Motivation for study
Hydrogen gas (H 2 ) is an attractive source for potential clean energy because it is most abundant in the universe as part of water, hydrocarbons, and biomass, etc
Utilizing hydrogen (H2) gas as an energy source significantly reduces environmental pollution, as it does not emit CO2 like fossil fuels Recent advancements in material-based hydrogen storage offer a promising solution for safe and efficient hydrogen storage in both transportation and stationary applications To fully harness hydrogen energy, it is essential to develop an integrated system encompassing production, storage, delivery, and fuel cell technologies One of the key challenges in this process is the low density of hydrogen gas, making the pursuit of advanced storage materials crucial for the success of hydrogen energy technology.
The U.S Department of Energy (DOE) has set 2025 targets for hydrogen storage, aiming for a gravimetric capacity of 1.8 kWh/kg (5.5 wt% H2) and a volumetric capacity of 1.3 kWh/L (40 g H2/L) under moderate conditions Various materials, including metal hydrides, carbon-based substances, zeolites, and metal-organic frameworks (MOFs), have been explored for hydrogen storage, with MOFs standing out due to their ultrahigh surface area, high porosity, and controllable structures, making them promising for commercial applications Despite the successful synthesis of thousands of MOFs, only a select few have been evaluated for hydrogen storage The MIL-88 series, known for its coordinatively unsaturated metal sites (CUS), offers a strategic advantage for enhancing gas storage capacity Additionally, MIL-88s exhibit high flexibility and thermal stability, positioning them as excellent candidates for long-term hydrogen storage solutions.
[18] Although MIL-88s structures have been assessed for catalyst [19], photo-catalyst
[20], NO adsorption [21], and CO 2 capture [22], they have not yet been explored for hydrogen storage
In this dissertation, vdW-DFT calculations are utilized to examine favourable adsorption sites of H 2 in the MIL-88s via the adsorption energy The interaction of the
The interaction of H2 molecules with the MIL-88 series is analyzed through various electronic structure properties, including electronic density of states (DOS), charge density difference (CDD), Bader charge, and the overlapping DOS and wave functions between the gas molecule and the metal-organic framework (MOF) Additionally, grand canonical Monte Carlo (GCMC) simulations quantitatively evaluate the H2 storage capability by examining the H2 adsorption isotherms of MIL-88s and determining the strength of the H2-MOF interaction via the isosteric heat of adsorption This research encompasses three main directions.
We selected a metal-organic framework (MOF) from the MIL-88s family for our research, specifically MIL-88A, which has the fewest atoms per primitive unit cell, facilitating faster density functional theory (DFT) calculations for the [H2 + MOF] system In Co-MIL-88A, hydrogen molecules preferentially adsorb at the hollow site of the metal trimers due to optimal overlap between the bonding (σ) state of H2 and the total density of states of Co-MIL-88A Furthermore, hydrogen adsorption isotherms were evaluated using grand canonical Monte Carlo simulations, revealing that Co-MIL-88A is a competitive candidate for efficient hydrogen storage materials.
By modifying the ligands in MIL-88 variants, including MIL-88A, MIL-88B, MIL-88C, and MIL-88D, researchers quantitatively estimated absolute and excess hydrogen (H2) uptake through Grand Canonical Monte Carlo (GCMC) simulations The findings revealed that MIL-88D exhibited the highest gravimetric hydrogen storage capacity, achieving 5.15 wt% at 100 bar and 4.03 wt% at 25 bar at 77 K, along with 0.69 wt% and 0.23 wt% at 100 bar and 298 K, respectively.
MIL-88A exhibits exceptional volumetric hydrogen storage capabilities, achieving a total storage of 50.69 g/L (44.32 g/L at 15 bar) at 77 K and 6.97 g/L (2.49 g/L) at 298 K, making it comparable to the best metal-organic frameworks (MOFs) for hydrogen storage Notably, the interaction between hydrogen and the MIL-88 series, particularly MIL-88C, is characterized by strong bonding, despite MIL-88C's lower storage capacity This interaction is primarily influenced by the bonding states of hydrogen with the p orbitals of oxygen and the d orbitals of iron atoms, highlighting the unique properties of the MIL-88 series in hydrogen adsorption.
MIL-88D is the best choice for hydrogen storage based on the commercialization, stability and high storage capacity However, if the volumetric H 2 storage is concerned, MIL-88A is noteworthy
To improve the hydrogen storage capacity of MIL-88s, it is essential to explore the substitution of metal centers in this MOF material MIL-88A, known for its high volumetric H2 storage, has been selected for this purpose, replacing its metal component with trivalent transition metals such as Sc, Ti, V, Cr, and Mn, in contrast to the previously studied Fe- and Co-based MIL-88A Calculations indicate that the binding energy of H2 with V-MIL-88A is the strongest at -17 kJ/mol in a side-on configuration However, Sc-MIL-88A exhibits the highest H2 adsorption isotherm and heat of adsorption (Qst) among the M-MIL-88A structures analyzed At 77 K, Sc-MIL-88A achieves maximum absolute and excess H2 loadings of 5.13 wt% at 50 bar and 4.63 wt% at 10 bar, while at 298 K and 100 bar, the loadings are 0.72 wt% and 0.29 wt%, respectively.
Structure of PhD dissertation
The structure of this dissertation consists of 6 chapters and the supporting contents, described as follows
- Introduction: introduce the motivation of the work and the outline of this dissertation
- Chapter 1: Literature review of metal-organic frameworks
In this chapter, an overview of the metal-organic framework, the main applications of MOFs, the overview of experimental and computational research methods in the literature are introduced
In this part, I introduce the theory of the computational methods that are density functional theory (DFT) using revPBE functional and Grand canonical
Monte Carlo (GCMC) simulations We also provide computational details for the concerns of this dissertation
- Chapter 3: Hydrogen adsorption in Co-MIL-88A
This chapter investigates the hydrogen adsorption properties of Co-MIL-88A, detailing the physical mechanisms behind the interaction between hydrogen (H2) and the material Initially, the most favorable adsorption sites for H2 are identified through adsorption energy calculations, followed by an analysis of the electronic properties using van der Waals density functional theory (vdW-DFT) Ultimately, the hydrogen adsorption isotherms for Co-MIL-88A are determined through grand canonical Monte Carlo (GCMC) simulations.
- Chapter 4: Hydrogen storage in MIL-88 series
The MIL-88 series, comprising MIL-88A, B, C, and D, is evaluated for its hydrogen storage capacity GCMC simulations are employed to quantitatively analyze the hydrogen uptake of these MIL-88 sorbents, utilizing H2 adsorption isotherms at a temperature of 77 K.
298 K with the pressures below 100 bar using the GCMC simulations The vdW-DF calculations elucidate the interaction between the H 2 molecule and the MIL-88 series
- Chapter 5: Effects of transition metal substitution in MIL-88A on hydrogen adsorption
This research evaluates the hydrogen storage capacity of MIL-88A and investigates the effects of trivalent transition metal substitution (Sc, Ti, V, Cr, Mn, Fe, and Co) on M-MIL-88A The study calculates the adsorption energies of H2 in side-on and end-on configurations near the metal centers using the vdW-DF approach to identify the most stable arrangements Additionally, the electronic properties of these stable adsorption configurations are analyzed GCMC simulations are employed to study the hydrogen adsorption isotherms at 77 K and 298 K, along with the isosteric heats of hydrogen adsorption across the M-MIL-88A series.
This chapter highlights the main findings, scientific contributions, and an outlook in the near future
LITERATURE REVIEW OF METAL-ORGANIC
Structural aspects of MOFs
Metal ions, serving as connectors, along with organic linkers or ligands, form the essential primary building units (PBUs) that create the porous three-dimensional structure of metal-organic frameworks (MOFs).
Transition metal ions are often used as versatile connectors in the construction of MOFs Common transition metal ions in MOFs are Zn 2+ , Co 2+ , Ni 2+ , Cu 2+ , Fe 2+ , Fe 3+ ,
Cr 3+ ions, as first-row transition metal ions, are commonly used in the construction of metal-organic frameworks (MOFs), alongside other metal ions such as alkali metal ions (Mg 2+), alkaline-earth metal ions (Al 3+), and rare-earth metal ions (In 3+, Ga 3+) The coordination numbers and geometries of these metal connectors are crucial, with transition metals typically exhibiting coordination numbers ranging from 2 to 6, while lanthanides can range from 6 to 12 These coordination geometries, which include linear, T- or Y-shaped, square-planar, tetrahedral, square-pyramidal, octahedral, and polyhedral forms, significantly influence the structural design of MOFs.
Figure 1.3 Several common coordination geometries of metal ions used for MOF construction The numbers indicate the numbers of functional sites [30] b Organic ligands
Organic ligands or linkers typically feature coordinating functional groups such as carboxylate, phosphate, sulfonate, amine, and nitrile Notable ligands used in metal-organic frameworks (MOFs) include benzene-dicarboxylate (BDC), benzene-tricarboxylate (BTC), polycarboxylate (BTB), as well as imidazole, pyrazole, triazole, and tetrazole, along with various mixed ligands, as detailed in Table A1 Examples of these organic ligands are illustrated in Figure 1.4.
Figure 1.4 Several common organic ligands used for MOF construction [28]
Organic linkers in metal-organic frameworks (MOFs) are interconnected through metal-oxygen-carbon clusters, known as secondary building units (SBUs), rather than solely through metal ions These SBUs possess inherent geometric properties that enhance the structural integrity and functionality of MOFs.
The topology of Metal-Organic Frameworks (MOFs) is illustrated in Figure 1.5, showcasing various common Secondary Building Units (SBUs) In this representation, metal polyhedrons are depicted in blue, while oxygen and carbon atoms are shown in red and black, respectively The red polygons or polyhedrons, defined by carboxylate carbon atoms, serve as extension points and highlight the arrangement of carboxylate carbons at the vertices of trigonal prismatic geometries.
Figure 1.5 Several common SBUs of MOFs [31, 32].
History of MOFs
Metal-organic crystals with open spaces were first identified by Alfred Werner, the pioneer of coordination chemistry, over a century ago, with the term "coordination compound" originally referring to Co(NH3)6Cl3 Early solid-state chemistry recognized these crystal structures for their potential in host-guest adsorption; however, their fragility often led to the collapse of their porous structures upon the removal of guest molecules In the 1990s, advancements in the field, particularly by Prof Omar Yaghi's research group at the University of California Berkeley, resulted in the development of metal-organic frameworks (MOFs), initially termed porous coordination networks (PCNs) and microporous coordination polymers (MCPs), which successfully addressed these limitations by creating stable frameworks capable of maintaining permanent porosity.
MOF-5, or Zn4O(BDC)3, is a widely recognized metal-organic framework (MOF) with a Brunauer-Emmett-Teller (BET) surface area of 3800 m²/g, making it the first MOF studied for hydrogen adsorption Since then, significant advancements have been made in the design and synthesis of various MOFs, leading to improvements in both their quality and quantity As illustrated in Figure 1.6, the Cambridge Structural Database (CSD) has documented the development of numerous 1D, 2D, and 3D MOFs up to 2015.
Figure 1.6 1D, 2D, and 3D MOF structures from 1970 to 2015 [35]
During the last two decades, MOFs continuously set new records in terms of specific surface areas, pore volumes, and gas storage capacities MOF-177 and MOF-
Two metal-organic frameworks (MOFs), NU-109 and NU-210, have been technically evaluated for their impressive capabilities in hydrogen (H2) storage and carbon dioxide (CO2) capture These MOFs demonstrate exceptionally high storage capacities at 77 K and operate effectively under relatively low pressures, specifically below 100 bar.
110 (Table A2) exhibited the highest experimental S BET with 7000 m 2 /g and 7140 m 2 /g (Figure 1.7), respectively [38] The theoretical researchers also estimated based on the
Metal-Organic Frameworks (MOFs) have demonstrated a remarkable potential for high surface areas, with theoretical maximum BET surface areas reaching approximately 14,600 m²/g or more Currently, there are thousands of different MOFs, which continue to evolve and develop In comparison to traditional inorganic materials like zeolites and silicas, which have average surface areas below 1,000 m²/g, and activated carbons under 2,000 m²/g, MOFs exhibit significantly larger specific surface areas Additionally, the pore volume of MOFs surpasses that of conventional porous materials, enhancing their gas storage and capture capabilities.
Figure 1.7 NU-110 structure with the highest BET surface area of MOFs reported until now
[23] with CCDC data taken from [38].
Figure 1.8 BET surface areas (m 2 /g) and pore volumes (cm 3 /g) of representativeMOFs, compared to conventional porous materials (zeolites, silicas, and activated carbons) From
Ref [29] Reprinted with permission from AAAS.
Nomenclature of MOFs
Metal-Organic Frameworks (MOFs) are typically named based on their isoreticular synthesis sequences, the order of their discovery, or the initials of the institutions where they were first synthesized.
Naming by the sequence of isoreticular synthesis
Certain metal-organic frameworks (MOFs) are classified by their shared network topologies, particularly those exhibiting a common cubic structure formed from identical organic ligands A notable example is the zinc-based isoreticular MOF (IRMOF) series, which includes a range from IRMOF-1 to IRMOF-16.
Naming by the sequential number of synthesis
The numerical designation of a Metal-Organic Framework (MOF) reflects the sequence of its discovery or synthesis, with examples including MOF-2, MOF-3, MOF-4, MOF-5, and MOF-177.
Naming by the initials of Institution or Laboratory of discovery
Metal-Organic Frameworks (MOFs) can also be named using acronyms derived from the institutions or laboratories where they were first synthesized For instance, HKUST stands for Hong Kong University of Science and Technology, MIL represents Material Institute Lavoisier, NENU is North East Normal University in China, VNU refers to Vietnam National University, and NU denotes Northwestern University.
Current research of MOFs in Vietnam
In Vietnam, various research groups have focused on Metal-Organic Frameworks (MOFs), notably the experimental research team led by Nam T S Phan at the Faculty of Chemical Engineering, HCMC University of Technology This group has successfully synthesized a range of MOFs, including MOF-5 (IRMOF-1), IRMOF-3, IFMOF-8, MOF-199, Cu(BDC), and Cu2(BDC)2.
Research on metal-organic frameworks (MOFs) such as Co2(BDC)2(DABCO) and Cu2(BPDC)2(DABCO) has highlighted their applications in heterogeneous catalysis Notable contributions have come from various institutions, including the Center for Innovative Materials and Architectures (INOMAR) and the Vietnam Academy of Science and Technology (VAST) The VNU MOF series, developed by the Vietnam National University, includes synthesized frameworks like VNU-10, VNU-15, and VNU-20, which have shown promise in applications such as heterogeneous catalysis and proton conduction Additionally, computational research by Dr Nguyen-Nguyen Pham-Tran and Dr Hung M Le has employed DFT and GCMC simulation methods to further investigate these MOFs.
Figure 1.9 Application areas of porous reticular materials (MOFs, ZIFs and COFs) [35]
Metal-Organic Frameworks (MOFs) are distinguished by their unique combination of organic and inorganic components, resulting in remarkable structural properties including large surface areas, high pore volumes, and ultrahigh porosity These characteristics enable complete exposure of metal sites and facilitate the high mobility of guest species within their nanopores Due to these advantages, MOFs are increasingly utilized in various applications, including catalysis, gas capture and storage, gas separation and purification, sensing, biological applications, and semiconductor technologies.
1.2.1 Gas storage, capture, and separation
Current gas storage techniques include cryogenic vessels, high-pressure tanks, and methods such as chemisorption and physisorption While these approaches have reached nearly practical storage capacities, significant improvements and cost reductions are still necessary for widespread adoption.
Pressurized tank-based hydrogen storage for electrical energy faces safety and economic challenges In contrast, the chemisorption method enhances gas storage density by forming chemical bonds between gases and storage materials, although it still struggles with kinetics, reversibility, and heat management Metal-organic frameworks (MOFs), a type of porous material, have been primarily studied for gas storage and capture through physical adsorption MOF-based storage technologies offer advantages such as rapid kinetics and complete reversibility, which can lower costs due to the ease of desorption and the reusability of MOFs.
Hydrogen gas is an environmentally friendly and non-toxic energy source that can address the impending global energy crisis However, its volatile nature necessitates high-pressure storage for onboard use, which raises safety concerns and costs Metal-organic frameworks (MOFs) are gaining attention for hydrogen storage due to their ultra-high surface areas and large pore volumes, making them advantageous for physisorption-based gas storage solutions.
Metal-organic frameworks (MOFs) have demonstrated superior hydrogen storage capacity compared to traditional methods like empty tanks and zeolite X13 The initial investigation into hydrogen storage in MOF-5, conducted in 2003, revealed gravimetric uptakes of 4.5 wt% at 78 K and 0.8 bar, and 1 wt% at 298 K and 20 bar, sparking significant interest and new research avenues in computational simulations In 2004, Hüber et al pioneered the use of MP2 calculations to analyze the interaction of hydrogen with benzene and naphthalene, reporting adsorption energies of 3.91 kJ/mol and 4.28 kJ/mol, respectively Additionally, the quantitative evaluation of hydrogen storage in MOF-5 was first calculated, further emphasizing the potential of MOFs in hydrogen storage applications.
2004 using GCMC simulations with the universal force field (UFF) by Ganz group
Recent research identified a high energy binding site at the corners of the metal-organic framework (MOF), achieving a hydrogen uptake of 1.27 molecules per Zn4O(BDC)3 formula unit, equivalent to 4.5 wt%, at 78 K Additionally, various research groups have utilized Grand Canonical Monte Carlo (GCMC) simulations to explore the hydrogen gas adsorption isotherms of MOF structures, employing different force fields such as DREIDING and the optimized potential for liquid simulations (OPLS), with further details available in the literature.
Figure 1.10 H 2 storage in MOFs relatively compared to zeolite and empty tank Reproduced with permission of Royal Society of Chemistry from Ref [74]
The experimental record in the highest total (or absolute) hydrogen storage capacity reported so far was found in MOF-210 with 17.6 wt% (176 mg/g) at 77 K and
80 bar, while in this same condition the excess uptake obtained 8.6 wt% (86 mg/g)
NU-100 exhibits the highest excess hydrogen uptake of 9.95 wt% at 56 bar and 77 K, with an absolute uptake of 16.4 wt% at 70 bar Other notable metal-organic frameworks (MOFs) include MOF-200, which shows 7.4 wt% excess and 16.3 wt% total uptake at 77 K and 100 bar, and MOF-205 with 7.0 wt% excess and 12.0 wt% total uptake at 77 K and 80 bar However, due to weak hydrogen-MOF interactions and low isosteric heat of hydrogen adsorption (4-13 kJ/mol), these frameworks demonstrate significant hydrogen uptake primarily at cryogenic temperatures, yielding only around 1 wt% excess and 2.3 wt% absolute uptake at room temperature and pressures below 100 bar None of the MOFs have achieved the Department of Energy's targets under moderate temperature and pressure conditions.
Absolute pressure conditions play a crucial role in developing new metal-organic frameworks (MOFs) with enhanced properties for hydrogen storage Strategies to improve hydrogen storage capacity at ambient temperatures include utilizing MOFs with open metal sites, which significantly boost hydrogen adsorption Additionally, achieving an isosteric heat of hydrogen adsorption between 15 and 25 kJ/mol is essential for effective onboard applications Recent advancements in computer simulations have enabled the prediction and design of innovative MOFs that can greatly enhance hydrogen uptake.
Table 1.1 High hydrogen uptakes in MOFs at 298 K and pressures below 100 bar
Methane (CH4) is a crucial hydrocarbon fuel known for its high energy density and low carbon emissions due to its favorable hydrogen-to-carbon ratio The initial research on CH4 storage in Metal-Organic Frameworks (MOFs) began in 1997 by Kitagawa's team, achieving limited uptake of 2.3 mmol/g at 30 atm and 298 K They developed coordination polymers with three-dimensional frameworks and large cavities, allowing for enhanced methane adsorption compared to traditional storage methods Since then, numerous MOFs have been synthesized, focusing on factors like high surface area, ligand functionalization, and open metal sites, which significantly improve CH4 adsorption capacity Computational simulations suggest that the presence of open metal sites enhances the binding strength of methane, increasing its affinity for the metals Recent studies by H Wu et al evaluated six promising MOFs for CH4 storage, revealing that HKUST-1 achieved the highest volumetric uptakes of 230 cc(STP)/cc at 298 K and 35 bar.
At 298 K and 65 bar, a specific metal-organic framework (MOF) has achieved a record methane uptake, meeting the Department of Energy's new volumetric target of 263 cc(STP)/cc Other MOFs, including NU-111, Ni-MOF-74, and PCN-14, have reached up to 70% of the DOE's gravimetric and volumetric goals However, in terms of the gravimetric target, MOFs have only achieved less than 0.5 g/g of the DOE's benchmark.
Methane storage has generally met the Department of Energy's (DOE) targets, but further advancements are needed for greater economic viability, while hydrogen storage targets remain elusive To enhance hydrogen storage capacity, various strategies have been developed, including the creation of open metal sites, metal ion doping, fabrication of metal nanoparticles to leverage the spillover effect, functionalization of ligands, and the catenation/interpenetration of frameworks These approaches have significantly improved gas storage capacity, positioning Metal-Organic Frameworks (MOFs) as the leading material for both methane and hydrogen storage.
Figure 1.11 CH4 storage in MOFs relatively compared to active carbon and empty tank Reproduced with permission of Royal Society of Chemistry from Ref [74]
Metal-organic frameworks (MOFs) have demonstrated significant potential for reducing atmospheric CO2 levels A systematic study conducted in 2005 on nine MOF structures revealed a correlation between surface area and CO2 uptake capacity under high pressures Notably, MOF-177 exhibited the highest Langmuir surface area of 5640 m²/g and a CO2 uptake of 33.5 mmol/g at 35 bar and room temperature, outperforming conventional porous materials like zeolites 13X and activated carbon (MAXSORB) Additionally, MOF-200 and MOF-210 achieved a remarkable CO2 uptake of approximately 2400 mg/g at 298 K and 50 bar, setting a new record for CO2 adsorption capacity among all porous materials.
Figure 1.12 Volumetric CO2 capacity of MOF- 177 relatively compared to zeolite 13X pellets, MAXSORB carbon powder, and pressurized CO 2 Reproduced with permission from
Ref [37] Copyright © 2005, American Chemical Society
Figure 1.13 CO 2 uptakes of MOFs at 298 K Reprinted from Ref [36] Reprinted with permission from AAAS
Research has focused on developing Metal-Organic Frameworks (MOFs) for both gas storage and separation applications Numerous MOFs have demonstrated significant potential for the selective separation of gas mixtures, achieved by systematically adjusting pore and window sizes.
Biomedical applications
Liposomes made from polymers, amorphous silica, and zeolites are commonly used for drug delivery but face challenges like low drug storage capacity, rapid release, and toxicity from heavy metals Metal-Organic Frameworks (MOFs) are emerging as superior drug carriers due to their large pore volume and high flexibility They offer several advantages: low toxicity from biocompatible metals, biodegradability, adjustable hydrophilicity/hydrophobicity, efficient drug uptake, and controlled release that mitigates the "burst effect." Additionally, MOFs can store high drug quantities, reducing the need for extensive carrier materials The combination of non-toxic metals and customizable linkers enhances MOFs' appeal for delivering drug molecules and biological gases.
Figure 1.14 Schematic diagram of the drug and biomedical gas delivery by MOFs [23]
Additionally, MOFs have other potential applications such as SO 2 , CO capture,
NO adsorption, catalysts, sensing/sensors, ion exchange, and magnetic materials Among potential applications of MOFs, we pay special attention to hydrogen storage for clean energy application
Overview of synthesis and research methods for MOFs
Metal-Organic Frameworks (MOFs) are synthesized through solvothermal reactions involving organic ligands and metal salts at temperatures below 300°C The reactants are dissolved in polar solvents such as water, dimethyl sulfoxide, and acetonitrile Key parameters influencing MOF synthesis include temperature, concentrations of metal salts and ligands, pH levels, and the solubility of reactants The characteristics of the ligands, including their lengths, bond angles, and chirality, significantly impact framework formation, while the tendency of metal ions to adopt specific geometries also plays a crucial role The solvothermal method is the most prevalent synthesis technique, accounting for up to 70% of MOF production, due to its simplicity and controllability, utilizing thermal energy to heat the mixture of organic ligands and metal salts in high boiling solvents.
The synthesis of fine particle powders typically takes between 48 to 96 hours and is often more efficient than conventional methods Various techniques are employed in this process, including microwave-assisted synthesis, sonochemical synthesis, slow evaporation synthesis, electrochemical synthesis, and mechanochemical synthesis.
Most synthesized Metal-Organic Frameworks (MOFs) utilize commercially available ligands, making it challenging to integrate desired functional groups during their synthesis To address this issue, post-synthetic modification (PSM) strategies have been developed, allowing for the addition of functional groups to pre-synthesized MOFs to meet specific application goals PSM can involve processes such as protonation or doping through non-covalent, coordinate, or covalent interactions This approach enables the creation of new MOFs without direct synthesis and offers significant advantages, including the introduction of multiple functionalities within the same framework and the production of diverse MOFs with identical topology but varying functionalities.
Figure 1.15 (a) Synthesis conditions commonly used for MOF preparation; (b) indicative summary of the percentage of MOFs synthesized using the various preparation routes [75]
Figure 1.16 Schematic representation of PMS [110].
Solvo the rm al Microwave Electrohemica l Mechanochemic al Sonochemical
Theoretical studies
Theoretical studies are now recognized as equally valuable as experimental techniques in material science, providing insights into experimental outcomes and guiding costly and time-consuming research Specifically, the use of computational methods to explore the hydrogen adsorption properties of Metal-Organic Frameworks (MOFs) is essential for developing efficient solutions to the hydrogen storage challenge.
Computational methods for studying molecular systems primarily fall into two categories: quantum mechanical (QM) calculations and classical physics simulations QM methods, such as Müler–Plesset perturbation theory, coupled cluster, and density functional theory (DFT), are essential for analyzing the energetics of hydrogen interactions with adsorbents These calculations reveal the strength and nature of these interactions, identify available binding sites in metal-organic frameworks (MOFs), and provide insights into the electronic properties of hydrogen-sorbent interactions In contrast, classical simulations, including Monte Carlo, molecular dynamics, and molecular mechanics, complement QM methods in exploring molecular behavior.
Classical simulations enable the assessment of hydrogen adsorption capacity in metal-organic frameworks (MOFs) under varying thermodynamic conditions, such as temperature and pressure These simulations can generate absorption isotherms and hydrogen uptake data that can be directly compared to experimental results A key advantage of classical simulations is their ability to handle larger systems due to their computational efficiency, unlike quantum mechanics (QM) methods Additionally, they account for the influence of pressure and temperature on the simulation outcomes However, a significant drawback is that the accuracy of classical simulations is contingent upon specific parameters.
The analysis of quantum and classical methods reveals that each has significant strengths and notable drawbacks Recently, a new approach known as the multiscale methodology has emerged, designed to effectively address large-scale systems while achieving the desired level of accuracy, as illustrated in Figure 1.17.
Figure 1.17 Multiscale methodology scheme, showing the different levels of theory and the corresponding size of systems under study Reproduced with permission of Royal Society of
The initial scale of the Metal-Organic Framework (MOF) comprises the fundamental building units of its skeleton At this level, we can utilize highly precise ab initio methods, such as Møller-Plesset perturbation theory (MP) and Coupled Cluster (CC) techniques, to accurately calculate the binding interactions of hydrogen molecules.
- The second scale contains larger molecular models (more than 50 atoms) or a unit cell of the periodic structure where DFT or mixed methods involving QM and
The QM/MM approach enables the prediction of essential data for developing an ab initio interatomic potential that characterizes the interaction between hydrogen molecules and metal-organic framework (MOF) pores In this study, the density functional theory (DFT) method is employed to determine the adsorption energy of H2 within MOFs.
In my dissertation, I conducted Grand Canonical Monte Carlo (GCMC) simulations to replicate isotherms for hydrogen (H2) adsorption across various thermodynamic conditions, enabling direct comparisons with experimental data Additionally, the third scale allows for Molecular Dynamics (MD) or Monte Carlo (MC) simulations that account for thousands of atoms, enhancing the accuracy of our findings.
Despite the synthesis of thousands of Metal-Organic Frameworks (MOFs), only a limited number have been assessed for hydrogen storage, particularly at ambient temperatures and low pressures The MIL-88 series stands out as it has been explored for various applications but remains untested for hydrogen storage Comprising trimeric building units formed by three metallic octahedra linked through a μ3-O atom, the MIL-88 series incorporates organic ligands such as fumarate (MIL-88A), terephthalate (MIL-88B), 2,6-naphthalenedicarboxylate (MIL-88C), and 4,4'-biphenyldicarboxylate (MIL-88D), resulting in a three-dimensional porous network The MIL-88 series is of particular interest due to its unique structural properties and potential for hydrogen storage applications.
For effective gas storage, Metal-Organic Frameworks (MOFs) must demonstrate stability in liquid environments to prevent structural collapse in humid conditions The MIL-88 series is particularly notable for its exceptional flexibility and stability These MOFs can undergo significant swelling when immersed in different liquids, exhibiting reversible changes in unit cell volume ranging from 85% for MIL-88A to 240% for MIL-88D, while maintaining their structural integrity and open framework topology.
To enhance gas uptake capacity, the most effective approach is to generate open metal sites In the case of MIL-88s, these exposed metal sites can be readily achieved by applying heat treatment to eliminate counter-anions at each metal position.
So far, no works have been performed to evaluate MIL-88 series for H 2 storage
This dissertation explores the MIL-88 series for hydrogen storage, employing advanced computational methods, specifically dispersion-corrected density functional theory and grand canonical Monte Carlo simulations The findings assess the potential of MIL-88 for effective hydrogen storage, providing valuable insights that serve as important references for experimental research and advancing the field of hydrogen storage in energy applications.
The Schrửdinger equation
The many-body problem is a significant challenge in physics that focuses on understanding the properties of microscopic systems composed of numerous interacting particles This complex issue is addressed through the time-dependent Schrödinger equation, which provides a general framework for analyzing such systems.
H r R t E r R t (2.1) where ˆH is the Hamiltonian operator, E is the total energy, and is the wave function of the system
For the N-atom system, the Hamiltonian operator ˆH is given by ˆ ˆ e ˆ N ˆ ee ( ) ˆ NN ( ) ˆ Ne ( , ),
H T T V r V R V r R (2.2) in which, r stands for a set of r r 1 , , 2 , r N e that are coordinates of N e electrons, and
R stands for a set of R R 1 , 2 , , R N that are coordinates of N nuclei The kinetic energy and potential energy operators of the system are listed below
(2.3) here, m e is the mass of the electron
(2.4) in which, M I is the mass of nucleus I (I 1, 2, ,N)
+ The electron-electron interaction (i.e the electron-electron repulsion energy):
(2.5) here r i is the coordinate of the electron i (i1, 2, ,N e )
+ The inter-nuclear interaction (i.e the nucleus-nucleus repulsion energy):
(2.6) where factor 1/2 in (2.5) and (2.6) equations implies the uniformity of electrons/nuclei;
Z I and R I are the charge and the coordinate of nucleus I, respectively
+ The electron-nucleus attraction energy operator:
Therefore, the total Hamiltonian of the N-atom system is
Here, I and J run over the N nuclei while i and j denote the N e electrons in the system
By substituting (2.2) into (2.1), we obtained the following equation
To solve equation (2.9), appropriate boundary conditions must be applied, such as decaying to zero at infinity for atoms or molecules, or adhering to periodic boundary conditions for infinite solids The solution yields the energy E and the wave function Ψ, from which the probability distribution function Ψ² can be derived However, solving this equation becomes increasingly complex as the number of particles N increases.
To effectively separate the equation (2.9) into distinct equations for electrons and nuclei, it is essential to apply the Born-Oppenheimer and adiabatic approximations These critical approximations will be discussed in detail in the following sections.
Born-Oppenheimer and adiabatic approximations
In electronic structure calculations, the mass of nuclei is significantly greater than that of electrons, resulting in much slower movement of the nuclei compared to the electrons This allows electrons to quickly adapt to the nuclei's motion, leading to the assumption that nuclei are approximately stationary Consequently, the movement of nuclei is considered not to induce excitations in the electronic system, allowing them to be treated as fixed points in space By applying these approximations, we can effectively separate the wave function of the system into two distinct components.
(2.10) where r R , and R t , are wave functions of electrons and nuclei, respectively The wave function of nuclei is more localized than that of electrons; therefore, we have the following approximation
On the basis of these approximations, we can obtain the decoupled adiabatic
Schrửdinger equations of electrons and nuclei
The Hamiltonian for electrons in the system is expressed as H ˆ e T ˆ e V ˆ ee V ˆ Ne, where V ˆ Ne represents the nucleus-electron interaction The electronic energy, E R , is dependent on the ionic positions R and is determined by solving the time-independent Schrödinger equation (2.12) for a specific configuration The most effective method for obtaining solutions to this equation is density functional theory (DFT), which is founded on the principles established by the Hohenberg and Kohn theorems.
Thomas-Fermi theory
Traditional methods in quantum mechanics focus on the wave function Ψ, which encapsulates all the information about a system However, Ψ is highly complex and cannot be experimentally measured, depending on 3N variables for N electrons, or 4N variables in spin-polarized systems.
The Thomas-Fermi model, introduced in 1927, represents the first density functional theory, grounded in the concept of a uniform electron gas Both Thomas and Fermi, in their respective works from 1927 and 1928, developed the initial framework for density functional theory, which includes a specific functional for calculating kinetic energy.
The energy of an atom is finally obtained using the classical expression for the nucleus-nucleus potential and the electron-electron potential
To find the appropriate density for inclusion in equation (2.15), the researchers utilized the variational principle They posited that the system's ground state correlates with the density ρ(r) that minimizes energy while adhering to specific constraints.
Hohenberg- Kohn theorems
In 1964, on the basis of the new DFT, Hohenberg and Kohn [119] showed that this principle could be generalized to any electronic system a Theorem 1
The first Hohenberg-Kohn theorem demonstrates that the electron density uniquely determines the Hamiltonian operator and thus all the properties of the system
If we consider specifically the ground state, the total energy must be extremal, that allows formulating a more strong condition:
The ground-state energy of a many-body system is a unique functional of the electron density o r o o
The energy functional E r has its minimum relative to variations r of the electron density at the equilibrium density o r ,
Variational condition and Levy constrained search formulation
Variational principle for the ground state
When a system is in the state , the average value of the expected energy is calculated by
The variational principle states that the energy computed from a guessed Ψ is an upper bound to the true ground-state energy E o :
The full minimization of the functional E[Ψ] with respect to all wave functions of
N e -electrons will give the true ground state Ψ o and energy E[Ψ o ] = E o :
Therefore, every eigenstate is an extremum of the functional E[Ψ] In other words, one can substitute the Schrửdinger equation by the variational principle that is
This equation is written to ensure the normalized condition of the wave function
A universal functional of the electron density is defined by the constrained search for a given N e -electron density r (Levy, 1979 [120])
(2.25) in which, the electron density satisfies r dr N , (2.26) and 1 2 r 2 dr (2.27)
These conditions remove the non-physical values of the electron density.
The Kohn-Sham equations
In 1927, the Thomas-Fermi model introduced the first density functional theory based on a uniform electron gas, but its effectiveness was limited due to inadequate kinetic energy approximations To address this issue, Kohn and Sham developed a new approach in 1965, building on the foundational concepts of Hohenberg and Kohn Their method reformulates the challenge of determining the correct electron density into a solvable set of equations, each focused on a single electron, assuming a Hamiltonian for a non-interacting N electron system.
(2.28) where u R r i is the reference potential The Schrửdinger equation for one electron in the system of non-interacting electrons has the form:
The eigenstates of the N e -electron system can be expressed in the form of the Slater determinant
The equation (2.29) is called the Kohn-Sham equation for one electron It is re- written as follows
H r r (2.31) in which, Hamiltonian of N e electrons is
The electron density is also determined by
The energy of N e electrons is
E r H T V V (2.34) or E r F r V ˆ Ne , (2.35) here F T ˆ e V ˆ ee is the universal functional consisting of the contributions of the kinetic energy and the classical Coulomb interaction of the electrons in the N e - electron system
Substituting of (2.30) into (2.37) and (2.38), via some steps, we determine:
Therefore, the universal density functional is written as follows
The correlation and exchange energy, denoted as E XC, is represented by the equation E XC[ρ] = E C[ρ] + E X[ρ] This energy is essential for correcting the deficiencies in the kinetic and potential energy of a homogeneous electron gas when transitioning from an interacting system to a non-interacting system.
By substituting F functional in the equation (2.42) into (2.35), we obtain the
Minimizing the energy functional (2.43), based on the variable principle
The chemical potential (μ) of a system of interacting electrons is defined as μNe = E, where Ne represents the total number of electrons, calculated using Ne = ∫ρ(r) dr In contrast, determining the energy of a system of non-interacting electrons is a straightforward process.
Applying the variable principle, we also determine
To transition from the interacting many-body problem to a non-interacting framework, it is essential that the electronic densities or chemical potentials of both systems are equivalent This relationship is established through equations (2.45) and (2.47).
(2.49) in which, the exchange-correlation potential
If u R r is determined by (2.49) (i.e V Ne r and XC r are determined), we can solve the Schrửdinger equation for one electron (i.e Kohn - Sham equation):
Exchange-correlation functional
The Kohn-Sham equation requires knowledge of the exchange-correlation functional for a solution, but its exact form remains unknown, leading to the use of approximations Two widely used approximations in this context are the Local Density Approximation (LDA) and the Generalized Gradient Approximation (GGA).
The exchange-correlation functional can be determined by the exchange functional and correlation functional [116]:
- The exchange functional was accurately described by Dirac's expression:
is the average distance between electrons, is the electronic density
- The correlation energy of an electron gas was parameterized by Perdew and Zunger (1981):
For the case r s 1, the expression was derived from the random phase approximation by Gell-Mann and Brueckner (1957)
The coefficients A, B, C, D, , 1 , and 2 were determined:
+ For non-spin-polarized systems:
The exchange-correlation functional of LDA is defined by:
E r r dr (2.56) here, XC LDA stands for the exchange-correlation energy density functional for the homogeneous electron gas
+ X LDA : the exchange energy density functional determined by (2.52);
+ C LDA : the correlation energy density functional determined by (2.53)
For the spin-polarized system, the electronic density was composed by two independent spin densities r with spin up and r with spin down Therefore, the equation (2.56) was written by
+ r r r : the spin-polarization (or magnetization) density;
+ XC LSDA , f U XC 1 f XC P : the density functional of exchange-correlation energy Herein, f was proposed by von Barth and Hedin:
The local density and spin density approximations (LDA and LSDA) are effective for systems with uniform electronic density, such as bulk metals, and also for less uniform systems like molecules, semiconductors, and ionic crystals While LDA provides reasonably accurate ionization energies, dissociation energies, and cohesive energies within 10-20%, it achieves remarkable precision in bond lengths of molecules and solids, often within 2% Despite this level of accuracy, LDA is generally insufficient for many chemistry applications and can struggle with systems dominated by strong electron-electron interactions, such as heavy fermions.
To address the non-homogeneity of true electron density, we can expand the density using gradient and higher order derivatives This leads us to express the exchange-correlation energy through the generalized gradient approximation (GGA).
The equation E = ∫ρr ε[ρr F[ρr ∇ρr ∇ρr]dr (2.60) illustrates how the exchange enhancement factor FX modifies the local density approximation (LDA) based on density variations near a specific point This enhancement factor, FX, can be expanded up to the fourth order gradient to provide a more accurate representation of the system's behavior.
(2.62) is the square of the density gradient, and
(2.63) is the Laplacian of the density
The coefficient D has not been calculated explicitly In numerical calculation, D can be chosen equal zero.
The basis sets
To effectively solve the Kohn-Sham equation, selecting an appropriate wave basis set for the Hilbert space is essential Various basis sets have been developed since the inception of quantum mechanics, categorized into four primary groups: i) Extended basis sets, which are delocalized and cover all space, making them suitable for condensed phases like solids and liquids but inefficient for molecular systems; ii) Localized basis sets, centered on atomic positions or bonds, primarily used for molecular systems but also applicable to periodic systems; iii) Mixed basis sets, which combine extended and localized functions to leverage the advantages of both, though they may encounter technical challenges such as over-completeness; and iv) Augmented basis sets, which enhance either extended or atom-centered sets with atomic-like wave functions around nuclei, offering high accuracy but increased complexity.
Pseudopotentials
The rapid oscillations of wave-functions in the core region make their computational representation costly, as a large number of plane waves are required to accurately expand tightly bound core orbitals near the nuclei Since the physical and chemical properties of materials are primarily influenced by valence electrons rather than core electrons, explicit calculations for core electrons are often deemed unnecessary To address this, the pseudopotential approach is introduced, which replaces the strong Coulomb potential of the nucleus and the effects of core electrons with a smoother, effective ionic potential that acts on the valence electrons.
The development of pseudo-potentials hinges on the principle of norm conservation, which is governed by specific requirements Firstly, for any given atomic configuration, the valence eigenvalues of both the all-electron potential and the pseudopotential must match Secondly, at the cutoff radius \( r_c \), the logarithmic derivatives and the energy derivative of the logarithmic derivative of the true all-electron eigenfunctions must align with those of the pseudo eigenfunctions Lastly, to ensure norm conservation, the integrated charge within the radius \( r_c \) for each wave function must be satisfied, while for \( r < r_c \), the pseudopotential and radial pseudo orbitals must also conform to these criteria.
differ from real all election system The integrated charge for the all electron radial orbitals r (rr c ) is the same for r
Figure 2.1 Schematic illustration for all-electron (solid lines) and pseudo wave potentials
(dashed lines) and their corresponding wave functions [117]
There are many types of pseudopotentials such as ultrasoft (VUS or US) pseudopotentials proposed by Vanderbilt [122] and Hamann, Schlüter, Chiang (HSC) pseudopotentials [123] and the projector augmented wave (PAW) pseudopotentials
[124] In our DFT calculations, we employed ultrasoft and PAW pseudopotentials proposed to describe the electron-ion interaction [125].
Self-consistent field methods
The only remaining task is to solve Kohn-Sham equation within DFT calculation computationally [117, 118]
As known, the Kohn-Sham equation is defined by (2.31), we have the following equation:
H r r , (2.65) in which, in is the input density Hamiltonian of N e -electrons described by
m (2.66) in which, the expression of vˆ KS as follows
ˆ ˆ KS in Ne in r XC in v V r dr r r r
By solving the Kohn-Sham equation, seen in Figure 2.2 for a self-consistent loop, we will obtain the output density out r :
Figure 2.2 Flow chart of a self-consistent loop of the Kohn Sham equation
Note that out r and in r (or KS in and KS out ) must satisfy the self- consistent condition, i.e they must be equal within a certain error limit in the energy.
Van der Waals density functional (vdW-DF) calculations
A comprehensive theory for solids and molecules must consider various interactions, including electrostatic interactions, covalent bonds, hydrogen bonds, and van der Waals interactions However, traditional Density Functional Theory (DFT) using Local Density Approximation (LDA) or Generalized Gradient Approximation (GGA) fails to accurately account for van der Waals interactions These London dispersion interactions play a crucial role in stabilizing a diverse range of systems, from biomolecules to molecules adsorbed on material surfaces.
DF approach has been received great attentions since it can be incorporated with the
Calculate the Kohn-Sham potential
Solve the Kohn-Sham equation
DFT framework The exchange correlation energy of vdW-DF calculation is given by
The equation E = E_X^GGA + E_C^LDA + E_NL represents the total energy in a system, where E_X^GGA denotes the exchange energy calculated using the Generalized Gradient Approximation (GGA), E_C^LDA refers to the short-range correlation energy derived from the Local Density Approximation (LDA), and E_NL signifies the nonlocal energy term that approximates nonlocal electron correlation effects.
Here, r i is the electron density at the coordinate r i , and is a function depending on r 1 r 2 , called vdW kernel and more details shown in Refs [127, 128]
The vdW-DF method enhances the accuracy of van der Waals interactions compared to traditional DFT using GGA and LDA By employing suitable exchange-correlation functionals and refining the E C NL vdW-DF calculations developed by Lee and colleagues, improved accuracy can be achieved for a diverse array of materials.
Computational details
In our study of favorable adsorption sites and electronic structure properties, we utilized the Vienna ab initio simulation package (VASP) for van der Waals dispersion-corrected density functional theory (vdW-DF, specifically revPBE-vdW) calculations We applied a plane-wave basis set with a cut-off energy of 700 eV, along with the revised Perdew-Burke-Ernzerhof (revPBE) functional for exchange-correlation energy, and employed the projector-augmented-wave method for electron-ion interactions Surface Brillouin-zone integrations were conducted using the Monkhorst and Pack k-point sampling technique, with a 4×4×4 mesh grid centered at the Gamma point, and we implemented Methfessel-Paxton smearing of order 1 for enhanced accuracy.
[134] was used for the geometry relaxation with the smearing width sigma of 0.1 eV
However, the linear tetrahedron method with Blửchl corrections [135] was employed for the calculations of total energy
The electronic structure properties were elucidated through the analysis of the density derived electrostatic and chemical (DDEC) net atomic charge [136], the Bader point charge [137], DOS, and CDD… a Adsorption energy
For computing the favourable adsorption sites of hydrogen molecule (H 2 ) in the MIL-88s, we calculated the adsorption energy (E ads ) of H 2 in the MOF by using the equation:
E is the total energy of a [MOF + H 2 ] system (i.e the total energy of
MIL-88A with an absorbed hydrogen molecule); E MOF and
E are the total energy of the pristine MOF, and the isolated hydrogen molecule, respectively b Electronic structure properties
The electronic structure properties were elucidated through the analysis of the Bader point charge [137], DOS [116], and CDD with smaller sigma, = 0.1 eV
The Density of States (DOS) quantifies the number of electron states available at a specific energy level within the range of (E, E + dE) Typically, DOS is illustrated through a graph that displays the number of states in relation to energy, normalized to the Fermi energy level.
For investigating the charge density exchange between H 2 molecule and the MOF, we use the following formula
are the charge density of the [MOF + H 2 ] system, pristine MOF, and isolated H 2 gas, respectively
Grand canonical Monte Carlo simulations
To effectively characterize material systems, quantum computational methods, particularly Density Functional Theory (DFT), are widely utilized However, for large systems where results need to align with empirical data, ab initio calculations are often impractical due to their high time and cost requirements In such cases, classical simulations, specifically Monte Carlo (MC) and molecular dynamics (MD) methods, provide a viable alternative These simulation techniques determine physical quantities by averaging over a broad range of microscopic states within the system.
Grand canonical Monte Carlo (GCMC) simulations are commonly employed to assess hydrogen uptake in metal-organic frameworks (MOFs) by calculating gas adsorption isotherms, which mimic experimental measurements In GCMC, the temperature, volume, and chemical potential are held constant while allowing the number of particles to fluctuate through random processes such as creation, deletion, and displacement, enabling the system to reach equilibrium It is crucial to note that experimental measurements reflect only excess adsorption; thus, comparisons with GCMC results must consider this factor The relationship between absolute adsorption (n_abs) and excess adsorption (n_ex) is expressed by the equation n_abs = n_ex + ρ_gas * V_p, where ρ_gas is the hydrogen density and V_p is the pore volume of the MOF When normalized by the unit cell mass, the total gravimetric uptake of hydrogen (g_H2/g_MOF) can be determined Typically, excess adsorption decreases at high pressures, while total gas uptake consistently increases due to the adsorbate's bulk density contributions.
When gas molecules interact with a sorbent surface, their adsorption is influenced by the partial pressure in the bulk The relationship between the amount of gas adsorbed and varying partial pressures at a constant temperature is depicted in an adsorption isotherm graph Various isotherm shapes exist in the literature, determined by factors such as the type of adsorbent (e.g., MOFs), the nature of the adsorbate (e.g., gas molecules), and the intermolecular interactions between the gas and the sorbent surface The foundational interpretation of gas-solid adsorption isotherms was pioneered by Brunauer, Deming, Deming, and Teller (BDDT) in 1940.
Isotherms are classified into five main types (Type I to IV) as illustrated in Figure 2.3, with an additional Type VI introduced later by Sing et al This brings the total to six types, completing the IUPAC classification, which has become the standard for modern adsorption isotherm classification The distinct shapes of these isotherms are crucial for understanding adsorption behavior.
Type I isotherms are indicative of microporous adsorbents, such as activated carbons, molecular sieve zeolites, COFs/MOFs, and certain porous oxides, which have pore sizes less than 2 nm and relatively small external surfaces In these materials, the limiting adsorption capacity is determined by the available micropore volume rather than the internal surface area.
- Types II and III describe adsorption on non-porous or macroporous adsorbents (> 50 nm) with strong and weak adsorbate – adsorbent interactions, respectively
Types IV and V adsorption isotherms are characteristic of mesoporous solids with pore sizes ranging from 2 to 50 nm and exhibit hysteresis due to capillary condensation These isotherms demonstrate a limiting uptake at high relative pressures (P/P₀), where the lower curve represents the adsorption process and the upper curve signifies desorption.
The type VI isotherm signifies layer-by-layer adsorption occurring on a highly uniform nonporous surface In this model, the step-height indicates the capacity of each adsorbed layer, while the sharpness of the step varies based on the specific system and temperature conditions.
Figure 2.3 The IUPAC classification for adsorption isotherms [143]
The accuracy of Grand Canonical Monte Carlo (GCMC) simulations for H2 molecule interactions with metal-organic frameworks (MOFs) relies heavily on the precision of the force fields used, particularly when compared to experimental data To achieve optimal GCMC simulations, it is essential to employ force fields that incorporate both dispersion and electrostatic interactions, parameterized consistently from Density Functional Theory (DFT) calculations However, developing comprehensive force fields that accurately describe all adsorbate-framework interactions remains a complex and lengthy challenge Currently, only a limited number of studies have undertaken quantum mechanical calculations to create new force fields for specific MOF-H2 interactions For ease of implementation, the parameterization of the Coulomb interaction is derived using the DDEC method based on van der Waals density functional (vdW-DF) calculations, while the dispersion term parameters are sourced from the RASPA software force fields.
Recent studies have utilized a combined approach for H2 adsorption simulations, employing generic force fields such as UFF, DREIDING, and OPLS Notably, potential parameters for metal atoms in metal-organic frameworks (MOFs) are sourced from UFF, as they are unavailable from DREIDING UFF has proven effective in evaluating the H2 adsorption capacity of MOFs, demonstrating reliability when compared to experimental data For further details, refer to the literature.
A multi-scale simulation method that integrates quantum-level calculations with Grand Canonical Monte Carlo (GCMC) simulations has been employed to predict the hydrogen uptake capacities of Metal-Organic Frameworks (MOFs) The interaction energies between hydrogen molecules and MOF atoms are derived from quantum mechanical calculations, which are then fitted to a suitable potential function for use as force fields in GCMC simulations For further information on the quantum-level calculations and the fitting of force field parameters, please refer to the relevant literature.
Figure 2.4 The scheme for the multi-scale simulation method [151]
The term multi-scale simulation can have a different meaning depending on the field in which it is used In our research, we use “multi-scale” to represent:
Density Functional Theory (DFT) calculations are employed to determine the binding and adsorption energies of hydrogen (H2) in Metal-Organic Frameworks (MOFs), allowing us to identify stable adsorption sites and understand the interactions between H2 and MOFs through electronic properties like Charge Density Distribution (CDD) and Density of States (DOS) Additionally, DFT is utilized to compute partial charges for MOF atoms, facilitating the assessment of electrostatic interactions in Grand Canonical Monte Carlo (GCMC) simulations On the atomic and molecular scale, GCMC simulations are conducted to evaluate adsorption isotherms, enabling us to infer the highest storage capacity of H2 and compare it with experimental data Furthermore, we calculate the isosteric heats of adsorption (Qst) for H2 in MOFs to assess the strength of H2-MOF interactions.
Computational details
GCMC simulations utilizing the RASPA software were conducted to calculate the gravimetric loadings of hydrogen gas in MIL-88s, a type of nanoporous material.
The study investigates VTμ ensembles at temperatures of 77 K and 298 K under pressures up to 100 bar, involving 10^5 equilibration cycles followed by 3×10^5 Monte Carlo steps for the random manipulation of hydrogen molecules within the simulation box The GCMC simulation setups for MIL-88A, B, C, and D are detailed in Table 2.1 Throughout the simulations, the framework remains rigid while hydrogen molecules move freely within the metal-organic framework (MOF) The interactions between hydrogen gas and the MOF atoms (carbon, oxygen, hydrogen, and metal) are modeled using the Lennard-Jones 6-12 potential and electrostatic potential.
4 ij ij i j ij ij ij ij ij
The potential energy (U) between two atoms i and j at a distance r_ij is influenced by the dielectric constant (ε_o) and the partial charge (q_i) of atom i, derived from DDEC atomic net charge calculations using the DFT method Additionally, the parameters associated with the Lennard-Jones (LJ) potential, including well depth and diameter, are determined using the Lorentz–Berthelot mixing rule.
Here, the LJ parameters for atom i ( i , i ) were taken from UFFs for MOFs, listed in Table 2.2 The LJ interaction is neglected beyond the cutoff radius of 12.8 Å
Table 2.1 The GCMC simulation box of MIL-88s
Number of atoms per unit cell
Size of GCMC simulation box (number of times of unit cell)
Number of atoms in GCMC simulation box
Partial charges for the atoms (q i , i = Sc, Ti, V, Cr, Mn, Fe, Ni, Co, Cu, H, O and
The DDEC method, based on DFT calculations, was utilized to compute the C atom charges in MIL-88s, as detailed in Tables 2.2 and 2.3 These point charges were assigned to atomic sites for the calculation of electrostatic interactions, which were managed using the Ewald summation technique with a cutoff radius of 12 Å For the hydrogen molecule, a single Lennard-Jones (LJ) interaction site model was employed at the center of mass (H com), with LJ parameters sourced from the TraPPE force field.
with hydrogen bond length of 0.74 Å, and its charge is -0.936 e (Figure A1)
Table 2.2 The LJ parameters for atom types and atomic partial charges used in GCMC simulation
Table 2.3 The atomic partial charges of MIL-88s (s = A, B, C and D) used in GCMC simulation
Atom type MIL-88A MIL-88B MIL-88C MIL-88D