# Thermodynamic assessment of the quaternary
Cr-Fe-H-Zr system

Tuan-Minh Vu, Caroline Toffolon-Masclet, Jean-Marc Joubert

► To cite this version:

Tuan-Minh Vu, Caroline Toffolon-Masclet, Jean-Marc Joubert. Thermodynamic assessment of the quaternary Cr-Fe-H-Zr system. Calphad, 2025, 90, pp.102861. 10.1016/j.calphad.2025.102861. hal-05193570

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# Thermodynamic assessment of the quaternary Cr-Fe-H-Zr system

Tuan-Minh Vu<fnref n="1" />,<fnref n="2" />, Caroline Toffolon-Masclet<fnref n="3" /> and Jean-Marc Joubert<fnref n="2" />

## Abstract

The thermodynamic assessments of the two ternary Cr-H-Zr and Fe-H-Zr systems are carried out using the CALPHAD approach. The hydrogenation properties of all the intermetallic compounds of these two systems are assessed. The formation enthalpies of the E1a-Zr₃FeH₇ and C16-Zr₂FeH₅ hydrides are evaluated by DFT calculations. The Pressure-Composition-Temperature diagrams of the Zr(CrₓFe₁-x)₂ (0.4 < x < 1.1) of the C14 Laves phase are optimized. The calculations and optimization are carried out under para-equilibrium conditions in which the complete system was treated like a pseudo-binary system.

**Keywords:** CALPHAD, DFT calculations, Hydrogenation, Para-equilibrium

## Introduction

Zirconium-based alloys are used as a material for the fuel claddings in pressurized water reactors thanks to their thermal, mechanical resistance and their low neutron absorption coefficient. During in-service lifetime and/or hypothetical accidental scenario like Loss Of Coolant Accidents (LOCA) of the fuel claddings, it is essential to apprehend and predict

<footnote n="1">¹ Université Paris-Saclay, CEA, Service de Recherche en Matériaux et Procédés Avancés, 91191 Gif Sur Yvette, France</footnote>

<footnote n="2">² Univ. Paris Est Créteil, CNRS, ICMPE, UMR 7182, 2 rue Henri Dunant, 94320 Thiais, France</footnote>

<footnote n="3">³ Université Paris-Saclay, CEA, Service de Recherche en Corrosion et Comportement des Matériaux, 91191 Gif Sur Yvette, France</footnote>
microstructural evolutions. As a result, a thermodynamic database containing all the elements of the fuel claddings Cr, Fe, Nb, Sn, Zr and O is being built by the CALPHAD approach [1,2]. At last, H is also included in order to take into account hydrogenation phenomena.

In literature [1–8], it was reported that hydrides could be formed from the different Cr-Fe-Zr intermetallic compounds. Thus, thermodynamic modeling of two ternary Cr-H-Zr and Fe-H-Zr systems appears to be essential for the thermodynamic database.

To the best of our knowledge, the thermodynamic modeling of the ternary Fe-H-Zr and Cr-H-Zr systems has not been undertaken. A detailed bibliography review of these ternary systems is presented. For the Laves C14 phase, the Pressure-Composition-Temperature (P-C-T) diagrams and hydrogenation properties are calculated within para-equilibrium using a pseudo-binary system to constrain the metallic atom ratio. At last, the assessments integrate the hydride formation enthalpies calculated by DFT.

# 1 Literature survey

## 1.1 Ternary Fe-H-Zr system

Several authors [9–14] found that Zr₂Fe compound (C16 structure, space group *I*4/mmc) can absorb up to 5 H/f.u. Neutron diffraction measurements indicate that the Zr₄ and Zr₃Fe tetrahedral sites are prefered. The hydrogen capacity varies as a function of the temperature. During dehydrogenation, a distortion in the structure, responsible for a space group change from *P*4/ncc to *I*4/mmm, was observed [10–14]. P-C-T diagrams from 200 to 500 °C are reported by Nobile *et al.* [14]. However, impurities were observed in the compound prior to hydrogenation. First-principles calculations on this hydride were carried out by Chattaraj *et*
al. [15]. The thermodynamic properties including the heat capacity ($Cp$ and $Cv$) of the hydride were reported.

The intermetallic $ZrFe_2$, which crystallizes with the Laves $C15$ structure, was reported to absorb up to 4 H/f.u at room temperature and very high pressure [8]. Because the Laves $C15$ phase possesses a non-stochiometric domain extending until a composition of $ZrFe_{2.7}$, the P-C-T diagram of $ZrFe_{1.9}$, $ZrFe_{2.0}$ and $ZrFe_{2.5}$ compositions were studied by Zotov et al. [16] and Sivov et al. [17].

A study conducted by Yartys et al. [11,12] focused on the hydride phase formed by the $Zr_3Fe$ intermetallic compound ($E1a$ structure). The hydrogen absorption was reported to be up to 6.7 H/f.u. By neutron diffraction, the hydrogen atoms are reported to occupy the $Zr_4$, $Zr_3Fe$ and $Zr_3Fe_2$ interstitial sites.

## 1.2 Ternary Cr-H-Zr system

In the Cr-Zr binary phase diagram, the intermetallic $ZrCr_2$ compound crystallizes with 3 different polymorphs. The cubic Laves $C15$ structure is stable at low temperatures, it transforms into the hexagonal Laves $C36$ at 1772 K, and the hexagonal Laves $C14$ is stable at high temperatures.

The P-C-T diagram of the Laves $C15$ phase was measured by Krupenchenko et al. [3], Andreev et al. [4] and Bodega et al. [5]. This compound can absorb 4 H/f.u at low pressures. The heat capacity of the Laves $C15$ type $ZrCr_2H_x$ ($0 \le x \le 0.5$) in the temperature range 5.3 - 293 K was measured by Skripov et al. [18]. An anomaly peak attributed to H ordering was observed. However, due to the limitation of the model used, the experimental measurement of the heat capacity at low temperatures ($T < 200$ K) can not be taken into account.
In the study conducted by Pebler and Gulbransen [7], a P-C-T diagram of C14-ZrCr₂ was reported. It was found that the hydrogen capacity was only 1.2 H/f.u. However, the measurement was carried out at high temperature and the pressure was limited to 1 bar.

Jacob *et al.* [19] reported that this intermetallic compound can absorb up to 4.5 H/f.u. Hirosawa *et al.* [20] and Skripov *et al.* [6] found that the hexagonal compound can reach a hydrogen content of 4.2 H/f.u. Bodega *et al.* [5] measured the P-C-T diagram of this compound in the temperature range of 348-423 K. It should be noted that their samples contained both Laves C14 and Laves C36 structures, making the quantification measurement challenging due to overlapping principal peaks on the XRD diffractogram. However, the Laves C14 structure was reported to be a major phase and no major difference is expected between the hydrogenation properties of the C14 and C36 phases, thus the measured data will be considered for the Laves C14 phase in our assessment.

To the best of our knowledge, no data was found in the literature concerning hydrogen solubility in α and β-Zr(Cr,Fe) solid solutions.

## 1.3 Hydrogenation of Zr(Cr<sub>x</sub>Fe<sub>1-x</sub>)<sub>2</sub> compound

Since the C14-ZrCr₂ compound can absorb hydrogen at low pressures, the substitution of Cr by Fe is promising for hydrogen storage due to its ability to increase the pressure plateau at atmospheric pressure and room temperature. Numerous studies have investigated the hydrides formed from the Zr(Cr<sub>x</sub>Fe<sub>1-x</sub>)₂ compound (x = 0.2 to 0.55). Both the host alloys and the hydrides adopt the Laves C14 structure. It has been reported that this compound exhibits hydrogen capacities similar to C14-ZrCr₂ (3.5 H/f.u) and demonstrates rapid absorption and desorption kinetics. Indeed, the crystal structure of the C14-ZrCr₂ compound possesses 3
sites: 4f occupied by Zr and 2a, 6h occupied by Cr. The synthesis of C14-Zr(CrₓFe₁₋ₓ)₂ compound is carried out by substituting the Cr in 2a and 6h sites with Fe. Canet et al. [21] and Soubeyroux et al. [22] reported the site occupancy of these substituted compounds (Figure 1). The occupations of each atom in both sites are equal.

<figure><img src="image_2.png" /><figcaption>Figure 1: Site occupation of Cr in C14-Zr(CrₓFe₁₋ₓ)₂ structure at 1000°C. △ Canet et al. [21] and ■ Soubeyroux et al. [22].</figcaption></figure>

Several P-C-T diagrams at different compositions have been reported. Yu *et al.* [23] presented a P-C-T diagram for the C14-Zr(Cr₀.₅Fe₀.₅)₂ composition at temperatures of 51, 101, and 152 °C. Ivey and Northwood [24,25] measured the P-C-T diagram at 20 °C for the C14-Zr(Cr₀.₂Fe₀.₈)₂, C14-Zr(Cr₀.₂₅Fe₀.₇₅)₂, and C14-Zr(Cr₀.₅₅Fe₀.₄₅)₂ compositions. Wallace *et al.* [26] reported a P-C-T diagram extending from 23 to 100 °C for the C14-Zr(Cr₀.₃Fe₀.₇)₂ composition. Zhang and Wallace [27] provided a P-C-T diagram for the C14-ZrCrFe composition. Qian and Northwood [28,29] measured the P-C-T diagram for the C14-Zr(Cr₀.₂Fe₀.₈)₂, C14-Zr(Cr₀.₃Fe₀.₇)₂, C14-Zr(Cr₀.₄Fe₀.₆)₂, and C14-Zr(Cr₀.₅Fe₀.₅)₂.
compositions. Park and Lee [30] reported a P-C-T diagram for the C14-ZrCr<sub>1-y</sub>Fe<sub>1+y</sub>
compositions with y values of 0.2, 0.3, and 0.5. Additionally, Grant et al. [31] reported a P-C-
T diagram for the C14-Zr(Cr<sub>0.25</sub>Fe<sub>0.75</sub>)<sub>2</sub> composition.

**2 Methodology**

2.1 DFT calculations

DFT calculations were carried out to determine the formation enthalpies of the C16-Zr₂FeH₅
and E1a-Zr₃FeH₇ compounds since they have ordered structures with the atomic positions
reported. The calculations were carried out using the Vienna Ab-Initio Simulation (VASP).
The Generalized Gradient Approximation (GGA) with the projector augmented wave (PAW)
pseudo-potentials of 600 eV were used. The formation enthalpies of the other stable
compounds in the system were also calculated to draw a stability envelope of the Fe-H-Zr
system. A dense grid of K-points in the Brillouin zone was optimized and used with 11×11×7
k-points meshing for E1a-FeZr₃ phase, 9×9×9 k-points meshing the Fe₂Zr phase, 5×5×3 k-
points meshing the γZrH phase, 7×7×7 k-points meshing the δZrH₂⁻ₓ phase, 11×11×11 k-
points meshing the εZrH₂ phase, 9×11×11 k-points meshing the C16-Zr₂FeH₅ phase and
9×9×5 k-points meshing the E1a-Zr₃FeH₇ phase.

The formation enthalpy is obtained by subtracting the DFT total energy of the calculated
structure to the molar fraction weighted sum of the DFT energies of the pure elements in
their Stable Element Reference (SER), i.e., ground-state structure for Fe (magnetic) in bcc, Zr
in hcp and H in gas.
## 2.2 The CALPHAD method

The CALPHAD method involves the optimization of the Gibbs energy for each phase in the system in order to match the available experimental data. The Gibbs energies are represented by equations that contain adjustable parameters. Through this assessment, phase equilibria and thermodynamic properties that have not been studied experimentally can be predicted. Detailed information about the CALPHAD method can be found in the Lukas et al. [32] textbook.

### Thermodynamic modelling

#### 2.2.1 Gas phase

Since the hydrides are formed at an intermediate pressure (below 100 bar), the gas phase has been considered as ideal. The Gibbs energy of the H₂ species in its stable-element reference (SER) state at a temperature of 298.15 K and pressure of 10⁵ Pa is obtained from the PURE database.

The Gibbs energy of pure hydrogen can be represented as:

$$G^{gas} = G_{H_2}^{SER} + RT\ln(P/P_0)$$

#### 2.2.2 Solid phases

The Gibbs energy function of pure elements is taken in the PURE database. Since the crystal structures do not change before and after hydrogenation, the distorsion observed in C16-Zr₂Fe hydride is not considered, the hydride phases are treated as solid solutions. Taking the example of the C15-Zr(CrₓFe₁₋ₓ)₂H₄ hydride, the compound is modeled with 4 sublattices corresponding to 4 cristal sites: (Zr)₄(Cr<sub>y<sub>Cr</sub></sub><sub>y<sub>Fe</sub></sub>)₂(Cr<sub>y<sub>Cr</sub></sub><sub>y<sub>Fe</sub></sub>)₆(H,Va)₁₄. The occupation fraction of A element (A = Cr or Fe) in the b (b = 2 or 3) sublattice is written y<sub>A</sub><sup>(b)</sup> in the hydride composition. The thermodynamic modeling of the Cr-Fe-Zr system was
defined by Lafaye [1]. The end-member energies and interaction parameters of intermetallic compounds from [1] will not be modified in this study. Three end-members $G_{Zr:Cr:Cr:Va}^{Laves\_C14}$, $G_{Zr:Fe:Fe:Va}^{Laves\_C14}$, $G_{Zr:Cr:Fe:Va}^{Laves\_C14}$, $G_{Zr:Fe:Cr:Va}^{Laves\_C14}$ and four interaction parameters $0L_{Zr:Cr,Fe:Cr:Va}^{Laves\_C14}$, $0L_{Zr:Cr,Fe:Fe:Va}^{Laves\_C14}$, $0L_{Zr:Cr,Fe:Cr,Fe:Va}^{Laves\_C14}$ were optimized. The Gibbs energy of this compound can be expressed as:

$$G^{Laves\_C14} = ref G^{Laves\_C14} + id G^{Laves\_C14} + ex G^{Laves\_C14} \#(1)$$

Where

$$\text{ref } G^{Laves\_C14} = y_{Cr}^{(2)} y_{Cr}^{(3)} y_{Va} G_{Zr:Cr:Cr:Va}^{Laves\_C14} + y_{Fe}^{(2)} y_{Fe}^{(3)} y_{Va} G_{Zr:Fe:Fe:Va}^{Laves\_C14} + y_{Cr}^{(2)} y_{Fe}^{(3)} y_{Va} G_{Zr:Cr:Fe:Va}^{Laves\_C14} \\ + y_{Fe}^{(2)} y_{Cr}^{(3)} y_{Va} y_{Va} G_{Zr:Fe:Cr:Va}^{Laves\_C14} + y_{Cr}^{(2)} y_{Cr}^{(3)} y_{H} G_{Zr:Cr:Cr:H}^{Laves\_C14} + y_{Fe}^{(2)} y_{Fe}^{(3)} y_{H} G_{Zr:Fe:Fe:H}^{Laves\_C14} \\ + y_{Cr}^{(2)} y_{Fe}^{(3)} y_{H} G_{Zr:Cr:Fe:H}^{Laves\_C14} + y_{Fe}^{(2)} y_{Cr}^{(3)} y_{H} G_{Zr:Fe:Cr:H}^{Laves\_C14} \#(2)$$

$$\text{id } G^{Laves\_C14} = 2RT(y_{Cr}^{(2)}lny_{Cr}^{(2)} + y_{Fe}^{(2)}lny_{Fe}^{(2)}) + 6RT(y_{Cr}^{(3)}lny_{Cr}^{(3)} + y_{Fe}^{(3)}lny_{Fe}^{(3)}) \\ + 14RT(y_{H}lny_{H} + y_{Va}lny_{Va}) \#(3)$$

$$\text{ex } G^{Laves\_C14} = y_{Cr}^{(2)} y_{Fe}^{(2)} y_{Cr}^{(3)} y_{Va} 0L_{Zr:Cr,Fe:Cr:Va}^{Laves\_C14} + y_{Cr}^{(2)} y_{Fe}^{(2)} y_{Fe}^{(3)} y_{Va} 0L_{Zr:Cr,Fe:Fe:Va}^{Laves\_C14} \\ + y_{Cr}^{(2)} y_{Fe}^{(2)} y_{Cr}^{(3)} y_{Fe}^{(3)} y_{Va} 0L_{Zr:Cr,Fe:Cr,Fe:Va}^{Laves\_C14} + y_{Cr}^{(2)} y_{Fe}^{(2)} y_{Cr}^{(3)} y_{H} 0L_{Zr:Cr,Fe:Cr:H}^{Laves\_C14} \\ + y_{Cr}^{(2)} y_{Fe}^{(2)} y_{Fe}^{(3)} y_{H} 0L_{Zr:Cr,Fe:Fe:H}^{Laves\_C14} + y_{Cr}^{(2)} y_{Fe}^{(2)} y_{Cr}^{(3)} y_{Fe}^{(3)} y_{H} 0L_{Zr:Cr,Fe:Cr,Fe:H}^{Laves\_C14} \\ + y_{Cr}^{(2)} y_{Cr}^{(3)} y_{H} y_{Va} 0L_{Zr:Cr:Cr:H,Va}^{Laves\_C14} + y_{Fe}^{(2)} y_{Fe}^{(3)} y_{H} y_{Va} 0L_{Zr:Fe:Fe:H,Va}^{Laves\_C14} \\ + y_{Fe}^{(2)} y_{Cr}^{(3)} y_{H} y_{Va} 0L_{Zr:Fe:Cr:H,Va}^{Laves\_C14} + y_{Cr}^{(2)} y_{Fe}^{(3)} y_{H} y_{Va} 0L_{Zr:Cr:Fe:H,Va}^{Laves\_C14} \\ + y_{Cr}^{(2)} y_{Cr}^{(3)} y_{H} y_{Va} (y_{H} - y_{Va}) 1L_{Zr:Cr:Cr:H,Va}^{Laves\_C14} \\ + y_{Fe}^{(2)} y_{Fe}^{(3)} y_{H} y_{Va} (y_{H} - y_{Va}) 1L_{Zr:Fe:Fe:H,Va}^{Laves\_C14} \#(4)$$

The thermodynamic model used to describe the hydride phases formed from intermetallic compounds in the Cr-Fe-H-Zr quaternary system is presented in Table 1:

<table><caption>Table 1. Thermodynamic model of hydride compounds in the Cr-Fe-H-Zr quaternary system</caption><tr><td>Phase</td><td>C16</td><td>Laves C15</td><td>Laves C14</td><td>E1a</td></tr></table>
<table><tr><td>Model</td><td>(Fe,Zr)(Fe,Zr)<sub>2</sub>(H,Va)<sub>5</sub></td><td>(Cr,Fe,Zr)<sub>2</sub>(Cr,Fe,Zr)(H,Va)<sub>4</sub></td><td>(Cr,Fe,Zr)<sub>4</sub>(Cr,Fe,Zr)<sub>2</sub>(Cr,Fe,Zr)<sub>6</sub>(H,Va)<sub>16</sub></td><td>(Fe,Zr)(Fe,Zr)<sub>3</sub>(H,Va)<sub>7</sub></td></tr></table>

## 2.3 Calculation and optimization of the para-equilibrium conditions

Para-equilibrium occurs when there are differences in the diffusion rates of the constituting elements at a given temperature. During the para-equilibrium state, the elements with higher diffusion rates tend to reach equilibrium, while the elements with slower diffusion rates remain relatively stationary. In the specific context of hydrogen storage materials, the measurements are often carried out close to room temperature where the metallic elements can be considered as frozen. This has two consequences. Firstly, hydrides of intermetallic compounds may be observed while the most stable state often corresponds to the decomposition into the hydrides of the pure elements. Secondly, when an equilibrium is observed between an intermetallic compound and its hydride, both have the same metallic compositions.

For the hydrides forming through a continuous solid solution, only the first problem is relevant and can be treated by suspending the other phases in the system to avoid the decomposition. However, for the hydrides forming by a phase transformation, whether because the structures are different or due to the presence of a miscibility gap, the observed para-equilibrium must be treated specifically. An additional constraint to fix the ratios of metallic elements should be introduced.

Pelton et al. [33] carried out the calculations in para-equilibrium by using virtual elements to apply a constraint to the composition. Nevertheless, the miscibility gap was not a focal point within their method. Recently, Hannappel et al. [34] made progress in the thermodynamic
modeling of multicomponent phases in para-equilibrium by creating an open-source Python tool that is integrated into the CALPHAD framework.

Joubert and Thiébaut [35] calculated P-C-T diagrams with the presence of the miscibility gap by treating the system as pseudo-binary, i.e. the intermetallic solid solution was treated by a pseudo-binary atom to ensure that the metallic composition remained constant throughout hydrogenation.

The same method is used in this work and explained in the following paragraph.

Hence, a pseudo-atom Z = (Zr)₄(Cr<sub>y<sub>Cr</sub></sub><sup>(2)</sup>Fe<sub>y<sub>Fe</sub></sub><sup>(2)</sup>)₂(Cr<sub>y<sub>Cr</sub></sub><sup>(3)</sup>Fe<sub>y<sub>Fe</sub></sub><sup>(3)</sup>)₆ representing the intermetallic part of hydride is introduced (with y<sub>Cr</sub> and y<sub>Fe</sub> the site occupation of Cr and Fe respectively).

The Gibbs energy for the solid phase in the pseudo-binary system can be written by the functions depending on the parameters in the ternary system:

$$G^{\text{Laves\_C14}} = \text{ref} G^{\text{Laves\_C14}} + \text{id} G^{\text{Laves\_C14}} + \text{ex} G^{\text{Laves\_C14}} \#(5)$$

where

$$\text{ref} G^{\text{Laves\_C14}} = y_{Va} G^{\text{Laves\_C14}}_{Z:Va} + y_{H} G^{\text{Laves\_C14}}_{Z:H} \#(6)$$

$$\text{id} G^{\text{Laves\_C14}} = RT(y_{H} \ln y_{H} + y_{Va} \ln y_{Va}) \#(7)$$

$$\text{ex} G^{\text{Laves\_C14}} = y_{H} y_{Va} \sum_{v=0}^{n} v L_{Z:H,Va}^{\text{Laves\_C14}} (y_{H} - y_{Va})^{v} \#(8)$$

The equations (1) and (5) have to be identical, the following relationships are given:
$$
\begin{aligned}
G_{Z:Va}^{Laves\_C14} = & y_{Cr}^{(2)} y_{Cr}^{(3)} G_{Zr:Cr:Cr:Va}^{Laves\_C14} + y_{Fe}^{(2)} y_{Fe}^{(3)} G_{Zr:Fe:Fe:Va}^{Laves\_C14} + y_{Cr}^{(2)} y_{Fe}^{(3)} G_{Zr:Cr:Fe:Va}^{Laves\_C14} \\
& + y_{Fe}^{(2)} y_{Cr}^{(3)} G_{Zr:Fe:Cr:Va}^{Laves\_C14} + y_{Cr}^{(2)} y_{Fe}^{(2)} y_{Cr}^{(3)} {}_0L_{Zr:Cr,Fe:Cr:Va}^{Laves\_C14} \\
& + y_{Cr}^{(2)} y_{Fe}^{(2)} y_{Fe}^{(3)} {}_0L_{Zr:Cr,Fe:Fe:Va}^{Laves\_C14} + y_{Cr}^{(2)} y_{Fe}^{(2)} y_{Cr}^{(3)} y_{Fe}^{(3)} {}_0L_{Zr:Cr,Fe:Cr,Fe:Va}^{Laves\_C14} \\
& + RT(y_{Cr}^{(2)} lny_{Cr}^{(2)} + y_{Fe}^{(2)} lny_{Fe}^{(2)} + y_{Cr}^{(3)} lny_{Cr}^{(3)} + y_{Fe}^{(3)} lny_{Fe}^{(3)}) \#
\end{aligned}
\quad (9)
$$

$$
\begin{aligned}
G_{Z:H}^{Laves\_C14} = & y_{Cr}^{(2)} y_{Cr}^{(3)} G_{Zr:Cr:Cr:H}^{Laves\_C14} + y_{Fe}^{(2)} y_{Fe}^{(3)} G_{Zr:Fe:Fe:H}^{Laves\_C14} + y_{Cr}^{(2)} y_{Fe}^{(3)} G_{Zr:Cr:Fe:H}^{Laves\_C14} \\
& + y_{Fe}^{(2)} y_{Cr}^{(3)} G_{Zr:Fe:Cr:H}^{Laves\_C14} + y_{Cr}^{(2)} y_{Fe}^{(2)} y_{Cr}^{(3)} {}_0L_{Zr:Cr,Fe:Cr:H}^{Laves\_C14} \\
& + y_{Cr}^{(2)} y_{Fe}^{(2)} y_{Fe}^{(3)} {}_0L_{Zr:Cr,Fe:Fe:H}^{Laves\_C14} + y_{Cr}^{(2)} y_{Fe}^{(2)} y_{Cr}^{(3)} y_{Fe}^{(3)} {}_0L_{Zr:Cr,Fe:Cr,Fe:H}^{Laves\_C14} \\
& + RT(y_{Cr}^{(2)} lny_{Cr}^{(2)} + y_{Fe}^{(2)} lny_{Fe}^{(2)} + y_{Cr}^{(3)} lny_{Cr}^{(3)} + y_{Fe}^{(3)} lny_{Fe}^{(3)}) \#
\end{aligned}
\quad (10)
$$

$$
\begin{aligned}
{}_0L_{Z:H,Va}^{Laves\_C14} = & y_{Cr}^{(2)} y_{Cr}^{(3)} {}_0L_{Zr:Cr:Cr:H,Va}^{Laves\_C14} + y_{Fe}^{(2)} y_{Fe}^{(3)} {}_0L_{Zr:Fe:Fe:H,Va}^{Laves\_C14} + y_{Fe}^{(2)} y_{Cr}^{(3)} {}_0L_{Zr:Fe:Cr:H,Va}^{Laves\_C14} \\
& + y_{Cr}^{(2)} y_{Fe}^{(3)} {}_0L_{Zr:Cr:Fe:H,Va}^{Laves\_C14} \#
\end{aligned}
\quad (11)
$$

$$
{}^1L_{Z:H,Va}^{Laves\_C14} = y_{Cr}^{(2)} y_{Cr}^{(3)} {}^1L_{Zr:Cr:Cr:H,Va}^{Laves\_C14} + y_{Fe}^{(2)} y_{Fe}^{(3)} {}^1L_{Zr:Fe:Fe:H,Va}^{Laves\_C14} \#(12)
$$

# 3 Results

## 3.1 DFT results

The formation enthalpies of the end-members of E1a-Zr₃Fe and C16-Zr₂Fe hydrides and stable end-members of binary phases are presented in Table 2. The calculated values in this work are close to the reported values in the literature.

<table><caption>Table 2: DFT calculation results for the Fe-H-Zr system and comparison with experimental values from literature</caption><thead><tr><th>Space group</th><th>Prototype</th><th>Configuration</th><th>Δ<sub>f</sub>H (this work)</th><th>Δ<sub>f</sub>H (in literature)</th></tr></thead><tbody><tr><td rowspan="4">P4/ncc(130)<br>(Distored structure)</td><td rowspan="4">Filled Al<sub>2</sub>Cu</td><td>Zr<sub>2</sub>FeH<sub>5</sub></td><td>-40.69</td><td>-39.20 [15], -37.65 [36]</td></tr><tr><td>ZrFe<sub>2</sub>H<sub>5</sub></td><td>76.42</td><td></td></tr><tr><td>Fe<sub>3</sub>H<sub>5</sub></td><td>15.25</td><td></td></tr><tr><td>Zr<sub>3</sub>H<sub>5</sub></td><td>-33.53</td><td></td></tr></tbody></table>
<table><tbody><tr><td rowspan="4">Cmcm(63)<br>(E1a)</td><td rowspan="4">Filled Re<sub>3</sub>B</td><td>Zr<sub>3</sub>FeH<sub>7</sub></td><td>-39.91</td><td>-36.14 [36]</td></tr><tr><td>ZrFe<sub>3</sub>H<sub>7</sub></td><td>6.85</td><td></td></tr><tr><td>Fe<sub>4</sub>H<sub>7</sub></td><td>13.11</td><td></td></tr><tr><td>Zr<sub>4</sub>H<sub>7</sub></td><td>-29.84</td><td></td></tr><tr><td>14/mmm(139)</td><td>ThH<sub>2</sub></td><td>εZrH<sub>2</sub></td><td>-55.16</td><td>-49.67 [37]</td></tr><tr><td>Fm<span style="text-decoration: overline">3</span>m(225)</td><td>CaF<sub>2</sub></td><td>δZrH<sub>2-x</sub></td><td>-54.23</td><td>-48.3 [37]</td></tr><tr><td>Cccm(66)</td><td>PtS</td><td>γZrH</td><td>-41.58</td><td>-58 [37]</td></tr><tr><td>Fd<span style="text-decoration: overline">3</span>m(227)<br>(Laves C15)</td><td>MgCu<sub>2</sub></td><td>ZrFe<sub>2</sub></td><td>-28.78</td><td>-35.03 [1]</td></tr><tr><td>14/mcm(140)</td><td>CuAl<sub>2</sub></td><td>Zr<sub>2</sub>Fe</td><td>-15.65</td><td>-17.35 [1]</td></tr></tbody></table>

The hydride stabilities were checked by drawing the stability envelope (Figure 2). This envelope is drawn by the convex hull algorithm [38]. The vertical axis is the formation enthalpy values of binary and ternary compounds obtained by DFT calculations and the horizontal plane is the composition plane.

<figure><img src="image_3.png" /><figcaption>Figure 2: Stability envelope of Fe-H-Zr system</figcaption></figure>
Both ternary hydrides are located above the envelope. The distances between the E1a-Zr₃FeH₇ and C16-Zr₂FeH₅ to the envelope are 5.06 kJ/mol.at and 0.47 kJ/mol.at respectively. Indeed, these hydrides are metastable. The stable equilibrium for these compositions is the Fe + H₂ + εZrH₂ equilibrium.

## 3.2 Fe-H-Zr system

In order to model the Fe-H-Zr system, three binary systems Fe-Zr, H-Zr and Fe-H are combined using the database of Lafaye [1], Zhong [39] and Zinkevich [40] respectively. The multiplicity of the interstitial sites within the HCP_A3 phase of the H-Zr system is modified from (Zr)(H,Va)₁ to (Zr)(H,Va)₂ and the one within the BCC_A2 phase of the Fe-H system is converted from (Fe)(H,Va)₁ to (Fe)(H,Va)₃ using Tai's model [41], [42]. The hydrides in this system are modeled as solid solutions. The multiplicity of hydrogen sublattices corresponds to the maximal hydrogen absorption capacity. The enthalpic parameters of all end-members of the hydrides formed from C16 and E1a structures are fixed to DFT calculated values. The entropy difference between a solid phase with gas and the hydride is approximatively equal to the entropy of the disappearing gas phase (~ 130 J.mol⁻¹ (H₂)). This approximative value is applied to the entropic parameters of all end-members of hydride phases. In order to optimize the heat capacity of C16-Zr₂FeH₅ hydride, supplementary parameters dependent on the temperature are adjusted. The enthalpic parameters of hydrides formed from Laves C15 ZrFeₓ are adjusted to optimize the pressure plateaus of ZrFeₓ-H (x = 1.9, 2.0 and 2.5) P-C-T diagrams.

Figure 3 represents the hydrogen capacity as a function of the temperature at atmospheric pressure, P-C-T diagram and the heat capacity as a function of the temperature
of the hydride C16-Zr₂FeH₅. The calculated lines are in acceptable agreement. However, the gap within the P-C-T diagram is explained by the impurities observed by Nobile *et al.* [14].

<figure><img src="image_4.png" /><figcaption>Figure 3: Hydride C16-Zr₂FeH₅: (a) hydrogen capacity as a function of the temperature at ambient pressure [9–13]; (b) P-C-T diagram [14], (c) heat capacity as a function of the temperature [15].</figcaption></figure>

Figure 4 represents the P-C-T diagrams of three compositions C15-ZrFe₁.₉-Hₓ, C15-ZrFe₂.₀-Hₓ and C15-ZrFe₂.₅-Hₓ. The calculated diagrams are in agreement with the experimental points.
<figure><img src="image_5.png" /><figcaption>Figure 4: P-C-T diagram of three compositions (a) C15-ZrFe<sub>1.9</sub>-H<sub>x</sub>, (b) C15-ZrFe<sub>2.0</sub>-H<sub>x</sub> and (c) C15-ZrFe<sub>2.5</sub>-H<sub>x</sub> compared to experimental measurements [16,17].</figcaption></figure>

The hydrogen capacity measurement for the E1a-Zr<sub>3</sub>Fe compound has been effectively implemented (Figure 5a). Finally, Figure 5b presents an isothermal section in full equilibrium at ambient temperature and atmospheric pressure. The three points on the figure are the compositions of three hydrides at their maximal hydrogen content. Indeed, at full equilibrium, these hydrides are unstable compared to the decomposition into Fe, H<sub>2</sub> and εZrH<sub>2</sub>.
the values obtained for the two end-members $G_{Fe:Fe:H}^{Laves\_C15}$ and $G_{Zr:Fe:H}^{Laves\_C15}$ applying the
difference obtained between $G_{Cr:Zr:H}^{Laves\_C15}$ and $G_{Fe:Zr:H}^{Laves\_C15}$ $G_{Fe:Fe:H}^{Laves\_C15}$ and $G_{Zr:Fe:H}^{Laves\_C15}$.

Figure 6 illustrates the formation enthalpies of end-members of the hydrides formed from
Laves C15 in both Fe-H-Zr and Cr-H-Zr systems.

<figure><img src="image_6.png" /><figcaption>Figure 6: Formation enthalpies of end-members of Laves C15 hydrides in Fe-H-Zr and Cr-H-Zr systems</figcaption></figure>

The calculated P-C-T diagram of the hydrides formed from Laves C15 and Laves C14
structures in Figure 7 is in agreement with the experimental points. On the P-C-T diagram of
Laves C14 hydride, the experimental points at high temperatures of Pebler and Gulbransen
[7] are optimized with less weight.
<figure><img src="image_7.png" /><figcaption>Figure 7: P-C-T diagrams of hydrides formed from ZrCr₂ compound of Laves C15 (a) and Laves C14 (b) structures</figcaption></figure>

## 3.4 Zr(CrₓFe₁₋ₓ)₂ hydride

In the Cr-H-Zr system, both Laves C14 and Laves C15 structures exist. The Laves C15 structure is predominantly present near the Cr-Zr and Fe-Zr binary systems. The Laves C14 phase, on the other hand, is situated at intermediate compositions between ZrCr₂ and ZrFe₂. The hydrogen capacity of these structures is similar (3.5 H/f.u) and their plateau pressure values at the same temperature are close. Therefore, the thermodynamic properties of the hydrides formed from this composition are similar for both structures. In the same way, the G<sub>Zr:Fe:Fe:H</sub><sup>Laves_C14</sup> and G<sub>Fe:Zr:H</sub><sup>Laves_C15</sup> parameters give the same composition: ZrFe₂H₄ in both structures.

The enthalpic and entropic parameters of these end-members should be approximatively equal. The enthalpic and entropic parameters of three end-members G<sub>Zr:Fe:Fe:H</sub><sup>Laves_C14</sup>, G<sub>Zr:Fe:Cr:H</sub><sup>Laves_C14</sup> and G<sub>Zr:Cr:Fe:H</sub><sup>Laves_C14</sup> are adjusted to optimize the pressure plateau of the entire composition range Zr(CrₓFe₁₋ₓ)₂. Regarding the metallic atom interaction parameters for hydrides, as the
metallic composition of the hydrides remains unchanged compared to the intermetallic compounds, these parameters are set to be identical to those of the intermetallic compounds:

$$ ^{0}L_{Zr:Cr,Fe:Cr,Va}^{Laves\_C14} = ^{0}L_{Zr:Cr,Fe:Cr,H}^{Laves\_C14} \quad ^{0}L_{Zr:Cr,Fe:Fe:Va}^{Laves\_C14} = ^{0}L_{Zr:Cr,Fe:Fe:H}^{Laves\_C14} \quad ^{0}L_{Zr:Cr,Fe:Cr,Fe:Va}^{Laves\_C14} = ^{0}L_{Zr:Cr,Fe:Cr,Fe:H}^{Laves\_C14} $$

The interaction parameters $^{0}L_{Zr:Fe:Cr,H,Va}^{Laves\_C14}$ and $^{0}L_{Zr:Cr:Fe:H,Va}^{Laves\_C14}$ are adjusted. The interaction parameters $^{0}L_{Zr:Fe:Fe:H,Va}^{Laves\_C14}$ and $^{1}L_{Zr:Fe:Fe:H,Va}^{Laves\_C14}$ are approximately equal to the interaction parameters $^{0}L_{Fe:Zr:H,Va}^{Laves\_C15}$ and $^{1}L_{Fe:Zr:H,Va}^{Laves\_C15}$.

<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /></figure>

<figure><img src="image_11.png" /></figure>
<figure><img src="image_12.png" /><figcaption>Figure 8: Calculated P-C-T diagrams of the hydride $C_{14}-Zr(Cr_xFe_{1-x})_2$ compared with experiments: ☒ [23], × [24], ◦ [25], △ [27], □ [28], ◦ [29], ▽ [30], ⊗ [26], ★ [31]</figcaption></figure>

The optimized diagrams (figure 8) are in acceptable agreement with the measured points.

Calculated and experimental diagrams have similar behaviors: the substitution of Fe by Cr induces the decrease of the plateau pressure at a fixed temperature.

The P-C-T diagram of the $C_{14}-Zr(Cr_xFe_{1-x})_2$ compound at room temperature is presented in Figure 9, along with the diagram showing the plateau pressure value for different compositions. Based on these figures, it is predicted that the composition **C14-**
$Zr(Cr_{0.32}Fe_{0.68})_2$ would be capable of absorbing hydrogen at room temperature and atmospheric pressure.

<figure><img src="image_13.png" /><figcaption>Figure 9: Plateau pressure at room temperature of the $Zr(Cr_xFe_{1-x})_2H_{3.5}$ Laves C14 hydride.</figcaption></figure>

## 4 Conclusion

The thermodynamic assessments of Fe-H-Zr, Cr-H-Zr systems and $C14-Zr(Cr_xFe_{1-x})_2H_4$ hydride have been carried out. The assessments required performing the calculations in para-equilibrium conditions. According to the temperature, the paraequilibrium or the full equilibrium should be used for the calculations. An atom representing the intermetallic compositions was added in order to have the constraint of the metallic ratio for the Laves phases. This method succeeded in calculating all the hydrogenation properties of the intermetallic compounds and in reproducing the behavior of the pressure plateau while substituting Cr by Fe in the $C14-Zr(Cr_xFe_{1-x})_2$ compound.
# Acknowledgments

The authors acknowledge Jean-Claude Crivello for the discussion on the DFT calculations. This work is funded by the projects GAINES and ICOMB within the framework of the French nuclear institute (I3P - CEA, EDF and Framatome).

**Appendice: Assessed thermodynamic parameters in this work.**

<table><thead><tr><th>Phase</th><th>Parameters (J.mol<sup>-1</sup>)</th></tr></thead><tbody><tr><td>BCC_A2<br/>(Fe)(H,Va)<sub>3</sub></td><td>G<sup>BCC_A2</sup><sub>Fe:H</sub> = +GFEH#<br/>0L<sup>BCC_A2</sup><sub>Fe:H,Va</sub> = 10.83T</td></tr><tr><td>HCP_A3<br/>(Zr)(H,Va)<sub>2</sub></td><td>G<sup>HCP_A3</sup><sub>Zr:H,Va</sub> = -91930 + 98.83T + GHSERZR#<br/>+ 2GHSERHH#<br/>0L<sup>HCP_A3</sup><sub>Zr:H,Va</sub> = -3.83T</td></tr><tr><td>Laves_C15<br/>(Cr,Fe,Zr)<sub>2</sub>(Cr,Fe,Zr)(H,Va)<sub>4</sub></td><td>G<sup>Laves_C15</sup><sub>Fe:Fe:H</sub> = +3GHSERFE# + 4GHSERHH# + 10000<br/>+ 260T<br/>G<sup>Laves_C15</sup><sub>Zr:Fe:H</sub> = +2GHSERZR# + GHSERFE# + 4GHSERHH#<br/>+ 70000 + 260T<br/>G<sup>Laves_C15</sup><sub>Fe:Zr:H</sub> = +2GHSERFE# + GHSERZR# + 4GHSERHH#<br/>- 146000 + 260T<br/>G<sup>Laves_C15</sup><sub>Zr:Zr:H</sub> = +3GHSERZR# + 4GHSERHH# - 156000<br/>+ 260T<br/>G<sup>Laves_C15</sup><sub>Cr:Cr:H</sub> = +3GHSERCR# + 4GHSERHH# + 144000<br/>+ 260T<br/>G<sup>Laves_C15</sup><sub>Zr:Cr:H</sub> = +2GHSERZR# + GHSERCR# + 4GHSERHH#<br/>+ 100000 + 260T<br/>G<sup>Laves_C15</sup><sub>Cr:Zr:H</sub> = +2GHSERCR# + GHSERZR# + 4GHSERHH#<br/>- 105531.644 + 260T<br/>0L<sup>Laves_C15</sup><sub>Fe:Zr:H,Va</sub> = +23000<br/>1L<sup>Laves_C15</sup><sub>Fe:Zr:H,Va</sub> = -12000<br/>0L<sup>Laves_C15</sup><sub>Cr:Zr:H,Va</sub> = +25000 - 15T</td></tr></tbody></table>
<table><thead><tr><th>Phase</th><th>Parameters (J.mol<sup>-1</sup>)</th></tr></thead><tbody><tr><td>Laves_C14</td><td>G<sub>Cr:Cr:Cr:H</sub><sup>Laves_C14</sup> = +12GHSERCR# + 16GHSERHH# + 300000 + 1040T</td></tr><tr><td>(Cr,Fe,Zr)<sub>4</sub>(Cr,Fe,Zr)<sub>2</sub><br>(Cr,Fe,Zr)<sub>6</sub>(H,Va)<sub>16</sub></td><td>G<sub>Zr:Cr:Cr:H</sub><sup>Laves_C14</sup> = +4GHSERZR# + 8GHSERCR# + 16GHSERHH#<br>-478175.567 + 1040T<br>G<sub>Cr:Zr:Cr:H</sub><sup>Laves_C14</sup> = +2GHSERZR# + 10GHSERCR# + 16GHSERHH#<br>-150000 + 1040T<br>G<sub>Zr:Zr:Cr:H</sub><sup>Laves_C14</sup> = +6GHSERZR# + 6GHSERCR# + 16GHSERHH# + 1040T<br>G<sub>Cr:Cr:Zr:H</sub><sup>Laves_C14</sup> = +6GHSERZR# + 6GHSERCR# + 16GHSERHH# + 1040T<br>G<sub>Zr:Cr:Zr:H</sub><sup>Laves_C14</sup> = +10GHSERZR# + 2GHSERCR# + 16GHSERHH# + 1040T<br>G<sub>Cr:Zr:Zr:H</sub><sup>Laves_C14</sup> = +8GHSERZR# + 4GHSERCR# + 16GHSERHH#<br>+400000 + 1040T<br>G<sub>Zr:Zr:Zr:H</sub><sup>Laves_C14</sup> = +12GHSERZR# + 16GHSERHH# - 624000 + 1040T<br>G<sub>Zr:Cr:Cr:H,Va</sub><sup>Laves_C14</sup> = +55000</td></tr><tr><td>E1a</td><td>G<sub>Fe:Fe:H</sub><sup>E1a</sup> = +4GHSERFE# + 7GHSERHH# + 144226.371 + 456T</td></tr><tr><td>(Fe,Zr)(Fe,Zr)<sub>3</sub>(H,Va)<sub>7</sub></td><td>G<sub>Zr:Fe:H</sub><sup>E1a</sup> = +GHSERZR# + 3GHSERFE# + 7GHSERHH# + 75366.5123 + 456T<br>G<sub>Fe:Zr:H</sub><sup>E1a</sup> = +GHSERFE# + 3GHSERZR# + 7GHSERHH# - 439016.836 + 456T + 43T × LN(T)<br>G<sub>Zr:Zr:H</sub><sup>E1a</sup> = +4GHSERZR# + 7GHSERHH# - 328219.196 + 456T<br><sup>0</sup>L<sub>Fe:Zr:H,Va</sub><sup>E1a</sup> = -80000</td></tr></tbody></table>
<table><thead><tr><th>Phase</th><th>Parameters (J.mol<sup>-1</sup>)</th></tr></thead><tbody><tr><td>C16</td><td>G<sup>C16</sup><sub>Fe:Fe:H</sub> = +3GHSERFE# + 5GHSERHH# + 122012.072 + 326T</td></tr><tr><td>(Fe,Zr)(Fe,Zr)<sub>2</sub>(H,Va)<sub>5</sub></td><td>G<sup>C16</sup><sub>Zr:Fe:H</sub> = +GHSERZR# + 2GHSERFE# + 5GHSERHH# + 611332.481 + 326T<br>G<sup>C16</sup><sub>Fe:Zr:H</sub> = +GHSERFE# + 2GHSERZR# + 5GHSERHH# - 325490.332 + 560T - 29.6551949TxLN(T) + 0.002T<sup>2</sup> + 554097.027T<sup>-1</sup><br>G<sup>C16</sup><sub>Zr:Zr:H</sub> = +3GHSERZR# + 5GHSERHH# - 268213.481 + 326T<br><sup>0</sup>L<sup>C16</sup><sub>Fe:Zr:H,Va</sub> = -100000 + 59T</td></tr><tr><td colspan="2">FUNCTIONGFEH = +213295.198 - 2916.37218T + 467.826T x LN(T) - 0.26955T<sup>2</sup> + GHSERFE# + 3GHSERHH#; 1811Y + 241181.85 + 72.9 * T; 6000N</td></tr></tbody></table>

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