Volume of solution of hydrogen in lanthanum hydrides

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33 (2008) 3447 – 3450

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Volume of solution of hydrogen in lanthanum hydrides J.L. Gervasonia,b,,1, H.L. Corsoa, H.A. Perettia, S. Seguia,1 a

Centro Ato´mico Bariloche, Comisio´n Nacional de Energı´a Ato´mica, 8400 S.C. Bariloche, Argentina Instituto Balseiro, Universidad Nacional de Cuyo and Comisio´n Nacional de Energı´a Ato´mica, 8400 S.C. Bariloche, Argentina

b

art i cle info

ab st rac t

Article history:

In this work the jellium model approach is used to calculate the volume of solution of

Received 1 August 2007

hydrogen in lanthanum di- and tri-hydrides. Results are compared with the corresponding

Received in revised form

values of hydrogen diluted in pure lanthanum. Similar behaviors are obtained, which

18 December 2007

would allow to propose a relation between the volume of solution of hydrogen in pure

Accepted 19 December 2007

elements and in hydrides. Within the same approximation, values for the heat of solution

Available online 7 March 2008

are also calculated for the considered systems, and compared with experimental data

Keywords: Volume of solution

available in the literature. & 2008 Published by Elsevier Ltd. on behalf of International Association for Hydrogen Energy.

Hydrides Density functional theory

1.

Introduction

One of the basic properties of metal-hydrogen systems is the volume of solution of hydrogen in hydrides, which are important due to their technological applications as well as for the scientific interest in the theory of alloys. From a basic point of view, the study of such systems gives information related to the stress and deformation fields generated by defects. From a technological point of view, these calculations are useful in a wide range of applications, since the absorption of hydrogen may alter the mechanical properties of structural alloys. Among these applications, let us cite the design of security systems in nuclear facilities, and the development of hydrogen storage devices. Within this framework, rare earths hydrides have worldwide interest for their potential application in storage of hydrogen as a source of energy. Hence, it is of fundamental importance to understand the basic processes of hydriding. In particular, the lanthanum (one of the rare earths precursor) presents a wide range of hydrogen solubility [1], being an important candidate in the search of new materials for

storage. Lanthanum hydrides can be obtained by fusion or from nanocompounds obtained through mechanical alloying of powders. In this work, we use the jellium model to obtain the screening density charge, the volume of solution and the heat of solution of hydrogen in lanthanum di- and tri-hydrides, and we compare the results with the corresponding hydrogen solution in pure lanthanum.

2.

Theoretical model

Hohenberg and Kohn (HK) [2] have shown that, for a system of N electrons in an external potential Vext ðrÞ: (1) Vext ðrÞ is completely determined (unless an additive constant) by the charge density ZðrÞ of the fundamental state; (2) the energy of the system is given by Z E½Z ¼ Vext ðrÞZðrÞ dr þ F½Z,

Corresponding author at: Centro Ato´mico Bariloche, Comisio´n Nacional de Energı´a Ato´mica, 8400 S.C. Bariloche, Argentina.

Tel.: +54 2944 445118; fax: +54 2944 445190. E-mail address: [email protected] (J.L. Gervasoni). 1 Also member of the Consejo Nacional de Investigaciones Cientı´ficas y Te´cnicas. 0360-3199/$ - see front matter & 2008 Published by Elsevier Ltd. on behalf of International Association for Hydrogen Energy. doi:10.1016/j.ijhydene.2007.12.047

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and B is the bulk modulus, or equivalently, the inverse of the compressibility, for which experimental values are available.

where F½Z ¼ hCðZÞjðT þ Vee ÞjCðZÞi is minimum for the density ZðrÞ of the fundamental state. T and Vee are the kinetic energy and interaction potential operators, respectively. CðZÞ is the wave function of the system. The HK theorems have as a consequence that all the thermodynamical properties of a system are determined by ZðrÞ, once the number of atoms N of the system has been fixed. Hence, it is crucial to start with a most accurate model for the charge density, since it will determine all the remaining characteristics of the system. In this work, we use the density functional theory (DFT) [3,4], within the jellium approximation to study the electronic behavior of hydrogen in lanthanum and its hydrides. In this approximation, the metallic structure is replaced by an uniform electron gas with a positive-charge background of density equal to the mean charge density of the material, in order to keep the system neutral. This density is defined as 3 3 rðrÞ ¼ n1 0 ¼ 4prs ,

(1)

where rs is the radius of the sphere corresponding to the volume occupied by an electron in the solid. It is worth noticing that the jellium model takes plane wave functions for the electronic states, being adequate for studying properties with spherical symmetry. Calculations were done assuming the system to be at a temperature of 0 K and negligible pressure. When an impurity is introduced in a material, it induces a screening charge density which in turn determines many of the microscopic and macroscopic properties of the system, such as the volume of solution of hydrogen, activation and interaction energies, heat of solution, diffusion properties and others. Starting with the value of DZ obtained from the jellium model, we obtained the value of the total screening charge Q, integrated over the space, QðRÞ ¼

Z

R

DZðrÞ4pr2 dr.

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2.1.

Heat of solution DH

If the jellium model were exact, the change in the total energy calculated by this method should be equal to the heat involved in the process of absorption of hydrogen in the material, at zero temperature and pressure, DH ¼ DE. The simplifications introduced by the model alter this equality; nevertheless, it is possible to relate both quantities through a linear relation, ~ DH ¼ a~ DE þ b.

(5)

In the present case of lanthanum and its hydrides, the values obtained for the parameters are a~ ¼ 41:29 kJ=mol-H eV and b~ ¼ 515:58 kJ=mol-H [6]. The change in total energy is calculated by the jellium code and is the sum of the kinetic energy of the non-interacting electron gas, the exchange and correlation energies and the electrostatic energy. This linear relation has been proposed to obtain the heat of solution for the case of dilution of hydrogen in pure hosts. In this work we use it for obtaining the heat of solution in hydrides in a very simple way, checking its validity in the considered cases.

(2)

0

We use our previous parameterization of this quantity valid for the metallic range [5], which for a hydrogenic impurity takes the form   1 n QðRÞ ¼ 1  2 þ R þ R2  mR3 þ R5 e2R 2 2 A B=2kF R e þ sinð2kF R þ fÞ. 2kF R

(3) Table 1 – Calculated and experimental values of VH , given

In the case of lanthanum, the parameters take the following values: A ¼ 1:97, B ¼ 1:33, m ¼ 0:144, n ¼ 0:060, kF ¼ 0:98 and f ¼ 1:81. Q is related to the volume of solution VH through VH ¼

S , 3B

(4)

where S is given by S ¼ 4pZ0 lim

R!1

Z 0

R

ð1  QðrÞÞr dr

Fig. 1 – Total charge Q as a function of the distance to the hydrogen atom in La, LaH2 and LaH3 . Notice the overscreening in hydrides compared to the pure element.

˚ in A

3

Material La LaH2 LaH3

˚) rs (A

VH (calc)

VH (exp)

3.14 2.91 2.82

4.90 4.53 4.07

4.25 4.76 4.18

Experimental data from Ref. [10]. Values of rs used in the calculations are also included.

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Fig. 2 – SEM picture of hydrided lanthanum ðLaH2 Þ showing the surface layer of hydride over the pure element.

3.

Results and discussion

As stated above, the calculations presented in this work were accomplished within the DFT formalism, using an effective jellium model [7]. Fig. 1 shows the integrated screening density charge Q (Eq. (2)), as a function of the distance to the origin where the impurity (hydrogen atom) is located. For this calculation, the values of rs are those shown in Table 1, taken from Ref. [8]. The three curves correspond to pure lanthanum and di- and tri-hydride. The qualitative behavior is similar for the three cases, but an overscreening is observed for the hydrides in the vicinity of the hydrogen, when compared with pure lanthanum. Such screening rapidly decreases when increasing the distance to the impurity. In order to obtain the volume of solution VH of the hydrogen in the studied materials we employ Eq. (2), which depends on the variation of the screening charge density DZ, and the characteristic parameters of the model. The obtained values are presented in Table 1, together with the corresponding experimental values found in the literature [9]. As can be observed, the value of VH obtained for pure lanthanum overestimates the experimental value; this feature has been observed in a previous work [7]. On the other hand, the values obtained for the hydrides show the opposite tendency, underestimating the experimental values. These discrepancies can be attributed to the fact that the hydrides form on the surface of a material (Fig. 2), while the present model considers the dilution of hydrogen in a bulk hydride. Since local variations of the induced density in an interface could be relevant, the present model must be improved to take into account these effects. For this purpose, it would be necessary to describe properly the interface, which is out of the scope of this work.

Table 2 – Heat of solution DH values calculated from expression (5) Material

DH (calc)

DH (calc) [10]

DH (exp)

La LaH2 LaH3

97.38 122.8 192.6

62.73 – –

96.49 104 187.8

Theoretical value for pure lanthanum from Ref. [11] and experimental values taken from Ref. [9].

Calculations for the heat of solution using Eq. (5) are given in Table 2, along with experimental values and a theoretical calculation for pure lanthanum [11]. As can be observed, the results obtained with the linear fitting (Eq. (5)) show an excellent agreement with experimental data.

4.

Conclusions

In this work the electronic behavior of hydrogen in lanthanum and its hydrides has been analyzed from a theoretical point of view, using the DFT formalism within the jelliumtype electronic system approximation. With this model some physical and physiochemical properties have been calculated, such as the screening charge density, the volume of solution and the heat of solution. The screening charge density shows similar characteristics for the three studied systems, with a strong overscreening near the hydrogen impurity in the case of hydrides. Regarding the volume of solution, the values obtained with this model

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show discrepancies with respect to the experimental values. These differences would possibly be corrected if interface effects were considered for the hydride systems. On the contrary, calculations for the heat of solution show a good agreement with experimental values. It is expected to continue this work by systematically comparing the electronic behavior of pure elements and their hydrides. For this sake, other systems are being studied.

Acknowledgments This work has been partially funded by ANPCYT (PICT 15047) and CONICET (PIP 6402), from Argentina. R E F E R E N C E S

[1] Vajda P. In: Gschneider Jr KA. Eyring L, editors. Handbook on the physics and chemistry of rare earths, vol. 20. Amsterdam: Elsevier; 1995. p. 207. [2] Hohenberg P, Kohn W. Inhomogeneous electron gas. Phys Rev 1964;136(3B):B864–71.

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[3] Kohn W, Sham LJ. Self-consistent equations including exchange and correlation effects. Phys Rev 1965;140(4A):A1133–8. [4] Stott MJ, Zaremba E. Quasiatoms: an approach to atoms in nonuniform electronic systems. Phys Rev B 1980;22(4):1564–83. [5] Gervasoni JL, Abriata JP, Ponce VH, Barrachina RO. Parametrization of the screening charge density induced by a proton in a metal. Radiat Eff Defects Solids 1997;140:133–40. [6] Zapata Herrera M, Gonza´lez RJ, Gervasoni JL. Int J Hydrogen Energy 2008, this issue, doi:10.1016/j.ijhydene.2007.11.006. [7] Marchetti JM, Segui S, Gervasoni JL, Juan A, Abriata JP. Volume and heat of solution of hydrogen in rare earths from proton screening charge. J Phys Chem Solids 2006;67:1692–6. [8] Griessen R. Heats of solution and lattice-expansion and trapping energies of hydrogen in transition metals. Phys Rev B 1988;38:3690–8. [9] Griessen R, Feenstra R. Volume changes during hydrogen absorption in metals. J Phys F 1985;15:1013–9. [10] Khatamian D, Stassis C, Beaudry BJ. Location of deuterium in a-yttrium. Phys Rev B 1981;23:624–7. [11] Marchetti JM, Corso HL, Gervasoni JL. Int J Hydrogen Energy 2005;30:627.