Bassam Abdallah

François Willot
email: francois.willot@mines-paristech.fr

Dominique Jeulin

Morphological modeling of three-phase microstructures of anode layers using SEM images

Keywords: Three-phase random sets, Boolean models, Plurigaussian 3D models, Fuel cells, SOFC

A general method is proposed to model 3D microstructures representative of three-phase anode layers used in fuel cells. The models are based on SEM images of cells with varying morphologies. The materials are first characterized using three morphological measurements: (cross-)covariances, granulometry and linear erosion. They are measured on segmented SEM images, for each of the three phases. Second, a generic model for three-phase materials is proposed. The model is based on two independent underlying random sets which are otherwise arbitrary. The validity of this model is verified using the cross-covariance functions of the various phases. In a third step, several types of Boolean random sets and plurigaussian models are considered for the unknown underlying random sets. Overall, good agreement is found between the SEM images and three-phase models based on plurigaussian random sets, for all morphological measurements considered in the present work: covariances, granulometry and linear erosion. The spatial distribution and shapes of the phases produced by the plurigaussian model are visually very close to the real material. Furthermore, the proposed models require no numerical optimization and are straightforward to generate using the covariance functions measured on the SEM images.

Introduction

The recent development of fuel cells has been driven by an increasing demand for new energy production, and technological advances [START_REF] Badwal | Review of progress in high temperature solid oxide fuel cells[END_REF][START_REF] Laguna-Bercero | Recent advances in high temperature electrolysis using solid oxide fuel cells: A review[END_REF]. Since the pioneering work of [START_REF] Grove | On voltaic series and the combination of gases by platinum[END_REF], fuel cells have been tremendously improved both in terms of performance and durability. The materials involved in the manufacturing process have become complex as per their composition and microstructure. In this regard, electrodes, made of an electrical conductor, an ionic conductor and pores are a key part of solid oxide fuel cells (SOFC). Sophisticated interactions between phases have been reported in commonly-used microstructures [START_REF] Joos | Representative volume element size for accurate solid oxide fuel cell cathode reconstructions from focused ion beam tomography data[END_REF] at the micro-and nano-scale. Thus, the characterization of anode microstructures is a critical point in the improvement of the whole cell. Several works have focused on the reconstruction of a 3D volume using FIB-SEM techniques [START_REF] Gostovic | Three-Dimensional Reconstruction of Porous LSCF Cathodes[END_REF][START_REF] Iwai | Quantification of SOFC anode microstructure based on dual beam FIB-SEM technique[END_REF][START_REF] Joos | Reconstruction of porous electrodes by FIB/SEM for detailed microstructure modeling[END_REF][START_REF] Shearing | 3D reconstruction of SOFC anodes using a focused ion beam lift-out technique[END_REF][START_REF] Wilson | Three-dimensional reconstruction of a solid-oxide fuel-cell anode[END_REF]. These however require expensive facilities for data acquisition. By contrast, the development of 3D microstructural models based on 2D images offers a low-cost easy alternative. Careful study of 2D images indeed provides estimates of 3D properties such as three-point boundaries [START_REF] Zhang | Unbiased characterization of three-phase microstructure of porous lanthanum doped strontium manganite/yttria-stabilized zirconia composite cathodes for solid oxide fuel cells using atomic force microscopy and stereology[END_REF] but cannot directly provide estimates of geometrical properties such as phases tortuosity, which are useful for estimating the cells's transport properties [START_REF] Masson | Morphological modeling of a metal foam SOFC configuration[END_REF]. These properties can be measured, however, on 3D models constructed from 2D images, without the need for expensive and time consuming FIB-SEM tomography. Furthermore, the computation of the effective transport properties of the materials, such as ionic diffusion, electronic conductivity and permeability are microstructure-sensitive and require accurate simulations of 3D microstructures [START_REF] Abdallah | Stokes flow through a Boolean model of spheres: Representative volume element[END_REF].

The present work is organized as follows. A short presentation of the anode samples is given in Sec. 2.1. The three-phase SEM images are segmented using specific techniques described in Sec. 2.2. The morphology is characterized in terms of various measurements in Secs. 3 and 4. A statistical characterization of the microstructure is provided in Sec. 5. It is shown that the morphological measurements support a strong probabilistic assumption which is especially useful to model the microstructure. Under this assumption, we attempt to model the Figure 1: Schematic principle of the fuel cell: Two composite electrode layers separated by a dense electrolyte. At the cathode, di-oxygen molecules (e.g. from air or H 2 syngas). are ionised into O 2-anions that go through the dense electrolyte in order to reach the anode side. On the anode compartment H 2 is injected to react with the oxygen anions and give water plus electrons. microstructure using Boolean models (Sec. 6) and plurigaussian truncated random functions (Sec. 7). Concluding remarks are in Sec. 8.

Anode cell layers 2.1 Anode samples

This work is concerned with the microstructural modeling of anode cell layers located at the top and bottom of an electrolyte substrate (see Fig. 1). The layers are heterogeneous materials made of two solid phases, gadoliniumdoped ceria (GDC) and lanthanum-doped strontium titanate (LST), and the pores. Specific transport properties are allowed by each of the three phases. The GDC phase is a molecule with high ionic conduction property (transport of 0 2-), LST is an electrical conductor. Fluid flow occurs in the material's porous phase, which results in an effective permeability. The morphology of each of the three phases is constrained by the two others, therefore increasing one of the material's effective properties without degrading the other two is difficult. One minimal requirement is that each of the three phases percolates to allow transport in the material. Additionally, electrochemical properties in anode layers have been found to be especially sensitive to contact surface areas [START_REF] Yurkiv | Towards understanding surface chemistry and electrochemistry of La0.1Sr0.9TiO3-α based solid oxide fuel cell anodes[END_REF].

The present work focuses on the morphological modeling of anode cells. The cells were produced at ISTEC (Instituto di Scienza e Tecnologia dei Materiali Ceramici, Italy). Additionally, a few anode cells prepared by the DLR (Deutsches Zentrum für Luft-und Raumfahrt e.V., Germany) and by the Center for Materials (Mines ParisTech, France) are used in the present study for comparison purposes. All cells were analyzed by the Center for Materials (Mines ParisTech, France). To achieve percolation of the two solid phases, the LST and GDC phases were prepared so that both phases have roughly the same volume fraction. The pores volume fraction is not known. The reader is referred to [START_REF] Masson | Morphological modeling of a metal foam SOFC configuration[END_REF] for details on material preparation. The present study is based on a set of SEM microscopy images of back-scattered electrons representing eight anode cells obtained at Center for Materials (again, the reader is refered to [START_REF] Masson | Morphological modeling of a metal foam SOFC configuration[END_REF] for details on image acquisition). Each anode cell is symmetrical: the microstructures at the top and bottom of the electrolyte should be very similar. Each of the eight samples have been prepared and processed in different manners. Furthermore, they have been used for varying periods of time. Accordingly, we expect differences in the morphology of each sample. A series of 9 to 14 images of the top (resp. bottom) parts of each sample are used hereafter. Examples of SEM images are shown in Fig. 2. Each SEM image contains 1024 × 883 pixels at resolution 12.4nm per pixel. A region of interest in the core of each layer, without electrolyte, is selected from each image. The size of the crop varies between 251 × 1024 pixels (39.5 µm 2 ) to 883 × 1024 pixels (139 µm 2 ). Hereafter we denote each ISTEC sample by a number in the range 14 to 32. The bottom layer is identified with the letter i, and the top layer with the letter s. For instance, 29-s refers to the top layer of sample 29.

Image segmentation

The SEM images are segmented in order to assign each pixel to one of the three phases. The most important steps of the segmentation are illustrated in Fig. (3). We first filter noise using a bilateral filtering [START_REF] Tomasi | Bilateral Filtering for Gray and Color Images[END_REF]. The latter is applied using the transform:

F(I; i, j) = 1 µ k,l∈W (i,j) I(i, j)f (|i -k|, |j -l|; σc)f ′ (|I(i, j) -I(k, l)|; σi), ( 1
)
where I is the input image, F(I) its transform, W (i, j) is a square-shaped window of side length a centered on a pixel with coordinates (i, j), σi and σc stand for the standard deviations of the centered Gaussian functions:

f (x, y; σ) = 1 2πσ 2 e -(x 2 +y 2 ) 2σ 2 , f ′ (x, σ) = 1 √ 2πσ e -x 2 2σ 2 . (2)
The filter does not remove the contrast at the interface between phases. All images have been acquired on the same microscope, therefore we fix the same set of parameters σi = 0.15, σc = 2.5 and a = 5 (pixels) for all images.

The normalizing parameter µ reads [START_REF] Tomasi | Bilateral Filtering for Gray and Color Images[END_REF]:

µ = k,l∈W (i,j) f (|i -k|, |j -l|; σc)f ′ (|I(i, j) -I(k, l)|; σi). (3)
A rough approximation of each phase is obtained by two thresholds on the output F(I). Noise-reduction is mainly visible when looking at enlargements (Fig. 3b), nevertheless it greatly improves the rest of the segmentation process.

In the rest of this study, for simplicity, the GDC, LST and porous phases are denoted respectively by the letter W (white), G (gray) and B (black).

The intensity of the signal in the image results from the scattering of electrons with the material. Therefore, phases containing atoms with the highest atomic number appear brighter than the other phases in the backscattered electron images. As a result, the GDC phase appear in white, the LST phase in gray and the pores in black. However, an unwanted effect should be first taken into account. Due to the halo artifact, the SEM images are not actual planar 2D sections. Parts of the structures under the plane of acquisition are visible in what should really be pores. The latter are located slightly behind the section along which the material was cut. Therefore, we fix the two threshold values that monitor the volume fractions of LST and GDC to minimize the halo effect (Fig. 3c). Note that thresholds are such that the volume fractions of the LST and GDC phases are equal, which leaves one free parameter. After thresholding, a relief effect is still observed around each phase, due to out-of-plane structures. Following [START_REF] Gillibert | 3D multiscale segmentation and morphological analysis of X-ray microtomography from cold-sprayed coatings[END_REF], a combination of simple mathematical morphology operators are used to remove this effect. We denote by δSE(A) and γSE(A) the morphological dilation and opening of a set A by a structural element SE. The halo effect is removed by considering the GDC phase (W ) and the LST phase (G) and applying the following:

W = W ∪ (δSE(W ) ∩ [G \ γSE (G)]) ,
with SE taken as a flat-disk of fixed radius. The radius of SE is chosen such as the out-of-plane structures are removed, in this case 3 pixels. This transformation provides the segmented image, such as image (d) in Fig. 3. In this image each pixel is assigned either to the porous phase (black), the LST phase (gray) or the GDC phase (white).

Morphological measurements

We now use the segmented image to measure various morphological criteria. The images are considered as realizations of a stationary random structure. Therefore all morphological criteria are measured separately on each image, and then averaged on all images that belong to the same sample. We first estimate the covariance and cross-covariance functions [START_REF] Matheron | Élements pour une théorie des milieux poreux[END_REF] as the probability:

Cij (h) = P {x ∈ i, x + h ∈ j} (4)
where i and j ∈ {W, G, B} are two phases, and h is a vector. The values of x in the above are taken so that both x and x + h are in the observation window. Numerical investigations indicate that the medium is isotropic with respect to the covariance so that Cij (h) depends only on the norm h = ||h|| of h. This is illustrated on Fig. 4 for the porous phase. Similar results hold for the other phases and all image sets. Accordingly, we identify Cij (h) with Cij (h). Note that the volume fractions of phase i is given by:

fi = P {x ∈ i} = Cii(0). (5)
We also estimate the following statistical properties:

Qi(ℓ) = P {L(ℓ) ⊂ i c }, Pi(ℓ) = P {L(ℓ) ⊂ i}, (6)
where L(ℓ) is a segment of length ℓ with unspecified orientation or location, i = W , G, B is one of the three phases and i c is the complementary set of i. In practice, the functions Qi and Pi are estimated by computing the area (or volume fraction) of the sets i and i c eroded by segments of length ℓ. The latter is related to the linear contact distribution. We (abusively) refer to Pi and Qi as "linear erosions" hereafter. For general Boolean models with convex grains, linear erosion functions are exponentials (see Sec. 6). This allows one to determine with few computations if a Boolean set might be appropriate to model a material. The grain size distribution of set i is estimated by the granulometry by opening Gi(S). Denoting by γ B(S) (i) the morphological opening of the set i by the compact set B(S) of area S, one has:

Gi(S) = 1 -P {x ∈ γ B(S) (i)|x ∈ i}. (7)
In our case B(S) is taken as a disk of surface S. This measure is used in Secs. 6 and 7 to validate models.

The representativity of all three-phase models M developed in this work is evaluated quantitatively by the following error criterion E (M ):

E (M ) = 1 3 i=W,G,B |Ci(h) -C M i (h)| 2 1/2 h + |Qi(ℓ) -Q M i (ℓ)| 2 1/2 ℓ + |Gi(S) -G M i (S)| 2 1/2 S , (8)
where averages, denoted by • , are taken in the intervals h <1.5 µm, ℓ <1.5 µm and S <1.5 µm 2 .

Sample 4 Representative volume element of the microstructure

f W ± ǫ fW f G ± ǫ fG f B ± ǫ fB 14-i 0.
The representativity of the microstructures in terms of volume fraction fi for the phase i is considered. The latter is estimated on 2D sections by estimating area fractions as explained below.

Consider a 2D section of area S0, either of the real material (SEM images) or of a 3D model M . The apparent area fraction of each phase is estimated on disjoint subdomains of surface S. The variance of the latter, D 2 f i (S) scales as shown in [START_REF] Kanit | Determination of the size of the representative volume element for random composites: statistical and numerical approach[END_REF][START_REF] Matheron | The theory of regionalized variables and its applications[END_REF]:

D 2 f i (S) D 2 f i ∼ A f i 2 S , S ≫ A f i 2 , (9)
where A f i 2 is the integral range and D 2 f i is the point variance for phase i, i.e.:

D 2 f i = fi(1 -fi). (10)
The integral range A f i 2 is the integral of the correlation function of phase i; we refer to [START_REF] Matheron | The theory of regionalized variables and its applications[END_REF] for a discussion of its main properties. Simply remark that when (9) holds, the domain S acts as n = S/A f i 2 independent sub-domains each of area A f i 2 . Eq. ( 9) is useful to estimate the representativity of a microstructure. For instance, for a given domain of area S the relative error ǫ f i on the volume fraction fi estimated for phase i is given by:

ǫ f i = 2D f i fi A f i 2 S (11)
In the next sections, the function D 2 f i (S) is used to compare simulations of the models and the SEM images. Tab. 1 gives estimates of the area fractions, and therefore the volume fractions fi of each phase, averaged over the 11 samples. The porosity is around fB ≈ 63% for all samples, while the volume fraction of GDC and LST lies between 15% ≤ fG,W ≤ 20%. The integral range A f i 2 is extracted numerically by fitting the left-hand side of (9) for large S. This value is used to estimate ǫ f i in Eq. 11. As seen in Tab. 1, the volume fractions of the two solid phases fG and fW differ by up to 10%, except for sample 14-i for which the relative difference is 25%. These differences are much higher than the relative errors ǫ f i , and are presumably an effect of the preparation techniques.

5 Three-phase media modelled using two underlying independent random sets

In this section, we assume that the three sets B, W and G are obtained from two independent random sets X and Y . To obtain three random sets Ai (i = 1, 2, 3), we consider the following combinations of the two sets X and Y :

A1 = X, A2 = X c ∩ Y, A3 = X c ∩ Y c . ( 12
)
Each set Ai (i = 1, 2, 3) is used to represent either B, W or G. The above construction is represented schematically in Fig. 5. We denote by Ci(h) and Cij (h) (i, j = 1, 2, 3) the covariance and cross-covariance, respectively, of the sets Ai and Aj as in (4). The covariance of the complementary set A c i , denoted by Ci(h), is given by: The covariances Ci(h) of the sets Ai read:

Ci(h) = 1 -2Ci(0) + Ci(h) (13)
C1(h) = CX (h), C2(h) = CX (h)CY (h), C3(h) = CX (h)CY (h), ( 14
)
where CX (h) and CY (h) are the covariances of the sets X and Y respectively, and CX (h) and CY (h) that of their complementary sets X c and Y c . The independency of X and Y implies, for the cross-covariance functions Cij(h) [START_REF] Jeulin | Modelling three phase random media[END_REF]:

C12(h) = C2(0) 1 -C1(0) [C1(0) -C1(h)] , (15)
C13(h) = C3(0) 1 -C1(0) [C1(0) -C1(h)] , (16)
C23(h) = C2(0) 1 -C1(0) C1(h) -C2(h). (17)
For instance, formula ( 15) is obtained as follows:

C12(h) = P {x ∈ A1, x + h ∈ A2} = P {x ∈ X, x + h ∈ X c ∩ Y } = P {x ∈ X, x + h ∈ X c }P {x + h ∈ Y } = [CX (0) -CX(h)]CY (0).
Eqs. ( 16) and ( 17) are derived in a similar way. This particular structure, resulting from the construction of the model, is useful to test the validity of the independence assumptions from experimental covariances. We emphasize that (15), ( 16) and ( 17) are valid if X and Y are independent. This assumption is tested by considering all possible choices for assigning any of the sets B, G or W to A1, A2 or A3. The 6 possibilities, labelled H k (1 ≤ k ≤ 6) are given in Tab. 2. We denote C H k ij (h) where (i, j) ∈ {B, G, W } the cross-covariance Cij defined in ( 15), ( 16) and ( 17) with the Ai specified by

H k (1 ≤ k ≤ 6).
For each pair of values i = j and each of the possible set of choices 15), ( 16), ( 17) and compared to its estimate directly measured on the SEM image. For instance, referring to Tab. 2, C H 1 12 (h) is the cross-covariance between sets B and G, which is compared to the right-hand-side of Eq. ( 15). The latter is computed using the covariance of the sets B and the volume fraction of the set G. Accordingly, hypothesis H1 implies that:

H k (1 ≤ k ≤ 6), the cross-covariance C H k ij (h) is computed according to (
CGB(h), C H 1 GB (h) = CG(0) 1 -CB(0) [CB(0) -CB(h)]
are equal. The cross-covariances C H k GW of G and W are plotted in Fig. 6 for sample 29-i, for all hypothesis H1, ..., H6, and compared to CGW .

In order to quantify the quality of each possible choice H k we compute the relative error:

E(k) = i,j 1 n + 1 n h=1 |C H k ij (h) -Cij (h)| Cij (h) (18)
for all image sets and set of choices H k (Tab. 3).

The configuration H6 minimizes the relative error ε k on 12 out of 16 image sets. Furthermore, E( 6) is close to the minimum value on 3 out of the 4 other samples. In the following, we assume that hypothesis H6 holds, i.e. GW computed using ( 15), ( 16) and ( 17). The latter are computed for each configuration k = 1, ..., 6 from Tab. (2).

Hypothesis A 1 = X A 2 = X c ∩ Y A 3 = X c ∩ Y c H 1 B G W H 2 B W G H 3 G B W H 4 G W B H 5 W B G H 6 W G B
Table 2: The 6 possible choices for assigning any of the sets B, G or W to A 1 , A 2 or A 3 .

the three phases are written as ( 12) with A1 = W , A2 = G and A3 = B as specified in Tab. 2. In particular, using the independency of X with respect to Y :

fW = fX , fG = (1 -fX )fY . ( 19
)
These equations give the volume fractions of X and Y for each sample (Tab. 1).

Herefter, we aim to determine the random sets X and Y , which can be generated independently from each other. Boolean (Sec. 6) and plurigaussian models (Sec. 7) are explored in the following.

Boolean models

The Boolean model is a set A ⊂ R 3 obtained by implantation of the primary grains A ′ (with possible overlaps) on points x k distributed according to a Poisson point process P with intensity θ [START_REF] Matheron | Élements pour une théorie des milieux poreux[END_REF]:

A = x k ∈P A ′ x k , (20)
where

A ′ x k = {x + x k ; x ∈ A ′ } is the translation of A ′ by x k .
To determine if the set X or Y might be approached by Boolean models, we measure the linear erosion Qi(ℓ) (see 6) for the three phases G, W and B on all data sets. Results are represented in Fig. 7 in lin-log scale for sample 29-s and show an exponential decrease log[Qi(ℓ)] ∼ ℓ. A similar behavior is observed on all samples. This scaling law is compatible with a Boolean model with convex 18)) between the measured cross-covariances C ij (h) and the ones computed using ( 15), ( 16) and ( 17) for all combinations H 1 , ..., H 6 , denoted C H k ij (h). For each sample, the minimum value of E(k), corresponding to the best configuration from Tab. 2, is highlighted in grey.

primary grain A ′ . Indeed, the covariance CA(h) and linear erosion QA(ℓ) of a Boolean model A read [START_REF] Matheron | Élements pour une théorie des milieux poreux[END_REF]:

CA(h) = q 2-r A ′ (h) , QA(ℓ) = q 1-ℓr ′ A ′ (0) , (21)
where q is the volume fraction of the complementary set of A, r A ′ (h) is the normalized geometrical covariogram of the primary grain A ′ and r ′ A ′ (h) its derivative. The latter is given by [START_REF] Matheron | Élements pour une théorie des milieux poreux[END_REF] in three dimensions:

r A ′ (h) = µ3(A ′ ∩ A ′ -h ) µ3(A ′ ) , ( 22
)
where µ3 is the Lebesgue measure in dimension 3. Note that the exponential scaling law is expected to hold for the linear erosions of the underlying random sets X and Y . Indeed:

QX (ℓ) = QW (ℓ), PB(ℓ) = QX (ℓ)QY (ℓ). ( 23
)
The above suggests that one uses Boolean random sets to model X and Y . We emphasize that the exponential decrease holds for general isotropic Boolean models built on a distribution of convex random primary grains A ′ with arbitrary shape and orientations. Accordingly, in the following, we consider various primary grains A ′ oriented uniformly on the unit sphere. The two models X and Y are determined by their primary grains A ′ X and A ′ Y , respectively, and by the Poisson intensity ΦX and ΦY . In [START_REF] Greco | The use of the texture analyser to study sinter structure: application to the morphology of calcium ferrites encountered in basic sinters of rich iron ores[END_REF], such a combination of Boolean models with parallelipiped primary grains was used to model iron ore sinter textures.

The round-shaped GDC particles and the elongated shapes observed for the LST phase suggest that one uses spheres of radius RX for A ′ X and cylinders of radius RY and height LY for A ′ Y . Taking into account the Poisson intensities ΦX and ΦY , the entire model, denoted M1, depends on five parameters. The normalized geometrical covariogram reads, for the spherical primary grain A ′ X :

rX(h) = 1 - 3h 4RX + h 3 16(RX ) 3 , h ≤ 2RX . ( 24
)
The normalized mean covariogram of cylinders takes the form of a one-dimensional integral which can be solved analytically using special functions [START_REF] Willot | Mean covariogram of a cylinder and applications to Boolean random sets[END_REF]. The covariograms of spheres and cylinders allow to compute the covariance CB,G,W (h) of B, G and W using ( 13), ( 14), and (21) without generating 3D Boolean models. The quadratic difference between the covariances of the model and that measured on the images is used as error criterion.

We minimize numerically the error criterion by optimizing the 5 parameters of the model using the algorithm of [START_REF] Nelder | A Simplex Method for Function Minimization[END_REF]. The optimal microstructure for model M1 is found for R

M 1 X = 14 µm, Φ M 1 X = 1.31 10 -5 µm -3 , R M 1 Y = 8µm, L M 1 Y = 147µm and Φ M 1 Y = 6
.70 10 -6 µm -3 for sample 29-s. Accurate volume fractions are recovered for the three phases: fB = 0.71%, fG = 16%, fW = 15% (SEM) compared to fB = 68%, fG = 16% and fW = 14% (model). Good agreement is found as well for the covariance functions and linear erosions (Fig. 9a and9b). However, the model is unable to reproduce the granulometries measured on the SEM images (Fig. 9c), which is much wider than the narrow distribution of sizes obtained with the model. The model is not visually satisfactory either. A 2D section of M1 is represented in Fig. 8b, which is quite different from the SEM images (Fig. 8a). Two other Boolean models are explored as an attempt to obtain wider granulometries for phases G and W. In the first one, denoted M2, the set X is a Boolean model containing two populations of spheres. The spheres have radius R M 2 X with probability s M 2 and R ′M2 X with probability 1s M and radii R M 3 X,Y of the sphero-cylinders. Sphero-cylinders are used in order to reproduce the smooth boundaries observed on the segmented image.

The parameters of models M2 and M3 are optimized, as before, on the covariances of the three phases. Again, the quadratic difference between covariances measured on the segmented images and computed on the models is used as error criterion. A different optimization is carried out for each sample. Using ( 21), the optimization of model M2 is carried out without generating the microstructures. Again, the algorithm of Nelder & Mead is used. For model M2, the optimal microstructure is found when Φ 2.42 10

M 2 X = 1.84 10 -7 µm -3 , D M 2 X = 75µm, Φ ′M2 X = (a) (b) (c)
-5 µm -3 , D ′M2 X = 21µm, Φ M 2 Y = 2.04 10 -5 µm -3 , D M 2 Y = 11µm, L M 2 Y = 113µm for sample 29-s. For model M3 we find R M 3 X = 8µm, R M 3 Y = 10µm, L M 3 X = 123µm, L M 3 Y = 75µm, Φ M 3 X = 8
.0 10 -6 µm -3 and ΦY = 8.2 10 -6 µm -3 for the same sample. 2D sections of the optimized models M2 and M3 are represented in Fig. 8c and Fig. 8d respectively. The models are not visually satisfactory. Quantitatively, only a small improvement on the granulometry is observed when using model M2, compared to model M1 (not shown). Similar results are obtained for model M3 (not shown).

Finally, we mention that we considered a fourth model that consists of a Boolean model of spheres for set X, with radii uniformly distributed from 0 to a maximum value, and a Boolean model of cylinders for set Y . Again however, the models were unable to reproduce the observed granulometry.

Plurigaussian model

In this section, we model sets X and Y by means of truncated Gaussian models [START_REF] Armstrong | Plurigaussian Simulations in Geosciences[END_REF][START_REF] Lantuéjoul | Geostatistical Simulation -Models and algorithm[END_REF]. This choice is led by the need to generate three-phase microstructures with smooth boundaries. Such models have been used successfully to represent food microstructures [START_REF] Bron | Modelling a food microstructure by random sets[END_REF]) and (binary) SOFC electrodes [START_REF] Lanzini | Microstructural characterization of solid oxide fuel cell electrodes by image analysis technique[END_REF].

The three-phase microstructures are represented by two underlying independent sets X and Y , as described in Sec. 5. The latter sets are both generated by the convolution of a Gaussian noise U with a weight function w. The resulting fields are thresholded. The value of the threshold is chosen to reproduce the measured volume fractions of the three phases. For set X, for instance, we use:

X = {x ∈ Ω; ZX (x) ≥ λX } , ( 25a
)
ZX (x) = (wX * UX )(x), UX (x) ∼ N (0, 1), ( 25b
)
where * is the convolution product, λX is the threshold and UX (x) follows the normal distribution N (0, 1) of mean 0 and variance 1. The expectation of the indicator function of the random set X reads:

fX = P {ZX (x) ≥ λX } = P {N (0, 1) ≥ λX}, (26)
P {N (0, 1) ≥ λX } is the probability that a variable following the normal distribution is greater than λX . This equation provides the value for the threshold λX . Indeed:

λX = F -1 (1 -fX ), ( 27
)
where F is the cumulative distribution function of the normal distribution and the volume fraction fX is given by ( 19) and by the measurements on segmented SEM images (Tab. 1). The weight function wX is normalized and symmetric: -3 ) E(M ) (see Eq. 8) computed on models M 1-2 and M 4-6 , for 11 samples. The best fit obtained for each sample is highlighted in grey.

Ω w 2 X (x)d x = 1, wX(x) = wX (-x), x ∈ Ω. ( 28
) Sample E(M 1 ) E(M 2 ) E(M 4 ) E(M 5 ) E(M
Figure 10: Weight functions w X and w y for sample 29-i. Each weight function is defined by (30) where ρ X,Y is obtained by numerical inversion of (29).

The latter is directly related to the covariance CX (h) of X by [START_REF] Lantuéjoul | Geostatistical Simulation -Models and algorithm[END_REF]:

CX (h) = 1 2π ρ X (h) 0 1 √ 1 -t 2 e -λ 2 X 1+t d t, (29)
where ρX(h) = * wX )(h). The function wX is obtained using:

wX = F F T -1 F F T {ρX} , (30)
where FFT and FFT -1 are the forward and backward discrete fast Fourier transforms. Once λX is known, the function ρ is provided by inverting numerically (29) and wX is given by (30). We emphasize that the covariance measured on the SEM image for the set X (i.e., W ) is reproduced exactly by the model. The same procedure is applied to determine Y . Recall that, under assumption H6, W = X, G = X c ∩ Y and B = X c ∩ Y c . Accordingly, the covariance of the set X is given by that of W whereas the covariance of Y (see Eq. 14) reads:

CX(h) = CW (h), CY (h) = CG(h) CW (h) . (31)
As an example, the weight functions wX and wY corresponding to X and Y are plotted in Fig. 10. A microstructure model of size 512 3 voxels (6.14 µm 3 ) is simulated for each of the 16 samples. A 2D section of the resulting model is represented in Fig. 13 and compared to the segmented image for sample 29-s. The covariance functions, linear erosions and cumulative granulometries of the three phases are estimated on realizations of the plurigaussian model. For a given model, 10 random 2D sections are selected. The morphological measurements are averaged over the 2D sections.

The plurigaussian model defined above is hereafter denoted M4. The contours of this model, represented in Fig. 11b, are very noisy, as previously observed by [START_REF] Bron | Modelling a food microstructure by random sets[END_REF]. In [START_REF] Bron | Modelling a food microstructure by random sets[END_REF], noise is Note that for the sake of comparison, all images are simulated using fixed Gaussian noises U X,Y .

removed using an analytical model for the covariance. This model reads [START_REF] Corson | Correlation functions for predicting properties of heterogeneous materials. I. Experimental measurement of spatial correlation functions in multiphase solids[END_REF]:

C(h) = f 2 + f (1 -f )e -ch n , (32)
where n ≤ 1 and c are two parameters. We follow this procedure and fit the right-hand sides of ( 31) with the Corson model (32). Values of parameters n and c are given in Tab. 5. These fits are used for the covariances CX,Y (h). Hereafter, this model is referred to as M5. Results are represented in Fig. 11c. Noise is not removed using the Corson model, and the method produces sub-optimal models (see Tab. 4, column 5). Indeed, the fit of the covariances using the Corson model is not satisfactory (not shown). A different approach is followed to remove noise on the contours, that preserves the information provided by the weight functions wX,Y . The plurigaussian model M4 is filtered using a low-pass normalized-centered Gaussian kernel. This filtering is applied on ZX,Y , after the convolution of the Gaussian noise UX,Y by the weight function wX,Y . Equivalently, the weight functions w M 6 X,Y in model M6 is replaced by:

w M 6 X,Y (x) = w M 4 X,Y * K (x), (33)
where K refers to the Gaussian kernel:

K(x) = e -||x|| 2 /σ 2 K . ( 34
)
The value of the variance in the Gaussian kernel is fixed to σK = 0.055 µm. This value removes most of the small-length artifacts while preserving the structure of the model. This filtered model is denoted M6. As shown The volume fraction of each phase in model M6 is given in Tab. 6 together with their relative error. The phase volume fractions of model M6 (Tab. 6) are close to that measured on the SEM images (Tab. 1). Other morphological measurements on the plurigaussian model M6 are represented in Fig. 12 and compared to that of the segmented image. As expected, a very good agreement is found for the two covariance functions CW and CG, which are directly used in the plurigaussian model. A good agreement between the plurigaussian model M6 and the SEM images is observed as well for the covariance CB, and for the linear erosions Qi and granulometry functions Gi, for the three phases. This result holds even though the linear erosion and granulometry were not used for the identification and the generation of the model. For certain samples, as the one represented in Fig. 12, however, a small misfit is observed for the covariance CB(h). In a few other samples, a small discrepancy appears regarding the granulometry GB (not shown) but overall, the model follows quite closely the measurements. A visual comparison between model M6 and the segmented SEM images (Fig. 13) confirms the good behavior of the model. The two images are close to one another.

Exactly the same procedure has been applied to compute M6 models corresponding to the DLR and Mines ParisTech samples. The latter are also visually close to the SEM images (Fig. 14). This is especially remarkable as the anode samples have quite different morphologies, due to the particular preparation techniques involved in the fabrication of the cells.

The specific surface areas of the SEM images and M6 models are compared in Tab. (7). The contact surface areas between G and W , denoted SV GW , is computed by estimating the derivative at 0 of the cross-covariance functions of the segmented images and models. The contact surface area between B and W , denoted SV W B , and between B and G, denoted SV BG , are estimated in the same way. The comparison is carried out for the ISTEC samples as well as 4 DLR anode samples (labeled A1 to A4) and one Mines ParisTech sample (labeled B1). The relative error on the measurement of the various surface areas is estimated by computing the derivative of the crosscovariance functions in different ways and studying representativity effects. Details are given in [START_REF] Abdallah | Analyse morphologique et modélisation pour l'optimisation structurelle d'éléctrodes[END_REF]. The relative error is estimated to be at most 15%. Overall, good agreement is found between the models and SEM images, except for the contact surface area between G and W for the DLR samples, which are underestimated by model M6 by a factor 2. Use of the parameter σK in (33) would presumably provide additional degrees of freedom for improving the model in this respect, as will be seen hereafter. 29), ( 30) and ( 31), for the anode layer 29-s (dots); fit w X (h) = e - √ h/2.0

(solid line). with parameter α X = 2.0 pixels ≈ 25 nm.

The model M6 is based on the covariance functions of X and Y , or equivalently, of B, W and G. Thus a parametrization of the latter functions is needed to obtain an analytical model for M6. Numerical investigations show that the weight functions wX,Y (h) for X and Y are close to the law:

wX,Y (h) = e -(h/α X,Y ) β X,Y , (35)
where, depending on the material, 0.5 ≤ βX ≤ 0.55 and 2.0 ≤ αX ≤ 2.07. The two parameters are not independent.

Because of the condition (28), they satisfy:

αX,Y = 2 1/β X,Y Γ 1 + 1 βX,Y -1 , ( 36
)
where Γ is the extended factorial function. A numerical fit of wX (h) is illustrated in Fig.

The expansion for CX,Y (h) with first-order correction ∼ √ h indicates an infinite specific surface area, which explains the roughness of the contours observed for model M4 (Fig. 11b). For model M6, as a result of the regularization (33) by a Gaussian kernel, one finds instead, for h → 0:

ρX,Y (h) ≈ 1 - h 2 4σ 2 , CX,Y (h) ∼ C(0) - h 2σ , (38)
so that smooth boundaries and a finite surface area are obtained for the random set. In that case, the specific surface areas of X and Y is given by the parameters σK (Eq. 34) and λX,Y as:

SV X = 1 σKπ e -λ 2 X /2 , SV Y = 1 σKπ e -λ 2 Y /2 . ( 39
)
The contact surface areas between any two phases are given by:

SV W G = fY SX , SV W B = (1 -fY )SX, SV BG = (1 -fX )SY . ( 40
)
The formula above match the numerical measures for the contact areas in the the model (Tab. 7).

Conclusion

A methodology has been presented for simulating 3D models of three-phase anode composites using 2D SEM images. Such images should first be segmented, which requires removing the "halo" (or 3D) effect present in porous materials. For moderate halo effects, a combination of dilation and openings is sufficient. Second, the segmented microstructures have been characterized by a set of stereological measurements, in particular the covariance and cross-covariance functions.

Third, a generic three-phase random model has been introduced, based on two underlying independent random sets. We showed that this generic model results in simple constraints on the covariance and cross-covariance functions that can be tested in practice. For the anode layers studied here, the covariance and cross-covariance functions were found to be roughly compatible with one of the possible combinations of the three-phase model. In a fourth step, several models, namely Boolean sets and plurigaussian fields, were explored for the underlying random sets defining the three-phase model. Good agreement was obtained with plurigaussian models in terms of covariance, granulometry and linear erosion functions and the plurigaussian models were found to be visually very close to the materials. Furthermore, the models can be simulated very easily and efficiently, using the covariance function of the material or an analytical model that we propose. Similar results were obtained on anode layers of different origins and microstructures, highlighting the versatility of the proposed method.

We emphasize that the methodology developed in this work is applicable to materials with an arbitrary number of phases. In the general case, an n-phases microstructure is modeled using n -1 underlying independent random sets. The independency assumption enforce a set of constraints on the covariance and cross-covariance functions of the material's phases, as studied in [START_REF] Jeulin | Morphologie Mathématique et Propriétés Physiques des agglomérés de minerais de fer et du coke métallurgique[END_REF]. This work demonstrates that such n-phases models are useful for simulating real materials.

Figure 2 :Figure 3 :

 23 Figure 2: SEM images of symmetric cells 14 (a; b) and 29 (c; d) manufactured by CNR ISTEC. Images are 1024 pixels (12.7 µm) wide and 883 pixels (11 µm) height for a spatial resolution of 12.4 nm/ pixel (images: D. Masson). Images (a) and (c) show the layer on top of the electrolyte, and (b) and (d) on the bottom. Black: pores; white: GDC; dark-gray: LST. The electrolyte is visible in images (a) and (b).

Figure 4 :

 4 Figure 4: Mean covariance C ii (h) of the porous phase over a sample, as a function of the vector h, parametrized by its norm h and angle with the x-axis θ. The angle θ takes on increasing values θ = 0, ..., π. The isotropy hypothesis is validated since the covariance remains the same regardless the direction of h.

Figure 5 :

 5 Figure 5: Schematic representation of a three-phase material A 1 (dark gray), A 2 (light gray), A 3 (crosshatch) modeled by two underlying independent random sets X (ellipses) and Y (triangles).

Figure 6 :

 6 Figure 6: Cross-covariance C GW between sets G and W measured on images of sample 29-s (solid line). Comparison with the cross-covariances C H kGW computed using (15), (16) and (17). The latter are computed for each configuration k = 1, ..., 6 from Tab. (2).

Figure 7 :Figure 8 :

 78 Figure 7: Linear erosion function Q i (ℓ) of each phase of the composite 29-s, averaged over all 8 images of the sample and over 10 directions, represented in lin-log scale. The exponential decrease is compatible with a Boolean model with convex grains (see Eq. 21).

  2 . The set Y , a Boolean model of cylinders, is left unchanged. The model M2 is accordingly parametrized by 7 variables: the Poisson intensities Φ M 2 X,Y of X and Y , the spheres radii R M 2 X and R ′M2 X , the probability s M 2 , the radius R M 2 Y and length L M 2 Y of the primary grains in model Y . In the second model, denoted M3, the two sets X and Y are Boolean models of sphero-cylinders. This model is parametrized by 6 variables: the intensities of the Poisson point processes Φ M 3 X,Y and the lengths L M 3 X,Y

Figure 9 :

 9 Figure 9: Sample 29-s: covariances C i (h) (a), linear erosion Q i (ℓ) (b) and granulometries G i (h) (c) of the segmented SEM image (solid lines) and that of the optimal Boolean model M 1 of mono-sized spheres and cylinders (dotted lines) for phases B (black), G (blue) and W (red).

Figure 11 :

 11 Figure 11: Sample 29-i: (a) crop of a segmented image, (b) plurigaussian model M 4 , (c) plurigaussian model M 5 using the Corson model for the covariance functions, (d) plurigaussian model M 6 with Gaussian low-pass filtering.Note that for the sake of comparison, all images are simulated using fixed Gaussian noises U X,Y .

Figure 12 :Figure 13 :

 1213 Figure 12: Sample 29-s: covariances C i (h) (a), linear erosion Q i (ℓ) (b) and granulometries G i (h) (c) of the segmented SEM image (solid lines) and that of the optimal Plurigaussian model M 6 (dotted lines) for phases B (black), G (blue) and W (red).

Figure 14 :Figure 15 :

 1415 Figure 14: SEM images (a, c) vs. microstructure model M 6 (b, d) for samples A4-i (a, b) and B1-i (c, d).

  (15). In this figure, the quantity log [-log wX (h)] is plotted as a function of log h. The form (35) leads to the expansions, for h → 0: ρX,Y (h) ≈ 1 -

Table 1 :

 1 Volume fractions f i of each phase i estimated on SEM images and their associated relative error ǫ fi .

	14-s 24-i 24-s 25-i 25-s 28-i 28-s 29-i 29-s 32-i	151 ± 0.009 0.203 ± 0.007 0.646 ± 0.009 0.151 ± 0.009 0.168 ± 0.008 0.681 ± 0.009 0.161 ± 0.008 0.161 ± 0.007 0.678 ± 0.009 0.179 ± 0.009 0.198 ± 0.007 0.624 ± 0.008 0.175 ± 0.009 0.177 ± 0.007 0.648 ± 0.009 0.171 ± 0.013 0.193 ± 0.010 0.636 ± 0.012 0.185 ± 0.006 0.196 ± 0.006 0.619 ± 0.006 0.197 ± 0.005 0.214 ± 0.005 0.589 ± 0.005 0.154 ± 0.007 0.162 ± 0.006 0.684 ± 0.007 0.187 ± 0.006 0.207 ± 0.005 0.605 ± 0.005 0.188 ± 0.009 0.197 ± 0.008 0.615 ± 0.009

Table 3 :

 3 Relative error E(k) (Eq. (

	Image set	H 1	H 2	H 3	H 4	H 5	H 6
	14-i	3.15	3.15	2.9	2.9 14.11	1.4
	14-s	4.14	3.99 3.84 3.86 6.55	2.07
	15-i	3.61	3.42	3.2 3.21 17.06 4.19
	15-s	3.94	3.97 2.52 2.52 19.46	1.96
	24-i	4.24	4.31 3.27 3.27 3.88	2.76
	24-s	3.62	3.84 3.33 3.31 7.64	2.38
	25-i	5.9	5.83 3.53 3.53 4.05	1.76
	25-s	4.96	4.84 4.34 4.35 7.45	1.87
	28-i	3.23	3.26 3.96 3.96 5.18	2.44
	28-s	2.18	2.16 3.74 3.74 5.99	2.43
	29-i	3.95	3.82 3.48 3.48 4.62	1.85
	29-s	3.17	3.16 3.96 3.96 7.16	2.42
	32-i	3.21	3.17 2.97 2.98 4.39	1.77
	32-s	3.28 3.31 3.64 3.63 7.57	3.43
	33-i	2.94	2.88 4.16 4.16 5.86	3.45
	33-s	2.75	2.71 2.97 2.97 4.15	2.17

Table 4 :

 4 Global error (×10

	6 )

Table 5 :

 5 Values of the parameters (c,n) (see Eq. 32) of the Corson model used for each of the 11 samples. For each sample two sets of parameters are identified, the first one (c X ,n X ) corresponds to the random set X and the other one (c X ,n X ) to the random set Y .

	Sample	c X	n X	c Y	n Y
	14-i	0.138 0.859 0.159 0.863
	14-s	0.135 0.850 0.162 0.851
	15-i	0.155 0.834 0.186 0.827
	15-s	0.153 0.813 0.168 0.844
	24-i	0.125 0.843 0.161 0.833
	24-s	0.120 0.867 0.150 0.859
	25-i	0.145 0.785 0.175 0.821
	25-s	0.147 0.795 0.176 0.831
	28-i	0.136 0.839 0.156 0.849
	28-s	0.206 0.798 0.228 0.813
	29-i	0.149 0.810 0.175 0.825
	29-s	0.146 0.812 0.164 0.840
	32-i	0.108 0.864 0.108 0.893

Table 7 :

 7 Specific surface areas S VGW , S VW B and S VGB (in µm -1 ) between any two phases measured on SEM images and on the corresponding M 6 model. Discrepancies between the models and SEM images are highlighted in dark and light gray when the model underestimates and overestimates, respectively, the measurement carried out on the SEM image.

		S VGW	S VW B	S VGB
	Sample M 6 SEM	M 6	SEM	M 6	SEM
	14-i	1.36 1.23	4.20	4.68	7.90	7.97
	14-s	1.02 1.07	4.43	4.88	6.99	6.93
	15-i	0.96 1.04	3.74	4.22	5.46 7.34
	15-s	1.42 1.47	3.91	4.80	6.34 8.43
	24-i	1.10 1.09	4.56	4.32	5.81	6.10
	24-s	1.41 1.42	4.47	4.74	6.75	6.99
	25-i	1.36 1.18	4.97	5.24	6.57	7.15
	25-s	1.46 1.23	4.83	5.31	7.10	7.78
	28-i	1.59 1.20	4.95	5.45	6.89	7.41
	28-s	1.84 1.42	5.05 8.20	7.30 10.55
	29-i	1.13 0.90	4.73	5.11	6.26	6.80
	29-s	1.70 1.33	4.99	5.61	7.27	7.89
	32-i	1.43 1.47	4.38	4.08	5.32	5.37
	32-s	1.59 1.78	4.32	4.14	5.64	5.89
	33-i	0.99 1.14	3.94	3.88	5.36	5.11
	33-s	1.31 1.38	4.33	4.14	5.62	5.22
	A1-i	0.56 1.42	3.00 2.25	4.99	4.69
	A1-s	0.54 1.21	2.81 2.32	4.69	4.56
	A2-i	0.93 1.79	3.77	3.47	5.57	4.84
	A2-s	1.26 2.22	4.13	3.74	5.90 4.76
	A3-i	0.98 1.74	3.88	3.60	5.19	4.92
	A3-s	1.23 2.05	4.20	4.18	5.27	4.78
	A4-i	0.71 1.07	3.64	3.55	5.05	4.70
	A4-s	1.05 1.16	4.02	4.22	5.40	5.20
	B1-i	3.85 3.96	5.28	5.78	7.90	8.00
	B1-s	4.06 4.59	5.46	5.78	7.67	7.19

Acknowledgements

The research leading to the results presented has received funding from the European Union Seventh Framework Programme (FP7/ 2007-2013) for the Fuel Cells and Hydrogen Joint Technology Initiative under grant agreement 303429. The authors are grateful to Alessandra Sanson, Elisa Mercadelli and Angela Gondoli for providing the ISTEC anode samples and to David Masson, Anthony Chesnaud and Alain Thorel for image acquisitions and Armines anode samples. The authors also thank Rémi Costa, Feng Han, Vitaly Yurkiv and Aitor Hornes for providing the DLR anode samples. Finally, the authors are grateful to two anonymous reviewers for helpful suggestions.


M Valdez

Lancinha Pereira

A Nicolle

D Berthout
email: david.berthout@ifpen.fr

Hydrothermal aging effects on Cu-zeolite NH 3 -SCR catalyst

After-treatment systems for Diesel vehicles face a challenge for upcoming emission regulations.

Regarding the level of nitrogen oxides (NOx) and soot required the global catalytic activity of the exhaust line needs to be improved. Hydrothermal aging conditions explored during particulate filter regeneration are very stringent for the deNOx catalyst and can strongly reduce its durability. Ammonia Selective Catalytic Reduction (NH 3 -SCR) systems mainly based on Copper or Iron exchanged zeolite catalysts have emerged as effective technologies to reduce NOx emissions with their durability being continuously improved. In this work, we evaluated the impact of different hydrothermal treatments on a commercial copper-exchanged zeolite catalyst. The goal was to link the catalytic material property changes with catalytic performance changes as a function of both aging duration and temperature. The evolution of the zeolite structure as well as the changes of the copper state were experimentally investigated. Based on a detailed material properties analysis of Cu-zeolite, a semi detailed kinetic model of has been built to quantitatively predicted adsorption and desorption of NH 3 as a function of hydrothermal aging conditions. This work is a first step to bridge the gap between lab analysis conclusions and global reactivity observed in real conditions.

1.

Introduction

Emissions standards in transportation have focused on nitrogen oxides (NOx) abatement in the past years. Removing NOx emitted by lean-burn gasoline and Diesel engines has become one of the main challenges for car manufacturers. One preferred solution for deNOx is ammonia selective catalytic reduction (NH 3 -SCR) based systems which have already been implemented by most automotive companies. The incorporation of these systems downstream of a particulate Filter (DPF) and more recently the combination of filtration function with SCR function in a same device (SCR on filter, SCR-F) subjects the catalytic material to very high temperatures, close to 1000°C [START_REF] Kawakami | High Porosity DPF Design for Integrated SCR Functions[END_REF], due to repeated filter regenerations. Consequently, NH 3 -SCR catalysts need to provide a great efficiency over a wide temperature range and to be durable at temperatures up to 900°C to maintain a good conversion over the complete life of the system. Metal-exchanged zeolite catalysts are commonly used in passenger and light duty diesel vehicles.

Various zeolites with different pore structures have been studied for mobile applications. Copperexchanged zeolites as Cu-MFI [START_REF] Sullivan | Conditions in which Cu-ZSM-5 outperforms supported vanadia catalysts in SCR of NO x by NH 3[END_REF][START_REF] Sjövall | Identification of adsorbed species on Cu-ZSM-5 under NH3 SCR conditions[END_REF], Cu-BEA [START_REF] Auvray | Local ammonia storage and ammonia inhibition in a monolithic copper-beta zeolite SCR catalyst[END_REF], Cu-CHA [START_REF] Kwak | Excellent activity and selectivity of Cu-SSZ-13 in the selective catalytic reduction of NOx with NH 3[END_REF], and iron-exchanged zeolites as Fe-MOR [START_REF] Ma | Propene Poisoning on Three Typical Fe-zeolites for SCR of NO x with NH 3 : From Mechanism Study to Coating Modified Architecture[END_REF], Fe-BEA [START_REF] Ma | Propene Poisoning on Three Typical Fe-zeolites for SCR of NO x with NH 3 : From Mechanism Study to Coating Modified Architecture[END_REF][START_REF] Coq | The simultaneous catalytic reduction of NO and N 2 O by NH 3 using an Fe-zeolite-beta catalyst[END_REF], Fe-MFI [START_REF] Ma | Propene Poisoning on Three Typical Fe-zeolites for SCR of NO x with NH 3 : From Mechanism Study to Coating Modified Architecture[END_REF][START_REF] Long | Superior Fe-ZSM-5 catalyst for selective catalytic reduction of nitric oxide by ammonia[END_REF], Fe-FER [START_REF] Malpartida | CO and NO adsorption for the IR characterization of Fe 2+ cations in ferrierite: An efficient catalyst for NOx SCR with NH3 as studied by operando IR spectroscopy[END_REF] or a mixture of Fe-and Cu-zeolites [START_REF] Stiebels | Cu-CHA/Fe-BEA mixed zeolite catalyst and process for the treatment of NOx in gas streams[END_REF] are evaluated for NH 3 -SCR applications. Initially, the studies in literature focused mostly on finding a catalyst with good NOx conversion as a function of the operating conditions. Currently, the target is to find catalysts combining high activity at low temperatures and durability at high temperatures.

Since the work of Iwamoto [START_REF] Sato | Cu-ZSM-5 zeolite as highly active catalyst for removal of nitrogen monoxide from emission of diesel engines[END_REF] on Cu-ZSM-5, an increasing number of papers has been published, showing an increasing interest of copper-exchanged zeolites for NH 3 -SCR applications. Park et al. [START_REF] Park | Hydrothermal stability of CuZSM5 catalyst in reducing NO by NH 3 for the urea selective catalytic reduction process[END_REF] investigated the hydrothermal stability of Cu-ZSM-5 catalysts at different copper contents. The worst deactivation was found to occur for the under-exchanged samples, even for an aging temperature as low as 600°C, the optimal copper loading lying in the range 3-4 wt%. Lambert et al. [START_REF] Cavataio | Enhanced durability of a Cu/zeolite based SCR catalyst[END_REF] evaluated an improved Cu-zeolite formulation under stringent hydrothermal conditions. The latest state-of-the-art catalyst maintained, a stable deNOx activity at low temperature after ageing at temperatures up to 950°C. Olsson et al. [START_REF] Wilken | Mechanistic investigation of hydrothermal aging of Cu-Beta for ammonia SCR[END_REF] observed no significant changes of the physical structure of their Cu-BEA catalyst after aging but reactivity was altered. They subsequently focused on the oxidation state variations of copper, attributing the activity of their catalyst to Cu 2+ and Cu + sites.

There is a growing interest in copper exchanged chabazites due to their high hydrothermal stability. Hydrothermal aging impacts the zeolite structure and its redox properties as a function of the zeolite structure and consequently leads to the modification of the catalytic activity [START_REF] Ma | Characterization of commercial Cu-SSZ-13 and Cu-SAPO-34 catalysts with hydrothermal treatment for NH 3 -SCR of NOx in diesel exhaust[END_REF] [16]. The presence of water combined to high temperature in exhaust gas promotes the loss of Brönsted acidity and dealumination [START_REF] Silaghi | Challenges on molecular aspects of dealumination and desilication of zeolites[END_REF]. The dealumination process has been found to involve either an hydrolysis mechanism of Al-O bonds [START_REF] Sano | Influence of crystallinity of HZSM-5 zeolite on its dealumination rate[END_REF] [START_REF] Lutz | Investigation and modeling of the hydrothermal stability of technically relevant zeolites[END_REF] or, alternatively, the removal of framework aluminum and migration of framework silicium followed by the self-healing of a silanol nest [START_REF] Marcilly | Acido-Basic Catalysis -Application to Refining and Petrochemistry[END_REF]. Brandenberger et al. [START_REF] Brandenberger | Hydrothermal deactivation of Fe-ZSM-5 catalysts for the selective catalytic reduction of NO with NH 3[END_REF] found that the dealumination mechanism of ironexchanged ZSM-5 zeolites starts with the migration of iron from the exchange positions and then proceeds with a hydrolysis mechanism of the Al-O bonds. Copper ions in exchange positions appear to protect the zeolite structure against dealumination [START_REF] Suzuki | Inhibition of dealumination of ZSM-5 zeolites by alkali metals[END_REF], preventing hydrolysis and structure collapse in the less aged states [START_REF] Grinsted | The relation between deactivation of CuZSM-5 in the selective reduction of NO and dealumination of the zeolite[END_REF], the structure being more flexible and less prone to changes in the presence of Cu 2+ and Al 3+ cations [START_REF] Kucherov | Rearrangement of cationic sites in CuH-ZSM-5 and reactivity loss upon high-temperature calcination and steam aging[END_REF]. However, if the cation is removed from the exchange position, the migration of metallic ions induces the formation of proton exchange sites, which are more easily dealuminated under wet conditions. NOx reduction activity is connected to active copper sites, as changes in its oxidation state were reported during deactivation [START_REF] Yan | Deactivation of Cu/ZSM-5 Catalysts for Lean NOx Reduction: Characterization of Changes of Cu State and Zeolite Support[END_REF]. It was found that during aging Cu 2+ ions can be hydrolyzed to oxo-cations [START_REF] Li | Characterization of Active Species in Cu-Beta Zeolite by Temperature-Programmed Reduction Mass Spectrometry (TPR-MS)[END_REF] and that copper migrates from exchanged positions to form CuO aggregates in the zeolite porosity [START_REF] Park | Hydrothermal stability of CuZSM5 catalyst in reducing NO by NH 3 for the urea selective catalytic reduction process[END_REF]. A loss of zeolite crystallinity was also observed [START_REF] Kharas | Deactivation in Cu-ZSM-5 lean-burn catalysts[END_REF]. As a last point, dealumination promotes the redistribution of the copper ions as they show strong interaction with the alumina clusters. This interaction appears to be the most important for highly deactivated and dealuminated samples and may lead to the formation of CuAl 2 O 4 [START_REF] Bolt | Formation of Nickel, Cobalt, Copper, and Iron Aluminates from α-and γ-Alumina-Supported Oxides: A Comparative Study[END_REF].

In this work, we evaluated the impact of different hydrothermal treatments on a commercial copper-exchanged zeolite catalyst. The objective of this study is to link more precisely the catalytic material modifications with catalytic performance changes as a function of both aging duration and temperature. The mechanism evolution of the zeolite structure, the changes of the copper state and sintering were experimentally investigated.

Experimental

Materials

Parent sample

A commercial NH 3 -SCR washcoated cordierite monolith of 400 cells per square inch (CPSI) cell density was used in this study. Some cores were extracted for aging and subsequent reactivity tests. The catalyst powder used for material characterization was extracted from the monolith by a specific process based on thermal shock in liquid nitrogen. The elemental composition of the sample was determined using X-Ray Fluorescence Thermo Fisher Scientific ARL AdvantX spectrometer. The structure of the zeolite has been determined by X-ray diffraction as BEA zeolite, having a Si/Al ratio of 12 with copper loading of 2.8% (w/w) [START_REF] Kieffer | Characterisation of a Commercial Automotive NH 3 -SCR Copper-Zeolite Catalyst[END_REF].

Hydrothermal treatment

The samples were hydrothermally aged with flowing gas containing 10% oxygen and 10% water vapor introduced at 12 000 h -1 . The aging was carried out in a muffle oven at 600°C for 2 h, at 750°C for 16 h and 64 h, at 850°C for 4 h, 16 h and 64 h, and at 950°C for 1 h and 4 h (see Table 1).

1h 2h 4h 16h 64h 600°C 600_2 750°C 750_16 750_64 850°C 850_4 850_16 850_64

950°C 950_1 950_4

Table 1 -Aged samples reference-names

Catalyst characterization

Micromeritics ASAP 2420 sorption analyzer was used to measure BET surface areas by a static volumetric technique, based on the amount of nitrogen adsorbed at 77K. Prior to the adsorption measurements, the catalyst samples were degassed at 150°C in vacuum. The surface area was calculated using the Brunauer-Emmett-Teller method [START_REF] Brunauer | Adsorption of gases in multimolecular layers[END_REF] and the consistency criteria proposed by Rouquerol et al. for microporous solids [START_REF] Rouquerol | Is the BET equation applicable to microporous adsorbents?[END_REF].

Powder X-Ray Diffraction (XRD) patterns for the catalysts were observed by using a Siemens D5005 X-ray diffractometer with in the Bragg-Brentano geometry with monochromated Cu Kα1 radiation (0,15406 Å).

The 27 Al MAS NMR and 29 Si MAS NMR spectra were measured at 12 kHz using a Bruker Avance 400 spectrometer with 4 mm CPMAS probe.

H 2 -TPR experiments were performed on Micromeritics AutoChem II 2920. The samples were firstly calcined with 20% oxygen in nitrogen at 550°C for 30 minutes to remove water and cooled down to 50ºC. Temperature programmed reduction was carried out in 3% hydrogen in argon, introduced at 15 ml/min, from 50ºC up to 550°C. The samples remained at the final temperature for 30 min. The temperature rate was of 10ºC/min.

CO chemisorption was performed in a DRIFT cell under dynamic conditions on

ThermoOptek Nicolet Nexus1 spectrometer. The sample was pre-treated at 550°C with air (50 ml/min) for 2 h and cooled down to room temperature (reference spectra). A blend of 10% CO (v/v) in air was then introduced to the reactor for 1 hour (CO spectra), followed by 30 min with air.

NO adsorption was performed in IRTF in situ cell under static conditions (vacuum) on

ThermoOptek Nicolet Nexus2 spectrometer. The sample was pre-treated at 450°C for 2 h and cooled down to room temperature under dynamic vacuum (reference spectra). The NO probe molecule was then introduced in increasing amounts in the cell.

The ammonia temperature programmed desorption was carried out on bench flow reactor system using small cylindrical monoliths of 1" diameter and 2" length at 150°C and 30000 h -1

Results and Discussion

Material changes analysis

N 2 isotherm adsorption -BET surface area

For a microporous material as Beta zeolite, the evolution of either the micro-porosity or the BET surface area turns out to be an attractive descriptor of the structure aging process. In order to adapt BET analysis to microporous materials, some authors [START_REF] Rouquerol | Is the BET equation applicable to microporous adsorbents?[END_REF][START_REF] Schneider | Adsorption isotherms of microporous-mesoporous solids revisited[END_REF][START_REF] Schneider | Pore-volume and surface area in microporous-mesoporous solids[END_REF] proposed to modify the classic BET equation or, alternatively, to choose a relevant pressure range for calculation. For a pressure range of 0.05 < P/P 0 < 0.3, Snurr et al. [START_REF] Bae | Evaluation of the BET method for determining surface areas of MOFs and zeolites that contain ultra-micropores[END_REF][START_REF] Walton | Applicability of the BET method for determining surface areas of microporous metal-organic frameworks[END_REF] tested the two consistency criteria proposed by Rouquerol et al. [START_REF] Rouquerol | Is the BET equation applicable to microporous adsorbents?[END_REF] on zeolites and metal-organic frameworks (MOF's), and concluded that the BET theory is acceptable to characterize porous materials with no similarities to the flat surfaces assumed upon the derivation of the BET theory. Consequently, we used these two criteria to compute the variations of surface area induced by hydrothermal treatment. The For a given temperature of aging, the surface area is always decreasing with the increasing time, and for a given duration of aging the surface area is always decreasing with the increasing temperature.

XRD

The XRD diffractograms of Cu-Beta before and after different hydrothermal treatments are shown in Fig. 2. In the 2θ region of 5-50°, diffraction peaks at around 7.6° and 22.4° can be observed, indicating the presence of the Beta phase.

Copper agglomerates as crystalline copper compounds can be detected by XRD if the copper content is not too low. In our patterns, no peak for CuO or Cu 2 O is detected even after strong aging but regarding the low content and high dispersion of copper in the catalyst, the detection limits are probably reached. This fact does not mean that the copper does not agglomerate after The intensity of the specific peaks of BEA zeolite decreases with the hydrothermal treatment showing a loss of crystallinity. It is interesting to point out that aging at temperatures below 800°C does not lead to the collapse of the zeolite structure. By considering the fresh sample as being 100% crystalline, the crystallinity evolution of the aged samples was determined using the two main peaks 2θ = 7,6° and 22,48°of the BEA zeolite. Results are reported in 29 Si MAS NMR Fig. 3 shows the 29 Si MAS NMR spectra of the catalyst for the fresh catalyst and four hydrothermal treatments. The evolution of the coordination of Si atoms and the Si/Al of the framework, provides interesting information on the lattice structure variations. The assignment of the chemical shift of the resonances as well as the Si/Al ratio calculation are based on the work of Camblor [START_REF] Camblor | Characterization of nanocrystalline zeolite Beta[END_REF]. which is assigned to Si(3Si, 1Al) and Si(3Si,1OH) in the zeolite. After hydrothermal treatment, the intensity of the resonances near 100 ppm decreases and the peaks at -111 ppm become smaller and broader. These spectral variations are a characteristic behavior of an increase of framework Si/Al ratio mainly due to dealumination phenomena. The spectra were deconvoluted using the Dmfit software [39] using five Gaussian/Lorentzian lines centered at -115, -111, -104, -100 and -95 ppm, to compute this variation of Si/Al ratio [START_REF] Camblor | Characterization of nanocrystalline zeolite Beta[END_REF]. Based on these results, the theoretical framework Al content of each sample was computed using the chemical formula of zeolite BEA ( 29 Si MAS NMR

27 Al MAS NMR

The spectrum of the fresh sample on Fig. 4-a exhibits a resonance centered near 54 ppm which can be attributed to tetrahedrally coordinated aluminum, Al (IV), in the zeolite framework [START_REF] Maier | Steaming of zeolite BEA and its effect on acidity: a comparative NMR and IR spectroscopic study[END_REF].

No other peak was detected. The catalysts patterns of aged samples presented also a peak at 54 ppm, but the relative intensity of this peak gradually decreased with increasing aging temperature. That means that alumina are removed from their framework tetrahedral positions.

For the samples aged at temperature larger than 750°C, an increasing contribution within aging severity can be noticed around 30 ppm, as can be seen on Fig 4-b. This peak can be attributed to pentahedral extraframework aluminum, Al(V) [START_REF] Yu | Insights into the Dealumination of Zeolite HY Revealed by Sensitivity-Enhanced 27Al DQ-MAS NMR Spectroscopy at High Field[END_REF] A contribution at 0 ppm is also visible for aged samples, increasing with aging severity and is assigned to the well-ordered octahedral extraframework aluminum, Al (VI) [START_REF] Maier | Steaming of zeolite BEA and its effect on acidity: a comparative NMR and IR spectroscopic study[END_REF]. A conclusion from these experiments is that the dealumination and the collapse of the structure seem to occur simultaneously based on XRD, and 29 Si, 27 Al MAS NMR results. However, the loss of microporous volume / surface area is not directly linked to the structural collapse for the low aging conditions (i.e. at 750°C ). The evolution of copper in the zeolite needs to be investigated to understand its interaction with the framework.

IR measurement and H 2 -TPR

The evolution of Cu + and Cu 2+ species in exchanged positions species was studied by IR-CO and IR-NO respectively. For the fresh sample and the low aging conditions : 2h at 600°C and 16h at 750°C, copper is present in exchanged positions. At room temperature, two main bands are noticeable at 2179 and 2152 cm -1 on IR-CO spectra (Fig. 6), these bands are assigned to symmetric and asymmetric vibrations of Cu + -(CO) 2 on isolated Cu + [START_REF] Kefirov | Stabilization of Cu + ions in BEA zeolite: Study by FTIR spectroscopy of adsorbed CO and TPR[END_REF][START_REF] Hadjiivanov | IR study of CO adsorption on Cu-ZSM-5 and CuO/SiO 2 catalysts: σ and π components of the Cu+-CO bond[END_REF]. For the low aged sample (16h at 750°C ), the intensity of these bands decreases showing a loss of copper (I) in exchanged position.

Fig. 6. IR spectra after 30 minutes under 10% CO / air mixture at ambient temperature On IR-NO spectra for high concentration of NO (Fig. 7), the main band is observable at 1902cm -1 assigned to isolated Cu 2+ [START_REF] Parvulescu | Effect of coexchanging with a second metal upon the interaction of nitric oxide with Cu-ZSM-5[END_REF] for the sample aged at 600°C during 2h. Another band visible at 1967cm -1 can be attributed to NO coordinated to associated Cu 2+ ions [START_REF] Chakarova | Coordination state of Cu + ions in Cu-[Al] MCM-41[END_REF]. Regarding the sample aged at 750°C during 16h, the band at 1967cm -1 disappeared, whereas two contributions at 1902cm -1 and 1886cm -1 are visible. The first one is attributed to isolated Cu 2+ and the second to small particles of CuO [START_REF] Hadjiivanov | Identification of neutral and charged N x O y surface species by IR spectroscopy[END_REF]. The presence of CuO seems to be also noticeable on the aged at 2h at 600°C sample and its proportion is increasing on the aged 16h at 750°C sample.

These IR measurements allows us to conclude that the ratio between Cu + and Cu 2+ is evolving in function of aging conditions. A reduction of the proportion of Cu + is visible with toughening aging conditions.

Fig.7. IR spectra for high concentration of NO at ambient temperature

In addition, temperature-programmed reduction of hydrogen was used to investigate the effects of hydrothermal aging on the reducibility of copper species. As shown in Fig. 8, H 2 consumption collected over fresh Cu/zeolites can be divided into two reduction peaks in the range of temperature tested. The first peak, centered at 190°C, can be attributed to the reduction of Cu 2+ to Cu + in exchanged positions and the second located at 350°C, to reduction of Cu + in exchanged positions to Cu 0 [START_REF] Sárkány | Redox chemistry in excessively ionexchanged Cu/Na-ZSM-5[END_REF][START_REF] Ribeiro | Characterisation of CuMFI catalysts by temperature programmed desorption of NO and temperature programmed reduction. Effect of the zeolite Si/Al ratio and copper loading[END_REF]. The low temperature peak is only present on the fresh sample and for the sample aged for 2 h at 600°C. For this last sample, a shoulder near 250°C is also present, meaning that the reduction at low temperature involves not only the reduction of Cu 2+ to Cu + , but also the reduction of CuO to Cu 0 . For the samples aged at 750°C, the low temperature peak is shifted to a temperature around 250°C and can be attributed to the reduction of small copper aggregates [START_REF] Kefirov | Stabilization of Cu + ions in BEA zeolite: Study by FTIR spectroscopy of adsorbed CO and TPR[END_REF]. By increasing temperature and duration of aging, the consumption peaks are shifted toward higher temperatures and become broader indicating a possible formation of copper aggregates and copper aluminate species that are reduced at high temperatures. This formation is indeed likely since dealumination promotes a redistribution of the Cu 2+ ions which can strongly bind to alumina clusters creating aluminate species [START_REF] Yan | Deactivation of Cu/ZSM-5 Catalysts for Lean NOx Reduction: Characterization of Changes of Cu State and Zeolite Support[END_REF]. Regarding our results, the hydrogen consumption decreases and the copper is not completely reduced at high temperatures, as the samples get more aged. This may be explained by the loss of zeolite crystalline structure, sintering and dealumination. The collapse of the zeolite structure damages its porosity and some copper can be sequestered inside the clusters or removed from the catalyst, hence being no longer available to be reduced. This is consistent with the results from Yezerets [START_REF] Li | Characterization of Active Species in Cu-Beta Zeolite by Temperature-Programmed Reduction Mass Spectrometry (TPR-MS)[END_REF], who reported that changes on the metal oxidation state of Cu-zeolite: hydrogen consumption at high temperatures was considered to be due to bulk copper oxide or copper aluminate, formed at high temperature treatment, which also contribute to the decrease easily reducible species content. They also suggested that some copper can be sequestered inside the collapsed zeolite structure, for the more severely aged samples, becoming inaccessible to be reduced.

Combining the results of IR measurements with H 2 -TPR and assuming that no copper oxide is present in the fresh sample, we can propose to compute the evolution of the copper in the different hydrothermal aged samples. The results obtained are shown on Fig. 9. Fig. 9. Evolution of the copper content in function for the different samples

Modeling the impact of hydrothermal aging on reactivity

Surface area hydrothermal aging model

The abovementioned experiments showed that the reactivity of the catalyst is significantly impacted by the different hydrothermal treatments. An important decrease of deNOx performance is observed with the increase of aging temperature. Thus, a model have been developed to link the kinetic constants variations to the catalyst composition evolution. In order to simplify the fitting of kinetic parameters, we limit their dependency to only one variable, the surface area, instead of two, temperature of aging and aging duration. To the best of our knowledge, this approach has not been attempted yet in the case of SCR semi-detailed kinetic modeling. Consequently, we built a model of surface area evolution as a function of hydrothermal aging condition for our experiment conditions.

The decrease of normalized BET surface area (S/S 0 ) appears to follow a power law, as shown in Fig. 10. This behavior is remarkably similar to that reported by Bartholomew on the sintering kinetics of supported noble metals [START_REF] Bartholomew | Sintering kinetics of supported metals: new perspectives from a unifying GPLE treatment[END_REF]. In fact, he suggested a general power law to describe the variation of normalized surface area of the zeolite as a function of aging temperature (T) and duration (t), following the equation :

- 𝒅 � 𝑺 𝑺 𝟎 � 𝒅𝒕 = 𝒌 𝒔 � 𝑺 𝑺 𝟎 - 𝑺 𝒆𝒒 𝑺 𝟎 � 𝒎
This expression allows a wide range of solutions depending on the power 'm'. In our experimental conditions, a value of m=2 provides the most accurate representation of aging. In that case, the normalized surface area writes :

𝑺 𝑺 𝟎 = 𝑺 𝒆𝒒 𝑺 𝟎 + (𝑪 𝟏 + 𝒌 𝒔 𝒕) -𝟏
The calibration in function of the temperature of aging, T, of C 1 , k s and S eq has been done on the 8 different aging conditions using least square method, and the setting has been validated against a supplementary sample hydrothermal aged in the same conditions at 800°C for 8 hours. The results obtained are presented on Fig. 10.

Reactor and kinetic Model

A model of reactor associated to a kinetic scheme was developed to quantify the impact of hydrothermal aging on the catalytic activity, using LMS Amesim software [START_REF] Mauviot | 0D Modelling: a promising means for after-treatment issues in modern automotive applications[END_REF]. The model is based on the work of Skarlis et al. [START_REF] Skarlis | Multisite modeling of NH 3 adsorption and desorption over Fe-ZSM5[END_REF] with a semi-detailed kinetic scheme [START_REF] Olsson | A kinetic model for ammonia selective catalytic reduction over Cu-ZSM-5[END_REF]. In our approach, the model of monolith was regarded as a series of continuously stirred tank reactors accounting for the impact of axial dispersion on macromixing without impairing CPU time.

The associated kinetic model takes into account ammonia adsorption-desorption. Five ammonia adsorption sites were proposed by Skarlis et al. [START_REF] Skarlis | Multisite modeling of NH 3 adsorption and desorption over Fe-ZSM5[END_REF] for Fe-ZSM-5, to simplify the model description only two sites were considered : σ 1 for weakly bound NH 3 , σ 2 for strongly bound NH 3 .

Table 4 -Global reactions used in the model

The reaction rates constants were described using Arrhenius formalism :

𝒌 = 𝑨 𝒆𝒙𝒑 �- 𝑬 𝒂 (𝜽, 𝑻) 𝑹𝑻 �
For the ammonia adsorption desorption on each site the global reaction rate follows the equation :

𝒓 𝒂𝒅𝒔-𝒅𝒆𝒔 = 𝒌 𝒂𝒅𝒔 𝑪 𝑵𝑯 𝟑 𝜽 𝝈𝒗𝒂𝒄𝒂𝒏𝒕 -𝒌 𝒅𝒆𝒔 𝜽 𝑵𝑯 𝟑
Adsorption was assumed to be a non-activated process, and the desorption was defined using Temkin approach widely used [START_REF] Skarlis | Multisite modeling of NH 3 adsorption and desorption over Fe-ZSM5[END_REF][START_REF] Olsson | A kinetic model for ammonia selective catalytic reduction over Cu-ZSM-5[END_REF][START_REF] Ciardelli | SCR-DeNOx for diesel engine exhaust aftertreatment: unsteady-state kinetic study and monolith reactor modelling[END_REF]:

𝑬 𝒂 𝒅𝒆𝒔𝒐𝒓𝒑 = 𝑬 𝒂 𝒅 𝟎 �𝟏 -𝜶 𝜽 𝑵𝑯 𝟑 �
The parameters adjusted were the ammonia storage capacity for each site, useful to compute NH 3 coverage, the pre-exponential factor for adsorption, the activation energy for desorption and the coverage dependence parameter.

NH 3 -TPD modeling

The reaction rate parameters for ammonia adsorption and desorption were calibrated with a least square method using NH 3 -TPD experiments for five different aging conditions : 2h at 600°C, 16h and 64h at 750°C, 4h at 850°C and 1h at 950°C. Fig. 11 shows the results of the model fitting on ammonia desorption during the temperature slope for these different aging temperatures. In our conditions, when the S/S 0 ratio is higher than 0.7, corresponding to the "low aging states", the reduction of NH 3 storage capacity seems to be directly linked to the decrease of the reducible copper content. For these low aging states, NH 3 storage capacity evolution can be computed using the following empirical equation :

𝑪 𝑵𝑯 𝟑 𝒂𝒈𝒆𝒅 = 𝑪 𝑵𝑯 𝟑 𝒇𝒓𝒆𝒔𝒉 �𝟑 𝑪𝒖 𝑯 𝟐 𝒂𝒈𝒆𝒅 𝑪𝒖 𝑯 𝟐 𝒇𝒓𝒆𝒔𝒉 -𝟐�
where 𝐶 𝑁𝐻 3 𝑎𝑔𝑒𝑑 , 𝐶 𝑁𝐻 3 𝑓𝑟𝑒𝑠ℎ are the storage capacities for the aged and fresh samples and, 𝐶𝑢 𝐻 2 𝑎𝑔𝑒𝑑 , 𝐶𝑢 𝐻 2 𝑓𝑟𝑒𝑠ℎ are the total reducible copper computed from H 2 TPR results for the aged and fresh samples.

The same trend is also visible on desorption energy and coverage factor for sites σ 1 and σ 2 , meaning that not only the number of adsorption sites is lower but also that the strength of sites is reduced.

In the "medium aging" zone, the activation energy values for desorption on σ 1 and σ 2 are kept constant, since the temperatures for maximum ammonia release are relatively constant during "medium aging", near 230°C for σ 1 and near 345°C for σ 2 . The activity decrease appears to mainly be related to the loss of adsorption capacity. For S/S 0 lower than 0.7, a simple correlation with the reduction of surface area, or active sites, can be established :

𝑪 𝑵𝑯 𝟑 𝒂𝒈𝒆𝒅 = 𝑲 � 𝑺 𝑺 𝟎 �
with K close to 300 µmol/g in our conditions.

Concerning the "strongest aging states", when S/S 0 is lower than 0.1, the values of desorption energy for σ 1 need to be reduced, meaning that globally the interactions between ammonia and adsorption site become weaker. The same reduction tendency is also observed for the coverage dependency factors, for both adsorption sites. This behavior is in accordance with the previous observations in material analysis, especially the loss of crystallinity observed on XRD patterns, which implies that the strongest adsorption sites for NH 3 are removed with the strongest hydrothermal treatment.

Conclusions

The aim of this work was to understand the global material changes phenomena appearing during deactivation, mainly surface loss, metal state changes and sintering in a Cu-beta catalyst used for NH 3 -SCR, in order to build a kinetic model accounting for deactivation.

Comparing to protonated zeolites, copper ions in exchanged positions contribute to a primary protection to dealumination. But with the increasing temperature of aging, they easily migrate due to changes in the electronic environment promoted by the presence of water and the hydrolysis of the zeolite structure induce a decrease of the crystallinity. The loss of copper species from exchanged positions promotes sintering that can occur, in the zeolite channel, or directly outside the zeolite framework. The copper sintering contributes to a loss of catalytic active sites, since the copper can be sequestered into large particles or removed from the catalyst.

This phenomenon was highlighted by the loss of reducible copper in H 2 -TPR analysis.

For the studied catalyst and under the investigated conditions, the BET surface area appears to be a relevant descriptor of the hydrothermal aging process. The BET surface area of metalexchanged zeolites can be modeled as a function of temperature and duration of hydrothermal aging by using a model initially developed to describe the evolution of metal dispersion over alumina or silica supports. The equation used for the variations of normalized surface area is applicable to our aging conditions and catalyst but its validity over a wider range of conditions needs to be assessed before it can be applied to every kind of aging process and any metal exchanged zeolite. Indeed, many parameters might have an impact on the form of the equation and parameter calibration, including the muffle oven used, the concentration of water, the structure of the zeolite, the Si/Al ratio and the metal content.

The conclusions from the material analysis have been compared to the parameters variations of a semi detailed kinetic modeling of NH 3 -TPD experiment. Two zones have been identified, the first one for the "low aging states", corresponding to the loss of reducible copper and the second "for the "medium" and "strong aging states" corresponding to the structure collapse evolution. In these zones, the global evolution of the parameters can be directly linked to material evolution.

However, the simple correlations established for ammonia storage capacity variations need to be further investigated, especially with respect to different zeolitic structures, Si/Al ratio and copper loading.

The predicted alteration of zeolite properties are well in accordance with those observed during the material analysis study, confirming the potential of this approach to deal with hydrothermal aging of Cu-zeolites for NH 3 -SCR.

As a last remark, one must keep in mind that this approach is a very simplified one, while hydrothermal aging of metal exchanged zeolites is a very complex process. Further improvement and validation of this method against a wider range of experimental conditions and smaller pore zeolites, such as Cu-SSZ-13, is part of on-going work in our group.

results obtained are shown on Fig. 1 :Fig. 1 .

 11 Fig.1. Normalized BET surface (S/S 0 ) variations in function of hydrothermal aging duration for

Fig. 2 .

 2 Fig.2. X-ray diffraction powder pattern of Cu-Beta for different hydrothermal treatments

Fig. 3 .

 3 Fig.3.29 Si MAS NMR spectra of fresh Cu-Beta sample and hydrothermal aged samples

Fig. 4 . 5 .Fig. 5 .

 455 Fig.4.27 Al MAS NMR spectra of (a) fresh and (b) aged at 850°C during 4h samples

Fig. 8 .

 8 Fig.8. Temperature-programmed reduction profiles of Cu-BEA catalysts for different

Fig. 10 .

 10 Fig.10. Normalized surface areas for different hydrothermal aging. Circle symbols correspond to

Fig. 11 .Fig. 12 .

 1112 Fig.11. Ammonia concentration during TPD experiment. Experimental data are plotted in solid

Table 2 .

 2

		22.48°	7.6°	S/S 0
	Fresh	1	1	1

Table 2 -

 2 Crystallinity variations computed from peaks 2θ = 7.6° and 22.48°, and normalized

BET surface (S/S 0 ) variations

3.1.3.

Table 3 ).

 3

	Hydrothermal aging	600°C 2h	750°C 16h	750°C 64h	850°C 16h	850°C 64h
	S/S 0	0.96	0.74	0.63	0.1	0.05
	Framework Al [µmol/g]	1110	980	790	180	160

Table 3 -

 3 Framework aluminum content variations computed from