Claire Lacour

Pascal Massart

Minimal penalty for Goldenshluger-Lepski method

Keywords: Nonparametric statistics, Adaptive estimation, Minimal penalty 2000 MSC: 62G07

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Introduction

Adaptive estimation is a challenging task in nonparametric estimation. Many methods have been proposed and studied in the literature. Most of them rely on some data-driven selection of an estimator among a given collection. Wavelet thresholding [START_REF] Donoho | Density estimation by wavelet thresholding[END_REF], Lepski's method [START_REF] Lepskiȋ | A problem of adaptive estimation in Gaussian white noise[END_REF], and model selection [START_REF] Barron | Risk bounds for model selection via penalization[END_REF] (see also [START_REF] Birgé | An alternative point of view on Lepski's method[END_REF] for the link between model selection and Lepski's method) belong to this category. Designing proper estimator selection is an issue by itself. From a constructive point of view, it is a crucial step towards adaptive estimation. For instance, selecting a bandwidth for kernel estimators in density estimation means that you are able to estimate the density without specifying its degree of smoothness in advance. Recently an interesting new estimator selection procedure has been introduced by [START_REF] Goldenshluger | Universal pointwise selection rule in multivariate function estimation[END_REF]. Assume that one wants to estimate some unknown function f belonging to some function space endowed with some norm . . Assume also that we have at our disposal some collection of estimators ( fh ) h∈H indexed by some parameter h, the issue being to select some estimator fĥ among this collection. The Goldenshluger-Lepski method proposes to select ĥ as a minimizer of B(h) + V (h) with

B(h) = sup{[ fh ′ -fh,h ′ 2 -V (h ′ )] + , h ′ ∈ H}
where x + denotes the positive part max(x, 0) and where fh,h ′ are auxiliary (typically oversmoothed) estimators and V (h) is a penalty term (called "majorant" by Goldenshluger and Lepski) to be suitably chosen. They first develop their methodology in the white noise framework (Goldenshluger andLepski, 2008, 2009), next for density estimation [START_REF] Goldenshluger | Bandwidth selection in kernel density estimation: oracle inequalities and adaptive minimax optimality[END_REF] and then for various other frameworks [START_REF] Goldenshluger | General selection rule from a family of linear estimators[END_REF]. Their initial motivation was to provide adaptive procedures for multivariate and anisotropic estimation and they used the versatility of their method to prove that the selected estimators can achieve minimax rates of convergence over some very general classes of smooth functions (see [START_REF] Goldenshluger | On adaptive minimax density estimation on R d[END_REF]. To this purpose, they have established oracle inequalities to ensure that, if V (h) is well chosen, the final estimator fĥ is almost as efficient as the best one in the collection. The Goldenshluger-Lepski methodology has already been fruitfully applied in various contexts: transport-fragmentation equations (Doumic et al., 2012), anisotropic deconvolution [START_REF] Comte | Anisotropic adaptive kernel deconvolution[END_REF], warped bases regression [START_REF] Chagny | Penalization versus Goldenshluger-Lepski strategies in warped bases regression[END_REF] among others (see also [START_REF] Bertin | Adaptive estimation of conditional density function[END_REF] which contains some explanation on the methodology). We cannot close this paragraph without citing the nice work of [START_REF] Laurent | Adaptive estimation of linear functionals by model selection[END_REF], who have independently introduced a very similar method, in order to adapt the model selection point of view to pointwise estimation.

In this paper we focus on the issue of calibrating the penalty term V . As we mentioned above the "positive" known results are of the following kind: the method performs well (at least from a theoretical view point) when V is well chosen. More precisely one is able to prove oracle inequalities only if V is not too small. But the issue is now: what is the minimal (or the optimal) value for V to preserve (or optimize) the performance of the method? Here we consider this issue from a theoretical point of view but actually it is a crucial issue for a practical implementation of the method. In this paper we focus on the (simple) classical bandwidth selection issue for kernel estimators in the framework of univariate density estimation. The main contribution of this paper is to highlight a phase transition phenomenon that can be roughly described as follows. For some critical quantity V 0 (that we call "minimal penalty") if the penalty term V is defined as V = aV 0 then either a < 1 and the risk E f -fĥ 2 is proven to be dramatically suboptimal, or a > 1 and the risk remains under control. This kind of phase transition phenomenon and its possible use for penalty calibration appeared for the first time in [START_REF] Birgé | Minimal penalties for Gaussian model selection[END_REF] in the context of Gaussian penalized model selection. It is interesting to see that the same phenomenon occurs for a pairwise comparison based selection method such as the Goldenshluger-Lepski method.

Proofs are extensively based on concentration inequalities. In particular, left tail concentration inequalities are used to prove the explosion result below the critical value for the penalty. Although the probabilistic tools are non asymptotic by essence, they merely allow us to justify that suprema of empirical processes are well concentrated around their expectations and the approximations that we make on those expectations are indeed asymptotic. Needless to say this means that our final results are (unfortunately) a bit of an asymptotic nature, at least as far as the identification of the critical value a = 1 is concerned. To be more concrete, we mean that for a given unknown density and a given sample size n, it is unclear that a phase transition phenomenon (if any) should occur at the critical value a = 1 as predicted by the (asymptotic) theory. But still, because of the concentration phenomenon, one can hope that some phase transition does occur (even non asymptotically) at some critical value even though it is not equal (or even close) to the (asymptotic) value a = 1. To check this, we have also implemented numerical simulations. These simulations allow us to understand what should be retained from the theory as a typical behavior of the method. In fact the simulations confirm the above scenario. It turns out that the phase transition does occur when you run simulations even though the critical point is not located at a = 1. This is actually what should be retained from the theory (at least from our point of view). The fact that some phase transition does occur is good news for the calibration issue because this means that in practice you can detect the critical value from the data (forgetting about the asymptotic value a = 1). Then you can hope to use this value to elaborate some fully data-driven and non asymptotic calibration of the method. We conclude the paper with providing some hints on how to perform that explicitly.

In Section 2 we specify the statistical framework and we recall the oracle inequality that can be obtained in the framework of density estimation. Then Section 3 contains our main theorem about minimal penalty. This result is illustrated by some simulations (Section 4). Finally, some proofs are gathered in Section 6 after some concluding remarks.

Kernel density estimation framework and upper bound on the risk

We consider independent and identically distributed real variables X 1 , . . . , X n with unknown density f with respect to the Lebesgue measure on the real line. Let . denote the L 2 norm with respect to the Lebesgue measure. For each positive number h (the bandwidth) we can define the classical kernel density estimator

fh (x) = 1 n n i=1 K h (x -X i ) (1)
where K is a kernel and K h = K(./h)/h. We assume here that the function to be estimated is univariate and we study the Goldenshluger-Lepski methodology without oversmoothing. This means that we do not use auxiliary estimators. We could actually prove the same results for the original method but the proofs are more involved and we decided to keep the proofs as simple as possible trying not to hide the heart of the matter.

To be more precise the procedure that we study is the following one: starting from some (finite) collection of estimators { fh , h ∈ H}, we set

B(h) = sup h ′ ≤h fh ′ -fh 2 -V (h ′ ) + with V (h ′ ) = a K h ′ 2 n (2)
with a being the tuning parameter of interest. Then the selected bandwidth is defined by ĥ = arg min

h∈H {B(h) + V (h)} . (3)
It is worth noticing that the penalty term V (h) which is used here is exactly proportional to the integrated variance of the corresponding estimator.

We introduce the following notation:

f h := E( fh ), h min := min H, h max := max H D(h) := max(sup h ′ ≤h f h ′ -f h , f -f h ) ≤ 2 sup h ′ ≤h f h ′ -f
We assume that the kernel verifies assumption

(K0) |K| = 1, K < ∞ and ∀ 0 ≤ x ≤ 1 K, K(x.) K 2 ≥ 1.
Assumption (K0) is satisfied whenever the kernel K is nonnegative and unimodal with a mode at 0. Indeed in this case K(xu) ≥ K(u) for all u ∈ R and x ∈ [0, 1]. This is verified for classical kernels (Gaussian kernel, rectangular kernel, Epanechnikov kernel, biweight kernel; see Lemma 4). This entails that for all

h ′ ≤ h, K h ′ -K h 2 ≤ K h ′ 2 -K h 2 .
This Pythagore type inequality is a one of the key properties that we shall use for proving our results.

Let us now recall the positive results that can be obtained for the selection method if a is well chosen.

Proposition 1. Assume that f is bounded and K verifies (K0). Let fĥ be the selected estimator defined by (1), ( 2), (3). Assume that the parameter a in the penalty V satisfies a > 1.

• There exist some positive constants C 0 > 0 and c > 0 such that, with probability larger than

1 -2 h∈H h ′ ≤h max(e -c √ n , e -c/h ′ ), the following holds fĥ -f ≤ C 0 inf h∈H D(h) + √ a K h √ n .
The values

C 0 = 1 + 2(1 + (a 1/3 -1) -1 ) and c = (a 1/3 -1) 2 min( 1 24 , K 2 6 f ∞ ) are suitable. • Moreover, if n -1 ≤ h min ≤ h max ≤ log -2 (n), there exists a positive constant C depending only on K and f ∞ such that E fĥ -f 2 ≤ 2C 2 0 inf h∈H D 2 (h) + a K h 2 n + Cn|H| 2 e -(a 1/3 -1) 2 C log 2 (n) (C = max(24, 6 f ∞ / K 2 , 4 f ∞ + 4 K 2 ) works).
We recognize in the right-hand side of the oracle type inequalities above the classical bias variance tradeoff. This oracle inequality shows that the Goldenshluger-Lepski methodology works when a > 1, at least for n larger than some integer depending on a and the true density. From a non asymptotic perspective this "positive result" should be understood with caution, it is clear from the analysis of the behavior of the constants involved with respect to a that these constants are worse when a is close to 1.

The proof of Proposition 1 is postponed in Section 6.1. It is based on the following concentration result (adapted from [START_REF] Klein | Concentration around the mean for maxima of empirical processes[END_REF]) and more precisely on inequality (4) below.

Lemme 2. Let X 1 , . . . , X n be a sequence of i.i.d. variables and

ν(t) = n -1 n i=1 [g t (X i ) -E(g t (X i ))] for t belonging to a countable set of functions F . Assume that for all t ∈ F g t ∞ ≤ b and Var(g t (X 1 )) ≤ v. Denote H = E(sup t∈F ν(t)). Then, for any ε > 0, for H ′ ≥ H, P(sup t∈F ν(t) ≥ (1 + ε)H ′ ) ≤ max exp - ε 2 6 nH ′2 v , exp - min(ε, 1)ε 24 nH ′ b (4) P(sup t∈F ν(t) ≤ H -εH ′ ) ≤ max exp - ε 2 6 nH ′2 v , exp - min(ε, 1)ε 24 nH ′ b (5) Moreover Var(sup t∈F ν(t)) ≤ v n + 4 bH n (6)

Minimal penalty

In this section, we are interested in finding a minimal penalty V (h), beyond which the procedure fails. Indeed, if a and then V (h) is too small, the minimization of the criterion amounts to minimize the bias, and then to choose the smallest possible bandwidth. This leads to the worst estimator and the risk explodes.

In the following result h min denotes the smallest bandwidth in H and is of order 1/n. Theorem 3. Assume that f is bounded. Choose H = {e -k , ⌈2 log log n⌉ ≤ k ≤ ⌊log n⌋} as a set of bandwidths. Consider for K the Gaussian kernel, the rectangular kernel, the Epanechnikov kernel or the biweight kernel. If a < 1, then there exists C > 0 (depending on f , a, K) such that, for n large enough (depending on f and K), the selected bandwidth ĥ defined by (2) and (3) satisfies P( ĥ

≥ 3h min ) ≤ C(log n) 2 exp(-(log n) 2 /C)
i.e. ĥ < 3h min with high probability. Moreover

lim inf n→∞ E f -fĥ 2 > 0.
This theorem is proved in Section 6.2 for more general kernels and bandwith sets. Here we have simplified the conditions on H for the sake of readability. Actually the real condition on H for Theorem 3 is that E H = min{h/h ′ ; h ∈ H, h ′ ∈ H, h > h ′ } does not depend on n and is larger than 1. It can be verified for the highlighted set

H = {e -k , a n ≤ k ≤ b n }, but for H = {c n + d n k, a n ≤ k ≤ b n } as well.
Mathematically, the proof of this result relies on two main arguments. The first argument is probabilistic: roughly speaking concentration inequalities which allow to deal with expectations of the pairwise square distances between estimators instead of the square distances themselves. The other argument is analytical: it essentially relies on proper substitutes to Pythagoras' formula for kernel smoothing. The phase transition phenomenon is actually easier to highlight in a context for which we have the actual Pythagoras' identity at our disposal, see the discussion on projection estimators for Gaussian white noise model in [START_REF] Lacour | Minimal penalty for goldenshluger-lepski method[END_REF].

Theorem 3 ensures that the critical value for the parameter a is 1. Beyond this value, the selected bandwidth ĥ is of order 1/n, which is very small (remember that for minimax study of a density with regularity α, the optimal bandwidth is n -1/(2α+1) ), then the risk cannot tend to 0.

Simulations

In this Section, we illustrate the role of tuning parameter a, the constant in the penalty term V . The aim is to observe the evolution of the risk for various values of a. Is the critical value a = 1 observable in practice? To do this, we simulate data X 1 , . . . , X n for several densites f . Next, for a grid of values for a, we compute the selected bandwidth ĥ, the estimator fĥ and the integrated loss fĥf 2 . We consider the following examples, see Figure 1:

Example 1 f is the Cauchy density Example 2 f is the uniform density U(0, 1) Example 3 f is the exponential density E(1) Example 4 f is a mixture of two normal densities 1 2 N (0, 1) + 1 2 N (3, 9) Example 5 f is a mixture of normal densities sometimes called Claw Example 6 f is a mixture of eight uniform densities
We implement the method for various kernels, but we only present results for Gaussian kernel, since the choice of kernel does not modify the results.

On the other hand, the method is sensitive to the choice of bandwidths set H: here we use

H = {e -k , 3 ≤ k ≤ 10} ∪ {0.002 + k × 0.02, 0 ≤ k ≤ 24}.
Note that the theoretical conditions on the bandwidths are asymptotic. Then, they have no real sense in our simulations with given n. In practice, this set must be rich enough for catching optimal bandwidths for a large class of densities, but small enough for the computation time. For our study, we choose equally distributed bandwidths for a good observation of the choice of ĥ, and we also add the set {e -k , 3 ≤ k ≤ 10} to have very small bandwidths avalaible, which are useful for irregular densities.

For n = 5000 and n = 50000, and several values of a, the Figure 2 plots

C 0 = Ẽ fĥ -f 2 min h∈H fh -f 2
where Ẽ means the empirical mean on N = 50 experiments. Thus smaller C 0 better the estimation. Moreover, we also plot on Figure 3 the selected bandwidth compared to the optimal bandwidth in the selection (for N = 1 experiment), i.e. ĥh 0 where fh 0f 2 = min h∈H fhf 2 .

We can observe that the risk (and then the oracle constant C 0 ) is very high for small values of a, as expected. Then it jumps to a small value, that indicates the method begins to work well. For too large values of a the risk finally goes back up. Thus we observe in practice the transition phenomenon that was announced by the theory. However, contrary to the theoretical results, the critical value may be not exactly at a = 1, especially for small values of n. As already mentioned above this is related to the asymptotic nature of the theoretical results that we have obtained. For irregular densities (examples 2, 5, 6), the optimal bandwidth is very low, then it is consistent to observe a smaller jump for the bandwidth choice. However the jump does exist and this is the interesting point. We can also observe that the optimal value for a seems to be very close to the jump point. That may pose a problem of calibration and this is what we would like to discuss now.

Discussion

To calibrate the penalty V , we face two practical problems: first, the optimal value for a seems to be extremely close to the minimal value; secondly, this latter value is not necessarily equal to the (asymptotic) theoretical value a = 1. In order to clearly separate the optimal value from the minimal, we propose to use some slightly different procedure, which depends on two possibly different penalty parameters instead of one as in the previous one.

B(h) = sup h ′ ≤h fh ′ -fh 2 -a K h ′ 2 n + , ĥ = arg min h∈H B(h) + b K h 2 n with b = a.
Of course this procedure is merely the one that we have previously studied when a = b. Our belief is that taking a and b to be different leads to a better and more stable calibration. A good track for practical purpose seems to use the procedure of Section 2 to find â where there is a jump in the risk (in practice this jump can be detected on the selected bandwidths) and then to choose b = 2â. Once again, what is important for practical calibration of the penalty is not that the jump appears at a = 1 (this value should be considered as some "asymptopia" which is never achieved) but that the jump does exist so that it becomes possible to use the calibration strategy that we just described. Proving theoretical results for this procedure is another interesting issue related to optimality considerations for the penalty that we do not intend to address here.

Proofs

Proof of Proposition 1

The first step is to write, for some fixed h ∈ H, fĥf ≤ fĥfh + fhf .

The last term can be splitted in fh -

f h + f h -f ≤ fh -f h + D(h).
Notice that for all h ′ ≤ h, using (2), fh ′ -fh 2 ≤ B(h) + V (h ′ ), which can be written , for all h, h ′ ;

fh ′ -fh 2 ≤ B(h ∨ h ′ ) + V (h ∧ h ′ ) where h ∨ h ′ = max(h, h ′ ) and h ∧ h ′ = min(h, h ′ ). Then, using (3), fĥ -fh 2 ≤ B(h ∨ ĥ) + V (h ∧ ĥ) ≤ B(h) + V (h) + max(B(h), V (h)).
We obtain, for any h ∈ H,

fĥ -f ≤ 2B(h) + 2V (h) + D(h) + fh -f h .
Thus the heart of the proof is to control B(h

) ′ ≤h fh ′ -fh 2 -V (h ′ )
+ by a bias term. First we center the variables and write

fh ′ -fh 2 ≤ (1 + ε) fh ′ -f h ′ -fh + f h 2 + (1 + ε -1 ) f h ′ -f h 2 ,
with ε some positive real to specified later. Moreover fh ′f h ′ -fh + f h = sup t∈B ν(t) where B is the unit ball in L 2 and

ν(t) = t, fh ′ -f h ′ -fh + f h = 1 n n i=1 g t (X i ) -E(g t (X i )) with g t (X) = (K h ′ -K h )(x -X)t(x)dx.
We shall now use the concentration inequality stated in Lemma 2, with F a countable set in B such that sup t∈F ν(t) = sup t∈B ν(t) (this equality is true for any dense subset of B for the L 2 topology, since ν is continuous). To apply result (4), we need to compute b, H and v.

• For all y ∈ R, since t ∈ B,

|g t (y)| = | (K h ′ -K h )(x-y)t(x)dx| ≤ K h ′ -K h t ≤ K h ′ -K h ≤ K h ′ so that b = K h ′ . We used assumption (K0) which implies, for h ′ ≤ h, K h ′ -K h 2 ≤ K h ′ 2 -K h 2 ≤ K ′ 2 . • Jensen's inequality gives H t∈F ν 2 (t)). Now sup t∈F ν 2 (t) = fh ′ -f h ′ -fh + f h 2 = 1 n n i=1 (K h ′ -K h )(x -X i ) -E((K h ′ -K h )(x -X i )) 2 E(sup t∈F ν 2 (t)) = Var( 1 n n i=1 (K h ′ -K h )(x -X i ))dx = 1 n Var((K h ′ -K h )(x -X 1 ))dx (7) ≤ 1 n E((K h ′ -K h ) 2 (x -X 1 ))dx (8) ≤ 1 n K h ′ -K h 2 ≤ 1 n K h ′ 2 Then H 2 ≤ n -1 K h 2 .
• For the variance let us write

Var(g t (X )) ≤ E (K h ′ -K h )(x -X)t(x)dx) 2 ≤ E |K h ′ -K h |(x -X)dx E |K h ′ -K h |(x -X)t 2 (x)dx ≤ K h ′ -K h 2 1 f ∞ t 2 ≤ 4 K 2 1 f ∞ t 2 since K h ′ -K h 1 ≤ 2 K 1 . Then v = 4 K 2 1 f ∞ = 4 f ∞ .
Finally, using (4), with probability larger than 1-

h ′ <h max(e -ε 2 ∧ε 24 √ n , e -ε 2 K 2 24 f ∞ 1 h ′ ) ∀h ′ ≤ h ∈ H fh ′ -f h ′ -fh + f h ≤ (1 + ε) √ n
where we choose ε such that a ≥ (1 + ε) 3 . Then, with probability larger than

1 -h∈H h ′ ≤h max(e -ε 2 ∧ε 24 √ n , e -ε 2 K 2 24 f ∞ 1 h ′ ) for any h, B(h) ≤ (1 + ε -1 )D(h) 2
In the same way, choosing 0 < ǫ ≤ √ a-1, we can prove that, with probability 1 -h∈H max(e -ǫ 2 ∧ǫ

24

√ n , e

-ǫ 2 K 2 6 f ∞ 1 h ), for any h, fh -f h ≤ (1 + ǫ) K h √ n ≤ V (h)
Finally, with high probability,

fĥ -f ≤ 2(1 + ε -1 )D(h) 2 + 2V (h) + D(h) + V (h) ≤ ( 2(1 + ε -1 ) + 1) D(h) + V (h)
To conclude we choose ε = ǫ = a 1/3 -1. Regarding the second result, note that the rough bound fh 2 ≤ K h 2 ≤ K 2 /h min is valid for all h. Then, denoting A the set on which the previous oracle inequality is verified,

E fĥ -f 2 ≤ E fĥ -f 2 1 A + 2( f 2 + K 2 /h min )P(A c ) with P(A c ) ≤ 2 h,h ′ ∈H max(e -c √ n , h ′ ) ≤ 2|H| 2 e -c/hmax

Proof of Theorem 3

We shall prove a more general version of the theorem, where several bandwidths sets H and kernels K are possible. We denote Crit(h) := B(h) + V (h) and E H = min{h/h ′ ; h ∈ H, h ′ ∈ H, h > h ′ }. We assume that E H does not depend on n and is larger than

1 (H = {e -k , a n ≤ k ≤ b n } suits with E H = e). Let us define φ(x) = K -2 K -K x 2 = 1 + 1 x -2 K, K(x.) K 2 .
We assume that the kernel K satisfies :

(K1) the function φ is bounded from over [E H , +∞), (K2) for 0 < µ < 1, the function φ(x) -µ x tends to +∞ when x → 0 and is decreasing in some neighborhood of 0, (K3) for 0 < µ < 1, the function φ(x) + µ

x is increasing for x ≥ 2. These assumptions are mild, as shown in the following Lemma, proved in Section 6.3.

Lemme 4. The following kernels satisfy assumptions (K0-K3):

a -Gaussian kernel: K(x) = e -x 2 /2 / √ 2π b -Rectangular kernel: K(x) = 1 [-1,1] (x)/2 c -Epanechnikov kernel: K(x) = (3/4)(1 -x 2 )1 [-1,1] (x) d -Biweight kernel: K(x) = (15/16)(1 -x 2 ) 2 1 [-1,1] (x)
The general result is:

Theorem 5. Assume (K0-K3) and that f is Assume that E H does not on n and h max → when n → ∞. We also assume that there exist θ 1 < θ 2 reals such that θ 2 ≥ 2, θ 1 .h min ∈ H and φ(θ 2 )φ(θ 1 ) ≥ 1/θ 1 -1/θ 2 . Then, if a < 1, there exists C = C( f ∞ ) > 0 such that, for n large enough (depending on f, H, K), This results implies Theorem 3, since under (K1),

P( ĥ ≥ θ 2 h min ) ≤ h∈H h ′ <h max(e -Cε 2 √ n , e -Cε 2 K h ′ -K h 2 ) where ε < 1 -a 1/3 . If H = {e -k , a n ≤ k ≤ b n }
K h ′ -K h 2 = K 2 h ′ φ(h/h ′ ) ≥ (min E H φ) K 2 h ′ as soon as h > h ′ , so that h∈H h ′ <h e -C K h ′ -K h 2 ≤ |H| 2 e -C ′ /hmax . Let ε ∈ (0, 1) such that a < (1 -ε) 3 and ε 3 + 3ε < φ(θ 2 ) -φ(θ 1 ) -a/θ 1 + a/θ 2 φ(θ 2 ) + φ(θ 1 ) (9)
(possible since a < 1 ≤ (φ(θ 2 )φ(θ 1 ))/(1/θ 1 -1/θ 2 )). Let us decompose

fh ′ -fh = ( fh ′ -f h ′ -fh + f h ) + (f h ′ -f h ) = S(h, h ′ ) + (f h ′ -f h ) with S(h, h ′ ) = 1 n n i=1 (K h ′ -K h )(x -X i ) -E((K h ′ -K h )(x -X i ))
and the bias term

f h ′ -f h ≤ sup h ′ ≤h K h ′ * f -K h * f = D(h). First write (1-ε) S(h, h ′ ) 2 - 1 ε -1 D(h) 2 ≤ fh ′ -fh 2 ≤ (1+ε) S(h, h ′ ) 2 + 1 + 1 ε D(h) 2
Now we shall prove that with high probability

(1 -ε) K h ′ -K h √ n ≤ S(h, h ′ ) ≤ (1 + ε) K h ′ -K h √ n .
First, we can prove as in Section 2 that for all h ′ < h

′ ) ≥ (1 + ε) K h ′ -K h √ n ≤ max exp - ε 2 ∧ ε 24 √ n , exp - ε 2 24 f ∞ K h ′ -K h 2 .
Next, we shall use (5) in Lemma 2 in order to lowerbound S(h, h ′ ) . Recall that S(h, h ′ ) = sup t∈B ν(t) where B is the unit ball in L 2 and ν

(t) = 1 n n i=1 g t (X i ) -E(g t (X i )) with g t (X) = (K h ′ -K h )(x -X)t(x)dx. With notations of Lemma 2, we have b = K h ′ -K h , H ′2 = n -1 K h ′ -K h 2 and v = 4 K 2 1 f ∞ . It remains to lowerbound H. First, remark, that (7) provides nE(sup t∈B ν 2 (t)) = K h ′ -K h 2 -(K h ′ -K h ) * f 2 . Next, using (6) E(sup t∈B ν 2 (t)) ≤ v n + 4 bH n + H 2 ≤ v n + H + 2b n 2 . Then n H + 2b n 2 ≥ nE(sup t∈B ν 2 (t))-v = K h ′ -K h 2 -(K h ′ -K h ) * f 2 -4 K 2 1 f ∞ which implies √ n H + 2b n ≥ K h ′ -K h 2 -4 K 2 1 ( f ∞ + f 2 ). Since b = K h ′ -K h , H ≥ K h ′ -K h 2 -4 K 2 1 ( f ∞ + f 2 ) n - 2 K h ′ -K h n Now, for h ′ < h H ≥ K h ′ -K h √ n 1 - 4 K 2 1 ( f ∞ + f 2 ) K h ′ -K h 2 - 2 √ n so H - ε 3 H ′ ≥ H ′ 1 - 4 K 2 1 ( f ∞ + f 2 ) K h ′ -K h 2 - 2 √ n - ε 3 . From (K1), K h ′ -K h = K 2 h ′ φ(h/h ) ≥ (min E H φ) 2 h ′ ≥ C hmax → ∞ and, in consequence, for n large enough H - ε 3 H ′ ≥ H ′ (1 -ε) .
Thus for n large enough

P S(h, h ′ ) ≤ (1 -ε) K h ′ -K h √ n (10) ≤ max exp - ε 2 ∧ (3ε) 24 × 9 √ n , exp - ε 2 24 × 9 f ∞ K h ′ -K h 2 Let δ(h, h) = 0 and, if h = h ′ , δ(h, h ′ ) = 2 max exp - ε 2 ∧ (3ε) 24 × 9 √ n , exp - ε 2 24 × 9 f ∞ K h ′ -K h 2 .
We just proved that for n large enough, with probability larger than 1 -

δ(h, h ′ ) (1 -ε) 2 K h ′ -K h 2 n ≤ S(h, h ′ ) 2 ≤ (1 + ε) 2 K h ′ -K h 2 n .
Next, with probability larger than 1

-h ′ ≤h δ(h, h ′ )    B(h) ≥ sup h ′ ≤h (1 -ε) 3 K h ′ -K h 2 n -a K h ′ 2 n + -1 ε -1 D(h) 2 B(h) ≤ sup h ′ ≤h (1 + ε) 3 K h ′ -K h 2 n -a K h ′ 2 n + + 1 + 1 ε D(h) 2 But, if h min small enough, for λ > a sup h ′ ≤h λ K h ′ -K h 2 n -a K h ′ 2 n + = λ K h min -K h 2 n -a K h min 2 n Indeed, for x = h ′ /h ≤ 1 λ K h ′ -K h 2 n -a K h ′ 2 n = λ K 2 nh 1 + 1 -a/λ x -2 K, K(x.) K 2 = λ K 2 nh φ(x) - a/λ x
and the function φ(x) -a/λ x tends to +∞ when x → 0 in some neighborhood of 0 (assumption (K2)). Then with probability larger than 1h h ′ ≤h δ(h, h ′ ), for all h    Crit(h) ≥ K 2 nh min -a + (1ε) 3 φ(h/h min ) + a h/h min -1 ε -1 D(h) 2 Crit(h) ≤ K 2 nh min -a + (1 + ε) 3 φ(h/h min ) + a h/h min + 1 + 1 ε D(h) 2 In particular, for h = θ 1 h min ,

Crit(θ 1 h min ) ≤ K 2 nh min -a + (1 + ε) 3 φ(θ 1 ) + a θ 1 + 1 + 1 ε sup h D(h) 2 .
(11) Moreover, since a < (1ε) 3 , (1ε) 3 φ(x) + a

x is increasing for x ≥ 2 (assumption (K3)). This implies that

∀h ≥ θ 2 h min , Crit(h) ≥ K 2 nh min -a + (1 -ε) 3 φ(θ 2 ) + a θ 2 - 1 ε -1 sup h D(h) 2 .
(12) Since (K h ) is an approximation to the identity, f -K h * f tends to 0 when h tends to 0. This implies that D(h) ≤ 2 sup h ′ ≤h f -K h ′ * f tends to 0 and sup h∈H D(h) tends to 0, as soon as h max tends to 0. Now (9) leads to ∆ := (1ε) 3 φ(θ 2 ) + a θ 2 -(1 + ε) 3 φ(θ 1 ) -a θ 1 > 0. Then, for n large enough, (2/ε) sup h D(h) 2 < K 2 nh min ∆ so that K 2 nh min -a + (1 + ε) 3 φ(θ 1 ) +

a θ 1 + 1 + 1 ε sup h D(h) 2 < K 2 nh min -a + (1 -ε) 3 φ(θ 2 ) + a θ 2 - 1 ε -1 sup h D(h) 2 (13)
Finally, combining (11) and ( 12) and (13) gives ĥ < θ 2 h min with probability larger than 1h h ′ ≤h δ(h, h ′ ).

Let us now prove the second part of Theorem 3, that is the lower bound on the risk. Let A n = { ĥ ≤ 3h min } and

B n = ∩ h∈H { f h -fh ≥ 1 2 K h √ n }.
We have just proved that P(A c n ) ≤ C(log n) 2 exp(-(log n) 2 /C). In the same way that (10), we can write for n large enough

P f h -fh ≤ (1 -ε) K h √ n ≤ max exp - ε 2 ∧ (3ε) 24 × 9 √ n , exp - ε 2 6 × 9 f ∞ K h 2
which implies P(B c n ) ≤ C (log n) exp(-(log n) 2 /C ) and then

P(A n ∩ n ) ≥ 1 -o(1).
Then we can write f -fĥ ≥ f ĥ -fĥ 1 An∩Bnff ĥ ≥ min

h≤3h min f h -fh 1 An∩Bn -max h f -f h ≥ min h≤3h min 1 2 K h √ n 1 An∩Bn -max h f -f h ≥ K 2 √ 3 1 √ nh min 1 An∩Bn -max h f -f h
But max h ff h → 0 (since h max → 0), and nh min → 1 when n → ∞. Hence

E f -fĥ ≥ K 2 √ 3 P(A n ∩ B n ) 1 + o(1) -o(1)
which proves that E f -fĥ ≥ K 4 √ 3 for n large enough.

Proof of Lemma 4

To prove Lemma 4, it is sufficient to do computations on integrals. We obtain: a -if K is the Gaussian kernel,

K, K(x.) K 2 = 2 1 + x 2 . b -if K is the rectangular kernel, K, K(x.) K 2 = 1 x ∧ 1. c -if K is the Epanechnikov kernel, K, K(x.) K 2 = 5 4 1 x ∧ 1 - x 2 5 1 x ∧ 1 5 .

Figure 1 :

 1 Figure 1: Plots of true density f for Examples 1-6

Figure 2 :

 2 Figure 2: Oracle constant C 0 as a function of a, for Examples 1-6

Figure 3

 3 Figure 3: ĥh 0 as a function of a, for Examples 1-6

  and the kernel is Gaussian, rectangular, Epanechnikov or biweight, θ 1 = e and θ 2 = 3 work.

d -if K is the biweight kernel:

These formulas permit to verify all the assumptions.


G Ozouf

C Beauger

Niobium-and antimony-doped tin dioxide aerogels as new catalyst supports for PEM fuel cells

Keywords: Tin dioxide, aerogel, Nb and Sb doping, catalyst support, Fuel cells

In order to tackle the problem of low durability, tin dioxide was studied to replace carbon black as a catalyst support in Proton Exchange Membrane Fuel Cells (PEMFCs). SnO 2 is a well-known n-type semi-conductor whose electronic conductivity can be improved by doping with hypervalent cations such as Nb 5+ or Sb 5+ . In addition, as a catalyst support, this material has to develop a high specific surface area with an adequate mesoporous morphology to allow both good dispersion and activity of the catalyst (Pt). To this end, our objective was to develop doped SnO 2 aerogels in order to gather in a same material both a high electronic conductivity and an adapted morphology. In this study SnO 2 xerogels and aerogels were successfully synthetized following an acid-catalyzed sol-gel route starting with metal alkoxides as precursors. Dried gels were calcined for 5 h at 600 °C in flowing air. The effect on both the structure and the morphology of the material resulting from doping with niobium or antimony was investigated by XRD, SEM and nitrogen sorption. The electronic conductivity of pure and doped SnO 2 materials was obtained from impedance spectroscopy and resistance measurements. Our materials showed a very interesting airy morphology adapted for the foreseen application: a reasonable specific surface area (80-90 m²/g) with a bimodal pore size distribution centered on around 25 nm and 45 nm. Moreover, all Sb-doped samples exhibited significant improvement in electronic conductivity. 5 at.% Sb-doped SnO 2 even showed an electronic conductivity of 1 S/cm, very similar to that of Vulcan XC-72 (4 S/cm) and representing a 5 orders of magnitude increase compared to that of pure SnO 2 .

Introduction

Fuel cells are environmentally friendly electrochemical energy converters that produce electricity and heat mainly from hydrogen and air [START_REF] Rabis | Electrocatalysis for Polymer Electrolyte Fuel Cells: Recent Achievements and Future Challenges[END_REF][START_REF] Costamagna | Quantum jumps in the PEMFC science and technology from the 1960s to the year 2000 Part I. Fundamental scientific aspects[END_REF][START_REF] Haile | Fuel cell materials and components[END_REF]. Among the many existing types of fuel cells, Proton Exchange Membrane Fuel Cells (PEMFCs) have the advantage of providing a large range of power, from a few W to hundreds of kW.

Operating at low temperatures (80 °C), PEMFCs allow quick starts what makes them particularly interesting for transport applications. Low temperature operation limits the choice of catalysts for electrochemical reactions (H 2 oxidation, HOR and O 2 reduction, ORR) to noble metals, mostly platinum. To reduce the amount of platinum while maintaining good performance, platinum nanoparticles are dispersed on a support. Because they have both a high electronic conductivity and a high specific surface area (respectively about 4 S/cm and 254 m²/g for Vulcan XC-72 [START_REF] Antolini | Carbon supports for low-temperature fuel cell catalysts[END_REF]), current electrocatalysts are based on carbon blacks (Vulcan XC-72R or TKK TEC10E40E for instance). However, the corrosion of carbon black on the cathode side in fuel-cell operating conditions (E = 1.2V, T = 80 °C, high relative humidity and strongly acidic environment) is responsible for contact losses between the conductive carbon support and the catalyst, thus limiting the durability of PEMFC. This phenomenon is particularly marked for transport applications, during start / stop cycles [START_REF] Antolini | Carbon supports for low-temperature fuel cell catalysts[END_REF][START_REF] Maass | Carbon support oxidation in PEM fuel cell cathodes[END_REF]. They may indeed cause a reverse current compared to the normal operation of the fuel cell, leading to an increase in the cathode potential of up to 1.44 V. In the presence of platinum nanoparticles, such high potential causes severe corrosion of the cathode following the reaction: C + 2H 2 O → CO 2 + 4H + + 4e - E 0 = 0.207 V versus NHE Degradation mechanisms include dissolution as well as coalescence of platinum particles, thus limiting their activity [START_REF] Yasuda | Platinum dissolution and deposition in the polymer electrolyte membrane of a PEM fuel cell as studied by potential cycling[END_REF].

According to thermodynamic calculations derived from Pourbaix pH-potential diagrams, tin dioxide is expected to be very stable in PEMFC operating conditions. Its utilization as a catalyst support should improve PEMFC durability [START_REF] Sasaki | Alternative Electrocatalyst Support Materials for Polymer Electrolyte Fuel Cells[END_REF]. Tin dioxide is an n-type semi-conductor (Eg = 3.6 eV). Doping SnO 2 with hypervalent donor components like Nb 5+ or Sb 5+ leads to a significant increase of the electronic conductivity [START_REF] Gokulakrishnan | Investigations on the structural, optical and electrical properties of Nb-doped SnO2 thin films[END_REF][START_REF] Szczuko | XPS Investigations of Surface Segregation of Doping Elements in SnO2[END_REF]. Due to its remarkable bulk and surface properties, this material is already used as a transparent conductive oxide (TCO) [START_REF] Bressand | Electrochromic device with controlled infrared reflection[END_REF], an electrode material in Li-ion batteries [START_REF] Megahed | Lithium-ion Battery for Electronic Applications[END_REF][START_REF] Courtney | In-situ Sn-119 Mossbauer effect studies of the reaction of lithium with SnO and SnO : 0.25 B2O3 : 0.25 P2O5 glass[END_REF], a catalyst for oxidation reaction [START_REF] Fuller | The Catalytic Oxidation of Carbon Monoxide on Tin(lV) Oxide[END_REF][START_REF] Matsui | Electrochemical oxidation of CO over tin oxide supported platinum catalysts[END_REF][START_REF] Herniman | An Investigation of the Relationship between the Bulk and Surface Composition of Tin and Antimony Mixed Oxide Catalysts and the Oxidative Dehydrogenation of I-Butene to Butadiene[END_REF] and a sensing material in gas sensors [START_REF] Viricelle | Gas Sensors Based on Tin Dioxide for Exhaust Gas Application, modeling of response for pure gases and for mixtures[END_REF][START_REF] Shukla | Synthesis and characterization of sol-gel derived nanocrystalline tin oxide thin film as hydrogen sensor[END_REF][START_REF] Barsan | Metal oxide-based gas sensor research: How to?[END_REF].

Several authors have shown that using pure SnO 2 as a catalyst support effectively improves the durability of PEMFC, mostly thanks to improved corrosion resistance [START_REF] Masao | Carbon-Free Pt Electrocatalysts Supported on SnO2 for Polymer Electrolyte Fuel Cells[END_REF][START_REF] Takabatake | Cycle durability of metal oxide supports for PEFC electrocatalysts[END_REF][START_REF] Zhang | Mesoporous Tin Oxide as an Oxidation-Resistant Catalyst Support for Proton Exchange Membrane Fuel Cells[END_REF]. Moreover, it reduces polymer membrane degradation by neutralizing the formation of H 2 O 2 formed during ORR [START_REF] Andersen | Tin Dioxide as an Effective Antioxidant for Proton Exchange Membrane Fuel Cells[END_REF]. It also allows better tolerance to CO than Pt / C electrocatalysts [START_REF] Matsui | Electrochemical oxidation of CO over tin oxide supported platinum catalysts[END_REF][START_REF] Matsui | Effect of reduction-oxidation treatment on the catalytic activity over tin oxide supported platinum catalysts[END_REF][START_REF] Okanishi | Chemical interaction between Pt and SnO2 and influence on adsorptive properties of carbon monoxide[END_REF]. This can be explained by the oxidation of CO adsorbed at the Pt surface by the hydroxyl groups covering the surface of SnO 2 [START_REF] Arenz | Carbonsupported Pt-Sn electrocatalysts for the anodic oxidation of H(2), CO, and H(2)/CO mixtures. Part II: The structure-activity relationship[END_REF]. However, electrochemical characteristics, like the electrochemical surface area (ECSA) or the oxygen reduction reaction (ORR) mass and specific activities of Pt/SnO 2 electrocatalysts, are relatively low compared to those of Pt/C, mostly due to their poor electronic conductivity and low specific surface area. Recent studies have shown the potential of doping SnO 2 by Nb 5+ [START_REF] Takasaki | Carbon-Free Pt Electrocatalysts Supported on SnO2 for Polymer Electrolyte Fuel Cells: Electrocatalytic Activity and Durability[END_REF][START_REF] Kakinuma | Characterization of Pt catalysts on Nb-doped and Sb-doped SnO2-delta support materials with aggregated structure by rotating disk electrode and fuel cell measurements[END_REF][START_REF] Cavaliere | Dopant-Driven Nanostructured Loose-Tube SnO2 Architectures: Alternative Electrocatalyst Supports for Proton Exchange Membrane Fuel Cells[END_REF][START_REF] Shahgaldi | The effect of low platinum loading on the efficiency of PEMFC's electrocatalysts supported on TiO2-Nb, and SnO2-Nb: An experimental comparison between active and stable conditions[END_REF] or Sb 5+ [START_REF] Takasaki | Carbon-Free Pt Electrocatalysts Supported on SnO2 for Polymer Electrolyte Fuel Cells: Electrocatalytic Activity and Durability[END_REF][START_REF] Kakinuma | Characterization of Pt catalysts on Nb-doped and Sb-doped SnO2-delta support materials with aggregated structure by rotating disk electrode and fuel cell measurements[END_REF][START_REF] Kakinuma | Synthesis and electrochemical characterization of Pt catalyst supported on Sn0.96Sb0.04O2-delta with a network structure[END_REF][START_REF] Gurrola | High surface electrochemical support based on Sb-doped SnO2[END_REF][START_REF] You | Platinum-antimony tin oxide nanoparticle as cathode catalyst for direct methanol fuel cell[END_REF][START_REF] Yin | Highly active and stable Pt electrocatalysts promoted by antimony-doped SnO2 supports for oxygen reduction reactions[END_REF][START_REF] Oh | Preparation of Mesoporous Sb-, F-, and In-Doped SnO2 Bulk Powder with High Surface Area for Use as Catalyst Supports in Electrolytic Cells[END_REF][START_REF] Senoo | Cathodic performance and high potential durability of Ta-SnO2-delta-supported Pt catalysts for PEFC cathodes[END_REF][START_REF] Avila-Vazquez | Electrochemical performance of a Sb-doped SnO2 support synthesized by coprecipitation for oxygen reactions[END_REF]. In addition to improving electronic conductivity, dopants are known to inhibit particle growth [START_REF] Leite | A new method to control particle size and particle size distribution of SnO2 nanoparticles for gas sensor applications[END_REF] and modify the material's morphology. Although improvements have been observed compared to pure SnO 2 , experimental results generally show lower performance than carbon black, particularly regarding mass catalytic activity. In some cases, depending on the SnO 2 morphology, mass catalytic activities were improved, even surpassing those of Pt/C electrocatalysts [START_REF] Kakinuma | Characterization of Pt catalysts on Nb-doped and Sb-doped SnO2-delta support materials with aggregated structure by rotating disk electrode and fuel cell measurements[END_REF][START_REF] Senoo | Cathodic performance and high potential durability of Ta-SnO2-delta-supported Pt catalysts for PEFC cathodes[END_REF]. This has been ascribed to the sometimes contested strong metal-support interaction (SMSI) [START_REF] Tauster | Strong Metal-Support Interactions -Occurrence among Binary Oxides of Groups IIA-VB[END_REF][START_REF] Kamiuchi | Nanoscopic observation of strong chemical interaction between Pt and tin oxide[END_REF]. It has been proposed that this strong interactions between SnO 2 and Pt may change the Pt electronic structure and lower the activation barrier for oxygen dissociation [START_REF] Spencer | Models of Strong Metal Support Interaction (SMSI) in Pt on TiO2 Catalysts[END_REF]. However a recent study has shown that steric effects are more likely the cause of this lower activation barrier [START_REF] Daio | Lattice Strain Mapping of Platinum Nanoparticles on Carbon and SnO2 Supports[END_REF].

The morphology of the support is primordial to limit concentration polarization and ease fluids management in volume electrode. Aerogels are to this end particularly interesting materials. Carbon aerogels have already proven their ability to efficiently support catalysts for PEMFC application [START_REF] Ouattara-Brigaudet | Carbon Aerogels as Catalyst Supports and First Insights on Their Durability in Proton Exchange Membrane Fuel Cells[END_REF][START_REF] Ouattara-Brigaudet | Influence of the carbon texture of platinum/carbon aerogel electrocatalysts on their behavior in a proton exchange membrane fuel cell cathode[END_REF]. Their morphology can be adjusted by playing on the sol-gel parameters for higher specific surface areas with adequate mesoporosity.

In this study, SnO 2 aerogels were thus synthesized to be used as a catalyst support for PEMFC in order to overcome the reported limitations regarding morphology and durability. The effect of doping SnO 2 with niobium or antimony on the electronic conductivity as well as on the morphology has also been investigated.

Experimental SnO 2 synthesis route

Metal oxides were synthesized following a sol-gel route, starting with metal alkoxide precursors: Sn(OiPr) Morphologies were analyzed by SEM using a Supra 40 with Gemini column operated at 3.00 kV. The oxide powders were deposited on adhesive conducting carbon tapes and coated with a 7 nm thick platinum layer, using a Quorum (Q150 T).

In addition, nitrogen sorption analysis was performed with a Micromeritics ASAP 2020. Samples were first degassed for 120 min at 100 °C and 10 μmHg. The Brunauer-Emmett-Teller (BET) model was applied to determine the specific surface area. The pore size distributions were calculated applying the Barret-Joyner-Halenda (BJH) method to the desorption branch of the isotherms. The assessments of microporosity were made from t-plot construction using the Harkins-Jura correlation.

Concerning the pore size distribution determination using this technique, we assumed that the samples were mechanically strong enough after calcination to withstand the pressure applied during the experiment.

The bulk chemical composition was analyzed by Energy Dispersive X-ray Spectroscopy (EDX) performed during SEM observation with a Philips XL30 operated at 15 kV.

The surface chemical composition was analyzed by X-ray Photoelectron Spectroscopy (XPS) using a Thermo Scientific K-Alpha system. The spectrometer was equipped with an AlKα monochromated source and a low energy flood-gun for charge compensation.

The ellipsoid spot size is about 350 µm x 700 µm.

The electronic conductivity was investigated by impedance spectroscopy and direct resistance measurements.

For impedance spectroscopy, the frequency ranged from 100 Hz to 1 MHz with a potential of 200 mV and an amplitude of 100 mV. Since pellet formation was difficult for SnO 2 samples, impedance spectroscopy measurements were only performed on xerogel sample parts. Due to their powder form, it was actually not possible to perform impedance spectroscopy measurements on aerogels. Next, a homemade conductivity cell was developed, made up of two copper electrodes (S = 0,785 cm²) surrounded with a Teflon ring. Samples were introduced between these two electrodes (see Fig. 1). A potentiostat was used to apply a current of 105 mA, -105 mA or 400 mA and the voltage was measured for each current. The homemade conductivity cell was placed in a press and measurements were made at room temperature, without and with pressure at 500 kg and 1000 kg (respectively 6.5 × 10 2 kg/cm 2 and 13 × 10 2 kg/cm 2 ).

The conductivity (σ) was calculated with the formula σ = e/(R×S electrode ), where e is the thickness of the sample, R the measured resistance and S electrode the surface of the electrode.

Results and discussion

Structural and morphological characterization

In this section, we present the structural and morphological characterization results. These results highlight specific properties such as crystallinity and specific surface areas or pore size distributions, which are crucial in selecting the most appropriate sample for the foreseen application.

Since the electronic conductivity in semi-conductors is strongly related to the periodic crystal structure, materials were calcined at different temperatures in air. The resulting XRD diagrams are shown in Fig. 2.

All calcined samples show characteristic peaks of rutile SnO 2 . After calcination at 400°C, the XRD peaks are relatively wide with very weak intensity, showing low crystallinity, with the material remaining partially amorphous. The crystallinity improves following calcination at 700°C. SnO 2 xerogels then exhibit good crystallinity, the peaks are intense and the Full Width at Half Maximum is small. Both the electronic conductivity and the specific surface area are important for catalyst support. Given that the higher the calcination temperature the better the crystallinity but the lower the specific surface area, we needed to find a compromise.

In order to avoid too severe a decrease in the specific surface area, materials were calcined at 600°C. Whatever the amount of dopant between 5 and 15 at.%, XRD diffractograms of doped SnO 2 aerogels also only showed the characteristic peaks of the SnO 2 rutile phase (see Fig. 3 for pure and 10 at% doped samples). No new crystal phase was detected and we observed a peak shift over a smaller angle. This shift is more pronounced for Sb-doped samples than for Nb-doped ones (insert in Fig. 3), resulting from a greater modification of the SnO 2 lattice parameters. Both the absence of a new phase and the observation of XRD peak shifts are good indications of proper doping. The morphology of pure SnO 2 aerogel (A6S) results from the agglomeration or aggregation of rather small particles (about 13 nm). The organization of such agglomerates results in the formation of large pores. Doping seems to inhibit particle growth during the sol-gel process. Nb-or Sb-doped aerogels are indeed made up of smaller particles than pure SnO 2 aerogels. Nucleation and growth control of SnO 2 particles with dopant has already been observed and reported [START_REF] Szczuko | XPS Investigations of Surface Segregation of Doping Elements in SnO2[END_REF][START_REF] Cavaliere | Dopant-Driven Nanostructured Loose-Tube SnO2 Architectures: Alternative Electrocatalyst Supports for Proton Exchange Membrane Fuel Cells[END_REF][START_REF] Leite | A new method to control particle size and particle size distribution of SnO2 nanoparticles for gas sensor applications[END_REF].

The overall picture of Nb-doped SnO 2 aerogel (A6SN100) is quite similar to that of pure SnO 2 (A6S) although the agglomerates look slightly smaller. On the contrary, the morphology of Sb-doped SnO 2 aerogels (A6SS050 and A6SS100) is significantly different.

They show a very interesting airy morphology lying on a three-dimensional network made up of interconnected particles. It is noteworthy that this morphology is quite similar to that of carbon aerogels previously synthesized in our group [START_REF] Ouattara-Brigaudet | Influence of the carbon texture of platinum/carbon aerogel electrocatalysts on their behavior in a proton exchange membrane fuel cell cathode[END_REF]. Sb doped SnO 2 aerogels are from this point of view more interesting than Nb doped ones.

As expected from the drying route, xerogel samples showed a very different morphology. X6SS050 and X6SS100, presented in Fig. 5, are much denser and show less apparent porosity than their aerogel counterparts. To obtain additional morphological data, we performed nitrogen sorption analysis. The results are reported in Table 1 and Fig. 6 for aerogels and in Table 2 and Fig. 7 for xerogels.

First of all, the type-IV isotherms obtained are representative of mesoporous materials.

Specific surface areas were calculated based on the BET model. Pore size distributions and mesoporous volumes were calculated based on the BJH model, assuming that all calcined samples are mechanically strong enough to withstand applied forces during analysis.

It is interesting to note that the doping rate impacts both the specific surface area and the mesoporous volume. A maximum was observed for 10 at.% doping rate. Values are actually relatively high for pure and 10 at.% doped materials, reaching almost 90 m²/g for the specific surface area, and 0.3 cm 3 /g for the mesoporous volume, whatever the dopant used (Sb or Nb).

The microporous volume of aerogel samples is negligible compared to their mesoporous volume. This is of utmost importance for the foreseen application, since most of the porosity will thus be accessible for Pt deposition and utilization. Here again, Sb-doped aerogels are more promising than pure or Nb-doped ones,

showing an even smaller microporous volume.

The pore size distribution, calculated using the BJH method from isotherms, shows, for pure SnO 2 aerogels, a multimodal distribution with quite a narrow peak at around 15 nm and broader ones centered on 20 and 40 nm (Fig. 6).

Looking at Fig. 6, N 2 sorption should be complemented by mercury porosimetry analysis to fully characterize the pore size distribution, especially in the macroporous domain. Due to the nature of the samples, such an analysis could not be performed. Still, N 2 sorption analysis allowed to highlight morphology differences between aerogels samples.

The pore size distribution observed for doped samples is notably modified. First, the main peaks at 15, 20 and 40 nm observed for pure SnO 2 have shifted to larger diameters, namely 20, 30 and 45 nm. Second, the contributions at 30 and 45 nm are significantly lower for Nb-and Sb-doped samples than for pure SnO 2 . The peak at 45 nm is even smaller for Sb doped samples. The decrease in porosity around 45 nm may be related to the SEM observations and the change in size of the particle agglomerates.

To sum up, a multimodal distribution with broad peaks centered around 25 nm and 45 nm was observed for Sb and Nb doped SnO 2 .

In line with SEM observations, xerogel samples showed smaller specific surface areas than aerogel samples and much smaller porous volumes (0.1 cm 3 /g vs 0.3 cm 3 /g). Accordingly, the microporous volume is also negligible, at two orders of magnitude smaller than that of aerogels.

In the case of xerogels, the specific surface area clearly increases with the Sb content. It is noteworthy that the 10 and 15 at.% doped xerogel samples showed surprisingly high specific surface areas of around 60 m²/g, close to the aerogel one.

Xerogel pore size distribution is also significantly different. Only one peak is observed for pure SnO 2 , centered at around 13 nm. This peak shifts toward smaller diameters with the increase in Sb content. It is centered at around 4 nm for SnO 2 : Sb 15 at.%. Such behavior is consistent with our SEM observations and literature that reports a decrease in particle size after doping.

So, despite negligible microporous volume and quite high specific surface area, due to pore size distributions centered on small mesopores (all < 15 nm), xerogels appear less appropriate for the application.

Chemical composition

Chemical composition obviously impacts electronic conductivity. Although the amount of dopant precursor can be perfectly controlled during the sol-gel process, the exact concentration of dopant and its repartition in the final material has to be analyzed. The homogeneity of the sol may indeed not be optimal and some precursors could also partially precipitate during the sol-gel transition, thus modifying the real dopant amount within samples.

Moreover, during calcination the thermal energy can induce some atomic diffusion of the dopant within the SnO 2 matrix.

Thus, both bulk and surface chemical compositions were measured by EDX and XPS respectively. These two methods are semi-quantitative and are used to compare different samples rather than to provide their real chemical composition. According to the EDX measurements (Table 3), the niobium bulk concentrations correspond to the expected values for 5 and 10 at.% doped SnO 2 samples. The surface niobium concentration, measured by XPS, is however slightly higher than the bulk concentration (respectively 7 and 13 at.% vs 5 and 10 at.% expected for A6SN050 and A6SN100). The chemical composition does not seem to be perfectly homogenous in all SnO 2 : Nb samples.

Such surface enrichments have already been reported by Szczuko et al. [START_REF] Szczuko | XPS Investigations of Surface Segregation of Doping Elements in SnO2[END_REF].

Regarding Sb-doped samples, the EDX peaks of Sn and Sb are very close due to the similar electronic structure of these two elements (Fig. 8). The doping rate calculation is thus more difficult. According to the EDX analysis, the Sb bulk concentration is higher than expected. This may however be a simple overestimation due to the Sn and Sb peaks overlapping. The trend is consistent for 5 and 10 at.% doped samples whose dopant concentration was calculated respectively to 7.5 and 12 at.%.

XPS survey spectra of SnO 2 : Nb and SnO 2 : Sb aerogels are shown on Fig. 9. Similar difficulties than for EDX analysis arose to quantify Sb rate at the surface, since the most intense peak of Sb (Sb 3d5) stands at the same binding energy as the O 1s peak. Thus, we performed measurements considering a less intense Sb peak (Sb 3d3) (Fig. 10). As for EDX, the characterization accuracy of Sb-doped samples is rather poor but comparison is still possible. As shown in Table 3, the Sb rate is also higher than expected on the surface, whatever the expected amount of Sb. The surface concentrations are still slightly larger than the bulk ones. They may also have been overestimated. As already observed [START_REF] Fabbri | Pt nanoparticles supported on Sb-doped SnO2 porous structures: developments and issues[END_REF][START_REF] Rabis | Catalyzed SnO2 Thin Films: Theoretical and Experimental Insights into Fabrication and Electrocatalytic Properties[END_REF], Sb seems also prone to segregation in our SnO 2 aerogels (about 10% more Sb measured on surface vs volume), however in a lesser extent than Nb (30% more Nb measured on surface vs volume).

If such segregations reveal to be problematic in fuel cell operating conditions, we will try to mitigate them through accurate selection or modification of metal precursors. Segregation may indeed result from different hydrolysis or condensation kinetics with the precursors used. In our case, tin isopropoxide may be more reactive than niobium and antimony isopropoxide. Segregation can probably be mitigated selecting or even synthesizing precursors with similar hydrolysis/condensation kinetics.

Electronic conductivity measurement

High electronic conductivity is essential for a material to be a good catalyst support in PEMFC. As metal oxides are by definition dielectric, high conductivity can only be obtained by modifying their electronic configuration.

Doping is one option, as followed in this work. It is of paramount importance to check how the dopant type and concentration influence the electronic conductivity of doped samples. In this section, the electronic conductivity was measured for different Nb and Sb concentrations and compared to that of pure SnO 2 on one hand and carbon on the other hand. All materials were analyzed in the same conditions. In order to limit the impact of interface resistance between particles, measurements were first performed on xerogel sample pieces. This allowed us to study the influence of both the dopant type and its concentration. Measurement errors are quite high due to surface roughness of the samples, which was not possible to improve.

Nb 5+ doping was first selected to increase the SnO 2 conductivity (Table 4). Typical impedance curves obtained for Nb-doped SnO 2 exhibit a characteristic half circle arc that is representative of a semi-conductor behavior (Fig. 11). The high frequency resistance is representative of bulk resistance, and the diameter of the half circle corresponds to that of interface resistances. We used both to analyze the conductivity of our samples.

The bulk conductivity of SnO 2 is considerably low (5×10 -4 S/cm). The interface resistance between particles is relatively high, resulting in lower global conductivity, even for doped samples.

Compared to pure SnO 2 , Nb-doped samples exhibit a significant improvement in bulk electronic conductivity.

5×10 -3 S/cm was reached for 10 at.% of Nb, i.e. one order of magnitude higher than that of pure SnO 2 . The improvement in global conductivity is very low compared to that of bulk conductivity, here again due to significant interface resistances. The doping inhibits grain growth during the sol gel process, as mentioned above and in agreement with SEM observations and pore-size distribution calculations. Hence, the number of interfaces should be greater for doped materials, thus partially accounting for the limited improvement in the global conductivity.

The bulk conductivities of Nb-doped samples are still very low and far from that of Vulcan XC-72. In agreement with other studies [START_REF] Szczuko | XPS Investigations of Surface Segregation of Doping Elements in SnO2[END_REF][START_REF] Gurrola | High surface electrochemical support based on Sb-doped SnO2[END_REF][START_REF] Yin | Highly active and stable Pt electrocatalysts promoted by antimony-doped SnO2 supports for oxygen reduction reactions[END_REF][START_REF] Avila-Vazquez | Electrochemical performance of a Sb-doped SnO2 support synthesized by coprecipitation for oxygen reactions[END_REF][START_REF] Muller | Highly Conducting Nanosized Monodispersed Antimony-Doped Tin Oxide Particles Synthesized via Nonaqueous Sol-Gel Procedure[END_REF][START_REF] Mulla | Deposition of Improved Optically Selective Conductive Tin Oxide-Films by Spray Pyrolysis[END_REF], an impressive improvement in conductivity was observed for Sb-doped SnO 2 (Table 5). X6SS100 (10 at.% Sb-doped SnO 2 xerogel) shows indeed the typical behavior of a conducting material (Fig. 12). No impact of grain boundaries can be observed on the Nyquist plot. In agreement with Bruneaux et al. [START_REF] Bruneaux | Structural, electrical and interfacial properties of sprayed SnO2 films[END_REF], this may result from a higher charge carriers density than in the case of Nb doping. Whatever the Sb rate between 2 at.% and 15 at.%, conductivities were improved by at least 4 orders of magnitude compared to pure SnO 2 . The electronic conductivity reached a maximum of about 1 S/cm for 5 at.% Sb, a value very close to that reported for Vulcan XC-72 (4 S/cm). Based on this first series of results, Sb was selected as the favored dopant to determine the conductivity of doped SnO 2 aerogels.

Aerogel materials are not monolithic enough after calcination to allow impedance spectroscopy measurements and it was difficult to prepare pellets. We thus produced a homemade cell to measure the resistance of aerogel materials, pressed between two electrodes inserted into a Teflon cylinder. For comparison purposes, we also measured the resistance of xerogels in the same conditions.

The evolution of conductivity with the pressure applied on the sample through the electrodes is shown in Fig. 13 for xerogels and Fig. 14 for aerogels.

As expected, the conductivity increases with the pressure applied. No significant difference was observed following a change in the current (+105 mA, -105 mA or 400 mA). In most cases, a plateau is reached from 6.5 × 10 2 kg/cm 2 of applied pressure.

Fig. 15 represents the evolutions of conductivity for aerogels and xerogels as a function of the antimony doping rate and for the two methods used in this study (1: EIS, 2: direct resistance measurement).

First, conductivity values obtained with the homemade cell (13 × 10 2 kg/cm 2 pressure, data referenced (2) on Fig. 15) are 4 times higher than those obtained with impedance spectroscopy (data referenced (1) in Fig. 15). This is probably due to better contact between particles resulting from the relatively high pressure applied.

Then, the conductivity of xerogels is roughly between one and two orders of magnitude higher than that of aerogels.

Finally, whatever the measurement technique, the conductivity reaches a maximum for 5 at.% Sb doping rate in the case of xerogels and between 5 and 10 at.% in the case of aerogels.

Discussion

The purpose of this study was to synthetize a new corrosion resistant catalyst support for PEMFC. Both the morphology and the electronic conductivity are of major importance considering the foreseen application. In a first part, the unique material morphology will be highlighted in comparison with those reported in the literature.

In a second part the electronic conductivity and its evolution with materials composition will be discussed.

Morphology

Considering the morphology, we have already stated that both the presence of meso and macropores are beneficial to fluid management within electrodes [START_REF] Ouattara-Brigaudet | Influence of the carbon texture of platinum/carbon aerogel electrocatalysts on their behavior in a proton exchange membrane fuel cell cathode[END_REF]. Moreover, since the optimum Pt particle size was reported to be between 3 and 4 nm in diameter, microporosity should be avoided and may become clogged after the deposition of platinum nanoparticles. Mesopores on the contrary allow good platinum nanoparticle dispersion while avoiding pore obstruction, still leaving room for high gas flow rates. Macropores are also beneficial to efficiently manage water coming from humidified gases and that produced during ORR.

Regarding such considerations, all SnO 2 aerogels show a very interesting morphology. Combining specific surface area close to 90 m²/g, adequate pore size distribution revealing medium and large mesopores (25 and 45 nm) and negligible microporous volume, they are particularly well adapted to the foreseen application.

Little research has been reported on such an interesting morphology, mixing a high specific surface area with wide pore size distribution. Our materials show higher values for SnO 2 than most of those reported in literature [START_REF] Takasaki | Carbon-Free Pt Electrocatalysts Supported on SnO2 for Polymer Electrolyte Fuel Cells: Electrocatalytic Activity and Durability[END_REF][START_REF] Kakinuma | Characterization of Pt catalysts on Nb-doped and Sb-doped SnO2-delta support materials with aggregated structure by rotating disk electrode and fuel cell measurements[END_REF][START_REF] Yin | Highly active and stable Pt electrocatalysts promoted by antimony-doped SnO2 supports for oxygen reduction reactions[END_REF]. Usually, high specific surface area is developed at the expense of wide pore size distribution and vice versa. Indeed, high specific surface area nanostructured Sb-doped SnO 2 materials have already been synthesized. Lee et al. [START_REF] Lee | Electrocatalytic activity and stability of Pt supported on Sb-doped SnO2 nanoparticles for direct alcohol fuel cells[END_REF] achieved 100 m²/g from chloride precipitation in basic conditions. Sol-gel route using template, dodecylamine [START_REF] Gurrola | High surface electrochemical support based on Sb-doped SnO2[END_REF][START_REF] Gurrola | Evaluation of the corrosion of Sb-doped SnO2 supports for electrolysis systems[END_REF] or tetradecylamine [START_REF] Oh | Preparation of Mesoporous Sb-, F-, and In-Doped SnO2 Bulk Powder with High Surface Area for Use as Catalyst Supports in Electrolytic Cells[END_REF] also provide high specific surface areas with respectively 216 m²/g and 263 m²/g. However, these materials show a narrow pore size distribution centered on few nm only.

Although the specific surface areas developed by our materials are still lower than that of Vulcan XC-72 (254 m²/g), a general carbon black reference, the difference in bulk density is in favor of SnO 2 . We may actually expect a similar developed surface area for the same volume. Since our aerogels crack after calcination, it was unfortunately not possible to measure their density. This advantage will be checked after electrochemical characterizations.

Thus, from a morphology point of view, the materials developed in this study are promising candidates for PEMFC catalyst support.

Electronic conductivity

High enough electronic conductivities where achieved only after doping with Sb. Doping with Nb allowed improving electronic conductivity by a factor of only 2 compared to undoped SnO 2 , what remains too low for the foreseen application. Doping may actualy have been less efficient with Nb than with Sb. On one hand, the degree of segregation seems to be larger for Nb (see table 3) and on the other hand the repartition of the dopant in the SnO 2 matrix may be less homogeneous, resulting in possible local Nb rich regions. Moreover, we cannot exclude the possibility of amorphous Nb oxide regions which would concentrate Nb at the expense of SnO 2 . Finally the association of defects due to the introduction of dopant in SnO 2 may be less prone to increase the electronic conductivity in the case of Nb than in the case of Sb.

Most of the conductivities reported in literature [START_REF] Szczuko | XPS Investigations of Surface Segregation of Doping Elements in SnO2[END_REF][START_REF] Yin | Highly active and stable Pt electrocatalysts promoted by antimony-doped SnO2 supports for oxygen reduction reactions[END_REF][START_REF] Muller | Highly Conducting Nanosized Monodispersed Antimony-Doped Tin Oxide Particles Synthesized via Nonaqueous Sol-Gel Procedure[END_REF][START_REF] Mulla | Deposition of Improved Optically Selective Conductive Tin Oxide-Films by Spray Pyrolysis[END_REF] were measured using different levels of pressure applied on samples. Since conductivity obviously depends on powder density (the higher the density, the higher the conductivity), it is quite difficult to compare reported conductivity values from one study to another.

However, the values obtained in our study for Sb-doped SnO 2 xerogels are among the highest reported ones.

The observed evolution of electronic conductivity with the antimony dopant rate and the presence of a maximum have already been reported [START_REF] Szczuko | XPS Investigations of Surface Segregation of Doping Elements in SnO2[END_REF][START_REF] Cavaliere | Dopant-Driven Nanostructured Loose-Tube SnO2 Architectures: Alternative Electrocatalyst Supports for Proton Exchange Membrane Fuel Cells[END_REF][START_REF] Gurrola | High surface electrochemical support based on Sb-doped SnO2[END_REF][START_REF] Yin | Highly active and stable Pt electrocatalysts promoted by antimony-doped SnO2 supports for oxygen reduction reactions[END_REF][START_REF] Muller | Highly Conducting Nanosized Monodispersed Antimony-Doped Tin Oxide Particles Synthesized via Nonaqueous Sol-Gel Procedure[END_REF][START_REF] Mulla | Deposition of Improved Optically Selective Conductive Tin Oxide-Films by Spray Pyrolysis[END_REF]. This may be related to the oxidation state of the dopant.

SnO 2 is known to be an oxygen-deficient non-stoichiometric oxide (SnO 2-δ ) [START_REF] Godinho | Energetic and Electronic Structure Analysis of Intrinsic Defects in SnO2[END_REF]. Its intrinsic electronic conductivity results from the presence of oxygen vacancies (V O

) which are compensated with electrons. Hence, the non-stochiometry of SnO 2 gives rise to an n-type conductivity according to the equilibrium (1):

2 O O x  O 2 (g) + 4 e' + 2 V O °° (1)
When SnO 2 is doped with Sb, a possible competition between the two antimony oxidation states has been suggested [START_REF] Terrier | Sb-Doped SnO2 Transparent Conducting Oxide from the Sol-Gel Dip-Coating Technique[END_REF][START_REF] Terrier | Electrical and optical properties of Sb:SnO2 thin films obtained by the sol-gel method[END_REF][START_REF] Lekshmy | Microstructure and physical properties of sol gel derived SnO2:Sb thin films for optoelectronic applications[END_REF][START_REF] Las | Influence of Additives on the Electrical-Properties of Dense SnO2-Based Ceramics[END_REF]. It has been shown that when Sb 3+ is incorporated into SnO 2 , it is oxidized during calcination in air [START_REF] Caldararu | Redox processes in Sb-containing mixed oxides used in oxidation catalysis I. Tin dioxide assisted antimony oxidation in solid state[END_REF]. This oxidation to Sb 5+ is favored in the presence of SnO 2 , which acts as a catalyst and a stabilizer of the Sb 5+ state. This results in an increased Sb 5+ /Sb 3+ ratio. The lower the Sb content, the higher the Sb 5+ /Sb 3+ ratio.

For low doping rates, the antimony is agreed to be stabilized in a +5 state after calcination. According to geometrical considerations, the substitution, in the SnO 2 lattice, of Sn 4+ ions with Sb 5+ ions is allowed thanks to compatible ionic radius. Such a substitution leads to the generation of positively charged point defects Sb Sn ° which can be compensated by creating tin vacancies what can be represented with the equation (2): →

These tin vacancies may then react with native oxygen vacancies, decreasing their concentration (at constant temperature and O 2 partial pressure). Such a decrease in oxygen vacancy concentration impacts the equilibrium (1) (right shift) resulting in a higher concentration of electrons in the lattice. Hence, the introduction of Sb 5+ results in increased conductivity, as observed up to 5 at.% with xerogel samples.

For higher doping rates, Sb 5+ is no longer favored and Sb 3+ is stabilized. It substitutes Sn 4+ as possibly represented with the equilibrium (3): →

The creation of oxygen vacancies according to equilibrium (3) now impacts equilibrium (1) in the opposite direction, making the electron concentration decrease. Consequently, the electronic conductivity decreases, as experimentally observed from 5at.% on xerogel samples.

This hypothesis could partially explain the observed conductivity trend. It is however incomplete because it does not account for different parameters, such as the charge carrier mobility, the change in morphology between samples with different doping rates, and the effect of Sn 4+ / Sn 2+ transition. It has also been reported that Sb 3+ cannot fit in Sn 4+ octahedral coordination as good as Sb 5+ , which induces charge defects [START_REF] Bruneaux | Structural, electrical and interfacial properties of sprayed SnO2 films[END_REF].

All conductivity values obtained here are not really representative of real catalytic layer conductivities. They are useful to compare samples and select the proper dopant and its concentration. The conductivity of aerogels measured here is notably lower than that of xerogels and carbon. It is however noteworthy that the doped SnO 2 materials developed in this study are intended to be used as platinum support for fuel cell application. Pt nanoparticles deposited at the surface of SnO 2 may then lower the depletion layer resulting from adsorbed oxygen species (O 2 -, O -or O 2-) limiting the electronic conductivity [START_REF] Barsan | Conduction model of metal oxide gas sensors[END_REF]. Thus as already reported [START_REF] Senoo | Cathodic performance and high potential durability of Ta-SnO2-delta-supported Pt catalysts for PEFC cathodes[END_REF][START_REF] Senoo | Improvements in electrical and electrochemical properties of Nb-doped SnO2-delta supports for fuel cell cathodes due to aggregation and Pt loading[END_REF] we might expect an improvement in the conductivity after Pt deposition. This will be checked in a further study.

Conclusion

Antimony and niobium-doped tin dioxide xerogels and aerogels were synthesized by a sol-gel route starting with metal alkoxide precursors. After calcination at 600 °C in air, only the rutile crystal phase of SnO 2 was detected by XRD for all samples, whether pristine or doped.

Due to their unique morphology, our Nb-and Sb-doped SnO 2 aerogels appear to be very promising catalyst supports for PEMFC cathode.

Aerogel samples show an interconnected particle network, particularly pronounced for Sb-doped samples, with bimodal mesoporous pore size distributions.

Doping SnO 2 induces airier morphologies and leads to quite high specific surface areas: 85 m 2 /g for both SnO 2 :

Nb (10 at.%) and SnO 2 : Sb (10 at.%).

As expected from the drying route, xerogels are denser and present only very small pore diameters (< 20 nm).

Highly doped xerogel samples, however, show surprisingly high specific surface areas (60 m²/g for 10 at.% Sb).

All Sb-doped samples exhibited an impressive improvement in electronic conductivity compared to pure SnO 2 .

An increase of at least a 4 orders of magnitude has been recorded after doping SnO 2 xerogels with Sb, to reach electronic conductivities of about 1 S/cm (SnO 2 : Sb (5 at.%)), one order of magnitude less for aerogels. This value is even very close to that of carbon Vulcan XC-72 (4 S/cm).

Only poor conductivities were achieved for Nb-doped samples (4.5 × 10 -5 S/cm for 10 at.% of Nb versus 2 × 10 -5 S/cm for pure SnO 2 ).

To conclude, with an adequate morphology (high specific surface area, bimodal pore size distribution centered on 25 and 45 nm, negligible microporous volume) and thanks to a high bulk electronic conductivity close to that of Vulcan XC-72, Sb-doped SnO 2 aerogel samples are very promising candidates as catalyst supports for PEMFC cathodes.

Future work will center on Pt deposition and catalytic performance assessment (activity and durability).

Fig 1

 1 Fig 1 Homemade conductivity cell picture assembled (left) and disassembled (right)

Fig. 2 X

 2 Fig. 2 X ray diffraction diagrams of pure SnO 2 xerogels before and after calcination at 200°C, 400°C, 600°C and 700°C for 5 h in air

Fig. 3 X

 3 Fig. 3 X ray diffractogram of aerogels: pure SnO 2 (A6S), SnO 2 : Sb (10 at.%) (A6SS100) and SnO 2 : Nb (10 at.%) (A6SN100). All samples were calcined for 5 h at 600 °C in air

Fig. 4

 4 Fig. 4 SEM pictures of aerogels: pure SnO 2 (A6S), SnO 2 : Nb (10 at.%) (A6SN100), SnO 2 : Sb (5 at.%) (A6SS050) and SnO 2 : Sb (10 at.%) (A6SS100) (All samples calcined for 5 h at 600 °C in air)

Fig. 5

 5 Fig. 5 SEM pictures of xerogels: SnO 2 : Sb (5 at.%) (X6SS050) and SnO 2 : Sb (10 at.%) (X6SS100). All samples calcined for 5 h at 600 °C in air

2 Fig. 6

 26 Fig. 6 Pore size distribution for pure and doped SnO 2 aerogels (A6S: pure SnO 2 , A6SN100: SnO 2 : Nb 10 at.%, A6SS100: SnO 2 : Sb 10 at.%)

Fig. 7

 7 Fig. 7 Pore size distribution for pure and doped SnO 2 xerogels (X6S: pure SnO 2 , X6SS010: SnO 2 : Sb 1 at.%, X6SS050: SnO 2 : Sb 5 at.%, X6SS150: SnO 2 : Sb 15 at.%)

Fig. 8 Fig. 9

 89 Fig. 8 EDX typical spectra obtained for SnO 2 : Nb (A6SN100) and SnO 2 : Sb (A6SS100) aerogels

Fig. 11

 11 Fig. 11 Nyquist plot obtained for SnO 2 : Nb 10 at.% xerogel

Fig. 12

 12 Fig. 12 Nyquist plot obtained for SnO 2 : Sb 10 at.% xerogel

Fig. 13 Fig. 14 Fig. 15

 131415 Fig. 13 Evolution of electronic conductivity with pressure applied on SnO 2 : Sb xerogels -(2 at.%-X6SS020, 5 at.%-X6SS050, 10 at.%-X6SS100 and 15 at.%-X6SS150)

  HNO 3 /Sn = 0.072 mol/mol) in water (H 2 O/Sn = 3.06 mol/mol) and isopropanol (iPrOH/Sn = 119 mol/mol), was also mixed under magnetic stirring. Solution B was then slowly dripped into solution A and a gel formed after a few minutes. The gel was then covered with isopropanol to prevent any drying and aged for 48 hours at room temperature before being washed with isopropanol three times a day for two days. After washing, the resulting gels were dried either by conventional evaporation to obtain xerogels, or under CO 2 in supercritical conditions (80 bars, 40 °C) to obtain aerogels. The resulting solids were heat treated under air for 5 h.After calcination, the sample color turned blue or black respectively for SnO 2 : Sb aerogels or xerogels. The SnO 2 : Nb xerogels turned light green whereas the SnO 2 : Nb aerogels remained white. Compared to aerogels, xerogels are likely to be darker in color due to a higher density resulting from the drying route.

	Samples were labeled according to a dedicated nomenclature. For example, the SnO 2 : Nb (10 at.%) aerogel
	sample is labeled A6SN100, where A stands for aerogel (X for xerogels), 6 for calcination at 600 °C in air, S for
	SnO 2 , N for Nb dopant (S for Sb) and 100 for 10 at.% doping.
	Characterization
	XRD measurements were performed using an X'Pert pro, Philips diffractometer (Cu Kα of λ = 1.5405 Å). The
	diffractometer was operated at 45 kV and 30 mA. Data were collected at steps of 0.05 ° from 20° to 90° in 2-θ
	mode with a Pixcel counter.

[START_REF] Antolini | Carbon supports for low-temperature fuel cell catalysts[END_REF]

, Nb(OiPr)

[START_REF] Maass | Carbon support oxidation in PEM fuel cell cathodes[END_REF]

and Sb(OiPr) 3 all purchased from Alfa Aesar. Two solutions (A & B) were prepared. Solution A, consisting of tin isopropoxide in isopropanol (Acros Organics, 99.5%) with a corresponding amount of the doping element, was placed under magnetic stirring. Solution B, consisting of nitric acid (Alfa Aesar, 2N) (

Table 1

 1 Nitrogen sorption measurement results -SnO 2 aerogel samples (calcined for 5 h at 600 °C in air)

	Sample	Dopant-Rate (at.%)	S BET (m²/g)	PSD (BJH) (nm)	Pore Volume (BJH) (cm 3 /g)	µpore volume (t-plot) (cm 3 /g)
	A6S	x-x	81	15-20-40	0.31	1.5 × 10 -2
	A6SN050	Nb-5	76	20-30-45	0.21	1.2 × 10 -2
	A6SN100	Nb-10	87	20-30-45	0.29	1.3 × 10 -2

Table 2

 2 Nitrogen sorption measurement results -SnO 2 xerogel samples (calcined for 5 h at 600 °C in air)

	Sample	Dopant-rate (at.%)	S BET (m²/g)	PSD(BJ H) (nm)	Pore Volume (BJH) (cm 3 /g)	µpore volume (t-plot) (cm 3 /g)
	X6S	x-x	21	13	0.10	1.6 × 10 -4
	X6SS010	Sb-1	20	13	0.09	1.0 × 10 -4
	X6SS020	Sb-2	23	11	0.09	1.7 × 10 -4
	X6SS050	Sb-5	32	7	0.09	1.6 × 10 -4
	X6SS100	Sb-10	57	6	0.09	1.6 × 10 -4
	X6SS150	Sb-15	63	4	0.08	1.6 × 10 -4

Table 3

 3 Chemical composition of aerogels from EDX and XPS measurements

	Sample	Dopant-rate (at.%)	x/(x+Sn) * EDX (at.%)	x/(x+Sn) * XPS (at.%)
	A6SN050	Nb-5	5.5	7
	A6SN100	Nb-10	10	13
	A6SS050	Sb-5	7.5	8
	A6SS100	Sb-10	12.0	13.5

* x = Nb or Sb

Table 4

 4 SnO 2 : Nb conductivities obtained from Impedance Spectroscopy measurements on xerogel samples

	Samples	Dopant-rate (at.%)	Bulk conductivity 10 -5 (S/cm)	Global conductivity 10 -5 (S/cm)
	X6S	-	53 ± 2	1.9 ± 1.3
	X6SN020	Nb -2	63 ± 11	1.0± 0.4
	X6SN100	Nb -10	490 ± 360	4.5 ± 3.6

Table 5

 5 SnO 2 : Sb conductivity obtained from Impedance Spectroscopy measurements on xerogel samples

	Samples	Dopant-rate (at.%)	Conductivity (S/cm)
	X6S	-	1.9 × 10 -5 ± 1.3 × 10 -5
	X6SS020	Sb -2	0.41 ± 0.16
	X6SS050	Sb -5	0.97 ± 0.14
	X6SS100	Sb -10	0.81 ± 0.4
	X6SS150	Sb -15	0.71 ± 0.39

ACKNOWLEDGMENTS

The authors wish to thank Pierre Ilbizian for supercritical drying, Frédéric Georgi for XPS analysis, Suzanne Jacomet for SEM/EDX analysis as well as Frédéric Fabry and Matthieu Houllé for their help in manufacturing the homemade conductivity measurement cell. This work was funded by the European Union's Seventh Framework Program for the Fuel Cells and Hydrogen Joint Technology Initiative under grant agreement n325239 (FCH-JU project Nano-CAT) and the French National Research Agency PROGELEC programme, (ANR-12-PRGE-007 project SURICAT). It was supported by Capenergies and Tenerrdis. The final publication is available at Springer via http://dx.doi.org/[10.1007/s10853-016-9833-7]

AUTHOR INFORMATION

Corresponding Author: christian.beauger@mines-paristech.fr Author Contributions: the manuscript was written through contributions of all authors.