Physicochemical and thermodynamic investigation of hydrogen absorption and desorption in LaNi3.8Al1.0Mn0.2 using the statistical physics modeling

Results in Physics 9 (2018) 1323–1334

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Physicochemical and thermodynamic investigation of hydrogen absorption and desorption in LaNi3.8Al1.0Mn0.2 using the statistical physics modeling Nadia Bouaziz, Marwa Ben Manaa, Abdelmottaleb Ben Lamine

T

⁎

Unité de Recherche de Physique Quantique, 11 ES 54, Faculté des Science de Monastir, Tunisia

A R T I C LE I N FO

A B S T R A C T

Keywords: Absorption-desorption isotherms Statistical physics model Metal hydride Thermodynamic potential functions

In the present work, experimental absorption and desorption isotherms of hydrogen in LaNi3.8Al1.0Mn0.2 metal at two temperatures (T = 433 K, 453 K) have been fitted using a monolayer model with two energies treated by statistical physics formalism by means of the grand canonical ensemble. Six parameters of the model are adjusted, namely the numbers of hydrogen atoms per site nα and nβ, the receptor site densities Nmα and Nmβ, and the energetic parameters Pα and Pβ. The behaviors of these parameters are discussed in relationship with temperature of absorption/desorption process. Then, a dynamic investigation of the simultaneous evolution with pressure of the two α and β phases in the absorption and desorption phenomena using the adjustment parameters. Thanks to the energetic parameters, we calculated the sorption energies which are typically ranged between 276.107 and 310.711 kJ/mol for absorption process and between 277.01 and 310.9 kJ/mol for desorption process comparable to usual chemical bond energies. The calculated thermodynamic parameters such as entropy, Gibbs free energy and internal energy from experimental data showed that the absorption/desorption of hydrogen in LaNi3.8Al1.0Mn0.2 alloy was feasible, spontaneous and exothermic in nature.

Introduction With the developing of fuel cell technology and the growth of hydrogen application domains, hydrogen storage has attracted more and more attention [1]. As a significant method for hydrogen storage, metal hydride [MH] is considered one of the pertinent ways. Among different alloys treated to store hydrogen, LaNi5 intermetallic compound and its substituted derivatives, due to their exceptional capacity to react reversibly with hydrogen at moderate temperature and pressure conditions, have been broadly studied [2,3]. As previously reported, the influences of partial substitution of nickel with other alloys (Al, Mn, Cu, Cr, Sn etc.) have been widely investigated [4,5]. Among them, a partial replacement of Ni through Al could enhance the absorption kinetics and cycling performance and diminish the plateau pressure. However it could reduce the hydrogen storage capacity [6,7]. Manganese substitution could also lower the plateau pressure of the hydride without impairing the hydrogen capacity although the hysteresis process was slightly enhanced [5–8]. Some previous experiments in La-Ni-Al-Mn have shown that LaNi3.8Al1.0Mn0.2 alloy has higher storage of hydrogen content and lower plateau pressure than other LaNi5 based metals [9,10]. Moreover, in order to demonstrate the influence of Mn replacements for Ni and Al on hydrogen absorption characteristics of these metals, Li et al. [11] have measured the hydrogen absorbed/

⁎

released quantity and the plateau pressure and revealed that they increase with Manganese content in LaNi3.8Al1.2−xMnx (x = 0.2, 0.4, 0.6) metals. In order to describe the behavior of real systems, many models were proposed by several authors through integrating a statistical physics method [12,13]. This methodology consists of fitting the P-C-T isotherms with analytical expressions. An empirical P-C-T equilibrium correlation is often established in the literature [14–16]. The models have been successfully applied to simulate the absorption/desorption isotherms of different storage materials such as LaNi5, LaNi5−xAlx etc… Indeed, the investigation of hydrogen storage using a statistical thermodynamics approaches in intermetallic compounds featuring stable hydrides is of great importance for their application in various domains, such as hydrogen storage units. The investigations reported in the present manuscript were developed by using a statistical physics analysis of the hydrogen absorption and desorption isotherms in LaNi3.8Al1.0Mn0.2 metal at T = 433 K and T = 453 K. The development of various model expressions which are susceptive to adjust the hydrogen experimental isotherms leads to choose an appropriate model where its parameters given by the term of this model will be attributed numerical adjusted values which can define microscopic states of absorption and desorption. The chosen best fitting model will provide new physical interpretations of hydrogen

Corresponding author at: Department of Physics, Faculty of Sciences of Monastir, University of Monastir, Boulevard de l'Environnement, 5019 Monastir, Tunisia. E-mail address: [email protected] (A. Ben Lamine).

https://doi.org/10.1016/j.rinp.2018.04.035 Received 31 January 2018; Received in revised form 13 April 2018; Accepted 13 April 2018 Available online 22 April 2018 2211-3797/ © 2018 Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/BY-NC-ND/4.0/).

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n H2 + M ⇄ MHn 2

absorption/desorption of LaNi3.8Al1.0Mn0.2 alloy at microscopic level. The physico-chemical parameters studies will be divided in two groups: first the steric parameters such as the numbers of hydrogen atom per site, the densities of interstitial sites and the hydrogen absorbed/released amount at saturation, second, the energetic parameters such as the absorption and desorption energies. Thanks to this model, we could evaluate the dynamics of absorption and desorption in terms of phase transformation and hysteresis after steric and energetic studies. We have compared the absorption and desorption processes, in order to find an explanation of the hysteresis encountered during desorption stage. A macroscopic study is also achieved by calculating from grand canonical partition function and investigating three thermodynamic functions such as the entropy, the internal energy and the free enthalpy of Gibbs.

(1)

n is a stoichiometric coefficient in the absorption reaction. MHn is the formed hydride. In general, n could be inferior or superior to 1. If n is superior to 1, it means that many atoms occupy one receptor site. So we can talk about an aggregate of hydrogen atoms. If n is inferior to 1, it means that a receptor site is occupied by a fraction of hydrogen atom. Since a hydrogen atom is not divisible, we can expect that this case do not occur for hydrogen. We will actually see that this is true posteriori and n will be always superior to 1. The departure point is the grand canonical partition function describing the microscopic states of the system depending on the physical condition in which this system is placed [17–21].

Zgc =

∑

e−β (εi− μ) Ni (2)

Ni

Materials and experimental procedure

where εi is the interstitial site absorption energy, µ is the chemical potential of an absorption site, Ni is the interstitial site occupation level: Ni is equal to zero if the site is vacant and is equal to one if the site is filled through an atom of hydrogen, and β is defined as 1/kBT where kB is the Boltzmann constant and T is the absolute temperature. The total grand canonical partition function is attached to Nm receptor sites per mass unit. If these sites are supposed to be identical and independent, so the total grand canonical partition function is expressed under the form of a simple product:

The starting materials have a purity of 99% for La, 99.9% for Ni, 99.7% for Al and 99.7% for Mn [9–11]. The preparation of LaNi3.8Al1.0Mn0.2 alloy were performed, under a safeguard of argon atmosphere, through introduction melting in a crucible of copper. For homogenisation tasks, the melting was iterated several times and the button ingot is inverted in the furnace each time. Next, to get a single phase, the as-cast buttons were annealed under vacuum for 6 h at T = 1323 K. We shivered and sieved the metal through 150 meshes for structure determination and phase analysis. Its absorption and desorption curves of hydrogen were measured in a temperature range of 433 K–493 K with a volumetric system. Then, we put around 1 g of the sample in the reaction chamber, which was then evacuated during 0.5 h at T = 286 K. Next, we allowed react the metal with hydrogen gas at a pressure of 1700 kPa. The absorbed amount of hydrogen was removed by heating and evacuating. After many cycles of absorption and desorption, if the quantity of hydrogen absorbed by the sample is still the same, the metal was considered to be perfectly activated. Then the PC isotherms of absorption/desorption measurements were performed at T = 433 K and T = 453 K. We used a hydrogen gas of 99.999% purity [9–11].

Zgc = (z gc ) NM

This total function is used to the calculation of the average site occupation number, which is given as below [13–21]

N0 = kB T

∂μ

(4)

Zgc = (z gcα ) Nmα (z gcβ ) Nmβ

In order to study absorption and desorption processes involving a statistical physics method we will need to assume some assumptions as a basis of the procedure.

• •

∂ln(z gc ) NM

In order to establish the monolayer model with two energies, we develop the expression of the absorbed quantities of hydrogen according to the pressure. Two variable numbers Na1 and Na2 of hydrogen atoms which are absorbed in two independent kinds of interstitial sites were taken into consideration and determined. The absorption density in the first type is Nmα and the density of the second type is Nmβ which were defined with the energy (−εα) and (−εβ), respectively. So, the total grand canonical partition function can be given as follow:

Theoretical background

•

(3)

(5)

zgcα and zgcβ are expressed by:

z gcα =

The investigation of these processes necessitates the employing of the grand canonical ensemble in statistical physics since it implies an exchange of particles from the free level to the absorbed one [17,18]. The hydrogen molecules are considered as an ideal gas [19] and the interaction of hydrogen molecules will be neglected in the free state. Each molecule is characterized by many internal degrees of freedom that are, in addition to translational, rotational, electronic and vibrational degree. To facilitate the problem, we have to take into account only the most important degrees of freedom, namely the degree of translation which has a typical translational temperature φtr = 10−15 K and the degree of rotation (φrot = 85.3 K) [20]. Thus, translational and rotational movement’s molecules can be released at ambient temperatures. This occurs because the electronic degree cannot be thermally existed [13]. Similarly, the vibrational degree could be neglected compared to the other degrees.

∑

e−

β (−ε − μ) N α i

= 1 + e β (εα + μ) (6)

Ni = 0,1

z gcβ =

∑

β e− (−εβ − μ) Ni

= 1 + e β (εβ + μ) (7)

Ni = 0,1

The equality of the various chemical potentials is represented by: µm = µ/n with µ is the chemical potential of the interstitial site, n is the number of atoms per site and µm is the chemical potential of one free H2 molecule which expression is given in gaseous state [20]:

N μm = kB T ln ⎛⎜ ⎞⎟ z ⎝ g⎠

(8)

where

z g = z gtr × z grot

(9)

zgtr is the translation partition function which expression is [18–22]: 3

z gtr = V ⎛ ⎝

We consider that a variable number Na of hydrogen atoms are absorbed into Nm receptor sites located on a mass unit of the absorbent. The overall reaction of hydrogen molecule with an alloy (M) to form a metal hydride (MHx) is expressed as:

2πmkB T ⎞ 2 h2 ⎠

(10)

m is the absorbed molecule mass and namely the hydrogen atom mass, V the volume of the studied system and h the Planck’s constant. The 1324

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Table 1 Partition functions of the tested models. Name of tested model

Partition function

Reference

Monolayer with one energy (model 1)

Zgc = (1 + e β (ε + μ) ) Nm

[18–21]

Monolayer with two energies (model 2)

Zgc = (1 + e β (ε1+ μ) ) N1m × (1 + e β (ε2+ μ) ) N2m

[18–21]

Monolayer with three energies (model 3)

Zgc = (1 + e β (ε1+ μ) ) N1m × (1 + e β (ε2+ μ) ) N2m × (1 + e β (ε3+ μ) ) N3m

[18–21]

Double layer with one energy (model 4)

Zgc = (1 + e β (ε + μ) + e 2β (ε + μ) ) Nm

[18–21]

Double layer with two energies (model 5)

Zgc = (1 +

e β (ε1+ μ)

translational partition function per unit of volume zgtr, for an ideal hydrogen gas, is formulated versus vaporization energy ΔEv and the saturated vapor pressure utilizing the expression [18–21]:

z gtr =

ΔE v βPvs e RT

( )

( )

with Pvs is the saturated vapor pressure: 3 2

−ΔE v RT

T 2θrot

Nmα 1+

Pα nα P

( )

1+

Adjustment In order to test the validity of the present model, the adjustment of the hydrogen absorption isotherms, through the different proposed analytical models is performed. The experimental data of hydrogen in LaNi3.8Al1.0Mn0.2 alloy were tested by five analytical models. These models are presented and developed in last reports [18–21]. In Table 1, the partition functions of the treated models are reported. Among the tested models, the monolayer model with two energies (model 2) that we already treated, furnished the best depiction of the absorption and desorption phenomena. The choice of this model based on the analysis of the residual root mean square error RMSE also known as the estimated standard error of the regression [24,25]. This technique is considered as the most broadly utilized method to define the optimum isotherms. Also, the best fitting of absorption isotherms is determined by the multiple correlation coefficients squared R2 [23]. The selection of an appropriate model that describes the experimental data is determined basing on the values of R2 and the values of RMSE shown in Table 2 for absorption process and in Table 3 for desorption process. It is known that the absorption and desorption isotherms present a great correlation with the chosen model in the case of the determination coefficient R2 is close to 1 and the values of RMSE are near to zero. According to the values of R2 (Tables 2 and 3), we can see that the numerical values of R2 with model 2 and 3 are very close to each other. So we must take into consideration the physics justification of the adequate model since the several parameters fitted in the monolayer model with three energies are not physically acceptable. Moreover we

Pβ nβ

( )

(14)

P

where nα and nβ are the numbers of atoms per site, Pα = kB T Zg e−βεnα and Pβ = kB T Zg e−βεnβ are the pressures at half saturation defining respectively the α and β sites, [25,26]. By exploiting Eq. (1) and the number of average site occupation N0, the average number of absorbed or desorbed atoms is:

Na = nα N01 + nβ N02

(15)

Finally, the formula of the hydrogen absorbed or desorbed quantity as a function of pressure can be represented by:

nα Nmα

Na = Na1 + Na2 =

1+

Pα nα P

( )

+

nβ Nmβ 1+

Pβ nβ

( )

(16)

P

The experimental absorbed and desorbed quantities are determined in term of H/M which indicates a number of H atoms per unit formula. The absorbed amount for each site equals to:

[H / M ]α =

[H / M ]β =

nα Nmα 1 Nmα + Nmβ 1 + Pα P

( )

nβ Nmβ Nmα + Nmβ 1 +

nα

1 Pβ nβ

( ) P

=

[H / M ]αsat 1+

=

Pα nα P

( )

(17) Table 2 Values of coefficient of determination R2 and the residual root mean square error (RMSE) for absorption process.

[H / M ]βsat 1+

Pβ nβ

( ) P

(20)

(13)

Nmβ

and N02 =

( )

(12)

where θrot is the rotational typical temperature of H2 molecule and T is the absolute room temperature. The average numbers of occupied sites are determined by the equations as follow:

N01 =

( )

⎞ ⎟ ⎟ ⎠

For this model, we defined six physico-chemical fitted parameters: the numbers of atoms per site for each type (nα and nβ), the densities of each type of interstitial sites (Nmα and Nmβ) and the pressures at halves saturation (Pα and Pβ) which will be used to fit and interpret the absorption process.

The rotational partition function is given as below [20,21]

z grot =

[18–21]

⎛ 1 ⎞ × ⎜ nα Nmα + nβ Nmβ [H / M ](P ) = ⎜⎛ ⎟ Pβ nβ Pα nα 1+ P ⎝ Nmα + Nmβ ⎠ ⎜ 1 + P ⎝ [H / M ]βsat [H / M ]αsat = + Pβ nβ P nα 1 + Pα 1+ p

(11)

2πmkB T ⎞ (kB T ) e Pvs = ⎛ ⎝ h2 ⎠

+

e β (ε1+ ε2+ 2μ) ) Nm

(18)

T(K)

where

Model 1

Model 2

Model 3

Model 4

Model 5

R 433 453

0.976 0.982

0.998 0.998

0.999 0.999

0.953 0.971

0.942 0.957

RMSE 433 453

0.034 0.024

0.002 0.001

0.01 0.03

0.066 0.038

0.083 0.058

2

[H / M ] = [H / M ]α + [H / M ]β and [H / M ]sat = [H / M ]αsat + [H / M ]βsat (19) Thus, the equation of the total absorbed/desorbed quantity per unit formula is defined by: 1325

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Table 3 Values of coefficient of determination R2 and the residual root mean square error (RMSE) for desorption process. T(K)

Model 1

Model 2

Model 3

Model 4

Model 5

R2 433 453

0.977 0.971

0.998 0.997

0.999 0.998

0.961 0.963

0.951 0.949

RMSE 433 453

0.035 0.036

0.002 0.003

0.01 0.02

0.059 0.047

0.074 0.066

eliminate the double layer models (model 4 and 5) because of many researches in literature [26,31] reported that the multilayer character of absorption phenomenon is dismissed since it would not be possible in such narrow receptor sites. In addition the values of RMSE for the monolayer model with two energies which are the most near to zero compared to the other treated models which confirm the choice of this model. So, we can note that the low RMSE values and the high R2 values show that the experimental data is successfully described using a monolayer model with two energies. Indeed, their six parameters, nα, nβ, Nmα, Nmβ, Pα and Pβ can be consistently appraised and the investigation about the hydrogen-metal interaction is possible and coherent at a molecular level. Several previous works [32–34] indicate that in many intermetallic AB5 compounds with the hexagonal CaCu5 structure which is the case of our metal (LaNi3.8Al1.0Mn0.2), atoms of hydrogen fill two types of sites which are a tetrahedral (Td) and octahedral (Oh) sites. So, the two energy levels of absorption introduced in the chosen model are detected through the presence of two phases (α and β) during hydrogen absorption/desorption processes [23–35]. The adjustment of the experimental hydrogen absorption/desorption data into LaNi3.8Al1.0Mn0.2 alloy at T = 433 K and 453 K is shown in Fig. 1. The values of the adjusted parameters are listed in Table 4. Results investigation Microscopic investigation of absorption and desorption phenomena

Fig. 1. Experimental data of (a) absorption- (b) desorption isotherms of hydrogen fitted by statistical physics “model 2” (illustrated by different continuous lines with red color) at T = 433 K and 453 K. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Effect of physico-chemical parameters on the absorption and desorption phenomena To better understand the role of the fitting parameters of the selected model through investigating the influence of the variation of each one on the absorption and desorption isotherms, we will study the effect of the stereographic parameters nα and nβ (numbers of atoms per site), Nmα and Nmβ (densities of interstitial site) and the energetic parameters Pα and Pβ.

The Eq. (24) is divided into two independent sites α and β:

Mα Hnα′ → Mβ Hnβ′ →

Effect of nα and nβ on absorption and desorption isotherms During hydrogen absorption and desorption, the overall reactions occurred are in general given by Eqs. (23) and (24):

n For absorption: i H2 + M → MHni 2 For desorption: MHni′ →

ni′ H2 + M 2

nβ 2

H2 + Mβ → Mβ Hnβ

nβ′ 2

H2 + Mβ

(25)

(26)

Since the bulk of the MH metal is defined as a host-guest system, H atoms can fill the host interstitial sites [31,32]. We assume that the bulk of hydrogen storage metal is composed of M unit cells, which can be either in the α phase (Mα) or β phase (Mβ) respectively. nα and nβ are the numbers of hydrogen atoms per site in the absorption phenomenon and nα′ and nβ′ during desorption process. The effect of the variation of the parameters nα and nβ on the absorbed quantity are presented in Figs. 2 and 3 respectively. The numbers of atoms per site are relevant in the behavior of hydrogen isotherms during absorption process. Concerning the evolution of the parameter nα shown in Fig. 2, we remark that the rise in the value of nα leads to the increase of the absorbed quantity in a wide pressure range. This can be explained by the fact that the majority of α receptor sites are empty and hydrogen atoms can easily be attracted by these α host sites already for very weak pressure about Pα due to energetic accounts.

(21) (22)

We assume that the two levels of energy can be assigned to the two phases known in literature α and β [23–32]. The numbers of H- atoms per site for both phases α and β are respectively the parameters nα and nβ. The Eq. (23) is divided into two different sites α and β.

nα H2 + Mα → Mα Hnα 2

nα′ H2 + Mα 2

(23) (24) 1326

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Table 4 Adjustment parameter values for absorption and desorption processes corresponding to the best fitting model (model 2). T(K)

nα

nβ

Nmα

Nmβ

Pα

Pβ

[H/M]αsat

[H/M]βsat

[H/M]sat

For absorption 433 453

1.26 1.32

10.51 11.09

2.09 1.1

0.56 0.25

44.3 76.3

70.54 89.45

0.99 1.07

2.22 2.05

3.21 3.12

For desorption 433 453

1.04 1.11

9.52 10.29

1.67 1.54

0.28 0.19

41.43 72.56

54.9 81.74

0.89 0.98

1.36 1.13

2.25 2.11

Fig. 2. Effect of the number of atoms per site nα on the absorption isotherm at T = 453 K.

Fig. 4. Effect of the density of the receptor site Nmα (mass unit) on the absorption isotherm at T = 453 K.

Effect of Nmα and Nmβ on the sorption isotherms Fundamentally, this parameter depicts the number of the occupied interstitial sites at saturation. According to Fig. 4 we notice that the absorbed quantity in the first phase α increases with the increase of Nmα. Evidently, the influence of this rise due to [H/M]αsat is observed in the overall absorbed quantity from very weak pressure about Pα. So the effect of this parameter is appeared at the first level of saturation. Fig. 5 shows that the rise of Nmβ gives an increase of the absorbed quantity of the second phase β which is a proportion of the global

Fig. 3. Effect of the number of atoms per site nβ on the absorption isotherm at T = 453 K.

In the case of nβ parameter, we note that, at low pressure, the rise of nβ leads to a slight decrease before Pβ. However an important increase is achieved after Pβ. This is due to the very weak probability of forming an agglomerate at low pressures. On the contrary, for higher pressure, the reverse happens. When nβ has a low value, the saturation necessitates a high pressure to be attained. This is related to the difficulty of hydrogen atoms to find an empty receptor sites near the saturation. This situation is inverted at low pressure. Also, at saturation the absorbed quantity increases with the increase of the agglomerate number nβ.

Fig. 5. Effect of the density of the receptor site Nmβ (mass unit) on the absorption isotherm at T = 453 K. 1327

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N. Bouaziz et al.

saturation quantity. However the effect of Nmβ is visible only for high pressure after Pβ. Obviously when P∈ [0,50 atm], the variation of Nmβ has a neglected effect on the evolution of absorbed quantity. Effect of Pα and Pβ From the fitted absorption energy parameters Pα and Pβ we can calculate the absorption energies ΔEα for α phase and ΔEβ for β phase which are given by the following relations:

ΔEα = RTLn

Pvs Pα

(27)

ΔEβ = RTLn

Pvs Pβ

(28)

We used the same equations to calculate the desorption energies changing Pα and Pβ by P′α and P′β. R = 8.314472 J/mol.K is the ideal gas constant. The expression of the hydrogen saturated vapor pressure Pvs is written as [33]:

PVS = exp ⎡12.69− ⎣

94.896 + (1.1125) Ln (T ) + (3.2915 × 10−4) T 2⎤ T ⎦

Fig. 7. Effect of the pressure Pβ (kPa) on the absorption isotherm at T = 453 K.

(29)

These two energies define the nature of the binding of hydrogen atoms with the interstitial sites. These parameters are the energetically coefficient that describe the dynamics of absorption phenomenon. They offer information about the type of interaction between H-atoms and the interstitial sites. From Fig. 6 we can remark that an increase of Pα does not significantly affect the formation of α phase. In fact, if Pα increases, the absorbed amount [H/M]α is slightly delayed toward high pressures. The higher the order of magnitude of Pα is, the lower the energy ΔEα becomes. We can notice also according to Fig. 7 that the decrease of Pβ leads to an increase of the absorbed quantity and the saturation is rapidly achieved for the phase β. This means that the absorption is more blocked with regard to pressure. So a variation of these parameters values does not change the absorption capacity which is expressed by a saturation quantity. However, a change of these parameters values brings forward or delays the absorption process relatively to the pressure.

during absorption stage. Now, if we compare the desorption parameters of those of absorption, we can deduce generally that if nα = n′α and nβ = n′β we could expect that the absorption phenomenon will be totally reversible and no hysteresis process would be observed. However, if nα ≠ n′α and nβ ≠ n′β a hysteresis process would exist. It‘s known that n′α and n′β cannot be higher than nα and nβ respectively for matter conservation causes. By comparing the values of nα and nβ, it may be noted that these steric parameters are not equal during absorption and desorption phenomena. A same site will be first emptied with a lower nβ′ during desorption because of weaker absorption energy of β sites. In desorption with a lower dis-anchorage coefficient (n′β < nβ), an amount of hydrogen atoms (nβ − n′β) remains retained by the site. Thus hysteresis phenomenon can be seen in desorption isothermal curves on LaNi3.8Al1.0Mn0.2 alloy. The hysteresis process is appeared through the non-superposition of the absorption and desorption curves. Concerning the receptor site density parameter, we note that although the sites of the second phase β is voluminous (higher nβ face to nα), nonetheless its density (Nmβ ≪ Nmα) is weak comparatively to the first density during absorption/desorption processes. The parameter [H/M] represents the absorbed amount of hydrogen per unit formula of the studied system. The expression of this parameter is divided into two terms one of the first phase [H/M]αsat given by the Eq. (17) and one of the second phase [H/M]β given by the Eq. (18) with [H / M ]total = [H / M ]αsat + [H / M ]βsat . We remark that for both phenomena [H/M]βsat > [H/M]αsat. Consequently, the hydrogen storage capacity is caused by the interstitial sites β which are by far the more voluminous than α sites and the atoms of hydrogen prefer a receptor site with the highest receptor hole size.

Comparison of absorption and desorption phenomena Firstly, for both absorption and desorption phenomena nβ ≫ nα for T = 433 K and 453 K. This behavior may be interpreted by the difference in volume between the α sites and those of β sites. So we can deduce that the size of interstitial sites of β phase is greater than that of α phase which explain the appearance of the aggregation process

Absorption and desorption energies Absorption and desorption energy values into LaNi3.8Al1.0Mn0.2 alloy has been calculated using Eqs. (29) and (30). Pα and Pβ are the halves-saturation pressures in the absorption process and P′α, P′β in desorption process correspondent to the two types of receptor sites for the adopted model. Pvs values are determined by Eq. (31) for the two temperatures. The calculated values of Pvs at:

T= 433K gives Pvs = 1431.49 × 1032Pa T= 453K gives Pvs = 5120.95 × 1034 Pa Table 5 summarizes the values of absorption and desorption energies at T = 433 K and 453 K. These energies give information about

Fig. 6. Effect of the pressure Pα (kPa) on the absorption isotherm at T = 453 K. 1328

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Table 5 Values corresponding to the energies of absorption and desorption. Process

T(K)

ΔEα(kJ/mol)

ΔEβ(kJ/mol)

Absorption

433 453

277.782 310.711

276.107 310.112

Desorption

433 453

278.023 310.900

277.010 310.451

the interaction between hydrogen atom and the site of metal. The order of magnitude of absorption/desorption energies values for the two temperatures indicated that the binding between the atoms and the site is a chemical type for both phases α and β. Another remark is that for both absorption and desorption phenomena |ΔEβ | < |ΔEα |. It is reported in ref [34] that we can justify this aspect by the fact that, after the first phase α, the dislocation density created is sufficient to assist both the establishing as well as the migration of incoherent surface during the β phase [34]. This needs some rearrangement of the dislocations previously present which necessitates less energy during the second phase. Thus, the reason of the variation between ΔEα and ΔEβ is due to the variation of sizes of α and β sites and the hindrance to hydrogen to be integrated or disintegrated into the receptor site. Moreover we can remark that the desorption energy values are greater than that of absorption energy for LaNi3.8Al1.0Mn0.2 alloy. This is related to the cracking energy which involves a greater porosity giving a greater interaction [35,36]. This is a property of hysteresis phenomenon showed through the difference of energy between the insertion and the dis-insertion of H-atoms. Finally, the increase of the absorption and desorption energies with the rise of temperature is due to the thermal agitation. The thermal agitation make more difficult for the H-atom to penetrate into the site volume and a greater pressure is needed to the H-atom to be stored in the same site. So the absorption energy is increased with temperature.

Fig. 8. Variation of percentages of [H/M]α and [H/M]β as function of pressure for T = 453 K: Absorption case.

Investigation of the dynamics of the different phases in the absorption and desorption processes The thermodynamic equilibrium conditions between the metal form and its hydride form depends on the temperature, the pressure and the hydrogen composition of the system. The absorption/desorption PCT isotherms of the investigated alloy LaNi3.8Al1.0Mn0.2 at two different temperatures (T = 433 K and 453 K) are represented by Fig. 1. These curves depict the global evolution of the absorption/desorption reaction [H/M] as function of hydrogen pressure. With the adopted model, we can describe the details of the evolution of the two phases α and β: a first solid solution supposed to be the phase α and a second hydride phase supposed to be the phase β. In other words, they describe all the reactions on all types of different sites. Before describing the absorption/desorption processes, we plot in the same figure the two contributions of the two term expressions of the model for both phenomena (Figs. 8 and 9). The fitting of the experimental curves through the monolayer model with two types of sites can facilitate the investigation of the hydriding and can best depict the hydrogen absorption/desorption by LaNi3.8Al1.0Mn0.2 with analytical physics method. The adequate model presents two energies of absorption which will be supposed to represent the two phases α and β during hydrogen storage. The first phase is defined by the parameters nα, Nmα and Pα. The second one is defined by the parameters nβ, Nmβ and Pβ.

Fig. 9. Variation of percentages of [H/M]α and [H/M]β as function of pressure for T = 453 K: desorption case.

Domain1: From P = 0 to P = Pαsat Since Pα is very low, it means that the absorption energy of the first insertion site is enough high to force H2 molecules to be dissociated. As the hydrogen pressure over the metal is increased, the concentration of H atoms in the alloy in equilibrium increased too proportionally. This hydrogen absorption is achieved around Pα till Pαsat where all α sites are completely filled. This is a limit in the amount of hydrogen that the αphase can store. The metal lattice dilates proportionally to the concentration of hydrogen by approximately 2–3 Å3 per hydrogen atom [37]. For this phase α, the number of hydrogen atoms nα = 1.26 at T = 433 K and nα = 1.32 at T = 453 K (nα are weak face to nβ) with a density Nmα. It means that a percentage of these Nmα sites are filled and other are empty. For example, for these values of nα situated between 1 and 2 and to simplify the problem, we can consider that x percentage of sites are filled with one hydrogen atom. The rest (1−x) are filled with two hydrogen atoms. We can write: x × 1 + (1−x) × 2 = 1.26. So we get% filled by one atom and% are filled with 2 atoms. Also the hydrogen is exothermically dissolved in the metal with high energy compared to that of β. In this interval of pressure the second term of β phase is almost null as it appears in Fig. 8. Lastly, nα is increased with the temperature. It means that the thermal agitation which dilates the site volume increases the interaction energy to attract

During absorption The two phases α and β contributions are presented in Fig. 8. In fact, these isotherms can be divided into three parts: 1329

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more hydrogen atoms to the α sites. So we can say the same for β sites.

Internal energy

Domain2: From P = Pαsat to P = Pβsat Once the α phase is saturated, the second phase, or the hydride phase (β phase), begins to be formed with a relatively large number nβ of absorbed atoms of hydrogen, which equals to 10.51 at T = 433 and 11.09 at T = 453 K, in the shape of agglomerates in β sites with a density Nmβ. This is achieved from Pα and around Pβ till the pressure Pβsat. The saturation level in the β is higher than α phase. There is also a limit in the amount of hydrogen that the metal can store: the maximum hydrogen concentration in the β phase is [H/M]βsat which is significantly higher than the maximum concentration [H/M]αsat. The final plateau of absorption saturation in the PCT curve corresponds to the coexistence of the α and β phases saturated at a time. The hydride phase (β-phase) nucleates and rises due to the very negative chemical potential inner the site leading to a condensation inner this potential simulating a solid state of the aggregated hydrogen atom. We can remark that the term correlated to the α-phase keeps in the saturated level. The term correlated to the β-phase completes and gives independently from α phase the corresponding absorption in the β phase. So, by our model, it is possible to affirm that there is not a reaction from the α to the β phase and there is no an eutectic phase α + β but the presence at a time of the α and β phases completely separated by their sites types. The storing energy of the α phase is higher than the energy of the β phase which indicates that the stability of the β phase is less than that of the α phase, but both energies are of chemical absorption (of order of 100 kJ/mol).

Internal energy (Eint) contains in forms of energy introduced into the system, due to current interactions between the particles of the system and the absorbent. However, here, we will limit only to absorption interaction form. The absorption internal energy is written as follow [39]:

Eint = −

∂LnZgc ∂β

+

μ ⎛ ∂LnZgc ⎞ ⎜ ⎟ β ⎝ ∂μ ⎠

(30)

with

μ = kB TLn

βP zg

(31)

Thus, the internal energy Eint is given by the expression as below:

Eint

nβ

( ) + N ( ) ( ( ) ) ⎛⎝1 + ( )

⎧⎛ βP ⎪ ⎜ Nmα = kB TLn Zg ⎨ ⎜ 1 + ⎪⎜ ⎩⎝

−kB T

⎧⎛ ⎪ ⎜ Nmα ⎨⎜ ⎪⎜ ⎩⎝

P nα Pα P nα

P Pβ

mβ

P Pβ

Pα

P nα P nα Ln P Pα α P nα

( )

( )

(1 + ( ) ) Pα

nβ

nβ

( ) Ln ( ) ⎛1 + ( ) ⎞⎠ ⎝

Nmβ +

⎞⎫ ⎟⎪ nβ ⎟ ⎬ ⎞ ⎟⎪ ⎠ ⎠⎭

P Pβ

P Pβ

P Pβ

nβ

⎞⎫ ⎟⎪ ⎟⎬ ⎟⎪ ⎠⎭

(32)

The expression (34) can be divided into two additive contributions for the first site and the second site:

Eint = Eintα + Eintβ

Domain3: From P = Pβsat to Pvs of hydrogen Finally, along the saturation plateau: P > Pβsat, the entire metal/ hydrogen alloy consists of the α and β phases (Fig. 8). At pressures higher than Pβsat till the saturated vapor of hydrogen, the hydride metal is saturated with hydrogen and it rests there. In this interval, we note the existence of the two phases with 31% for α phase and 69% for β phase in mass proportion.

Eintα = kB TLn

Eint2 During desorption A desorption isotherm is generated through decreasing the pressure. The equilibrium hydrogen pressure for desorption process is in general smaller than that of absorption for the same absorbed amount. Thus, the behavior of hysteresis would permit H-atoms to be absorbed at a determined pressure but released at a lower pressure than in storing. This lower pressure is needed to dis-anchor hydrogen atoms which present a stronger link at desorption process than at absorption process. Flanaganet al [38] have affirmed that pressure hysteresis of metal hydride is due to a plastic degradation mechanism achieved by the cracking phenomenon done during a notable fitting of absorbing sites. The most relevant fact is that the energy of desorption is increased relatively to that of absorption due to the higher porosity resulted after the irreversible cracking process leading to the hysteresis phenomenon. Moving towards the pressure Pβsat where Pβsat(des) is inferior than Pβsat(abs), the atoms of hydrogen of the second phase β begin to disanchor and then the α phase. So, the retained quantity β and then for α lessens till tends to zero.

βP ⎛ Nmα ⎜ Zg ⎜ 1 + ⎝

⎛ Nmβ βP = kB TLn ⎜ Zg ⎜ ⎜1 + ⎝

(33)

( ) ( ) P Pα

nα

P nα Pα nβ

() () P Pβ

P Pβ

nβ

( )

⎞ ⎛ Nmα PP α ⎟−kB T ⎜ ⎟ ⎜ 1+ ⎠ ⎝

nα

Ln

( ) P Pα

nα

P nα Pα

( ) nβ

nβ

( ) Ln ( ) ()

⎞ ⎛ Nmβ P Pβ ⎟−kB T ⎜ ⎟⎟ ⎜⎜ 1+ ⎠ ⎝

P Pβ

P Pβ

nβ

⎞ ⎟ ⎟ ⎠ ⎞ ⎟ ⎟⎟ ⎠

(34)

(35)

Figs. 10 and 11 present the evolution of internal energy with pressure for the two temperatures during absorption and desorption phenomena. We can see that the internal energy values are negative for both processes which means that the system must give energy to absorb H-atoms and must receive energy to release absorbed particles indicating the exothermic character of the absorption process and the endothermic character of the desorption process. Fig. 12 shows the

Macroscopic study of the absorption/desorption through thermodynamic potentials Thermodynamic potential functions are determined utilizing the analytical expression of the established best fitting model or more precisely the partition function related to the adequate model. These functions can give macroscopic properties of absorption or desorption process from a microscopic description by different involved model parameters. Free enthalpy of Gibbs, internal energy and entropy are examples of microscopic depiction of the system.

Fig. 10. Evolution of the internal energy as function of pressure for the absorption process. 1330

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equilibrium chemical potential of gas state is equal to that of absorbed state. We can recognize here the well-known definition of the chemical potential µ as the free enthalpy per atom: Ga/ Qa . It is given as follow:

βP μ = kB TLn ⎛⎜ ⎞⎟ ⎝ Zg ⎠

(36)

Finally the free enthalpy expression is given by:

Ga = kB TLn

⎛ nβ Nmβ βP nα Nmα ⎜ + Pβ nβ Zg ⎜ 1 + Pα nα + 1 P P ⎝

( )

( )

⎞ ⎟ ⎟ ⎠

(37)

This can be divided into two contributions for the first site and the second site.

Ga = Gaα + Gaβ

Fig. 11. Evolution of the internal energy as function of pressure for desorption process.

Ga1 = kB TLn

Ga2 = kB TLn

(38)

βP ⎛ nα Nmα ⎜ Zg ⎜ 1 + Pα nα P ⎝

⎞ ⎟ ⎟ ⎠

(39)

⎛ n N βP β mβ ⎜ Zg ⎜ 1 + Pβ nβ P ⎝

⎞ ⎟ ⎟ ⎠

(40)

( )

( )

All values of the free enthalpy during absorption and desorption process, illustrated in Figs. 13 and 14, are negative in a large pressure range indicating the spontaneous nature of both phenomena. In the case of desorption, the investigation occurs in the direction of reducing pressure by gradually decreasing the pressure. According to Fig. 15, we notice that the free enthalpy of the first phase α appears firstly. However, the free enthalpy Gaβ of the second type is greater than of type 1 since it potentially is the most energetic site. At high pressure, Gaα remains constant where its dynamics is finished as Gaβ begins to appear. Entropy The studies of the variation of the entropy during the absorption phenomenon depending on the pressure offer details about the degree of freedom of the system and accounts its disorder [41–42] and also about spontaneity of the phenomenon. The entropy is determined through exploiting the grand potential J and the grand canonical partition function Zgc:

Fig. 12. Evolution of the absorbed internal energy of the first, the second site and the overall versus the pressure for T = 453 K.

different contributions of the two sites. We remark that at the beginning of the absorption process, this energy is null. When the pressure increases, the internal energy Eint appears. The contribution of α sites is greater than that of β sites despite of the hydrogen storage of β sites is greater than α sites. This is due to the high absorption interaction of α sites. Moreover, we observe that the internal energies values of desorption are higher in modulus than that of absorption. This indicates that the system necessitates more energy to release hydrogen due the hysteresis phenomenon.

J = −kB TLnZgc

(41)

The free enthalpy of Gibbs The free enthalpy of Gibbs G, which is defined from the internal energy by: G = Eint−TS + PV , gives details about the exchange of the open system. It describes the spontaneity of the system [45]. The free enthalpy incorporates the interaction of the studied system concerning the exchange of particles with the exterior environment. The definition of G gives the following expression of the absorption free enthalpy Ga [44]:

Ga = μ N0 n = μ Qa where Qa is the absorbed hydrogen atoms quantity and µ is the chemical potential which formula is calculated in the gas state since at

Fig. 13. Evolution of the free enthalpy versus the pressure at absorption. 1331

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⎡ nβ n ⎢ P α P ⎛ ⎞ Sa = kB ∗ ⎢Nmα Ln ⎜⎛1 + ⎛ ⎞ ⎟⎞ + Nmβ Ln ⎜1 + ⎜⎛ ⎟⎞ ⎟ Pβ ⎠ ⎝ Pα ⎠ ⎠ ⎢ ⎝ ⎝ ⎝ ⎠ ⎢ ⎣ ⎜

⎧ P nα P ⎪ Nmα · Pα ·Ln Pα − P nα ⎨ 1+ P ⎪ α ⎩

( )

⎟

( )

nα

+

( ())

nβ

nβ

( ) ·Ln ( ) ⎛1 + ( ) ⎞⎠ ⎝

Nmβ ·

P Pβ

P Pβ

P Pβ

nβ

⎫⎤ ⎪⎥ ⎥ ⎬⎥ ⎪⎥ ⎭⎦

(45)

This can be divided into two additive contributions for the first site and the second site.

Sa = Saα + Saβ

Saα

Fig. 14. Evolution of the free enthalpy versus the pressure at desorption.

Saβ

∂

Since J is expressed as: J = E −μN −TS and E −μN = − ∂β lnZgc So the grand potential of absorption is:

∂ LnZgc−T ·Sa ∂β

( )

n

( )

α P P ⎡ ⎧ n P ⎞ α ⎞ ⎪ Nmα · Pα ·Ln Pα ⎢ ⎛ ⎛ = kB ∗ ⎢Nmα Ln ⎜1 + ⎟− P nα ⎝ Pα ⎠ ⎠ ⎨ ⎝ 1+ P ⎢ ⎪ α ⎩ ⎣ ⎜

⎟

nα

( ()) nβ

nβ

( ) ·Ln ( ) ( ) ⎞⎠

⎡ ⎧N · P nβ mβ P ⎢ β P ⎛ ⎞ ⎪ = kB ∗ ⎢Nmβ Ln ⎜1 + ⎜⎛ ⎟⎞ ⎟− ⎨ P β ⎢ ⎝ ⎠ ⎠ ⎪ ⎛1 + ⎝ ⎢ ⎝ ⎩ ⎣

P Pβ

P Pβ

nβ

⎫⎤ ⎪⎥ ⎬⎥ ⎪⎥ ⎭⎦

(47)

⎫⎤ ⎪⎥ ⎥ ⎬⎥ ⎪⎥ ⎭⎦

(48)

The Figs. 16 and 17 show the evolution of the entropy versus the pressure of the hydride at T = 433 K and T = 453 K for both processes. It is obvious that the entropy has two maxima corresponding to the α and β sites, as it is detailed in Fig. 18 about the contributions of each type of sites. The disorder Sa presents two behaviors below and above Pα and Pβ (halves saturation pressures). At low pressure, the entropy Saα for the first phase (Fig. 18) reaches a maximum in the proximity of the first half saturation (at P = Pα) showing a rise of the disorder and the movement of H-atoms around Pα of the phase α which corresponds to the greatest reactivity. This is related to the probability of the H-atoms to choose several vacant sites among Nmα free sites. This probability increases with pressure till a maximum at Pα. Beyond the half saturation Pα and after the first maximum, the probability and so the entropy diminishes through the influence of the reduction of the empty α sites until the saturation of these α sites where no more free sites exist and Saα tends to zero. After that, Saβ begins to increase and to reach a second maximum at P = Pβ. Beyond the second half saturation Pβ, H-atoms has low probability to select free receptor sites. Moreover, the entropy expressed by Saβ decreases since the surface tends toward the saturation and thus tends toward the order. When the saturation is attained, the disorder can reach zero for both phases α and β (Fig. 18). For desorption phenomenon, we remark that the development of entropy presents nearly the same behavior for the two temperatures

Fig. 15. Evolution of the absorbed free enthalpy of the first, the second site and the overall versus the pressure for T = 453 K.

J=−

(46)

(42)

Then, the entropy can be obtained by:

TSa = −

∂ LnZgc + kB T ·LnZgc ∂β

Sa = −βkB

∂LnZgc ∂β

+ kB LnZgc

(43)

(44)

Thus the absorption or desorption entropy expression Sa is written as below:

Fig. 16. Evolution of the absorption entropy according to the pressure. 1332

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with two types of sites in the case of a system with two levels of energies [31–50]. Conclusion By means of statistical physics formalism employing the grand canonical ensemble and applying some simplifying hypotheses, we could describe the absorption and desorption processes of hydrogen in LaNi3.8Al1.0Mn0.2 at T = 433 K and 453 K at a microscopic and macroscopic level: 1) The hydrogen absorption and desorption in LaNi3.8Al1.0Mn0.2 alloy is a monolayer chemisorption with two levels of energy which are supposed corresponding to α and β phases 2) The formation and the evolution of both phases α and β are described using our monolayer model with two energies of statistical physics formalism. We noticed a simultaneous coexistence of the two different phases α and β but jutting out each from the other with pressure, all over the range of temperature and pressure for absorption and desorption phenomena. 3) An energetic study has been carried out by means of the parameters Pα and Pβ deduced from the experimental curves fit using the monolayer model. The values of absorption and desorption energies are a direct proof that the absorption involved in our study is a chemical adsorption. These energies are increased with temperature. 4) The disorder of the absorption and desorption system expressed by the entropy of the system has been investigated. We also treated the thermodynamic characteristics of absorbate-absorbent system by the variation of the free enthalpy of Gibbs and the internal energy during the absorption/desorption process.

Fig. 17. Evolution of the desorption entropy according to the pressure.

Acknowledgements The authors would like to thank Professor Demin Chen from Institute of Metal Research, Chinese Academy of Sciences, Shenyang 110016, China, for kindly providing the hydrogen absorption and desorption isotherms data. Fig. 18. Evolution of the absorption entropy of the α phase, the β phase and the overall according to the pressure at T = 453 K.

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compared to the absorption disorder with a variation of the equilibrium pressures which are the halves saturation pressures Pα and Pβ during the liberation of hydrogen became lower than that of absorption. This behavior is one of the characteristics of hysteresis process. Discussion A comparison of the present work results with literature should emphasize the novelty of this work. Concerning the energies involved in modeling we note that many articles interested by the absorption and desorption energies gave energies values of the same order of our energies which are between 276 and 310 kJ/mol. Guoliang Liu et al. [43] indicated that the LaNi3.8Al1.0Mn0.2 presents an absorption energy equal to 258.8 kJ/mol which is in good agreement with our calculated energies. Also, they gave the value 228.3 kJ/mol for LaNi3.8Al0.8Mn0.4 alloy which is slightly smaller than ours due to the increase of pressure because of the substitution of manganese. Concerning the choice of the fitting model itself, the monolayer model with two energies which depicts two sites with two dissimilar energies of absorption and desorption, is in agreement with literature [44]. It has been affirmed that within AB5 metals there are two sets of minima that the H-atom may fill, tetrahedral (Td) holes where the Hatom is fourfold-coordinated and octahedral (Oh) holes where the H atom is sixfold-coordinated. Also many authors have used the model 1333

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