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Thermodynamic Modeling of the Ni–H System Natacha Bourgeois, Jean-Claude Crivello, Arkapol Saengdeejing, Ying Chen, Pierre Cénédèse, and Jean-Marc Joubert J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.5b06393 • Publication Date (Web): 04 Oct 2015 Downloaded from http://pubs.acs.org on October 13, 2015

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The Journal of Physical Chemistry

Thermodynamic Modeling of the Ni–H System Natacha Bourgeois,† Jean-Claude Crivello,† Arkapol Saengdeejing,‡ Ying Chen,‡ Pierre Cenedese,† and Jean-Marc Joubert∗,† Université Paris Est, ICMPE (UMR 7182), CNRS, UPEC, F-94320 Thiais, France, and 6-6-11, Aramakiaza Aoba, Aoba-ku, Sendai, Miyagi 980-8579, Japan E-mail: [email protected]

∗

To whom correspondence should be addressed Université Paris Est, ICMPE (UMR 7182), CNRS, UPEC, F-94320 Thiais, France ‡ 6-6-11, Aramakiaza Aoba, Aoba-ku, Sendai, Miyagi 980-8579, Japan †

1 ACS Paragon Plus Environment


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Abstract A new thermodynamic assessment of the Ni–H system has been carried out, providing a complete description valid up to 6×109 Pa of this system at the center of hydrogen storage issues. The study includes the hydride formation reaction and the presence of a miscibility gap in the fcc interstitial solid solution of hydrogen in nickel. In addition to a complete literature review, first principles calculations were carried out and combined with other modeling approaches. The Cluster Expansion Method allowed to describe the energetic interactions between atoms leading to the miscibility gap in the fcc solid solution. Besides, the compressibility at high pressure was characterized for each phase including the condensed phases. For this purpose, a specific high pressure model was assessed with the contribution of quasi-harmonic phonon calculations. The obtained consistent model allows to characterize entirely the thermochemical behavior of the Ni–H system.

1

Introduction

Hydrogen production and storage are considered as promising technologies to soften the fluctuations on the electric network induced by sustainable energy forms, whose intermittency still prevents large-scale development. In this context, the natural ability of several transition metals to react reversibly with hydrogen represents an attractive and secure way to store hydrogen and use it as an energy carrier. 1–5 However, in order to optimize the storage properties, it is crucial to understand the thermochemistry of these compounds. It includes for instance their reaction conditions, their hydrogen stoichiometry and the properties of the hydride compounds. Besides, basic bulk properties of the metal-hydrogen systems have been extensively studied also from a fundamental viewpoint 6 because of interesting phenomena linked with hydrogen absorption like electronic properties modifications 7–9 or optical transitions. 10


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Thermochemical databases constructed within the framework of the Calphad approach (CALculation of PHAse Diagrams) 11 are widely used to predict the properties of multicomponent systems from the properties of simple systems. This technique can be pertinently applied to predict the reactions in metal–hydrogen systems 12 and quantify the heat of reactions which are crucial quantities for hydrogen storage applications.

As nickel is involved in many promising hydrogen storage alloys, 13 for instance combined with magnesium (Mg2 NiH4 ), lanthanum (Haucke phase LaNi5 H6 ) or vanadium (V–Ni solid solutions), an accurate thermodynamic description of the Ni–H system is required. The Ni–H binary system is also characterized by the high pressure of formation of the hydride (in the range of 1×108 Pa). This compound has the NaCl structure corresponding to the Ni fcc structure in which all the octahedral interstitial sites are occupied by hydrogen. The solubility of hydrogen in pure nickel is very low and the formation of the hydride occurs across a wide miscibility gap.

Up to now, the only Calphad assessment of this system by Zeng et al. 14 was limited to moderate pressure and did take into account neither the hydride formation nor the phase separation. That is why the present model extends the description to higher pressures in order to describe the chemical reaction yielding the hydride formation as a function of temperature and pressure. For this purpose, new experimental data were therefore used, completed with the following results obtained from computational chemistry: • The enthalpy of reaction corresponding to the hydride formation was calculated within the framework of the Density Functional Theory (DFT). 15 • The enthalpy of solution was obtained using the Cluster Expansion Method (CEM). 16 • The high pressure behavior was evaluated after computation of the bulk modulus and thermal expansion (quasi-harmonic phonon calculations). 3 ACS Paragon Plus Environment


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Finally, a complete characterization of the thermochemistry of this important and prototype fcc metal–hydrogen system was achieved.

2

Methodology

2.1

DFT calculations

The DFT method was implemented using the Vienna Ab-initio Simulation Package (VASP). 17,18 The generalized gradient approximation (GGA) was used for the exchange and correlation energy functional with the Perdew-Burke-Ernzerhof functional (PBE). 19,20 Calculations were carried out with spin polarization. A high energy cutoff of 800 eV was used for the plane wave basis set and a dense grid of k-points in the irreducible wedge of the Brillouin zone was generated with the Monkhorst-Pack scheme. 21 The pseudopotential used included the Ni p-electrons.

The heat of reaction corresponding to the hydride formation at 0 K was calculated from the total energies of H2 molecule, pure Ni and NiH in the NaCl structure, according to equation 1. The enthalpy of reaction was corrected with the zero-point energy (ZPE) contribution. This energy is associated with the fluctuations of atoms positions at 0 K and was determined with phonon calculations 22 in the harmonic approximation (HA). 1 tot for tot = ENiH − ENi − EHtot2 4HNiH 2

2.2

(1)

CEM calculations

A CEM study was carried out using the Alloy Theoretic Automated Toolkit (ATAT) 23 code coupled with VASP calculations. This approach is based on the expression of the internal energy as a function of independent configurational functions {ξ}α , each of them related with a specific atomic cluster α (pairs, triplets,... ). Within this framework, each lattice site 4 ACS Paragon Plus Environment

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i is associated with an occupation variable σi whose value characterizes the type of atom occupying the site (+1 or -1 in Ising Model). The total energy is represented as a polynomial function of the configurational variables:

E(α) =

X

να ξα

(2)

ξα = < σi1 σi2 . . . σiNα >

(3)

α

where the correlation function ξα associated with the cluster α of Nα sites, is the average of the occupation variables {σ}i over all the sites of the cluster. The variables να called effective cluster interactions (ECI) represent the energetic contribution of each cluster α, including their multiplicity. They represent the parameter to be optimized. In the present study they have been refined on the basis of a set of DFT calculated enthalpies of relevant ordered configurations. 24 The correlation functions of the ordered structures being known, the ECI can be determined by inverting the matrix of the correlation functions in order to obtain a linear equation system whose unknown parameters are the ECI. They can be determined by fitting the calculated enthalpies. On the basis of an ECI set, structures of low enthalpy can be predicted, whose calculated enthalpies allow further to refine the initial ECI values. Thus, iteratively, the lowest energy curve can be constructed, linking the most stable structures for each composition and representing the enthalpy of solution at 0 K of the solid solution. This enthalpy was determined in the present work for the solid solution of H occupying the octahedral sites of the Ni fcc parent lattice.

2.3

Phonon calculations

HA-phonon calculations within the supercell approach were performed to determine the ZPE correction of pure nickel and nickel hydride. First, DFT calculations were used in order to fully relax the lattice parameter of both Ni and NiH and reduce the residual inter-atomic forces under 10−8 eV·˚ A−1 . 2×2×2 supercells with relevant atomic displacements were then



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built (Phonopy code 22 ) and the sets of resulting inter-atomic forces were determined (VASP). Finally, the Phonopy code was used to determine from the set of forces, the frequencies of the eigenmodes of vibration. Within the framework of the quasi-harmonic approximation (QHA), this above procedure was repeated for several cell volumes in order to include the pressure dependence of the eigenmodes. QHA-phonon calculations were carried out to calculate the temperature dependence of the volume, the thermal expansion coefficient and the bulk modulus of Ni and NiH.

2.4

The Calphad method

The semi-empirical Calphad method is based on the description of the Gibbs energy of each phase by equations with adjustable parameters. These parameters are refined by a least squares method to fit experimental or theoretical data and thus reproduce correctly the thermodynamic behavior of the system. One may find methodology description given in some reviews. 25,26 Details on the models are given in section 4.

3

Literature review and selection of data

The list of experimental data present in the literature on the Ni–H system is given in Table 1.

3.1

Pure nickel high temperature and pressure data

A consistent set of molar volume values of fcc nickel VmNi,fcc as a function of temperature was collected up to 1670 K by dilatation method and X-ray diffraction. 27–30 The thermal expansion coefficient was measured up to the melting point. 28,31,32 The variation of VmNi,fcc with pressure was studied by synchrotron X-ray diffraction up to 7.4×109 Pa. 33 The coefficient of isothermal compressibility was reported at room temperature 34 (5.4×10−12 Pa−1 ). Liquid nickel molar volume VmNi,liq temperature dependence was measured 35–47 up to 2419 K. Nickel melting point variation with pressure was measured with diamond anvil cell experiments up 6 ACS Paragon Plus Environment

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to 6.3×1010 Pa. 48–51

3.2

Hydrogen solubility data

From the large panel of hydrogen solubility data present in the literature, 52–67 only a consistent set of isothermal and isobaric measurements was conserved for the optimization. The experimental data can be altered by surface-related experimental effects, 56 oxygen contamination 59 or a non-reached equilibrium. 59 Each dataset was carefully compared with the others after being converted using Sieverts’ law when necessary. Only the one confirmed by external studies were retained. It has been noted that the solubility at 873 K measured by Smittenberg 59 was slightly higher than that measured by Armbruster 66 for identical pressures. Only Sieverts 52 measured the solubility in the liquid phase. Fukai et al. 68,69 reported solubility measurements in the hydrogen rich fcc phase at high pressure.

3.3

Pressure-composition isothermal curves

The nickel hydride can be synthesized using two methods: electrochemically 70–74 or by applying very high hydrogen pressure. 75–77 Absorption and desorption pressure-composition isotherms were measured 74,78–80 from ambient temperature to 651 K, showing an significant hysteresis. Indeed, the average formation/decomposition plateau pressures of NiH at ambient temperature are respectively 6.23×108 Pa and 3.40×108 Pa. Antonov et al. 81 measured the plateau pressure on a temperature range from 375 to 651 K.

3.4

Other phase diagram data

Serdyuk et al. 82 reported the temperature and H composition of the solid-liquid-gas equilibrium under hydrogen pressure at 108 Pa. Shapovalov and Serdyuk 53 measured the temperature of this 3-phase equilibrium up to 5.5×107 Pa as a function of hydrogen pressure, highlighting its decrease during the absorption process. Fukai et al. 68 confirmed this trend



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with measurements at higher hydrogen pressure (2.4 to 6.2×107 Pa). In addition, the Curie temperature decrease with increasing hydrogen pressure 81 was reported up to 284 K at 1.26×109 Pa. The compositions on both sides of the miscibility gap were measured by Antonov et al. 81 and Fukai et al. 68

3.5

Thermodynamic data

Despite the difficulties to carry out calorimetric measurements at high pressure, three papers report the molar enthalpy of hydride formation and/or decomposition reaction. Tkacz and Baranowski 79 found ∆H = −5.0 kJ/mol-H at 6.2×108 Pa and ∆H = −3.1 kJ/mol-H at 3.4×108 Pa during the formation and decomposition respectively. The previous values correspond to the quantity of heat. But at these pressures, the work of the pressure forces represents a non negligible term which should be added to obtain the enthalpy. Tkacz and Baranowski made the assumption that the main part of it is due to gaseous hydrogen. They extrapolated the volume change from experimental values and determined the pressure work by integration. The calculated enthalpies including the pressure work are -8.1 kJ/mol-H (formation) and -4.5 kJ/mol-H (decomposition). Baranowski and Czarnota 83 report a molar enthalpy measured at ambient temperature and pressure during the decomposition with a differential calorimeter (-5.0 kJ/mol-H), consistent with the corrected enthalpy of Tkacz and Baranowski of -4.5 kJ/mol-H. Besides, Czarnota and Baranowski 84 realized a set of calorimetric measurements during the spontaneous decomposition at normal pressure of several samples prepared at different compositions, from 0.02 to 0.36 hydrogen molar fraction x(H). The trend (∆H = −4.2 kJ/mol-H at x(H) = 0.36) is in agreement with the other experimental values . Furthermore, the partial enthalpy values derived from the plateau pressures of Antonov 81 are -5.1 kJ/mol-H (formation) and -8.2 kJ/mol-H (decomposition).


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Table 1: Compilation of experimental data. Conditions

Refs.

used

Temperature dependence of fcc nickel molar volume. 27

T =493 to 1671 K

only for comparison (comp.)

T =500 to 788 K

28

comp.

T =298 K

29

comp.

T =89 to 297 K

30

comp.

Temperature dependence of nickel thermal expansion coefficient. T =381 to 1102 K

28

comp.

T =292 to 1499 K

31

comp.

T =2.8 to 1729 K

32

comp.

Nickel molar volume pressure dependence. P =1×105 to 7.4×109 Pa

33

comp.

Nickel isothermal compressibility. 34

T =298 K

comp.

Temperature dependence of the liquid nickel density. T =1726 to 2151 K

36

yes

T =2173 to 2419 K

37

yes

T =1855 to 2225 K

38

yes

T =1728 to 2133 K

39

yes

T =1774 to 1932 K

40

yes

T =1873 K

41

yes

T =1726 and 1973 K

42

yes

T =1737 and 2369 K

43

yes



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Conditions

Refs.

used

Melt. point

44

yes

T =1773 K

45

yes

T =1823 K

46

no

(discrepan-

cies) 47

Melt. point

no

(discrepan-

cies) 85

T =1730 to 1974 K

no

(discrepan-

cies) fcc nickel melting point pressure dependence. P =1.0×108 to 6.3×1010 Pa

48

no (higher than the other data)

P =1.0×105 to 5.8×1010 Pa

49

yes

P =1.0×105 to 5.4×1010 Pa

50

yes

P =8.7×109 to 5.1×1010 Pa

51

yes

Isobaric hydrogen solubility data. P =1.013×105 Pa - T =485 to 1873 K

52

yes

P =1.013×107 Pa - T =818 to 1636 K

53

yes

P =1.000×105 Pa - T =779 to 1650 K

54

comp.

P =1.010×105 Pa - T =623 to 1495 K

55

comp.

P =1.013×105 Pa - T =357 to 554 K

56

yes

P =1.013×105 Pa - T =999 to 1516 K

86

comp.

P =1.013×105 Pa - T =723 to 1223 K

57

comp.

P =1.013×105 Pa - T =579 to 1479 K

58

comp.

P =1.333×101 Pa - T =873 to 573 K

59

comp.

P =1.013×105 Pa - T =1007 to 1494 K

60

no

(non-

agreement with the other 10 ACS Paragon Plus Environment

data)

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Conditions

Refs.

used

P =1.013×105 Pa - T =658 to 892 K

61

no

(non-

agreement) P =1.013×105 Pa - T =596 to 934 K

62

no

(non-


63

no

(non-


64

no

(non-


65

no

(non-

agreement) Isothermic hydrogen solubility data. P =6.00×10−1 to 2.35 ×101 Pa - T =573-873 K

59

yes

P =1.01×10−1 to 2.40×102 Pa - T =673-773-873 K

66

yes

P =1.21×104 to 1.24×105 Pa - T =1073 K

67

no

P =2.50×106 to 9.37×107 Pa - T =873-1073-1473-

54

comp.

53

yes except the

1623 K P =2.56×106 to 6.29×107 Pa - T =873-1073-14731623 K

data at 873 K Hydrogen solubility data in the hydrogen rich phase.

P =1.1×109 Pa to P =5.4×109 Pa - T =293 to

68

1062 K

comp.

(non-

defined temperature and pressure)



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Conditions

Refs.

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used

Pressure-composition isothermal curves. T =298 K

74

comp.

T =298 K

75

comp.

T =298 K

77

comp.

T =298-338 K

76

comp.

Decomposition plateau pressure. T =298 K

75

comp.

T =298 - 340 K

78

comp.

T =298 - 338 K

76

comp.

T =298 K

77

comp.

T =298 K

74

comp.

T =298 K

79

comp.

T =298 K

80

comp.

Formation plateau pressure. T =298 - 340 K

78

comp.

T =298 - 338 K

76

yes

T =298 K

77

comp.

T =298 K

79

comp.

T =375 to 651 K

81

yes

T =298 K

87

no agreement)

Melting point of nickel under hydrogen pressure. P =1.00×105 to 5.47×107 Pa

53

comp.

P =2.45×109 to 6.18×109 Pa

68

comp.

P =1.00×108 Pa

82

yes


(non-

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Conditions

Refs.

used

Curie temperature of nickel under hydrogen pressure. P =1.00×105 to 1.26×109 Pa - T =628 to 557 K

81

comp.

Solubility branches of the miscibility gap. T =472 to 624 K

81

comp.

T =296 to 629 K

68

comp.

Enthalpies of formation/decomposition of NiHx . T =298 K - formation

84

comp.

T =298 - formation and decomposition

79

comp.

T =298 K - decomposition

83

comp.

T =298 K - decomposition

79

comp.

T =275 to 651 K - calculated (Van’t Hoff) for for-

81

comp.

mation and decomposition

4

Thermodynamic modeling

4.1

Gas phase

The gas phase is composed of the species H2 and Ni. The standard Gibbs energy of gaseous dihydrogen and nickel at P0 = 105 Pa were taken from the PURE database 88 and SSUB3 database, 89 respectively. The non ideal behavior of hydrogen gas at high pressure was taken into account using the equation of state for hydrogen gas developed by Joubert. 90 Concerning gaseous nickel, a simple model was adopted, assuming that the asymptotic limit of the molar volume at high pressure is equal to that of solid nickel. Besides, we assumed that there is no interaction between dihydrogen molecules and nickel.



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4.2

Solid and liquid phases

The interstitial solid solution was described with a two-sublattice model in agreement with the nickel hydride fcc crystal structure: • the first sublattice corresponds to the fcc host lattice, fully occupied by nickel atoms • the second sublattice corresponds to the octahedral interstitial sites, occupied by vacancies and hydrogen. The solid phase was therefore described in the Compound Energy Formalism (CEF) 91 by: (Ni)1 :(Va,H)1 . In the model of Zeng et al., 14 the liquid phase was described using a substitutional solution model. The Gibbs energy of the hypothetical end member hydrogen liquid was fixed to a constant value, refined with experimental data from Ni–H and Mg–H systems. This value was found incompatible with the present model, in particular at high pressure. Consequently, it was decided to describe the liquid with an interstitial sublattice model, identical to the solid phase description. The validity of this model relies on the hypothesis that hydrogen composition in the liquid is far from reaching one hydrogen per nickel atom. This assumption is justified by the very low solubility of H measured in liquid 52 (2.2×10−3 hydrogen molar fraction at 1873 K and 1.0×105 Pa), almost 250 times lower than the limit allowed by the model. The advantage of this model compared to the substitutional one is to avoid the definition of metallic H in the liquid state with no reasonable physical meaning but which should be identical for any M –H system. Instead, the defined metastable end-member NiH has an impact only in Ni–H system. The present description is thus made compatible with other Calphad models of M –H systems using the same interstitial sublattice model to describe the liquid phase.


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The Gibbs energy of the phase ϕ (ϕ = fcc, liquid) is given by: ϕ

ϕ

ϕ

Gϕ = ref G + id G + ex Gϕ + phys G

(4)

with ϕ

•

ref

•

id

•

ex

•

phys

4.2.1

G : reference Gibbs energy ϕ

G : ideal Gibbs energy Gϕ : excess Gibbs energy ϕ

G : physical Gibbs energy.

Reference Gibbs energy

The reference Gibbs energy can be written as:

ref

ϕ

G = yVA GϕNi + yH GϕNi:H

(5)

GϕNi were taken from the PURE database. GϕNi:H are described by the following equations. 1 SER SER fcc Gfcc + bfcc · T + cfcc · T 2 Ni:H = GNi + GH2 + a 2 1 SER liq SER Gliq + bliq · T Ni:H = GNi + GH2 + a 2

(6) (7)

with Vi parameters to be optimized. 4.2.2

Ideal Gibbs energy

The ideal Gibbs energy results from the random mixing of H and vacancy on the second sublattice:

id

ϕ

ϕ

G = −T · conf S = RT (yVA ln yVA + yH ln yH ) 15 ACS Paragon Plus Environment

(8)


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4.2.3

Page 16 of 46

Excess Gibbs energy

The excess Gibbs energy, representing the interaction between hydrogen and vacancies on the interstitial sites, is described with the Redlich-Kister polynomial: 92

ex

Gϕ = yH yVA

n X i,ϕ

LNi:H,VA (yH − yVA )i

(9)

i=0

Li,ϕ Ni:H,VA can be made temperature dependent following: i,ϕ Li,ϕ + bi,ϕ · T Ni:H,VA = a

(10)

with ai,ϕ and bi,ϕ are the parameters to be optimized.

4.2.4

Physical Gibbs energy

Two contributions were added to the total Gibbs energy: the first one from magnetism mag

Gfcc and the second comp Gϕ representing the pressure-dependent part of the Gibbs energy.

phys

Gfcc =

phys

Gliq =

mag

Gfcc + comp Gfcc

comp

Gliq

(11) (12)

Magnetic contribution The magnetic contribution is present only in the fcc phase and can be described by:

mag

Gfcc = RT f (τ ) ln(β + 1)

(13)

where f (τ ) is an empirical function proposed by Inden 93 and adapted by Hillert and Jarl, 94 β is the Bohr magneton number (composition dependent) and τ is T /TC with TC


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the Curie Temperature. The physical quantities β and TC depend linearly on hydrogen composition. In the single fcc phase, they are not sensible to a pressure variation whereas this is the case in a multi-phase equilibrium including the fcc and gas phases, as hydrogen composition in fcc is pressure dependent.

Pressure-dependent contribution In the considered pressure range, a model where only the gas phase depends on pressure is no longer valid. Inconsistencies appear like an important destabilization of the gas at high pressure, leading to a dramatic increase of the boiling point of nickel. Furthermore, the condensed phases have non negligible compressibility and the work associated with the volume change under high pressure may not be neglected. The resulting contribution

comp

4Gϕ has thus to be considered, expressed as:

comp

4 Gϕ =

Z P

V (T, P )dP

(14)

P0

Within this framework, we used the model of Lu et al., 95 implemented in Thermo-Calc. The temperature and pressure dependencies of the molar volume of both solid and liquid phases are described on the basis of an empirical relationship between molar volume and bulk modulus: 96

Vm (T, P ) = x + y ln

B Pref

(15)

with x and y functions of temperature, characteristic of the considered material. After integration the Gibbs energy derived from this equation of state can be written as: "

comp

!

#

c(T ) Vm (T, P ) − Vm (T, P0 ) 4G = exp −1 κ(T, P0 ) c(T ) ϕ

with c(T ) the adjustable parameter. For each end member, the four following parameters were assessed:


(16)


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• V0 [m3 .mol−1 ]: molar volume of reference at a reference temperature T0 , P =1 bar; • VA (T ) [-]: integrated thermal expansivity with respect to the temperature between T0 and T VA (T ) =

Z T

ΨdT

(17)

T0

with 1 Ψ= Vm

∂Vm ∂T

!

(18) P,ni

The molar volume can thus be written as follows:

Vm (T ) = V0 exp(VA )

(19)

The parameters V0 and VA (T ) allow to describe the temperature variation of the molar volume at constant pressure; • VC (T ) [m3 .mol−1 ]: adjustable parameter c(T ) in eq 16; • VK (T ) [Pa−1 ]: isothermal compressibility at 1 bar κ(T, P0 ) (eq 16) the inverse of bulk modulus, representing the molar volume variation with pressure.

5

Results and discussion

5.1

DFT calculations

The electronic structure of pure nickel corresponds to ferro-magnetic state with a magnetic moment M = 0.62 A·m2 . Whereas in the hydride phase, the additional electron, given by the hydrogen atom, leads to the complete filling of the Ni-3d states. Thus, the hydride turns to a paramagnetic state (density of state not shown). No magnetic model was therefore considered in the Calphad description of this end member. The following heat of reaction associated with the hydride formation was calculated:


Page 19 of 46

NiH Efor = −7.0 kJ/mol-H. Taking into account the ZPE at 0 K, the corrected value is −7.8 kJ/mol-

H. As shown in Figure 1, the computed value agrees the VASP-GGA calculations of Wolverton et al. 97 but is not fully in agreement with the experiments. Since NiH is associated to a low heat of formation, its estimation is dominated by pure elements energy contributions whose calculation is not entirely satisfactory for H in our DFT description. Indeed, this inaccuracy is partly related to the inappropriate plane-wave basis set to describe the H2 dimer. Thus, low weight was associated to the DFT formation

( k J /m o l)

enthalpy value in the Calphad assessment.

0 -1 -2 -3

M o la r e n th a lp y o f fo r m a tio n o f N iH

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


-4 .2

-5

-5

-6 -7 -8 -9 -1 0 -1 1 -1 2 -1 3 -1 4

-2 .5

-3 .1

-4

-4 .1

-4 .5

-5

-3 .4

-7 -7 .8 c a lc u c a lc u c a lc u [c z a 6 [b a r6 [tk a 8 [tk a 8

-7 .5

-8 .1 la te la te la te 6 ] 4 ] 5 ] 5 ] -

d (D d (D d (C d e c d e c fo r ( d e c

F T ) - th is w o r k F T ) w ith Z P E c o r . - th is w o a lp h a d ) - th is w o r k c a c a V a h ig h p r e s s u r e ) V a ( h ig h p r e s s u r e ) c a

rk lc lc n n lc

u la u la 't H 't H u la

te d te d o ff o ff te d

[tk [tk [a n [a n (D

a 8 5 a 8 5 t7 7 t7 7 F T )

] - fo r ] - d e c ] - fo r ] - d e c - [w o l0 4 ]

-1 5

Figure 1: Calculated enthalpy of reaction at 0 K - comparison with raw and derived experimental data obtained during the formation (for) or decomposition (dec) process.

Besides, Ni and NiH molar volume relation with mechanical pressure was investigated by calculating the external pressure on cells whose lattice parameters has been reduced. As shown in Figure 2, the data for Ni are in agreement with the experimental measurements 33 in a pressure range up to 7.4 × 109 Pa. The results were used to assess the parameters of the 19 ACS Paragon Plus Environment


high pressure model of Lu et al.

6 .8 0 x 1 0

-6

6 .6 0 x 1 0

-6

6 .4 0 x 1 0

-6

6 .2 0 x 1 0

-6

6 .0 0 x 1 0

-6

5 .8 0 x 1 0

-6

5 .6 0 x 1 0

-6

5 .4 0 x 1 0

-6

5 .2 0 x 1 0

-6

5 .0 0 x 1 0

-6

4 .8 0 x 1 0

-6

4 .6 0 x 1 0

-6

4 .4 0 x 1 0

-6

) -3

(m V m

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 20 of 46

c a lc u la te d ( C a lp h a d ) [z h a 0 7 ] c a lc u la te d ( D F T )

3 .0 x 1 0

1 0

6 .0 x 1 0

1 0

9 .0 x 1 0

1 0

1 .2 x 1 0

1 1

1 .5 x 1 0

1 1

1 .8 x 1 0

1 1

P (P a )

Figure 2: Molar volume of pure Ni as a function of pressure calculated by DFT at 0 K (open squares) and by Calphad assessment at 298 K (solid line). Experimental values of Zhang et al. 33 at 298 K (filled squares).

5.2

CEM calculations

The calculated enthalpies of solution (squares) and the ones predicted by the model (circles) are both represented in Figure 3 as a function of hydrogen site fraction y(H). The enthalpies are normalized with the enthalpies of the references Ni and NiH. Presenting a cross validation score1 of 2.7×10−3 eV, the 21 calculated enthalpies are satisfactorily reproduced by the model which considers the contributions up to clusters of four atoms. The enthalpies of solution allow to quantify the interactions between H and vacancies 1

23 The predictive Pn power ofˆ the2 fit can be quantified by the cross-validation (CV) score, defined as 2 −1 (CV ) = n i=1 (Ei − E(i) ) , where n is the number of structures, Ei the calculated enthalpy of a ˆ structure i and E(i) the predicted enthalpy for the structure i, obtained by a least-square fit realized on the set of structures without the ith one.


Page 21 of 46

on the interstitial sublattice. In the Ni–H system, these interactions were found repulsive and result in positive enthalpies of solution, highlighting the instability of the solid solution. Indeed, a miscibility gap is experimentally observed in the fcc phase. This phase separates into two phases with the same structure but different H compositions. The results were used in the present Calphad model to determine the excess enthalpy at 0 K of the Ni–H solid solution. The good agreement between the CEM and Calphad models is highlighted in Figure 3.

7 0 0 0

E n th a lp y o f m ix in g ( J /m o l - H )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


c a lc u la te d ( D F T ) T = 0 K C E M T = 0 K c a lc u la te d ( C a lp h a d ) T = 2 9 8 K

6 0 0 0 5 0 0 0 4 0 0 0 3 0 0 0 2 0 0 0 1 0 0 0 0 0 .0

0 .2

N i

0 .4

0 .6

0 .8

y (H )

1 .0

N iH

Figure 3: Calculated enthalpies of solution (DFT) of the Ni–H solid solution (squares) as a function of hydrogen site fraction y(H) compared with the ones determined by the CEM model (circles). Calculated enthalpy of solution (Calphad) at 298 K.

5.3

Phonon calculations

Quasi-harmonic phonon calculations allowed to calculate three sets of data on Ni and NiH: • the molar volume as a function of the temperature (Figure 4)



• the thermal expansion coefficient defined as α =

1 1 3 Vm

∂Vm ∂T P,ni

directly determined

from the first derivative of the first data set • the bulk modulus B = −V

∂P ∂V

T,ni

.

The calculated thermal properties for pure Ni are in excellent agreement with the experimental studies (Figure 4) and were used to assess the parameters of the model of Lu et al. for both Ni and NiH. From the QHA-phonon calculations, the NiH structure has been found intrinsically unstable at temperatures above 600 K.

4 .6

7 .2

N i 7 .1

N iH

4 .4

(1 0

4 .3

-6

m

6 .8

(1 0

c a lc c a lc [s u h [y o u [s u h [b a n

4 .0 m

V 3 .9 3 .8

u la u la 8 8 8 6 8 8 7 7

te d ( C a lp h a d ) - th is w o r k te d ( p h o n o n ) - th is w o r k ]

6 .7

/m o l a to m )

3

3

-6

m

6 .9 4 .2 4 .1

N iH

N i

7 .0

m

/m o l a to m )

4 .5

c a lc u la te d ( p h o n o n ) - th is w o r k c a lc u la te d ( C a lp h a d ) - th is w o r k

V

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 22 of 46

] 6 .6 ] ] 6 .5

0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0

T (K )

Figure 4: Molar volumes of Ni (blue, right-hand scale) and NiH (black, left-hand scale) as a function of temperature determined by phonon calculations and Calphad modeling (this work). Experimental data from Suh et al., 27 Yousuf et al., 28 Von Batchelder and Raeuchle 29 and Bandyopadhyay and Gupta. 30

5.4

Calphad results

The description has been optimized using the Parrot module 98 of Thermo-Calc. 99 One may find the corresponding thermodynamic database (TDB file format) in supporting informa22 ACS Paragon Plus Environment

Page 23 of 46

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


tion. First, the Thermo-Calc variables V0, VA, VC and VK, corresponding to V0 , VA , VC and VK were refined for each end member of the condensed phases independently. The volume temperature and pressure dependence parameters of the solid end members Ni and NiH were optimized with the same procedure, using only DFT and quasi-harmonic phonon data (Sections 5.1 and 5.3). These calculated data were essential for NiH, as no experimental data was available in the literature. The high pressure parameters of liquid nickel were optimized by the exclusive use of experimental data. As short range order in liquids is optimally fitted by a bcc configuration, the molar volume of reference V0 of the liquid end member NiH was estimated equal to the solid NiH molar volume, multiplied by the bcc/fcc compressibilities ratio. VA , VC and VK were set at the same values as for pure liquid nickel, assuming that the thermal and pressure behaviors are not significantly affected by hydrogen solubility. Attention was paid to keep the parameters and their corresponding physical quantities to relevant values, on the complete set of temperatures and pressures. After optimization, a satisfactory agreement with the available experimental and theoretical data could be reached as shown in Figures 4, 5 and 6 for pure nickel. The calculated coefficient of isothermal compressibility of 5.2×10−12 Pa−1 agrees properly with the experimental one (5.4×10−12 Pa−1 ).



2 8 0 0

2 6 0 0

L 2 4 0 0

T (K )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 24 of 46

2 2 0 0

2 0 0 0

c a [la [e [ja [e

S 1 8 0 0

lc u la te d ( th is w o r k ) z 9 2 ] rr0 1 ] p 0 5 ] rr1 3 ]

1 6 0 0 0 .0

2 .0 x 1 0

1 0

4 .0 x 1 0

1 0

6 .0 x 1 0

1 0

8 .0 x 1 0

1 0

P (P a )

Figure 5: Calculated P -T phase diagram of pure Ni. Experimental data from Lazor et al., 48 Errandonea et al., 49 Japel et al. 50 and Errandonea. 51


-6

8 .0 x 1 0

-6

7 .8 x 1 0

-6

7 .6 x 1 0

-6

7 .4 x 1 0

-6

7 .2 x 1 0

-6

7 .0 x 1 0

-6

6 .8 x 1 0

-6

6 .6 x 1 0

-6

c a lc u la te d ( p h o n o n ) - th is w o r k c a lc u la te d ( C a lp h a d ) - th is w o r k [v e r6 4 ] [s a i7 0 ] [k ir 6 3 ] [s h ir 6 4 ] [w a t7 1 ] [o g i7 9 ] [n a s 9 8 ] [d ro 8 1 ] [lu c 6 0 ] [ta v 6 5 ] S

(m

-3

8 .2 x 1 0

V m

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


)

Page 25 of 46

8 0 0

1 0 0 0

1 2 0 0

1 4 0 0

1 6 0 0

1 8 0 0

L

2 0 0 0

2 2 0 0

2 4 0 0

2 6 0 0

T (K )

Figure 6: Calculated Ni molar volume increase at the melting point. Experimental data from , Tavadze, 45 Saito and Sakuma, 37 Kirschenbaum and Cahill, 38 Ogino et al., 41 Drotning, 43 Lucas, 44 Nasch and Manghnani, 42 Watanabe et al., 40 Shiraishi and Ward 39 and Vertman et al. 36

The high pressure model for the liquid phase allows to minimize the gas relative destabilization in the high pressure range. The boiling point variations with pressures are thus more reasonable in the conditions where hydrogen is in its gaseous state (Figure 7).



2 5 0 0 0

in itia l 2 0 0 0 0

T (K )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 26 of 46

1 5 0 0 0

1 0 0 0 0

w ith h ig h p re s s u re m o d e l

5 0 0 0

0 1 0

6

1 0

7

1 0

8

1 0

9

P (P a )

Figure 7: Calculated P -T phase diagram of pure Ni with (dark solid line) and without (red dashed line) specific high pressure model for condensed phases).

Finally, the complete set of parameters for the system Ni–H was assessed. The starting values for GϕNi:H (ϕ = fcc, liq) and L0,liq Ni:H,VA were evaluated in agreement with Zeng’s model. L0,fcc Ni:H,VA was defined as a function of temperature. The asymmetry of the miscibility gap was described using a temperature-independent sub-regular interaction term. The entire set of parameters in the fcc phase were assessed considering the selected experimental and calculated data. To describe properly the curvature of the experimentally measured H solubility at 105 and 107 Pa, it has been found necessary to add a T 2 term to describe Gfcc Ni:H , corresponding to the slope of the heat capacity as a function of temperature. This parameter was thus refined in accordance with the trend of the calculated heat capacity curve. As little information was available on the liquid phase, only a constant regular interaction parameter was considered. Three parameters for the liquid phase (two for Gliq Ni:H and one for the interaction parameter) were refined on the basis of the 3-phase equilibrium data 53,82 and the solubility data of Sieverts. 52 Finally, all parameters were optimized simultaneously with all the selected experimental and theoretical data. Their final values are 26 ACS Paragon Plus Environment

Page 27 of 46

summarized in Table 2. The model could reproduce satisfactorily the available isobaric and isothermal solubility measurements in the whole pressure range from very low (1 to 240 Pa) (Figure 8) to moderate pressures (105 and 107 Pa) (Figure 9). It can be noted that the present model underestimates the solubility measurements of Smittenberg. 59 This discrepancy was already pointed out by Zeng and explained by experimental inaccuracies on the pressure, which value may be slightly higher than 13.3 Pa. Indeed, our calculated curve around 17.5 Pa is nearly superimposed to the experimental points. This assumption is confirmed by comparing the isothermal measurements of the same author with the ones of Armbruster. 66

1 4

1 0

1 /2

)

1 2

(P a

c a lc c a lc c a lc c a lc [s m [s m [a rm [a rm [a rm

1 /2

8

P

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


6 4 2

u la u la u la u la i3 4 i3 4 4 3 4 3 4 3

te d te d te d te d ] - 5 ] - 8 ] - 6 ] - 8 ] - 7

- 5 7 3 - 6 7 3 - 7 7 3 - 8 7 3 7 3 K 7 3 K 7 3 K 7 3 K 7 3 K

K K K K

0 0 .0

4 .0 x 1 0

-6

8 .0 x 1 0

-6

1 .2 x 1 0

-5

1 .6 x 1 0

-5

x (H )

Figure 8: Calculated hydrogen solubility in Ni at low pressure and constant temperatures. Experimental solubility measurements of Smittenberg 59 and Armbruster 66 at constant temperature.



L + G L

1 8 0 0 F C C 1 6 0 0

G

+ F C C

1 4 0 0

T (K )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 28 of 46

c a lc u la te d a t 1 0 c a lc u la te d a t 1 0 [s h a 7 9 ] [s e r8 0 ] [s u z 8 6 ] [le l8 4 ] [le l7 3 ] [s ta 7 4 ] [lu c 3 2 ] [s ie 2 9 ]

1 2 0 0 1 0 0 0 8 0 0 6 0 0 4 0 0 0 .0 0 0

0 .0 0 2

0 .0 0 4

0 .0 0 6

0 .0 0 8

5

P a 7

P a

0 .0 1 0

x (H )

Figure 9: Calculated Ni–H phase diagrams at moderate pressure: 1.0×105 Pa (blue solid line) and 1.0×107 Pa (black dashed line). Experimental isobaric measurements of Sieverts, 52 Shapovalov and Serdyuk, 53 Serdyuk and Chuprina, 54 Suzuki and McLellan, 55 Mc Lellan and Sutter, 56 McLellan and Oates, 86 Stafford and McLellan 57 and Luckemeyer-Hasse and Schenck. 58

Besides, a difficult point was the description of the solid-liquid-gas equilibrium temperature as a function of pressure measured by Shapovalov and Serdyuk, 53 particularly in the region between 2×107 and 4×107 Pa where the model overestimates the equilibrium temperature (Figure 10). Probably related with the simple model used for the liquid phase, the discrepancy reduces however at higher pressures.


Page 29 of 46

1 7 4 0 1 7 3 5

c a lc u la te d [s h a 7 9 ]

1 7 3 0 1 7 2 5 1 7 2 0 1 7 1 5

T (K )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1 7 1 0 1 7 0 5 1 7 0 0 1 6 9 5 1 6 9 0 1 6 8 5 1 6 8 0 1 6 7 5 0

1 x 1 0

7

2 x 1 0

7

3 x 1 0

7

4 x 1 0

7

5 x 1 0

7

6 x 1 0

7

7 x 1 0

7

8 x 1 0

7

9 x 1 0

7

P (P a )

Figure 10: Solid-liquid-gas equilibrium temperature as a function of pressure compared with the data of Shapovalov and Serdyuk. 53

At low hydrogen content, the solution of hydrogen in nickel is known to be an endothermic process. While increasing hydrogen content, the reaction heat decreases from positive to negative values. This transition from an endothermic to an exothermic reaction of Ni with H was theoretically predicted, 100–102 notably by the mean-field lattice gas model of Driessen et al. and corroborated by the calorimetric measurements of Czarnota and Baranowski. 84 The present model predicts also this characteristic feature of the Ni–H system, highlighted in Figure 11 where several calculated pressure-composition isotherms at different temperatures are represented: for low H content, the reaction is endothermic and a temperature increase yields a lower equilibrium pressure for a given H composition. Conversely, for higher H content, the reaction is exothermic and an increasing temperature increases the equilibrium pressure. A crossing of the isotherms was therefore anticipated and is here shown for the first time in a Calphad model.



9

1 0

P (P a )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

[s h [s h [s h [s h [s h [s e [s e [s e [s e [s e

8

1 0

1 0

Page 30 of 46

c a lc c a lc c a lc c a lc c a lc 7

0 .0 5

u la u la u la u la u la

te d te d te d te d te d

- 8 7 - 1 0 - 1 2 - 1 4 - 1 6

3 K 7 3 7 3 7 3 2 3

0 .1 0

K K K K

a 7 9 a 7 9 a 7 9 a 7 9 a 7 9 r8 0 r8 0 r8 0 r8 0 r8 0

0 .1 5

] - 8 ] - 1 ] - 1 ] - 1 ] - 1 ] - 8 ] - 1 ] - 1 ] - 1 ] - 1

7 3 K 0 7 3 2 7 3 4 7 3 6 2 3 7 3 K 0 7 3 2 7 3 4 7 3 6 2 3

K K K K K K K K

0 .2 0

x (H )

Figure 11: Pressure-composition isotherms at different temperatures from 873 to 1623 K. The crossing is characteristic of the transition from an endo to exothermic reaction as a function of hydrogen composition. Experimental solubility measurements of Shapovalov and Serdyuk 53 and Serdyuk and Chuprina 54 at constant temperature.

The plateau pressure was assessed following the curve measured during NiH decomposition 81 as shown in Figures 12 and 13.


Page 31 of 46

9

1 0

P (P a )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1 0

c a lc u la te c a lc u la te [b a r7 8 ] [b a r7 8 ] [tk a 0 1 ] [b a r7 2 ] [m a j6 7 ] 8

0 .0

0 .1

0 .2

0 .4 8

d - 3 3 8 K d - 2 9 8 K 3 3 8 2 9 8 2 9 8 2 9 8 2 9 8

0 .4 9

0 .5 0

x (H )

Figure 12: Pressure-composition isotherms at 298 and 338 K. Experimental measurements of Baranowski, 75,76 Tkacz 77 and Majchrzak. 74


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

P (P a )


2 .0 x 1 0

9

1 .8 x 1 0

9

1 .6 x 1 0

9

1 .4 x 1 0

9

1 .2 x 1 0

9

1 .0 x 1 0

9

8 .0 x 1 0

8

6 .0 x 1 0

8

4 .0 x 1 0

8

2 .0 x 1 0

8

Page 32 of 46

c a lc u la te d [b a r7 2 ] [b a r8 0 ] [b a r7 8 ] [tk a 0 1 ] [m a j6 7 ] [tk a 8 5 ] [a n t7 7 ] [b a r6 5 ]

0 .0 1 .5 x 1 0

-3

2 .0 x 1 0

-3

2 .5 x 1 0

-3

1 /T (K

3 .0 x 1 0 -1

-3

3 .5 x 1 0

-3

4 .0 x 1 0

-3

)

Figure 13: Calculated plateau pressure in the Ni–H system as a function of the inverse temperature from 298 K to 651 K. Experimental plateau pressure of Baranowski and Czarnota, 75 Baranowski and Tkacz, 78 Baranowski, 76 Tkacz, 77 Majchrzak, 74 Tkacz and Baranowski, 79 Antonov et al., 81 Baranowski and Bochenska. 80

At high pressure, the model highlights the formation of a large miscibility gap (Figure 14). The scarce available experimental data on the solubility branches 68,81 are correctly reproduced. Besides, the predicted Curie temperature variation with hydrogen composition was found in agreement with the experimental values of Antonov et al. 81


Page 33 of 46

2 4 0 0 2 2 0 0

L

2 0 0 0

c a lc c a lc c a lc c a lc

u la u la u la u la

te d te d te d te d

[fu k 0 [fu k 0 [fu k 0 [fu k 0 [s e r9 [a n t7 [a n t7

- 1 E 8 P a - 2 .4 E 9 P a - 3 .1 E 9 P a - T c

L + F C C

1 8 0 0 1 6 0 0

T (K )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


4 ] 4 ] 4 ] 4 ] 2 ] 7 ] 7 ] -

1 .5 1 .9 1 .5 1 .1

T

E 9 to E 9 to E 9 to E 9 to

2 .4 2 .2 2 .4 2 .1

E 9 P E 9 P E 9 P E 9 P

a a a a

L + G c

F C C

1 4 0 0

G

+ F C C

1 2 0 0 1 0 0 0 8 0 0 6 0 0

F C C + F C C 4 0 0 0 .0

0 .1

0 .2

0 .3

0 .4

0 .5

x (H )

Figure 14: Ni–H phase diagrams at several pressures highlighting the miscibility gap formation. Experimental solubility branches of the miscibility gap of Antonov et al. 81 and Fukai et al., 68 solubility data in the hydrogen rich fcc phase 68 measured on a hydrogenated sample during cooling (empty symbols) or heating (filled symbols), 3-phase equilibrium of Serdyuk et al. 82 and Curie temperatures (Tc ) of Antonov et al. 81



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Table 2: Results of the assessment. Gas phase GGAS = GGAS,SER + RT ln P + 6.5836.10−6 · P − 6.5836.10−1 Ni Ni GGAS see ref. 12 H2 Interstitial solid solution: sublattice model (Ni)1 (H,Va)1 −3 SER · T2 Gfcc + 21 GSER H2 − 9154 + 69.77 · T − 5.824 × 10 Ni:H = GNi 0,fcc LNi:H,VA = 1.446 × 104 − 4.452 · T L1,fcc Ni:H,VA = −701.0 fcc V0,Ni:VA = 6.568 × 10−6 fcc VA,Ni:VA = 3.164 × 10−5 · T + 8.215 × 10−9 · T 2 fcc = 1.315 × 10−6 + 1.035 × 10−10 · T VC,Ni:VA fcc VK,Ni:VA = 4.646 × 10−12 + 1.846 × 10−15 · T fcc V0,Ni:H = 7.923 × 10−6 fcc VA,Ni:H = 6.223 × 10−5 · T + 6.708 × 10−8 · T 2 fcc VC,Ni:H = 1.513 × 10−6 + 1.049 × 10−10 · T fcc = 5.186 × 10−12 + 5.850 × 10−15 · T VK,Ni:H Liquid interstitial solution: sublattice model (Ni)1 (H,Va)1 1 SER 4 SER Gliq Ni:H = GNi + 2 GH2 + 1.769 × 10 + 41.01 · T 0,liq LNi:H,VA = 491.5 liq = 6.264 × 10−6 V0,Ni:VA liq VA,Ni:VA = 1.067 × 10−4 · T liq = 5.478 × 10−7 + 2.380 × 10−10 · T VC,Ni:VA liq VK,Ni:VA = 6.814 × 10−12 + 6.356 × 10−15 · T liq = 8.6221 × 10−6 V0,Ni:H liq VA,Ni:H = 1.067 × 10−4 · T liq = 5.478 × 10−7 + 2.380 × 10−10 · T VC,Ni:H liq VK,Ni:H = 6.814 × 10−12 + 6.356 × 10−15 · T

6

Conclusion

The present Calphad optimization provides a new description of the Ni–H system including the NiH hydride formation at high pressures and the miscibility gap. Coupled with QHAphonon calculations, the DFT contributed to offset the scarceness of available experimental data on NiH. The CEM method provided essential information to understand and model the phenomena leading to the phase separation in the fcc solid solution. It appeared as a very useful 34 ACS Paragon Plus Environment

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method to determine enthalpies of solution in a metastable solid solution which were used in the present model to improve the description of the excess Gibbs energy in the present Calphad model. Two phenomenological models were effectively used to take into account the pressure effect on solid, liquid and gaseous phases. First, the model of Joubert 90 allowed to consider the non ideality of gaseous hydrogen. Second, the model of Lu et al. 95 was applied to both liquid and solid phases, improving the reliability of high pressure extrapolations. Within this framework, the available experimental data could be satisfactorily reproduced over the entire pressure range. Besides, the new sublattice model adopted for the liquid phase makes the system more easily compatible if one wants to built a database with other M –H systems and facilitates the development of a future database centered on hydrogen.

Acknowledgement This work has benefited from a French government grant managed by ANR within the framework of the national program Investments for the Future ANR-11-LABX-022-01. DFT calculations were performed using HPC resources from GENCI-CINES (Grant 2014-96175).

Supporting Information Available TDB (Thermo-Calc Database Format), plain-text file format including the thermodynamic database on the Ni–H system obtained from the present assessment. Input file for thermodynamic software packages, it allows to compute thermodynamic quantities, phase equilibria and phase diagrams. This material is available free of charge via the Internet at http://pubs.acs.org/.



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