Yang Ji
email: yang.ji@inria.fr

Samuel Deslauriers-Gauthier
email: samuel.deslauriers-gauthier@inria.fr

Rachid -2781-121x]

Deriche

Structure-Function Mapping via Graph Neural Networks

Keywords: Brain connectivity mapping, Graph neural network, Graph convolutional network, Graph transformer network, Autoencoder

Understanding the mapping between structural and functional brain connectivity is essential for understanding how cognitive processes emerge from their morphological substrates. Many studies have investigated the problem from an eigendecomposition viewpoint, however, few have taken a deep learning viewpoint, even less studies have been engaged within the framework of graph neural networks (GNNs). As deep learning has produced significant results in several fields, there has been an increasing interest in applying neural networks to graph problems. In this paper, we investigate the structural connectivity and functional connectivity mapping within a deep learning GNNs based framework, including graph convolutional networks (GCN) and graph transformer networks (GTN). To our knowledge, this original GTN based framework has never been studied in the context of structure-function and brain connectivity mapping. To achieve this goal, we use a GNNs based encoder-decoder system, where the encoder takes structural connectivity (SC) matrix as input and generates a latent representation of each node in a lower dimension, then the decoder uses the latent representation to reconstruct or predict the associated functional connectivity (FC) matrix. Besides comparing different encoders for node embedding, we also demonstrate that a decoder, which projects lower dimension vectors onto higher dimensional space, can improve the model performance. Our experiments demonstrate that both GCN encoder and GTN encoder combined with the proposed decoder can provide better results on our data than the previously proposed GCN autoencoder model. GTN encoder is also shown to be much more effective when it comes to noisy data and outliers.

Introduction

Studying the relationship between structural connectivity and functional connectivity is essential to understand how brain function emerges from their un-derlying structural substrate. A better characterization of the link between SC and FC will provide insights into how lesions to the structural substrate affect brain function [START_REF] Sporns | The human connectome: A structural description of the human brain[END_REF][START_REF] Alstott | Modeling the impact of lesions in the human brain[END_REF][START_REF] Váša | Effects of lesions on synchrony and metastabistability in cortical networks[END_REF], with great potential clinical applications. Previous studies have shown that functional connectivity and structural connectivity patterns are correlated [START_REF] Honey | Predicting human resting-state functional connectivity from structural connectivity[END_REF][START_REF] Deco | The dynamics of resting fluctuations in the brain: metastability and its dynamical cortical core[END_REF][START_REF] Hermundstad | Structural foundations of resting-state and task-based functional connectivity in the human brain[END_REF]. Therefore, it is reasonable to expect that FC matrices can be predicted from SC matrices.

Many models have been proposed to map structure to function [START_REF] Deligianni | A framework for inter-subject prediction of functional connectivity from structural networks[END_REF][START_REF] Abdelnour | Network diffusion accurately models the relationship between structural and functional brain connectivity networks[END_REF][START_REF] Meier | A mapping between structural and functional brain networks[END_REF][START_REF] Liang | Structure-function network mapping and its assessment via persistent homology[END_REF][START_REF] Becker | Spectral mapping of brain functional connectivity from diffusion imaging[END_REF]. A subset of these models are specifically based on the link between the eigenvalues and eigenvectors of SC and FC matrices. However, this approach is based on the assumption that there exists a correspondence between structural and functional eigenmodes that can be captured by the models. Moreover, they are usually computationally expensive when dealing with large data. In contrast, deep learning (DL) approaches provide a more general approach to learn the mapping between SC and FC. Deep learning has produced significant results in several fields, including computer vision, speech recognition and natural language processing [START_REF] Lecun | Deep learning[END_REF]. To integrate non-Euclidean graph data, graph neural networks (GNNs) have been developed to deal with graph problems [START_REF] Wu | A Comprehensive Survey on Graph Neural Networks[END_REF]. For example, graph convolutional networks (GCN), which were proposed by [START_REF] Kipf | Semi-supervised classification with graph convolutional networks[END_REF], are motivated by the graph convolution operator [START_REF] Bruna | Spectral networks and locally connected networks on graphs[END_REF]. At a higher level, GCNs use the graph Fourier transform to aggregate neighboring nodes' features, thus exploiting the topology of the graph during the learning process. GCNs have been studied on structurefunction mapping in [START_REF] Li | Mapping Brain Structural Connectivities to Functional Networks Via Graph Encoder-Decoder With Interpretable Latent Embeddings[END_REF], where a graph encoder-decoder based on GCNs was proposed to recover the SC-FC mapping. However, the decoder in this model is mainly an inner product operation with no trainable parameters. The linear encoder used by the model can also be sensitive to outliers presented in data.

A second example are graph transformer networks (GTN) [START_REF] Yun | Graph Transformer Networks[END_REF] are analogous to Spatial Transformer Networks [START_REF] Jaderberg | Spatial transformer networks[END_REF]. The most advantageous point of GTNs over GCNs is that they consider heterogeneous graph structures, and can therefore be used to predict FC from several different SC matrices. This feature makes GTN more robust when dealing with outliers or noisy data, where GCN is much less effective.

In this work, we revisit structure-function mapping via deep graph neural networks. First, we investigate the importance of a trainable decoder in the GCN architecture of [START_REF] Li | Mapping Brain Structural Connectivities to Functional Networks Via Graph Encoder-Decoder With Interpretable Latent Embeddings[END_REF] and show that it improves the prediction of FC from SC. Second, we propose a GTN architecture to predict FC from several diffusion derived SC matrices and show that is more robust to outliers. To our knowledge, this is the first time GTNs have been studied in the context of structure-function and brain connectivity mapping. All of the models are tested on 1050 subjets of the Human Connectome Project, therefore providing reproducible results.

The rest of the paper is composed of three parts. The first part is Preliminaries, where we state the main problem and then introduce the mathematical theories of GCN, GTN and encoder-decoder system. Then we propose numerous models based on GNNs. The second part is Experiments, where we present how SC/FC matrices are extracted from Magnetic resonance imaging(MRI), and de-tails of the implementation. The final part is Results and conclusion, where we compare and discuss in detail the performance of each proposed model.

Preliminaries

In this section, we first state the problem and the objective of this paper and review some GNNs architectures including GCN, GTN, and the main framework Autoencoder that includes them all. At the end of each architecture description, a model is proposed.

Problem statement

Given an undirected SC matrix, which is extracted from diffusion MRI, we define A ∈ R N ×N as its symmetric adjacency matrix, D ii = N j=0 A ij as the diagonal degree matrix which denotes the number of connections of each node, N as the number of nodes and in our case it is the number of regions of the atlas. The normalized Laplacian of A is defined as

L = I N -D -1 2 AD -1 2 = U ΛU T
, where Λ denotes its eigenvalues and U as its eigenvector. The brain connectivity mapping is between the SC matrix and the FC matrix. The objective is to identify a mapping that takes A as input and predicts a FC matrix Z ∈ R N ×N , such that Z is close to the empirical FC matrix. The problem can be reformulated as

minimize 1 K K k=1 f (A k ) -Z k 2 F ( 1
)
where f is the mapping function (in our case a DL architecture), K is the number of subjects, and • F is the Frobenius norm.

Autoencoder

The idea of autoencoder [START_REF] Rumelhart | Parallel distributed processing: Explorations in the microstructure of cognition[END_REF] is to use an encoder to learn a latent representation of the data, then use a decoder to reconstruct its input. Here, we use the autoencoder as our main structure, but we alter it into reconstructing the associated FC matrix instead of SC matrix. An encoder generates a latent vector representation y i for each node, then a decoder uses these representation vectors to compute the prediction of the FC matrix, denoted as

Z = f (Y ) with Y ∈ R N ×F
where F is the dimension of the latent node representation. In this work, we test both GCN (as in [START_REF] Li | Mapping Brain Structural Connectivities to Functional Networks Via Graph Encoder-Decoder With Interpretable Latent Embeddings[END_REF]) and GTN as encoders. A diagram of the model is presented in Figure 1.

The decoder mainly serves to transform latent nodes embedding Y to target symmetric FC prediction Z. In [START_REF] Li | Mapping Brain Structural Connectivities to Functional Networks Via Graph Encoder-Decoder With Interpretable Latent Embeddings[END_REF], the authors proposed a decoder based on inner product, defined as: where tanh(x) = e x -e -x e x +e -x and ReLU(x) = max(0, x) are activation functions. Z is a symmetric matrix whose values are in the range [0, 1], like the observed FC matrix. However, apart from reconstructing a prediction that fits the FC matrix' symmetry and range property, it is not trainable. In the rest of the article, this decoder is referred to as DC1.

Z = tanh(ReLU(Y Y T )) (2)
To overcome the disadvantage of DC1, we propose a decoder that is trainable, defined as

Y = Y W d 0 + B d , Y = tanh(Y * W d 1 ), Z = ReLU(Y Y T ) (3)
where

W d 0 ∈ R F ×N is a trainable parameter which augments the dimension of the latent embedding Y , B d ∈ R N ×N is a bias parameter, W d 1 ∈ R N ×1
is also a trainable parameter, initialized with constant 1 to keep the output's range close to [0, 1] from the start. It balances the dynamic of the inner product operation. The * operator denotes a row-wise multiplication. To predict a semidefinite positive FC matrices, for example when FC matrices are not thresholded, it suffices to omit ReLU(•). In the rest of the article, this decoder is referred to as DC2.

Graph Convolutional Networks(GCN)

A spectral graph convolution operator, as defined by [START_REF] Defferrard | Convolutional neural networks on graphs with fast localized spectral filtering[END_REF], is given by

g θ x = U g θ U T x (4)
with the filter g θ = diag(θ), θ ∈ R N is the parameter. Therefore g θ can be considered as a function of the eigenvalues of the Laplacian Λ. In order to avoid expensive computation of eigendecomposition of graph Laplacian L, [START_REF] Hammond | Wavelets on graphs via spectral graph theory[END_REF] proposed using Chebyshev polynomials to approximate g θ (Λ). Given the recursively defined Chebyshev polynomial as

T k (x) = 2xT k-1 (x) -T k-2 (x), with T 0 = 1, T 1 = x, g θ (Λ) is reformulated as g θ (Λ) ≈ K k=0 θ k T k ( Λ) (5)
where Λ = 2 λmax Λ -I N , λ max is the largest eigenvalue of L. θ ∈ R K is the coefficient vector of Chebyshev polynomial. Combining equations ( 4) and [START_REF] Deco | The dynamics of resting fluctuations in the brain: metastability and its dynamical cortical core[END_REF], a new graph convolution operator can be defined as

g θ x ≈ K k=0 θ k T k ( L)x (6)
where L = 2 λmax L -I N . In [START_REF] Kipf | Semi-supervised classification with graph convolutional networks[END_REF], a simple graph convolution operator is proposed, which is motivated by a first-order approximation of Chebyshev polynomials [START_REF] Hammond | Wavelets on graphs via spectral graph theory[END_REF]. By limiting K = 1 and setting λ max = 2, θ 0 = -θ 1 = θ in equation ( 6), we have the graph convolution operator that is motivated by the first order of Chebyshev

g θ x ≈ θ(I N + D -1 2 AD -1 2 )x = θ D-1 2 Ã D-1 2 x (7)
where à = A + I N is a symmetric self-looped adjacency matrix, Dii = N j=0 Ãij denotes the degree matrix of the Ã. Finally the first order approximation GCN is defined as

Y (l+1) = σ D-1 2 Ã D-1 2 Y (l) W (l) 0 , (8)
where Y 0 = X, Y (l) denotes the lth hidden layer, W

0 is the weight parameter of lth layer. σ(•) is an activation function.

In our implementation, the GCN encoder is a single layer GCN defined as

Y = ReLU( ÂXW 0 ), (9)
where  = D-1 2 à D-1 2 , à was defined in equation ( 7), W 0 ∈ R N ×F is a trainable parameter of the encoder, N and F are previously defined.

Graph Transformer Networks(GTN)

Graph Transformer Networks [START_REF] Yun | Graph Transformer Networks[END_REF] take heterogeneous graphs as multi-channel input and use these channels to compute multi-channel meta-path graph tensors. Then, GTNs apply a single layer GCN to each channel of the meta-path tensor and generate a node representation for each channel. Finally, a perceptron layer is added on top of the concatenation of the node representations of each channel.

Consider A ∈ R N ×N ×C a set of adjacency matrices of heterogeneous graphs, C denotes channel number, which in our case A =[ Â1 , Â2 , Â3 ] is composed by the first order approximation of three types of SC matrices, whose nature is detailed in Section 3.1. First the Graph Transformer (GT) layer generates two intermediate adjacency matrices, Q 1 ∈ R N ×N ×C and Q 2 ∈ R N ×N ×C , then performs a 1x1 convolution as

Q i = φ(A, softmax(W φ )), (10)
where φ is a graph convolution, W φ ∈ R 1×1×K is the parameter of φ, softmax(•) is an activation function defined as softmax(x i ) = exp(xi) j exp(xj ) . Second, the GT layer combines the intermediate adjacency matrices Q i together to compute a normalized new graph structure. For the

A (1) = D -1 (1) Q 1 Q 2 , D ∈ R N ×N ×C is the multi channels degree tensor of Q 1 Q 2 .
For multi layers GTN, we repeat the procedure from Equation ( 10) by generating Q l+1 , then we compute the new graph

A (l) = D -1 (l) A (l-1) Q (l+1)
. Finally, a graph convolution is performed on each channel of the new structure A (l) before combining all channels' node embedding. Instead of concatenating all channels' embedding as in the original paper [START_REF] Yun | Graph Transformer Networks[END_REF], we found that summing them performs better. Thus the intermediate output is

Y = C i=1 σ D-1 i Ã(1) i XW 0 ( 11
)
where

Ã(l) i = A (l)
i + I. C denotes the number of channels, W 0 ∈ R N ×N is a trainable weight parameter. In our implementation of GTN, we set the layer number to l = 1 and the channel number to C = 3. Upon the output defined in Equation ( 11) in section 2.4, an additional layer is added in order to obtain lower dimension embedding. The ultimate GTN encoder is defined as

Y = ReLU Y W 1 + B 1 ( 12
)
where W 1 ∈ R N ×F is a trainable parameter, B 1 ∈ R F ×F denotes bias.

Experiments

Data

The SC and FC dataset, which contain 1050 subjects, is provided by Human Connectome Project5 (HCP). The SC and FC data are extracted from diffusion MRI (dMRI) and functional MRI (fMRI) using the same pipelines as in [START_REF] Deslauriers-Gauthier | A Unified Framework for Multimodal Structure-function Mapping Based on Eigenmodes[END_REF]. We used the streamlines produced by the diffusion pipeline to compute three types of SC matrices, Count, SIF T 2, Length. The first one Count(i, j) represents the total number of streamlines between regions i and j. The second one SIF T 2(i, j) was computed by summing the SIFT2 weights of streamlines connecting i and j [START_REF] Smith | SIFT2: Enabling dense quantitative assessment of brain white matter connectivity using streamlines tractography[END_REF]. The third one Length(i, j), denotes the reciprocal average length of the streamlines connecting i and j. Both SC and FC adjacency matrix contain N = 68 rows and columns, each row and column corresponds to a brain region of the Desikan-Killiany atlas.

The SC matrix A extracted from dMRI is not directly used as model input. First, we set the diagonal values of A to be zero. Then, as explained in Section 2.3, we use first order approximation of Chebyshev polynomial on A to obtain the input Â, defined in Equation [START_REF] Lecun | Deep learning[END_REF]. As Length data has infinite values (between disconnected brain regions), for computational stability, we set its diagonal values as 1, then compute 1/Length(i, j) beforehand. The FC originally ranges between -1 and 1, but in our implementation, the negative values in FC are set to 0 as in previous researches dealing with functional brain connectivity [START_REF] Power | The development of human functional brain networks[END_REF][START_REF] Rubinov | Complex network measures of brain connectivity: uses and interpretations[END_REF]. After our processing pipeline, the average correlation between the FC matrix and the Count, Length, and SIF T 2 matrices is 0.24, 0.28, and 0.24, respectively.

Implementation

The GCN uses only the Count matrix, while GTN uses all three SC matrices mentioned previously. The dataset is split into three parts: 80% for training, 10% for validation and 10% for test. Cross-validation (10 folds) and early stop are applied. We use Adam as optimizer [START_REF] Kingma | Adam: A method for stochastic optimization[END_REF], set learning rate to 0.001; batch to 25 for GCN, 1 for GTN; dropout rate to 0.05; encoder's output dimension to F = 32 for all models. The weight parameters are initialized using [START_REF] Glorot | Understanding the difficulty of training deep feedforward neural networks[END_REF]. The models are implemented via Pytorch [START_REF] Paszke | Automatic Differentiation in PyTorch[END_REF]. We use Mean Square Error(MSE) as all models' loss function, however, it is only applied on the lower triangular part of the predicted FC matrix, since FC matrix is symmetric and whose diagonal values are always 1. We reference it as Tril-MSE in the following. We also use Pearson Correlation as a supplementary measure to double evaluate the quality of predictions, although we do not train our models on it. Same as Tril-MSE, we apply it only on the lower triangular part of output matrices, we reference it as TPearson for the rest of the paper.

Results and discussion

We evaluate the average and median Tril-MSE test error, as well as the average and median TPearson between predicted FC and empirical FC. It is also important to evaluate the median test error because outliers are often present in data and difficult to identify beforehand. A model with a lower median test error is more robust when it comes to a dataset with outliers, an important consideration in clinical applications.

As presented in Table 1, the first architecture "GCN+DC1" from [START_REF] Li | Mapping Brain Structural Connectivities to Functional Networks Via Graph Encoder-Decoder With Interpretable Latent Embeddings[END_REF] is the baseline for comparison, while the rest are the new models proposed in this work. We observe that both GCN and GTN encoder combined with DC2, defined in Equation (3) have obtained better scores in all measures than when they are combined with DC1, defined in Equation (2). This improvement can be explained by the additional trainable parameters of the decoder which projects the latent space to the FC prediction. When comparing the median Tril-MSE, we can notice that GTN increases the performance by up to 18% when compared to GCN. This significant change in the median error highlights the robustness of GTN to outliers. In addition, GTN also outperforms GCN at the supplementary measure TPearson. Also note that the GTN always provides a better FC prediction regardless of the decoder selected, showing that important information is captured in the Length, and SIF T 2 matrices not exploited by the GCN architecture.

Conclusion

A better understanding of the link between SC and FC has the potential to provide insights into how lesions to the structural substrate affect brain function. In this work, we have revisited the structure-function mapping via graph neural networks. We investigated the importance of a trainable decoder in a previously proposed GCN architecture and proposed a novel GTN architecture to predict FC from several diffusion derived SC matrices. We presented prediction results on 1050 subjects of the Human Connectome Project, showing that trainable parameters in the decoder improve the performance of the models. In addition, our results indicated that GTNs, which simultaneously exploit different realisations of the SC matrix, outperformed GCNs. Overall, the preliminary results presented in this work highlight the potential for graph neural networks in structure-function mapping.

Fig. 1 :

 1 Fig. 1: Model pipeline. The first step is to process SC matrix to obtain Â. Then the encoder(GCN or GTN) takes  as input and generate embedding Y . Finally, the decoder(DC1 or DC2) takes Y as input and reconstruct FC matrix Z . A loss function(Tril-MSE) calculates the error between Z and the empirical FC.

Table 1 :

 1 Average and median test error for the graph convolutional and transformer networks using different decoders: DC1 and DC2 are defined in Equation (2) and (3), respectively.

	Results	Average	Median	Average	Median
		Tril-MSE	Tril-MSE	TPearson	TPearson
	GCN+DC1	0.0430±0.0016	0.0431	0.6628±0.0077	0.6671
	GCN+DC2	0.0404±0.0018	0.0402	0.7003±0.0073	0.7069
	GTN+DC1	0.0401±0.0020	0.0351	0.7160±0.0077	0.7228
	GTN+DC2 0.0397±0.0018	0.033	0.7154±0.0061	0.7231

6 Acknowledgment

 6 This work has been supported by the ERC under the European Union's Horizon 2020 research and innovation program (ERC Advanced Grant agreement No 694665:CoBCoM : Computational Brain Connectivity Mapping) and by the French government, through the 3IA Côte d'Azur Investments in the Future project managed by the National Research Agency (ANR) with the reference number ANR-19-P3IA-0002. Data were provided by the Human Connectome Project, WU-Minn Consortium (Principal Investigators: David Van Essen and Kamil Ugurbil; 1U54MH091657)

http://www.humanconnectomeproject.org/

The authors are grateful to Inria Sophia Antipolis -Méditerranée https:// wiki.inria.fr/ClustersSophia/Usage_policy"Nef" computation cluster for providing resources and support.

The authors are grateful to the OPAL infrastructure from Université Côte d'Azur for providing resources and support.


Omar Bouhidel

Sandrine Pons

Richard Souktani

Roland Zini

Alain Berdeaux

Bijan

Bouhidel Omar

Pons Sandrine

Souktani Richard

Zini Roland

Berdeaux Alain

Ghaleh Bijan

Myocardial ischemic postconditioning

Keywords: AMP-Activated Protein Kinases, metabolism, Animals, Disease Models, Animal, Dual Specificity Phosphatase 6, metabolism, Male, Mice, Mice, Inbred C57BL, Mice, Obese, Mitogen-Activated Protein Kinase 1, metabolism, Mitogen-Activated Protein Kinase 3, metabolism, Myocardial Infarction, etiology, metabolism, pathology, prevention & control, Myocardial Ischemia, complications, metabolism, pathology, therapy, Myocardial Reperfusion Injury, etiology, metabolism, pathology, prevention & control, Myocardium, enzymology, metabolism, pathology, Obesity, complications, metabolism, pathology, PTEN Phosphohydrolase, metabolism, Phosphoprotein Phosphatases, metabolism, Phosphorylation, Proto-Oncogene Proteins c-akt, metabolism, Ribosomal Protein S6 Kinases, 70-kDa, metabolism, Signal Transduction Infarction, Obesity, postconditioning, kinase, phosphatase

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INTRODUCTION

Ischemic postconditioning has been described for the first time in dogs by Zhao et al. ( ). These authors demonstrated that a series of 42 repetitive cycles of brief reperfusion and reocclusion of the coronary artery applied at the onset of reperfusion after a prolonged ischemic event reduced infarct size. This finding has been confirmed in numerous investigations performed in mice ( , ), rats ( , , ), 17 [START_REF] Kaljusto | Postconditioning in rats and mice[END_REF] 25 35 38 rabbits ( , , , ), pigs ( ) as well as in humans ( , , ). To date, different mechanisms have been associated with this 2 6 7 40 18 8 31 33 cardioprotective strategy ( , ). Among them, a pro-survival signaling pathway so-called the reperfusion injury salvage kinase (RISK) 13 41 pathway has been demonstrated to play a key role in postconditioning ( , ). This pathway includes several pro-survival kinases such as 7 38 phosphatidyl-inositol-3-OH-kinase (PI K)-Akt, p42/44 extra-cellular signal-regulated kinases (ERK1/2)orp70S6K1 ( ).

3 [START_REF] Hausenloy | New directions for protecting the heart against ischaemia-reperfusion injury: targeting the Reperfusion Injury Salvage Kinase (RISK)-pathway[END_REF] The majority of these studies investigating postconditioning have been conducted in healthy animals and there is a paucity of experimental investigations with associated risk factors such as hyperlipidemia, diabetes or obesity. Importantly, these pathological situations as well as aging can jeopardize the effectiveness of well established cardioprotective strategies in healthy animals ( , , , ).

3 19 22 37 To date, obesity is a major health issue in the western countries and its importance is constantly growing. This pathology constitutes a major risk factor for myocardial infarction and cardiovascular disease ( ). Therefore, the effect of any developed cardioprotective 16

strategy should be investigated in the context of this co-morbidity.

Accordingly, the aim of the present study was to investigate the effects of postconditioning against myocardial infarction and subsequent activation of the RISK pathway in a mouse model of obesity. For this purpose, we chose the leptin-deficient so-called ob/ob mice, characterized by hyperphagia, decreased energy expenditure and early onset of obesity ( ).

METHODS

Animals

Male 8 10 weeks old wild-type C57BL/6J (WT) and ob/ob mice were used (R. Janvier, Le Genest St Isle, France). Mice were housed in an air-conditioned room with a 12 h light-dark cycle and received standard rodent chow and drinking water The experiments ad libitum.

were performed in accordance with the official regulations of the French Ministry of Agriculture.

Plasma measurements

Plasma glucose, triglycerides, cholesterol and non-esterified fatty acids (NEFA) levels were measured using a multiparametric automat (Olympus automat type AU-400, Rungis, France).

Experimental protocol

Mice were anesthetized by intraperitoneal injection of pentobarbital sodium (50 mg/kg), intubated and ventilated mechanically. The body temperature was maintained at 37 C. A left thoracotomy was performed in order to realize the sequences of coronary artery occlusion

°(CAO)
followed by reperfusion (CAR) with an 8-0 Prolene thread placed around the left coronary artery as previously described ( ).

28

Myocardial ischemia was confirmed by ST segment deviation of the electrocardiogram and the occurrence of regional cyanosis.

Reperfusion was confirmed by visualisation of hyperemic response and the chest was closed in layers.

The experimental protocol is shown in . Both WT and ob/ob mice were randomly subjected to 30 min CAO (Control-WT and Figure 1 Control-ob/ob, respectively) or 30 min CAO followed by 6 cycles of 10 s CAR/10 s CAO (IPCD-WT and IPCD-ob/ob, respectively).

Infarct sizes were measured at 24 h of CAR. Additional animals underwent the same protocol but were sacrificed at 10 min of CAR in order to perform Western Blot experiments. Finally, hearts were also obtained from WT and ob/ob mice that did not underwent any intervention in order to perform Western blot analysis at baseline.

Myocardial infarct size determination

At the end of the 24 h reperfusion period, mice were reanesthetized, the coronary artery was reoccluded at the previous site of occlusion and the heart was excised after Evan s blue perfusion. The area at risk was identified using Evan s blue staining and the infarct

' '
area was identified by 2,3,5-triphenyltetrazolium chloride (TTC) staining. The area at risk was identified as the non-blue region and expressed as a percentage of the left ventricle weight. The infarcted area was identified as the TTC negative zone and expressed as a percentage of the area at risk.

Tissue preparation

Fresh ventricular tissues were placed in medium containing 220 mM mannitol, 70 mM sucrose, 10 mM HEPES, 1 mM EGTA, 1 mM PMSF, 5 l/ml of protease inhibitor cocktail, 1 mM sodium orthovanadate, 5 mM sodium fluoride, 1 mM sodium Na -glycerol phosphate

μ 2 β
(Sigma-Aldrich, St. Louis, MO, USA), pH 7.4 at 4 C. The tissues were scissor minced and homogenized on ice using a Teflon Potter °homogenizer.

The homogenate was centrifuged at 17600 for 30 min at 4 C to collect the cytosol. The cytosolic protein concentration g °was determined by the BCA Protein Assay Kit (Pierce, Rockford, IL, USA).

Western Blot analysis

Extracted protein samples were denatured at 95 C for 5 min. Proteins (30 g/lane) were separated by SDS-polyacrylamide gel °μ electrophoresis, transferred to nitrocellulose membranes, and probed with primary antibodies for phospho-Akt (Serine 473), phospho-ERK 1/2 (Threonine 202/Tyrosine 204), phospho-p70S6K1 (Threonine 398), phospho-p70S6K1 (Threonine 421/Serine 424), phospho-AMPK (Threonine 172), total Akt, total ERK1/2, total p70S6K1, total AMPK, PTEN (all from Cell Signaling Technology, Danvers, MA, USA), MKP-3 (Epitomics, Burlingame, CA, USA) and PP2C (Abcam, Cambridge, UK). Blots were washed and then incubated with goat anti-rabbit IgG secondary antibody conjugated to horseradish peroxidase (Santa Cruz Biotechnology, Santa Cruz, CA, USA). Blots were finally incubated using the ECL detection system (ECL Western Blotting Substrate, Pierce, IL, USA) and exposed to ECL chemiluminescence film (Hyperfilm ECL, Amersham, Buckinghamshire, UK). Bands of interest were scanned and quantified in a blinded manner using gel analysis software ImageJ-1.37 (National Institute of Health, Bethesda, ML, USA).

Statistical analysis

All values are expressed as mean SEM. Comparisons were performed using Kruskall-Wallis analysis followed if necessary by ± Mann-Whitney test. Statistical significance was defined as a value of <0.05. p

RESULTS

Biological and morphological parameters

The biological characteristics of WT and ob/ob mice are summarized in . Serum glucose and cholesterol levels were Table 1 significantly higher in ob/ob as compared to WT mice. Similar concentrations of triglycerides and NEFA were observed in the serum of both strains.

As shown in

, body weight was significantly increased by 1.9 fold in ob/ob as compared to WT mice. Left ventricular weights Table 2 were modestly ( 13 ) but significantly greater in ob/ob than in WT mice and this mild left ventricular hypertrophy was confirmed by + % calculation of the left ventricular weight to tibia length ratio (also 13 ).

+ %

Myocardial infarct sizes

The areas at risk were similar among all groups excepted for the ob/ob mice subjected to postconditioning (44 3 , n 9; 40 3 , n 9

± % = ± % = and 41 4 , n 8 for Control-WT, IPCD-WT and Control-ob/ob as compared to 27 2 , n 7 for IPCD-ob/ob, respectively). The area at risk

± % = ± % =
of postconditioned ob/ob mice was intentionally reduced. Indeed, in preliminary experiments, all ob/ob mice subjected to postconditioning with an area at risk averaging 40 died during reperfusion.

%

As illustrated in , postconditioning significantly reduced infarct size by 58 in WT mice (33 1 14 3 for Control-WT Figure 2 % ± % vs ± % and IPCD-WT, respectively). In contrast, this strategy failed to protect ob/ob mice against myocardial infarction (53 4 56 5 for ± % vs ± % Control-ob/ob and IPCD-ob/ob, respectively). Interestingly, in control conditions, infarct size was significantly increased by 61 in ob/ob % as compared to WT mice.

Baseline analysis

As illustrated in , phospho-Akt (Ser 473)/Akt, phospho-ERK 1/2(Thr 202/Tyr 204)/ERK 1/2 and phospho-AMPK (Thr Figure 3 172)/AMPK ratios were significantly increased in ob/ob (n 5) as compared to WT (n 5) mice at baseline, ., in the absence of

= =
i.e ischemia-reperfusion. Concomitantly, PTEN, MKP-3 and PP2C were significantly decreased in ob/ob as compared to WT mice.

Effect of IPCD on PTEN, MKP-3, PP2C and the phosphorylation of Akt, ERK 1/2, p70S6K1 and AMPK

The effects of IPCD on the phosphorylation states of pro-survival kinases Akt, ERK 1/2 and p70S6K1 as well as AMPK were investigated in Control-WT (n 3), IDPC-WT (n 3), Control-ob/ob (n 4) and IDPC-ob/ob (n 4) mice.

= = = =
As illustrated in , IPCD induced a significant increase in the phospho-Akt (Ser 473)/Akt ratio ( 77 ) in WT mice. Figure 4A + %

Similarly, in WT mice, IPCD significantly increased the phosphorylated ERK 1/2 (Thr 202/Tyr 204)/ERK 1/2 ratio ( 41 ) ( ). + % Figure 4B In contrast, these effects were not observed in ob/ob mice. In control conditions ( ., in the absence of IPCD), these ratios were i.e significantly greater in ob/ob as compared to WT mice.

As p70S6K1 is known to be phosphorylated at Threonine 389 by activated Akt as well as at both Threonine 421 and Serine 424 by activated ERK 1/2, we examined these phosphorylations. As illustrated in , IPCD induced significant increases in phosphorylated Figure 5 p70S6K1 (Thr 389)/p70S6K1 ( 153 ) and phosphorylated p70S6K1 (Thr 421/Ser 424)/p70S6K1 ratios ( 57 ) in WT mice. This strategy

+ % + %
failed to elicit similar effects in ob/ob mice.

As illustrated in , the phosphorylated AMPK (Thr 172)/AMPK ratio was significantly increased by IPCD ( 64 ) in WT but Figure 6 + % not in ob/ob mice.

Concerning phosphatases, IPCD induced significant decreases in PTEN, MKP-3 and PP2C levels in WT mice. Interestingly in ob/ob mice, PTEN, MKP-3 and PP2C were significantly increased with IPCD as compared to their respective control. In control conditions ( ., i.e in the absence of postconditioning), the three phosphatase levels were significantly lower in ob/ob as compared to WT mice.

DISCUSSION

The present study demonstrates for the first time that postconditioning fails to limit infarct size in obese mice. This lack of efficacy was accompanied by an impaired activation of the PI K-Akt and ERK 1/2 signaling pathways. To date, there is a consensus indicating that 3 the postconditioning stimulus must be applied immediately within the first minutes after the onset of reperfusion ( , ). Concerning the 25 40 algorithm, the number and the duration of cycles are of major importance ( ). The postconditioning protocol used in the present study 39

was determined on the basis of preliminary experiments and literature. We rather chose very short durations as their beneficial effects have been demonstrated in rats, ., periods of 10 s were more potent than those of 30 s ( ). Concerning the number of cycles, a study i.e 35 performed in isolated mice hearts reported that 6 postconditioning cycles were more effective in improving post-ischemic systolic and diastolic functions than 3 postconditioning cycles ( ). Accordingly, we investigated the cardioprotective effects of 6 cycles of 10 s 24 reperfusion/10 s occlusion. This postconditioning protocol conferred a rather strong cardioprotection in WT mice as infarct size was significantly reduced by 58 . It is therefore unlikely that the failure of postconditioning to protect ob/ob mice against myocardial % infarction is related to the algorithm. Similar alterations of cardioprotection with postconditioning have been reported in hypercholesterolemic rabbits and during aging ( , ). Paradoxically, in our study, postconditioning tended to exacerbate the consequences 3 19

of ischemia-reperfusion as the postconditioned ob/ob mice elicited a greater mortality when the area at risk was similar to the other groups of mice. So, we had to intentionally reduce this parameter in ob/ob mice subjected to postconditioning.

It is well known in normal animals that the activation of the so-called RISK pathway is essential for cardioprotection by postconditioning. It includes the PI K-Akt and ERK 1/2 cascades as well as their common downstream kinase target p70S6K1 ( , ). In 3 7 38 the present study, postconditioning significantly increased the phosphorylated state of Akt and ERK 1/2 in WT mice. Consequently, we found that phosphorylation of p70S6K1 was increased at Thr 389 (the site phosphorylated by Akt) and Thr 421/Ser 424 (the sites phosphorylated by ERK 1/2). In contrast, postconditioning failed to increase the phosphorylation of Akt, ERK 1/2 or p70S6K1 in ob/ob mice, comforting the infarct size results. We also investigated AMPK, a serine/threonine protein kinase, which acts as a fuel sensor responsible for mediating the cellular adaptation to environnemental stress ( ). The phosphorylation of its threonine 172 site is essential 14

for AMPK activation ( ) which has been demonstrated to limit ischemic injury during postischemic reperfusion ( ). Our results 34 30 demonstrated that postconditioning activated AMPK by increasing its phosphorylation state in WT mice. Once again in ob/ob mice, postconditioning failed to elicit a similar effect.

The mechanisms responsible for this failure of postconditioning to activate the RISK pathway and to reduce infarct size in ob/ob mice remain unclear. Nevertheless, some hypothesis can be raised. At baseline ( ., in the absence of ischemia-reperfusion) and in control i.e conditions ( ., without postconditioning), phosphorylation of most of the investigated kinases was significantly greater in ob/ob as i.e compared to WT mice. This might be the consequence of left ventricular hypertrophy which has been reported to activate these signaling pathways ( ). In addition, such increased phosphorylation of Akt has been previously reported in hepatic and muscular tissues of obese 10 rats ( ). This was accompanied by a concomitant and significant decrease in phosphatase levels such as PTEN which dephosphorylates 23

PIP to PIP ( ), MKP-3, a member of the dual-specificity phosphatases which plays a major role in ERK 1/2 dephosphorylation ( ) and 3 2 26 5 PP2C which is known to be a major enzyme that inactivates AMPK ( , ). Nevertheless, this phenomenon should theoretically end up 9 27

with a reduced infarct size in ob/ob as compared to WT mice in the absence of postconditioning. However our experiments revealed the opposite as infarct size was significantly increased by 61 in ob/ob as compared to WT mice, in agreement with previous studies % performed in diabetic mice and obese rats ( , , ). During postconditioning, we did not observe any increase in phosphorylation of 11 20 22 Akt, ERK 1/2, p70S6K1 or AMPK in ob/ob mice. Our results show for the first time that PTEN, MKP-3 and PP2C are significantly increased with postconditioning in ob/ob mice while these phosphatases are significantly decreased in WT mice in the same situation.

Interestingly, such reduction in PTEN protein has been previously reported with preconditioning in normal mice ( ). Although we did not 4 directly measure phosphatase activities, one could speculate that an enhanced phosphatase activity would be responsible for the lack of enhanced phosphorylation of protecting kinases and therefore has contributed to the lack of beneficial effect of postconditioning.

Interestingly, alterations in phosphatase activity or expression limit the efficacy of both preconditioning and postconditioning during aging ( , ). Finally, we could also suggest that protein kinases and phosphatase determinants for infarct tolerance and transmission of 12 29 cytoprotective signals may be different between C57BL/6J and ob/ob mice.

It should be stressed that our results could be explained by the lack of leptin which exhibits cardioprotective properties by activating the RISK pathway ( ). Indeed, increase in infarct size in control conditions ( ., in the absence of postconditioning) could result from the 32 i.e loss of protection afforded by leptin in ob/ob mice. In addition, although ob/ob mice exhibit the characteristics of obesity, it should be acknowledged that leptin mutation does not represent all causes of obesity in humans. Nevertheless, one would expect a decrease in Akt or ERK phosphorylation in ob/ob mice. In fact, we observed the opposite as these phosphorylations were increased along with a greater infarct size in ob/ob as compared to WT mice. Interestingly, administration of exogenous leptin in ob/ob mice has been demonstrated to worsen infarct volume following ischemia-reperfusion in the brain and this effect was associated with an increased inflammatory response ( ). Furthermore, postconditioning has been shown to be abrogated with hypercholesterolemia (cholesterol-enriched diet rabbits) ( ), 36 19 i.e

., a model independent of leptin signaling disruption. Thus, although we cannot rule out that our results could be the consequence of the lack of leptin, we believe that the abrogation of postconditioning in ob/ob mice is rather due to obesity.

In conclusion, the present study demonstrates that obesity impairs the ability of postconditioning to protect the heart against myocardial infarction. Further studies are necessary to determine how to restore the cardioprotective effect of postconditioning in this context of obesity.

Ackowledgements:

This study was supported by grants from the Fondation de France (2005-005338 and 2008-002688). Omar Bouhidel was a recipient from the Minist re de l Enseignement Sup rieur et de la Recherche. The authors wish to thank CEFI (IFR02, Universit Denis Diderot-Paris7, Paris è ' é é F-75018, France) for blood analyses.

References:

Figure 1

 1 Figure 1Experimental protocols used for investigating the effects of ischemic postconditioning (IPCD) in wild-type and ob/ob mice (CAO, coronary artery occlusion; CAR, coronary artery reperfusion).

Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7

 234567 Figure 2 Infarct size (expressed as percentage of the area at risk) measured in wild-type (WT) and ob/ob mice subjected to 30 min of coronary artery occlusion and 24 h of reperfusion in the absence or presence of postconditioning (Control and IPCD, respectively). Open circles represent individual values and closed circles indicate the average. <0.05 Control; <0.05 respective WT.*p

Table 1

 1 Glycemia, triglyceridemia, cholesterolemia and NEFA measurements in WT and ob/ob mice

	±	=	=	*p	vs	†p	vs

±

Values are expressed as mean SEM.

Table 2

 2 General parameters in WT and ob/ob mice.

	WT	ob/ob
	Control (n 9)

= IPCD (n 9) = Control (n 8) = IPCD (n 7) = ± 17.1 0.2 ± 17.3 0.2 ± 17.3 0.3 ± Left ventricular weight/tibial length (mg/mm) 4.6 0.1 ± 4.4 0.2 ± 5.1 0.2

[START_REF] Fenton | Inhibition of phosphatase activity enhances preconditioning and limits cell death in the ischemic/reperfused aged rat heart[END_REF]

[START_REF] Fenton | Inhibition of phosphatase activity enhances preconditioning and limits cell death in the ischemic/reperfused aged rat heart[END_REF]

[START_REF] Fenton | Inhibition of phosphatase activity enhances preconditioning and limits cell death in the ischemic/reperfused aged rat heart[END_REF]

[START_REF] Fenton | Inhibition of phosphatase activity enhances preconditioning and limits cell death in the ischemic/reperfused aged rat heart[END_REF]

[START_REF] Fenton | Inhibition of phosphatase activity enhances preconditioning and limits cell death in the ischemic/reperfused aged rat heart[END_REF]

[START_REF] Fenton | Inhibition of phosphatase activity enhances preconditioning and limits cell death in the ischemic/reperfused aged rat heart[END_REF]

[START_REF] Fenton | Inhibition of phosphatase activity enhances preconditioning and limits cell death in the ischemic/reperfused aged rat heart[END_REF]

[START_REF] Fenton | Inhibition of phosphatase activity enhances preconditioning and limits cell death in the ischemic/reperfused aged rat heart[END_REF]

[START_REF] Fenton | Inhibition of phosphatase activity enhances preconditioning and limits cell death in the ischemic/reperfused aged rat heart[END_REF]

[START_REF] Fenton | Inhibition of phosphatase activity enhances preconditioning and limits cell death in the ischemic/reperfused aged rat heart[END_REF]


Ihou Hilaire

Krunoslav Mouko

Mahdi Romanjek

Mostafa Mejri

Soufiane Oulfarsi

Pauline El Oualid

Yohann Malinconi

Benoît Thimont

Claude Malard

Nicolas Estournès

David

H Ihou Mouko

K Romanjek

M Mejli

M Oulfarsi <l

S El

Oualid <l

P Malinconi

Y Thimont

B Malard

C Estomnès C

N David

A Dauscher

Manufacturing and performances of silicide-based thermoelectric modules

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

Introduction

The search for new non-polluting energy sources, substitutes of pri maiy fossil energies, has become a major challenge for our modern world, especially after the signing of the Kyoto Protocol in 1995 (COP 1). Knowing that a large pait of the fossil energies used is even lost, generally as waste heat, for instance internai combustion engines of cai•s only convert about 30% of the prin1aiy energy while the other pait is lost as heat in both the exhaust gases or the coolant [START_REF] Saidur | Technologies to recover exhaust heat from internal combustion engines[END_REF], it becomes inter esting to think on how it could be recycled. The use of thermoelectric (TE) devices appea1-s as an elegant way to convert directly this waste heat into useful electricity using the Seebeck effect, with the advantage of having no moving paits insming silence, scalability, and robustness for a long-tem1 use. They have been used for yea1-s for space applications to power spatial probes for a mission far from the Sun [START_REF] Cataldo | Space radioisotope power Systems and Applications: Past, Present and Future[END_REF], and more recently the Cmiosity and Pel'Severance revers launched on Mai• s [START_REF] Holgate | Increasing the efficiency of the multi-mission radioisotope thermoelectric generator[END_REF].

• Corresponding author.

E-mail address: hilaire.ihou-mouko@hotblock.fr (H. !hou Mouko). https://doi.org/10.1 0l 6/j.enconman.2021. 114304 TeITestrial applications concem the heat recoveiy for instance in the automotive [START_REF] Brignone | Impact of novel thermoelectric materials on automotive applications[END_REF] or aircraft sectors [START_REF] Bode | Potential of future thermoelectric energy recuperation for aviation[END_REF], the cement industries [START_REF] Qi | A Thermoelectric Waste-Heat-Recovery System for Portland Cement Rotary Kilns[END_REF], and quite recently the sector of micro-generation for powei'ing wireless sensors networks [START_REF] Hudak | Small-scale energy harvesting through thermoelectric, vibration, and radiofrequency power conversion[END_REF] and Intei•net of Things [START_REF] Haras | Thermoelectricity for IoT -A review[END_REF].

Conventional it-shaped TE devices are constituted of a plurality of TE couples (or legs) made of nand p-type semiconducting matelials con nected electrically in series and the1mally in parallel. The TE legs are usually sandwiched between two metalized cerantic plates and brazed or soldei•ed on the electrical tracks in order to provide a good electrical and them1al transfei• and to insure a high mechanical cohesion of the whole set. The manufacturing and development of efficient and long-tem1 stable TE modules is however a great challenge, especially for ntiddle or high opei•ating tempei•atures, due both to the TE matei'ials themselves (chentical and mechanical stability problen1s) and to the presence of numerous interfaces ( diffusion, elecnical and them1al contact re sistances, expansion coefficient compatibilities, .... problen1s). Vaifous

Experimental

Thermoelectric materials and legs preparation

The raw powders used in this study were produced in large quantities, ten kilograms per batch for each type of material, with the targeted n-type composition Mg 2 Si 0.6 Sn 0.4 doped with 1.35 wt% Sb (further called MSS) and p-type composition MnSi 1.77 (further called HMS). They were obtained from the melting of commercial powders of Mg (99%), Si (99.9%), Sn (99.9%), Sb (99.9%) and Mn (99%) followed by ball milling. The powders were then densified by spark plasma sintering (SPS) in graphite dies under argon atmosphere. The materials were held at 750 • C during 5 min under 50 MPa and at 1030 • C during 5 min under 55 MPa for the n-and p-types, respectively. The obtained pellets (60 mm in diameter and about 5.5 mm in thickness) were further annealed in an oven under argon atmosphere for 2 h at 450 • C and 900 • C for the n-and p-type materials, respectively. They were rectified mechanically under oil down to a thickness of 4.60 ± 0.02 mm, corresponding to the height of the thermoelectric legs, using a AC-530 Peter-Wolters machine. A metallic multi-layer constituted of Ti (30 nm)/Au (335 nm) was deposited on the lower and upper faces of the rectified pellets by magnetron sputtering (ALCATEL DP28). Finally, the pellets were cut in TE legs of dimensions 3 × 3 × 4.6 mm 3 with a wire saw under oil and with a micro-chain saw for the MSS and HMS materials, respectively. About 250 legs can be produced in one sintered pellet. Prior to their integration in a thermoelectric module, the resistance of all the legs was measured in order to eliminate any leg showing too much variation with regard to the mean value, resulting for instance from an unexpected contact resistance. For the HMS material, the mean resistance value is 8.3 mΩ, with a standard deviation of 2%, while it is 2.1 mΩ with a standard deviation of 13% for the MSS material. The lower resistance value results from the lower intrinsic resistivity (as will be seen later) while the higher standard deviation is due to the presence of some composition inhomogeneities in the material as it has already been shown by Mejri et al. [START_REF] Mejri | Thermal stability of Mg 2 Si 0.55 Sn 0.45 for thermoelectric applications[END_REF]. Finally, all the legs issued either from a HMS or a MSS pellet have been qualified. Knowing the position of the leg in the pellet, neither edge-center nor shading effects have been observed, showing the reliability of the SPS fabrication process, even for a diameter as large as 60 mm.

Module design and assembly

This TE module model produced by HBOB is composed of 24 TE legs (Fig. 1a), i.e. of 12n(MSS)/p(HMS) pairs, prepared as previously described. The originality of these modules lies in their 'half-skeleton' structure with the cold side conventionally linked to a ceramic plate but with the hot side linked to a thin silver foil acting as the electrode. This design ensures flexibility at the hot side such as to counterbalance the mismatch of thermal expansion coefficients between HMS and MSS, as mentioned in the introduction. In order to facilitate the study of reliability and repeatability tests, modules constituted of only two n/p couples have been fabricated in this study (Fig. 1b). Twenty-two of such modules have been manufactured according to the process flow described in Fig. 2.

Prior to the assembly, the n and p metallised TE legs, the AlN ceramic plates and the silver electrodes have been cleaned with acetone and absolute ethanol in an ultrasonic bath. The first assembly step consists in applying a commercial silver brazing paste by screen printing on the cold side electrodes made of nickel-coated copper tracks deposited on an aluminium nitride (AlN) ceramic substrate by the direct bonding copper (DBC) technique. Thanks to a pick & place machine, the legs are placed on the silver braze (step 2). Silver braze is then applied on the top of the TE legs (step 3), and finally the top silver electrode foils are automatically placed on the legs (step 4), consisting in the hot side of the module. The whole module is further placed in a thermo-press at 250 • C under 43 MPa for 30 min in air to sinter the brazing paste and ensure the final TE modules architectures and designs, as well as manufacturing methods have been proposed owing to the improvement in their mechanical robustness, chemical stability and electrical performance [see the recent reviews 9-11, and the references therein].

Industrial terrestrial needs are mainly limited to near room (0-200 • C) and middle (200-500 • C) temperature ranges [START_REF] Champier | Thermoelectric generators: a review of applications[END_REF]. For near room temperatures applications, the well-established TE modules are based on bismuth and antimony telluride alloys [START_REF] Goldsmid | Bismuth telluride and its alloys as materials for thermoelectric generation[END_REF]. However, for medium temperature range applications, very few TE modules are available on the market today, while the need is real and the progress on the materials performance (expressed as the dimensionless TE figure of merit ZT: ZT S 2 T/ρκ, where T is the absolute temperature, S the Seebeck coefficient, ρ the electrical resistivity, and κ the thermal conductivity) is effective. Indeed, several new promising families of TE materials have been identified during these last two-three decades in this temperature range such as for instance skutterudites [START_REF] Rogl | Skutterudites, a most promising group of thermoelectric materials[END_REF], Half-Heuslers [START_REF] Poon | Recent advances in thermoelectric performance of Half-Heusler compounds[END_REF], SnX (X Te, Se) [START_REF] Moshwan | Eco-friendly SnTe thermoelectric materials: Progress and future challenges[END_REF][START_REF] Li | Advances in environment-friendly SnTe thermoelectrics[END_REF] or silicide-based materials [START_REF] Fedorov | Thermoelectric silicides: past, present and future[END_REF][START_REF] Pandel | A review of the Mg 2 (Si, Sn) alloy system as emerging thermoelectric material: experimental and modeling aspects[END_REF]. More details on the performances of all these materials, as well as others, can be found in recent general reviews [START_REF] Beretta | From history, a window to the future[END_REF][START_REF] Schierning | Concepts for medium-high to high temperature thermoelectric heat-to-electricity conversion: a review of selected materials and basic considerations of module design[END_REF]. Among the silicides family, p-type Higher Manganese Silicide (HMS) and n-type Mg 2 (Si,Sn) solid solutions are particularly promising candidates as they are constituted of non-toxic, Earth-abundant, light, and low cost elements [START_REF] Amatya | Trend for thermoelectric materials and their earth abundance[END_REF][START_REF] Leblanc | Material and manufacturing cost considerations for thermoelectrics[END_REF], while they exhibit high TE performance [START_REF] Fedorov | Thermoelectric silicides: past, present and future[END_REF][START_REF] Pandel | A review of the Mg 2 (Si, Sn) alloy system as emerging thermoelectric material: experimental and modeling aspects[END_REF]. The objective of much of the recent investigations has clearly been to improve the figure of merit ZT by simultaneously decreasing the electrical resistivity and the thermal conductivity, which is a difficult task, by appropriate doping, band structure and/or microstructure engineering [24,25 and the references inside]. For n-type Mg 2 (Si,Sn) solid solutions, the maximum published ZT values are up to 1.5 around 500 • C, when the materials are doped with Sb [START_REF] Liu | Optimized thermoelectric properties of Sb-doped Mg 2(1+z) Si 0.5-y Sn 0.5 Sb y through adjustment of the Mg content[END_REF][START_REF] Gao | Transport and mechanical properties of high-ZT Mg 2.08 Si 0.4 x Sn 0.6 Sb x thermoelectric materials[END_REF][START_REF] Bernard-Granger | Microstructure investigations and thermoelectrical properties of an N-type magnesium-silicon-tin alloy sintered from a gas-phase atomized powder[END_REF]. The p-type Mg 2 (Si,Sn) materials unfortunately possess lower TE performance, a maximum of 0.6 having been nevertheless recently achieved at 380 • C with the Li element [START_REF] De Boor | Thermoelectric performance of Li doped, p-type Mg2(Ge,Sn) and comparison with Mg2(Si,Sn)[END_REF]. It is why the n-type Mg 2 (Si-Sn) materials are often combined with p-type HMS (MnSi 1+x ) materials for the fabrication of TE modules, even if the performance of the HMS is less as they exhibit only ZT values ranging between 0.6 and 0.8 at around 530 • C [START_REF] Fedorov | Thermoelectric silicides: past, present and future[END_REF][START_REF] Luo | Rapid synthesis of high thermoelectric performance higher manganese silicide with in-situ formed nanophase of MnSi[END_REF][START_REF] Luo | Improved thermoelectric properties of Al-doped Higher Manganese Silicide prepared by a rapid solidification method[END_REF][START_REF] Bernard-Granger | Microstructure investigations and thermoelectrical properties of a P-type polycrystalline higher manganese silicide material sintered from a gas-phase atomized powder[END_REF]. However, problems of thermomechanical stability may arise from the mismatch between the coefficient of thermal expansion of both materials (12 10 -6 K -1 and 17 10 -6 K -1 for the n and p-type materials, respectively [START_REF] Mejri | Characterization of the thermomechanical properties of p-type (MnSi1.77) and n-type (Mg 2 Si 0.6 Sn 0.4 ) thermoelectric materials[END_REF]), as it has already been shown for such modules [START_REF] Skomedal | Design, assembly and characterization of silicide-based thermoelectric modules[END_REF]. To try to solve this problem, Skomedal et al. [START_REF] Skomedal | Design, assembly and characterization of silicide-based thermoelectric modules[END_REF] have manufactured TE modules using electrical contacts with springs. The springs provide pressure on the TE legs to maintain thermal and electrical contact and ensure flexibility of the whole. Nemoto et al. have developed unileg TE modules based on n-type Mg 2 Si [START_REF] Nemoto | Development of an Mg 2 Si unileg thermoelectric module using durable Sb-doped Mg 2 Si legs[END_REF][START_REF] Nemoto | Improvement in the durability and heat conduction of uni-leg thermoelectric modules using n-type Mg 2 Si legs[END_REF]. This architecture is certainly more robust and reliable than a two-leg architecture since there is no problem of thermal expansion mismatch between the two TE materials, but the achieved TE performance is of course less.

In this paper, we present the manufacturing of silicide-based TE modules from TE materials prepared by an industrial process. We introduce a new design to minimize the thermomechanical issues, by developing 'half-skeleton' modules. Twenty-two such TE modules made of two pairs (n/p) legs were manufactured, and their performance tested in term of output power. The repeatability module to module was excellent while the achieved areal output power lies within the state of the art compared to other modules fabricated either with silicides or other TE materials working in the same temperature range (skutterudites, half-Heuslers, …). Among all these materials, the silicides have the lowest cost/kg [START_REF] Amatya | Trend for thermoelectric materials and their earth abundance[END_REF][START_REF] Leblanc | Material and manufacturing cost considerations for thermoelectrics[END_REF] which gives a competitive advantage for our modules. The TE performances of our TE modules as function of temperature are also compared with analytical and numerical models. assembly of the different parts (step 5).

Fig. 3 shows a general scheme of the final structure of the fabricated TE modules., with the different interfaces in presence and their thickness.

Microstructural and TE characterizations of the materials

The apparent densities of the SPS sintered HMS and MSS samples were measured using the Archimedes method, in distilled water and absolute ethanol, respectively. The quality of the interfaces at the different leg/electrode junctions of a polished assembled TE module was checked by scanning electron microscopy (SEM) in the back-scattered electron mode (BSE) and elemental X-ray mapping on a Quanta FEG 650 (FEI) instrument.

The TE properties of the HMS and MSS materials were investigated in the 25-700 • C temperature range on different sintered samples. Samples of appropriate shape were cut from the non-metallized SPS pellets. The electrical conductivity (σ) and the Seebeck coefficient (S) were measured simultaneously with a four-probe method using a ZEM-3 (ULVAC GmbH) apparatus on parallelepipedic-shaped samples of 3 × 3 × 12 mm 3 in size under helium atmosphere. The thermal conductivity (κ) was calculated from experimental data of the thermal diffusivity (D), specific heat (C p ), and density (d) using the equation κ DC p d. The thermal diffusivity was measured by a laser flash method (LFA 457 MicroFlash, NETZSCH GmbH) on square-shaped samples of 10 × 10 and 1 mm 3 in thickness, under argon. The specific heat was measured by differential scanning calorimetry (DSC 404 F1 PEGASUS, NETZSCH GmbH) on disk-shaped samples of 5.2 mm in diameter and 1 mm in thickness, under argon atmosphere. The dimensionless thermoelectric figure of merit ZT was estimated from the data of the three previous parameters. The measurement error of all the transport properties is no more than ± 5%.

Module performance measurement

The performance of each of the twenty-two fabricated TE modules was measured on a dedicated test bench [START_REF] Romanjek | Highperformance silicon-germanium-based thermoelectric modules for gas exhaust energy scavenging[END_REF] developed in collaboration with OMICRON (Eybens, France). The measurement takes place inside a vacuum chamber allowing controlling the atmosphere during the test. A controlled hot temperature is applied at the top side of the TE module thanks to a heating ceramic element and a cold temperature at the bottom side thanks to a water cooling system. The hot temperature is measured with a K-type thermocouple directly in contact with the top side of the TE module. The cold temperature is measured with a K-type thermocouple inside a copper buffer located between the cold side of the module and the water cooler chuck. Our measurement tests have been carried out under an atmosphere of 10 mbar of nitrogen while applying a mechanical pressure of 1 MPa on the module with the help of a pneumatic piston in order to lower the thermal contact resistance. As well, a thermal grease is used between the module, the copper buffer and the water cooler chuck so as to ensure minimal thermal contact resistance. The Seebeck coefficient of MSS is negative in the entire temperature range (Fig. 4a), indicating a n-type conduction, while that of HMS is positive (Fig. 5a), indicating a p-type conduction, both in good agreement with what was expected. The Seebeck coefficient values of our MSS material increases in absolute value from 81 µV/K at 20 • C to 216 µV/K at 500 • C, reaching the values achieved by Bernard-Granger et al. [START_REF] Bernard-Granger | Microstructure investigations and thermoelectrical properties of an N-type magnesium-silicon-tin alloy sintered from a gas-phase atomized powder[END_REF] and Gao et al. [START_REF] Gao | Transport and mechanical properties of high-ZT Mg 2.08 Si 0.4 x Sn 0.6 Sb x thermoelectric materials[END_REF] but around 6% lower than those of Liu et al. [START_REF] Liu | Optimized thermoelectric properties of Sb-doped Mg 2(1+z) Si 0.5-y Sn 0.5 Sb y through adjustment of the Mg content[END_REF]. For the HMS material, the Seebeck coefficient increases from 130 µV/K at 20 • C to 231 µV/K at 500 • C. These values are very close to the values reported by Bernard-Granger et al. [START_REF] Bernard-Granger | Microstructure investigations and thermoelectrical properties of a P-type polycrystalline higher manganese silicide material sintered from a gas-phase atomized powder[END_REF] and slightly above that of Luo et al. [START_REF] Luo | Rapid synthesis of high thermoelectric performance higher manganese silicide with in-situ formed nanophase of MnSi[END_REF][START_REF] Luo | Improved thermoelectric properties of Al-doped Higher Manganese Silicide prepared by a rapid solidification method[END_REF].

The temperature dependence of the electrical conductivity of both MSS (Fig. 4b) and HMS (Fig. 5b) materials exhibit the typical behavior of a degenerated semiconductor. As can be observed, the electrical conductivity values of both our materials are lower compared to the values reported in the literature. These discrepancies may be due to the difference of chemical composition with regard to our alloys. Indeed, the ntype alloy used by Gao et al. [START_REF] Gao | Transport and mechanical properties of high-ZT Mg 2.08 Si 0.4 x Sn 0.6 Sb x thermoelectric materials[END_REF] has a higher tin content compared to our MSS materials. On the other hand, the HMS alloy used in our work is not doped while the best materials reported in the literature are usually Al or Ge-doped [START_REF] Luo | Rapid synthesis of high thermoelectric performance higher manganese silicide with in-situ formed nanophase of MnSi[END_REF][START_REF] Luo | Improved thermoelectric properties of Al-doped Higher Manganese Silicide prepared by a rapid solidification method[END_REF][START_REF] Bernard-Granger | Microstructure investigations and thermoelectrical properties of a P-type polycrystalline higher manganese silicide material sintered from a gas-phase atomized powder[END_REF].

The behavior of the thermal conductivity with temperature follows similar trends as those previously reported in the literature, for both MSS (Fig. 4c) and HMS (Fig. 5c).

The maximum values of the dimensionless TE figure of merit ZT (Figs. 4d and5d) are 1.26 and 0.5 at 500 • C for the MSS and HMS materials, respectively. For the MSS material, this value is very close to the highest values ever reported in the literature for the Mg 2 Si 0.6 Sn 0.4 (Sb doped) composition. Even if in the case of the HMS material, the The cold temperature has been fixed to 50 • C. Once the temperature is stabilized, current is injected (Keithley 2651 A, 0.1% accuracy) and the voltage is measured to give the characteristic voltage versus current curve. The open circuit voltage V OC , internal resistance R int , and maximum output P parameters have been determined for hot side temperatures from 100 • C up to 450 • C, with a step of 50 • C. The measurements were quickly recorded (50 points in 1 s), avoiding thus any parasitic Peltier effect.

Results and discussion

Characterization of the TE materials

The sintered samples are dense, in agreement with the relative density values obtained of 5.1 g/cm 3 (96% of the theorical value) and 2.3 g/cm 3 (99% of the theorical value) for the HMS and MSS materials, respectively. The chemical composition and homogeneity, the crystalline structure and the microstructure as well as the thermomechanical properties of the SPS consolidated pellets of both the HMS and MSS materials used within this work have been already studied [START_REF] Mejri | Characterization of the thermomechanical properties of p-type (MnSi1.77) and n-type (Mg 2 Si 0.6 Sn 0.4 ) thermoelectric materials[END_REF][START_REF] Mejri | Thermal stability of Mg 2 Si 0.55 Sn 0.45 for thermoelectric applications[END_REF].

The measurement of the TE properties were carried out on several samples cut from different HMS and MSS sintered pellets, in order to check the reproducibility pellet to pellet. For both materials, the differences in between the pellets are very small, showing again the good control of the SPS process. The temperature dependences of the electrical conductivity, Seebeck coefficient, thermal conductivity, and figure of merit ZT of the MSS and HMS materials are shown in Figs. 4 and5, respectively. They are compared with the best data reported so far in the literature for TE materials of similar composition. different elements has been observed.

TE performance of the modules

In Fig. 8 are reported typical current-voltage and current-power curves measured on the 22 fabricated TE modules for a cold side temperature of 50 • C and hot side temperatures from 100 • C up to 500 • C (with 50 • C steps). The voltage (U) follows Ohm's law according to:

U V oc R int •I ( 1
)
where I is the current, R int the internal electrical resistance and V oc the open-circuit voltage. Hence, the electrical power (P U.I) is given by:

P R int •I 2 + V oc •I (2)
The obtained voltage-current and power-current measured curves follow the expected behaviors given by Eqs. ( 1) and ( 2), linear and parabolic, respectively. From these measured curves are extracted the open circuit voltage V OC (ordinate at the origin of the voltage-current curve), the internal electrical resistance R int (slope of the voltagecurrent curve), and the maximum electrical power output P max (maximum value of the power on the power-current curve). In Fig. 9 are reported the measured V oc , R int , and P max parameters versus the hot Fig. 6. SEM micrographs (BSE mode) of the interfaces of (a) hot side and (c) cold side of a HMS leg in a TE module. Zoom on the interfaces of (b) hot and (d) cold sides with overlays of elemental X-ray mappings of the elements mentioned on the images. maximum ZT achieved is slightly lower than the best values reported in this family of materials (ZT max 0.6-0.7 at 500 • C, [START_REF] Luo | Rapid synthesis of high thermoelectric performance higher manganese silicide with in-situ formed nanophase of MnSi[END_REF][START_REF] Luo | Improved thermoelectric properties of Al-doped Higher Manganese Silicide prepared by a rapid solidification method[END_REF][START_REF] Bernard-Granger | Microstructure investigations and thermoelectrical properties of a P-type polycrystalline higher manganese silicide material sintered from a gas-phase atomized powder[END_REF]), it is still highly acceptable regarding the way of fabrication. Indeed, going from laboratory production (some grams of material) to mass production (some kilograms) is often unfavorable for the TE performance. Our materials, manufactured using three simple metallurgic methods (melting, ball milling and SPS), offer the advantage to combine mass production with TE performance fully compatible with state of the art performance.

Characterization and performance of the modules 3.2.1. Module interfaces

In Figs. 6 and7 are reported SEM micrographs of the interfaces between the electrode layer/brazing layer/TE legs on the cold and hot sides of TE modules just after fabrication, for HMS and MSS TE legs, respectively. The brazing layers have a thickness of about 10 to 15 µm. Nor porosity, nor cracks or oxygen could be found at the different interfaces. The interfaces can be considered as perfect on both hot and cold sides of the module, corresponding perfectly to the scheme of Fig. 3, as it is highlighted on the overlays of all elemental X-rays mappings, whatever the material is (Figs. 6b,d,7b,7d). No diffusion in between the temperature for all 22 fabricated TE modules. One can see that each parameter exhibits a remarkable reproducibility module to module.

The open circuit voltage is the voltage generated by a module by the Seebeck effect. It is ideally equal to:

V oc N leg SΔT (3)
where N leg is the number of legs, S the mean measured value of the Seebeck coefficient between the n and p-type TE materials and ΔT the temperature difference between the hot and cold sides.

If we compare the V oc values obtained by this analytical model and the experimental values, a very good match is observed (Fig. 9a), meaning that our test bench does not add any significant thermal contact resistance, thanks to the pneumatic piston system and the use of thermal grease. Moreover, it shows that there is no significant temperature jump between both the DBC substrate and the TE legs on the cold side and the silver electrodes and the TE legs on the hot side of the TE module. Our TE module assembly process is thus validated in terms of thermal functionality.

In a similar way, the internal electrical resistance is directly linked to the electrical resistivity, and is ideally equal to:

R int N leg ρ h leg A leg (4)
where h leg is the leg height, A leg is the leg cross section and ρ the mean measured value of the electrical resistivity between the n-and p-type TE materials.

A good match between the ideal case and the measured values is also observed for the R int parameter (Fig. 9b), meaning that the electrical resistance between the TE legs and the electrical tracks is very low. The TE module assembly is also thus validated in terms of electrical functionality.

Taking into account Eqs. ( 3) and ( 4), the Ideal maximum generated electrical power is equal to:

P max N leg 4 S 2 ΔT 2 ρ A leg h leg (5)
A good match between the ideal values and the measured value is also obtained for the P max parameter (Fig. 9c). Hence, no significant losses are to be deplored and almost all the heat conversion performed by the TE material is available in the TE module. The TE modules generate a maximum power of 0.35 W corresponding to a power areal density of 0.95 W/cm 2 (total area of the TE legs 36 mm 2 ) for a temperature difference of 400 • C (cold side at 50 • C). Table 1 provides a statistic overview of the generated power at different temperature conditions.

Finite elements performance modelling

To predict the performance of the fabricated modules, modelling by three-dimensional finite-elements analyses implemented in the COM-SOL Multiphysics software has been used. The employed method has been already described [START_REF] El Oualid | Innovative design of bismuth-telluride-based thermoelectric micro-generators with high output power[END_REF]. However, as the temperature differences studied here are greater than those described in the paper where bismuth telluride-based modules were studied, we introduced in the modelling the actual measured temperature-dependent values of the material properties, fitted using third-order polynomials. The simulations were conducted on modules whose dimensions correspond to the fabricated ones. The boundary conditions were set to 50 • C at the cold side and from 100 to 450 • C at the hot side. Electrical contact resistances on the hot and cold sides (between the Ti/Au metallization and the TE material) were introduced into the simulation, estimated to be 38 μOhm⋅cm 2 and 17 μOhm⋅cm 2 for the p-and n-type materials, respectively. These electrical contact resistance values were calculated through the shift of the measured electrical resistance of individual legs with and without metallization. An external load resistance, connected to the generator to realize closed-circuit operation, was varied giving rise to various electric currents.

From the results of the simulation, it is possible to represent the temperature and voltage distributions in the studied generators as well as the current density and heat flux fields obtained for a load resistance that we have chosen equal to 65 mΩ (Fig. 10). The isotherms and equipotentials are obtained in the thermoelectric materials parallel to their surface and therefore only depend on the variable of dimension z. Likewise, the load and heat flows are directed through the legs in the z direction.

The evolution of the electrical output power as a function of the load resistance (which corresponds to a variation in current) is plotted in Fig. 11a. The maximum power is obtained when the load resistance is equal to the internal resistance of the generator. This maximum is shifted slightly towards a higher load resistance as the temperature increases, linked to a greater internal resistance (Figs. 4a and5b). The electrical power increases steadily with the temperature at the hot side, reaching a maximum value of 0.70 W for a temperature difference of 500 • C (0.43 W for a temperature difference of 400 • C). In Fig. 11b are compared the experimental mean values of the electrical power determined from the twenty-two modules, with data obtained either by numerical (ideal condition) or analytical modelling. The simulated values are higher than the experimental values, the difference increasing with the temperature of the hot side. For T hot 450 • C, the power loss is of the order of 25%. This loss is assumed to be due to the electrical contact between the solder and metallized TE legs. These results show that there are still some improvements to be made to have a perfect electrical contact at all solder interfaces, which is still a challenge for the experimenter.

Comparison with state of the art

The performance of our thermoelectric modules is compared with the state-of-art in Fig. 12. The chosen parameter is the generated electrical power (P max ), which is the key parameter for a ThermoElectric Generator system, when waste heat is used. All values have been normalized with respect to the cross-sectional area of the thermoelectric elements in order to have an accurate comparison between modules with different designs. The comparison is performed with the best published results describing thermoelectric modules working at a medium temperature range (200-500 • C) including different nonconventional thermoelectric materials (silicides [START_REF] Skomedal | Design, assembly and characterization of silicide-based thermoelectric modules[END_REF][START_REF] Nakamura | Powergeneration performance of a pi-structured thermoelectric module containing Mg 2 Si and MnSi 1.73[END_REF], skutterudites [START_REF] Ochi | Power-generation performance and durability of a skutterudite thermoelectric generator[END_REF][START_REF] Prado-Gonjal | Skutterudite thermoelectric modules with high volume-power density: Scalability and reproducibility[END_REF] and Half-Heuslers [START_REF] Zhang | High-temperature and high-power-density nanostructured thermoelectric generator for automotive waste heat recovery[END_REF][START_REF] Zillmann | Thermoelectric generators fabricated from large-scale-produced Zr-/Hf-based Half-Heusler compounds using Ag sinter bonding[END_REF]). The performances of our TE modules are in line with previous published results, compared either with silicide-based modules or with other materials-based modules, even using an industrial process for the TE materials production and the module assembly. Among all these materials, the silicides have the lowest cost/kg [START_REF] Amatya | Trend for thermoelectric materials and their earth abundance[END_REF][START_REF] Leblanc | Material and manufacturing cost considerations for thermoelectrics[END_REF] which gives a competitive advantage for our modules. A mean value of 4.4% has been estimated for the TE efficiency at a temperature difference of 400 • C (hot side temperature of 450 • C). Decreasing the electrical contact resistance remains a great challenge in order to achieve better performance.

Fig. 1 .

 1 Fig. 1. Configuration of the TE modules made of (a) 12 HMS/MSS couples and (b) 2 HMS/MSS couples.

Fig. 2 .

 2 Fig. 2. Manufacturing process of the 2 HMS/MSS couples TE modules.

Fig. 3 .Fig. 4 .

 34 Fig. 3. Expanded view of a TE module (one n/p couple) as built according to the process flow described in Fig. 2.

Fig. 5 .

 5 Fig. 5. Temperature dependences of a) Seebeck coefficient, b) Electrical conductivity, c) Thermal conductivity and d) Figure of merit of HMS TE materials.

Fig. 7 .

 7 Fig. 7. SEM micrographs (BSE mode) of the interfaces of (a) hot side and (c) cold side of a MSS leg in a TE module. Zoom on the interfaces of (b) hot and (d) cold sides with overlays of elemental X-ray mappings of the elements mentioned on the images.

Fig. 8 .

 8 Fig. 8. Typical voltage-current (left) and power-current (right) characteristics measured on one of the 22 fabricated TE modules (T cold = 50 • C, T hot from 100 to 450 • C by 50 • C step).

Fig. 9 .

 9 Fig. 9. a) Measured open-circuit voltage (V oc ), b) internal electrical resistance (R int ), and c) maximal power output (P max ) as a function of the hot side temperature for all the twenty-two TE modules.

Fig. 10 .

 10 Fig. 10. Temperature (a) and voltage (b) distributions calculated by three-dimensional finite element analyses on two couples of n-MSS and p-HMS for an illustrative T hot = 723 K and R load = 65 mΩ. The arrows on the schemes correspond to the direction of heat flow and current density. Their length on the figures is proportional to the represented parameter.

Fig. 11 .

 11 Fig. 11. (a) Power output as a function of the load resistance for different hot side temperatures (the arrow crosses the maximums), (b) mean maximal power output (P max ) as a function of temperature obtained from all twenty-two fabricated TE modules compared with numerical and analytical modelling.

Fig. 12 .

 12 Fig. 12. TE modules performance versus the state-of-art in the medium temperature range (200-500 • C).

Table 1

 1 Statistic values for the power output compiled for the twenty-two fabricated TE modules.

	Temperature ( • C)		P max (W/cm 2 )
	T Cold	T Hot	ΔT	Min	Max	Mean	Standard deviation
	50	100	50	0.009	0.012	0.011	0.001	5.8%
		150	100	0.039	0.047	0.044	0.002	5.4%
		200	150	0.093	0.113	0.103	0.006	5.3%
		250	200	0.176	0.214	0.196	0.010	5.3%
		300	250	0.292	0.354	0.324	0.017	5.2%
		350	300	0.441	0.539	0.490	0.026	5.3%
		400	350	0.630	0.767	0.698	0.037	5.3%
		450	400	0.853	1.036	0.949	0.050	5.3%

Acknowledgements

This work was supported by the French National Research (ANR-16-CE05-0012-03), within the RELIATEG (Reliable Thermoelectric Generators) project. It allowed funding two PhDs in Materials Sciences; Mahdi MEJRI at Université Toulouse III -Paul Sabatier (Toulouse, France) and Mostafa OULFARSI at Université de Lorraine (Nancy, France).

In this work, silicide-based TE modules have been manufactured on the base of a new design, called 'half skeleton', with the aim to manage the differences of thermal expansion between the n-type (Mg 2 Si 0.6 Sn 0.4 ) and p-type (MnSi 1.77 ) TE materials thanks to the presence of a flexible silver electrode located at the hot side of the modules. Indeed, no deterioration is to be deplored during the performance tests carried out up to 450 • C on any of the twenty-two modules made of two (n/p) couples tested. Such very promising result was achieved thanks to the optimization of each step of the process fabrication allowing mass production of TE modules. Special emphasis must be placed on the SPS step that allowed obtaining more than 200 perfectly reproducible TE legs. SEM analyses show that the different interfaces between the electrodes /brazing layer/TE legs are free of any defect (crack, diffusion, …) whatever the side or the material is. An average maximum power output of 0.37 W, corresponding to a power density of 0.95 W/cm 2 , has been achieved for a temperature difference of 400 • C (hot side temperature of 450 • C). This performance lies our modules within the state of the art, albeit the TE materials have been produced by 10 kg batches, for foreseen applications in the mid-range temperature (200-500 • C). The COMSOL modelling has shown that still some amelioration can be brought by solving the delicate problem of electrical contact resistance as the maximum power extracted represents around 86% of the expected value. Long-term stability tests at a given temperature and under given cycling conditions are to be performed in different working atmospheres in order to further validate our design. Such tests are underway, as well as the search for innovative coatings.

CRediT authorship contribution

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.