Elise Bayraktar
email: elise.bayraktar@univ-eiffel.fr

Monte-Carlo simulation results in estimating a pure-jump Cox-Ingersoll-Ross process

Keywords: Lévy process, Stable process, Cox-Ingersoll-Ross process, Parametric inference, Estimating functions, Monte-Carlo simulations

. We first present a numerical scheme to simulate this process. Next, we describe the challenge presented by the non-symmetric stable Lévy process when computing its density and its derivatives. We finally implement the estimators and carry out simulations to show good estimation accuracy.

Introduction

Many fields such as finance and neurosciences use models based on stochastic equations with jump. When modelling interest rates, this led to an extension of the classical Cox-Ingersoll-Ross process (CIR process) introduced in [START_REF] Cox | A theory of the term structure of interest rates[END_REF] to the α-stable CIR processes (see Jiao et al. [START_REF] Jiao | Alpha-CIR model with branching processes in sovereign interest rate modeling[END_REF] and [START_REF] Jiao | The alpha-Heston stochastic volatility model[END_REF]) described by

X t = x 0 + at -b t 0 X s ds + σ t 0 X s dB s + δ t 0 X 1/α s-dL α s t ≥ 0
where (B t ) t≥0 is a standard Brownian motion and (L α t ) t≥0 a non-symmetric α-stable Lévy process with α ∈ (1, 2). We assume that the characteristic function of L α 1 is given by E(e izL α 1 ) = exp -|z| α (1 -i tan πα 2 sgn(z)) .

From Theorem 14.15. in Sato [START_REF] Sato | Lévy processes and infinitely divisible distributions[END_REF], such a process is strictly self-similar ie L α t L = t 1/α L α 1 . We denote by φ α the density of L α 1 . This paper follows previous work [START_REF] Bayraktar | Estimation of a pure-jump stable Cox-Ingersoll-Ross process[END_REF], which focused on the estimation of the drift parameters a and b, the scaling parameter δ and the jump activity α from high-frequency observations of the process on a fixed time period [0, T ]. The estimation of the drift on a finite time interval is not feasible in the presence of a Brownian motion, so we considered a pure-jump α-stable CIR process (σ = 0 in the previous equation)

dX t = (a 0 -b 0 X t )dt + δ 0 X 1/α0 t-dL α0 t , X 0 = x 0 > 0 (0.1)
with a 0 > 0, b 0 ∈ R, δ 0 > 0. We know from Fu and Li [START_REF] Fu | Stochastic equations of non-negative processes with jumps[END_REF] that (0.1) admits a strong solution and that this solution is positive. The estimation proposed in [START_REF] Bayraktar | Estimation of a pure-jump stable Cox-Ingersoll-Ross process[END_REF] is based on estimating equations. We build a quasi-likelihood by approximating the conditional distribution of X t+h given X t by the stable distribution appropriately centered and rescaled (see Masuda [START_REF] Masuda | Non-Gaussian quasi-likelihood estimation of SDE driven by locally stable Lévy process[END_REF] and Clément and Gloter [START_REF] Clément | Joint estimation for SDE driven by locally stable Lévy processes[END_REF]). We also use the power variation method described by Todorov [START_REF] Todorov | Power variation from second order differences for pure jump semimartingales[END_REF] to build non-rate optimal estimators, then correct them using the one-step method. The onestep method was described by Masuda [START_REF] Masuda | Optimal stable Ornstein-Uhlenbeck regression[END_REF] for the estimation of Ornstein-Ulhenbeck type processes. Brouste and Masuda [START_REF] Brouste | Efficient estimation of stable Lévy process with symmetric jumps[END_REF] also used the one-step improvement for a stable Lévy process. We notice that the previous study [START_REF] Brouste | Efficient estimation of stable Lévy process with symmetric jumps[END_REF] was conducted for a symmetric Lévy. In this paper, we are concerned with a non-symmetric Lévy process. Numerically, this makes the computation of the density and its derivatives harder, and the one-step correction more difficult to implement.

The paper is organised as follows. Section 1 presents the numerical scheme used to simulate the process that solves (0.1). Section 2 describes the challenge presented by the computation of the density of the nonsymmetric stable Lévy process and its derivatives. Section 3 recalls the estimation method proposed in [START_REF] Bayraktar | Estimation of a pure-jump stable Cox-Ingersoll-Ross process[END_REF] as well as the asymptotic properties of estimators. Next, we describe in this section how to build the estimators in practice and we confirm numerically the theoretical convergences.

Simulating the process

Numerical scheme

We want to generate the process (X t ) t∈[0,1] that solves equation (0.1) by using a discretisation scheme. We know that the solution X is positive with the choice of parameters a 0 > 0 and δ 0 > 0, we therefore want a scheme that preserves this positivity. We use the discretisation scheme proposed by Li and Taguchi [START_REF] Li | On a positivity preserving numerical scheme for jump-extended CIR process: the alpha-stable case[END_REF]. For technical reasons, this scheme uses a truncation and considers a bounded jump coefficient, as the authors were not able to prove consistency without a truncated jump coefficient. Denoting by X H t the solution of

dX H t = (a 0 -b 0 X H t )dt + δ 0 h(X H t-)dL α0 t , X H 0 = x 0 > 0
where h(x) = min{|x| 1/α0 , H} for some arbitrarily large constant H > 1, they consider the following positivity preserving scheme : X H,n 0 = x 0 and for i ≥ 0

X H,n i+1 n = |X H,n i n + a0 n + δ 0 h(X H,n i n )∆ n i L α0 | (1 + b0 n ) . (1.1)
Assuming α 0 ∈ ( √ 2, 2), they get in Corollary 2.9. a strong rate of convergence. For any p ∈ (0, α 2 0 -2), setting H = n l for l = (2/α0-1)/(4α0) 1+p(1-α0/2) , there exists C T,p > 0 such that

sup t≤T E(|X t -X n l ,n t |) ≤ C T,p n -L (1.2)
where L = (2/α 0 -1)/(4α 0 ) -l and C T,p converges to infinity as p increases to α 2 0 -2. In practice, H = n l leads to a very large H for the values of n that we will use (n will be larger than 100 000), hence we will use H = +∞ which corresponds to no truncation or h(x) = |x| 1/α0 .

Generation of a α-stable random variable

The previous discretisation scheme relies on simulating the independent increments of the Lévy process

∆ n i L α0 = L α0 i n -L α0 i-1 n L = L α0 1/n . Using the self-similarity property of (L α0 t ) t∈[0,1] , we know that ∆ n i L α0 L = n -1/α0 L α0 1 .
Therefore, we only need to simulate independent copies of L α0 1 . We use the method presented in Weron [START_REF] Weron | Computer simulation of Lévy α-stable variables and processes[END_REF] to generate L α,β 1 a α-stable random variable, α ∈ (0, 2) and α ̸ = 1, of skewness parameter

β ∈ [-1, 1], with characteristic function Φ α,β (z) = E e izL α,β 1 = exp -|z| α 1 -iβ tan πα 2 sgn(z) . (1.3)
This method is based on the simulation of V a uniform random variable on (-π 2 , π 2 ) and an independent exponential random variable W with mean 1. Using these variables, we get a α-stable random variable L α,β

1 by L α,β 1 = S α,β × sin(α(V + B α,β )) (cos(V )) 1/α × cos(V -α(V + B α,β )) W (1-α)/α , (1.4) 0 0.2 0.4 0.6 0.8 1 0 2 4 (a) X0 = 1, a = 1, b = 1, δ = 1, α = 1.8 0 0.2 0.4 0.6 0.8 1 0 1 2 3 (b) X0 = 1, a = 1, b = -1, δ = 1, α = 1.5 0 0.2 0.4 0.6 0.8 1 0 1 2 3 (c) X0 = 1, a = 1, b = 0, δ = 1, α = 1.5 0 0.2 0.4 0.6 0.8 1 0 1 2 3 (d) X0 = 1, a = 1, b = 1, δ = 1, α = 1.2
B α,β = arctan(β tan απ 2 ) α , S α,β = 1 + β 2 tan 2 απ 2 1/(2α)
.

(

In this paper, we will consider α-stable variables with β = 1, α ∈ (1, 2).

Using independent uniform and exponential random variables, we are now able to simulate independent copies of L α 1 = L α,1 1 . We can now simulate the process (X t ) t∈[0,1] that solves (0.1) using the numerical scheme (1.1) with H = +∞. For the rest of the simulations, we will consider high-frequency observations (X i n ) 1≤i≤n . To have a more accurate simulation of our process, we use the numerical scheme with step (1000n) -1 . We show in Figure 1 some simulated trajectories for different parameter values.

Computing the density of a strictly α-stable process

To estimate the parameters, we use a quasi-likelihood method based on the density φ α of a strictly α-stable random variable L α 1 = L α,1 1 . We now explain how to compute this density and its derivatives efficiently.

Density of a strictly α-stable process

The characteristic function of L α 1 has the following expression

Φ α,1 (z) = E e izL α 1 = exp -|z| α 1 -i tan πα 2 sgn(z) . (2.1)
We could compute the density φ α using a Fourier inversion of the characteristic function, but this would result in an integral on R. We instead use the expression presented in Nolan [START_REF] Nolan | Numerical calculation of stable densities and distribution functions[END_REF], with an integral on a bounded interval which allows for an easy numerical computation of this density. In [START_REF] Nolan | Numerical calculation of stable densities and distribution functions[END_REF], Nolan gives a computation of the density of a α-stable random variable Y with characteristic function

E e izY = exp -|z| α 1 + iβ tan πα 2 sgn(z)(|z| 1-α -1) (2.2) = E e iz(L α 1 -β tan( πα 2 )) .
For such a random variable, and denoting by f its density function, the following representation holds :

f (x, α, β) =                α(x-ξ) 1/(α-1) π|α-1| π/2 -B α,β V (θ, α, β) exp(-(x -ξ) α/(α-1) V (θ, α, β))dθ x > ξ Γ(1+1/α) cos(B α,β ) π(1+ξ 2 ) 1/(2α) x = ξ f (-x, α, -β) x < ξ (2.3) where ξ = -β tan πα 2 , Γ(a) = ∞ 0 x a-1 e -x dx, B α,β is defined in (1.5) and V (θ, α, β) = (cos(αB α,β )) 1/(α-1) cos θ sin(α(θ + B α,β )) α/(α-1) cos(αB α,β + (α -1)θ) cos θ .
As highlighted by the characteristic functions (2.2) and (2.1), we have Y = L α 1 + ξ. Hence, we get φ α by

φ α (x) = f (x + ξ, α, 1), ∀x ∈ R. (2.4)
This density is available in the Python package Scipy [START_REF] Virtanen | SciPy 1.0 Contributors. SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python[END_REF] with the scipy.stats.levy stable function, or in the R package RStableDist [START_REF] Wuertz | Rmetrics core team members. stabledist: Stable Distribution Functions[END_REF]. Figure 2 gives the graph of the density for different values of α. We can also check that this distribution is coherent with simulations done in Section 1.2. We generate independent copies of L α 1 according to equation (1.4) and compare the obtained distribution to the expected density as computed in (2.4). Figure 3 compares the density to the histogram of the obtained density when simulating n = 100000 independent copies of L α 1 .

-8 -6 -4 -2 0 2 4 6 8 10 0 0.1 0.2 0.3 α = 1.2 α = 1.5 α = 1.8
For the rest of the simulations, we will use the scipy density because it has been optimised and makes the computations faster, but the explicit expression of the density is useful later when trying to compute the derivatives.

Derivatives of the density

The correction that we will apply to our first-step estimator uses the density φ α and its derivatives. For symmetric α-stable distributions, a method for efficient evaluation of the derivatives of the density has been studied in Matsui and Tamekura [START_REF] Matsui | Some improvements in numerical evaluation of symmetric stable density and its derivatives[END_REF]. Similarly to what is done for the symmetric process, we use the 3) to get the derivative with respect to x of the density. For x > ξ, we have

f ′ (x, α, β) = αx 1/(α-1) π|α -1| 1 x(α -1) π/2 -B α,β V (θ, α, β) exp(-x α/(α-1) V (θ, α, β))dθ - α α -1 x 1/(α-1) π/2 -B α,β V (θ, α, β) 2 exp(-x α/(α-1) V (θ, α, β))dθ ,
and for x < ξ f ′ (x, α, β) = -f ′ (-x, α, -β).
We conclude using that φ ′ α (x) = f ′ (x + ξ, α, 1).

(2.5)

We confirm this expression of the derivative by comparing it to a two-point estimate and see that it agrees.

In Figure 4, we give the graph of the derivative of the density for two values of α. However, contrary to the symmetric case, the bound of the integral given by the B α,β parameter defined in (1.5) depends on α. This makes the exact computation of ∂ α φ too costly, and leads in practice to very unstable results. We therefore use a two-point estimate

∂ α φ(x, α) = φ(x, α + h) -φ(x, α -h) 2h .

Monte-Carlo estimators

Estimation method

This paper is a follow-up to the previous article [START_REF] Bayraktar | Estimation of a pure-jump stable Cox-Ingersoll-Ross process[END_REF] in which we propose estimators of the parameter θ = (a, b, δ, α) and study their asymptotic properties. Following [START_REF] Bayraktar | Estimation of a pure-jump stable Cox-Ingersoll-Ross process[END_REF] we propose a first-step estimator, which is fairly easy to compute but is not rate optimal. This estimator will then be corrected to reach an efficient estimator.

We first estimate the jump activity coefficient α 0 by the power variation method described in Todorov [START_REF] Todorov | Power variation from second order differences for pure jump semimartingales[END_REF]

αn = log 2 2 log(V 2 n (1/2, X)/V 1 n (1/2, X)) 1 V 1 n (1/2,X)̸ =V 2 n (1/2,X) (3.1)
where

V 1 n (p, X) = n i=2 |∆ n i X -∆ n i-1 X| p with ∆ n i X = X i n -X i-1 n , V 2 n (p, X) = n i=4 |∆ n i X -∆ n i-1 X + ∆ n i-2 X -∆ n i-3 X| p .
This estimator is easy and quick to compute numerically.

The scaling parameter δ 0 is next estimated, using again a power-variation method

δn =    1 m 1/2 (α n ) 1 n n i=2 n 1/2 αn X i n -2X i-1 n + X i-2 n 1/2 X 1/2 αn i-2 n    2 (3.2) with m 1/2 (α) = E(|2 1/α S α 1 | 1/2
) where S α 1 has a symmetric α-stable distribution. From Masuda [START_REF] Masuda | Joint estimation of discretely observed stable Lévy processes with symmetric Lévy density[END_REF], we have

∀ q ∈ (-1, α), E|S α 1 | q = 2 q Γ( q+1 2 )Γ(1-q α ) √ πΓ(1-q 2 )
. Hence m 1/2 (α n ) is easy to compute, and so is δn .

To estimate the drift parameters (a, b), we consider the quasi-likelihood function given by (recalling that θ = (a, b, δ, α))

L n (θ) = n i=1 log   n 1/α δX 1/α i-1 n φ α (z n i (θ))   with z n i (θ) = n 1/α X i n -X i-1 n -a n + b n X i-1 n δX 1/α i-1 n . (3.3)
When δ 0 and α 0 are known, we maximise the quasi-likelihood function (a, b) → L n (a, b, δ 0 , α 0 ) to build the estimators of the drift (â n , bn ). Similarly, when α 0 and δ 0 are estimated by αn and δn , we maximise (a, b) → L n (a, b, δn , αn ) and we denote by (ã n , bn ) the resulting estimators. We introduce the notations

h α (x) = φ ′ α φα (x), k α (x) = 1 + xh α (z), f α (x) = ∂αφα φα (x)
and

I 11 (θ) =   1 δ 2 1 0 1 X 2/α s ds E(h 2 α (L 1 )) -1 δ 2 1 0 Xs X 2/α s ds E(h 2 α (L 1 )) -1 δ 2 1 0 Xs X 2/α s ds E(h 2 α (L 1 )) 1 δ 2 1 0 X 2 s X 2/α s ds E(h 2 α (L 1 ))   . (3.4)
We summarise in the next result the asymptotic properties of these estimators.

Theorem 3.1. (i) αn given by (3.1) converges in probability to α 0 and √ n(α n -α 0 ) stably converges in law.

(ii) δn given by (3.2) converges in probability to δ 0 and √ n ln(n) ( δn -δ 0 ) is tight. (iii) For α 0 and δ 0 known, we have the stable convergence in law

n 1/α0-1/2 ân -a 0 bn -b 0 L-s ----→ n→∞ I 11 (θ 0 ) -1/2 N (3.5)
where N is a standard Gaussian variable independent of I 11 (θ 0 ).

(iv) For αn and δn estimating α 0 and δ 0 , n 1/α 0 -1/2 ln(n) 2

ãn -a 0 bn -b 0 is tight.

(i) is proven in Todorov [START_REF] Todorov | Power variation from second order differences for pure jump semimartingales[END_REF]. (ii) is proven in Theorem 4.3 of [START_REF] Bayraktar | Estimation of a pure-jump stable Cox-Ingersoll-Ross process[END_REF]. (iii) and (iv) are proven in Theorem 4.2. of [START_REF] Bayraktar | Estimation of a pure-jump stable Cox-Ingersoll-Ross process[END_REF].

We compute a preliminary estimator by successively estimating αn from (3.1) and δn from (3.2). Then, we estimate the drift parameters (a, b) by maximising the quasi-likelihood function restricted to the drift (a, b) → L n (a, b, δn , αn ) with L n defined in (3.3) and we denote by (ã n , bn ) the resulting estimators. We obtain a preliminary estimator θ0,n = (ã n , bn , δn , αn ). For Theorem 3.1 this estimator is consistent but not rate optimal as we only prove the tightness of ln(n) -2 u -1 n ( θ0,n -θ 0 ), where

u -1 n =    n 1/α0-1/2 Id 2 0 0 √ n 1 δ0 ln(n) α 2 0 0 1    , Id 2 = 1 0 0 1 . (3.6)
We correct this non-efficient preliminary estimator θ0,n with a one-step procedure :

θ1,n = θ0,n -J n ( θ0,n ) -1 G n ( θ0,n ), (3.7)
where G n = -∇ θ log L n (θ) and J n = ∇ θ G n . We define

I(θ) = I 11 (θ) (I 21 (θ)) T I 21 (θ) I 22 (θ) (3.8)
where I 11 (θ) is defined in (3.4), (I 21 (θ)) T denotes the transpose of I 21 (θ) and

I 21 (θ) = 1 δ 1 0 ds X 1/α s E(h α k α (L α 1 )) -1 δ 1 0 Xs X 1/α s dsE(h α k α (L α 1 )) I 21 21 (θ) I 22 21 (θ) , I 22 (θ) = E(k 2 α (L α 1 )) I 22 21 (θ) I 22 21 (θ) I 22 22 (θ)
where

I 21 21 (θ) = - 1 δα 2 1 0 ln(X s ) X 1/α s dsE(h α k α (L α 1 )) - 1 δ 1 0 ds X 1/α s E(f α h α (L α 1
))

I 21 22 (θ) = 1 δα 2 1 0 ln(X s )X s X 1/α s dsE(h α k α (L α 1 )) + 1 δ 1 0 X s X 1/α s dsE(f α h α (L α 1 )) I 22 21 (θ) = - 1 α 2 1 0 ln(X s )dsE(k 2 α (L α 1 )) -E(f α k α (L α 1 )) I 22 22 (θ) = 1 α 4 1 0 ln(X s ) 2 dsE(k 2 α (L α 1 )) + 2 α 2 1 0 ln(X s )dsE(f α k α (L α 1 )) + E(f 2 α (L α 1 )).
Corollary 3.1. We have the stable convergence in law, with I(θ 0 ) defined in (3.8)

u -1 n ( θ1,n -θ 0 ) L-s ----→ n→∞ I(θ 0 ) -1/2 N . (3.9)
This was proved in Corollary 4.1. of [START_REF] Bayraktar | Estimation of a pure-jump stable Cox-Ingersoll-Ross process[END_REF].

Simulation results for the first-step estimator

Estimation of α

In Table 1, we present results of numerical simulations conducted with the true value of the parameter α 0 = 1.3. We let the number of data points n range in the set {128, 256, 512, 1024, 2048, 4096}. The process (X t ) is simulated according to the scheme (1.1) with step (1000n) -1 . We show a Monte-Carlo evaluation, based on n M C = 1000 replications, for the mean and standard deviation of the estimator αn given in (3.1).

Estimation of δ

In Tables 23, we present results of numerical simulations for the estimator δn given in (3.2) conducted with the true value of the parameter δ 0 = 1, respectively for α 0 = 1.3 known and for α 0 estimated by αn as described in 3.2.1. We again let n range in the set {128, 256, 512, 1024, 2048, 4096} and simulate (X t ) using (1.1) with step (1000n) -1 . We give an estimation by Monte-Carlo of the mean of the estimators together with their standard deviations. In these Monte-Carlo experiments, we used n M C = 1000 replications. As expected, δn converges to δ 0 both when α 0 is known and when it is estimated, but the convergence is slower when α 0 is unknown and has to be estimated. This corresponds to the loss of rate ln(n) that happens when estimating δ and α simultaneously. This quasi-likelihood function is concave. We see that the maximum in (a, b) is reached near the true value (a 0 , b 0 ). Maximising with respect to the two parameters using the Python Scipy package, we get (â n , bn ) = (3.047, 5.068). We can see that there is an area where the quasi-likelihood doesn't vary much. This can lead to difficulties when maximising it, especially later for estimated values of δ and α.

The maximisation of L n is conducted using quasi-Newton methods implemented in Python Scipy package. It necessitates to compute numerically the values of the quasi-likelihood function as detailed in Section 2, and thus involves the numerous evaluations of (φ α (z n i (θ))) i=1,...,n . To make the estimation faster, these In order to check the convergence (3.5), in which the asymptotic law is conditionally Gaussian, we will check that for Σ = I 11 (θ 0 ) -1

n 1/α0-1/2 (â n -a 0 ) L-s ----→ n→∞ Σ 11 N and n 1/α0-1/2 ( bn -b 0 ) L-s ----→ n→∞ Σ 22 N .
We estimate Σ by Σ n = I 11 n (θ 0 ) -1 with

I 11 n (θ 0 ) = 1 n n i=1      1 δ 2 0 1 X 2/α 0 i-1 n E(h 2 α0 (L α0 1 )) -1 δ 2 0 X i-1 n X 2/α 0 i-1 n E(h 2 α0 (L α0 1 )) -1 δ 2 0 X i-1 n X 2/α 0 i-1 n E(h 2 α0 (L α0 1 )) 1 δ 2 0 X 2 i-1 n X 2/α 0 i-1 n E(h 2 α0 (L α0 1 ))     
where E(h 2 α0 (L α0 1 )) is calibrated once with a Monte-Carlo method with a very large n M C = 100000. The proof of Theorem 3.1. in [START_REF] Bayraktar | Estimation of a pure-jump stable Cox-Ingersoll-Ross process[END_REF] shows that I 11 n (θ 0 ) converges in probability to I 11 (θ 0 ) > 0 which allows us to conclude From Table 4, we see that assuming δ 0 and α 0 known the joint estimation of drift parameters works well, but with a relatively slow rate of convergence n 1/α0-1/2 . Hence this estimation works better for a value of α 0 closer to 1. This is also highlighted by Figure 6, where the convergence is a lot quicker for α 0 = 1.1 where n 1/α0-1/2 ≈ n 0.41 , than for α 0 = 1.3 where n 1/α0-1/2 ≈ n 0.27 .

((Σ n ) 11 ) -1/2 n 1/α0-1/2 (â n -a 0 ) L ----→ n→∞ N and ((Σ n ) 22 ) -1/2 n 1/α0-1/2 ( bn -b 0 ) L ----→ n→∞ N . n Mean
In Table 5, we see that the asymptotic behaviour of the estimator is as predicted from the theoretical study, the rate of estimation for (â n , bn ) is n 1/α0-1/2 and the asymptotic rescaled standard deviations are very close to the theoretical one.

In Figure 7, we plot the distributions of the rescaled errors of estimation together with their Gaussian limits. We see that the empirical distributions fit very well the theoretical ones.

((Σn) 11 ) -1/2 n 1/α 0 -1/2 (ân -a 0 ) ((Σn) 22 ) -1/2 n 1/α 0 -1/2 (

Simulation results for the one-step correction

In the following, we aim to show that the joint estimation of θ = (a, δ, α) is feasible in practice, assuming b 0 = 5 known, and that the one-step improvement leads to a rate-efficient estimator. The gradient and Hessian of the quasi-likelihood function G n and J n will be computed using finite differences, for reasons explained in Section 2.2. For this estimation, the theoretical variance is Σ = I(θ 0 ) -1 with where the integrals are estimated by Riemann sums, and the expectations are calibrated once with a Monte-Carlo method with a very large n M C = 100000. For instance

I(θ 0 ) =    1 δ 2 0 1 0 1 X 2/α 0 s dsE(h 2 α0 (L α0 1 )) 1 δ0 1 0 ds X 1/α 0 s E(h α0 k α0 (L α0 1 )) I 21 21 (θ) 1 δ0 1 0 ds X 1/α 0 s E(h α0 k α0 (L α0 1 )) E(k 2 α (L α 1
(I n (θ 0 )) 11 = 1 δ 2 0 1 n n i=1 1 X 2/α0 i-1 n E(h 2 α0 (L α0 1
)).

We have proved in [START_REF] Bayraktar | Estimation of a pure-jump stable Cox-Ingersoll-Ross process[END_REF] that I n (θ 0 ) converges in probability to I(θ 0 ) > 0. Hence from (3.8) we have the convergences in law

((Σ n ) 11 ) -1/2 n 1/α0-1/2 (â 1,n -a 0 ) L-s ----→ n→∞ N , ((Σ n ) 33 ) -1/2 √ n ln(n) α 2 0 δ 0 ( δ1,n -δ 0 ) L-s ----→ n→∞ N , ((Σ n ) 33 ) -1/2 √ n( α1,n -α 0 ) L-s ----→ n→∞ N .
We consider larger values of n than we did previously, as smaller values such as n = 512 lead to mixed results. We have once again simulated the process (X t ) using the scheme described in (1.1) with step (1000n) -1 . From Tables 6 and7, we can check the impact of the one-step improvement. While the mean converges and the standard deviation decreases as n increases for both the preliminary and corrected estimators, we can see that the corrected estimators have much lower standard deviations. This significant improvement is also very clear in Figure 8. While both estimators are centered around the true value of the parameters, the lower variance of the one-step estimator leads to a more accurate estimation. In Figure 9, we plot the distributions of the rescaled errors of estimation of the one-step estimator, together with the standard Gaussian. We see that the empirical distributions of the one-step estimator fit the theoretical ones. Remark 3.1. When estimating α in Section 3.2.1, we already had a rate of convergence √ n, which is the same as the efficient rate. The improvement visible in Figure 8 comes from the better variance. This is also true for the estimation of δ. On the other hand, the estimation of the drift gains in rate with this improvement.

Remark 3.2. We also know from Section 5.6. in [START_REF] Bayraktar | Estimation of a pure-jump stable Cox-Ingersoll-Ross process[END_REF] that, for some neighbourhood of θ 0 V So we could try using (u T n ) -1 I( θ0,n )u -1 n instead of J n ( θ0,n ) in our one-step procedure. However, numerical computations show similar results in both cases. Moreover, we would need to estimate I( θ0,n ) by I n ( θ0,n ) using Riemann sums to estimate the integrals and to compute the expectations, for instance E(h αn (L αn 1 )). As the matrix I n ( θ0,n ) is full, the resulting Monte-Carlo simulation drastically slows down the estimation. In conclusion, this method would be significantly longer for similar results.

However, this method would be worth considering when working with a symmetric stable process S α such as in Brouste and Masuda [START_REF] Brouste | Efficient estimation of stable Lévy process with symmetric jumps[END_REF]. In this case a lot of terms in the matrix I n ( θ0,n ) are 0, leaving us with only a few expectations to compute which makes this technique viable.

Fig 1 :

 1 Fig 1: Simulation of trajectories of the α-CIR process

Fig 2 :

 2 Fig 2: Graphs of the α-stable density φ α (β=1) for different values of α

Fig 3 :

 3 Fig 3: Distribution of simulated α-stable random variable (α = 1.5, β = 1, n = 100000 variables) and comparison with the expected density φ α

8 Fig 4 :

 84 Fig 4: Graph of φ ′ α for α 0 = 1.5 and α 0 = 1.8

3. 2 . 3 .

 23 Estimation of the drift for δ 0 and α 0 known To have an idea of what the quasi-likelihood function (3.3) looks like, we simulate one sample path of observations (X i n ) i=0,...,n with n = 1000, a 0 = 3, b 0 = 5, δ 0 = 1 and α 0 = 1.3, and we plot in Figure 5 the graph of [0.5, 5] × [3, 8] → R (a, b) → L n (a, b, δ 0 , α 0 ).

Fig 5 :

 5 Fig 5: Views of the quasi-likelihood

Fig 6 :

 6 Fig 6: Impact of α 0 on the convergence of the drift (a 0 = 3, b 0 = 5, δ 0 = 1, n=2048)

  restriction of I(θ 0 ) (defined in (3.8)) to the estimation of (a, δ, α). We estimate Σ = I(θ 0 ) -1 by Σ n = I -1 n

Fig 7 :

 7 Fig 7: Distribution of the rescaled errors of estimation and comparison with N (0, 1) (a 0 = 3, b 0 = 5, δ 0 = 1, α 0 = 1.3, n = 2048)

  n u T n J n (θ)u n -I(θ 0 ) → 0.

Fig 8 :Fig 9 :

 89 Fig 8: Distribution of estimation before and after one-step improvement (a 0 = 3, b 0 = 5, δ 0 = 1, α 0 = 1.3, n = 5000)

Table 1 .

 1 Mean and standard deviation of αn with 1000 replications

	n	Mean of αn	Standard deviation of αn
	128	1.473	3.65 * 10 -1
	256	1.396	1.98 * 10 -1
	512	1.351	1.27 * 10 -1
	1024	1.333	8.88 * 10 -2
	2048	1.319	6.25 * 10 -2
	4096	1.310	4.65 * 10 -2

Table 2 .

 2 Mean and standard deviation of δn for α 0 = 1.3 known with 1000 replications

	n	Mean of δn	Std of δn
	128	0.859	4.93 * 10 -1
	256	0.899	3.79 * 10 -1
	512	0.932	3.52 * 10 -1
	1024	0.941	2.89 * 10 -1
	2048	0.959	2.31 * 10 -1
	4096	0.978	1.94 * 10 -1

Table 3 .

 3 Mean and standard deviation of δn for αn estimating α 0 with 1000 replications

Table 4 .

 4 Mean and standard deviation of the drift estimators with δ 0 = 1 and α 0 = 1.3 known, with 1000 replications

Table 5 .

 5 Mean and standard deviation of the rescaled errors

	bn -b 0 )

Table 6 .

 6 Mean of the estimators before and after correction,

	n	Mean of ãn	Mean of â1,n	Mean of δn	Mean of δ1,n	Mean of αn	Mean of α1,n
	2048	3.017	2.854	0.948	0.984	1.322	1.289
	4096	3.011	2.913	0.962	0.989	1.314	1.294
	8192	3.020	2.932	0.976	0.991	1.307	1.298
				with 1000 replications
	n	Std of ãn	Std of â1,n	Std of δn	Std of δ1,n	Std of αn	Std of α1,n
	2048	1.15	0.875	2.22 * 10 -1	1.19 * 10 -1	6.18 * 10 -2	4.43 * 10 -2
	4096	7.99 * 10 -1	5.27 * 10 -1	1.87 * 10 -1	7.55 * 10 -2	4.67 * 10 -2	2.24 * 10 -2
	8192	8.45 * 10 -1	4.37 * 10 -1	1.44 * 10 -1	5.07 * 10 -2	3.16 * 10 -2	1.21 * 10 -2

Table 7 .

 7 Standard deviation of the estimators before and after correction, with 1000 replications


R Troncy

G Bonnet
email: gbonnet@univ-lr.fr

F Pedraza
email: cfpedraza@univ-lr.fr

Synthesis of self-regenerating NiAl-Al 2 O 3 composite coatings

Keywords: Coating, Aluminothermic reaction, NiAl-Al 2 O 3, Microstructure, Self-regenerating coating

. The present work now investigates the formation of potential selfregenerating coatings using the aluminothermic reaction between nickel oxide and aluminum. The use of this exothermic reaction allows the simultaneous formation of both alumina and nickel aluminides. The coatings were obtained by first pre-oxidizing the nickel substrate, then spraying an aluminum slurry before applying an adequate heat treatment. The most promising self-regenerating coatings display a first composite layer, made of an intermetallic compound (Al 3 Ni or NiAl) and alumina, and a second layer with a higher concentration of oxide. The first layer is expected to play the role of an Al reservoir coating and the second one that of a diffusion barrier to slow down the Al flux to maintain the Al reservoir in the coating. The effects of preoxidation time and of temperature on the resulting coating microstructures are discussed as a function of time.

Introduction

Aluminum diffusion coatings manufactured by CVD-related techniques have long and widely been used for the protection of turbine blades made of Ni-based superalloys exposed to oxidative and corrosive conditions [START_REF] Goward | Current research on the surface protection of superalloys for gas turbine engines[END_REF][START_REF] Stringer | High-temperature corrosion of superalloys[END_REF]. Some recent studies focused on slurry aluminizing via aqueous routes due to their milder ecological impact and much lower cost. The oxidation resistance of the slurry aluminide coatings was similar the one observed with traditional ones [START_REF] Mc Mordie | Oxidation resistance of slurry aluminides on high temperature titanium alloys[END_REF][START_REF] Rannou | Comparative isothermal oxidation behavior of new aluminide coatings from slurries coating Al particles and conventional out-of-pack aluminide coatings[END_REF]. Pedraza et al. described the mechanisms of formation of such aluminide coatings on pure Ni [START_REF] Pedraza | Potential thermal barrier coating systems from Al microparticles. Mechanisms of coating formation on pure nickel[END_REF] and Galetz and collab. [START_REF] Galetz | The role of combustion synthesis in the formation of slurry aluminization[END_REF] revealed the high exothermicity associated during the reaction of molten Al and Ni, which allowed the impressively quick formation of the coatings on different Ni-based superalloys [START_REF] Galetz | The role of combustion synthesis in the formation of slurry aluminization[END_REF][START_REF] Biswas | A study of self-propagating hightemperature synthesis of NiAl in thermal explosion mode[END_REF].

However, irrespective of the fabrication mode (CVD or slurry), the Al from the diffusion coatings is mostly lost by diffusion into the substrate and diluted by the upward flow of the alloying elements upon exposure to high temperatures [START_REF] Perez | Interdiffusion analysis for NiAl versus superalloys diffusion couples[END_REF]. As a result, the Al reservoir of the coatings is depleted in particular with high activity coatings because of the significant Al gradient between the coating and the substrate [START_REF] Rannou | Comparative isothermal oxidation behavior of new aluminide coatings from slurries coating Al particles and conventional out-of-pack aluminide coatings[END_REF]. While different approaches to slow down the Al inward diffusion have been proposed in the open literature, they all rely on placing diffusion barriers between the coating and the substrate mostly by using CVD [START_REF] Müller | Chemical vapor deposition of smooth α-Al 2 O 3 films on nickel base superalloys as diffusion barriers[END_REF], PVD [START_REF] Guo | Yttria partially stabilized zirconia as diffusion barrier between NiCrAlY and Ni-base single crystal René N5 superalloy[END_REF], electrodeposition [START_REF] Bouchaud | Slurry aluminizing mechanisms of Ni-based superalloys incorporating an electrosynthesized ceria diffusion barrier[END_REF][START_REF] Liu | Improving cyclic oxidation resistance of Ni 3 Albased single crystal superalloy with low-diffusion platinum-modified aluminide coating[END_REF][START_REF] Yang | Microstructure and cyclic oxidation of a Hfdoped (Ni,Pt)Al coating for single-crystal superalloys[END_REF], and even by explosive welding, forming and heat treatment [START_REF] Shmorgun | Phase composition of heat-resistance layered coatings of the Al-Cr-Ni system[END_REF]. Such diffusion barriers can result in different coating compositions, microstructures and thicknesses that can be tailored as a function of the final application. Yet, none of these studies investigated the simultaneous formation of a hybrid coating like the ones derived from the well-known aluminothermic exothermic reactions that result in composite coatings made of nickel aluminides and alumina. For instance, Padmavardhani et al. sintered functionally graded NiAl-Al 2 O 3 composite compounds by reactive hot compaction of powders [START_REF] Padmavardhani | Synthesis and microstructural characterization of NiAl-Al 2 O 3 functionally gradient composites[END_REF]. Hot-pressing of pre-oxidized NiAl powder was also employed to form NiAl-Al 2 O 3 bulk samples with adequate bending strength and toughness [START_REF] Hwang | Microstructure and mechanical properties of NiAl/Al 2 O 3 composites[END_REF]. One of our previous works highlighted the different reactive processes when mixing pre-oxidized Ni and metal Al vs. stacking metal Al onto pre-oxidized Ni. It was concluded that the particle size and preoxidation time governed the growth of the multilayered bulk samples made of nickel aluminides of different stoichiometries with various dispersions of the Al 2 O 3 issued from the aluminothermic reaction [START_REF] Troncy | Microstructural characterization of NiAl-Al 2 O 3 composite materials obtained by in situ aluminothermic reduction of NiO for potential coating applications[END_REF]. Therefore, the goal of this paper is to extrapolate our previous findings to bulk samples to form potential self-regenerating coatings. The idea is to exploit the in situ exothermicity (with no external compaction) between the powders deposited from water-based slurries onto a pure Ni substrate to sinter the composite Ni x Al y -Al 2 O 3 coatings. In addition, the high heat released during the aluminothermic reaction will be absorbed by the high thickness of the nickel, which will avoid the emission of gas or the projection of molten metal, as the work of K. Woll et al. as shown [START_REF] Woll | The utilization of metal/metal oxide core shell-powders to enhance the reactivity of diluted thermite mixtures[END_REF]. The mechanisms of formation of these original coatings are discussed for the first time in the literature.

Experimental procedure

Bulk Ni samples (99.98 wt. %, Goodfellow) were cut from 12.7 mm diameter cylindrical rods into coupons of about 1.5 mm thickness, polished with SiC P180 papers and then rinsed with deionized water and cleaned in ethanol before pre-oxidation at 1100°C for 2 h in air. After this pre-oxidation step, an Albased slurry was sprayed on the coupons. The slurry formulation is based on a 1/10 PolyVinyl Alcohol (PVA)/deionized water solution to which 45 wt. % of spherical Al powders (4±3 µm, Hermillon Poudres, France) were added [START_REF] Bouchaud | Slurry aluminizing mechanisms of Ni-based superalloys incorporating an electrosynthesized ceria diffusion barrier[END_REF]. After blending, the slurries were deposited by airbrush on the pre-oxidized coupon samples. After each deposit, the samples were systematically dried in a stove at 50 °C for 30 min. The samples were weighed before and after slurry deposition with a 10 -4 g precision balance to control the deposited slurry quantity.

The dry coatings were then annealed in a TGA apparatus to form the diffusion coating, under a flow of argon (20 mL/min) to prevent any significant oxidation of the substrate and of the Al particles [START_REF] Galetz | The role of combustion synthesis in the formation of slurry aluminization[END_REF]. The heat treatment was adapted to consider the thermal events observed in the differential scanning calorimetry (DSC, SETARAM Labsys Evo 1600) measurements [START_REF] Troncy | Microstructural characterization of NiAl-Al 2 O 3 composite materials obtained by in situ aluminothermic reduction of NiO for potential coating applications[END_REF]. This gave us three steps: at 620°C for 2 h to foster the solid/solid reduction between Al and NiO; 700°C for 2h to promote the liquid/solid aluminothermic reduction and the Al inward diffusion and a final step at 1080°C to terminate the aluminothermic reduction and foster the outward diffusion of Ni [START_REF] Pedraza | Potential thermal barrier coating systems from Al microparticles. Mechanisms of coating formation on pure nickel[END_REF].

Two series of short (2 h) and long (12 and 24 h) heat treatment times were investigated (Fig. 1). In all of them, the heating and cooling rates were set at 5°C/min. In the short annealing series (Fig. 1a), temperature increments from 620 to 700 and finally to 1080°C were also investigated. In the long annealing series (Fig. 1b), the same temperatures were employed but the dwell times were longer (24h at 620 and at 700 °C and 12 h at 1080 °C). In the following, each heat treatment will be named through the last applied stage, for example: 620°C/2 h, 700°C/2 h or 1080°C/2 h, or 620°C/24 h. Following these six different tests, the most promising heat treatments (700°C/24 h and 1080°C/2 h) were performed on the nickel substrate pre-oxidized for different times (30 min, 1 h, 1.5 h, 2 h), with a fixed quantity of deposited slurry (10±2 mg/cm²) to examine the influence of the effects of the nickel oxide layer. The analyses and observations were done with a SEM (FEI QUANTA 200F) coupled to an EDAX detector for chemical analysis, at 20 kV under low vacuum (0.9 mbar). Raman micro-spectrometry (Jobin Yvon LabRam HR800, λ=632.82 nm) was also carried out to identify the different oxide phases. In addition, the crystal structure was characterized by X-ray diffraction (XRD) in a Bruker AXS D8 Advance with Cu k α1 radiation (λ=0.15406 nm). The cross-sections were polished down to 1 µm water-based diamond suspension (Struers).

In addition, the SEM surface images of the pre-oxidized samples were processed with Photoshop to remove the oxide ridges. This step was required to reduce noise in the Image J treatment for the calculation of the areas and grain sizes of the oxides.

Results

Pre-oxidation of bulk Ni

The Ni substrates were pre-oxidized for 30 min, 1, 1.5 and 2 h at 1100 °C in air. The SEM micrographs taken on the surface and in the cross-section are shown in Fig. 2 in the secondary electron (SE) and backscattered electron (BSE) modes. The main results retrieved from image analysis are summarized in Table 1. Two morphologies of NiO (ridge-like and compact grains) can be observed at the surface of the samples for pre-oxidation times shorter than 1.5 h. For low temperature oxidation, the nickel oxides exhibit a cellular or platelet morphology. As the temperature increases, the morphology of the nickel oxide layer becomes ridge-like. In the literature [START_REF] Peraldi | High Temperature Corrosion and Materials Chemistry[END_REF][START_REF] Peraldi | Correlations between growth kinetics and microstructure for scales formed by high-temperature oxidation of pure nickel, I. Morphologies and Microstructures[END_REF][START_REF] Peraldi | Correlations between growth kinetics and microstructure for scales formed by high-temperature oxidation of pure nickel, II. Growth Kinetics[END_REF][START_REF] Haugsrud | On the high-temperature oxidation of nickel[END_REF], two mechanisms have been proposed to explain the formation of this morphology in the nickel oxide layers. One mechanism relates to the increased growth above grain boundary regions as a consequence of the rapid outward grain boundaries diffusion. Another suggested mechanism is that the ridges form at grain boundaries due to strong internal stress generated by internal oxide formation [START_REF] Peraldi | High Temperature Corrosion and Materials Chemistry[END_REF]. The oxide then releases by plastic deformation compressing the oxide and forming ridges. In addition, the amount of ridges would decrease with increasing oxidation time which in turn increases the thickness of the nickel oxide layer Fig. 2 and Table 1. Thicker oxide layers decrease the cation outward diffusion, while the surface diffusion would remain constant and disperse the nickel oxide on the surface. Overall, the surface roughness is reduced. In our case, the amount of ridges decreases with increasing the pre-oxidation time (Table 1). Increasing the pre-oxidation times allows the oxide to expand from the grain boundaries towards the grain surface, which thus lowers the outward cation diffusion and decreases the growth stresses. In a similar vein, the size of the compact oxide grains increases with the exposure time (Fig. 3). Their growth is well reported in the literature to occur by outward lattice diffusion of Ni via singly or doubly charged Ni vacancies that leads to the columnar morphology [START_REF] Haugsrud | On the high-temperature oxidation of nickel[END_REF] also observed in our study

Surface

2.2: Influence of the annealing temperature

The microstructures of the pre-oxidized (2 h) bulk Ni samples after reaction with Al at 620, 700 and 1080°C for 2 and 24 h are shown in Fig. 5. The compositions of the metallic elements in the observed layers are also given. The compounds formed and the thicknesses of the different layers are summarized in Table 2. They have been assessed by XRD, EDS and the Raman spectroscopy (Fig. 6).

Heat treatment at 620°C for 2 h and 24 h

After 2 h at 620°C (Fig. 5.a), only the outermost layer of NiO has reacted with Al to result in a porous Nirich layer of about 2.0±0.9 µm that also contains some oxide inclusions (Fig. 5.a, layer 1). The layer below corresponds to the part of NiO that did not react during the heat treatment (Fig. 5.a, layer 2). Indeed, the EDS analyses indicate that very little aluminum has diffused into this layer (Al-Ni 1-99 at. %). Increasing the time to 24 h results in a similar microstructure than after 2 h but now Al has diffused markedly (Fig.

5.b).

Heat treatment at 700°C for 2 h and 24h

The additional treatment at 700°C for 2 h brings about significant changes of the layers and of the microstructure (Fig.

Heat treatment at 1080°C for 2 and 12h

With the additional last heat treatments at 1080°C, the morphologies of the coatings are dramatically different from what was observed at lower temperatures. After 2 h, the first layer consists of NiAl intermetallic compound with Al 2 O 3 inclusions (Fig. 5.e, layer 1) and surmounts a thin and compact layer of Al 2 O 3 (Fig. 5.e, layer 2). Then, the fine layer of Ni appears above the layer of NiO and the substrate (Fig. 5.e, layers 3 and 4). Finally, the pattern of the NiAl grains of the first layer (Fig. 5.e layer 1) seems to be close to those observed for the nickel oxide layer (Fig. 5.e layer 4).

After 12 h, the coating is mainly composed of NiAl but three inhomogeneous sub-layers can be observed (Fig. 5.f). The first sub-layer is characterized by the presence of numerous pores (Fig. 5.f, layer 1). The second one is quite similar to the one observed after 24 h at 620 and 700 °C, with the presence of Al 2 O 3 clusters (Fig. 5.f, layer 2). The third layer is very uneven and contains large oxidized cavities extending into the 4 th layer that is composed of NiAl (Fig. 5.f, layer 4). One can also note the delamination between the coating and the substrate probably due to a difference of their thermal expansion coefficients.

Spot

Influence of the pre-oxidation time of the nickel substrate

Following the results presented in section 2.2, the most promising heat treatments for a possible selfregenerating coating application appear to be 1080°C/2 h and 700°C/24 h. Therefore, the study of the influence of the pre-oxidation of the nickel substrate was exclusively conducted with such heat treatments. The pre-oxidation times were thus set at 30 min, 1 h and 1.5 h. The microstructures and metal contents in the coatings after the heat treatments are given in Fig. 7. Table 3 summarizes the crystallographic compounds assessed by XRD, EDS and Raman spectroscopy (Fig. 8), and the thicknesses of the different layers (from SEM images).

Spot

Heat treatment of 1080°C/2h

For the heat treatment 1080°C/2 h applied on a Ni sample pre-oxidized for 30 min, the coating presents a non-homogeneous structure because the areas made of "diffusion islands" alternate with more homogeneous zones. The diffusion islands are composed of a NiAl (Fig. 7.a label 1) and Ni 3 Al core (Fig. 7.a label 1) surrounded by aluminum oxide (Fig. 7.a label 3). The more homogeneous area is made up of a first layer of NiAl, embedding alumina particles that surmounts a thick layer of aluminum oxide, nickel and NiO. It can be noted that only a few discontinuous areas of NiO remain (Fig. 7.a label 5), i.e. most of the NiO has been consumed.

With 1 h of pre-oxidation, the diffusion islands can no longer be observed and the coating exhibits a more homogeneous aspect. Like with the 30 min of pre-oxidation, the first layer consists of an intermetallic compound (Fig. 7.c label 1) that surmounts an alumina layer (Fig. 7.c label 2) whose thickness is similar to the previous one. Finally, Ni resulting from NiO reduction (Fig. 7.c label 3) and the remaining NiO can be observed closer to the substrate (Fig. 7.c label 4).

For a pre-oxidation time of 1.5 h, the coating shows a very different structure, except for the first layer (Fig. 7.e label 1) (NiAl embedding alumina particles). Below, a composite structure comprising Al 2 O 3 and NiAl is found whose thickness is uneven and delimited by a dense layer of alumina (Fig. 7.e label 2). Underneath, a Ni layer (Fig. 7.e label 3) with NiO islands (Fig. 7.e label 4) appears over the Ni substrate (Fig. 7.e label 6) containing aluminized areas (Fig. 7.e label 5) and alumina particles (Fig. 7.e label 7).

Heat treatment of 700°C/24h

For the heat treatment 700°C/24 h, the coatings appear rather similar to the one shown in 3). In addition, the coating contains no trapped particles when the preoxidation is conducted for just 30 min (Fig. 7.b label 1).

Discussion

3.1: Influence of the heat treatment on the formation mechanisms of coatings

Concerning the heat treatment at 620°C, the observed phenomena are identical when the heat treatment is of 2 or 24 h and barely any aluminothermic reaction occurred. In contrast, significant differences appear when the temperature is raised to 1080°C (short times) or the times are extended (24 h) at intermediate temperature (700°C).

For the short times (2 h at 620 and at 700°C) till 1080°C/2 h, a sequence for the mechanisms of formation of the coatings is schematically depicted in Fig. 9. When the temperature reaches 517±15°C [START_REF] Troncy | Microstructural characterization of NiAl-Al 2 O 3 composite materials obtained by in situ aluminothermic reduction of NiO for potential coating applications[END_REF], an aluminothermic reaction between the Al microparticles and NiO starts at their interface (eq. 1). This solid/solid reaction of reduction results in the formation of a thin layer (2.9±1 µm) of metallic Ni after 2 h at 620°C (Fig. 9.a).

Then, at a temperature of about 660°C, Al melts and begins to dissolve the Ni product near the interface and provokes a competition reaction between the formation and the dissolution of the eutectic Al 3 Ni [START_REF] Galetz | The role of combustion synthesis in the formation of slurry aluminization[END_REF]. During this step, no aluminothermic reaction takes place [START_REF] Troncy | Microstructural characterization of NiAl-Al 2 O 3 composite materials obtained by in situ aluminothermic reduction of NiO for potential coating applications[END_REF][START_REF] Yang | Microstructure and cyclic oxidation of a Hfdoped (Ni,Pt)Al coating for single-crystal superalloys[END_REF].

When the temperature increases and approaches 700°C, the formation of the Al 3 Ni intermetallic is assumed to be sufficiently fast (eq. 2) to overcome its dissolution (3 µm after 2 h (Fig. 9.b)).

At higher temperature (1080°C), the Ni content locally increases and causes the transformation of Al 3 Ni into solid Al 3 Ni 2 (eq. 3) and then into NiAl (eq. 4) (Fig. 9.c). The alumina formed during this step gets stuck at the grain boundaries of the NiO, depicting the pattern of nickel oxide shown in Fig. 5.e and 5.f.

One shall note that these reactions are assumed to be fostered by the local temperature increase related to the high exothermicity involved in the dissolution of Ni in molten Al [START_REF] Galetz | The role of combustion synthesis in the formation of slurry aluminization[END_REF][START_REF] Biswas | A study of self-propagating hightemperature synthesis of NiAl in thermal explosion mode[END_REF]. Therefore, the alumina formed by aluminothermic reaction (eq. 5) rapidly segregates at the interface between the intermetallic and the NiO, further reducing matter exchange, which stops the aluminothermic reaction (Fig. 9.d). The combined thickening of alumina and the increased Ni content of the intermetallic lower the solid-state diffusion and the overall activity of the system [START_REF] Woll | The utilization of metal/metal oxide core shell-powders to enhance the reactivity of diluted thermite mixtures[END_REF][START_REF] Vrel | Reactions kinetics and phase segregation in the 3NiO + 2Al → 3Ni+ Al 2 O 3 Thermite system[END_REF][START_REF] Vrel | SHS reactions in the NiO-Al system: influence of stoichiometry[END_REF]. In contrast, the evolution of the coating is quite significant at 700°C/24 h (Fig. 10.a). Initially, the solid/solid aluminothermic reactions followed by melting of Al and the formation/dissolution of the eutectic Al 3 Ni occur (Fig. 10.b). However, two different microstructures of the coatings have been observed.

A first microstructure (Figure 10.c) results from the aluminothermic reaction between molten Al and NiO (eq. 5). This reaction brings about the formation of Al 2 O 3 that segregates at the Al/NiO interface. While compact, the alumina layer is relatively discontinuous and therefore restricts the diffusion phenomena between the species without blocking them fully. As a result, the thickness of alumina increases till a threshold value (~5 µm) over which interdiffusion is arrested and the remaining NiO cannot be reduced.

In the second coating microstructure (Fig. 10.d), the initial development is assumed to be quite similar. However, the Al The coating microstructures obtained after 24 h at 700°C on the substrate pre-oxidized for different times do not differ markedly. In particular, the thin NiO scale formed after just 30 min of pre-oxidation remains almost identical (Fig. 11). In contrast, a greater permeability of Al through the NiO layer to form Al 3 Ni is obtained with longer pre-oxidation times (Fig. 11.a). This can be interpreted on the basis of the vacancy concentration in the oxide. Indeed, NiO is a p-type semiconductor, which implies that thickening of the oxide takes place by outward cation diffusion hence leading to a porous scale (Fig. 12.a). In addition, different studies reported that the NiO oxide scale develop a continuous network of pores and cracks that reach the substrate [START_REF] Haugsrud | On the high-temperature oxidation of nickel[END_REF][START_REF] Atkinson | Transport processes in the oxidation of Ni studied using tracers in growing NiO scales[END_REF][START_REF] Khoi | Growth and structure of nickel oxide on nickel crystal faces[END_REF][START_REF] Atkinson | Diffusion of 18 O tracer in NiO grain boundaries[END_REF][START_REF] Rapp | The high temperature oxidation of metals forming cation-diffusing scales[END_REF][START_REF] Kim | Microstructural evidence for short-circuit oxygen diffusion paths in the oxidation of a dilute Ni-Cr alloy[END_REF][START_REF] Verma | Chemical metallurgy: a tribute to Carl Wagner[END_REF] due to many different reasons including the dissociation of the oxide along the defects (grain boundaries, porosity, etc.) [START_REF] Brückman | Investigation of the high-temperature oxidation Fe-Cr alloys by means of the isotope 18 O[END_REF][START_REF] Brückman | The mechanism of transport of matter through the scales during oxidation of metals and alloys[END_REF][START_REF] Mrowec | On the mechanism of high-temperature oxidation of metals and alloys[END_REF][START_REF] Yurek | Deviation from local thermodynamic equilibrium during interdiffusion of CoO-MgO and CoO-NiO[END_REF], cracking induced by the internal stresses [START_REF] Atkinson | Transport processes in the oxidation of Ni studied using tracers in growing NiO scales[END_REF][START_REF] Atkinson | Transport of nickel and oxygen during the oxidation of nickel and dilute nickel/chromium alloy[END_REF][START_REF] Robertson | Criteria for formation of single layer, duplex, and breakaway scales on steels[END_REF][START_REF] Evans | The mechanical properties of nickel oxide and their relationship to the morphology of thick scales formed on nickel[END_REF][START_REF] Kofstad | On the formation of porosity and microchannels in growing scales[END_REF] and opening of the microcracks related to different diffusion rates within the oxide scale (Fig. 12.b) [START_REF] Kofstad | On the formation of porosity and microchannels in growing scales[END_REF]. In this study, we hypothesize that the molten Al and Al 3 Ni can pass through these channels by capillarity. This is supported by the fact that the 1.5 h pre-oxidized nickel does not form homogeneous clusters of Al 2 O 3 after annealing the coatings till 1080°C/2 h and 700°C/24 h (Fig. 12.c). In contrast, the Al

 1080°C/2 h

As opposed to 700°C/24 h, the pre-oxidation time exerts a great influence on the aluminothermic reaction and on the aluminizing of the nickel substrate. Indeed, for the samples pre-oxidized 30 min and 1 h, the Al 2 O 3 resulting from the aluminothermic reaction blocks the reaction between Al and Ni. As a result, the reaction is stopped and the substrate cannot be aluminized. For longer pre-oxidation times (1.5 h), the Al 2 O 3 mimics the shape of the former NiO grains. This suggests that molten Al and/or Al 3 Ni permeates through the defects of NiO and that reduction of half of the NiO thickness has occurred (Fig. 11.b). Finally, for a preoxidation time of 2h, the dark contrast of layer 1 shown in the backscattered electron mode of Fig. 5.e corresponds to Al 2 O 3 surrounding the grains of NiAl. Therefore, Raman spectra in these local areas and in the continuous layer 2 were conducted (Fig. 6). The α-Al 2 O 3 polymorph is clearly inferred from the doublet in the fluorescence region. Atkinson and Taylor [START_REF] Atkinson | The diffusion of Ni in the bulk and along dislocations in NiO single crystals[END_REF][START_REF] Atkinson | The diffusion of 63 Ni along grain boundaries in nickel oxide[END_REF] demonstrated that the thickness of the grain boundaries is around 70 Å. It can be assumed that during the heat treatment, the aluminum oxide would segregate where there is vacant space.

Conclusions

The formation of potential self-regenerating coatings with Al-rich Ni x Al y phases and Al 2 O 3 dispersion has been investigated using aluminothermic reactions on pre-oxidized nickel substrate. It has been found that one key parameter to achieve even coatings is the duration of the preoxidation of the substrate. The short pre-oxidation times (< 1 h) grow a "ridge-like" morphology of NiO that impedes the homogeneous aluminization of the substrate. The longer pre-oxidation times (1<t<2 h) leads to the growth of compact grains that allows the aluminothermic reaction to occur, hence to obtain even aluminized layers. It is believed that the increase in the permeability of NiO to Al is due to the many defects present in the oxide layer caused by the major diffusion of Ni 2+ cations to the oxide surface.

One of the promising coatings was sintered by pre-oxidizing the Ni substrate for 2 h followed by a slurry aluminizing at 700°C for 24 h. The coating consists of a first layer of Al 3 Ni and alumina microspheres and a second layer with a composite microstructure between the alumina and the intermetallic compound. However, an unreacted NiO layer remained over the substrate even after 24 h of aluminizing. It thus appears that the control of the thickness of the NiO oxide needed for the reduction could be achieved by electroplating nickel layers onto the substrates to be coated, which shall be investigated. Similarly, the interdiffusion behavior should be studied to assess the regenerating effect of the coatings Al 3 Ni + Al 2 O 3 .

Fig. 1 :

 1 Fig. 1: Schematic drawing of the annealing treatments (a) short (dwell of 2h) and (b) long (dwells of 12 and 24 h) times of treatment.

Fig. 2 :

 2 Fig. 2: SEM (SE and BSE mode) of surface and cross-section images on Ni samples pre-oxidized at 1100 °C under air for 30 min, 1, 1.5 and 2 h.

Fig. 3 :

 3 Fig. 3: Size and dispersion of the NiO compact grain sizes for the different pre-oxidation times at 1100 °C.

Fig. 4 :

 4 Fig. 4: Cross-section of the sample after application of the slurry on the pre-oxidized Ni substrate (1100 °C for 2 h).

  5.c). Indeed, a large part of the reduced Ni layer has reacted with Al to form an Al 3 Ni intermetallic compound (Fig. 5.c, layer 1). However, a thin layer of metallic Ni still remains below the Al 3 Ni layer (Fig. 5.c, layer 2). Then, the layer of NiO and the substrate are observed (Fig 5.c, layer 3). As opposed to 620°C/2h, a significant amount of Al has now diffused all across the layer of NiO (13 at. % Al) towards the substrate (9 at. % Al). By increasing the reaction time to 24 h (Fig. 5.d), three layers can be distinguished over the substrate. The thickest (16±3 µm) most external layer is composed of an intermetallic/oxide composite, with an Al 3 Ni intermetallic matrix and Al oxide shells surrounding the original Al particles (3.8±1.5 µm) (Fig. 5.d, layer 1). The central layer (thickness 8.7±1.8 µm) is composed of a cluster of oxides (Fig. 5.d, layer 2). This layer is separated from a 7.5±1.7 µm layer of unreacted NiO by a very thin layer of metallic Ni (Fig. 5.d, layers 3 and 4). In addition, the EDS analyses indicate that a significant Al has diffused through NiO to the substrate, with Al amounts of 23 and 16 at. % respectively.

Fig. 5 :

 5 Fig. 5: SEM (BSE mode) cross-sections of the samples pre-oxidized at 2h for 1100°C and subsequent annealing of the Al-slurry coating at 620°C (a and b), 700°C (c and d) and 1080°C (e and f) for 2, 12 or 24 h.

Fig. 6 :

 6 Fig. 6: Raman spectra on the cross-sections of 2 h pre-oxidized samples after the heat treatments 1080°C/2h (a), 700°C/24 h and 1080°C/12 h (b).

Fig. 7 :

 7 Fig. 7: SEM (BSE mode) cross-sections of the pre-oxidized samples at 30 min, 1 h and 1.5 h for 1100 °C after TGA treatment at 1080°C/2h and 700°C/24 h.

Fig. 8 :

 8 Fig. 8: Raman spectra on the cross-sections of samples pre-oxidized for 30 min, 1 and 1.5 h made after the heat treatments 1080°C/2 h (a) and 700°C/24 h (b).

Fig. 5

 5 .d (section 2.2) irrespective of the pre-oxidation time. Four different layers can again be observed. The top layer made of the intermetallic compound Al 3 Ni (Fig. 7.b,d,f label 1) surmounts an aluminum oxide cluster (Fig. 7.b,d,f label 2), then a thin layer of Ni resulting from the NiO reduction (Fig. 7.b,d,f label 3) and finally, the remaining NiO (Fig. 7.b,d,f label 4). Their thicknesses and compositions are however somewhat different (see Table

Fig. 9 :

 9 Fig. 9: Schematic drawing of the aluminothermic reduction and reaction between a pre-oxidized Ni substrate and Al particles heated by stages till 1080°C.

  2 O 3 grains issued from the aluminothermic reactions form clusters that do not slow down sufficiently direct contact of molten Al with the newly formed Al 3 Ni. This exothermic reaction brings about the dissolution of the intermetallic component and the Al 2 O 3 clusters to float over the melt. The subsequent aluminothermic reaction will keep on producing the Al 2 O 3 clusters dispersed in the intermetallic layer. The end of the reactions can occur by either an insufficient amount of Al that would stop the aluminothermic reaction, but this does not seem to be the case, or by the increased amount of Al 2 O 3 in the coating, which appears in line with the experimental observations because Al 2 O 3 both blocks interdiffusion of the species and lowers the local temperature given the fact that Al 2 O 3 is an inert heat absorber [24].

Fig. 10 : 3 . 2 :

 1032 Fig. 10: Schematic drawing of the aluminothermic reduction and reactions between a preoxidized Ni substrate and Al particles heated at 700°C during 24 h

  2 O 3 seems to segregate at the channels and grain boundaries of NiO (Figs. 12.c). Furthermore, the Al-rich flux does not reach the Ni/NiO interface because (i) the defects in the oxide are too discontinuous to allow a regular flow of Al and/or diffusion of Ni to form the Al 3 Ni, (ii) the Al activity lowers significantly, hence limiting the aluminothermic reaction and the interdiffusion of the species and (iii) the Al 2 O 3 layer grows over the critical threshold and blocks the direct contact of Al with Ni. Very likely, a combination of the three phenomena occurs but (iii) appears more plausible and is coherent with the significant amounts of Al 2 O 3 shown in the coatings and with the fact that Al particles get trapped by outward diffusion of Ni (Fig. 12.d).

Fig. 11 :

 11 Fig. 11: Histogram of the thicknesses of the different formed layers for a heat treatment 700°C/24 h (a) and 1080°C/2 h (b) applied after different Ni substrate pre-oxidation times.

Fig. 12 :

 12 Fig. 12: Schematic drawing of the aluminothermic reduction and reactions between pre-oxidized Ni substrate and Al particles heated at 700°C during 24 h.

Table 1 :

 1

Summary table of the ratio, thickness and size of the grains for different times of pre-oxidation.

Time

Compact grain area ratio (%)

Size of the compact grains (µm) Thickness of the oxide layer (µm) 30 min

		40.6	2±1	8±1.1
	1h	72.8	2±1	12±1
	1.5h	100	2±1	14±2
	2h	100	3±1	15±2

Table 2 :

 2 Summary table of the different compounds and thicknesses of the samples after the heat treatment.

	Figure	6.a

Al (at %) Ni (at%) Spot Al (at %) Ni (at%) 1080°C/2 h 700°C/24 h 30 min

		1	59 41		77 23
		2	84 16		85 15
	1h
		3	10 90		75 25
		4	6	94		11 89
		1	47 53		23 77
		2	70 30		79 21
	1.5h	3	11 89		22 78
		4	10 90		15 85
		5	27 73		94	6
		6	15 85
		1	45 55	1	74 26
		2	35 65	2	82 18
		3	63 37	3	28 72
		4	24 76	4	17 83
		5	9	91	5	76 24

Table 3 :

 3 Summary table of the different compounds and thicknesses of the coating layers resulting from the heat treatments of the slurry after TGA treatment (1080°C/2 h and 700°C/24 h).

			1080°C/2 h			700°C/24 h
	Figure			7.a			7.b
	Layer	1	2	3	4	1	2	3	4
	Compounds	NiAl	Al 2 O 3	Ni	NiO	Al 3 Ni	Cluster of Al 2 O 3	Ni	NiO
	Thickness (µm) 1.1±0.5 1.6±0.3	1±0.2	1.1±0.5	2.1±0.5	3±1.3	0.4±0.3	7.8±1.4
	Figure			7.c			7.d
	Compounds	NiAl	Al 2 O 3	Ni	NiO	Al 3 Ni	Cluster of Al 2 O 3	Ni	NiO
	Thickness (µm) 1.7±0.7 2.6±0.3	1.7±0.3	9±0.8	5±1.5	2.4±1.2	0.4±0.3	9.5±1.8
	Figure			7.e			7.f
	Compounds	NiAl	Cluster of Al 2 O 3	Ni	NiO	Al 3 Ni	Cluster of Al 2 O 3	Ni	NiO
	Thickness (µm) 3.4±1.3 9.6±3.4	2.3±3.5	2.8±1.8	3.7±1.9	3.9±1.4	0.3±0.2	10±2


Germain Boissonnet
email: germain.boissonnet@univ-lr.fr

Jean-Luc Grosseau-Poussard

Gilles Bonnet

Fernando Pedraza

Development of thermal barrier coating systems from Al microparticles -Part II: Characterisation of mechanical and thermal transport properties

Keywords: slurry coating, thermal barrier coating, thermal diffusivity, scratch tests

Introduction

Finding alternative coating technologies to the standard thermal barrier coatings (TBCs) to extend the protection to the lower stages of the turbine is of growing interest as current technologies implies expensive and complex processes. As demand for efficiency and reduction of fuel consumption and greenhouse gas emission is a mandatory concern for the future design of gas turbine engines, the temperature will thus rise in low pressure turbine (LPT) section as well. Among the different options, the formation of a complete thermal barrier coating using the slurry technique appears to be a convincing alternative for the protection of materials of this LPT section due to its relative low cost and ease of application [START_REF] Pedraza | Potential Thermal Barrier Coating Systems from Al Microparticles. Mechanisms of Coating Formation on Pure Nickel[END_REF]. This technology uses an aqueous slurry that contains Al microparticles which form, after a proper annealing, a foam of hollow alumina microspheres on top of an aluminide coating [START_REF] Pedraza | Potential Thermal Barrier Coating Systems from Al Microparticles. Mechanisms of Coating Formation on Pure Nickel[END_REF][START_REF] Boissonnet | Development of Thermal Barrier Coating Systems from Al Microparticles. Part I: Influence of Processing Conditions on the Mechanisms of Formation[END_REF][START_REF] Galetz | The Role of Combustion Synthesis in the Formation of Slurry Aluminization[END_REF][START_REF] Montero | A Single Step Process to Form In-Situ an Alumina Foam/Aluminide TBC System for Alloys in Extreme Environments at High Temperatures[END_REF][START_REF] Mollard | Comparative Degradation of Nickel Aluminized by Slurry and by Pack Cementation under Isothermal Conditions[END_REF][START_REF] Brossard | Influence of Water Vapour on the Oxidation Behaviour of a Conventional Aluminide and a New Thermal Barrier Coating System Sintered from a Slurry: Water Vapour Oxidation of Aluminide Coatings[END_REF][START_REF] Kolarik | Factors Influencing the Formation of a Diffusion Zone and the Adherence of the Top Coat of High Temperature Coatings from Micro-Sized Spherical Aluminium Particles[END_REF]. In a previous study, the resulting 40 μm-thick foam coating was shown to confer equivalent insulative properties than those of a conventional 400 μm YSZ TBC deposited by plasma spray [START_REF] Pedraza | Rheological Behaviour, Synthesis and Performance of Smart Thermal Barrier Coating Systems Based on Hollow Alumina[END_REF]. However, this light foam coating was also observed to be particularly brittle and not sufficiently resistant to be a convincing solution. Therefore, in the first part of this work which was published earlier, the goal was to improve the mechanical strength of the top foams by controlling the atmosphere of the annealing in order to promote the growth of the oxide shells of the original microparticles while enabling enough Al to react with the Ni substrate to form the diffusion coating [START_REF] Boissonnet | Development of Thermal Barrier Coating Systems from Al Microparticles. Part I: Influence of Processing Conditions on the Mechanisms of Formation[END_REF]. In this previous study, the authors investigated the influence of Ar, synthetic air, water vapour as well as hybrid atmosphere on the mechanisms of coating formation and showed that different microstructures were achieved. While a fast consumption of Al occurred in inert Ar atmosphere to form the aluminides, synthetic air and water vapour promoted the strengthening of the microparticles oxide shell but also hampered the diffusion of the Al toward the substrate resulting in heterogeneous aluminide coatings. Hybrid treatments allowed compromises between the thickening of the shells and the providing of Al for the diffusion coating by delaying the introduction of the oxidizing atmosphere. The different annealing in single and hybrid atmospheres resulted in various microstructures of the coating systems with, for instance, different top foam and oxide shell thickness, homogeneity of the aluminide and the presence of oxide and un-emptied Al microparticles in the top coating.

However, the mechanical strength, adhesion and thermal insulation properties were not investigated in this previous study as it mostly focused on the coating formation mechanisms. Therefore, this work aims at filling this purpose to evaluate the relevance and applicability of such coatings.

Experimental procedure

Materials and microstructural characterization

The coatings were performed on pure Ni (99.98% purity, Goodfellow) discs samples of 12.7 mm diameter. The coatings were achieved by the heat treatment of slurries made from Al micro-sized particles that were previously sprayed on the surface of the Ni substrates. Two different types of Al powders were used: ones with a homogeneous size distribution and that possessed a slightly thicker passive shell (HS), and others of a more dispersed size distribution with thin passive shells (DS).

After the deposition of the slurry (approximately 9±1 mg•cm -2 ), the samples were dried in a fume cupboard for 1h and, annealed in a thermobalance following a threestep heating program. With heating ramps fixed at 5°C•min -1 , the samples were first cured at 400°C to remove the organic binder (polyvinyl alcohol), then heated at 700°C for 2h to allow the Al to melt and to diffuse into the substrate, and finally annealed at 1100°C for 2h to stabilise the oxide into α-Al2O3 and the diffusion coating into a β-NiAl phase. The thermal treatments were conducted into different atmospheres to optimize the thickening of the shells by oxidizing the leaking Al coming out of the micro-sized particles while allowing enough Al to reach the substrate for the aluminisation to occur. For this purpose, Ar and synthetic air were employed to perform the thermal treatments. Moreover, hybrid atmosphere heat treatments were investigated where the Ar was replaced with synthetic air at different temperature of the annealing during the heating ramp from 400°C to 700°C.

The Table 1 shows a summary of the samples with the different annealing performed. The analysis of surfaces and cross sections of the samples were carried out using scanning electron microscopy (SEM FEI Quanta 200F) coupled to an energy dispersive x-ray spectroscopy detector (EDS EDAX) for elemental analysis. Each sample was mounted in an epoxy resin and cross sections were polished till a fineness of 1 µm using 1µ diamond paste (Nap B1 STRUERS).

Scratch tests and micro-indentation

Micro-indentation was performed in order to assess the hardness of the coatings of the study and to compare them with standard thermal barrier coatings made by plasma-spray or EB-PVD that are presented in other studies [START_REF] Boissonnet | Evolution of Thermal Insulation of Plasma-Sprayed Thermal Barrier Coating Systems with Exposure to High Temperature[END_REF][START_REF] Boissonnet | Phase Stability and Thermal Insulation of YSZ and Erbia-Yttria Co-Doped Zirconia EB-PVD Thermal Barrier Coating Systems[END_REF]. The normal force applied for the penetration of Vickers indents was recorded as a function of the depth of penetration till reaching 5 μm into the coating. This depth was chosen in accordance with the different thickness of the measured coatings.

While micro-indentation is the common method to measure the hardness of the coatings, the scratch test is a well-known technique that is used to assess the coating/substrate adhesion of thin hard coating on different types of substrates [START_REF] Bull | An Overview of the Potential of Quantitative Coating Adhesion Measurement by Scratch Testing[END_REF].

This adhesion is measured from an indent that is drawn across the coated surface with an increasing load until a defined failure occurs at a load known as the critical load, LC. Depending on multiple factors that are usually reduced to the hardness of both substrate and coating, different modes of failures that include coating detachment, through-thickness cracking, plastic deformation or cracking in the coating or substrate may occur during the scratch test. Therefore, the adherence of the TBCs made from the slurry route was appraised using the scratch test method.

Scratching experiments were performed using a scratch tester (ANTON-PARR STEP E400) with a Rockwell-M-120 diamond indenter with a 100 μm tip radius coupled to an acoustic emission sensor. The range load was set to 0.03 to 10 N over a length of 5 mm with a speed rate of 2.5 mm•min -1 and an acquisition rate of 30 Hz.

One shall note that a prescan at 0.03 N was performed prior to each scratch or microindentation test to identify the original surface of the sample and thus, was already damageable to the weakest coatings as it could already penetrate them. Therefore, the measured depth must be considered carefully. Additional indentation at 0.03 N using the same indenter were performed on the different coatings to assess the impact of the prescan.

Thermal insulation assessment

While the thermal conductivity is usually employed in the literature to describe the thermal insulation potential of thermal barrier coatings, it is derived from the thermal diffusivity , the density ( ) and the specific heat capacity ( ) following the equation 1:

(1)

However, as the coatings present various porous microstructures, it is very complex to measure the specific heat capacity or the density of the coatings with the precision needed to calculate the true thermal conductivity. Therefore, as stated in other studies, it is often preferable to discuss the thermal diffusivity of the coatings in a comparative manner to assess the thermal insulation potential of such coatings. The high-temperature thermal diffusivity was determined using a laser-flash apparatus (LFA 1600 LINSEIS) working under vacuum (~1.10 -2 mbar) with the measurements being made every 100°C from room temperature till 1100°C. Prior to the laser-flash measurements, the thickness of the samples was assessed using a digital calliper (±10 -

3 ) and both surfaces of the disc samples were coated with a 4-5 um graphite layer to increase absorption of the laser pulse on the back side as well as the emissivity of the heat signal on the front side to be measured with the InSb infrared detector. For postprocessing multi-layer calculations (2-layer model) in order to extract the contribution of the coating, the thickness of each layer was measured from the crosssection images.

Results

Microstructure of the thermal barrier systems

The Figure 1 and Figure 2 gathers respectively the surfaces and the cross sections of the coatings that were synthesized using the different atmospheres. As expected, the different atmospheres resulted in a wide variety of coating microstructures due to the difference in type of Al microparticles and oxidation potential that depend on the duration of the exposition to the synthetic air (full Ar < air introduced at 700°C < 650°C < 600°C < 550°C < full synthetic air).

In Ar atmosphere, both DS and HS microparticles resulted in thin top coatings made of the collapsed hollow particles, i.e. ~8 µm for DS-Ar and ~20 µm for HS-Ar (a and f from Figures 1 and2). The dark grey aspect of the coating heat treated under Ar in To highlight the impact of the increasing oxidation potential of the different heat treatment, top coating thickness as well as the thickness of the shell of the emptied microparticles were also plotted for the different samples on Figure 3 and Figure 4 respectively. For both types of microparticles, the top coating thickness as well as the thickness of the walls from the microspheres tend to increase with greater exposition to the synthetic air during the heat treatment. Figure 3 shows that the oxide shell of the HS particles is greater than the DS ones for all the different heat treatment while the growth rate of both particles is similar with respect to the increase of exposure to synthetic air. However, the top coating thickness evolution displayed in Figure 4 shows very different behaviour between the two types of particles. While the HS microparticles lead to the slow growth of the coating thickness, the DS microparticles show an abrupt increase in the growth of the top coating when considering the two samples in the most oxidizing heat treatment (with air introduced at 600°C and in the full synthetic air atmosphere). Figure 4 -Top coating thickness of the samples measured using SEM micrographs.

-9-

Mechanical tests

Figure 5 shows the penetration depth of the indent as a function of the applied force across the length of the scratch line while Table 2 gathers the penetration depth measured in different locations. As the coating of the HS-600 sample had already spalled from the surface, no scratch was performed on its surface. Similarly, only the coating of the DS-600 samples spalled during the scratch test (see also Figure 6). This is observed on the Figure 5b as a sharp drop of the penetration depth was registered in the early stage of the scratch. All the other coatings remained adherent to the underlying coating after the scratch tests. The recorded depths against the applied load (Figure 5) follows a two-stage sequence: (1) Over the first microns of the scratch, the indent tends to penetrate the whole thickness of the coatings as highlighted by the early drop of the measured depth. The maximum penetrated depth (Dmax in

Table 2) appears to be similar and, in some cases, even higher than the thickness of the top coatings of the different samples. Moreover, the measured depth at 0.5 N (D0.5N in Table 2) corresponds to values close to the total coating thickness. As mentioned in §2.2, it is necessary to acknowledge the contribution of the prescan that was performed at 0.03 N (see Figure 6b). As a matter of fact, the coating depth penetrated with only 0.03 N (D0.03N on Table 2) shows that the penetration of the indent is already quite significant compared to the coating thickness and thus, this D0.03N value should be added to the Dmax and D0.5N depths to obtain the effective penetrated depth. It is also observed that the D0.03N tends to decrease as the top coatings are more oxidized (700 < 650 < air). In contrast, the coatings annealed in Ar exhibit smaller penetration at 0.03 N as coating thicknesses are in the same range as the penetrated depth. In addition, for the most oxidised and thickest coatings, the measured depth follows a stable stage after the sharp drop until a given value of applied force is reached. (2) Afterwards, the penetration of the indent tends to stabilize and steadily increases as it starts to reach the diffusion coating. The Figure 6 shows an example of a scratch that was observed by SEM. Except for the DS-600 sample where the coating completely detached from the substrate, every scratch performed looks like the one presented for DS-650 sample in Figure 6c. As the micrograph from Figure 6b shows, the indenter penetrates and reaches the substrate surface (the aluminide coating) in the early stage of the scratch where the applied force is less than 0.5 N. In addition, the Figure 6c shows no evidence of large spallation of the coating around the scratch. As these observations are the same for the different scratches of the coatings, at the exception of the DS-600 sample that lost its coating after the scratch and of the HS-600 sample where the coating was already detached from the substrate, this means that the adhesion strength of the coating to the substrate is greater than the cohesive strength of the microspheres from the foam.

To further assess the difference in terms of mechanical strength, micro-indentation tests were performed on the different top coatings produced. The force needed to penetrate the first 5 µm of the different coatings is gathered in Table 3. Except for the thin top coatings performed in Ar atmosphere where the underlying substrate is rapidly reached, the force required to penetrate the 5 µm increases with increasing the oxidation of the microparticles from the top coating. The micro-indentations performed on the APS and EB-PVD YSZ coatings show that there is a significant difference between the mechanical strength of the standard TBCs and the Al slurry ones. As a matter of fact, the indentations in the plasma-sprayed coatings resulted in an average load that was 10 times higher (~200 mN) than the Al slurry coatings performed in synthetic air. The indentations performed on the EB-PVD coatings were even higher but showed a wide dispersion of the values due to the inherent heterogeneity of the surface of these columnar coatings.

Thermal diffusivity

The thermal diffusivity of the top coatings was then calculated via laser-flash measurements of the complete system (substrate + aluminide + top coating) and the results are presented in Figure 7. All the coatings present thermal diffusivity values ranging between 0.8•10 -7 to 5.0•10 -7 m 2 .s -1 with trends that decrease as a function of the temperature. By comparing the two different types of microparticles, the thermal diffusivity of the HS microparticles -based coatings display higher thermal diffusivity than the DS ones with ranges going from 1.4•10 -7 to 5.0•10 -7 m 2 .s -1 and 0.8•10 -7 to 3.3•10 -7 m 2 .s -1 respectively. Independent of the type of microparticles (DS or HS), the coatings that are the less oxidized (Ar) display the lowest thermal diffusivity and the top coatings that were heat treated in synthetic air present lower thermal diffusivities than the ones heat treated in Ar + air-650°C.

Discussion

As stated previously, the formation of the coatings is fully discussed in a previous work [START_REF] Boissonnet | Development of Thermal Barrier Coating Systems from Al Microparticles. Part I: Influence of Processing Conditions on the Mechanisms of Formation[END_REF]. Therefore, the microstructural aspects of the coatings obtained in the different annealing conditions are only briefly presented in this work.

Annealing in a quasi-inert Ar atmosphere lead, for both HS and DS aluminium microparticles, to the fast consumption of the Al to form the aluminide coating and the collapsing of the remaining emptied shells from the top coating due to a lack of strength of the oxide shells. This collapsing was explained in other studies to be due to the shrinkage of the thin alumina shells when γ-Al2O3 transform into the α-Al2O3 after exposure to temperatures above 1000°C [START_REF] Boissonnet | Development of Thermal Barrier Coating Systems from Al Microparticles. Part I: Influence of Processing Conditions on the Mechanisms of Formation[END_REF][START_REF] Kolarik | Factors Influencing the Formation of a Diffusion Zone and the Adherence of the Top Coat of High Temperature Coatings from Micro-Sized Spherical Aluminium Particles[END_REF][START_REF] Hasani | The Oxidation Mechanism of Pure Aluminum Powder Particles[END_REF][START_REF] Hasani | A Tension Analysis During Oxidation of Pure Aluminum Powder Particles: Non-Isothermal Condition[END_REF]. The fact that the collapsing of the foam top coating was more obvious for the DS-Ar than for the HS-Ar sample is probably due to a thinner passive shell for the DS microparticles, (see a and f from Figures 1 and2). The oxide shell thickness measured after the heat treatments for the DS particles is very thin (~0.2 µm) (Figure 4). Therefore, when Al starts to melt (around 660°C), the shells are not thick enough to avoid collapsing when the γ to α-Al2O3 transition occurs (see Figure 2f). The HS particles that display thicker shells, i.e. ~0.3 µm after the heat treatment, are able to withstand a bit more the transition and seem to keep a quasi-spherical shape (see cross sections on Figure 2a). In addition, the scratch tests performed on these thin coatings did not cause the detachment of the foam top coatings but rather resulted in the crushing of the hollow microspheres. As the coatings of this study are made of the agglomeration of small hollow oxide spheres on top of a hard aluminide coating, plastic deformation is supposed to be minimal as fracture should dominate the scratch response [START_REF] Bull | An Overview of the Potential of Quantitative Coating Adhesion Measurement by Scratch Testing[END_REF].

Therefore, as the strength of the oxide shells built under Ar atmosphere is very poor, the scratch only showed the brittleness and the poor cohesive strength of the foam coating. The micro-indentation tests performed on the DS-Ar and HS-Ar even showed higher value of the load needed to penetrate 5 microns (see Table 3) than the one of the coatings annealed in the other conditions. As the indenter rapidly reached the aluminide surface, it was not possible to properly measure the contribution of the top foam coating itself. Such poor mechanical strength was also observed in previous works and, is at the origin of the study that seek to improve the strength of the top foam [START_REF] Boissonnet | Development of Thermal Barrier Coating Systems from Al Microparticles. Part I: Influence of Processing Conditions on the Mechanisms of Formation[END_REF].

The use of a heat treatment in synthetic air to improve the mechanical strength of the top foam coating resulted in the oxidation of the substrate that hampered the diffusion of the aluminium from the microparticles to form the aluminide coating (Figure 2e and2j). However, such oxidation also resulted in an increase of the shell thickness of the microparticles (~0.5 µm and ~0.7 µm for DS and HS particles respectively) and a reduction of the collapsing of the emptied microspheres that led to thicker top foam coatings for the heat treatment in air compared than in Ar (see Figures 3 and4). The scratch tests showed the good adhesion of the foam coating to the substrate despite the increase of the stiffness of the foam implied by the increase of the shell thickness and the sintering caused by the oxidation of the aluminium of the microparticles. This increase in hardness was highlighted by microindentation (see Table 3) as the HS-air and DS-air showed higher resistance than the coatings with introduction of the synthetic air during the heat treatment.

By introducing the synthetic air during the annealing, different improvements are achieved in comparison with the treatment performed under Ar or synthetic air only.

Delaying the introduction of air from 700°C to 650°C and then to 600°C allowed longer exposition of the coating under the oxidation potential of the synthetic air and thus, allow the achievement of thicker oxide shells and thicker top coating than with a heat treatment under Ar only (see Figures 3 and4). With the introduction of air at 700°C, almost no improvement was made on the thickness of the microsphere walls in comparison with the full Ar heat treatment, but the introduction of air allowed to slightly reduce the collapsing of the microspheres as the top coating achieved are thicker than the one under Ar atmosphere. With the air introduced at 600°C, the thickness of the shells and of the top coatings for both types of particles are greatly improved. However, similarly than with the heat treatment performed under air, the substrate oxidation hampers the formation of a homogeneous aluminide coating. It is only with the introduction of air at 650°C that a good compromise between the aluminide formation and the achievement of thick shells and top foam is found. The indentations performed on the different coatings (Table 3) also showed that the thickening of the oxide shells (Figure 3) lead to a greater resistance to the penetration of the indent as time of exposure to the synthetic air increase, e.g. from 8.5-9.6 N with air introduced at 700°C to 12-12.6 N with air introduced at 650°C needed to penetrate 5 μm into the coating. However, when compared with bulk APS and EB-PVD YSZ coatings, the porous foam coating achieved in this study is still 20 to 75 less resistant than the previous two others. However, the porosity of the foam coatings cannot be compared with these bulk coatings that can reach, for instance, a maximum of ~25% porosity for the standard plasma sprayed YSZ coatings [START_REF] Boissonnet | Evolution of Thermal Insulation of Plasma-Sprayed Thermal Barrier Coating Systems with Exposure to High Temperature[END_REF].

The high porosity of the foam coatings achieved in this study thus lead to great thermal insulation properties. The coatings heat treated in Ar present the lower thermal diffusivities since the shell of the microparticles that trap air is very thin. As a matter of fact, heat conduction is greater in alumina (λair ~5-35 W.m-1.K-1 at RT) than in the gas filled pores (λair ~0.025 W.m-1.K-1 at RT) and could decrease even further when the pores are smaller than the gas mean free path (typically pores ~1 μm) where the heat conduction of the gas falls below the conduction of the free gas to ~0.01 W.m-1.K-1 due to gas molecule-wall collisions (Knudsen conduction) [START_REF] Brossard | Influence of Water Vapour on the Oxidation Behaviour of a Conventional Aluminide and a New Thermal Barrier Coating System Sintered from a Slurry: Water Vapour Oxidation of Aluminide Coatings[END_REF][START_REF] Kolarik | Factors Influencing the Formation of a Diffusion Zone and the Adherence of the Top Coat of High Temperature Coatings from Micro-Sized Spherical Aluminium Particles[END_REF][START_REF] Pedraza | Rheological Behaviour, Synthesis and Performance of Smart Thermal Barrier Coating Systems Based on Hollow Alumina[END_REF].

However, the top coatings that were heat treated in air present lower thermal diffusivities than the one heat treated in Ar + air-650°C despite the thicker shells of the former. This could be explained by the fact that a thick thermal oxide grew at the interface between the top coating and the diffusion coating for the samples heat treated in synthetic air. As a matter of fact, the contribution of the oxide layer that was not considered in the multilayer calculation via the laser-flash technique was previously shown to be responsible for adding a non-negligible resistance to heat transport and hence, leading to an underestimation of the thermal diffusivity [START_REF] Boissonnet | Evolution of Thermal Insulation of Plasma-Sprayed Thermal Barrier Coating Systems with Exposure to High Temperature[END_REF][START_REF] Boissonnet | Phase Stability and Thermal Insulation of YSZ and Erbia-Yttria Co-Doped Zirconia EB-PVD Thermal Barrier Coating Systems[END_REF].

Moreover, the Al_HS coatings exhibit higher thermal diffusivity values than the Al_DS ones. This can be attributed to several factors that derive from the initial differences in particle size. Indeed, for the same coating thickness, the number of interfaces increases with decreasing the size of the particles. Therefore, and because interfaces are known to scatter the heat through ceramic coatings [START_REF] Boissonnet | Evolution of Thermal Insulation of Plasma-Sprayed Thermal Barrier Coating Systems with Exposure to High Temperature[END_REF][START_REF] Boissonnet | Phase Stability and Thermal Insulation of YSZ and Erbia-Yttria Co-Doped Zirconia EB-PVD Thermal Barrier Coating Systems[END_REF][START_REF] Bull | An Overview of the Potential of Quantitative Coating Adhesion Measurement by Scratch Testing[END_REF], the greater the number of interfaces, the lower the thermal diffusivity. In addition, since the heat flow could also be enhanced when the ratio of material vs. void is more significant, the Al_HS particles whose particle shells are thicker than those of the Al_DS will lead to higher heat conduction. As also shown in another study, the thermal insulation potential of the foam coating is significant when compared with the standard APS and EB-PVD coatings that reaches thermal diffusivities between 1•10 -7 to 3•10 -7 m 2 .s -1

for APS YSZ and 3•10 -7 to 6•10 -7 m 2 .s -1 for EB-PVD YSZ coatings while the range of thermal diffusivity for the coatings in this study goes from 0.6•10 -7 to 5•10 -7 m 2 .s -1 [START_REF] Boissonnet | Evolution of Thermal Insulation of Plasma-Sprayed Thermal Barrier Coating Systems with Exposure to High Temperature[END_REF][START_REF] Boissonnet | Phase Stability and Thermal Insulation of YSZ and Erbia-Yttria Co-Doped Zirconia EB-PVD Thermal Barrier Coating Systems[END_REF].

Conclusions

-Great thermal insulation properties achieved for the foam coatings.

-Increase of the mechanical resistance with increasing the time of exposure to air during the heat treatment -What about the relevance of the scratch test to perform adhesion assessment for such foam coatings?

Figure 1

 1 Figure1results from the poor oxidation of the top coatings, i.e. the hollow microparticles have such thin shells that it is possible to see the substrate colour through. In synthetic air, the aspect is also grey due to the presence of metallic Al trapped in some of the particles (see e and f from Figures1 and 2). In air, the resulting top coatings are thick due to the early oxidation of the particles, i.e. ~60 µm for DS-air and ~50 µm for HS-air, but the diffusion coatings obtained are heterogeneous and display a NiO layer on top of a thick alumina layer. By modifying the temperature of introduction of synthetic air as a replacement of the Ar atmosphere during the heat treatment, various structures were achieved. Introducing the synthetic air at 600°C led to the important oxidation of the coatings. However, in contrast with the complete heat treatment under synthetic air, the switch led to spallation of the top coating for the HS-600 sample (Figure1d and 2d) and to cracks and voids at the interface between the diffusion coating and the top coating for the

Figure 1 -

 1 Figure 1 -SEM surfaces of the coatings of the study; a) HS-Ar, b) HS-700, c) HS-650, d) HS-600, e) HS-air, f) DS-Ar, g) DS-700, h) DS-650, i) DS-600 and j) DS-air.

Figure 2 -

 2 Figure 2 -SEM cross-sections of the coatings of the study; a) HS-Ar, b) HS-700, c) HS-650, d) HS-600, e) HS-air, f) DS-Ar, g) DS-700, h) DS-650, i) DS-600 and j) DS-air.

Figure 3 -

 3 Figure 3 -Shell thickness of the samples top coatings measured using SEM micrographs.

Figure 5 -

 5 Figure 5 -Depth as a function of the applied load FN during scratch tests.

Figure 6 -

 6 Figure 6 -Macrographs of DS-650 and 600 after the scratch tests and corresponding SEM micrographs in SE mode of a) the bottom of the scratch, b) the beginning of the scratch and c) the scratch in full length.

Figure 7 -

 7 Figure 7 -Thermal diffusivity as a function of the temperature for (a) the HS foam coatings and (b) the HS foam coatings.

Table 1 -

 1 Summary of the different heat treatments performed in TGA.

	Type of Al	Heat	Starting	Introduction of	Sample ID
	microparticles	treatment	atmosphere	synth. air
	DS	400°C-1h	Ar	-	DS-Ar / HS-Ar

Table 2 -

 2 Depth of penetration measured after the scratch tests. Dmax and D0.5N correspond to the maximum depth and the depth reached at 0.5 N of applied force respectively. D0.03N* was measured using additional indentations at 0.03 N.

	Atmosphere of	Dmax (µm)		D0.5N (µm)		D0.03N* (µm)
	heat treatment	HS	DS	HS	DS	HS	DS
	Ar	19.0	14.7	4.5	3.9	5.9 ± 1.4	6.5 ± 2.4
	Switch 700°C	23.4	25.7	10.2	4.8	11.0 ± 5.0	8.1 ± 1.1

Table 3 -

 3 Results of the micro-indentation tests with the average load (FN,av) applied to penetrate 5 microns into the different coatings.

	Atmosphere of heat	FN,av (mN)
	treatment	HS	DS
	Ar	19.6 ± 5.1	21.3 ± 9.3
	Switch 700°C	8.5 ± 0.7	9.6 ± 2.3
	Switch 650°C	12.0 ± 1.7	12.6 ± 4.5
	Synthetic air	17.6 ± 7.4	27.3 ± 4.5
	Other standard thermal barrier coatings
	APS YSZ	201.3 ± 22.0
	EB-PVD YSZ	897.3 ± 229.0

ACKNOWLEDGEMENTS

The authors gratefully acknowledge…