# Usable hydrogen-storage capacities of Li-decorated borophene nanopores in charge-discharge cycles

Alexandre Lebon, B. Calvez, M.B. Torres, L.J. Gallego, A. Vega

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Alexandre Lebon, B. Calvez, M.B. Torres, L.J. Gallego, A. Vega. Usable hydrogen-storage capacities of Li-decorated borophene nanopores in charge-discharge cycles. Journal of Energy Storage, 2024, 92, pp.112172. 10.1016/j.est.2024.112172. hal-04586465

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Research Papers

# Usable hydrogen-storage capacities of Li-decorated borophene nanopores in charge-discharge cycles

A. Lebon<sup>a, *</sup>, B. Calvez<sup>a</sup>, M.B. Torres<sup>b</sup>, L.J. Gallego<sup>c</sup>, A. Vega<sup>d</sup>

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## ARTICLE INFO

**Keywords:**

Hydrogen storage

2D materials

Boron-based materials

Density-functional theory

Statistical physics

## ABSTRACT

By combining density-functional theory and a quantum-thermodynamic model, we compute the usable hydrogen storage capacities of slit pores formed by two Li-decorated layers of the strongly anisotropic Pmmn8 borophene sheet. Our results show the important role played by the rotational degree of freedom of the hydrogen molecule in determining the confining potential within the slit pores and their hydrogen storage capacities. A remarkable finding is that for small pore widths of around 6 Å, the usable volumetric capacity of these borophene-based adsorbents at room temperature reaches the target stipulated by the U.S. Department of Energy for a quite low loading pressure of about 6 MPa when the depletion pressure is set to 0.1 MPa, which could be very useful for practical applications. Our results also show that a good usable gravimetric capacity can be reached only when the pore widths are of at least 30 Å, and operating at very large loading pressures of at least 60 MPa and temperatures lower than 100 K.

## 1. Introduction

Hydrogen storage with sufficient gravimetric and volumetric capacity is one of the main problems for the realization of fuel-cell power hydrogen vehicles. The conditions stipulated by the U.S. Department of Energy (DOE) for 2025 are a reversible usable gravimetric capacity of at least 5.5 wt % and a reversible usable volumetric capacity of at least 0.040 kg/L at room temperature and moderate pressure [1]. The identification of adsorbents meeting all these requirements simultaneously is still a challenging problem, in spite of the advances that have already been made in this field in the last years (see, e.g., Refs. [2–9]). Because hydrogen is a pollution-free energy vector, its use can contribute to the decarbonation of the transport and other sectors, such as the industry, the power generation, etc. In fact, great efforts are currently being made in many countries around the world to make the “hydrogen economy” a widespread reality [10–14].

So far, several hydrogen storage methods have been proposed, including the conventional high-pressure storage in gas form or the cryogenic storage in liquid form and the hydride-based storage in solid form. However, these methods have several inconveniences: molecular

hydrogen gas requires a large volume, liquid hydrogen evaporates easily, and metal hydrides add significant weight to the systems [12,15,16]. For these reasons, other media have been explored, including carbon-based nanostructures, which have low density, high surface area and high thermal and chemical stability (see, e.g., Refs. [17–28]).

Slit graphene pores (i.e., two parallel graphene sheets separated by a distance in the range of nanometer) are simple models for describing the pores existing in porous carbon-based nanostructures, which allows to investigate the effect of confinement on the hydrogen storage capacity. In fact, the pioneering experiments by Franklin [29] and subsequent experimental studies on nanoporous carbons [30,31] showed that these materials are mainly composed by slit-shaped pores with a pore width distribution in the range of several to hundred angstroms. Moreover, theoretical calculations performed by Cabria et al. confirm that the local environment of porous carbon materials is well captured by means of slit pores [24,32,33]. Those theoretical calculations were performed using a quantum-thermodynamic method that improves the model proposed by Patchkovskii et al. [34] and takes into account the quantum effects of the motion of the hydrogen molecules in the confining potential of the

<footnote><sup>a</sup> Laboratoire de Chimie Electrochimie Moléculaire et Chimie Analytique, Université de Brest, UMR CNRS 6521, F-29285 Brest, France</footnote>

<footnote><sup>b</sup> Departamento de Matemáticas y Computación, Escuela Politécnica Superior, Universidad de Burgos, ES-09006 Burgos, Spain</footnote>

<footnote><sup>c</sup> Área de Física de la Materia Condensada, Departamento de Física de Partículas, Facultad de Física, Universidad de Santiago de Compostela, ES-15782 Santiago de Compostela, Spain</footnote>

<footnote><sup>d</sup> Departamento de Física Teórica, Atómica y Óptica, Universidad de Valladolid, ES-47011 Valladolid, Spain</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: alexandre.lebon@univ-brest.fr (A. Lebon).</footnote>
pores.

The quantum-thermodynamic model used by Cabria et al. in their study of graphene slit pores [24,32,33] can be extended to compute the hydrogen storage capacities of slit pores formed by anisotropic 2D sheets. This is what we have recently done in a study on the hydrogen storage capacity of Li-decorated borophene slit pores [35]. Borophene, a strongly anisotropic mono-elemental 2D sheet that has a very stable Pmmn8 symmetry [36] with 8 atoms in its unit cell generated by two interpenetrating lattices, was synthesized several years ago [37], and some borophene-based 3D porous structures have theoretically been predicted to be dynamically, thermally, and mechanically stable [38]. However, in our previous study on Li-decorated borophene slit pores we provided information only about the adsorbed hydrogen storage capacities (mainly because our goal was to compare our results with those reported in previous studies on undoped graphene slit pores [24,32]), but not the usable capacities, which are those to which the DOE criterion refers for vehicular hydrogen storage. The usable capacity is defined as the difference between the capacity of the full tank at maximum pressure (the loading pressure) and the capacity remaining in the tank at the depletion, back or minimum pressure required to run the fuel cell (the delivery pressure) [1,39–41]. On the other hand, in Ref. [35] we did not investigate the role played by the rotational degree of freedom of the hydrogen molecule in determining the confining potential within the slit pores and their hydrogen storage capacities, something that we have shown to be crucial in a recent study of slit pores formed by two parallel layers of the binary, but planar, graphene-like BC₃ sheet [42].

Accordingly, in this work we present the results of additional investigations on the hydrogen storage capacities of Li-decorated borophene slit pores, with special emphasis on the two aspects mentioned above, not considered in our previous study [35]. By doing this, we give a more complete information on the hydrogen storage capacities of these kinds of pores, which can also be useful in the study of slit pores formed by other anisotropic 2D sheets.

## 2. Sketch of the methodology employed

The methodology used in our calculations was described in Refs. [24,32–35,42]. It combines density-functional theory (DFT) and a quantum-thermodynamic model. The basic equations of the quantum-thermodynamic model are indicated below (Eqs. 2–5). The DFT is used to obtain the structures and the 3D interaction potential energy between a hydrogen molecule and one layer of the Li-decorated borophene, defined as

$$V(x, y, z) = E(H_2@Li-borophene) - E(H_2) - E(Li-borophene) \quad (1)$$

where $E(H_2@Li-borophene)$, $E(H_2)$ and $E(Li-borophene)$ are the energy of a hydrogen molecule at $(x, y, z)$ on the Li-decorated borophene sheet, the energy of an isolated hydrogen molecule and the energy of the Li-decorated borophene sheet, respectively. Our DFT calculations were performed here in the same conditions as in our previous work [35], i.e., we employed the VASP code [43,44] with the non-local optB88-vdW functional of Klimeš et al. [45], which includes dispersion interactions. We note that although the expression of the potential given by Eq. 1 does not contain explicit information on the orientational degree of freedom of the H₂ molecule, it can be implicitly taken into account as described in detail in Sec. 3.

The left-hand panel of Fig. 1 is a schematic representation of the slit pore used in our investigation, consisting in two parallel Li-decorated borophene sheets separated by a distance $d$, the distance between the Li atom layers. The volume of the slit pore is $V_{pore} = Sd$, where S is the surface of the sheet. As shown in Ref. [46], the Li atoms occupy the furrow sites of the buckled geometry of the borophene sheet. In the right-hand panel of Fig. 1 we show the phases of hydrogen stored inside the pore. The (pore-width dependent) interaction potential energy of the hydrogen molecule inside the slit pore is the sum of the interaction potentials with the two parallel layers, i.e., $V_{slitpore}(x, y, z) = V(x, y, z) + V(x, y, d-z)$, where the two terms are obtained by DFT calculations as indicated above. We would like to stress that the $(x, y, z)$ coordinates of Eq. 1 refer to a generic position of the center of mass of the H₂ molecule respect to the already Li-decorated borophene layer.

Since the slit pore creates a confining region of nanometric dimensions for the molecular hydrogen, quantum confinement effects arise, i.e., the energy of the hydrogen molecule is quantized. The energy spectrum is calculated by solving the Schrödinger equation in the corresponding confining potential, determined as indicated in detail in Sec. 3 (illustrative examples of the quantum states of the hydrogen molecule in a confining potential can be found in Ref. [24] for the case of graphene slit pores). Following the work by Patchkovskii et al. [34] (see also Refs. [24,32,33,35,42]), the equilibrium constant, $K_{eq}$, between the two phases of hydrogen that are in the pore (physisorbed molecules and non-adsorbed molecules compressed by an external pressure exerted on them) can be written as

$$K_{eq} = \frac{Z_{ads}}{Z_{com}} \quad (2)$$

<figure><img src="image_4.png" /><figcaption>Fig. 1. Schematic representations of a Li-decorated borophene slit pore of width $d$ with some hydrogen molecules (left-hand panel) and of the masses involved in the slit pore (right-hand panel).</figcaption></figure>
where $Z_{ads}$ is the partition function of the adsorbed hydrogen phase, obtained as

$$Z_{ads} = \sum e^{\frac{-\epsilon_i}{k_B T}} \qquad (3)$$

$k_B$ being the Boltzmann constant and $\epsilon_i$ the eigenvalues of the Schrödinger equation of the hydrogen molecule in the confining potential (we used a method akin to that given in the Supporting Information of Ref. [47] to solve the Schrödinger equation). $Z_{com}$ is the partition function of the compressed or free hydrogen phase, which in the classical approximation is given by [24,33,35,42].

$$Z_{com} = (d - 2d_{exc}) \sqrt{\frac{2\pi m k_B T}{\hbar}} \qquad (4)$$

where $m$ is the mass of one hydrogen molecule and $d_{exc}$ is an exclusion distance due to the steep repulsive part of the interaction potential near the pore layers. In the thermodynamic equilibrium, the equilibrium constant $K_{eq}$ is related to the pressures of the compressed (or external) and adsorbed phases, $P_{ext}$ and $P_{ads}$, through the exact equation [24,32,33,35,42].

$$\ln(K_{eq}) = \frac{1}{RT} \int_{P_{ext}}^{P_{ads}} v_{mol}(P, T) dP \qquad (5)$$

where $R$ is the gas constant and $v_{mol}(P, T)$ is the molar volume of the compressed hydrogen. The integral that appears in Eq. 5 represents the decrease in the Gibbs free energy in the transfer of one mole of the hydrogen molecules from the free phase to the adsorbed phase. In the original model by Patchkovskii et al. [34], $v_{mol}$ was calculated using the ideal gas equation of state, $v_{mol} = RT/P$, which is only valid for low hydrogen densities. Subsequently, Cabria et al. improved the treatment by considering the Mills-Younglove equation of state [24,32], which includes the interactions between the $H_2$ molecules, and the same approach has been used in the present work. By solving the Mills-Younglove equation of state, we obtain $P_{ads}$ for each value of $P_{ext}$, and from the equation of state we find the molar volume of the adsorbed hydrogen phase, $v_{mol}(P_{ads}, T)$. The masses of the two phases are calculated when the thermodynamic equilibrium between them is reached. We note that the inclusion of a real equation of state for the molecular hydrogen (the Mills-Younglove equation of state, which results from a fit to a polynomial function with 750 parameters, enables an accurate description in wide ranges of temperature and pressure [24]) introduces, in an indirect way, the interactions between the hydrogen molecules in the free or compressed phase.

Although the Mills-Younglove equation of state takes into account the effect of having multiple interacting $H_2$ molecules, one could think that the interaction potential that we obtain by ab initio DFT calculations (Eq. 1) is only a first approximation that could be improved by considering its dependence with the density of the $H_2$ molecules, which in turns has to do with the pressure. As the pressure increases, it seems logical to think that each $H_2$ molecule will be screened by other molecules. However, the results of previous calculations for graphene slit pores [24,32,33] confirm that the quantum-thermodynamic model used in our calculation is a reliable method to compute the hydrogen storage capacities of slit-shaped pores. This is indicative of the fact that the main contribution of the $H_2-H_2$ interactions is well captured by the Mills-Younglove equation of state, and that the interaction potential used for describing the interaction between the $H_2$ molecule and the sheets of the pore is also satisfactory.

The only hydrogen storage capacities computed in Ref. [35] were the “adsorbed capacities”, which are defined as follows. The volumetric capacity, $v_c$, of the adsorbed hydrogen phase is calculated, in grams of $H_2$ per liter (L), by means of the equation

$$v_c(P, T) = \frac{\text{mass}_{H_2,ads}(P, T)}{V_{\text{pore}}} \qquad (6)$$

where mass$_{H_2,ads}$ is the mass of the adsorbed hydrogen in grams and $V_{pore}$ is the volume, in liters, of the pore. The mass of the adsorbed hydrogen is obtained as

$$\text{mass}_{H_2,ads}(P, T) = \frac{M(H_2)}{v_{mol}(P_{ads}, T)} V_{ads} \qquad (7)$$

where $V_{ads}$ is the volume, in liters, of the adsorbed phase, $M(H_2)$ is the molar mass of the molecular hydrogen in g/mol, 2.01588 g/mol, and $v_{mol}(P_{ads}, T)$ is the molar volume of the adsorbed phase in L/mol. $V_{ads} = Sw_{ads}$, where S is the surface of the layer and $w_{ads}$ is the width of the adsorbed phase, which is given by

$$w_{ads} = z_r - z_l \text{ if } z_r - z_l \le 2L, \text{ otherwise } w_{ads} = 2L \qquad (8)$$

where $z_r$ and $z_l$ are the distances 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, which must be chosen in an appropriate way (see Ref. [35] and the discussion below).

The gravimetric capacity, $g_c$, of the adsorbed hydrogen phase is calculated, in wt %, as

$$g_c(P, T) = \frac{100\text{mass}_{H_2,ads}(P, T)}{\text{mass}_{H_2,ads}(P, T) + \text{mass}_{\text{adsorbent}}} \qquad (9)$$

where mass$_{adsorbent}$ is the mass of the adsorbent material (i.e., that of the slit-shaped pore) in grams. The slit pore is composed by two sheets, and this is considered in the calculation of mass$_{adsorbent}$ and $g_c$.

On the other hand, the total volumetric and gravimetric capacities are given by similar equations to those of the adsorbed hydrogen phase, but using the total mass of the stored hydrogen, mass$_{H_2, total}$, i.e.,

$$v_{c,total}(P, T) = \frac{\text{mass}_{H_2,total}(P, T)}{V_{\text{pore}}} \qquad (10)$$

$$g_c.total(P, T) = \frac{100\text{mass}_{H_2,total}}{\text{mass}_{H_2,total}(P, T) + \text{mass}_{\text{adsorbent}}} \qquad (11)$$

where mass$_{H_2,total}$ includes the mass of the adsorbed hydrogen phase and that of the compressed hydrogen phase, not included in the adsorbed phase. The latter is obtained as the product of the volume occupied by the compressed hydrogen, $V_{compressed}$, and the free-hydrogen density.

The total capacities defined above give the total amount of hydrogen that can be stored at a given pressure and temperature. However, for practical purposes the relevant quantities are the usable capacities, which are those to which the DOE 2025 conditions refer [1,39–41]. The usable mass of hydrogen stored at a given pressure $P$ (loaded state) and temperature $T$ is the difference between the total mass of hydrogen stored at these $P$ and $T$ and the total mass stored at the depletion pressure and the same temperature $T$. The usable volumetric and gravimetric capacities are defined using the usable mass of stored hydrogen.

It should be pointed out that the results of the quantum-thermodynamic model are quite sensitive to the values of the exclusion distance due to the steep repulsive part of the interaction potential near the pore layers, $d_{exc}$ (Eq. 4), and to the parameter $L$, which determines the width of the adsorbed hydrogen phase, $d_{ads}$ (Eq. 8). The former corresponds to a limit in the energy for which the $H_2$ molecule cannot approach the layer, and in the calculations performed in this work it was chosen to be 1 eV, as in previous studies [33,35,42]. As regard the parameter $L$, we chose the value 2.89 Å, which corresponds exactly to the kinetic diameter of a $H_2$ molecule [48].

### 3. Determination of the confining potential

We now note that in the case of graphene the interaction potential
between a hydrogen molecule and the sheet depends so little on the (x, y) sites that one can consider that the approximations $V(x, y, z) = V(z)$ for a single layer and $V(z) + V(d-z)$ for a slit pore are reasonable. In contrast, in the case of Li-decorated borophene the interaction potentials have a strong dependence on the sites (x, y) due to the high structural anisotropy, so that we have to discard the 1D approximation assumed in the case of graphene and use instead 3D potentials for the Li-decorated borophene sheet and the Li-decorated borophene slit pore, $V(x, y, z)$ and $V(x, y, z) + V(x, y, d-z)$, respectively. In Ref. [35], we computed the interaction potential between a hydrogen molecule and the Li-decorated borophene sheet by imposing a fixed orientation of the molecule on the whole Li-decorated sheet as a first approximation (the most stable orientation on the relaxed sheet using the optB88-vdW functional [45]). In those calculations, we chose a (x, y) grid with $\Delta x = \Delta y = 0.25$ Å. This implied 936 grid points in the xy-plane. For each one of these (x, y) points or regions, we placed the hydrogen 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 $\Delta z = 0.125$ Å. Then, we made calculations for each one of these (x, y, z) grid points, and consequently we obtained a numerical 3D $V(x, y, z)$ interaction potential. The values of the interaction potential energy obtained in the DFT calculations at the (x, y, z) grid points were fitted to a piece-wise function. This piece-wise function was used in conjunction with the quantum-thermodynamic model to obtain the adsorbed hydrogen storage capacities [35].

It should now be pointed out that the inclusion of the rotational degree of freedom of the hydrogen molecule can have a strong influence on the interaction potential between the molecule and a sheet, and consequently on the hydrogen storage capacities of the corresponding

slit-shaped pores. This is what we have recently found that occurs in the case of Li-decorated slit pores formed by two layers of the binary, but planar, graphene-like BC₃ sheet [42], and similar effect could occur in the case of slit pores formed by the strongly anisotropic borophene sheet. To analyze this question, we performed now grid-point calculations similar to those indicated above, but allowing to relax the orientation of the hydrogen molecule around its center of mass over the (x, y) points of the grid. The optimal spherical polar and azimuthal angles at each point were determined using the Nelder-Mead simplex algorithm. This produces noticeable effects in the interaction potential $V(x, y, z)$: the actual local minima of the potential are captured, but the interaction potential energy is more binding. This is illustrated in the left-hand panels of Fig. 2, where we show results for the interaction potential energy between a H₂ molecule and the Li-decorated borophene sheet for both fixed and non-fixed orientations of the molecule (note that the potential is always binding in the case of the non-fixed orientation). In the right-hand panel of Fig. 2 we show the interaction potential energy curves at the deepest and shallowest (x, y) points as functions of the distance from the center of mass of the H₂ molecule to the layer of the Li atoms, for both fixed and non-fixed orientations of the H₂ molecule. The effect is more pronounced for the shallowest curve, where the position of the minimum is displaced, and the potential is more negative after releasing the rotational degree of freedom. All these features show that the inclusion of the rotational degree of freedom of the H₂ molecule has a strong influence of the confining potential of the Li-decorated borophene slit pores, $V(x, y, z) + V(x, y, d-z)$, what may also affect their hydrogen storage capacities. We note that even for points in which the approximation of a fixed orientation of the H₂ molecule is reasonable

<figure><img src="image_5.png" /><figcaption>**Fig. 2.** Left-hand panels: interaction potential energy between a single H₂ molecule in the plane at z = 0.75 Å from the Li layer, projected on the xy plane; the upper and lower panels correspond to the cases of fixed and non-fixed orientations of the H₂ molecule, respectively. Right-hand panel: the interaction potential energy between a H₂ molecule and a single Li-decorated borophene sheet at the deepest and shallowest xy points, as a function of the distance from the H₂ molecule to the layer of the Li atoms, for both fixed and non-fixed orientations.</figcaption></figure>
because it is close to the optimal orientation, a gain of energy of a few meV is found when optimizing the molecular orientation. This can be appreciated by comparing the two red curves in the right-hand panel of Fig. 2.

The fact that the interaction potential V(x, y, z) is more binding when the orientational degree of freedom of the H₂ molecule is included in the calculations has an effect on the accessible volume of the H₂ molecule. In fact, the exclusion distance due to the steep repulsive part of the interaction potential, taken at 1 eV as indicated above, is further away from the layer of the Li atoms in the case in which the orientation is fixed, which produces a reduction of the available volume that the H₂ molecule can occupy in the adsorption region.

As indicated in the Introduction, the role played by the rotational degree of freedom of the hydrogen molecule in determining the confining potential within the slit pores and their hydrogen storage capacities was found to be crucial in a recent study of slit pores formed by two parallel layers of the graphene-like BC₃ sheet [42]. In fact, imposing a fixed orientation of the H₂ molecule on the whole sheet prevented the location of the actual minima of the confining potential. The physical reason behind this fact is general: when a degree of freedom that initially is kept fixed in a system is allowed to relax and dissipate stress, it becomes more stable. In our case, the H₂ molecule becomes more bonded to the pore layers, and therefore the potential becomes more confining and, what is important, all the actual minima of the potential are located.

## 4. Results and discussion

Fig. 3 shows the calculated adsorbed hydrogen storage capacities of Li-decorated borophene slit pores as a function of the pore width at low and room temperatures, 80.15 K and 298.15 K, and low and high pressures, 0.5 MPa and 25 MPa. For the sake of comparison, we also show in the same figure the adsorbed hydrogen capacities as computed when the orientation of the hydrogen molecule is kept fixed. The curves have dependencies with the pore width qualitatively similar in both cases: they have maximum values and then decrease, in the case of the gravimetric capacities tending to asymptotic values. The highest adsorbed hydrogen storage capacities of Li-decorated borophene slit pores are obtained at the lowest temperature and the highest pressure. But the important point here is that the capacities obtained without imposing a fixed orientation to the hydrogen molecule are notably higher than those obtained by assuming a fixed orientation. As indicated in the previous section, the confining potential when the orientational degree of freedom of the hydrogen molecule is included in the calculations is more binding, and the exclusion region which is close to the Li-decorated borophene sheet is thinner. Therefore, there is a higher accessible volume in the pore for the H₂ molecule, which explains the increase of the volumetric capacity. As the mass of the adsorbed hydrogen increases, the gravimetric capacity increases as well. We note in Fig. 3 that, at low temperature, a small external pressure is enough to induce a high volumetric capacity if the confining potential is properly computed by taking into account the orientational degree of freedom of the H₂ molecule.

<figure><img src="image_6.png" /><figcaption>Fig. 3. Adsorbed volumetric and gravimetric hydrogen storage capacities of Li-decorated borophene slit pores, as a function of the pore width, at T = 80.15 or 298.15 K and P = 0.5 or 25 MPa. The volumetric and gravimetric capacities are shown for both fixed and non-fixed orientations of the H₂ molecule.</figcaption></figure>
The results of Fig. 3 show two important features. The first is that the maxima of the adsorbed hydrogen storage capacities are reached in all cases for small pore widths in the range of 5–10 Å. The second feature is that the difference between the adsorbed hydrogen storage capacities at 25 MPa and 0.5 MPa is much larger at room temperature than at cryogenic temperature. This means that the contribution of the adsorbed hydrogen phase to the usable capacities is expected to be greater at room temperature, a desirable trend from the technological point of view. It is important to realize that a low storage capacity is required at low pressure (delivery pressure) to be able to extract as much hydrogen as possible when depressurizing the adsorbent, a characteristic that increases its usable capacity. In this respect, the high adsorbed hydrogen storage capacities predicted so far for a variety of adsorbents by means of DFT calculations at T = 0 K and no pressure is a drawback more than an advantage for their efficiency in charge-discharge cycles.

We now focus on the usable hydrogen storage capacities as functions of temperature, pressure and loading pressure. We remind that both the adsorbed and the compressed phases of hydrogen contribute to the usable capacities of the nanopores. These quantities were not considered in Ref. [35]. In Fig. 4 we show two-dimensional color maps in which one of those parameters is kept fixed at a typical value and the other two vary in a reasonable range. The results were obtained when the depletion pressure was set at 0.5 MPa. The upper values in the color scales correspond to those established by the U.S. DOE. Those maps allow to recognize what are the best operational values of these kinds of porous adsorbents. For instance, by looking at the upper panels, we infer that if we wanted to work at a loading pressure of 25 MPa, the material would be efficient in terms of the volumetric capacity either for pore widths in the range from 5 to 10 Å working at temperatures above 200 K or for larger pore widths of about 25 Å working at a temperature lower than 100 K (see the upper left-hand panel of Fig. 4). If we wanted to operate at room temperature, as the loading pressure is reduced the pore width should be reduced for it to be efficient as an adsorbent material (see the middle left-hand panel). However, in terms of the gravimetric capacity, the right-hand panels of Fig. 4 indicate that such absorbent would be useless unless the pore width was large (close to 30 Å or more) and operating at very large pressures (larger than about 60 MPa) and very low temperatures (lower than about 100 K).

Perhaps the most interesting goal in the field of solid-state adsor-
bents for vehicular hydrogen storage is to find a material that could
work at room temperature and very low loading pressures, since this
would improve the security and reduce costs and weight of the sophis-
ticated tanks used at present for high pressure storage. The middle left-
hand panel of Fig. 4 is useful in this respect. One sees that a reasonably
good volumetric capacity could be achieved for a loading pressure of
only 20 MPa if the material is made of nanopores with a width of about
6 Å. We remark that the usable capacities plotted in Fig. 4 were calcu-
lated at a depletion pressure of 0.5 MPa. The loading-delivery pressure
window can be modified to scan different sorption-desorption condi-
tions. In Fig. 5, we show the usable capacities of a Li-decorated bor-
ophene slit pore of width 6 Å at room temperature as a function of the
loading pressure for two different values of the depletion pressure, 0.1
MPa (room pressure) and 0.5 MPa. It can be seen that the usable volu-
metric capacity reaches the DOE 2025 target at a loading pressure as
small as about 6 MPa when the delivery pressure is 0.1 MPa. This is a
relevant finding, since it means that working with these types of ad-
sorbents would require to exert small pressure on the gases as compared
with the pressure of 70 MPa currently used in tanks. The usable gravi-
metric capacities, on the contrary, do not reach the DOE 2025 target at
those conditions.

5. Conclusions

In this work, we have used a quantum-thermodynamic model to
compute the hydrogen storage capacities of Li-decorated borophene slit
pores. The quantum-thermodynamic model is based on that proposed by

Patchkovskii et al. [34], which was used to investigate the hydrogen storage capacity of graphene slit pores by considering the hydrogen molecules as an ideal gas of featureless particles and describing the H₂-graphene interaction by means of a modified form of the Lennard-Jones potential. The theoretical model used in our calculations has three new main ingredients: the 3D interaction potential energy between a hydrogen molecule and the layers of the Li-decorated borophene slit pore are computed by means of DFT calculations; the H₂-H₂ interactions are implicitly taken into account by using a real equation of state for the molecular hydrogen such as the Mills-Younglove equation of state [24]; and the confining potential is calculated by taking into account the orientational degree of freedom of the hydrogen molecule (i.e., the most favourable orientation of the H₂ molecule at each point of the potential energy landscape). We show that the inclusion of this degree of freedom, which was not considered in our previous study of Li-decorated borophene slit pores [35], has a great influence in the computed hydrogen storage capacities (Fig. 3). On the other hand, in the present work we computed the usable hydrogen storage capacities, which are those to which the DOE criterion refers for vehicular hydrogen storage [1].

As regards the possible use of borophene-based porous systems as
hydrogen storage media, our results for borophene slit pores show that,
after Li doping, they can have a good usable gravimetric capacity for
large pore widths at low temperature, and a good usable volumetric
capacity for smaller pore widths at room temperature. A relevant result
is that, for small pore widths, the usable volumetric capacity can reach
the DOE 2025 target operating at room temperature and a loading
pressure as small as about 6 MPa when the depletion pressure is 0.1
MPa. This could be useful for practical applications, where low charging
pressures are desirable for security reasons as well as for reducing the
energy required to pressurize the tank. In contrast, the usable gravi-
metric capacities do not meet the DOE 2025 target at those conditions.
As indicated in the Introduction, some borophene-based 3D porous
structures have theoretically been predicted to be dynamically, ther-
mally and mechanically stable [38]. However, it is well known that the
synthesis of adsorbents decorated with atoms that can accommodate
several H₂ molecules is very difficult in general, and this is likely to be
happen in the particular case of borophene-based porous structures
(nanoporous boron frameworks containing many regions of borophene
slit-shaped pores could perhaps be formed by using zeolite as a sacrifi-
cial template, just as zeolite-template carbons are formed [49], as
indicated in Ref. [35]). In spite of that, there are strong experimental
evidences that metal-doping, in general, improves the hydrogen storage
capacities of solid-state adsorbents, as predicted by theoretical calcula-
tions (see, e.g., Ref. [40]).

As it is well known, the identification of adsorbents meeting all the
targets established by the U.S. DOE is a challenging problem. In this
respect, borophene-based porous structures do not appear to fulfill
simultaneously all the necessary requirements for vehicular hydrogen
storage, but are good candidates when the volumetric capacity is the
main target. Experimentalists can find our results useful to make in-
vestigations on these kinds of adsorbents. On the other hand, the
methodology used in this paper, which takes into account both the in-
teractions between the hydrogen molecules (through a real equation of
state, such as the Mills-Younglove equation [24]) and the orientational
degree of freedom, can be a useful guide for theoretical studies on the
hydrogen storage of slit pores formed by other anisotropic 2D materials.
It can also be useful in the investigation on adsorption of non-hydrogen
molecules, such as CO₂ or CH₄. Work on this line is currently underway
and will be published elsewhere.

CRediT authorship contribution statement

**A. Lebon:** Writing – original draft, Supervision, Project administra-
tion, Methodology, Investigation, Formal analysis, Data curation,
Conceptualization. **B. Calvez:** Methodology, Conceptualization. **M.B.**
**Torres:** Writing – original draft, Methodology, Data curation. **L.J.**
<figure><img src="image_7.png" /><figcaption>**Fig. 4.** Usable volumetric and gravimetric hydrogen storage capacities of Li-decorated borophene slit pores as functions of the pore width and temperature at 25 MPa (upper panels), of the pore width and pressure at 298.15 K (middle panels) and of the pressure and temperature for a pore width of 6 Å (lower panels).</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>**Fig. 5.** Usable volumetric and gravimetric hydrogen storage capacities of a Li-decorated borophene slit pore of width 6 Å at 298.15 K, as functions of the loading pressure and the depletion pressure. The 2025 DOE targets are shown with horizontal dashed lines.</figcaption></figure>

**Gallego: Writing – original draft, Supervision, Investigation. A. Vega:**
Writing – original draft, Supervision, Conceptualization.

Declaration of competing interest

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.

Data availability

Data will be made available on request.

Acknowledgments

AL and BC acknowledge the use of the high performance computing equipment of the Pole de Calcul Intensif pour la Mer (DATARMOR, IFREMER/UBO Brest). This research was financed 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 (GRC ED431C 2020/10).

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