Van-Truong Tran

Chu-Chun Fu
email: ǂchuchun.fu@cea.fr

Anton Schneider

Effective interaction model for coupled magnetism and phase stability in bcc Fe-Co systems

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Body-centered cubic (bcc) Fe-Co alloy in the ferromagnetic (FM) state has been found as the highest magnetic system among all known alloys [START_REF] Sourmail | Near equiatomic FeCo alloys: Constitution, mechanical and magnetic properties[END_REF]. Additionally, it is demonstrated experimentally that the Fe-Co based systems exhibit high Curie temperatures, up to about 1250 K [START_REF] Abbasov | A calorimetric study of lead-thallium alloys[END_REF][START_REF] Massalski | Binary alloy phase diagrams[END_REF]. All these outstanding features make this alloy extremely attractive for various magnetic by Prasad et. al. [START_REF] Prasad | Journal of Magnetism and Magnetic Materials Study of the effect of magnetic ordering on order -disorder transitions in binary alloys[END_REF] to investigate thermodynamic properties considering the magnetism. These two interaction models adopted the Ising-like magnetic interactions, and were treated within the mean-field approach. The latter approach provided a description in good agreement with experimental transition temperatures, but it presented magnetic exchange-coupling parameters dependent on the alloy concentration. However, please note that, as revealed by experiments [START_REF] Pierron-Bohnes | Magnetism and local order in dilute FeCo alloys[END_REF] and in previous DFT studies [START_REF] Tran | Predicting magnetization of ferromagnetic binary Fe alloys from chemical short range order[END_REF], the magnitude of Fe magnetic moment depends sensitively on the chemical long and short range orders. Therefore, magnetic interactions in Fe-Co cannot be correctly captured by the conventional Ising and Heisenberg models.

In this work, we aim at proposing an effective interaction model (EIM) for the study of magnetism, chemical-phase stability and their coupling in bcc Fe-Co structures. This EIM contains two parts: (i) a non-magnetic part describes the chemical interaction between atoms; and (ii) a magnetic-interaction part with a Heisenberg formalism. But, the spin magnitude of each Fe atom varies according to its local chemical environment, following a simple rule, determined by our previous DFT calculations [START_REF] Tran | Predicting magnetization of ferromagnetic binary Fe alloys from chemical short range order[END_REF]. In order to keep the model as simple as possible, and efficient for Monte Carlo simulations, we only include pair-interaction terms, at variance with the MCEs. Also, unlike the previous approach in Ref. [START_REF] Collins | The magnetic moment distribution in some transition metal alloys[END_REF], a unique set of interaction parameters is determined for the whole range of alloy concentration considered (from 0 to 70% Co). The EIM is therefore highly transferable. In addition, we also elucidate correlations between the magnetic state and the relative stability of chemically ordered and disordered phases for various temperatures and Co concentrations.

The paper is organized as follows: the proposed effective interaction model for bcc Fe-Co systems is described in Sec. 2, where we first present the Hamiltonian (Sec. 2.1) and then give the details about the parameterization (Sec. 2.2). The used DFT and Monte Carlo simulation methods are given in Sec. 3, together with the expression of various considered order parameters.

Sec. 4 is devoted to some results from the EIM and discussions. First, the accuracy of the model is discussed in Sec. 4.1. Then, finite-temperature magnetic and thermodynamic properties are predicted in Sec. 4.2, and, effects of an external magnetic field are discussed in Sec. 4.3. Finally, conclusions are given in Sec. 5.

The effective interaction model (EIM)

The model

The effective interaction model is written as

1 1 2 2 ij ij i j i j i j H V J M M = + ∑∑ ∑∑ uuu ruuur , (1)
here the first part accounting for non-magnetic contributions, describes chemical interactions between atoms. ij V are the chemical pair interaction parameters between i-th and j-th atoms. The second part of Eq. ( 1) corresponds to a Heisenberg-like model with ij J being the exchangecoupling parameter for the magnetic moments of i-th and j-th atoms. However, unlike the traditional Heisenberg model in which the amplitude of an atomic moment is fixed, the magnetic part of the model (1) takes into account the longitudinal spin fluctuation due to the effect of the local environment as observed by the DFT calculations [START_REF] Tran | Predicting magnetization of ferromagnetic binary Fe alloys from chemical short range order[END_REF], i.e., for a Fe atom, the magnetic moment can be described by a vector in the spherical coordinate system ( )

, ,

Fe Fe M M ϕ θ = uuuur with ϕ and θ being polar and azimuthal angles, respectively. The amplitude Fe M is defined as

2 3 2 3 0 1 1 2 1 3 1 1 2 2 2 3 2 2 2 1 2 1 1 2 2 1 2 Fe M P A N A N A N B N B N B N C N N D N N D N N = + × + × + × + × + × + × + × + × + × (2)
This equation is obtained from the analyses of the effect of the local environment on magnetic moments in bcc Fe-Co systems that are presented in detail in Ref. [START_REF] Tran | Predicting magnetization of ferromagnetic binary Fe alloys from chemical short range order[END_REF]. In Eq. ( 2), 1 N and 2 N are the numbers of Co atoms in the first and second neighboring shells of the central Fe atom. The fitting parameters P0, A1, A2, A3, B1, B2, B3, C, D1 and D2 were taken from Ref. [START_REF] Tran | Predicting magnetization of ferromagnetic binary Fe alloys from chemical short range order[END_REF].

For Co atoms, it is observed from the DFT calculations [START_REF] Tran | Predicting magnetization of ferromagnetic binary Fe alloys from chemical short range order[END_REF] and experiments [START_REF] Collins | The magnetic moment distribution in some transition metal alloys[END_REF] that the average moment of Co atoms stays almost constant at the value of 1.76 B µ . Hence, the magnetic moment vector of a Co atom writes ( )

1.76, , Co Co M M ϕ θ = = uuuur (3)
Thus, the model allows the magnetic moment vector of atoms to vary in 3D space, making spin polarization fully compatible with the non-collinear treatment.

It is worth noting that we assume the amplitude of a moment being insensitive to temperature.

Although there is no clear experimental evidence to support this assumption, thermal longitudinal fluctuations of spins in these systems were suggested to be negligible in previous theoretical studies [START_REF] Hasegawa | A spin fluctuation theory of degenerate narrow bands-finitetemperature magnetism of iron[END_REF][START_REF] Kakehashi | Patoary M 2011 First-Principles Dynamical Coherent-Potential Approximation Approach to the Ferromagnetism of Fe, Co, and Ni[END_REF].

Parameterization

The model (1) was first parameterized for ij J of the magnetic part by using a set of energy differences between two configurations of distinct magnetic states but with the same atomic configuration, i.e. E ∆ which contains only the magnetic contribution.

( )

1 . 2 ij i j i j E J M M ∆ = ∆ ∑∑ uuu ruuur , (4)
here, ( )

i j M M ∆
uuu ruuur is the difference of the scalar product i j M M uuu ruuur between two different magnetic states. By performing DFT calculations for different bcc Fe, Co, and Fe-Co structures, a set of Eq. ( 4) was constructed, and a least squares fitting was performed to obtain the ij J parameters.

First, JFe-Fe (JCo-Co) parameters were determined for magnetic properties in pure Fe (Co) systems.

Then these parameters were used in equations for Fe-Co systems to determine JFe-Co.

After obtaining the exchange-coupling parameters ij J , the chemical interaction parameters ij V were deduced by considering the mixing enthalpies of some Fe-Co structures, including special quasi-random structures which are useful to describe the equilibrium state of Fe-Co at high temperatures and the B2 phase which is known as the most stable structure at low temperatures.

These mixing enthalpies are defined as [START_REF] Amara | Aluminum and vacancies in α-iron: Dissolution, diffusion, and clustering[END_REF] mix .

. It is worth mentioning that in this work, to make an efficient parameterization using equations ( 4) and ( 5), we selected a set of important (most relevant) structures that could be enough to represent the overall behavior of most structures. There is a total of 48 configurations (atomic + magnetic) used for the fitting which include primitive unitcells, double layers and triple layers structures for pure Fe and Co; For Fe-Co structures: SQSs, ordered structures (B2, B32, cI16, D03) were selected; And these structures were considered with different magnetic states: ferromagnetic, anti-ferromagnetic, and certain disordered-magnetic structures."

FeCo Co Co Fe Fe E X X ε ε ε = - - , (5)
In the EIM Hamiltonian (1), the magnetic interactions (via ij J ) are considered up to the 5-th neighboring shell of each atom, which is demonstrated to be sufficient for the precise description of the magnetic properties [START_REF] Lavrentiev | Magnetic cluster expansion model for random and ordered magnetic face-centered cubic Fe-Ni-Cr alloys[END_REF]. Thus, it requires to know of 15 ij J parameters corresponding to Fe-Fe, Co-Co and Fe-Co pair interactions. The chemical-interaction range is commonly shorter than that of magnetic interactions. Therefore, to keep the model as simple as possible in order to reduce the computational cost of MC simulations, the chemical interaction was limited to only the first and second nearest neighbor shells. Please note that, in bcc structures, the first and second neighbor distances are very similar. And thus, there is a total of 6 ij V parameters.

In summary, all ij V and ij J parameters of the Hamiltonian (Eq. 1) were defined by fitting to the DFT data of a variety of bcc Fe, Co, and Fe-Co structures and in different magnetic states to ensure that the model has the capacity to describe physics at both 0 K and high temperatures. The set of parameters is given in Table I.

First-principles calculations and Monte Carlo simulations

Density functional theory methodology

The DFT calculations were carried out using plane-wave basis sets and projector augmented wave (PAW) potentials, as implemented in the Vienna Ab-initio Simulation Package (VASP).

[ [START_REF] Kresse | Ab initio molecular dynamics for liquid metals[END_REF][START_REF] Kresse | Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set[END_REF][START_REF] Kresse | Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set[END_REF][START_REF] Kresse | From ultrasoft pseudopotentials to the projector augmentedwave method[END_REF] The GGA-PBE functional was utilized [START_REF] Perdew | Generalized Gradient Approximation Made Simple[END_REF]. The calculations are spin polarized within the collinear approximation. The criterion of 10 -6 eV was employed in electronic self-consistent loops. The studied structures were fully relaxed until the forces on atoms being less than 0.01 eV/Å and stresses on the cells being smaller than 5 kbar. The plane-wave energy cut-off of 400

eV was chosen to have a good convergence of energetic, magnetic and structural properties. The Methfessel-Paxton smearing function with a width of 0.1 eV was used. The Brillouin zones were sampled by the Monkhorst-Pack method [START_REF] Monkhorst | Special points for Brillouin-zone integrations[END_REF] with a k-mesh of 16×16×16 for the simple cubic unit cell and an equivalent grid for a supercell: for example, a k-mesh grid of 4×4×4 was used for a supercell of 4×4×4 containing 128 atoms.

In this study, special quasi-random structures (SQS) were employed in DFT calculations to mimic disordered systems. The method used to generate the SQSs is described in detail in Ref. [START_REF] Tran | Predicting magnetization of ferromagnetic binary Fe alloys from chemical short range order[END_REF]. All the SQSs and the partially ordered structures of bcc Fe-Co are represented by 128-atom supercells, while the cells used for the ordered phases of Fe-Co and pure bcc Co, Fe are case dependent, for example, conventional unit cells of 2 atoms were used for pure bcc Co, Fe and a bcc Fe-Co B2.

Monte Carlo simulation setup

In this study, MC simulations were performed using the EIM (Eq. 1) for studies at finite temperatures. Two types of equilibrations were considered with MC simulations: (i) Spin equilibration at a given atomic configuration, which is referred as spin MC simulations; (ii)

Coupled magnetic and chemical equilibrations (barely spin-atom MC) which allow both spin variations and the exchange of position between pairs of atoms.

Both spin variations and atom exchanges are controlled by the Metropolis algorithm assuming the Boltzmann statistics.

In MC simulations, we used bcc systems of 16000 atoms (20×20×20 supercells) which were checked to be sufficiently large to ensure the convergence of energy and magnetization. Such a size of bcc systems for MC simulations was also used in previous studies [START_REF] Lavrentiev | Magnetic cluster expansion model for random and ordered magnetic face-centered cubic Fe-Ni-Cr alloys[END_REF][START_REF] Lavrentiev | Magnetic cluster expansion model for high-temperature magnetic properties of iron and iron-chromium alloys[END_REF][START_REF] Lavrentiev | Magnetic cluster expansion simulations of FeCr alloys[END_REF]. The initial disordered structures (the A2 phase) were generated by randomly replacing Fe atoms by Co atoms in the bcc box. All the obtained A2 configurations show chemical SROs smaller than 10 -4 for all neighbour shells and the chemical LRO almost equal to zero. We also verified that various random configurations at a given Co concentration lead to the same results in terms of energy and magnetization.

For the spin equilibration, about 2.0×10 8 MC steps (number of spin variation attempts) were used to get convergence at each temperature.

Regarding the spin-atom equilibrations, two-stage simulations were carried out: (i) a spin equilibration was performed for the initial atomic configuration, (ii) a spin-atom equilibration runs with about 1.5×10 6 to 3.0×10 6 MC atom exchange steps. After each exchange attempt of a pair of atoms, about 10 3 spin MC steps were performed some globally in the supercell and others locally to ensure the equilibrium state of the spin configuration corresponding to the new atomic configuration.

It is worth mentioning that MC simulations were performed at a given temperature to find the corresponding equilibrium state, and then the simulation at the next temperature starts from the converged spin and atomic configurations of the previous temperature. This procedure helps to speed up the reach-to-equilibrium of the system at successive temperatures.

Order parameters

In this study, several concepts are used to quantify the magnetic and chemical order of the studied systems. Chemical short-range order (chem-SRO) is widely employed and particularly useful to characterize solid solutions where chemical long-range ordering (chem-LRO) is missing.

However, the temperature evolution of chem-LRO is used to monitor the chemical phase transition from an ordered to a disordered state or vice versa. Similarly, magnetic short-range order (mag-SRO) and long-range order can also be defined to characterize magnetic structures.

The latter is actually the magnetization for ferromagnetic systems.

Chemical short-range order:

To determine the chem-SRO parameters of an alloy composed of two species A and B with a given arrangement of atoms, first, a local order parameter is defined by the Warren-Cowley formula: [START_REF] Cowley | An approximate theory of order in alloys[END_REF][START_REF] Martinez | Simulations of Decomposition Kinetics of Fe-Cr Solid Solutions during[END_REF] ( ) α over all the B atoms:

1 1 i i n A B n B n B Z Z X α - = - - , (6)
1 B i N n B n i B B N α α = = ∑ (7)

Chemical long-range order:

Chem-LRO is simply defined as follows [START_REF] Prasad | Journal of Magnetism and Magnetic Materials Study of the effect of magnetic ordering on order -disorder transitions in binary alloys[END_REF] 1 2

A A A LRO c c = -, (8)
where

A LRO is the chem-LRO of the species A, 1 1 1 / A A c N N = and 2 2 2 / A A c N N =
are the concentrations of the A atoms in two inequivalent sub-lattices 1 and 2 of the bcc crystal structure,

respectively. 1 2 / 2 N N N = =
where N is the total number of sub-lattice sites. It is also worth noting that

B A LRO LRO = .

Magnetic short-range order:

We define the local magnetic ordering parameter of a Fe atom as follows

( ) ( ) 1 / . if 0 0 if =0 n Fe Fe i i j i j i i i i i N Fe Fe Fe Fe n n Fe Fe Fe Fe n j Fe Fe n n Fe Fe Fe Fe M M M M N N N β β - - - = - - -   = ≠    =   ∑ uuuuruuuuu r uuuur uuuuu r , (9)
where i n Fe Fe β -is the magnetic ordering parameter of Fei. It represents the spin-spin correlation between the i-th Fe atom and its Fe neighboring atoms in the n-th shell.

i n Fe Fe N -is the number of Fe neighboring atoms in the n-th shell of the atom Fei. Mag-SRO of Fe-Fe interactions is then defined as

1 Fe i n N Fe Fe n Fe Fe i Fe N β β - - = = ∑ , ( 10
)
with Fe N is the number of Fe atoms in the system.

A similar definition of the mag-SRO is applied for Fe-Co and Co-Co magnetic couplings.

For example, the ordered B2 structure has chem-SRO equal to -1 and 1 for the first and second shells, respectively, while its chem-LRO is equal to 1. The ideal SQSs have both chem-SRO and chem-LRO equal to zero. While the ferromagnetic systems have mag-SRO equal to 1 for every shell, and the perfectly paramagnetic (PM) ones have mag-SRO equal to 0.

EIM-based results and discussions

Accuracy of the parameterization

The quality of the model (Eq. 1) parameterization is firstly verified by checking its capacity to reproduce the energy difference ( E ∆ ) between any two distinct magnetic states of a same atomic configuration. A comparison between the E ∆ obtained by the model and from DFT is displayed in Fig. 1. Most of the data points follow the diagonal line, showing an overall accuracy of the magnetic part. Only a couple of points deviate from the diagonal line, which indicates an overestimation by the model of the energy of a few high-energy magnetic states such as the antiferromagnetic and certain disordered-magnetic systems. It is worth mentioning that in our DFT calculation, disordered-magnetic structures are systems that contain some random (but still collinear) distribution of spins with the two possible orientations, they are commonly high energy systems compared to the magnetic ground state (FM). They may be relevant to some partially disordered-magnetic states that occur only at high temperatures in bcc Fe-Co.

Then, the parameterization of the model including both the magnetic and chemical parts is validated by comparing the mixing enthalpies of bcc Fe-Co from DFT and the model for 0 to 70% Co. To this end, we used the same DFT predicted atomic magnetic moments for the EIM. It is shown in Fig. 2 that the mixing enthalpies of the disordered structures (SQSs) by the model (red symbols) are in very good agreement with the DFT data (black symbols). Moreover, the results for the B2 structure are in excellent agreement between the two methods. Although the ordered structure D03-Fe3Co was not included in the fitting of the chemical part, we checked that its mixing enthalpy obtained by the model in combination with spin MC simulation at 1 K is -0.0653 eV/atom, which is in very good agreement with DFT result of -0.0625 eV/atom. This suggests the reliability of the model on the prediction of thermodynamic properties of the Fe-Co alloys at various temperatures that will be discussed in more detail below. As can be seen, both the DFT and the model exhibit negative mixing enthalpies of Fe-Co structures, which demonstrate a strong mixing tendency of Fe and Co atoms. Additionally, the B2 phase is more stable than the disordered one at 0 K.

Please note that in this paper, the atomic percentage is used to denote the composition of an alloy, and for the sake of simplicity, the prefix "at." is not explicitly written.

Concerning the magnetic properties, the model can be further validated by combining with spin MC simulations to obtain the magnetization of the Fe-Co structures at very low temperature (for example, at 1 K) and comparing them with the DFT results at 0 K. In Fig. 3, the average moment of each species Fe and Co as well as the total magnetization are shown for the disordered Fe-Co structures, which are often addressed as the A2 phase in the phase diagram. As can be seen, the results obtained by the model are in very good agreement with the DFT ones. Please note that the average magnetic moments of disordered structures obtained by DFT calculations have been confirmed to be in agreement with experiments in our previous study [START_REF] Tran | Predicting magnetization of ferromagnetic binary Fe alloys from chemical short range order[END_REF]. As observed from Fig.

Prediction of finite-temperature properties

The use of the EIM in MC simulations allow us to predict a variety of properties at finite temperatures. In this section, several magnetic and thermodynamic properties of bcc Fe-Co systems are examined and discussed.

Magnetic and thermodynamic properties at a given atomic configuration

Magnetic transition and the Curie temperature

The magnetization of the disordered Fe-Co structures at high temperatures is predicted by the EIM and MC and shown in Fig. 4(a), while the average moments of Fe and Co species are represented in Fig. 4(b). As can be observed, the average moments of Fe and Co drop at high temperatures, leading to a lower magnetization. Such results could be understood as at high temperatures spins fluctuate strongly and the Fe-Co system transforms from the ferromagnetic state to the paramagnetic one.

By quantifying the magnetization as a function of temperature, the Curie temperature Tc of a system can be determined when a sharp drop of magnetization is observed.

In Fig. 5, the magnetization curves are shown for bcc Fe, bcc Co, and bcc Fe-Co B2 and A2 at 50% Co. At first glance, within this model, the Curie points of the bcc Fe and Co systems are very close to each other and around 1100 K. While, the Fe-Co structures exhibit higher Curie temperatures, and the Tc of the B2 is predicted to be larger than that of its A2 counterpart.

The determination of the Curie temperature may involve a significant uncertainty, because, due to the finite-size of the simulation box, the obtained magnetization does not drop sharply to zero at Tc. Several methods have been proposed, either using the inflection point of the magnetizationtemperature curve [START_REF] Chui | Investigating the magnetovolume effect in isotropic bodycentered-cubic iron using spin-lattice dynamics simulations[END_REF][START_REF] Xie | Unambiguous determining the Curie point in perovskite manganite with second-order phase transition by scaling method[END_REF] or the peak of the temperature evolution of the heat capacity [START_REF] Lavrentiev | Magnetic cluster expansion simulations of FeCr alloys[END_REF][START_REF] Connelly | Specific Heat of Nickel near the Curie Temperature[END_REF].

However, these methods require a very dense sampling of the magnetization around the magnetic transition, therefore, computationally demanding. Alternatively, the fitting method using a power law has been widely used to determine Tc [START_REF] Carvalho | Room temperature structure and multiferroic properties in Bi 0.7La0.3FeO3 ceramics[END_REF][START_REF] Yang | Realization of high Curie temperature ferromagnetism in atomically thin MoS2 and WS2 nanosheets with uniform and flower-like morphology[END_REF][START_REF] Fuh | Newtype single-layer magnetic semiconductor in transition-metal dichalcogenides VX2 (X = S[END_REF]. All the data points of the magnetization before Tc are less affected by the finite-size effect and can be used to fit the following expression:

( ) ( ) ( )

1 0 c M T T m T M T β   = = -     , (11)
where, ( )

0 M and
( )

M T are the magnetizations at 0 K and the temperature T, respectively. A non-linear fitting can be used to define β and Tc.

By using this power law (Eq. 11), Tc = 1065 K is determined for pure bcc Fe in very good agreement with the experimental value of 1043 K [START_REF] Bozorth | Ferromagnetism[END_REF]. Although the bcc phase of Co does not exist in nature, the model predicts that Tc of this system is about 1077 K, which is lower than the Tc (1400 K) of the fcc Co system [START_REF] Bozorth | Ferromagnetism[END_REF].

For Fe-Co systems, Tc are obtained to be respectively 1316 K and 1461 K for the 50% Co A2 and B2 structures. Similar calculations were also performed for other Co concentrations. The results allow us to obtain the magnetic phase diagram of bcc Fe-Co alloys (Fig. 6). The experimental data extracted from Ref. [START_REF] Jacobs | Raising the Curie point in the iron-cobalt-(aluminum) system[END_REF] containing data from Bozorth [START_REF] Bozorth | Ferromagnetism[END_REF], Hansen [START_REF] Hansen | Constitution of Binary Alloys[END_REF] and Forrer [START_REF] Forrer | Le problème des deux points de Curie[END_REF] were also added for a comparison. However, please note that these experimental Curie temperatures may not come from direct measurements, but obtained by data extrapolations [START_REF] Forrer | Le problème des deux points de Curie[END_REF].

They roughly coincide with the bcc-fcc transition temperatures for a large range of Co concentrations. As can be seen, the Curie points obtained by our model via MC simulations are in good agreement with the experimental data, with the highest discrepancy not exceeding 100 K, which is within a reasonable margin of error of an effective model for magnetic transitions. The obtained result is also fully compatible with the equilibrium phase diagram, in which the bcc (B2 and A2) phases present a FM state while the fcc phase presents a PM state. Our predicted Tc is found to increase significantly, from around 1060 K to above 1220 K, in the range from 0 to 20% Co, then it increases to the maximum value of about 1340 K at 40% Co before decreasing.

Mixing enthalpies and magnetic moments at finite temperatures

Mixing enthalpy at finite temperatures is also examined. In Fig. 7, the mixing enthalpies of the disordered Fe-Co structures are presented for several temperatures: namely low (1 K), intermediate (700 K) and high (1000 K, 1500 K) temperatures. As can be seen, the mixing enthalpies decrease slightly when temperature increases from 1 K to 700 K. At higher temperatures, the variation is more visible as the mixing enthalpies drop faster when temperature varies from 700 K to 1000 K. Interestingly, the trend is inversed as the mixing enthalpies increase with temperature from 1000 K to 1500 K, in the paramagnetic regime. It is also worth noting that the mixing enthalpy curve presents a minimum around 40% Co at 1 K and this minimum slightly shifts to a lower concentration of Co when temperature increases. For all the studied temperatures, the mixing enthalpies remain negative.

In order to better understand these variations, in Fig. 8(a), the mixing enthalpy of several disordered (A2) and an ordered (B2) Fe-Co structures are shown as a function of the temperature.

The mixing enthalpies are found to decrease just slightly at low temperatures, below 700 K, and then drop more rapidly in the range from 700 K to 1100 K. Above 1100 K, the enthalpies increase back. As can be observed, all the curves reach a minimum at about 1100 K. In fact, this can be understood as at this point, the energy curves of Fe and Co systems change the curvature as can be seen from the lines with red and blue symbols in the inset of Fig. 8 (b). The change of total energy of each system (pure Fe, pure Co and Fe-50%Co A2) with respect to the respective 1 K value is shown in the main panel of Fig. 8(b). At low temperatures, the change of energy in all the three systems is almost the same, it leads to the almost unchanged mixing enthalpy as defined by Eq. ( 5). In the range from 700 K to 1100 K, the change in the energies of the pure Fe and pure

Co is faster than that in the Fe-Co system due to the strong magnetic disordering near the magnetic phase transition in the pure systems (1065 K for Fe and 1077 K for Co). As a result, the mixing enthalpy decreases (Fig. 8(a)). Above 1100 K, the energies of the pure Fe and Co systems tend to saturate, while the one of the Fe-Co grows up more rapidly as this system goes closer to its Curie point and thus the mixing enthalpy increases back. Similar results are observed in disordered Fe-Co alloys with other Co contents.

B2-A2 phase transition

In this section, the spin-atom MC simulations considering both spin variations and exchanges of atom positions are applied. Such treatment allows us to consider the interplay between the magnetic and the chemical order/disorder, and in particular the role of magnetism in the chemical phase transition.

Mixing tendency of Fe-Co systems

As revealed by experiments [START_REF] Wang | Mixing enthalpy of liquid Fe-Co-Ti alloys[END_REF] and from the negative mixing enthalpies predicted by DFT and the present model at 0 K (Fig. 2), Fe and Co atoms have a strong mixing and ordering tendency.

In order to provide a correct thermodynamic description, we first verify that the coupled spinatom MC simulations at low temperatures are able to capture these two features of Fe-Co systems.

To observe the mixing tendency via the MC simulations, one of the most convenient starting structures is the Fe-Co bi-phase system as illustrated in Fig. 9 (a). In the following subsections, the spin-atom MC simulations are applied to search for the equilibrium chemical and spin configurations for a given Co concentration at finite temperatures.

The chemical order-disorder transition can be determined by following the variation of chemical-LRO as a function of temperature.

B2-A2 phase transition within a fixed magnetic state

First, we examine the role of different magnetic states in the B2-A2 ordering phase transition.

The spin-atom MC simulations were performed with imposing a magnetic state: either the ferromagnetic, the paramagnetic or the nonmagnetic state. For all temperatures, such states were obtained by constraining the direction of the magnetic moments to be all parallel (FM) or fully random (PM). The nonmagnetic state was obtained by simply setting all the moments equal to zero. We focus on the case of Fe-50%Co.

Within the ferromagnetic (FM) state

In Fig. 10, the curve (1) was obtained by constraining the magnetic state at the FM one, which is the magnetic ground state. Within this state, the chemical phase transition temperature TB2-A2 is found to be 800 K. It is worth mentioning that the chemical-LRO curves are not well fitted to the power law (Eq. 11). Therefore, the TB2-A2 was determined using the inflection point of each curve.

This order-disorder transition temperature predicted by the model and MC simulations is found to be lower than the experimental one (about 1000 K) [START_REF] Massalski | Binary alloy phase diagrams[END_REF]. However, it is worth mentioning that this result is consistent with an estimation based on our DFT data, in which the B2-A2 transition point can be approximately determined by the crossing temperature of the Gibb free energies of formation of B2 and A2 structures with 50% of Co. The DFT results are shown as an inset in Fig. 10 and the crossover is found at 720 K. The DFT calculations were carried out within the perfect ferromagnetic state and the Gibbs free energy of formation is defined as

. f f f G H T S ∆ = ∆ -∆ , ( 12
)
where f H ∆ , f S ∆ are respectively the enthalpy and the entropy of formation. The latter includes the ideal configurational and the vibrational contributions. Although the crossover temperature is independent of the reference states used, the FM bcc Fe and hcp Co systems were adopted. The calculated vibrational entropies per atom at room temperature for bcc Fe, hcp Co, bcc Fe-Co B2 and A2 (50% Co) are 3.05 kB, 3.33 kB, 3.25 kB, and 3.31 kB, respectively which are in good agreement with experimental values of 3.18 kB (bcc Fe), 3.27 kB (hcp Co), 3.18 kB (B2) and 3.20 kB. (A2 50% Co) [START_REF] Lucas | Effects of chemical composition and B2 order on phonons in bcc Fe-Co alloys[END_REF]. Here kB is the Boltzmann constant.

Two main features may be at the origin of the rather small difference between the DFT and the model predicted transition temperature (80 K): (i) some chemical disorder (resp. order) can appear in the B2 (resp. A2) phase just below (resp. above) the transition temperature, and (ii) the presence of non-ideal configurational entropy. Both of them are not included in the DFT but in the model + MC determination. On the other side, we propose that the more significant underestimation of the B2-A2 transition temperature by both the DFT and the MC simulations with an imposed ferromagnetic state is mainly because some magnetic entropy, already emerged around 800-1000 K (Fig. 7), are not considered in these approaches.

Within the paramagnetic (PM) state

In Fig. 10, the curve (2) was obtained by constraining the magnetic state at a fully paramagnetic one. As shown, the chemical-LRO is found to be visibly lower than 1 even at very low temperatures (near 0 K). This indicates that the Fe-50%Co B2 phase is not stable within the paramagnetic state. By comparing the curves ( 2) and (1) near 0 K, it suggests that the ferromagnetic order stabilizes the B2 phase at low temperatures and it, in fact, stems from a lower magnetic energy of this phase compared to that of the A2 one at temperatures near 0 K as seen in Fig. 8(a).

Within the non-magnetic (NM) state

Another prediction from the model and MC simulations is given for the case of non-magnetic structures. In this case, only the non-magnetic part of the EIM plays a role in the chemical phase transition, that is, with magnetic energy and entropy are missing. The resulting transition temperature is around 1150 K for Fe-50%Co (curve (3)), which is higher than the one obtained by experiments (1000 K). This result is in qualitatively agreement with the previous study by Prasad et. al. [START_REF] Prasad | Journal of Magnetism and Magnetic Materials Study of the effect of magnetic ordering on order -disorder transitions in binary alloys[END_REF].

B2-A2 phase transition with spin equilibration

All the three curves (1), ( 2) and (3) in Fig. 10 reveal that the magnetic state and the magnetochemical coupling strongly influence the chemical transition. Thus, it is necessary to do a full equilibration of both chemical and spin degrees of freedom in order to accurately predict the transition temperature.

The (4)-green curve in Fig. 10 represents the obtained chemical-LRO. The transition temperature TB2-A2 is found at 1050 K for the Fe-50%Co alloy, in very good agreement with the experimental data of about 1004 K. To ensure that the obtained transition temperature does not depend on the starting atomic and spin configurations, we performed simulations from the B2 phase at a low temperature (1 K) towards high temperatures, and then from a perfectly random A2 phase at a very high temperature (1900 K) towards low temperatures (blue curve). The slight mismatch of the two curves in Fig. 10 is due to a numerical error.

It is worth mentioning that the present model and other interaction models do commonly not include either explicitly or implicitly the contribution of the lattice vibration. Nevertheless, our model still provides a satisfactory prediction of the transition temperature. This is because in Fe-Co systems, the vibrational entropies of the B2 and the A2 structures are found to be only slightly different. For instance, 3.25 kB, per atom for the B2 and 3.31 kB for the A2 phase, at 50% Co.

Thus, the effect of the lattice vibration can be negligible in the B2-A2 phase transition of the bcc Fe-Co structures.

By carrying out spin-atom equilibrations for other Co concentrations, the transition temperature is also determined as a function of the Co concentration and compared to experiment data [START_REF] Massalski | Binary alloy phase diagrams[END_REF]. An excellent agreement can be seen in Fig. 11. Together with the predicted Curie temperatures shown in Fig. 6, the present model, although rather simple, is able to predict simultaneously very satisfactory magnetic and chemical phase diagrams for a broad range of temperature and the alloy concentration.

Magnetism with coupled spin-atom equilibration

Fig. 12 (a) presents the magnetization with a fully coupled spin-atom equilibration as a function of temperature (blue line). The magnetization curves of the fixed 50%-Co B2 and A2 structures are also shown for comparison. As can be seen from the blue curve in Fig. 10 (a), at the chemical transition temperature TB2-A2 = 1050 K, the magnetization is still significantly large. Indeed, at this temperature, Both B2 and A2 structures are still in the ferromagnetic state. As expected, at low temperatures, the blue curve almost coincides with the B2 one while above the order-disorder transition temperature, this curve is well identical to the one of the A2 structure. Interestingly, the second crossover of the magnetization curves of the B2 and A2 phases (see Fig. 5) is identical to the chemical transition temperature obtained from the full spin-atom equilibration.

Mag-SRO is also examined to understand local correlations between magnetic moments. In Fig.

12 (b), the mag-SRO parameters of the Fe-Fe, Fe-Co and Co-Co pairs for the 1 st and 2 nd neighboring shells are shown. They are all positive (ferromagnetic interaction) for the temperatures considered, even beyond the Curie temperature. The 1 st shell Fe-Fe interactions for all the temperatures and the 2 nd shell Co-Co interactions below the B2-A2 transition present the highest values. Interestingly, at the chemical transition temperature, all the mag-SRO curves present either a kink or a change of curvature. In addition, the behavior of the 4 th shell is similar to the one of the 1 st shell, while the behaviour of the 3 rd and 5 th shells are similar to that of the 2 nd shell, for this and all other Co concentrations studied.

The effect of an external magnetic field on phase transitions

An external magnetic field is widely used as a versatile excitation to tune magnetic properties of materials [69,70]. In this section, the magnetic field is introduced into the EIM to investigate the impact of the field on the magnetic and chemical phase transitions of Fe-Co systems.

In the presence of a magnetic field, the Hamiltonian (Eq. 1) is modified by adding the Zeeman term as

' Zeeman H H H = + , (13)
with

Zeeman B i i H g B M µ = - ∑ ur uuu r , ( 14
)
in which g is the Landé factor and takes the value of 2.0 [START_REF] Vernyhora | Monte Carlo investigation of the correlation between magnetic and chemical ordering in NiFe alloys[END_REF]. Here we consider the case of a constant field.

4.3.1

The effect around the zero-field magnetic transition at a given atomic configuration First, we consider the effect of the magnetic field around the Curie temperatures of pure bcc Fe and disordered Fe-Co with 50% Co. It is noted that Chui et al. [START_REF] Chui | Investigating the magnetovolume effect in isotropic bodycentered-cubic iron using spin-lattice dynamics simulations[END_REF] have examined this effect for pure bcc Fe using spin-lattice dynamics. In Fig. 13 (a), the magnetization of the pure Fe system under different constant magnetic fields is shown. It is clearly observed that the magnetic field increases the magnetization at all finite temperatures and the effect is stronger at high temperatures, particularly just above the zero-field Curie point. As expected, a stronger field induces a higher magnetization at the same temperature. Such behaviors are consistent with the fact that magnetic fields tend to align the magnetic moments to the direction of the field, thus, they impact more strongly on magnetically disordered than on ferromagnetic states. This observation is similar to what obtained in the spin-lattice dynamics study [START_REF] Chui | Investigating the magnetovolume effect in isotropic bodycentered-cubic iron using spin-lattice dynamics simulations[END_REF]. We also find a similar effect on the A2 (50% Co) Fe-Co structure as displayed in Fig. 13 (b). The enhancedferromagnetism is also seen from the magnetic SRO parameters. Fig. 13 (c) displays the firstneighbor mag-SRO for Fe-Fe, Fe-Co and Co-Co interactions in the Fe-Co A2 structure. It reveals that the mag-SRO is enhanced at all temperatures and the most remarkable effect is observed around the Curie point (1316 K with zero field). The same behavior is also seen for farther neighbor shells and in pure bcc Fe.

The field-induced enhancement of magnetization is found to be almost linear with respect to the increase of the field strength as shown in Fig. 13(d). Interestingly, it is found at all temperatures that with the same field strength, the average moment of the Fe atoms in the Fe-Co system increases more strongly than that of the Co atoms in the same systems, as can be observed from the lines with square (Fe) and triangle (Co) symbols in Fig. 13 (d). This result may be associated to the electronic properties of Fe and Co atoms in pure and Fe-Co systems where Co has an essentially saturated magnetism, while Fe can still enhance its magnetism in the presence of excitations, by for example, shifting down the highest peak in the majority-spin states of Fe atoms towards the Fermi level in Fe-Co structures [START_REF] Tran | Predicting magnetization of ferromagnetic binary Fe alloys from chemical short range order[END_REF].

It is also worth noting that, below the Curie temperature of pure Fe (e.g. at 1000 K), Fe moments in pure Fe increase faster than those in the Fe-Co systems with increasing field strength, as seen in Fig. 13 (d), from the dot black line with the filled diamond symbols. This feature can be understood as the field induces a stronger impact on the more disorderly magnetic system and it is also consistent with the magnetic disordering in pure Fe and Fe-Co observed in Figs. 13 (a), (b) and (c). In contrast, at high temperatures near the Curie point of the Fe-Co systems, e.g. at 1500

K the enhancement in the moment of the Fe atoms in pure bcc Fe is slower than that of the counterpart in the Fe-Co systems, which is reflected by the solid black line with the filled diamond symbols in Fig. 13 (d).

The effect on the B2-A2 chemical transition temperature

As shown above in Fig. 10, magnetism plays a vital role in the order-disorder transition temperature TB2-A2. While a magnetic field has a significant influence on the magnetic order of the structure. Therefore, it is worth investigating the effect of magnetic fields on the chemical phase transition in Fe-Co alloys.

To this end, we performed MC simulations for Fe-50%Co systems, starting with the B2 structure at low temperatures. Both the spin and chemical configurations are equilibrated at each temperature in the presence of a magnetic field. The resulting chemical long-range-order evolution is presented in Fig. 14. Interestingly, the chemical transition is almost unaltered by the considered fields. This observation is similar to the finding in Fe-Ni alloys subjected to a magnetic field [START_REF] Vernyhora | Monte Carlo investigation of the correlation between magnetic and chemical ordering in NiFe alloys[END_REF]. In fact, this result can be understood as the order-disorder transition temperature (1050 K) is sufficiently lower than the Curie point, where the effect of the magnetic fields on the magnetization is weak (Fig. 13 (b)).

Conclusion

We have proposed an ab-initio based effective interaction model for the study of magnetism, thermodynamics, and their interplay in body-centered cubic Fe-Co alloys, with Co content from 0 to 70%. This model consists in a pairwise magnetic and a chemical-interaction term. For the former, a Heisenberg formalism is adopted. But, we go beyond the conventional formalism to account for the variation of the spin magnitude of each Fe atom as a function of its local chemical environment, as predicted by DFT.

The current model has successfully reproduced the energetic landscape and the ground-state magnetic moments of both ordered (B2) and disordered (A2) Fe-Co systems, as obtained by DFT. Moreover, in combination with atomic Monte Carlo simulations, the obtained Curie and chemical B2-A2 phase transition temperatures were found to be in good agreement with available experimental data for a wide range of concentration (0 to 70% Co). A strong dependency of the chemical transition temperature on the magnetic configuration is evidenced. In particular, the stability of the B2 phase at low and intermediate temperatures are clearly favored by the ferromagnetism. In addition, we pointed out a much more important effect of magnetic rather than vibrational entropy on the chemical phase transition temperatures. The above-mentioned findings attest the ability of this model to predict diverse magneto-energetic properties in stable and metastable bcc Fe-Co structures at any given temperature, which is useful to compare with and to interpret experimental results. Furthermore, the present model serves as a basis of a kinetic model which includes the presence of vacancies, in order to predict Fe and Co diffusion coefficients as a function of temperature and Co concentration via the vacancy mechanism. The approach is proved to be very promising in dilute Fe-Cu systems in a previous study [71]. Please note that the significant magnitude increases of Fe magnetic moments, next to a vacancy, can be easily captured by the present Heisenberg-like magnetic model.

In addition, we found a significant effect of an external magnetic field to enhance the magnetization and the magnetic short-range orders, even below the Curie point, But, the B2-A2 transition temperatures are unaltered with fields with a commonly accessible strength. This finding is similar to the case of order-disorder transition in fcc Fe-Ni alloys.

  in which FeCo ε is the energy per atom of the Fe-Co structure and , Co Fe ε ε are the energies per atom of the pure bcc Co, Fe systems, respectively. Co X and Fe X are the atomic fractions of Co and Fe in the Fe-Co structure.

α

  is the parameter calculated for the n-th neighboring shell of a B atom, atoms in the n-th neighboring shell of the B atom. n Z is the total number of atoms in the n-th shell. In bcc structures,1 8 Z = , 2 6 Z = and so on. B X is the global concentration of the B atoms in the alloy. The chem-SRO parameter (on the B atoms) is defined as the average of the individual i n B

Fig. 9 (

 9 b) is a snapshot of the atomic configuration after 400 atom exchange steps and Fig.9(c) is the one obtained after 5400 atom exchange steps where Fe and Co atoms are clearly observed to be fully mixed and the disordered A2 phase is formed. After 205400 atom exchange steps (Fig.9(d)), a highly ordered structure is observed with chem-SRO equal to -0.8024 and 0.8013 for the first and second neighboring shells, respectively which are close to those of the B2 structure (-1 and 1). Thus, the coupled spin-atom equilibration via MC simulations are confirmed to capture correctly the mixing and ordering tendency in Fe-Co structures.

[ 69 ] 7 [

 697 Alzate-Cardona J D, Barco-Ríos H and Restrepo-Parra E 2018 Dynamic phase transitions in La2/3 Ca1/3 MnO3 manganites: A Monte Carlo simulation study Phys. Lett. A 382 792-70] Shen S Q and Zhang F C 2002 Antiferromagnetic Heisenberg model on an anisotropic triangular lattice in the presence of a magnetic field Phys. Rev. B -Condens. Matter Mater. Phys. 66 1-4 [71] Schneider A, Fu C C, Soisson F and Barreteau C 2020 Atomic diffusion in alpha-iron across the Curie point: an effcient and transferable ab-initio-based modelling approach Phys. Rev. Lett.

Fig. 1 :Fig. 2 :

 12 Fig.1: Energy difference between two systems with different magnetic configurations but the same atomic structure: comparison between DFT and EIM results. For each system, the EIM adopted the DFT predicted magnetic moments.

Fig. 3 :Fig. 4 :

 34 Fig. 3: Magnetization of disordered structures obtained by the model (EIM) + MC simulations at 1 K, and compared with the DFT data.

Fig. 5 :

 5 Fig. 5: Magnetization as a function of temperature for different systems: pure bcc Fe (black), pure bcc Co (blue), bcc Fe-50%Co A2 (green) and B2 (orange).

Fig. 6 :

 6 Fig. 6: Curie temperature of the A2 phase as a function of Co concentration: the EIM + MC simulations versus experimental data [64].

Fig. 7 :Fig. 8 :Fig. 9 :Fig. 10 :

 78910 Fig. 7: Mixing enthalpy of the A2 phase as a function of concentration predicted for several temperatures by the EIM + MC simulations.

Fig. 11 :

 11 Fig. 11: B2-A2 transition temperatures obtained for different Co concentrations: the EIM + MC simulations versus experimental data [3].

Acknowledgments

This work was supported by the French-German ANR-DFG MAGIKID project. The DFT calculations were performed using the DARI-GENCI CPU resources within the A0050906020 project, and the CINECA-MARCONI supercomputer within the SISTEEL project.


Abdelali Zaki
email: abdelali.zaki@umontpellier.fr

Andrea Gutierrez
email: andrea.gutierrezrojas@dlr.de

Matthias Schmidt

Daniel Bielsa

Marc Linder

Abdessamad Faik

Investigation of Ca12Al14O33 Mayenite for hydration/dehydration thermochemical energy storage

Keywords: Thermochemical heat storage, Gas-solid reaction, Hydration/dehydration reaction, Calcium aluminates, Mayenite

Thermal energy storage using a reversible chemical reaction is a key parameter for increasing the storage capacity especially for medium and high temperature applications.

The reversible hydration/dehydration reaction of calcium hydroxide Ca(OH)2 to calcium oxide CaO is among the most investigated system due to its reaction cyclability and interesting operating temperature above 450 ºC suitable for high temperature applications.

However, the indicated reaction temperature was considered too high for some industrial applications which limit a further development of thermal energy storage technologies using this promising reaction system. In this regard, the material development is a key point to broaden the application of sustainable technologies. In this work, the investigation of Mayenite with general formula Ca12Al14O33 as a thermochemical energy storage material was conducted. Thereby, it was confirmed the possibility of the pair "Ca12Al14O12(OH)42"/Ca12Al14O33 to store and release the heat through chemical reaction thanks to a novel activation process of the pristine Mayenite that is presented for the first time in this work. This study might help to identify new pairs based on materials that have already been classified as inactive for the hydration/dehydration reaction. In addition, it was

Introduction

The use of renewable energy sources nowadays is vital in order to decrease the consumption of fossil resources and to decrease the carbon dioxide emissions partly responsible for the greenhouse effect. Energy storage is a promising solution which allows adjusting the time-discrepancy between supply and demand of energy. It is mainly applied for the utilization of intermittent energy sources, such as solar energy, variable energy load, and excessive energy uses in industrial process that would be wasted. Energy can be stored in different ways, such as mechanical energy, electric energy, chemical energy and thermal energy, where the last one is potentially capable of improving the efficiency of the concentrate solar power (CSP) plants and upgrade of the recovered low grade heat in industrial processes, among other applications [START_REF] Yan | A review of promising candidate reactions for chemical heat storage[END_REF].

There are three well known methods for thermal energy storage (TES), sensible, latent and chemical. The sensible and latent TES are the most developed technologies, where the first system stores the heat by a temperature change of the storage material; while the second system stores the heat by a phase change of the storage material. However, the storage density depends, in the first case, on the mass, the specific heat capacity of the storage medium and the applied temperature change [START_REF] Dincer | Thermal energy storage. Systems and applications[END_REF]. The least developed TES technology is the thermochemical storage, which is based on the mass, the temperature range and the enthalpy of the reversible chemical reaction. This technology presents the highest theoretical energy density compared to that of sensible and latent heat storages [START_REF] Afflerbach | Semipermeable encapsulation of calcium hydroxide for thermochemical heat storage solutions[END_REF].

In this perspective, thermochemical storage is considered as a very promising technology not only due to its higher theoretical storage energy, but also for the possibility of a longterm storage applications, as the reaction products can be stored at room temperature without any significant energy losses [START_REF] Tescari | Thermochemical solar energy storage via redox oxides: Materials and reactor/heat exchanger concepts[END_REF][START_REF] Cot-Gores | Thermochemical energy storage and conversion: A-state-of-the-art review of the experimental research under practical conditions[END_REF]. In the literature, different types of reversible reactions have been considered where the solid-gas reactions were extensively investigated due to some extra advantages such as easy separation of the reactants and, the dependence between the equilibrium temperature and gas pressure, which allows adjusting the reaction temperature of the system. For this purpose, many types of solid-gas reactions have been studied such as reduction/oxidation [START_REF] Karagiannakis | Monolithic Ceramic Redox Materials for Thermochemical Heat Storage Applications in CSP Plants[END_REF][START_REF] Carrillo | Thermochemical heat storage based on the Mn 2 O 3 /Mn 3 O 4 redox couple: influence of the initial particle size on the morphological evolution and cyclability[END_REF][START_REF] Carrillo | Thermochemical heat storage at high temperatures using Mn2O3/Mn3O4 system: Narrowing the redox hysteresis by metal co-doping[END_REF], decarbonation/carbonation [START_REF] André | Evaluation and performances comparison of calcium, strontium and barium carbonates during calcination/carbonation reactions for solar thermochemical energy storage[END_REF][START_REF] Bagherisereshki | Investigation into SrO/SrCO3 for high temperature thermochemical energy storage[END_REF] and hydration/dehydration [START_REF] Sakellariou | Calcium oxide based materials for thermochemical heat storage in concentrated solar power plants[END_REF][START_REF] Schaube | De-and rehydration of Ca(OH)2 in a reactor with direct heat transfer for thermo-chemical heat storage. Part A: Experimental results[END_REF][START_REF] Schaube | De-and rehydration of Ca(OH)2 in a reactor with direct heat transfer for thermo-chemical heat storage. Part B: Validation of model[END_REF]. Among these systems, hydration/dehydration systems present the most interesting temperature level that is suitable for high temperature TES applications such as those in the current CSP technologies and some industrial processes [START_REF] Pardo | A review on high temperature thermochemical heat energy storage[END_REF][START_REF] Tescari | Thermochemical solar energy storage via redox oxides: Materials and reactor/heat exchanger concepts[END_REF][START_REF] Alonso | First experimental studies of solar redox reactions of copper oxides for thermochemical energy storage[END_REF][START_REF] Azpiazu | Heat recovery from a thermal energy storage based on the Ca(OH)2/CaO cycle[END_REF]. In this sense, a suitable material is needed that is able not only to run the endothermic dehydration reaction, by absorbing heat when is available, but also to conduct the reversible exothermic hydration reaction by fully releasing absorbed heat when is needed. The most widely investigated system is the calcium hydroxide/calcium oxide Ca(OH)2/CaO pair following the reversible hydration/dehydration reaction (eq. 1). Ca(OH)2 + 104.4 kJ/mol CaO + H2O (eq. 1)

This system has been selected as the best candidate for its suitable equilibrium temperature (450 ºC -500 ºC), the low cost and non-toxicity of the materials, relatively high energy density (450 kWh/m 3 ), fast kinetics, reversibility and cycling stability [START_REF] Schaube | De-and rehydration of Ca(OH)2 in a reactor with direct heat transfer for thermo-chemical heat storage. Part A: Experimental results[END_REF][START_REF] Schaube | A thermodynamic and kinetic study of the de-and rehydration of Ca(OH) 2 at high H 2O partial pressures for thermo-chemical heat storage[END_REF]. Most of the studies have been focused on explaining the reaction mechanisms and to evaluate the impact of the physical bulk properties of the material on the operation characteristics of the storage systems [START_REF] Sakellariou | Calcium oxide based materials for thermochemical heat storage in concentrated solar power plants[END_REF]. Those works were focused on studying the reaction from a thermodynamic perspective by using small sample masses in thermogravimetric apparatus and the development of simulation models to understand the behavior of the materials [START_REF] Nagel | The influence of gas-solid reaction kinetics in models of thermochemical heat storage under monotonic and cyclic loading[END_REF][START_REF] Schmidt | Power generation based on the Ca(OH)2/ CaO thermochemical storage system -Experimental investigation of discharge operation modes in lab scale and corresponding conceptual process design[END_REF]. Other research activities aimed to enhance the reaction rate by improving the flowability of the material by the addition of nanoparticles as spacers or by the impregnation of calcium hydroxide into inert materials forming robust porous matrices [START_REF] Roßkopf | Improving powder bed properties for thermochemical storage by adding nanoparticles[END_REF]. In addition, some research groups were devoted on mechanical cycling stability studies of the materials by means of encapsulation of the storage material (calcium hydroxide) in a permeable shell. [START_REF] Afflerbach | Semipermeable encapsulation of calcium hydroxide for thermochemical heat storage solutions[END_REF] Recently, a step forward had been made in order to investigate the behavior of the system in large scale in fixed bed configuration [START_REF] Schmidt | Experimental results of a 10 kW high temperature thermochemical storage reactor based on calcium hydroxide[END_REF] and moving bed concept [START_REF] Schmidt | Power generation based on the Ca(OH)2/ CaO thermochemical storage system -Experimental investigation of discharge operation modes in lab scale and corresponding conceptual process design[END_REF]. Despite those efforts the technology is still under development and in an early research state.

One major drawback revealed by the experimental investigations is, that even when the system is operated at very low vapor pressures (e. g. 10 kPa), a minimum charging temperature of 450 °C is required in order to achieve technically relevant charging powers.

The high charging temperature limits the efficient application of the storage system into the process. The main advantage of the new reaction pair proposed here is the possibility to adjust the reaction temperature. This offers the feasibility to apply it to other applications than power tower CSP technologies that are operated at lower heat transfer fluid and energy storage medium temperatures such as the parabolic troughs and linear Fresnel concepts or even to thermal storage applications in e.g. process industry. Hence, a lower dehydration temperature of the reaction would greatly enhance the efficient process integration of the storage system for various applications [START_REF] Schmidt | Experimental results of a 10 kW high temperature thermochemical storage reactor based on calcium hydroxide[END_REF][START_REF] Schmidt | Thermochemical energy storage with CaO/Ca(OH)2-Experimental investigation of the thermal capability at low vapor pressures in a lab scale reactor[END_REF]. Therefore, more efforts are required in order to develop new materials with equilibrium temperatures adapted to the final applications. In this regard, the investigation of new materials of magnesium hydroxideporous matrix composites was conducted aiming to overcome the aforementioned limitations [START_REF] Shkatulov | Composite material " Mg(OH)2/vermiculite": A promising new candidate for storage of middle temperature heat[END_REF].

Mayenite has been widely used as a major constituent of calcium aluminate cements. Furthermore, this material has attracted the attention of the scientific and technological communities to be used as a transparent conductive oxide, as a catalyst for the combustion of volatile organic compounds or as an ionic conductor [START_REF] Lee | Defect chemistry of the cage compound, Ca12Al14O33-δ-understanding the route from a solid electrolyte to a semiconductor and electride[END_REF]; but, to the best of our knowledge, this is the first work dealing with the valorization of that material for thermochemical energy storage applications. The preparation of the pure Ca12Al14O33 material was obtained by Evaporation-Induced Self-Assembly (EISA) method. The crystallographic structure of the material and the size and morphology of the particles were investigated by means of X-ray powder diffraction (XRD) and scanning electron microscopy (SEM), respectively. The simultaneous thermal analysis (STA) was used to investigate the thermophysical characteristics, the equilibrium reaction temperatures, conversion rate and the re-hydration of the material.

Techniques and Materials

Material synthesis

The Evaporation-Induced Self-Assembly (EISA) method [START_REF] Brinker | Evaporation-Induced Self-Assembly: Nanostructures Made Easy[END_REF] was used for the synthesis of mixed calcium/aluminum oxide with the general formula Ca12Al14O33. The aluminum isopropoxide C9H21O3Al (98%) and calcium nitrate tetrahydrate Ca (NO3)2.4H2O (98%) are used, as received from Sigma-Aldrich, as aluminum and calcium precursors, respectively.

Typically, for 3 g of Ca12Al14O33 material preparation, 6.12 g of C9H21O3Al is dissolved in a pre-mixed solution of 60 mL of ethanol and 4.5 mL HNO3 (67 wt. %) by vigorous stirring for 1 hour. Then, to this homogenous solution, 6.072 g of Ca (NO3)2.4H2O were added and stirred continuously for another 5 hours. Then, the mixture was dried at the oven at 80°C during 1 day in air under static condition. To study the effect of the calcination temperature on the material crystallinity and particle size, four samples were calcined successively at 600, 800, 1000 and 1100 °C, with a heating/cooling rate of 1 ºC/min and a dwell for 10 hours. The thermodynamic behavior of the four samples was investigated in order to investigate the effect of the crystallinity and particle size on the hydration and dehydration processes.

In order to facilitate the understanding of the results the Mayenite samples were identified as shown in Table 1 and the details about the activation process will be explained later in section 3.

Activation process

The samples prepared at 600, 800, 1000 and 1100 ºC were initially hydrated using a hydrothermal set-up. However, 100 mg of dry powders were introduced in alumina crucibles, which, then each crucible was introduced in a bottle of Teflon of 250 mL filled with 15 mL of distilled water. The systems were hermetically closed and introduced in a furnace at 170 ºC for 30 minutes.

Structural and morphology characterizations

The X-ray powder diffraction (XRD) and scanning electron microscopy (SEM) were conducted to determine the structural characteristics and the particle size of the samples prepared at 600, 800, 1000 and 1100 ºC.

Bruker D8 Advance diffractometer equipped with a LYNXEYE detector using CuKα1 radiation (λ= 1.5418 Å) and θ-2θ geometry was used for XRD analysis. The data were collected at room temperature between 10° and 80° with a step size of 0.02° and a counting time of 8 seconds per step. The Rietveld refinement of the crystallographic structure was performed using the WinPlotr/FullProf package [START_REF] Rodríguez-Carvajal | Recent advances in magnetic structure determination by neutron powder diffraction[END_REF]. The peak shape was described by a pseudo-Voigt function, and the background level was modeled using a polynomial function. The refined parameters were: background coefficients, scale factor, lattice constants, atomic positions, isotropic independent atomic displacement parameters, zero shift, peak profile and asymmetry parameters.

The samples were imaged by means of a SEM Quanta 200 FEG operated in high vacuum mode at 30 kV featured with a secondary electron detector (ETD). To perform the SEM analysis, small amount of the powder was placed on a graphite holder using double-sided carbon tape.

Thermal analysis

Thermogravimetric analysis studies were carried out by using simultaneous thermal analysis (STA409, Netzsch). The measurements were carried out on about 20 mg of the samples placed in an alumina crucible with a volume of 100 µL at a constant heating/cooling rate of 2 K/min. The reversibility of dehydration/hydration reactions and the thermodynamic properties, namely temperatures of dehydration/hydration and cycle stability, were studied using a simultaneous thermal analyzer (STA-449C Jupiter®, Netzsch). The set-up was equipped with a water vapor generator (aDrop, Bronkhort®) and with a water vapor furnace. A thermogravimetric (TG) sample carrier with a thermocouple Type S and an accuracy of ± 1 K was used. Sample sizes of ~6 mg were measured in Al2O3 crucible (3.4 mL). In addition, a thermogravimetric-differential scanning calorimetric (TG-DSC) sample carrier with a thermocouple Type K and an accuracy of ± 1 K was used. With these results the relative enthalpy of dehydration was calculated for each of the materials using Ca(OH)2 as a reference material. Sample sizes of about ~6 mg were measured in open platinum crucibles (95 μL). The accuracy of the thermo-balance was ± 0.1 µg. The absolute pressure inside the furnace was equal to the atmospheric pressure (96 kPa). Nitrogen was used as protective and purges gas with a volume flow for both of 50 N-mL/min. The atmosphere surrounding the sample was kept inert using 100 N-mL/min of nitrogen flow and water vapor. Either pure nitrogen was used or a mix of nitrogen and water vapor in order to obtain different partial pressures of water vapor. Partial vapor pressures of 68 kPa or 96 kPa ± 1 kPa were set for the samples rehydration. Dynamic experiments were set to study the reversibility of reactions, with heating and cooling rates of 5 K/min and -5 K/min, respectively. The relative enthalpies of dehydration were determined performing also dynamic experiments, with heating rates of 1 K/min. The cycle stability was evaluated at isothermal conditions defined as a function of the temperatures of hydration determined in the previous experiments. Ten cycles of hydration and dehydration were performed alternating inert and humid (pH2O = 68 kPa) atmosphere.

Results and Discussion

Characterization of the samples.

The final synthesis temperature is a key parameter for obtaining pure samples with different levels of crystallinity and particle size. Figure .1 presents the XRD diffractograms obtained at room temperature for Ca12Al14O33 samples calcined at 600, 800, 1000 and 1100 ºC. The samples prepared at 600 and 800 ºC show amorphous nature, where almost no peaks were observed; while, the sample treated at 1000 ºC shows almost pure crystalline phase corresponds to Mayenite structure with the presence of small peaks at around 28º, 30º and 31º corresponding to unidentified impurities. The sample treated at 1100 ºC demonstrates a good crystallinity without the presence of any extra peaks related to some impurities. In order to check the sample purity and the crystallographic information of this material, a

Rietveld refinement was conducted using as a starting model the published Mayenite structure [START_REF] Palacios | Crystal Structures and in-Situ Formation Study of Mayenite Electrides[END_REF]. The refinement results are presented in Figure 2, where a good agreement between the experimental and calculated XRD patterns was obtained, which demonstrates the purity of the sample treated at 1100 ºC. The obtained structure of Mayenite, Ca12Al14O33, has a cubic crystal system with the space group I-43d and a lattice parameter constant of 11.982(1) Å, containing two molecules per unit (1720.24(2) Å 3 ). Figure 3a illustrates the Mayenite structure that can be represented as a tri-dimensional network of 12 sub-nanocages, where, each cage is elongated and contains two crystallographic positions for the oxide ion and the two so-called extra-framework oxide ions are randomly distributed in those cages [START_REF] Lee | Defect chemistry of the cage compound, Ca12Al14O33-δ-understanding the route from a solid electrolyte to a semiconductor and electride[END_REF][START_REF] Hayashi | Microporous Crystal 12CaO ' 7Al 2 O 3[END_REF][START_REF] Medvedeva | Hopping versus bulk conductivity in transparent oxides : 12 Ca O • 7 Al 2 O 3 Hopping versus bulk conductivity in transparent oxides : 12CaO[END_REF]. The effective diameter of nanocages is ranging between 0.44 nm and 0.56 nm depends on the oxide ion contained inside [START_REF] Lee | Defect chemistry of the cage compound, Ca12Al14O33-δ-understanding the route from a solid electrolyte to a semiconductor and electride[END_REF][START_REF] Boysen | Anion Diffusion Processes in O-and N-Mayenite Investigated by[END_REF]. The extra-framework oxide ions can be replaced by monovalent species, such as OH -group [START_REF] Kim | Metallic State in a Lime -Alumina Compound with Nanoporous Structure[END_REF]. There is a main difference between the Mayenite which possesses an oxide ion inside the nanocages and the Zeolite structures is the charge polarity of the lattice framework; cationic species are introduced to compensate for the negative charge caused by the substitution of Al 3+ ion by Si 4+ ion in the Zeolites, which, in this case, does not allow to the oxide ions in the cages to react with water molecules. The Zeolite structure counts with two big cages, α-cage and β-cage with respectively internal diameter of 1.1 nm and 0.8 nm, which, in this case, allows to Zeolite to adsorb a big quantity of water molecules [START_REF] Wang | Adsorption Equilibrium of Binary Mixtures of Carbon Dioxide and Water Vapor on Zeolites 5A and 13X[END_REF]. This difference leads to an hydration thermodynamics for Mayenite and not to water uptake compared to conventional Zeolite structure.

The hydration reaction can be described as follow:

Ca12Al14O33 + x H2O ⇌ Ca12Al14O33-x(OH)2x + ΔH (eq.2) The particle size and morphology of the oxide materials obtained at 600, 800, 1000 and 1100 ºC were investigated by SEM analysis. Figure 4 shows the obtained SEM micrographs where a notable increase of the particles size (~ 1.5 µm) was obtained for the sample treated at 1100 ºC, while a similar small size (less than 200 nm) was obtained for the rest of the samples. For the material prepared at 600 ºC, we observe two types of particles, rods and spherical shape that can be explained by the non-sufficient reaction temperature where the reaction was carried out in order to form the crystalized mixed oxide of Mayenite. The particles morphology becomes more homogeneous for the materials treated at 800, 1000 and 1100 ºC, where it was observed that the mixed phase becomes dominant for both samples.

Thermal analysis

Activation of the materials

The hydration/dehydration analysis shows that the hydration process of the pristine materials prepared at 1000 and1100 ºC is very hard to be conducted at low partial pressures of water vapor (96 kPa) as shown in Figure 5. In the case of the pristine material prepared at 1100°C, the sample showed about 2 wt% of chemical reversibility during the first dehydration. However, in the following cycles, this reversibility is slightly higher; reaching up to 2.2 wt% during the last hydration. A similar tendency is observed for the pristine material prepared at 1000°C. Nevertheless, the mass difference due to the hydration and dehydration of M-1000-NA was more significant from the first cycle in comparison with the M-1100-NA. In the first cycle a chemical reversibility close to 3 wt% was observed.

While in the last cycle the chemical reversibility is more than 4.5 wt%, even exceeding the initial mass which corresponds to 100 wt% sample mass. Therefore, and based on the comparison of different analyses it was demonstrated that the hydration process becomes harder when the crystallinity of the material increases. In addition, in the case of the material calcined at 1100 ºC, due to the big size of the particles which refers to the high crystallinity of the sample as confirmed by XRD pattern shown in Figures 1 and5, the incorporation of the water molecule becomes more difficult in order to convert the oxide to hydroxide form. For that reason, we carried out an ex-situ hydration under severe conditions aiming to activate the materials.

Thereafter, the high pressure of water vapor created by a hydrothermal set-up, forces the hydration process for the samples prepared at 600, 800 and 1000 ºC and generates a hydration/dehydration reversible behavior for these materials as it was confirmed by thermogravimetric analyses.

The activated samples showed a reversible behavior of hydration/dehydration under operating temperatures between 125°C and 300°C (Figure 6). Which are considerably lower than calcium oxide (CaO), for which theoretical hydration temperatures close to 250°C (pH2O < 2 kPa) and dehydration temperatures close to 380°C (pH2O < 1 kPa) have been previously reported [START_REF] Schaube | A thermodynamic and kinetic study of the de-and rehydration of Ca(OH) 2 at high H 2O partial pressures for thermo-chemical heat storage[END_REF]. In addition, the sample prepared at 600°C (M-600-A) showed a continuous mass increase during the isotherm exceeding the initial mass (at t=0 min); this could be a consequence of the low temperature of preparation that does not allow the formation of the mixed structured (this was previously mentioned in section 3.1).

Therefore, after the first dehydration the anhydrous phase undergoes not only the reaction of (re)hydration but seems to continuously adsorb water on the surface of the powder sample. In contrast, the other two samples, M-800-A and M-1000-A, reached a constant mass increase during the isotherm. This demonstrates that the reaction has reached equilibrium, and hence, that no higher chemical conversion can be reach under these operatin9g conditions (125°C and pH2O=96 kPa). However, this new finding might remove huge barriers in order to expand the thermochemical energy storage systems with water vapor to new mixed materials which cannot be considered for that purpose due to their initial inability to react with water molecules (inactive materials). This pre-hydration process is considered as an activation process for the materials to facilitate their further hydration/dehydration cycling.

On the other hand, unsuccessful activation process was obtained for the material prepared at 1100 ºC due to the big size particles and the high crystallinity as it is shown in Figure 7 in which, the material prepared at temperature of 1000 ºC exhibits a change of the particle morphology generating hexagonal shape that confirms the formation of the hydrated form.

Thermodynamic study

The relative enthalpies of dehydration were calculated by graphical integration of the DSC curves for the active samples, taking the well-known ∆RHCa(OH)2 [J/g] as a reference and the results obtained are showed in Table 2. It can be seen that the sample with the highest relative enthalpy of dehydration is the sample prepared at 1000°C (M-1000-A). This is in good agreement with the TG-Results showed in Figure 6 where it can be seen that the Sample M-1000-A is the one with the higher ∆mD (~27 wt%), hence a greater amount of water is involved in the dehydration reaction. (eq. 3)

It is important to mention that the hydrated product of the Mayenite material is amorphous where its XRD Diffractogram shows no peak. However, as no structural refinement of this material is possible, its general formula "Ca12Al14O12 (OH)42" is introduced between quotation marks. Figure 8 shows the XRD diffractograms obtained for the M-1000-A after the 1 st and the 10 th hydration/dehydration cycles. As it can be observed, both materials show patterns similar to the pristine one which confirms the cyclability of the material where the Mayenite structure was maintained at least after 10 hydration/dehydration cycles.

On the other side, the broadening and intensity of the peaks clearly indicate the decrease of the crystallinity of the material when the number of the cycles increases. The decrease in crystallinity also leads to a decrease in chemical conversion over the cycles, as can be seen in the TG signals shown in Figure 9. Therefore, the energy released due to the hydration of the solid phase also lowers over the cycles (See Figure 9, DSC Signals). However, in this work the reaction reversibility was confirmed, while the mitigation of reaction degradation with the cycles is ongoing in our laboratories following two approaches where the first one aims the reactivation of the material after a certain number of cycles and the second one is based on investigating the stabilization of the crystalline structure with some dopants.

Conclusions

The preparation of Ca12Al14O33 Mayenite structure was successfully performed by using the Evaporation-Induced Self-Assembly (EISA) method. Four oxide samples, i.e. M-600-NA, M800-NA, M-1000-NA and M1100-NA, were obtained by the calcination of the primary reactants at 600, 800, 1000 and 1100 ºC. XRD and SEM analyses reveal that the M-600-NA and M800-NA show an amorphous nature without the formation of the mixed structure for the first sample. The samples M-1000-NA and M1100-NA show a good crystallinity with the formation of Mayenite structure. The hydration process of these two materials was very hard to be conducted at low partial pressures of water vapor (96 kPa). However, a novel activation process of these materials was developed within this work that allows the activation of the sample prepared at 1000 ºC, which can be expanded for the activation of other materials that in literature were considered inactive.

The Finally, the reaction reversibility of Mayenite was confirmed at least for 10 hydration/dehydration cycles, but on the other side the material crystallinity becomes lower which needs a regeneration step after a certain number of cycles in order to keep the reaction performance.

In particular the low dehydration temperature of 228 °C offers many application possibilities for this storage material. For example current molten salt based CSP-plants operate at a temperature difference between 280 and 560 °C. Consequently the complete temperature difference of the heat transfer fluid could usefully be incorporated into the storage system. Additionally the larger enthalpy of reaction of thermochemical materials leads to less required storage material thus decreasing the cost per kWh of storage capacity.

In order to further elaborate the application potential of the material, the reaction should be investigated in lab scale storage reactors under process relevant boundary conditions.

Table 1 .

 1 Identification

Figure 1 .

 1 Figure 1. XRD patterns obtained at room temperature for Ca12Al14O33 samples prepared at 600, 800, 1000 and 1100 ºC. (*) corresponds to unidentified impurities.

Figure 2 .

 2 Figure 2. Experimental (symbols) and calculated (line) XRD patterns obtained for Ca12Al14O33 prepared at 1100 ºC. The bars in the lower part of the graphics represent the Bragg peak positions.

Figure 3 .

 3 Figure 3. (a) Mayenite crystal structure showing the nanocages. (b) Zeolite crystal structure illustration α-cage and β -cage.

Figure 4 .

 4 Figure 4. SEM micrographs of Ca12Al14O33 prepared at (a) 600, (b) 800, (c) 1000 and (d) 1100 ºC.

Figure 5 .

 5 Figure 5. Chemical reversibility over 5 cycles (pH2O = 96 kPa) of Mayenite prepared at 1100 °C and at 1000 °C.

Figure 6 .

 6 Figure 6. Chemical reversibility of activated samples

Figure 7 .

 7 Figure 7. SEM micrographs Ca12Al14O33 materials calcined at 1000 ºC (a) anhydrous and (b) hydrated; and at 1100 ºC (c) anhydrous and (d) hydrated.

Figure 8 .

 8 Figure 8. XRD diffractograms of Mayenite, the pristine (bottom) and after the 1 st (middle) and 10 th (top) hydration-dehydration cycles.

Figure 9

 9 Figure 9 Chemical conversion of re-hydration of M-1000-A over 10 cycles, performed at 125°C and 68 kPa

Table 2 .

 2 Relative enthalpies of hydration and dehydration of activated samples

	Sample	∆RH1D, Mayenite X / ∆RH1D, Ca(OH)2
		pH2O = 0 kPa
	M-600-A	0.25
	M-800-A	0.45
	M-1000-A	0.67
	Based on the amount
	According to the equation 2 and the ∆mD (~27 wt%) obtained for M-1000-A, the
	dehydration reaction which took place under these conditions is:
	Ca12Al14O33+ 21 H2O ⇌ "Ca12Al14O12 (OH)42" + 0.67•∆RHD, Ca(OH)2

  Ca 12 Al 14 O 33 Mayenite exhibits a dehydration process to be transformed to "Ca12Al14O12(OH)42" hydroxide at 228 ºC with which is much lower compared to CaO/Ca(OH)2 pair reaction temperatures between 300 and 500 ºC. Furthermore, Ca 12 Al 14 O 33 /"Ca12Al14O12(OH)42" pair counts with a relative enthalpy of dehydration 0.67•∆RHD, Ca(OH)2.

Acknowledgments

The authors acknowledge the financial support from the European Commission through the H2020 program under Grant agreement number 657690. The authors express their sincere thanks to Yagmur Polat and Cristina Luengo for their technical support. The authors would like to thank Andrea Hanke at the DLR-Institute of Engineering Thermodynamics for her technical support.