}{P_0}}=exp\left(-{\displaystyle \frac{\Delta H}{RT}}+{\displaystyle \frac{\Delta S}{R}}\right)$$

(15)

The pressure at equilibrium for H₂ is denoted as P_H2, with the ambient pressure labeled as P₀, the gas constant as R, the absolute temperature as T, and the changes in enthalpy and entropy that occur during the hydrogen desorption reaction represented by ΔH and ΔS, respectively. The desorption energy of molecular hydrogen in our calculation’s desorption reaction is represented by ΔH, and the hydrogen pressure is comparatively small. Since the typical pressure in the atmosphere (P_H2 = P₀ = 1 atm) is more closely associated with actual operating circumstances, we are particularly interested in it. In these circumstances, the desorption temperature can be easily calculated using the formula: Td = ΔH/ΔS.; The change in entropy, ΔS, is estimated to be 130 J mol⁻¹ K⁻¹ for gaseous H₂ in the context of metal hydrides [34].

The estimated desorption temperature for undoped MgH₂ is 487.37 K, consistent with previous theoretical studies [8,37]. The estimated desorption temperatures for the doped systems at 1 atm are 391 K for both Ti- and Zr-doped MgH₂, and 383 K for V-doped MgH₂. These lower temperatures indicate that adding Ti, V, and Zr significantly reduces the energy barrier for dehydrogenation.

Figure 10 illustrates the change in hydrogen desorption temperature (Td) as a function of equilibrium pressure (ranging from 0.01 ➔ 10 atm) for both undoped MgH₂ and MgH₂ doped with Ti, V, and Zr. As expected, T_d increases with increasing pressure. Notably, the curves for the three doped compounds lie below the curve for pure MgH₂, confirming that doping with Ti, V, and Zr lowers the desorption temperature. The relative effectiveness of these dopants in reducing Td follows this order: Mg_0.9455Ti_0.0555H₂ > Mg_0.9455Zr_0.0555H₂ > Mg_0.9455V_0.0555H₂. These trends are consistent with previous findings reported in Ref. [8].

3.2. Macroscopic Investigation: System-Scale Modeling Results

At the system level, the proposed lumped model is developed based on energy and mass balances to simulate the real dynamic behavior of the MgH₂-based hydrogen storage reactor under various doping conditions. Furthermore, this model is completed by the hydrogen kinetic reaction expression to predict the hydrogenation/dehydrogenation process. Figure 11 shows the evolution of the hydrogenation capacity and the reactor temperature during the hydrogenation process of the MgH₂-based hydrogen storage reactor. It should be noted that a thermal management system is integrated to regulate the reactor temperature with a control setpoint fixed at 500 K.

It can be shown clearly that at the onset of the hydrogenation process, the reactor temperature rises rapidly until reaching a temperature of 649 K where it becomes nearly stable. This temperature increase is attributed to the exothermic nature of the hydrogen absorption reaction, which generates significant thermal energy. The used heat transfer fluid effectively absorbs and dissipates this heat, maintaining the reactor temperature within the desired operating range. This result also illustrates the evolution of the hydrogenation process and indicates the time needed to fully charge the hydrogen rector. In this MgH₂-based reactor, the hydrogenation time is about 3.5 min under the specified operating condition.

Through analyzing the instantaneous removed thermal power from the MgH₂-based reactor (Figure 12), it can be observed that high amount of thermal power is removed during the beginning of the hydrogenation process and it about 1.12 kW. This high thermal load corresponds to the initial stage of hydrogen absorption, where the exothermic reaction rapidly generates heat. This removed thermal power decreases with the progress of the hydrogenation process until the end of the process where the reactor bed temperature becomes equal to the heat transfer fluid temperature. This dynamic behavior influences the thermal heating/cooling energy consumption as shown in the same figure. The cumulated removed thermal power from the MgH₂-based reactor increases linearly with the hydrogenation process and the final thermal energy consumption of this system is about 55 Wh to store 1.575 g of hydrogen under the specific operation conditions. It is important to note that this energy value accounts only for the heat removed from the reactor and does not include the energy required to heat up or maintain the heat transfer fluid at the setpoint temperature of 500 K. Based on this, the specific thermal energy removal is about 34.9 Wh/g H₂.

Figure 13 presents the time-wise variation in the reactor temperature during the hydrogenation process for both pure and doped MgH₂ systems using Ti, V, and Zr as doping elements. The reference case, corresponding to pure MgH₂, exhibits a sharp temperature increase due to the highly exothermic nature of the absorption reaction, reaching a peak temperature of approximately 653 K. In contrast, all doped systems show significantly lower maximum temperatures: around 585 K for Ti-doped, 575 K for Zr-doped, and 563 K for V-doped MgH₂. This trend reflects an improved thermal behavior, which is attributed to reduced reaction enthalpy and enhanced hydrogen sorption kinetics due to doping.

These macroscopic observations are in good agreement with micro-scale (DFT simulations) which showed that doping MgH₂, particularly with vanadium, leads to a substantial decrease in both the reaction enthalpy and the activation energy required for hydrogen absorption (see Table 1). This reduction is translated in the macro-scale dynamic model by the reduction in the maximal reached temperature in the hydrogen storage reactor, during the hydrogenation process. Moreover, the use of doped MgH₂ materials results in a lower required operating temperature of the heat transfer fluid (HTF) to manage the thermal load. While the pure MgH₂ system requires the HTF to be maintained at 500 K, the necessary HTF temperatures are reduced to 391 K, 390 K, and 379 K for the Ti-, Zr-, and V-doped systems, respectively.

To show the impact of the doping element on the hydrogenation process, the hydrogen capacity evolution of both pure and doped MgH₂ is given in Figure 14. This result clearly shows that doping the MgH₂ reactor is advantageous and leads to accelerating the hydrogenation kinetics, thereby reducing the total charging time of the reactor.

Compared to the reference case of pure MgH₂, the hydrogenation time is reduced by approximately 21% with Ti doping, 40% with Zr doping, and 42% with V doping. This improvement in reaction kinetics is consistent with the lower activation energies and enhanced thermodynamic behavior of the doped materials, as previously predicted by DFT simulations (see Table 1). The accelerated hydrogen uptake observed at the system scale confirms the beneficial role of doping elements in improving both the reaction rate and operational performance of MgH₂-based hydrogen storage systems.

Figure 15 illustrates the influence of doping on the consumed thermal energy during the hydrogenation process. It is evident that using pure Mg as the metal hydride for hydrogen storage leads to increase significantly the thermal energy consumption by the thermal regulation system. In contrast, the incorporation of doping elements such as Ti, Zr, and V significantly reduces the energy needed to regulate the reactor temperature throughout the absorption phase.

Among the evaluated cases, Vanadium-doped MgH₂ exhibits the lowest energy demand. Specifically, the thermal energy required to complete the hydrogenation process is approximately 55 Wh for the pure MgH₂ reactor, compared to 47 Wh, 46.5 Wh, and 45.5 Wh for the Ti-, Zr-, and V-doped systems, respectively. This energy is primarily used to maintain optimal thermal conditions during the exothermic absorption process and to ensure full hydrogen storage, with a maximum capacity of 1.575 g of H₂ in each case. As a result, the specific thermal energy removal, which is defined as the energy removed per gram of stored hydrogen, decreases from 34.9 Wh/g H₂ for pure MgH₂ to 29.8 Wh/g H₂ with Ti doping, 29.5 Wh/g H₂ with Zr doping, and 28.8 Wh/g H₂ with V doping. These reductions highlight the benefit of doping in improving the overall thermal efficiency of metal hydride-based hydrogen storage systems.

These obtained key performance indicators, including hydrogenation time, percentage reduction, thermal energy consumption, and specific thermal energy removal, are summarized in

Table 2. Summary of system-level hydrogenation performance indicators for pure and doped MgH₂ Reactors.

Material	Hydrogenation Time (min)	Reduction in Time (%)	Thermal Energy Consumption (Wh)	Energy Reduction (%)	Specific Thermal Energy Removal (Wh/g H2)
Pure MgH2	3.5	—	55.0	—	34.9
Ti-doped MgH2	2.76	21%	47.0	14.5%	29.8
Zr-doped MgH2	2.10	40%	46.5	15.5%	29.5
V-doped MgH2	2.03	42%	45.5	17.3%	28.8

. This comparison highlights the beneficial effect of Ti, Zr, and V doping on improving both the thermal and kinetic performance of MgH₂-based hydrogen storage reactors.

It is important to acknowledge that while the present study highlights improvements in hydrogenation kinetics and thermal management through transition metal doping, other critical aspects such as cycling stability, phase stability, and long-term reversibility were not explicitly investigated in this work. These factors are indeed crucial for the commercial adoption of MgH₂-based storage materials, as repeated hydrogenation/dehydrogenation cycles may lead to particle coarsening, structural degradation, or loss of capacity. Experimental studies in the literature have consistently identified cycling degradation and phase transformations as key barriers to practical implementation. The current modeling framework can be extended in future work to incorporate degradation phenomena, for example, by introducing cycle-dependent kinetic or thermodynamic parameters, thereby enabling a more comprehensive assessment of material and system performance under realistic operating conditions.

4. Conclusions

The multi-scale investigation indicates that magnesium hydride’s (MgH₂) ability to store hydrogen is greatly improved by doping it with transition metals including Ti, Zr, and V. By lowering the energy barriers related to hydrogen absorption and desorption, these dopants enhance the thermodynamic properties of MgH₂ at the atomic scale, as confirmed by DFT analysis. The doped materials show enhanced hydrogenation kinetics and improved thermal management at the system level. With the lowest specific thermal energy removal of 28.8 Wh/g H₂ and a 17% reduction in energy consumption, V-doped MgH₂ outperforms the others in terms of thermal efficiency. The importance of the suggested multi-scale modeling technique for the design and optimization of enhanced hydrogen storage systems is demonstrated by these results, which show a strong link between atomic-scale alterations and macroscopic performance improvements.

The findings highlight the critical role of doping in the development of tailored materials and the system-level optimization of hydrogen storage technologies. Future research could focus on exploring a wider range of dopant elements, their concentrations, and the potential synergistic effects of co-doping. In fact, the proposed multi-scale framework can be readily extended to investigate co-doping and multi-component nanocomposites, which are expected to provide synergistic improvements in hydrogen storage performance and will be the subject of future studies. To translate these results into practical and commercially viable storage systems, it will also be essential to combine experimental validation with long-term cycling performance and to investigate the scalability of the approach under diverse operating conditions.