Kévin Cortial
email: kevin.cortial@doctorant.uca.fr

Adélaïde Albouy-Kissi

Frédéric Chausse
email: frederic.chausse@uca.fr

Graph learning methods to analyze and support industrial resilience

Keywords: Graph learning, Knowledge graph, Industrial resilience JEL codes: C45-Neural Networks and Related Topics; C15-Statistical Simulation Methods

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

-Introduction

Graphs, also called networks, can model various real-world relationships between entities. Many disciplines use graphs to formulate relationships between entities. For example, social networks, chemical and biological networks, computer networks, are examples that use graphs to model connections between certain pairs of objects. Technically, a graph is often defined by a set of nodes and a set of edges between pairs of nodes. A directed graph is a graph in which edges have an orientation. The degree of a node is the number of entering or leaving edges in each node. A directed graph is a graph in which the edges have an orientation.

Generally, graph encoding involves creation of an adjacency matrix whose values 𝑎 !" are the weights of the edges linking node 𝑖 to node 𝑗. If the graph is non-oriented, adjacency matrix is symmetric. However, this structuring does not allow to store multiple data in nodes and edges. To build a knowledge graph of industrial ecosystems, we will use database graphs. In addition to storing the data freely, we will be able to query data without the need for multiple expensive joins that would be required in traditional relational databases. Thus, allow for very efficient updates, even for a very large data set. Triplestores store only one data's type. Triplet, consisting of start node, edge node and finish node. We do not need to create tables as in a relational database. Moreover, a triplestore is optimized for storing many triples and to retrieve these triples using the SPARQL query language.

Machine learning is a field of study in artificial intelligence that relies on mathematical and statistical approaches to give computers ability to "learn" from data. However, machine learning usually works with structured data (time series, images with pixels, texts, etc.). Classical machine learning methods will not work with graphs, as they can have arbitrary size, multimodal features, and complex topology.

Graphs can be embedded in a low-dimensional vector space where structural's information, attributes of nodes and edges can be preserved. These vectors are input data for graph learning methods. Vectorization methods can be various, depending on raw data typology of the graph (images, texts, etc.). These methods will be detailed in next chapters. According to nodes attributes, edges and subgraphs, graph learning tasks can be divided into three categories, respectively based on three ones.

(i) Classification of nodes will allow identification of key players in economic network. In fact, during node classification task, algorithm must determine labelling of a node by looking his embedding.

(ii) Grouping and detection of communities with similar properties. Technically, process of grouping together nodes that appear based on some similarity measure to be closer to each other. Identifying characteristics of each group is needed, this is the cluster analysis stage. For industrial resilience graph, clustering could identify stakeholder groups which may be economic and industrial sectors.

(iii) Prediction of links that do not yet exist between two nodes. Predicting a connection between two entities could be seen as a recommendation system. In a context of industrial resilience graph, link prediction can recommend partnerships between local industrial actors.

-Related Work

There are several studies on deep and machine learning methods on graphs. In contrast to these studies, our objective is to provide a comparison of graph learning methods, which can be applicable in economic context: in particular, knowledge extraction from graphs that model the industrial ecosystem. For our knowledge, this application is not covered by other literature's reviews. Xia and al. compared graph learning methods with other methods which are more statistical models, such as graph signal processing, random walk, and matrix factorization. Nickel and al. also discuss these statistical models, but their use is focused on knowledge graphs and their automatic construction. Zhou and al. only offer a comparison of deep learning models for GNN graphs and their variants by mentioning several fields of application without mentioning economics. Kosasih and Brintrup present GNN, however, in the case of supply chains graph the purpose is to detect unknown potential links. In addition, they use gradient embedding to improve explainability by highlighting the input features which influence the decisions of their deep learning model. Cai and al. summarized graph embedding methods, this work allows us to preprocess our graphs for the models we will present in the following chapters.

In summary, no existing studies provide a comprehensive overview graph learning's application to the learning of economic graphs modelling an industrial ecosystem. Originality of our paper is the application of methods in a field where is a lack of literature review on the application of graph learning in this economic context. Our paper will present the most recent machine learning techniques for graph data that could be applied to this context.

-Methods

In this section, several methods which allow to analyze knowledge graphs of industrial ecosystems modeling will be introduced. These statistical or machine learning tools extract knowledge from graph topologies. In a first step, the most classical methods of statistical analysis of graphs will be presented. Then, this chapter will ended by presenting deep learning methods that have been recognized as the most powerful techniques for graph's analysis with complex data in nodes and edges [START_REF] Zhang | Deep Learning on Graphs: A Survey[END_REF].

a) Modularity

Modularity is a measure of the quality partitioning of the nodes for a graph. This principle implies many intracommunity edges and a few inter-community edges. In other words, there are more connections between nodes of the same community than between nodes of different communities [START_REF] Newman | Modularity and Community Structure in Networks[END_REF]. The modularity score compares, for a group of nodes, the number of actual edges to the number of expected edges (in an equivalent graph where the edges are randomly placed like the Erdös-Rényi model). If there are more actual edges than expected, then one could have a community. This gives a division score of a graph defined by the following formula:

𝑄 = 1 2𝑚 )*𝐴 !" -𝑃 !" .δ*𝐶 ! , 𝐶 " . !,"
Where A $% , is the value 𝑖𝑗 of the adjacency matrix. 𝑚 is the number of edges. 2𝑚 is the total number of half-edges. δ is the Kronecker delta: δ*𝐶 ! , 𝐶 " . = 1 if 𝑖 and 𝑗 are in the same community 𝐶, i.e., 𝐶 ! = 𝐶 " . Otherwise 𝛿*𝐶 ! , 𝐶 " . = 0. P $% is the probability of the number of connections between 𝑖 and 𝑗, under a null Erdös-Rényi model, which produces a homogeneous graph.

Thus, maximizing the modularity 𝑄 means looking for groups of nodes with an abnormally high number of connections between them.

Modularity is the fundamental basis of community extraction applicable to graphs. the Louvain method [START_REF] Blondel | Fast Unfolding of Communities in Large Networks[END_REF]) is particularly well suited to handle large volumes of data.

The first phase of the Louvain method consists in finding small communities by local optimization of modularity on each node. In a second step, the nodes of the same community are grouped into a single node. First phase is repeated on the newly obtained network. Iterations are repeated until no modularity's increase is possible.

b) Stochastic block model (SBM)

SBMs are generative models for random graphs. These models produce graphs containing communities, with subsets of nodes characterized by their connections to each other with edge densities. To generate a random graph, the SBM needs three parameters:

-𝑛 the number of nodes.

-A partition of the set of nodes into communities according to a multinomial distribution.

𝑍 ! ∼ ℳ*1, 𝛼 = (𝛼 & , 𝛼 ' , … , 𝛼 ( ).
where 𝐾 is the number of communities and 𝛼 is the probability belongs to the community. -The probability matrix 𝜇 )* where diagonal elements correspond to the probability that nodes connect internally, within the community. Other elements correspond to the probability of connection between communities [START_REF] Faskowitz | Weighted Stochastic Block Models of the Human Connectome across the Life Span[END_REF].

Graph generation with an SBM model is simple. However, estimating all parameters of the SBM model such as 𝜇 )* and 𝛼 is a challenge. Indeed, standard optimization algorithms, such as the expectation maximization algorithm (to obtain maximum likelihood parameters), cannot be derived. To solve this problem, variational and stochastic approximations exist, such as variational Bayesian methods (Latouche, Birmele, and Ambroise 2010)

SBM methods and modularity score can create communities, but their results may be different. Indeed, modularity can illustrate communities whose nodes are strongly connected internally. On the other hand, there are few connections between nodes of different communities. The SBM identifies different clusters providing new information as hubs, which have many degrees per node. These nodes are loosely connected to each other but regularly connected to others forming star graphs. The strength of the SBM model is to provide probabilities of connections 𝜇 )* between different nodes according to their cluster assignment. Probabilities of connections between groups of nodes can be considered in multiobjective recommendation systems such as those of Pachot and al.

c) Graph Neural Networks (GNN)

Deep learning tools usually work on structured data. However, graphs could have arbitrary size, multimodal features, and complex topology. Thus, Graph Neural Networks (GNN) are deep learning algorithms which induce features from graph data. These inferences may be used directly for predictions, classifications, or other approaches. [START_REF] Wu | A Comprehensive Survey on Graph Neural Networks[END_REF] To exploit graph's data, GNNs have made the hypothesis that many pieces of node's information reside in its neighborhoods. To store this data, we use "Node Embedding" which gathers the neighborhoods information with multilayer neural networks (MLPs).

Thus, a GNN uses an MLP on each node's neighbor: we call this a GNN layer. In the Figure 2, there are therefore two layers: a first layer for A neighbor's, which in turn is derived from a second layer concerning the neighbors of the neighbors. For the node information set (its embedding) we apply the MLP (grey/black blocks in the Figure 2) and get a new learned node vector. The same will be done for the edges when they contain information (several edge types, values, weights, etc.).

Figure 2 : GNN encoding with neighborhood aggregation methods [START_REF] Hamilton | Representation Learning on Graphs: Methods and Applications[END_REF] The following formula indicates how the input information (from the target node ℎ ! + and its neighbors ℎ " + ) will be aggregate by neural networks and produce a new representation ℎ ! +,& . This is the mechanism found within the gray and black blocks in the Figure 2.

ℎ ! +,& = 𝜎 Cℎ ! + 𝑊 + ) 1 𝑐 !" "∈.(!) ℎ " + 𝑈H
ℎ ! + is the initial vector containing the information of the node ℎ ! . ℎ ! +,& is the updated vector containing the information of the neighbors but also graph's topology. This gives more information and a better representation of the ℎ ! node within the graph.

-𝑊 is the matrix of weights from a neural network (MLP) that identifies most important elements to keep within the initial vector ℎ ! + . -𝑈 is also a weight matrix from a neural network but will process vectors from neighboring nodes ℎ " + .

-

∑ & 1 !" "∈.(!)
is an aggregation function, this sum normalizes the neighboring representations 𝑁(𝑖). This requires a function that is invariant by permutation of its variables because we must be insensitive to the order of the neighbors to have the same result, whatever the order of the neighbors.

-𝜎 is an activation function.

Applying this function iteratively provides better representations of the nodes within their environment, i.e., in the graph. Thanks to these new embeddings, link prediction between nodes, clustering of nodes or subgraphs could be established. This method was popularized by the work of Kipf and Welling 2017.

d) Graph Attention Networks (GAT)

Like GNN, Graph Attention Networks (GAT) use data contained in these neighbors to create embeddings for each node. The difference in GATs is the use of the attention mechanism [START_REF] Vaswani | Attention Is All You Need[END_REF] to select the importance to be given to each neighbor.

Attention (𝑄, 𝐾, 𝑉) = 𝑠𝑜𝑓𝑡𝑚𝑎𝑥 R 𝑄𝐾 2 S𝑑 ) U 𝑉
Computation of attention coefficient is done by the above formula, where the three vectors are multiplied by the embedding of the input sequence. Here, 𝑄 is the search query, 𝐾 would be the features (texts, images, etc.) associated with the best fitting results 𝑉. The attention weights are divided by the square root of the dimension of key vectors S𝑑 ) to stabilize gradients during training, and then passed through a SoftMax function that normalizes weights to select the 𝑉 values to retain. These steps are the heads of attention which have queries and keys as input. In practice, the attention function is computed on a set of queries simultaneously, grouped in a 𝑄 matrix. Keys and values are also grouped in matrices 𝐾 and 𝑉 [START_REF] Vaswani | Attention Is All You Need[END_REF].

Advantages of multi-head attention are that it provides a diversification of the results during several selfattention/attention heads (ℎ times), as the vectors 𝑄, 𝐾 and 𝑉 are randomly initialized. The chaining of these attention's heads makes learning more robust and parallelizable.

GAT aggregates information present in the neighborhood of a node by a weighted sum as an attention mechanism. First, GAT calculates an attention coefficient for each neighbor of the node in question (node features). Then, GAT must integrate edge data according to their number (centrality). Finally, if the nodes position in the graph is important, GAT will consider this position data (spatial encoding). Indeed, for our knowledge graph which models industrial ecosystems, the geographical location of the companies (nodes) must be considered. The adaptation of the GAT architecture, presented above, requires the addition of all graph's information at self-attention layer [START_REF] Ying | Do Transformers Really Perform Bad for Graph Representation?[END_REF].

Figure 3 : Illustration of the encoding of graph information in the attention mechanism [START_REF] Ying | Do Transformers Really Perform Bad for Graph Representation?[END_REF])

-Experimentations and results

There are several dataset and benchmarks used to evaluate performance of graph learning approaches for various tasks, such as link prediction and node classification. For subgraph clustering or community detection, evaluation's metrics depend on results interpretation according to study contexts. Thus, in this section, we compare the results of models for link prediction and node classification.

For link prediction, the FB15k dataset contains knowledge base relation triples and textual mentions of Freebase entity pairs. It has a total of 592,213 triplets with 14,951 entities and 1,345 relationships. FB15K-237 is a variant of the original dataset where inverse relations are removed, since it was found that many test triplets could be obtained by inverting triplets in the training set (Bordes and al. 2013) For node classification, the Cora dataset consists of 2708 scientific publications classified into one of seven classes.

The citation network consists of 5429 links (Mccallum 2000). The best model for classification into the seven Cora classes is the model of Izadi and al. They used GNN algorithms using natural gradient information in the optimization process.

-Conclusion and perspectives

This paper provides an overview of graph learning and compares most recent graph learning methods. We have presented existing graph learning methods including modularity, SBM, GNN and GAT. Originality of our work is the application of graph learning methods to support industrial resilience. Indeed, by learning knowledge graphs that represent an industrial ecosystem, we could contribute to a more resilient and ecological production. Experiments and results from different benchmarks show that deep learning models (GNN and GAT) have better performances on clustering, classification, and link prediction. These results confirm our intention to use graph learning methods for our future research on knowledge graphs to model industrial ecosystems. We will work on the graph constructed with potential productive links between firms of Pachot and al. Graph learning is currently a very rapidly evolving field. We hope that this study will help researchers and practitioners in their research and development's works on graph learning and related fields.

Figure 1 :

 1 Figure 1 : Visualization of the steps of the Louvain method (Blondel and al. 2008) Creation of these communities would make possible to identify industrial sectors. We could use knowledge graph of productive skills of Pachot and al. to group local productive partners and secure their supplies, thus promote distributed manufacturing.

  . On this dataset,Wang and al. proposed a variant of GAT. Their work on a GAT based on attenuated attention shows best results in comparison with other state-of-the-art methods on FB15k-237 for link prediction.WN18RR is a link prediction dataset contains 93,003 triples with 40,943 entities and 11 relation types. WN18RR is a link prediction dataset contains 93,003 triples with 40,943 entities and 11 relation types. It is designed to produce an intuitively usable dictionary and thesaurus. Its entities correspond to word senses, and relationships define lexical relations between them(Bordes and al. 2013). On this dataset, Li, Yi, and He proposed a variant of BERT model[START_REF] Devlin | BERT: Pre-Training of Deep Bidirectional Transformers for Language Understanding[END_REF]) for link prediction. This BERT model uses attention mechanism on the textual data of the nodes and edges of this dataset to predict links. The work of Li, Yi, and He had the best state-of-the-art results on the WN18RR data. They improved by 5% over previous state-of-the-art result on the WN18RR dataset.


J L Bobet

Jacques Huot
email: jacques.huot@uqtr.ca

J.-Gabriel Roquefere

Sebastien Baduel

Catalytic effects of pseudo AB 2 phases on hydrogen sorption

Keywords: Hydrogen sorption, Magnesium, Catalyst, Ternary compounds

The compound GdNi 4 Mg in its amorphous and crystalline state has been investigated as catalyst for hydrogen sorption of magnesium. Addition of only 0.3 at.% of catalyst by ball milling resulted in an important improvement of hydrogen sorption kinetics. Optimum milling time has been found to be 5 h. The amorphous and crystalline GdNi 4 Mg compounds give almost identical kinetics. Both absorption and desorption have been improved and the kinetics obtained are almost the same as other metal catalysts. No evidences of decomposition during ball milling or hydrogenation have been found so that it should be concluded that the intermetallic itself acts as a catalyst.

Introduction

It is now well-known that the accumulation of CO 2 in the atmosphere is the principal cause of global warming. As long as carbon will be present in any form in the fuel used, the accumulation of CO 2 in the atmosphere cannot be stopped. This leaves us with hydrogen as the only choice as it is the only known carbon free combustible material.

However, the "hydrogen economy" will start only if the problems of hydrogen mass production and storage are solved. In the case of storage, the safety problem of pressurized tank and the high cost of liquefaction bring an interest in an alternative solution: alloys forming metal hydrides. In this case, the storage is based on the reversibility of the reaction with hydrogen of certain metal (Mg, V, Pd, . . .) or intermetallic (LaNi 5 , TiMn 2 , TiFe, Mg 2 Ni, . . .). Amongst all the metal or intermetallics, Mg attracts a lot of efforts because of its low density (1.74 g/cc), high hydrogen capacity (7.6 wt.%) and low cost (less than 5D /kg). However, the reaction with hydrogen is very slow (up to few days to hydride bulk magnesium) and takes place at high temperature (i.e. 300 • C). During the last decades, progress in hydrogenation kinetics of magnesium has been realized by ball milling [1][START_REF]Topics in Applied Physics, Hydrogen in Metals I, Basic Properties[END_REF][START_REF] Bobet | [END_REF][4][5][6] (in order to increase the number of defects) and addition of some catalysts [7][8][9][10] (for the dissociation of hydrogen molecule). The best results are obtained with oxides [8,[11][12][13] as a catalyst but the catalytic effect is not fully understood at the present time. Magnesium-based intermetallics that can be hydrided under ambient conditions have been studied but the results reported were never convincing. Recently, some interest has been focussed on 'pseudo-AB 2 ' compounds of general formula RENi 4 Mg [14]. In this work, it has been found that GdNi 4 Mg absorbs reversibly hydrogen at room temperature but no significant hydrogen absorption was noticed at 300 • C. Despite this lack of reaction at high temperature, it can be of interest to study the potential catalytic effect of this compound on magnesium hydride formation. In this paper, the effects of milling a mixture of magnesium hydride and 0.3 at.% of GdNi 4 Mg in its crystalline and amorphous state are reported. Structural and hydrogen sorption properties have been investigated as a function of milling time. The catalytic effect of GdNi 4 Mg and its possible origin have been studied.

Experimental details

The magnesium hydride powder MgH 2 (95 wt.% MgH 2 , 5 wt.% Mg) was purchased from Th. Goldschmidt AG. The compound GdNi 4 Mg was prepared by ball milling stoichiometric amount of Gd, Ni, and Mg powders under argon (p≈0.11 MPa) on a Fritsch Pulverisette 5 for 10 h with a disc rotation speed of 200 rpm. All the handlings were carried out in a glove box under purified argon to avoid any air contamination. In the following, the compounds obtained after ball milling will be identified as GdNi 4 Mg (a) (where "a" stands for amorphous) and the one obtained after subsequent heat treatment under Ar at 650 • C for 1 h will be identified as GdNi 4 Mg (c) (where "c" stands for crystallized). More details about the synthesis of the intermetallic compounds can be found in Ref. [14].

The MgH 2 powder was mixed with 0.3 at.% (i.e. 5 wt.%) of GdNi 4 Mg in a stainless steel container hermetically closed. Samples were prepared by ball milling the different compounds with a high-energy Spex 8000 Mixer/Mill using a stainless steel milling jar and three hardened steel grinding balls (two balls of 11.1 mm diameter and one ball of 15.9 mm diameter) under argon atmosphere. The initial ball-to-powder mass ratio was 10:1. All handlings of samples were performed inside a glove box with a continuously purified argon atmosphere in order to avoid the oxidation of samples.

The crystalline structure of as-milled powders was characterized by X-ray powder diffraction (XRD) using a Rigaku D-max diffractometer with Cu K␣ radiation. The diffraction patterns were analyzed by a whole pattern fitting procedure using the program Fullprof [15] in order to determine the lattice parameters, phase abundance, crystallite size, and strain.

The hydrogen sorption kinetics were measured on a HERA volumetric apparatus [16]. Sorption kinetics were measured at 573 K under 1000 kPa of hydrogen pressure for absorption and 12 kPa for desorption. Other characterizations such as Electron Probe Micro Analyzer (EPMA) (CAMECA SX-100 Instrument), particles size analysis (Malvern Mastersizer Hydro 2000S), and Secondary Electron Microscopy SEM (Jeol 840) have been done before and after hydrogen cycling.

Results and discussion

Crystal structure

The X-ray diffraction patterns of the mixtures MgH 2 + GdNi 4 Mg(a) ball milled for 1 and 5 h are presented in Fig. 1. Due to the amorphous nature of the catalyst and it's low content, it is not possible to detect it by X-ray analysis. After 1 h, the mixture is still well crystallized with an average crystallites size for magnesium hydride (determined from Scherrer formula) of more that 20 nm. After 5 h of ball milling, the magnesium hydride crystallite size decreases to about 9 nm. Moreover, it can be seen that the proportion of the ␥-MgH 2 phase slightly increases. The peaks relative to the magnesium (coming from the commercial MgH 2 ) are clearly visible after 1 h of milling and almost disappear after 5 h of milling. This could be due to amorphization [17] or to a large reduction of crystallite size. The X-ray diffraction patterns of the mixtures MgH 2 + GdNi 4 Mg(c) ball milled for 1, 2 and 5 h are presented in Fig. 2. As the crystallized compound was used, it could be expected to be able to detect it by XRD analysis. However, even before milling (i.e. just manual mixing in a mortar), it was not possible to see the major peak of the GdNi 4 Mg(c) (e.g. peak (3 1 1) at 2θ = 42 • ). This result can be explained by the fact that only 0.3 at.% of catalyst was used and that moreover, the catalyst is more dense and the particles size are smaller that MgH 2 . Then, the Mg particles are more present on the surface and as 99% of the X-ray beam is absorbed after 10 m in Mg, the catalyst cannot be detected. As previously observed with GdNi 4 Mg(a), the crystallite size decreases and the relative amount of the ␥-MgH 2 phase increases with milling time. From 5 to 10 h of milling, the magnesium hydride crystallite size is constant but the relative amount of the ␥ phase is still increasing. It is also surprising to see that the remaining unreacted magnesium is still present even after 5 h of milling meaning that the crystallite size reduction is less efficient than when the amorphous catalyst is used.

Hydrogen sorption properties

It could be expected that the structural changes due to milling gives some modification in the hydrogen sorption properties. In the case of the amorphous catalyst, the sorption properties of MgH 2 + GdNi 4 Mg(a) are presented in Fig. 3a. For the mixtures ball milled for 1 h, the full absorption takes approximately 20 min and desorption is slow (5 wt.% desorbed in 10 min but the full desorption takes about 2 h). When the ball milling is carried on up to 5 h, the sorption properties are greatly improved. The full absorption is reached in about 1 min and desorption is almost complete in 7.5 min. Such results suggest a further increase of milling time but the mixtures ball milled for 10 h (not presented here) exhibit the same properties as the one ball milled for 5 h. Then, it should be concluded that the optimum milling time is close to 5 h. Fig. 3b represents the result obtained when the crystallized catalyst is used. It is clear from that figure that the effect of ball milling time is very small. The full absorption is reached after 10 min for the mixture milled for 1 h as it is reached in 1 min for the mixture milled for 5 h. The desorption behaves in the same way as after 7 min the mixture ball milled for 5 h is fully desorbed while the mixture ball milled for 1 h desorb only 90% of its hydrogen in the same time. In the same way as for the amorphous catalyst, milling for more than 5 h does not lead to any more increase of the sorption properties.

Comparison with other catalysts

In order to estimate the catalytic effect of GdNi 4 Mg(a) or GdNi 4 Mg(c), a comparison with pure MgH 2 and other catalysts is judicious. As the GdNi 4 Mg forms a stable hydride under ambient conditions, it is interesting to compare with another catalyst that has similar behavior. In this respect, vanadium is a good choice because it is a well-known catalyst for magnesium and it Fig. 5. Comparison between the effects of the "elemental" dopants and "compounds" dopants. All mixtures are ball milled for 5 h. forms a stable hydride. Thus, a mixture of MgH 2 + V(0.3 at.%) was ball milled for 5 h under argon atmosphere in the same conditions as previously. Pure magnesium hydride was also milled for 5 h in argon in order to see the effect of milling on the sorption kinetics. In Fig. 4, the sorption properties of ball milled pure MgH 2 and mixture MgH 2 + V(0.3 at.%) compared to ball milled magnesium hydride doped with GdNi 4 Mg(a) and GdNi 4 Mg(c) are presented. It shows that complete absorption is reached in 1 min for magnesium hydride doped with GdNi 4 Mg(a) or GdNi 4 Mg(c), but it takes more than 10 min for MgH 2 + V(0.3 at.%) and pure MgH 2 .

The discrepancy between vanadium and GdNi 4 Mg doped magnesium hydride is even more remarkable for the desorption process. The full desorption takes only 7-8 min for the MgH 2 + GdNi 4 Mg mixtures while it requires 3 h for MgH 2 + V and even 5 h for pure MgH 2 . Fig. 4b shows that, for the mixtures milled for an optimum time of 5 h, the difference between the mixtures containing the crystallized or the amorphous compounds is small for both absorption and desorption processes. Only a small difference can be seen on the desorption process where the mixtures with GdNi 4 Mg(c) shows slightly better kinetics.

In order to get a better understanding of the catalytic effect, mixtures containing the same atomic content of elemental catalysts element than the alloy (i.e. MgH 2 + 0.3 at.% Mg, MgH 2 + 0.3 at.% Gd and MgH 2 + 1.2 at.% Ni) were ball milled for 5 h under the same conditions. The hydrogen sorption properties are presented in Fig. 5. The mixtures containing Gd or Mg as a catalyst behave almost on the same way. The full absorption is reached after 20 min (but 6 wt.% of hydrogen is already reached after 10 min) but desorption is a very slow process. After the first hour, only 0.5 wt.% of hydrogen has been desorbed from the mixtures. The mixture doped with 1.2 at.% Ni behaves in a very similar way to the mixture containing GdNi 4 Mg. On absorption, the first minute is strictly comparable and beyond the mixture with 1.2 at.% Ni is slightly better. On the desorption process, the behavior is comparable during the first 3 min and beyond it is the mixture with GdNi 4 Mg which is better.

As both 1.2 at.% Ni and 0.3 at.% GdNi 4 Mg have the same catalytic effect it is worth checking the structural properties. The XRD patterns of both mixtures are presented in Fig. 6. It is clear from this figure that in the mixtures MgH 2 + 1.2 at.% Ni, nickel stays well crystallized and does not form intermetallic compound with magnesium or magnesium hydride. In the case of MgH 2 + 0.3 at.% GdNi 4 Mg mixture there is no evidence of elemental nickel.

A possible explanation of the similar catalytic effect of 1.2 at.% Ni and 0.3 at.% GdNi 4 Mg may be that the GdNi 4 Mg compound decomposes during milling or during subsequent hydrogenation. The X-ray powder diffraction of Fig. 6 does not support decomposition due to ball milling. To check the decomposition due to hydrogen cycling, a powder diffraction pattern was taken on a dehydrogenated sample. Fig. 7 is the diffraction pattern of ball milled MgH 2 + GdNi 4 Mg(c) cycled 4 times and in a dehydrogenated state. All peaks are indexed to pure magnesium except for a small peak near 40 • which is due to hydrided magnesium. Another small peak near 45 • is close to the (1 1 1) Bragg peak of pure nickel. However, as shown in the inset, the nickel reflection does not correspond to the main diffraction peak. Moreover, the second inset demonstrate that the (2 0 0) nickel reflection is totally absent. A further test of the decomposition hypothesis have been performed by SEM analysis as well as EPMA. The SEM (backscatter radiation) micrograph shown in Fig. 8 reveals the presence of only 2 phases: one corresponding to the Mg matrix and the other corresponding to the catalyst GdNi 4 Mg. The catalyst is well dispersed after 5 h of milling and particle size range from 0.1 to almost 5 m. The EPMA investigation revealed a difference between mixtures containing amorphous and crystallized catalysts: After 1 h of milling the amorphous catalyst is not as well dispersed as the crystallized one and some large agglomerates can be observed.

Tests of sorption kinetics at different temperatures for magnesium hydride doped with 1.2 at.% Ni and 0.3 at.% GdNi 4 Mg gave identical kinetics for absorption and desorption. Even if the catalytic mechanism for nickel and GdNi 4 Mg is the same, the fact that the kinetics of both mixtures are identical should still be explained. One possible explanation, given the fact that the kinetics are very fast, could be that the rate limiting step is no longer due to the catalyst but is now heat transfer. One advantage of using GdNi 4 Mg as catalyst instead of nickel is the fact that upon cycling nickel will form the intermetallic Mg 2 Ni while in the case of GdNi 4 Mg our preliminary results, for a limited number of cycles, indicates that the compound does not decompose.

Experiments with 50 wt.% of GdNi 4 Mg catalyst shows that it is not hydrided during the sorption process which is in agreement with the thermodynamic data of GdNi 4 Mg-H system. Then, it should be concluded that the catalytic effect of GdNi 4 Mg is of the same type as the one of nickel: GdNi 4 Mg should act as a catalyst for the decomposition of dihydrogen. It could be of interest to test GdNi 4 Mg as a catalyst in some organic reaction where the Raney nickel is usually used in order to definitively validate our hypothesis.

Conclusion

Alloys GdNi 4 Mg (a and c) that can form a hydride under ambient conditions but do not absorb at high temperature have been used as a catalyst for the sorption of hydrogen in magnesium. To reach the best performances of the mixtures a ball milling time of 5 h is needed. Shorter and longer milling times give slower kinetics. For the optimized milling time mixtures, no significant difference between the mixtures containing the amorphous and the crystallized compounds can be highlighted. The catalytic activity of GdNi4Mg is comparable to the one of Ni (used in the same atomic proportion) and probably has the same mechanism (i.e. dissociation of molecular hydrogen).

These results clearly open some new research approach of new catalyst for both the hydrogen sorption of magnesium but also for the organic chemistry.

Fig. 1 .

 1 Fig. 1. XRD patterns of MgH 2 + GdNi 4 Mg(a) ball milled for 1 and 5 h.

Fig. 2 .

 2 Fig. 2. XRD patterns of MgH 2 + GdNi 4 Mg (c) ball milled for 1, 5 and 10 h.

  Finally, the broad peak around 2θ = 42 • is due to MgO, coming from the unreacted Mg present in the commercial MgH 2 .

Fig. 3 .

 3 Fig. 3. Sorption kinetics for MgH 2 + GdNi 4 Mg amorphous (on left) and crystallized (on right) milled for different duration.

Fig. 4 .

 4 Fig. 4. Comparison between the sorption kinetics of MgH 2 , MgH 2 + V and MgH 2 + GdNi 4 Mg (a) and (c) ball milled for 5 h. The right part is a zoom on the first 10 min.

Fig. 6 .

 6 Fig. 6. XRD patterns of MgH 2 + 4 Ni ball milled for 5 h. For a matter of comparison, the XRD of MgH 2 + GdNi 4 Mg BM in the same conditions is given on the bottom.

Fig. 7 .

 7 Fig. 7. XRD patterns of MgH2 + GdNi4Mg BM ball milled for 5 h, after four hydrogenation/dehydrogenation cycle in the dehydrogenated state.

Fig. 8 .

 8 Fig. 8. SEM observation of the MgH2 + GdNi4Mg (c) ball milled for 5 h.