Experimental and Mathematical Investigation of Hydrogen Absorption in LaNi₅ and La_0.7Ce_0.1Ga_0.3Ni₅ Compounds

Abstract

In the present study, the hydrogen-absorption properties of the LaNi₅ and the La_0.7Ce_0.1Ga_0.3Ni₅ compounds were determined and compared. This work is therefore divided into two parts: an experimental part that presents and discusses the kinetics and isotherms of hydrogen absorption in the two compounds at two different temperatures (298 K and 318 K). In addition, the temperature variations inside the hydride bed were determined. In the second section, the experimental isotherms were compared to a numerical model processed using statistical physics. Following that, thanks to the perfect agreement between the experimental data and the proposed model, the stereographic and energetic parameters associated with the hydrogen absorption reaction, such as the number of hydrogen atoms per receptor site (n₁, n₂), the densities of the sites (N_m1, N_m2), the half-saturation pressures (P₁, P₂) and the absorption energies (ΔE₁, ΔE₂) for each receptor site, were calculated. All of these parameters are acquired by making numerical adjustments to the experimental data. Thermodynamic functions, such as internal energy and Gibbs energy, which regulate the absorption process, were then identified using these parameters. For both compounds, all of the aforementioned were compared and discussed in relation to initial temperature and pressure. The results demonstrated that the hydrogen-storage properties in LaNi₅ are enhanced by more than 30% of stored mass and kinetics when Ce and Ga are substituted at the La sites.

1. Introduction

The world is going through an unprecedentedly deep and complicated global energy crisis [1]. As a result, decarbonization, electrification, and renewable energy sources are major themes in the World Energy Outlook for 2024, signaling a clear shift in the energy industry [2]. Hydrogen stands out among the many potential remedies as a novel, renewable and extremely potent energy vector that makes it possible to produce clean and silent electricity when it is needed [3]. However, hydrogen has a very low volume density despite having a very-high-mass energy density (1 kg of hydrogen has the same energy as around 3 kg of oil). Therefore, before being used, it must be stored in a usable volume under secure conditions [4]. By the way, a key market segment for hydrogen energy is storage [5]. Although various methods have been explored for hydrogen storage, including gas and liquid forms [5,6], solid storage in the form of metal hydrides appears to be a very promising solution, which is particularly interesting for transport and stationary applications [7,8]. Moreover, among the numerous families of metal hydrides [9,10], the AB₅ hydride possesses intriguing and alluring features for reversibly storing hydrogen, including simple activation [11], high mass storage capacity [12], good recyclability [13] and storage under standard pressure and temperature conditions [14]. LaNi₅ is specifically recognized as a hydrogen prototype for AB5 intermetallic compounds. Furthermore, several of its derivatives were among the first to be used electrochemically in nickel–metal hydride (Ni-MH) batteries [15]. They have been used as the negative electrode in Ni-MH batteries [16]. However, there are two main technical challenges related to solid-state hydrogen storage in LaNi₅ [17]: The first concern is the thermal management during absorption and desorption processes since M-H has limited thermal conductivity; as a result, bulk MH-bed systems frequently have a Heat Transfer System (HTS) built into them. Another concern is efficient and effective scaling up. Actually, the poor hydrogen storage density of LaNi5-based compounds explains why Li-ion batteries are more common in portable electronics and rechargeable batteries for electric and hybrid cars [16].

Thus, one of the key techniques used to improve the hydrogen-storage characteristics inside the LaNi₅ compound is the partial substitution of La and/or Ni sites [18,19]. Thereby, recent and innovative research has shown that partial or total substitution can significantly improve the cycle stability and high-rate capacity of the electrodes [20]. For example, we have taken a peek at an experimental work conducted by Zhida et al. [21], whereby the kinetic performances and hydrogen storage abilities of LaNi_5−xCo_x alloys (x = 0, 0.25, 0.50, and 0.75) were examined through experimentation. The findings demonstrated that when the Co content increases, the cell volume increases as well, lowering the equilibrium plateau pressure and promoting a more stable hydride phase. The addition of Co is also found to increase mass and reaction kinetics. In another research work envisaged by Jingjing L et al. [22], hydrogen storage in the LaNi_5−xAl_x alloy (x = 0, 0.25, 0.5) is studied experimentally. The research results demonstrated that Al substitution increases the stability of the alloy’s crystal lattice by limiting atomic migration during cycling and reducing the rate of lattice volume expansion during hydrogenation. This improves the cycling stability of Al-doped alloys by around 10%.

Changpeng W. et al. [23] examined a combination substitution at the La and Ni sites in more recent research. By adjusting the Al concentration, this study attempted to enhance the (La_0.33Y_0.67)₅Ni_18.1−xMn_0.9Al_x alloys’ overall electrochemical hydrogen-storage capabilities. The findings demonstrated that Al doping leads to a 69.80% improvement in desorption capacity and a decrease in equilibrium plateau pressure. Moreover, this study demonstrates the alloy’s potential (La_0.33Y_0.67)₅Ni_17.5Mn_0.9Al_0.6) as a high-performing Ni-MH battery anode (La_0.33Y_0.67)₅Ni_17.5Mn_0.9Al_0.6. Similarly, the (LaCeCa)₁(NiMnAl)₅ alloy series was examined in a study planned by Xuqi L et al. [13] under the framework of a combined multi-substitution. Experiments have revealed that the maximum hydrogen-storage capacity of the alloy series (LaCeCa)₁(NiMnAl)₅ is approximately 0.3 weight percent higher than that of LaNi5, despite the fact that structural analysis by X-ray revealed that the substituted intermetallic crystallizes in a structure similar to that of the LaNi₅ compound (CaCu₅ structure).

Thus, building on earlier research, we propose investigating the hydrogen storage properties of the compound La_0.7Ce_0.1Ga_0.3Ni₅ both experimentally and numerically in this paper. Generally, elemental substitution at La sites aims to improve the overall qualities of hydrogen storage, such as the mass, kinetics, and stability of the hydride produced, while also reducing the cost of alloy production. Consequently, selecting doping materials requires significant thought for practical use in hydrogen storage systems. Studies have indicated that partial substitution of Ce for La in LaNi₅ results in simpler activation, an increase in equilibrium plateau pressure, and an increase in reaction kinetics [24,25,26]. Furthermore, metallic Ce is less expensive than La. Thus, substitution can lower costs and encourage widespread commercial adoption [27,28]. Likewise, the Ga element has been used in several studies as a dopant in hydrogen storage materials [29,30]. These materials showed optimal hydrogen storage capacities and significant stability of the formed hydride. Thus, in this study, a combined substitution on the sites of La using the elements Ce and Ga is proposed. This study involves comparing experimentally and numerically the hydrogen-storage properties of LaNi₅ and La_0.7Ce_0.1Ga_0.3Ni₅. For this reason, two parts will be presented in this paper: an experimental part, for which the absorption kinetics and isotherms of the two compounds will be traced. Then, the experimental isotherms will be compared to a numerical model using the statistical physics approach [31]. This model must be used to access the many macroscopic and microscopic phases of the system by providing insight into the molecular processes involved in the absorption reaction [32]. The suggested model’s analytical expressions will be utilized to adjust the experimental isotherms pertaining to LaNi₅ and La_0.7Ce_0.1Ga_0.3Ni₅ compounds at varying temperatures (298 K and 318 K). Consequently, for a receptor site i, two categories of parameters are inferred from the adjustment: (i) stereographic parameters, which include the number of hydrogen atoms per receptor site (n_i) and the densities of each receptor site (N_mi). (ii) Energy parameters, which include the half-saturation pressures for each receptor site (P_i). Ultimately, these parameters will be used to compute the absorption energies ∆E_i, the concentrations of hydrogen per unit of metal [H/M]_i, and the thermodynamic functions governing the absorption.

Despite the various studies that have been carried out on these alloys, no numerical study using statistical physics has been considered to better describe and interpret the effect of substitution with the elements Ce and Ga on the Ni sites. Therefore, this research aims to provide new molecular investigations into the doping effect through theoretical work. The selected model was used to calculate the thermodynamic functions that control the hydrogen-storage process as well as the stereographic features that interfere with it.

2. Experimental Setup

The experimental device used is a Sievert apparatus (Figure 1a). It consists of a cylindrical metal hydride reactor M-H-R (Figure 1b) equipped with internal radial fins and connected to a gaseous hydrogen line via connection valves. A cold water bath is used to feed the hydride bed with cooling water during the absorption reaction through a helical heat exchanger. The reactor was filled with 30 g of purchased metal. A pressure sensor is installed between the reactor and the hydrogen tank, controlling the variation in hydrogen pressure during absorption processes. As well, a thermocouple is installed inside the reactor, controlling the internal temperature during absorption. These sensors are installed on an acquisition card and a microcomputer [33].

3. Results

3.1. Experimental Results

• Absorption kinetics

Hydrogen absorption reaction kinetics refers to the rate at which hydrogen is absorbed by a material. Therefore, the task at hand is to depict how the mass of hydrogen absorbed changes over time. Figure 2 presents the variation of the mass of hydrogen absorbed as a function of time at T = 298 K and at T = 318 K for the LaNi₅ and the La_0.7Ce_0.1Ga_0.3Ni₅ compounds.

Firstly, we observe that the mass of hydrogen steadily rises with time at a specific temperature until it reaches equilibrium. It is shown that, at a given temperature, the doped compound reaches equilibrium faster than LaNi₅ with more stored hydrogen. This means that doping with Ce and Ga on the La sites improves the absorption kinetics and mass. We also point out that raising the initial temperature harms the absorption response by decreasing the mass of hydrogen absorbed and the absorption kinetics. For example, at T = 298 K, the mass of hydrogen absorbed at equilibrium for the LaNi₅ is 7 g of hydrogen atoms, while at T = 318 K, it is 5 g of hydrogen atoms. For the La_0.7Ce_0.1Ga_0.3Ni₅ at T = 298 K, the mass of hydrogen absorbed at equilibrium for the LaNi₅ is 8 g of hydrogen atoms while at T = 318 K it is 6.5 g of hydrogen atoms. This is explained by the exothermic nature of the hydrogen-absorption reaction.

• Internal temperature

Temperature significantly affects absorption kinetics. It is important to regulate the internal temperature of an exothermic absorption reaction for multiple reasons. The process of exothermic absorption releases heat, which may raise the system’s temperature. The integrity of the absorbent material, the safety of the system, and the response kinetics may all be adversely impacted by improper management of this heat. A thermocouple was inserted on the acquisition card and inside the hydride bed to regulate the interior temperature.

Figure 3 depicts the variation of the temperature inside the hydride bed during absorption as a function of time at T_i = 298 K (a) and T_i = 318 K (b).

It is noted that in the initial stages of the absorption response, the internal temperature rises, which confirms the exothermic nature of the reaction. Then, the temperature decreases following the external cooling process by the cold water coming from the thermostatic bath, and it stabilizes at a temperature close to that set by the thermostatic bath (T_i). We see that La_0.7Ce_0.1Ga_0.3Ni₅ exhibits greater exothermic hydrogen absorption than does LaNi5. Since more heat is created during the doped compound’s absorption reaction, more hydrogen atoms are put into the La_0.7Ce_0.1Ga_0.3Ni₅’s interstitial sites under the same starting conditions.

• Absorption isotherms

Pressure-composition isotherms (P-C-T) are one of the most significant considerations to assess thermodynamic properties for an alloy that absorbs hydrogen. It is feasible to track the development of the reversible storage reaction (absorption/desorption) thanks to these isotherms. The hydrogen absorption isotherms for the compounds LaNi₅ and La_0.7Ce_0.1Ga_0.3Ni₅ at T = 298 K and T = 318 K are displayed in Figure 4.

It is shown, in Figure 4, that at a given temperature, an absorption isotherm is made up of three parts [34]:

α phase: This phase is characterized by a significant increase in pressure vs. 0 absorbed mass.

Hysteresis measurement. This indicates that the hydrogen molecules have not yet dissociated into atoms and are still present on the surface of the absorbent material.

α+β phase: This phase is associated with the plateau of equilibrium. It is distinguished by a pressure stabilization against an increasing rise in the hydrogen absorbed mass. This indicates that hydrogen can exist in both gaseous and solid forms.

β phase: This is related to the solid phase, in which the absorbent substance completely absorbs the hydrogen atoms.

First, it is demonstrated that raising the temperature causes the equilibrium plateau for both compounds to climb. This is explained by the exothermic nature of the hydrogen absorption reaction, which suggests that more pressure is required to store hydrogen at high temperatures. Secondly, through comparison, it is shown that, at a given temperature, the equilibrium plateau for the compound La_0.7Ce_0.1Ga_0.3Ni₅ is situated above that for LaNi₅, and that the compound La_0.7Ce_{0. 1}Ga_0.3Ni₅ has a greater total mass absorbed than the compound LaNi₅ by 0.2 wt%. In summary, doping increases the bulk capacity of stored hydrogen, even if it can destroy the stability of the hydride by slightly raising the plateau pressure. Consequently, doping with Ce and Ga elements on the La sites results in 1.6 weight percent of the storage capacity. Thus, this value represents, for example, theoretical storage capacity of the TiFe hydride, with better safety indicated by the normal working conditions of temperature and pressure [7].

• Hysteresis measurement

The term “hysteresis” describes how the current state of a system is influenced by its past. When the hydrogen absorption and desorption curves on an isotherm do not overlap, a hysteresis phenomenon occurs. Understanding hysteresis is essential in developing materials with the best possible hydrogen-storage features since it shows structural changes and other phenomena within the material throughout the process of reversible hydrogen storage. The hysteresis cycles for the compounds LaNi₅ and La_0.7Ce_0.1Ga_0.3Ni₅ are shown in Figure 5.

Consequently, we can use the following equation to estimate a hysteresis rate for each compound based on the experimental values of the equilibrium pressures (P_eq) [35]:

$$Hys=\ln ({\displaystyle \frac{{P_{eq}}^{abs}}{P_{eq}^{des}}})$$

(1)

Table 1. Hysteresis rate calculation.

Compound	Temperature (K)	Peq (Absorption) (bar)	Peq (Desorption) (bar)	Hys
LaNi5	298	4.50	4.00	0.11
	318	5.60	5.20	0.07
La0.7Ce0.1Ga0.3Ni5	298	4.88	4.45	0.09
	318	6.05	5.75	0.05

presents the hysteresis rate values for each compound versus temperatures.

First of all, the hysteresis rate calculation reveals that hysteresis falls with temperature. When considering the La_0.7Ce_0.1Ga_0.3Ni₅ compound, this reduction is much more impressive. In fact, the hysteresis is reduced by almost 50% when the temperature goes from 298 K to 318 K.

This can be explained by the prospect that, as the temperature rises, the desorption reaction will take place at temperatures that are closer to those of absorption, which will reduce the hysteresis. Then, it is shown that at a given temperature, the La_0.7Ce_0.1Ga_0.3Ni₅ compound has a lower hysteresis rate than the LaNi₅ compound. Given that hysteresis and cyclability are interrelated aspects of material performance, low hysteresis improves efficiency, leading to high cyclability, which ensures long-term use. Thus, low hysteresis indicates that the material is working more efficiently, with less energy lost during absorption and desorption cycles. This increased efficiency can improve cyclability by reducing energy losses and preventing heat buildup or other negative effects that could degrade the material. Therefore, it is observed that doping the compound with Ce and Ga on the La sites increases its cyclability as well as its ability to regularly absorb and desorb hydrogen without degrading it, thus extending its useful life.

In the subsequent section, we will use the statistical physics formalism to physically examine the aforementioned results, starting at the microscopic level and working our way up to the macroscopic level. Given that the absorption process entails the penetration of atoms into the absorbent material’s volume, as opposed to the adsorption process, which involves the attachment of adsorbed molecules to the material’s surface, studies have demonstrated the energy similarity between these two processes because they arise from the same phenomenon of diffusional equilibrium [36]. Therefore, theoretical research by Wjihi et al. [32] suggests that adsorption and absorption phenomena can be modeled using the same statistical physics model. Thus, it is assumed in this study to solve the problem of hydrogen absorption in LaNi₅ and La_0.7Ce_0.1Ga_0.3Ni₅ in the same way adopted for the adsorption process.

3.2. Modeling

• Description of the model

The application of the statistical physics model to study the phenomenon of hydrogen storage in intermetallics and their derivatives is a fascinating subject because this model contains important physical parameters in its expressions that are different from those of empirical models like Langmuir and Freundlich, the parameters of which generally have no direct physical meaning [37,38]. Thereby, to use a statistical physics approach to examine the absorption process, three assumptions need to be stated [39]. First, the statistical grand canonical ensemble must be used because it calls for an interchange of particles from the free level to the absorbed level. Second, hydrogen must be regarded as an ideal gas. Third, all internal degrees of freedom of hydrogen—aside from translation and rotation—will be disregarded.

On the one hand, the general equation that describes the reaction of intermetallics’ reversible hydrogen storage is

$$\mathrm{M}+{\displaystyle \frac{\mathrm{n}}{2}}{\mathrm{H}}_2\to \mathrm{M}{\mathrm{H}}_{\mathrm{n}}$$

(2)

where M is the absorbent intermetallic, n is the stoichiometric coefficient, H_n is defined as an aggregate of n hydrogen atoms in a receptor site, and the generated hydride is described by MH_n.

On the other hand, the grand canonical partition function Z_gc for one interstitial site is formulated in the following general formula [40]:

$${\mathrm{z}}_{\mathrm{g}\mathrm{c}}={\displaystyle \sum _{{\mathrm{N}}_{\mathrm{j}}}{\mathrm{e}}^{-\mathsf{\beta }(-{\mathsf{\epsilon }}_{\mathrm{j}}-\mu ){\mathrm{N}}_{\mathrm{j}}}}$$

(3)

where −(−ε_j) represents the absorption energy of one receptor site, N_j is the occupation state of one receptor site and is either 0 or 1, µ denotes the chemical potential of one absorbed site, and β is expressed as follows:

$$\beta ={\displaystyle \frac{1}{K_{B}T}}$$

(4)

where T is the absolute temperature, and K_B is the Boltzmann constant.

Assuming that the hydrogen absorption reaction occurs in this case on N_m independent and identical receptor sites, the grand canonical partition function can then be written as follows:

$$Z_{gc}={(Z_{gc})}^{N_m}$$

(5)

As mentioned by Belkhiria et al. [40], the expression of the average site occupancy number N_m is determined as

$$N_0=k_{B}T{\displaystyle \frac{\partial \ln (Z_{gc})}{\partial \mu }}=N_m{k}_{B}T{\displaystyle \frac{\partial \ln (z_{gc})}{\partial \mu }}$$

(6)

Therefore, considering that a variable quantity of N_a hydrogen atoms is absorbed in the N_mi receptor sites distributed throughout the mass unit of the intermetallic, the above formulations make it possible to consider a certain number of modeling methods based on the formalism of statistical physics, such as the single-layer or double-layer model with identical sites, two types of sites, three sites and four sites. Thus, the choice of the appropriate model is based on the model, which gives a better fit of the experimental isotherms and better-adjusted coefficient R². Applying the mono-layer model with two different types of receptor sites yields the optimal adjustment in the present case. Furthermore, the researchers have selected this model to represent the absorption process in intermetallics [41].

We suppose that the two energy levels can be allocated to phases 1 and 2 in the single-layer model with two distinct types of receptor sites. The values n₁ and n₂, respectively, represent the number of H atoms per site for phases 1 and 1. The two locations, 1 and 2, in Equation (2) are separated as follows:

$${\displaystyle \frac{{\mathrm{n}}_1}{2}}{\mathrm{H}}_2+\mathrm{M}\to \mathrm{M}{\mathrm{H}}_{{\mathrm{n}}_1}$$

(7a)

$${\displaystyle \frac{{\mathrm{n}}_2}{2}}{\mathrm{H}}_2+\mathrm{M}\to \mathrm{M}{\mathrm{H}}_{{\mathrm{n}}_2}$$

(7b)

First, we keep in consideration that the absorption is assumed to involve two distinct types of sites, n₁ and n₂. Second, we take into account that N_1m provides the density of sites n₁, and N_2m provides the density of sites n₂. Third, we employ ∆ε₁ and ∆ε₂ to determine the absorption energy at sites n₁ and n₂, respectively. Therefore, Z_gc is represented as follows:

$${\mathrm{Z}}_{\mathrm{g}\mathrm{c}}={({\mathrm{z}}_{1\mathrm{g}\mathrm{c}})}^{{\mathrm{N}}_{1\mathrm{m}}}\times {({\mathrm{z}}_{1\mathrm{g}\mathrm{c}})}^{{\mathrm{N}}_{2\mathrm{m}}}$$

(8)

where the partition functions of the first and second types of sites are denoted by Z_1gc and Z_2gc, respectively:

$$z_{1gc}={\displaystyle \sum _{Nj=0,1}{e}^{\frac{(-{\epsilon }_1-\mu )N_j}{k_{B}T}}}=1+e^{\beta ({\epsilon }_1+\mu )}$$

(9a)

$$z_{2gc}={\displaystyle \sum _{Nj=0,1}{e}^{\frac{(-{\epsilon }_2-\mu )N_j}{k_{B}T}}}=1+e^{\beta ({\epsilon }_2+\mu )}$$

(9b)

Therefore, using Equations (6) and (7), N₀ can be expressed as follows:

$$N_0=N_{1m}{k}_{B}T{\displaystyle \frac{\partial \ln (z_{1gc})}{\partial \mu }}+N_{2m}{k}_{B}T{\displaystyle \frac{\partial \ln (z_{2gc})}{\partial \mu }}={\displaystyle \frac{N_{1m}}{\left(1+e^{-\beta \left({\epsilon }_1+\mu \right)}\right)}}+{\displaystyle \frac{N_{2m}}{\left(1+e^{-\beta \left({\epsilon }_2+\mu \right)}\right)}}$$

(10)

Considering in the one hand, that hydrogen is an ideal gas, the chemical potential of a free hydrogen molecule state is given as follows:

$${\mu }_m=k_{B}T\ln \left({\displaystyle \frac{N}{z_g}}\right)$$

(11a)

Thus, at thermodynamic equilibrium, we have:

$${\mu }_m={\displaystyle \frac{\mu }{n}}$$

(11b)

μ presents the chemical potential of the receptor site, and n presents the number of atoms per one site.

In the other hand, as previously noted, just the degrees of freedom of translation and rotation are considered; therefore Z_gc is expressed as follows [40,41]:

$$z_g=z_{gtr}\times z_{grot}=V{\left({\displaystyle \frac{2\pi m{k}_{B}T}{h^2}}\right)}^{3/2}+{\displaystyle \frac{T}{2{\theta }_{rot}}}$$

(12)

where m is the mass of a hydrogen atom, V is the system’s volume, h is the Planck’s constant (h = 6.6260 × 10⁻³⁴ m² kg s⁻¹), and θ_rot is the dihydrogen molecule’s typical rotational temperature.

The relationship between µ and µ_m is governed by Equation (3). Thus, in the thermodynamic equilibrium state (equilibrium of pressure and temperature), the law of mass action providing the equilibrium constant K is written as follows:

$$K\left(T\right)={\displaystyle \frac{\left[M{H}_n\right]}{\left[M\right]\times {\left[H_2\right]}^{\left(\frac{n}{2}\right)}}}\text{↔}\left[M{H}_n\right]=K\times \left[M\right]\times {\left[H_2\right]}^{\left({\displaystyle \frac{n}{2}}\right)}$$

(13)

where [MH_n], [M], and [H₂] present the hydride, metal, and hydrogen concentrations, respectively. Thus, Equation (10) imposes the following:

$$\mu \left(M\right)={\displaystyle \frac{n}{2}}{\mu }_m\left(H_2\right)$$

(14)

And

$${\mu }_m\left(H_2\right)={\displaystyle \frac{\mu \left(M\right)}{\left(\frac{n}{2}\right)}}={\displaystyle \frac{2}{n}}\mu \left(M\right)$$

(15)

Consequently, the dihydrogen molecule’s absorption energy, ε_m, can also be represented as follows:

$${\epsilon }_m\left(H_2\right)={\displaystyle \frac{\epsilon \left(M\right)}{\left(\frac{n}{2}\right)}}={\displaystyle \frac{2}{n}}\epsilon \left(M\right)$$

(16)

Assume that a fully absorbed molecule from the first and second site defined by n₁/2 and n₂/2 has energies of ε_1m and ε_2m, respectively. In this case, ε_1m and ε_2m are expressed by the following expressions:

$${\epsilon }_{1m}={\displaystyle \frac{{\epsilon }_1}{\left(\frac{n_1}{2}\right)}}\mathrm{and}{\epsilon }_{2m}={\displaystyle \frac{{\epsilon }_2}{\left(\frac{n_2}{2}\right)}}$$

(17)

As a result, based on Equation (9), the average number of the occupied sites N₀ can be expressed by

$${\mathrm{N}}_{\mathrm{O}}={\displaystyle \frac{{\mathrm{N}}_{1m}}{1+{{\mathrm{e}}^{-\mathsf{\beta }}}^{{{(\mathsf{\epsilon }}_{1\mathrm{m}}+{\mathsf{\mu }}_{\mathrm{g}})}^{\left(\frac{{\mathrm{n}}_1}{2}\right)}}}}+{\displaystyle \frac{{\mathrm{N}}_{2m}}{1+{\mathrm{e}}^{-\mathsf{\beta }{{(\mathsf{\epsilon }}_{2\mathrm{m}}+{\mathsf{\mu }}_{\mathrm{g}})}^{\left(\frac{{\mathrm{n}}_2}{2}\right)}}}}={\displaystyle \frac{N_{1m}}{1+{\left(e^{-\beta {\epsilon }_{1m}}\right)}^{\left(\frac{n_1}{2}\right)}{\left(\frac{z_g}{\beta P_1}\right)}^{\left(\frac{n_1}{2}\right)}}}+{\displaystyle \frac{N_{2m}}{1+1+{\left(e^{-\beta {\epsilon }_{2m}}\right)}^{\left(\frac{n_2}{2}\right)}{\left(\frac{z_g}{\beta P_2}\right)}^{\left(\frac{n_2}{2}\right)}}}$$

(18)

where P₁ and P₂ present the pressures at halves saturation defining, respectively, the first and the second type of sites. They are expressed as follows:

$$P_1=K_{B}T{Z}_g\exp (-\beta {\epsilon }_{1m})\mathrm{and}{P}_2=K_{B}T{Z}_g\exp (-\beta {\epsilon }_{2m})$$

(19)

As a result, N₀ is involved as

$${\mathrm{N}}_{\mathrm{O}}=N_{01}+N_{02}={\displaystyle \frac{{\mathrm{N}}_{1m}}{1+{\left(\frac{P_1}{P}\right)}^{n_1}}}+{\displaystyle \frac{{\mathrm{N}}_{2m}}{1+{\left(\frac{P_2}{P}\right)}^{n_2}}}$$

(20)

Considering that the average number of hydrogen atoms absorbed is as follows:

$${\left[{\displaystyle \frac{H}{M}}\right]}_{\mathrm{O}}=n_1{N}_{01}+n_2{N}_{02}$$

(21)

Thus, this gives

$$\begin{array}{l}\left[{\displaystyle \frac{H}{M}}\right]={\left[{\displaystyle \frac{H}{M}}\right]}_1+{\left[{\displaystyle \frac{H}{M}}\right]}_2={\displaystyle \frac{n_1{N}_{1m}}{\left(N_{1m}+N_{2m}\right)\left(1+{\left(\frac{P_1}{P}\right)}^{\left(\frac{n_1}{2}\right)}\right)}}+{\displaystyle \frac{n_2{N}_{2m}}{\left(N_{1m}+N_{2m}\right)\left(1+{\left(\frac{P_2}{P}\right)}^{\left(\frac{n_2}{2}\right)}\right)}}\\ ={\displaystyle \frac{{\left[\frac{H}{M}\right]}_{1sat}}{1+{\left(\frac{P_1}{P}\right)}^{\left(\frac{n_1}{2}\right)}}}+{\displaystyle \frac{{\left[\frac{H}{M}\right]}_{2sat}}{1+{\left(\frac{P_2}{P}\right)}^{\left(\frac{n_2}{2}\right)}}}\end{array}$$

(22)

Lastly, the following formula provides the total amount of hydrogen absorbed as a function of pressure:

$$\left[{\displaystyle \frac{H}{M}}\right]\left(P\right)=\left({\displaystyle \frac{1}{N_{1m}+N_{2m}}}\right)\times \left({\displaystyle \frac{n_1{N}_{1m}}{1+{\left(\frac{P_1}{P}\right)}^{\left(\frac{n_1}{2}\right)}}}+{\displaystyle \frac{n_2{N}_{2m}}{1+{\left(\frac{P_2}{P}\right)}^{\left(\frac{n_2}{2}\right)}}}\right)$$

(23)

4. Results and Discussions

4.1. Adjustment of Experimental Isotherms

The two-site monolayer model previously developed was used to adapt the experimental isotherms for the LaNi₅ and La_0.7Ce_0.1Ga_0.3Ni₅ compounds (Figure 6).

Figure 5 illustrates the excellent agreement between the suggested numerical model and the experimental data. Hence, the stereographic and energetic parameters involved in the hydrogen absorption reaction by the LaNi₅ and the La_0.7Ce_0.1Ga_0.3Ni₅ are deduced using adjustment (

Table 2. Fitting parameters of the LaNi₅- and La_0.7Ce_0.1Ga_0.3Ni₅-doped compounds.

LaNi5 Compound
T (K)	n1	n2	n	Nm1	Nm2	P1	P2	R-Square
298	31.81	3.88	17.84	0.04	0.022	4.52	1.80	0.98
318	22.29	0.17	11.23	0.06	0.22	5.62	0.26	0.98
La0.7Ce0.1Ga0.3Ni5 Compound
298	1.50	44.65	23.08	0.09	0.03	2.71	4.92	0.98
318	0.54	27.62	14.08	0.12	0.05	0.87	6.07	0.99

).

The subsequent discussion will initially focus on these parameters in relation to temperature, providing an understanding of the microscopic level of the hydrogen absorption reaction via both the doped and parent compounds. The macroscopic level will then be accessible as they will be employed to determine the thermodynamic functions governing the process of hydrogen absorption.

Occupancy status: number of atoms per site.

A fundamental physicochemical parameter for understanding the absorption process is the number of atoms per site. Thus, n (= ${\displaystyle \frac{{\mathrm{n}}_1+{\mathrm{n}}_2}{2}})$ denotes the absorption reaction’s stoichiometric coefficient. It represents the stoichiometric coefficient presented in Equation (3) [35]. Additionally, the degree of atoms’ aggregation is estimated by this coefficient.

Figure 7 depicts the number of hydrogen molecules absorbed on each type of site n₁ and n₂ as well as the average value of the molecules in the two types of sites n.

First, it is demonstrated that for both the parent (LaNi₅) and doped (La_0.7Ce_0.1Ga_0.3Ni₅) compounds, n₁ and n₂ drop as temperature rises. As a result, n (=${\displaystyle \frac{n_{1+}{n}_2}{2}}$) decreases. This is explained by the exothermic nature of the hydrogen absorption process. Thus, the interstitial sites are blocked by the rise in temperature. Subsequently, the results indicated that, in the case of LaNi₅, at a given temperature, n₁ > n₂. This indicates that more hydrogen atoms are being taken up by n₁-type sites in the case of LaNi₅. Moreover, n₂ takes values close to 1. However, n₁ takes values much greater than 1. This means that the phenomenon of the agglomeration of hydrogen atoms is more important at the n₁ sites than n₂. On the other hand, in the case of the La_0.7Ce_0.1Ga_0.3Ni₅ compound, the results indicated that n₂ > n₁. This indicates that n₂-type sites are activated, and n₁-type sites are blocked by Ce and Ga doping on the La sites. Furthermore, the fact that n₁ takes values close to 1 while n₂ takes values much higher than 1 indicates that the agglomeration phenomenon originates at n₂ sites in the case of the doped compound. In addition, we notice that, at a given temperature, n takes on larger values in the case of the La_0.7Ce_0.1Ga_0.3Ni₅ compound than the LaNi₅ compound. This means that the average number of hydrogen atoms absorbed at the La_0.7Ce_0.1Ga_0.3Ni₅ sites is greater than that absorbed at the LaNi₅ sites. This gives a good explanation of the experimental findings. As previously noted, Figure 4 illustrates that the mass percentage of hydrogen stored is higher in the case of the La_0.7Ce_0.1Ga_0.3Ni₅ compound than the percentage of hydrogen stored in the case of the LaNi5 compound.

Interstitial sites refer to empty locations in a crystal structure where additional atoms or molecules can fit. While n in the LaNi5 and its derivatives is theoretically equal to 6, some statistical physics investigations have determined that n is much greater than 6 [35,42,43,44]. This is can indeed be attributed to the phenomena of confined agglomeration and decrepitating, which lead to the formation of additional sites for hydrogen absorption. These phenomena increase the material’s hydrogen-absorption capacity beyond the standard crystallographic interstitial sites.

The absorption energy associated with the bonds formed in these interstitial sites depends on several factors, such as the size of the interstitial sites, the nature of the atoms or molecules inserted, and the intermolecular or inter atomic interactions involved. In the section that follows, we propose determining the absorption energy of the hydrogen atoms in the n sites of LaNi₅ as well as La_0.7Ce_0.1Ga_0.3Ni₅.

4.2. Energy Analysis: Absorption Energies

The study of absorption energies is crucial to better understand the phenomenon of hydrogen absorption in metal hydrides since it reveals the type of bond between the hydrogen atom and metal. In fact, hydrogen atoms are absorbed with energy ∆E₁ on n₁ sites and ∆E₂ on n₂ sites, respectively. ∆E₁ and ∆E₂ are calculated using the energetic parameters P₁, P₂, and the saturated vapor pressure of gaseous hydrogen P_vs. The relations provide them in the following manner [45]:

$$\Delta E_1=RT\ln {\displaystyle \frac{P_{vs}}{P_1}}$$

(24a)

$$\Delta E_2=RT\ln {\displaystyle \frac{P_{vs}}{P_2}}$$

(24b)

where P₁ and P₂ are determined from the model by fitting the hydrogen absorption isotherms.

The saturated vapor pressure is expressed as follows:

$$P_{vs}=\exp \left[12.69-{\displaystyle \frac{94.896}{T}}+\left(1.1125\right)\ln \left(T\right)+\left(3.2915\times {10}^{-4}\right)T^2\right]$$

(25)

Thus, absorption energies over n₁ and n₂ sites are presented in Figure 8.

The absorption energy values show that the bond between the hydrogen atoms and the intermetallics is of a chemical type. This implies that the bond formed between the hydrogen atoms and the interstitial sites (n₁ and n₂) of the intermetallics considered is relatively strong. It is observed that if the temperature increases, the absorption energy also increases. This suggests that higher temperatures require more energy due to the exothermic nature of the absorption reaction. Additionally, it is noted that the doped compound’s absorption energies are somewhat higher than those of LaNi₅. This means that the amount of hydrogen stored in the doped compound is slightly greater than that stored in LaNi₅, leading to a slightly higher energy expenditure. In addition, it is demonstrated that for the LaNi₅ compound, ∆E₁ > ∆E₂, while for the La_0.7Ce_0.1Ga_0.3Ni₅ molecule, ∆E₂ > ∆E₁. This is explained by the fact that, as shown in Figure 8, the n₁ sites are more occupied by hydrogen atoms in the case of LaNi₅, whereas the n₂ sites are more occupied in the case of La_0.7Ce_0.1Ga_0.3Ni₅.

It should be noted that the enthalpy change indicates the thermodynamic favorability of the absorption process, which is controlled by the system’s relaxation and stabilization, whereas the absorption energy shows the overall energy input required to absorb the hydrogen. The difference between the initial, transition, and final states of the system, as well as the additional energy needed to overcome activation barriers and create lattice distortions, can all be taken into account when explaining the discrepancy between absorption energies and absolute enthalpy change during hydrogen absorption.

• Thermodynamic properties: Internal and Gibbs energies

Internal energy and Gibbs energy are two thermodynamic concepts crucial to understanding physical and chemical systems.

• Internal energy

The internal energy (U) presents the sum of all microscopic energies. Thus, it is the kinetic and potential energies of the molecules or atoms that make up the system. It also includes the energy of chemical and physical interactions between molecules. A system’s pressure and temperature have a big impact on its internal energy. The internal energy of hydrogen absorption is expressed as follows [42]:

$$U_{\mathrm{int}}=-{\displaystyle \frac{\partial \ln (z_{gc})}{\partial \beta }}+{\displaystyle \frac{\mu }{\beta }}({\displaystyle \frac{\partial \ln (z_{gc})}{\partial \mu }})$$

(26)

where

$$\mu =K_{B}T\ln \left({\displaystyle \frac{\beta P}{z_g}}\right)$$

(27)

Thus,

$$\begin{array}{l}{U}_{\mathrm{int}}=k_B\ln ({\displaystyle \frac{\beta P}{z_g}})\left({\displaystyle \frac{N_{1m}{(\frac{P}{P_1})}^{n_1}}{1+{(\frac{P}{P_1})}^{n_1}}}+{\displaystyle \frac{N_{2m}{(\frac{P}{P_2})}^{n_2}}{1+{(\frac{P}{P_2})}^{n_2}}}\right)\\ -k_{B}T\left({\displaystyle \frac{N_{1m}{(\frac{P}{P_1})}^{n_1}}{1+{(\frac{P}{P_1})}^{n_1}}}+{\displaystyle \frac{N_{2m}{(\frac{P}{P_2})}^{n_2}}{1+{(\frac{P}{P_2})}^{n_2}}}\right)\end{array}$$

(28)

Figure 9 depicts the variation of the internal energy as a function of pressure when T = 298 K and T = 318 K.

It is shown that the internal energies for both compounds are negative. This indicates that energy is released throughout the process of hydrogen absorption. Therefore, the exothermic nature of the hydrogen-absorption process explains this. We additionally observe that when the starting temperature rises, the internal energy’s absolute value rises as well. This demonstrates how an increase in starting temperature causes an increase in energy dissipation. Furthermore, a comparison of the results at a given temperature revealed that the LaNi5 compound has a slightly lower absolute value of internal energy than that of La_0.7 Ce_0.1Ga_0.3Ni₅. This can be explained by the fact that the hydrogen atoms are more active in the interstitial sites of the compound La_0.7Ce_0.1Ga_0.3Ni₅ (as shown by experimental results), which boosts the system’s heat release.

• Gibbs Energy

The Gibbs energy combines internal energy with pressure–volume effects (via enthalpy) and entropy, making it a powerful tool for predicting the spontaneity of processes at constant pressure and temperature. It is expressed as follows [42]:

$$G_a=\mu \times n\times N_0=\mu {\left[{\displaystyle \frac{H}{M}}\right]}_0$$

(29)

Therefore, G_a can be expressed as a function of the fitted parameters as follows:

$$G_a=K_B\ln \left[{\displaystyle \frac{\beta P}{z_g}}\left({\displaystyle \frac{n_1{N}_{1m}}{1+{\left(\frac{P_1}{P}\right)}^{\left(\frac{n_1}{2}\right)}}}+{\displaystyle \frac{n_2{N}_{2m}}{1+{\left(\frac{P_2}{P}\right)}^{\left(\frac{n_2}{2}\right)}}}\right)\right]$$

(30)

Figure 10 depicts the variation of the Gibbs energy as a function of pressure when T = 298 K and T = 318 K for both compounds.

First, we observe that, for a given temperature and pressure, the Gibbs energy takes negative values. This is a blatant sign that the intermetallics’ spontaneous and thermodynamically advantageous process of absorbing hydrogen is occurring. This means that the system will naturally evolve toward this state, often by releasing energy. Next, it is demonstrated that, up to a critical pressure, P_c, the absolute value of the Gibbs energy declines logarithmically as a function of pressure before rising and stabilizing. P_c corresponds to the establishment of thermodynamic equilibrium. This means that beyond P_c, the difference in chemical potential between the gaseous hydrogen and the absorbed hydrogen reaches equilibrium, where the absorption rate becomes zero. Then, the absorption of hydrogen stops, regardless of the increase in pressure. This pressure typically indicates that the material is getting close to its maximum absorption capacity and is discovered when the PCT curve starts to increase quickly after the plateau (P_eq). According to the experimental results (Figure 5), we see that this pressure varies between 6 bar and 9 bar, which confirms the results of our model. At a given temperature, it is observed that P_c is the same for both compounds. This means that they start to absorb at the same pressure. This is because the energy required for hydrogen to insert into the interstitial sites is similar for both compounds, resulting in identical absorption pressures.

We observe that if the starting temperature rises, this pressure will increase even more. In fact, P_c = 6 bar at T = 298 K and P_c = 9 bar at T = 318 K. This indicates that in order to achieve thermodynamic equilibrium, a greater pressure is needed as the temperature rises.

We also see that the parent and doped compounds require the same pressure (P_c) to reach equilibrium at a given temperature.

This means that the doping did not disturb the thermodynamic equilibrium of the absorption reaction. Nonetheless, we note that the doped compound has a higher absolute value of Gibbs energy than the parent compound. This indicates that in the case of the compound La_0.7Ce_0.1Ga_0.3Ni₅, hydrogen absorption occurs more spontaneously. As a result, the La_0.7Ce_0.1Ga_0.3Ni₅ compound’s metallic matrix can hold more hydrogen atoms. This explains why doping with Ce and Ga on La sites enhances hydrogen storage efficiency.

5. Conclusions

In this work, an experimental and numerical study of the substitution effect on the sites of La in the compound LaNi₅ by Ce and Ga was carried out. Therefore, the hydrogen-storage properties of the compounds LaNi₅ and La_0.7Ce_0.1Ga_0.3Ni₅ were compared. The experimental results showed that doping with Ce and Ga elements on La sites improves the kinetics and the mass of stored hydrogen by more than 30%. Additionally, doping improves its cyclability and capacity to routinely absorb and desorb hydrogen without causing degradation and extends its usable life, resulting in a reduction in hysteresis of over 28%.

The simulated results demonstrated that for the two compounds, absorption occurs when hydrogen atoms are inserted into two different types of interstitial sites (n₁, n₂). In fact, these interstitial sites are more occupied by the hydrogen atoms in the doped compound. The numerical calculation of the Gibbs energy and internal energy further demonstrates that the hydrogen atoms are introduced into the interstitial sites by means of an exothermic and spontaneous reaction.