Iván Cabria
email: ivan.cabria@uva.es

Alexandre Lebon

M Begoña

Torres Luis

Javier Gallego

Andrés Vega

Roberto C Longo
email: roberto.longo@utdallas.edu

Ignacio L Garzón
email: garzon@fisica.unam.mx

Roberto Robles
email: roberto.robles@ehu.eus

M B Torres

L J Gallego

borophene and pristine graphene slit pores: A combined ab

Keywords: Hydrogen storage, 2D materials, boron-based materials, Li-decorated materials, statistical physics Hydrogen storage, 2D materials, boron-based materials, Li-decorated materials, statistical physics

Hydrogen storage capacity of Li-decorated

Introduction

Although the current fleet of hydrogen-powered vehicles is still very limited, they are considered the next-generation of electric vehicles and firm candidates to replace the conventional fossil fuel vehicles in the coming years or, at least, be an efficient alternative within a world of presumable multiple solutions to meet the energy, the pollution and the climate challenges. So far, only some companies offer models, but quite expensive, and others are currently working in hydrogen car projects. Hydrogen is abundant and cheap and its combustion produces three times as much energy as an equivalent quantity of petrol; moreover, the result of this combustion is clean water (i.e., no CO 2 and no other harmful gases are produced), so that it is environmentally benign. Despite its advantages, the use of hydrogen involves technical challenges and rise questions about efficiency. Hydrogen extraction in the most efficient way requires a chemical process, such as water-splitting electrolysis or the reaction of hydrocarbon chains at high temperature. Although there exist non-chemical ways to produce hydrogen, like by means of algae and solar light [START_REF] Kanygin | Rewiring photosynthesis: a photosystem I-hydrogenase chimera that makes H 2 in vivo[END_REF][START_REF] Krishna | Photosystem ratio imbalance promotes direct sustainable H 2 production in Chlamydomonas reinhardtii[END_REF][START_REF] Rumpel | Enhancing hydrogen production of microalgae by redirecting electrons from photosystem I to hydrogenase[END_REF], they are very slow, compared with the chemical ways. Anyway, the hydrogen storage still remains a most critical problem [START_REF] Broom | Concepts for improving hydrogen storage in nanoporous materials[END_REF][START_REF] Allendorf | An assessment of strategies for the development of solid-state adsorbents for vehicular hydrogen storage[END_REF][START_REF] Ahn | Storage materials based on hydrogen physisorption[END_REF][START_REF] Pukazhselvan | High capacity hydrogen storage: Basic aspects, new developments and milestones[END_REF][START_REF] Gross | Recommended best practices for the characterization of storage properties of hydrogen storage materials[END_REF][START_REF] Broom | Hydrogen sorption properties of materials[END_REF][START_REF] Liu | Advanced materials for energy storage[END_REF][START_REF] Schlapbach | Hydrogen storage materials for mobile applications[END_REF].

Currently, hydrogen is stored in high-pressure tanks or in its liquid state at cryogenic facilities. Both ways present risks and inconveniences, which has made necessary the search for alternative methods of storage. Adsorption on light materials with large surface/volume ratio is being considered as the next step for hydrogen storage. Within this context, carbon-based nanostructures such as graphene or nanotubes have emerged as the most promising alternative due to their remarkable thermal and mechanical properties (see, e.g., Refs. [START_REF]Nanoporous Materials for Gas Storage[END_REF][START_REF] Hirscher | Are carbon nanostructures an efficient hydrogen storage medium?[END_REF][START_REF] Arellano | Density functional study of adsorption of molecular hydrogen on graphene layers[END_REF] and those cited therein). To be used in hydrogen-powered vehicles, a material requires a gravimetric storage capacity of 4.5 wt % and a volumetric storage capacity of 0.030 kg/L of H 2 at room temperature and moderate pressure, as established by the U.S. Department of Energy (DOE) [15,[START_REF]DOE technical targets for onboard hydrogen storage for light-duty vehicles[END_REF].

To achieve reversible storage of H 2 at room temperature and moderate pressure, the binding energy needs to be 0.1-0.3 eV per molecule (i.e., in a range between physisorption and chemisorption), as has been shown for a number of surfaces [START_REF] Bhatia | Optimum conditions for adsorptive storage[END_REF][START_REF] Li | Theoretical evaluation of hydrogen storage capacity in pure carbon nanostructures[END_REF]. Low-dimensional carbon nanotubes and graphene-based systems do not meet such H 2 -binding energy requirement. However, it has been found that doping or decoration (since these are non bulk-like structures) with carefully selected species can substantially increase their hydrogen storage ability. A pioneering work in this area was performed by Yildirim and Ciraci [START_REF] Yildirim | Titanium-decorated carbon nanotubes as a potential high-capacity hydrogen storage medium[END_REF], who used density functional theory (DFT) to show that Ti-decorated single-walled carbon nanotubes are promising candidates for high-capacity hydrogen storage media. A similar conclusion was also obtained by Lebon et al. [20] for Ti-decorated zigzag graphene nanoribbons.

Cabria et al. [START_REF] Cabria | Simulation of the hydrogen storage in nanoporous carbons with different pore shapes[END_REF][START_REF] Cabria | The optimum average nanopore size for hydrogen storage in carbon nanoporous materials[END_REF] applied a quantum-thermodynamic model to calculate the storage capacities of nanoporous carbons in a range of pressures and temperatures as a function of the size and shape of the carbon pores. The model, which is an improvement of the model proposed by Patchkovskii et al. [START_REF] Patchkovskii | Graphene nanostructures as tunable storage media for molecular hydrogen[END_REF], takes into account the quantum effects of the motion of H 2 in the confining potential of the pores. The storage capacities calculated at room temperature are far too short to reach the DOE capacity targets. The reason is that, in spite of the confinement effect, which can be varied by modification of the pore shape and size, the interaction between H 2 and the carbon surfaces is not attractive enough.

Concurrently with the research on carbon-based nanostructures, there has been a growing interest in recent years in two-dimensional (2D) materials beyond graphene (see, e.g., Refs. [START_REF] Ersan | Twodimensional pnictogens: A review of recent progresses and future research directions[END_REF][START_REF] Li | Graphene and related twodimensional materials: Structure-property relationships for electronics and optoelectronics[END_REF][START_REF] Tan | Recent advances in ultrathin two-dimensional nanomaterials[END_REF][START_REF] Zhang | Emergent elemental two-dimensional materials beyond graphene[END_REF][START_REF] Gupta | Recent development in 2D materials beyond graphene[END_REF]). Theoretical studies seem to indicate that hundreds of 2D novel compounds should exist, which have since led to the successful synthesis and characterization of many of them. As one can easily imagine, 2D materials display a large diversity of chemical, mechanical, electronic and optical properties and a wide range of potential applications, such as nanoelectronics, spintronics, optoelectronics or even in nanomedicine. Furthermore, some of those novel 2D materials have been proposed as electrodes in novel batteries or as novel hydrogen storage solid state devices. Among the large family of novel 2D compounds, borophene has attracted a great interest because of its intriguing and diverse properties, which arise from a large variety of geometrical structures and bonding patterns. Borophene has also been proposed as a suitable material for a wide range of applications, including metal-ion batteries, supercapacitors, sensors, catalytic devices, and others in medicine, to say the least (see a recent review in Ref. [START_REF] Wang | Review of borophene and its potential applications[END_REF].

We note that although the existence of the boron counterpart of graphene was theoretically predicted few years ago, it was not synthesized until recently using Ag(111) as underlying substrate [START_REF] Mannix | Synthesis of borophenes: anisotropic, twodimensional boron polymorphs[END_REF]. DFT predictions for the structure of borophene went from initial single-atomic layers made of triangular and hexagonal arrangements, the so-called α-sheet, [START_REF] Tang | Novel precursors for boronnanotubes: the competition of two-center and three-center bonding[END_REF] to novel nonplanar phases that are thermodynamically more favorable [START_REF] Zhou | Semimetallic two-dimensional boron allotrope with massless dirac fermions[END_REF]. One of such phases displays a complex buckled geometry with Pmmn symmetry and eight atoms in the unit cell. In this work we will refer to it as the Pmmn8 phase or the β-sheet. The boron monolayer synthesized on Ag(111) [START_REF] Mannix | Synthesis of borophenes: anisotropic, twodimensional boron polymorphs[END_REF] shows the same Pmmn symmetry, but first-principles relaxation of such a monolayer results in loss of corrugation along one of the in-plane directions, while keeping the buckled structure along the other [START_REF] Mannix | Synthesis of borophenes: anisotropic, twodimensional boron polymorphs[END_REF]. The resulting free standing borophene has only two atoms per unit cell. We will denote it as Pmmn2 or simply the γ-sheet. Phonon spectra calculations [START_REF] Carrete | Physically founded phonon dispersions of few-layer materials and the case of borophene[END_REF][START_REF] García-Fuente | What will freestanding borophene nanoribbons look like? An analysis of their possible structures, magnetism and transport properties[END_REF], by forcing translational symmetry and rotational symmetry, reveal that the most stable phase of borophene is the β-sheet.

The three phases of borophene indicated above have been investigated as candidates for hydrogen sto-rage. DFT-GGA calculations show that the Li-decorated α-sheet can contain up to 10.7 wt % of molecular hydrogen [START_REF] Er | DFT study of planar boron sheets: a new template for hydrogen storage[END_REF], with an average binding energy of 0.15 eV, which exceeds the capacity of the graphenebased counterpart. Moreover, the Li-decorated γ-sheet can reach a capacity of 13.7 wt % and H 2 -binding energies that also satisfy the binding energy requirement, as obtained using GGA-DFT and van der Waals (vdW)-corrected DFT calculations [START_REF] Li | The high hydrogen storage capacities of Li-decorated borophene[END_REF]. Recently, Lebon et al. [START_REF] Lebon | Li-decorated Pmmmn8 phase of borophene for hydrogen storage. A van der Waals corrected density-functional study[END_REF] investigated the Li-decorated β-sheet, using the non-local optB88-vdW functional proposed by Klimeš et al. [START_REF] Klimeš | Chemical accuracy for the van der Waals density functional[END_REF]. Satisfactory H 2 -binding energies were reported, but the obtained gravimetric density was of 3.84 wt %, slightly lower than the 4.5 wt % DOE threshold. However, all these results must be taken cautiously as they correspond to ground state calculations at T = 0 K; in other words, finite temperature and pressure are not accounted for.

In order to get a more realistic information on the hydrogen storage capacity of borophene, in the work described here we use the quantum-thermodynamic model of Cabria et al. [START_REF] Cabria | Simulation of the hydrogen storage in nanoporous carbons with different pore shapes[END_REF][START_REF] Cabria | The optimum average nanopore size for hydrogen storage in carbon nanoporous materials[END_REF] to investigate the temperature and pressure-dependencies of the gravimetric and volumetric hydrogen storage capacities of borophene nanostructures under doping and extreme conditions of confinement. Specifically, we considered a slit pore consisting of two parallel Li-decorated Pmmn8 borophene sheets (i.e., in the most stable configuration of borophene), separated by a certain distance, with hydrogen molecules in between. Hence, our aim goes beyond the standard (T = 0 K) DFT calculations in this field. We note that the system considered in this paper is much more complex than that studied by Cabria et al. [START_REF] Cabria | Simulation of the hydrogen storage in nanoporous carbons with different pore shapes[END_REF][START_REF] Cabria | The optimum average nanopore size for hydrogen storage in carbon nanoporous materials[END_REF], not only for the complexity of the Pmmn8 structure of the borophene walls, but also for the fact that it is not in pristine form, but doped with Li atoms.

We also note that nanoporous boron frameworks, containing many regions of borophene slit-shaped pores, could perhaps be formed by using zeolite as a sacrificial template, just as zeolite-template carbons (ZTC) are formed [START_REF] Nishihara | Zeolite-templated carbons -three-dimensional microporous graphene frameworks[END_REF].

The following sections of this paper are organized as follows. In Section 2 we give the essential technical details of the computational method used, the geometry of the slit pore of Li-decorated β borophene sheets and of the general features of the quantum-thermodynamic model employed to obtain the temperature and pressure dependent figures of merit. In Section 3 we present and discuss our results, with two subsections devoted to obtain the interaction potential confining the hydrogen molecules in the slit pore, and to investigate the gravimetric and volumetric hydrogen storage capacities of this nanostructure as functions of the pore width, the temperature and the pressure. The interaction potential and the storage capacities of Lidecorated borophene and graphene slit pores are also compared. Finally, in Sec. 4, we summarize our main conclusions.

Research methodology

Computational method

The quantum-thermodynamic model employed in this work (briefly outlined in a subsection below) accounts for quantum-confinement effects due to the nanometric dimensions of the pores in which the H 2 molecules are stored. These effects lead to the quantization of the energy spectrum of the molecule within the interaction potential of the slit pore. Those energy states are required in the quantum-thermodynamic model and, therefore, solving the Schrödinger equation for the Hamiltonian of the H 2 molecule in the interaction potential of the slit pore is the first step. Previously, we had to accurately determine the three-dimensional interaction potential of H 2 in the slit pore. This was accomplished by performing calculations within the DFT as implemented in the VASP package [START_REF] Kresse | Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set[END_REF][START_REF] Kresse | Ab initio molecular dynamics for liquid metals[END_REF]. This package solves the Kohn-Sham equations within the projector-augmented wave (PAW) approach [START_REF] Blöchl | Projector augmented-wave method[END_REF]. For the plane-wave basis, a cut-off energy of 500 eV was used. In the calculations we employed the non-local optB88-vdW functional of Klimeš et al. [START_REF] Klimeš | Chemical accuracy for the van der Waals density functional[END_REF], which includes the dispersion interactions. A k-point space of about 0.2 Å -1 was used for integrating Brillouin zones. The width of the Gaussian smearing used was 0.01 eV. The interactions between periodic images of the sheet along the normal to the surface are negligible, because they were separated by a distance of 23 Å.

As in Ref. [START_REF] Lebon | Li-decorated Pmmmn8 phase of borophene for hydrogen storage. A van der Waals corrected density-functional study[END_REF], we used a 2 × 2 supercell of the β borophene sheet made of 32 B atoms. The sheet was decorated with 4 Li atoms on both sides, and the resulting hybrid nanostructure was relaxed using the conjugated-gradient method until the total force on each atom was smaller than 0.01 eV/Å; the accuracy to convergence for the total energies was 10 -6 eV/unit cell. The relaxed structure of the Li-decorated β sheet is schematically represented in Fig. 1, which shows that the preferential adsorption sites of the Li atoms on the β sheet are the furrow sites. In the same figure also shown is the location of an H 2 molecule adsorbed on the Li-decorated β sheet in its most stable configuration.

The interaction potential energy, V(x, y, z), of a hydrogen molecule located at point (x, y, z) with the Li-decorated β borophene sheet is defined as

V(x, y, z) = E(H 2 @Li-borophene) -E(H 2 ) -E(Li-borophene) , (1)
where E(H 2 @Li-borophene), E(H 2 ) and E(Li-borophene) are the energy of H 2 at (x, y, z) on the Li-decorated β borophene sheet, the energy of an isolated H 2 molecule and the energy of the Li-decorated β borophene sheet, respectively, and z is the distance between the H 2 molecule and the sheet surface.

We calculated the physisorption energies (or V(x, y, z)) of a single H 2 molecule (at the most stable orientation) on the surface of the relaxed Li-decorated β borophene sheet using the optB88-vdW functional [START_REF] Klimeš | Chemical accuracy for the van der Waals density functional[END_REF].

Thus we obtained the interaction potential energy between the H 2 molecule and the Li-decorated borophene sheet, V(x, y, z). In our calculations we chose a xy grid with ∆x = ∆y = 0.25 Å. This implies 936 grid points in the xy-plane. Then, for each one of these xy points or regions, we placed the H 2 molecule at different values of z, ranging from 0.8 to 7.6 Å, near to the surface of the sheet, with a step between points ∆z = 0.125 Å. Then, we made calculations for each one of these (x, y, z) grid points, and consequently we obtained a numerical three-dimensional V(x, y, z) interaction potential. A slice of the three-dimensional interaction potential energy V(x, y, z) is plotted in left and middle panels of Fig. 2. It corresponds to the interaction potential energy in two planes, z = 2.58 and 3.33 Å. It can be noticed that the deepest regions correspond to the neighbourhood of the Li atoms. In right panel of Fig. 2, the potential is shown for two constant pairs of (x, y) grid points, those where the resulting curves V(z) are the deepest and the shallowest ones among the 936 V(z) curves obtained in the calculations. The latter figure also contains the interaction potential V(z) between a H 2 molecule and a graphene layer in the isotropic approximation mentioned later, as obtained using the optB88-vdW functional.

Geometry of the slit pores formed by two parallel Li-decorated borophene sheets

We have used the geometry of the Li-decorated β borophene sheet to conceptually build slit-shaped pores.

According to experiments [START_REF] Park | CO 2 adsorption characteristics of slit-pore shaped activated carbon prepared from cokes with high crystallinity[END_REF], nanoporous carbons contain many regions that are flat, graphitic-like parallel surfaces separated by a distance of some nanometers. Those regions are called slit pores. Graphene slit pores were studied in Ref. [START_REF] Cabria | Simulations of volumetric hydrogen storage capacities of nanoporous carbons: Effect of dispersion interactions as a function of pressure, temperature and pore width[END_REF] using the optB88-vdW functional, with the aforementioned approximation for the interaction potential of H 2 with graphene. The volumetric capacities, v c , of graphene slit pores published in Ref. [START_REF] Cabria | Simulations of volumetric hydrogen storage capacities of nanoporous carbons: Effect of dispersion interactions as a function of pressure, temperature and pore width[END_REF] will be plotted in this work, for comparative purposes. Other results presented here for graphene slit pores (the gravimetric capacities, g c , and the interaction potentials) are entirely new, although related with the results reported in that reference. The upper panel of Fig. 3 shows the model of the slit borophene pore used in our investigation of the hydrogen storage capacity of these nanostructures, consisting in two parallel Li-decorated β borophene sheets separated by a distance d.

Quantum-thermodynamic model

The quantum-thermodynamic model has been explained in detail in Refs. [START_REF] Cabria | Simulation of the hydrogen storage in nanoporous carbons with different pore shapes[END_REF][START_REF] Cabria | The optimum average nanopore size for hydrogen storage in carbon nanoporous materials[END_REF][START_REF] Cabria | Simulations of volumetric hydrogen storage capacities of nanoporous carbons: Effect of dispersion interactions as a function of pressure, temperature and pore width[END_REF]. The model is based on considering the thermodynamic equilibrium between the adsorbed and compressed phases of hydrogen gas inside a slit pore. These phases have been drawn schematically in Fig. 3 (lower panel). The H 2 molecules stored by physisorption on the pore surface form the adsorbed phase (the blue and purple regions in the lower panel of Fig. 3) and the H 2 molecules stored only by compression, without interacting with the pore surface, form the compressed phase (the region with red balls in the lower panel of Fig. 3).

The model uses the interaction potential energy V(x, y, z) of one single layer (Li-decorated borophene, graphene, etc.), where z is the distance to the layer surface. The interaction potential energy of a slit pore of width d, or slit pore potential, is the sum of the potentials of the corresponding two parallel layers: ,y,dz). The slit pore is confined in the z direction. The sheets or layers of the slit pore are located at z = 0 and z = d.

V slit pore (z; d) = V(x, y, z) + V(x
In the case of a graphene slit pore, the potential energy of one single layer depends very little on the sites (x, y), and hence the approximations V(x, y, z) = V(z) for a single layer and V(z) + V(dz) for a slit pore are sound approximations. As regards to the Li-decorated borophene slit pore, the potential energy of a single layer has a strong dependence on the sites (x, y). Therefore, we have used 3D potentials for a Li-decorated borophene single layer, V(x, y, z), and for a Li-decorated borophene slit pore, V(x, y, z) + V(x, y, dz).

The equation of the equilibrium between the two phases inside a pore is, in the quantum-thermodynamic model,

ln(Z ads /Z com ) = 1 RT P ads P com v mol (P, T )dP , (2)
where Z ads and Z com are the partition functions of the adsorbed and compressed hydrogen phases, respectively, P com and P ads are the pressures of the adsorbed and compressed phases, respectively, and v mol (P, T )

is the molar volume of hydrogen. The partition functions are given by

Z ads = e -ǫ i /k B T , (3)
Z com = (d -2d excl ) 2πmk B T/h 2 . ( 4
)
In those equations, k B is the Boltzmann constant, ǫ i are the eigenenergies of the quantum states of a single H 2 molecule in the slit pore potential V slit pore (z; d), obtained by solving the corresponding Schrödinger equation, m is the mass of a hydrogen molecule, h is the Planck's constant, d is the slit pore width and d excl is an exclusion distance due to the repulsive region of the interaction potential energy of a single layer. The repulsive region is the region close to the atoms that form the layer.

The exclusion distance is defined as the location where the repulsive region of the interaction potential energy of a single layer is 1 eV. If the interaction potential energy of a single layer depends on (x, y, z), then there is an exclusion distance for each site (x, y) and the model uses the average exclusion distance. If the interaction potential energy of a single layer depends only on z, then the exclusion distance is that of the V(z) potential energy.

The mass of hydrogen stored in the pore is the sum of the masses of the adsorbed and compressed hydrogen. In this investigation, we will calculate and compare only the storage capacities due to the adsorbed phase, because the differences between the hydrogen storage capacities of Li-decorated β borophene and graphene slit pores come from the different interaction with the pore surfaces, which impacts only on the adsorbed phase. The g c of Li-decorated β borophene and graphene slit pores are also different because of the different masses of the adsorbent sheets of the pores.

The quantum-thermodynamic model calculates the molar volume and the mass of the adsorbed phase and uses them to obtain the storage capacity. The model obtains the pressure of the adsorbed phase, P ads , from the equation of the thermodynamic equilibrium between the two phases (Eq. 2). This pressure is used to calculate the molar volume of the adsorbed hydrogen phase, v mol (P ads , T ), in L/mol. The molar volume of the adsorbed phase is that given by the Mills-Younglove equation of state of H 2 [START_REF] Cabria | The optimum average nanopore size for hydrogen storage in carbon nanoporous materials[END_REF] at the pressure of the adsorbed phase: v mol (P ads , T ) = v molMills-Younglove (P ads , T ). This molar volume will be used to calculate the volumetric and gravimetric capacities.

The volumetric capacity, v c , of the adsorbed hydrogen phase is calculated, in kg of hydrogen/L, by means of the equation

v c = M(H 2 ) v mol (P ads , T ) V adsorbed V pore , (5)
where M(H 2 ) is the molar mass of H 2 in kg/mol, 0.00201588 kg/mol, and V pore and V adsorbed are the volume of the pore and the volume of the adsorbed hydrogen phase, respectively.

An explanation of the calculation of the volume of the adsorbed phase, V adsorbed , follows. First, it must be taken into account that the adsorbed phase inside a slit pore can be composed by one or two layers. In the lower panel of Fig. 3 the adsorbed hydrogen is the blue and violet regions. In that figure, there are two layers of adsorbed hydrogen. If the slit pore is narrow, then there is only one layer. If the slit pore is wide, then there are two layers and each layer is attached to one of the sheets of the slit pore.

The volume of the adsorbed phase is V adsorbed = S w ads . The surface of one sheet is S and w ads is the width of the adsorbed phase. If there is only one layer, then w ads is the width of the layer. If there are two layers, then w ads is the sum of the widths of the two layers. The width of the adsorbed phase is given by

w ads =          z r -z l z r -z l ≤ 2L 2L z r -z l > 2L , (6)
where z r and z l are the points at which V slit pore (z; d) = 0, z r > z l and L is the width of a layer adsorbed on a single and isolated sheet.

The value of L is 3 Å. This value was selected because is approximately the kinetic diameter of a H 2 molecule, 2.89 Å [START_REF] Ismail | Gas Separation Membranes: Polymeric and Inorganic[END_REF], the van der Waals diameter of a H 2 molecule, 2.76 Å [START_REF] Weast | Chemical Rubber Company, Handbook of Chemistry and Physics: A Ready-reference Book of Chemical and Physical Data[END_REF], and also the equilibrium distance of most of the interaction potentials V(z) of a physisorbed H 2 molecule on a single and isolated graphene and on other adsorbent surfaces.

To calculate g c it is necessary to obtain the mass of the adsorbed phase, mass H adsorbed . This mass is obtained, in kg, as the product of the density in kg/L and the volume in liters of the adsorbed phase:

ρ adsorbed V adsorbed . The density of the adsorbed phase is given by a/v mol (P ads , T ). This amount was already used to calculate the volumetric capacity.

The g c of the adsorbed hydrogen phase is calculated, in wt %, as g c = 100 mass H adsorbed mass H adsorbed + mass adsorbent material ,

where mass adsorbent material is the mass of the adsorbent material of the slit-shaped pore.

Interaction potential energy of the graphene slit pore

Although it is a huge computational task, a good sampling of the interaction potential energy landscape is crucial to accurately determine the adsorption capacities of the slit pore. We note that in previous works for graphene slit pores [START_REF] Cabria | Simulation of the hydrogen storage in nanoporous carbons with different pore shapes[END_REF][START_REF] Cabria | The optimum average nanopore size for hydrogen storage in carbon nanoporous materials[END_REF][START_REF] Patchkovskii | Graphene nanostructures as tunable storage media for molecular hydrogen[END_REF], only a one-dimensional potential, along the normal direction, V(z), corresponding to the most stable physisorption site of H 2 on the surface, was calculated and assumed to be the same along the whole graphene unit cell, thus neglecting anisotropic effects; the hydrogen storage capacities

were calculated according to that potential. This is a first approximation, but as we will see below, deep and shallow regions decorate the potential energy landscape, which means that results based on an isotropic potential in x, y, equal to that corresponding to the most stable configuration, slightly overestimate both the g c and v c . In other words, the confining potential used in previous works for carbon based slit pores is a bit more confining that the real one.

In our comparison of the capacities of the Li-decorated borophene slit pores with those of graphene slit-pores we have used a correction factor to account for the overestimation of the storage capacities calculated in a previous work [START_REF] Cabria | The optimum average nanopore size for hydrogen storage in carbon nanoporous materials[END_REF]. We performed additional calculations for graphene slit pores to estimate such correction factor. On that previous work, the H 2 adsorption energy was calculated for the nine main configurations of H 2 on graphene [START_REF] Cabria | Searching for DFT-based methods that include dispersion interactions to calculate the physisorption of H 2 on benzene and graphene[END_REF]. A configuration is a combination of site and orientation of the molecule.

The three main sites are on top of an hexagon (H), on top of a carbon atom (A) and on top of a C-C bond (B) (see Fig. 4). The three main orientations are perpendicular to the graphene surface (⊥), parallel to the graphene surface and parallel to two C-C bonds of a graphene hexagon ( ), and parallel to the graphene surface and perpendicular to two C-C bonds of a graphene hexagon ( ⊥). We obtained that the most stable configuration is H . In the present work, we have calculated the binding energy in more sites of graphene, with the H 2 molecule in the orientation . Specifically, in the sites A, B, H, AB, BH, AH and ABH indicated in Fig. 4 (site AB is located at the middle of the line joining sites A and B, and so on for the other new sites studied).

The results are gathered in Table 1. The binding energies of H 2 on all the studied sites of graphene are very close, being the largest difference 0.0066 eV. The most stable site is H and the less stable site is A.

We have calculated the interaction potential energy V(z) of H 2 on graphene for those two configurations, H and A , and then we have used the potentials V(z) to calculate the corresponding storage capacities. The results are shown in Figs. [START_REF] Allendorf | An assessment of strategies for the development of solid-state adsorbents for vehicular hydrogen storage[END_REF].

The capacities with the H 2 molecule on the hexagon and on the carbon atom site are very similar for large pore widths. The difference between the H and A capacities is larger at narrow pores. The capacities of the exact three-dimensional potential of a graphene slit-pore should be between the H and A capacities.

Results and discussion

Using the quantum-thermodynamic model described above, we have investigated the hydrogen storage ability of slit pores of Li-decorated β borophene by performing extensive calculations of their g c and v c as functions of pore width, temperature and pressure. The storage capacities depend on the interaction potential V(x, y, z) between the hydrogen molecules and the confining Li-decorated borophene sheets. The computed g c and v c of these slit pores will be compared with those obtained for graphene slit pores using the same DFT functional (optB88-vdW). The results for the hydrogen storage capacities of Li-decorated borophene slit pores are plotted in the next figures as red solid lines, and those of graphene slit pores are plotted in black.

Two isotherm curves are shown for each type of capacity, gravimetric and volumetric; they correspond to room temperature, T = 298.15 K, and to a low temperature of T = 80.15 K (most of the experiments of hydrogen storage on solid nanoporous materials at low temperatures are performed at 77 K [START_REF] Broom | Concepts for improving hydrogen storage in nanoporous materials[END_REF][START_REF] Grünker | A new metal-organic framework with ultra-high surface area[END_REF][START_REF] Gogotsi | Importance of pore size in high-pressure hydrogen storage by porous carbons[END_REF][START_REF] Schmitz | Heat of adsorption for hydrogen in microporous high-surface-area materials[END_REF] and a few at 80.15 K [START_REF] Ismail | Synthesis and characterization of MnPS 3 for hydrogen sorption[END_REF], and can be representative for certain applications like in spacecrafts). The width of the pores has been changed from 4 to 20 Å and the pressure from 0.1 and 25 MPa.

Dependence of the hydrogen storage capacities on pore width

Figure 6 shows the calculated hydrogen storage capacities of Li-decorated borophene and graphene slit pores for the maximum value of the studied pressure, 25 MPa, and at the two considered temperatures, 80.15 K and 298.15 K. The g c and v c of both kinds of pores exhibit a similar dependence with the pore size: the curves have a maximum and decrease towards a constant value. The g c of the slit pore tends towards a constant value that is twice the g c of the corresponding single layer under the same conditions of pressure and temperature. On the other hand, the v c decays to zero because the volume ratio V adsorbed /V pore decreases as the pore width increases.

Although the variations of the hydrogen storage capacities of the Li-decorated borophene and graphene slit pores with the pore width have similar general features, important differences arise. Specifically, the maxima of the gravimetric and volumetric curves of Li-decorated borophene slit pores are located at pore widths smaller than those of the corresponding maxima of the graphene counterparts (see Table 2). On the other hand, the highest g c at any temperature corresponds always to the graphene slit pore. However, the highest v c at 80.15 K corresponds to the Li-decorated borophene slit pore, while at 298.15 K the result is in favour of the graphene one (although in the latter case the differences are small: 0.0371 and 0.0340 kg/L for graphene and Li-decorated borophene slit pores, respectively; see Table 2).

An important result that can be noticed in Fig. 6 is that, at a given temperature, the hydrogen storage : for a pore width below a crossing point, the hydrogen storage capacity of the Li-decorated borophene slit pore is larger than the corresponding capacity of the graphene slit pore, and hence the former is more appropriate for hydrogen storage. The specific value of the crossing point depends on the temperature and the type of capacity, gravimetric or volumetric, as it is shown in Table 3. Above the crossing points, the situation is the opposite. It is also worth to point out that, below 5.4 Å, the volumetric capacity of Li-decorated borophene slit pores at room temperature (298.15 K) is even larger than that of graphene slit pores at a temperature such small as 80.15 K (this is a relevant result since the decrease of the temperature is a way to increase the hydrogen storage capacity of porous materials, in general). Although we are comparing different slit pores, this result reflects the great ability of narrow Li-decorated borophene slit pores for hydrogen storage. In particular, the high v c of these narrow slit pores at low temperature could be important for applications in outer space vehicles. We note that nanoporous boron frameworks, containing many regions of borophene slit-shaped pores, could perhaps be formed by using zeolite as a sacrificial template, just as zeolite-template carbons (ZTC) are formed [START_REF] Nishihara | Zeolite-templated carbons -three-dimensional microporous graphene frameworks[END_REF], and these structures could be subsequently doped with Li atoms to obtain solid porous materials with optimal hydrogen storage capacities. The large capacities of Li-decorated borophene slit pores of narrow widths can be explained by analyzing the potential V(x, y, z) + V(x, y, dz) confining the H 2 molecules for different pore widths d (see upper panel of Fig. 3). Figure 7 shows this potential, V(z) + V(dz), at the deepest and shallowest regions in the xy plane for six pore widths: 5.2, 5.6, 6.4, 7, 12 and 20 Å. Results for graphene slit pores (obtained with the same optB88-vdW functional) are also shown. The confining potential (in the isotropic approximation) has a single deep minimum for narrow pores and two separated minima for larger pore widths, that become two independent potentials when the pore width is large. We note that the confining interaction potential of a Li-decorated borophene slit pore contains xy points or regions of different depths, and that the real influence of the whole interaction potential is some average between the two extremes cases considered, the deepest and the shallowest regions. The exact or real three-dimensional interaction potential of graphene slit pore also contains regions of different depths. However, we have showed before that the graphene slit pore potential in the isotropic approximation, V slit pore (z; d), is a good or reasonable approximation to the real three-dimensional potential.

The confining interaction potential at the deepest region of the Li-decorated borophene slit pore is much deeper than that of the graphene slit pore for the narrower pores of widths 5.2 and 5.6 Å (see Fig. 7).

Accordingly, in that region Li-decorated borophene slit pores can store much more hydrogen than graphene slit pores. At many other regions of the xy plane (between the deepest and the shallowest regions), Lidecorated borophene slit pores of those widths will store also more hydrogen than graphene slit pores.

Thus, one can understand that this is what will happen on average. This explains the larger hydrogen storage

We note that for pore widths above 7-8 Å, the volumetric capacities, v c , of Li-decorated borophene and graphene slit pores are similar for the same P and T , but the gravimetric capacities, g c , of graphene slit pores are about two times larger than those of Li-decorated borophene slit pores (see Fig. 6). This difference has its origin in the larger surface mass density of Li-decorated borophene with respect to that of graphene (6.75 u/Å 2 against 4.59 u/Å 2 ). This larger surface mass density implies that Li-decorated borophene slit pores have a larger mass of adsorbent material than graphene slit pores for a fixed pore volume. Hence, according to Eq. 7, if the mass of hydrogen (or equivalently the density of hydrogen and pore volume) is the same, Li-decorated borophene slit pores will have smaller g c than graphene slit pores.

Dependence of the hydrogen storage capacities on the pressure

Figure 8 shows the calculated g c and v c of Li-decorated borophene and graphene slit pores as functions of the pressure at a constant temperature (T = 80.15 or T = 298.15 K) for different pore widths. For a pore width of 7 Å (the width for which the v c of the graphene slit pore at 298.15 K is maximum; see Table 2), the g c is bigger for graphene slit pores at the two considered temperatures. Note that when the pressure decreases, the changes in the g c are more important at room temperature, especially for the graphene slit pore below 10 MPa. The v c of the pores of width 7 Å show similar features to those of the g c at room temperature; however, at low temperature the behavior is quite different: the v c of the Li-decorated borophene slit pore is bigger than that of the graphene slit pore for all studied pressures.

For a width of 6.4 Å (the width at which the v c of the Li-decorated borophene slit pore is maximum at 80.15 K; see Table 2), the gravimetric and volumetric curves are similar to those obtained for the pores with a width of 7 Å. However, for a pore width of 5.6 Å (the width at which the Li-decorated borophene slit pore has the maximum v c at room temperature; see Table 2), the behaviour changes drastically: the v c and g c of Li-decorated borophene slit pores become larger than those of graphene slit pores for any value of the pressure and temperature. It should be noted that, for this pore width, the v c of Li-decorated borophene pores at 80.15 K are almost twice those of graphene slit pores. Finally, for pores of width 5.2 Å, the trends observed at the pore width of 5.6 Å are amplified: the capacities of Li-decorated borophene slit pore are larger and even much larger than those of the graphene slit pore at any value of the pressure and temperature. Above 4

MPa and for a pore width of 5.2 Å, the v c of the Li-decorated borophene slit pore at 298.15 K is even larger than that of the graphene slit pore at low temperature, 80.15 K.

Conclusions

In this work, we performed extensive vdW-corrected DFT calculations, in conjunction with a quantumthermodynamic model, to investigate the gravimetric and volumetric hydrogen storage capacities of Lidecorated borophene slit pores as functions of width, temperature and pressure. Our results are compared with those obtained for graphene slit pores, which were partially investigated previously by some of us to analyze the effect of confinement on the hydrogen storage capacity of undoped carbon nanostructures. The main result of our calculations is that narrow slit pores of the strongly anisotropic β (Pmmn8) Li-decorated borophene sheet have an excellent volumetric hydrogen storage capacity, especially at low temperature. The volumetric capacity of narrow Li-decorated borophene slit pores at room temperature is even larger than that of narrow graphene slit pores at low temperature, 80.15 K. Li-decorated borophene slit pores store hydrogen in very narrow pores, below pore widths of 5 Å, while the graphene slit pores do not store hydrogen in those very narrow pores.

The volumetric storage capacities at room temperature of Li-decorated borophene slit pores with pore widths in the range 5-6 Å are similar to the volumetric capacities at room temperature of graphene slit pores with larger pore widths, in the range 6-7 Å. At low temperature and for pore widths below 9 Å, the volumetric storage capacities of Li-decorated borophene slit pores are larger than the volumetric storage capacities of graphene slit pores.

We hope that these findings will stimulate the design of nanoporous boron frameworks, composed by many regions of borophene-like parallel surfaces, with optimal hydrogen storage capacities.

Fig. 1 .

 1 Fig. 1. Front and side views of the relaxed 2 × 2 supercell of the β (Pmmn8) sheet decorated with 4 Li atoms at furrow sites on each side of the sheet. Li atoms are shown with purple balls. Also shown is an adsorbed H 2 molecule in its most stable configuration (with H atoms in white).

Fig. 2 .

 2 Fig. 2. Interaction potential energy between a single Li-decorated β borophene sheet and a H 2 molecule in the planes z = 2.58 and 3.33 Å plotted in 3D (left panel) and its projections in the xy plane (middle panel). The bottom and upper 3D surfaces (planes) of the left (middle) panel correspond to the planes z = 2.58 and 3.33 Å, respectively. Right panel: Interaction potential energy between a H 2 molecule and a single graphene layer, and a single Li-decorated β borophene sheet at the deepest and shallowest xy points, as a function of the H 2 molecule-surface distance.

Fig. 3 .

 3 Fig. 3. Upper panel: Geometrical structure of a Li-decorated borophene slit pore of width d, made of two parallel Li-decorated β borophene sheets. Lower panel: Schematic representation of the phases of hydrogen stored inside a slit pore.

Fig. 4 .

 4 Fig. 4. Sites of H 2 on graphene.

Fig. 5 .

 5 Fig. 5. Gravimetric and volumetric hydrogen storage capacities of graphene slit pores for 80.15 K and 298.15 K and two configurations, H and A . Upper panel: Capacities as functions of the pore width, for a pressure of 25 MPa. Lower panel: Capacities as functions of the pressure, for a pore width of 7 Å.

Fig. 6 .

 6 Fig. 6. Gravimetric and volumetric hydrogen storage capacities of slit pores of Li-decorated β borophene (red curves), as functions of the pore width, for 80.15 K and 298.15 K, and a pressure of 25 MPa. Capacities for graphene slit pores are also shown (black curves).

Fig. 8 .

 8 Fig. 8. Gravimetric (upper panels) and volumetric (lower panels) hydrogen storage capacities of the Li-decorated borophene slit pore (red curves), as a function of pressure, at 80.15 K and 298.15 K, for pore widths of 7, 6.4, 5.6 and 5.2 Å. Capacities of graphene slit pores are also shown (black curves).

Table 1 :

 1 Binding energy of H 2 in eV, E b , and equilibrium H 2 -surface distance in Å, d e , on different sites of graphene, as obtained by optB88-vdW calculations. The H 2 molecule is in the orientation .

	Configuration d e	E b	Configuration d e	E b
	A	3.2 -0.0686	AB	3.2 -0.0692
	B	3.2 -0.0696	BH	3.1 -0.0723
	H	3.1 -0.0752	AH	3.1 -0.0713
	ABH	3.2 -0.0704

Hence, a sound approximation consists on calculating the capacities of the graphene slit pores as the average of the capacities obtained with the H 2 molecule on the configurations H and A : c = (c H + c A )/2. The storage capacities of graphene slit pores presented and discussed in the next sections are these average capacities.

Table 2 :

 2 Maxima of the hydrogen storage capacities of Li-decorated borophene and graphene slit pores at 25 MPa, as functions of the pore width.

	Slit pore	Capacity	80.15 K	298.15 K
	Graphene	g c	11.1 Å 4.23 wt %	7.9 Å 1.61 wt %
	Graphene	v c	8.4 Å 0.0700 kg/L 7.0 Å 0.0342 kg/L
	Borophene-Li	g c	8.8 Å 2.70 wt %	6.8 Å 0.93 wt %
	Borophene-Li	v c	6.4 Å 0.0835 kg/L 5.6 Å 0.0340 kg/L
	capacities of Li-decorated borophene slit pores are larger than those of graphene ones for narrow or very
	narrow pores. In fact, there are crossing points in that figure

Table 3 :

 3 Crossing points (slit pore widths) in Å of the hydrogen storage capacities of Li-decorated borophene and graphene slit pores at 25 MPa and for two temperatures, 80.15 and 298.15 K.

	Capacity 80.15 K 298.15 K
	g c	6.1	5.9
	v c	9.0	6.4

Acknowledgments

This research was financially supported by the Spanish MICINN (Grant PGC2018-093745-B-I00), the Junta de Castilla y León (Project No. VA124G18), the University of Valladolid, Spain, and the Xunta de Galicia (ED431E 2018/08 and GRC ED431C 2016/001). We also acknowledge the use of the high performance computing equipment of the Pole de Calcul Intensif pour la Mer (DATARMOR, Brest) and the Centro de Proceso de Datos -Parque Científico (UVa).

capacities of Li-decorated borophene slit pores of narrow widths, as obtained in our calculations (see Fig. 6).

For pores of medium widths, 6.4 and 7.0 Å, the interaction potentials at the deepest regions of Li-decorated borophene slit pores are as deep as the interaction potentials of graphene slit pores (see Fig. 7). So, one can understand again, at least qualitatively, that for pores of these widths the two types of pores can store similar amounts of hydrogen. Finally, at larger pore widths, 12 and 20 Å, the interaction potential at the deepest region of the Li-decorated borophene slit pore is again deeper than the interaction potential of graphene slit pore. However, on average, the graphene potential attracts more hydrogen than the Li-decorated borophene potential for those larger pore widths.

Declaration of interests

☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:

To Applied Surface Science Editor Dear Editor:

We hereby submit our manuscript entitled "Hydrogen storage capacity of Li-decorated borophene and pristine graphene slit pores: A combined ab initio and quantumthermodynamic study", by I. Cabria et al., as candidate to be published in Applied Surface Science as a full-length article. In this work, we have theoretically investigated the hydrogen storage capacity of Li-decorated borophene slit pores and pure graphene slit pores. What makes our work stand out from most theoretical studies on the hydrogen storage ability of nanostructures is that our combined quantum-thermodynamical approach allows the volumetric and gravimetric storage capacities to be determined as functions of temperature and pressure for different pore sizes. Thus, our results could be directly compared with experiments on these kinds of systems in realistic conditions.

We must point out that our paper was originally submited for publication in Journal of Power Sources, but it was rejected on the basis of the report of an experimental referee whose main objections were: i) that our theoretical predictions for the hydrogen storage capacity of Li-decorated borophene slit pores do not give excelent storage performance, and ii) that our paper is more appropriate to a journal dedicated to the theoretical/computational methods. We must point out that there are other theoretical predictions on the hydrogen storage capacity of borophene and other nanostructures that give very high gravimetric densities, but, as indicated in the Introduction of our paper, they are not realistic, because they are based on standard density functional calculations, only valid at zero temperature and no pressure. On the other hand, although our main goal in this work concerns Li-decorated borophene slit pores, the same quantum-thermodynamic methodology that we propose can be useful for investigating the hydrogen storage capacity of confined nanostructures doped with appropiated species in general. These theoretical calculations could be a useful guide for the design of new nanoporous systems, composed by many regions of parallel surfaces, with optimal hydrogen storage capacities.

The version of the paper that we now submit for publication to Applied Surface Science is a slightly modified version of the one we submit for publication to Journal of Power Sources, in which we have introduced some modifications in the light of the comments of the above-mentioned referee, especially in the Conclusions section. The Editor of Journal of Power Sources suggested us the possibility to transfer our paper to Applied Surface Science We are sure that our predicted results shall become a fundamental reference for future research on borophene (and hydrogen-storage nanostructures in general), a material synthesized a few years ago by Mannix et Li-decorated borophene slit pores are promising materials for hydrogen storage Combined DFT and quantum-thermodynamic study outperforms standard DFT calculations Nanoporous boron frameworks could be ideal for hydrogen storage in spatial equipment Volumetric capacity at 298 K larger than that of narrow graphene slit pores at 80 K Li-decorated borophene slit pores store hydrogen in very narrow pores Highlights (for review)