G Zheng

D Boutat

J-P Barbot

Geometrical Conditions for Output Depending Observability Normal Form

Keywords: Algebraic/geometric methods, Nonlinear systems, output feedback and observers

We give geometrical conditions, which guarantee the existence of a diffeomorphism in order to transform a nonlinear system without inputs into a canonical normal form depending on its output. Moreover we extend our results to a class of systems with inputs. We end this paper by some examples and its simulations to highlight the proposed algo rithm.

I. INTRODUCTION

In order to use the Luenberger's observer [START_REF] Luenberger | An introduction to observers[END_REF] for nonlinear systems, the so-called problem of the Nonlinear Observer Canonical Form (NOCF) was born. In [START_REF] Krener | Linearization by output injection and nonlinear observer[END_REF] authors give sufficient and necessary conditions which guarantee the existence of a diffeomorphism and of an output injection to transform a single output nonlinear system without inputs into NOCF form. For a multi-output nonlinear system without inputs, it was solved in [START_REF] Krener | Nonlinear observer with linearizable error dynamics[END_REF] and [START_REF] Xia | Nonlinear observer design by observer error linearization[END_REF]. Another approach was introduced for the analytical systems in [START_REF] Kazantzis | Nonlinear observer design using Lyapunov's auxiliary theorem[END_REF] by assuming that the spectrum of the linear part must lie in the Poincaré domain and it was generalized in [START_REF] Xiao | Nonlinear observer design in the Siegel domain through coordinate changes[END_REF] by assuming that the spectrum of the linear part must lie in the Siegel domain. Other approaches using quadratic normal forms were given in [START_REF] Boutat-Baddas | Quadratic observability normal form[END_REF] and [START_REF] Chabraoui | Observability quadratic characteristic numbers[END_REF]. Moreover, [START_REF] Respondek | Time scaling for observer design with linearization error dynamics[END_REF] gave the sufficient and necessary geometrical conditions to transform a nonlinear system into a so-called output-dependent time scaling linear canonical form, while [START_REF] Guay | Observer linearization by output diffeomorphism and output-dependent time-scale transformations[END_REF] gave the dual geometrical conditions of [START_REF] Respondek | Time scaling for observer design with linearization error dynamics[END_REF].

In this paper, we study the geometrical conditions to guarantee the existence of a local diffeomorphism z = φ (x) which transforms the locally observable dynamical system into a NOCF form depending on the output [START_REF] Zheng | Output dependent observability linear normal form[END_REF]. More precisely, let us consider the following system:

ẋ = f (x), y = h(x), (1)
where x ∈ U ⊂ R n and h : U ⊂ R n → R are sufficiently smooth. We are interested in deducing necessary and sufficient geometrical conditions to guarantee the existence of a local diffeomorphism z = φ(x) which transforms system (1) into the following form

ż = A(y)z + β(y), y = z n = Cz, (2)
G. where

A(y) = 0 B B @ 0 • • • 0 0 α1(y) • • • 0 0 . . . . . . • • • . . . 0 • • • αn-1(y) 0 1 C C A , β(y) = 0 B B @ β1(y) β2(y) . . . βn(y) 1 C C A ,
and α i (y) = 0 for y ∈ ]-a, a[ and a > 0. This kind of linearization is called Single Output Dependent Observability normal form (SODO normal form).

For dynamical systems in the form (2), some types of high-gain observer [START_REF] Hammouri | Bilinearization up to the output injection[END_REF], [START_REF] Gauthier | Deterministic observation theory and applications[END_REF], [START_REF] Hammouri | Observer synthesis for state-affine systems[END_REF] can be synthesized. Here we may apply the following high gain observer [START_REF] Busawon | A simple observer for a class of nonlinear systems[END_REF]:

ż = A(y)ẑ + β(y) -Γ -1 (y) R -1 ρ C T (C ẑ -y), 0 = -ρR ρ -ĀT R ρ -R ρ Ā + C T C. (3)
where

Γ (y) is the n × n diagonal matrix Γ (y) = diag n-1 i=1 α i (y), n-1 i=2 α i (y), • • • , α n-1 (y), 1
and Ā is the n × n matrix defined as follows

Ā =      0 • • • 0 0 1 • • • 0 0 . . . . . . . . . . . . 0 • • • 1 0      .
Indeed, here the output of system (2) is considered as an input of [START_REF] Chabraoui | Observability quadratic characteristic numbers[END_REF]. Setting e = z -ẑ, the observation error can be obtained as follows:

ė = A (y) -Γ -1 (y) R -1 ρ C T C e.
And the convergence of such observer is proved in [START_REF] Busawon | A simple observer for a class of nonlinear systems[END_REF], thus in section 4 we simply highlight the design of such observer for systems in the form (2). This paper is organized as follows. The next section addresses basic notations. In section 3, we present our main theorem to guarantee the studied transform for nonlinear system without inputs. Section 4 is devoted to the generalization of our results to a class of systems with inputs, and some practical particular cases are studied in the same section.

II. NOTATIONS AND TECHNICAL RESULTS

Throughout this article, L i-1 f h for 1 ≤ i ≤ n denotes the (i -1) th Lie derivative of output h in the direction of f, and set θ i = dL i-1 f h as its differential. Assume that system (1) is locally observable, thus

θ = θ 1 , • • • , θ n T is
a basis of the cotangent bundle T * U of U . Then, we also consider the vector field τ 1 defined in [START_REF] Krener | Linearization by output injection and nonlinear observer[END_REF] as follows

θ i (τ 1 ) = 0, f or 1 ≤ i ≤ n -1, θ n (τ 1 ) = 1, (4)
and by induction we define

τ k = (-1) k-1 ad k-1 f (τ 1 ) , f or 2 ≤ k ≤ n. (5)
It is clear that

{τ 1 , • • • , τ n } is a basis of the tangent bundle T U of U .
Let us recall a famous result from [START_REF] Krener | Linearization by output injection and nonlinear observer[END_REF]. Theorem 1: The following conditions are equivalent i) There exist a diffeomorphism and an output injection which transform system (1) into normal form (2) with

α k (y) = 1 for 1 ≤ k ≤ n -1. ii) [τ i , τ j ] = 0 for 1 ≤ i, j ≤ n.
If for some 1 ≤ k ≤ n-1 the functions α k (y) in the form (2) are not constant, then ii) of Theorem 1 is not fulfilled. In [START_REF] Respondek | Time scaling for observer design with linearization error dynamics[END_REF], authors considered the case of α i (y) = s(y) for 1 ≤ i ≤ n -1. Firstly, for j ≥ 2, they defined

l j = j(j -1) 2 + 1
Then, for the non-trivial case n ≥ 2, according to the following equality:

d(θ n (τ 2 )) = l n λθ 2 modspan {θ 1 }
we can uniquely determine the smooth function λ. In the following, we calculate s(y) as follows:

s = exp σ
where σ is a solution of the following equation [START_REF] Respondek | Time scaling for observer design with linearization error dynamics[END_REF] can be stated as follows:

L τj+1 σ = 0 if 0 ≤ j ≤ n -2 λ if j = n -1 By defining τ 1 = s n-1 τ 1 τ i+1 = [ τ i , f s ] for 1 ≤ i ≤ n -1, then Theorem 2 in
Theorem 2: The following conditions are equivalent i) There exist a diffeomorphism and an output injection which transform system (1) into normal form (2) with

α k (y) = s(y) for 1 ≤ k ≤ n -1. ii) [ τ i , τ j ] = 0 for 1 ≤ i, j ≤ n.
This paper is for the purpose of studying the case where α i (y) for 1 ≤ i ≤ n -1 is neither constant, nor scalar function of the output, i.e. α i (y) can be not equal to α j (y) for 1 ≤ i, j ≤ n -1 and i = j. The main result is given in the next section by assuming that a family of functions α i (y) for 1 ≤ i ≤ n -1 can be uniquely determined. The details on calculation of such family of functions is exhaustively discussed in [START_REF] Zheng | Single output-dependent observability normal form[END_REF].

III. MAIN RESULT

If there exists a diffeomorphism which transforms system (1) into form (2), then Proposition 2.3 of [START_REF] Zheng | Single output-dependent observability normal form[END_REF] gives all α i for 1 ≤ i ≤ n -1. Therefore, let us consider a new family of vector fields defined as follows:

τ 1 = n i=1 α i (y)τ 1 and τ i+1 = 1 α i [ τ i , f ] (6)
for

1 ≤ i ≤ n -1. Set θ( τ 1 , • • • , τ n ) =          0 0 • • • 0 1 0 . . . • • • π n-1 l 2,n . . . • • • . . . • • • . . . . . . π 2 • • • • • • . . . π 1 l n,2 • • • • • • l n,n          := Λ,
where

l k,j = θ k ( τ j ) f or 2 ≤ k ≤ n and n -k + 2 ≤ j ≤ n. Consider the following R n -valued form ω ω = Λ -1 θ := (ω 1 , ω 2 , • • • , ω n ) T , (7)
where, for 1 ≤ s ≤ n, we have

ω s = n m=1 r s,m θ m . (8)
Then, the following algorithm gives all the components of ω.

Algorithm 1:

f or 1 ≤ j ≤ n, r n,j = • • • = r n-j+2,j = 0 and r n-j+1,j = 1. f or 2 ≤ k ≤ n -1 and 1 ≤ j ≤ n, r n-k,j = - k i=2 l k,n-k+i-(j-1) r n-k+i-(j-1),j ,
and then, equation ( 8) becomes:

ω s = n-s+1 m=1 r s,m θ m .
Theorem 3: The following conditions are equivalent 1) There exists a diffeomorphism which transforms system (1) into a SODO normal form (2).

2) There exists a family of functions α i (y) for 1 ≤ i ≤ n -1 such that the family of vector fields τ i for 1 ≤ i ≤ n defined in (6) satisfies the following commutativity conditions

[ τ i , τ j ] = 0, f or 1 ≤ i, j ≤ n. (9)
3) There exists a family of functions α i (y) for 1 ≤ i ≤ n -1 such that the R n -valued form ω defined in [START_REF] Hammouri | Observer synthesis for state-affine systems[END_REF] satisfies the following condition

dω = 0. ( 10
)
Proof: Assume that there exists a diffeomorphism which transforms system (1) into form (2), then we compute α i (y) for 1 ≤ i ≤ n -1 according to Proposition 2.3 in [START_REF] Zheng | Single output-dependent observability normal form[END_REF]. Thus, it is easy to show that τ 1 = 1 π1 ∂ ∂z1 which yields that τ 1 = ∂ ∂z1 and then, by construction we obtain

τ i = ∂ ∂zi for 2 ≤ i ≤ n. Consequently, we have [ τ i , τ j ] = 0 for 1 ≤ i, j ≤ n.
Reciprocally, assume that there exist α i > 0 for 1 ≤ i ≤ n -1 such that [ τ i , τ j ] = 0 for 1 ≤ i, j ≤ n, then it is wellknown ( [START_REF] Isidori | Nonlinear control systems, 2nd edtion[END_REF], [START_REF] Van Der | Nonlinear dynamical control systems[END_REF]) that we can find a local diffeomorphism φ = z such that

φ * ( τ i ) = ∂ ∂z i . As φ * ( τ i ) = ∂ ∂zi is constant, hence ∂ ∂z i φ * (f ) = φ * ([ τ i , f ]) = α i φ * ( τ i+1 ) = α i ∂ ∂z i+1 , thus ∂ ∂zi φ * (f ) = α i ∂ ∂zi+1 for 1 ≤ i ≤ n -1.
Consequently, by integration we obtain: φ * (f ) = A(y)z + β(y).

Moreover, as dh

• τ i = 0 for 1 ≤ i ≤ n-1 and dh• τ n = 1, we obtain h • φ -1 = z n .
Finally, in order to prove that in Theorem (3) Condition 2) is equivalent to Condition 3), it is sufficient to prove that equation ( 9) is equivalent to equation [START_REF] Respondek | Time scaling for observer design with linearization error dynamics[END_REF].

Recall that for any two vector fields X, Y, we have

dω(X, Y ) = L X (ω(Y )) -L Y (ω(X)) -ω([X, Y ]).
Setting X = τ i and Y = τ j , we obtain

dω( τ i , τ j ) = L e τi ω( τ j ) -L e τj ω( τ i ) -ω([ τ i , τ j ]).
As ω( τ j ) and ω( τ i ) are constant, then we have

dω( τ i , τ j ) = -ω([ τ i , τ j ]).
Because ω is an isomorphism and ( τ i ) 1≤i≤n is a basis of T U, then equation ( 9) is equivalent to equation [START_REF] Respondek | Time scaling for observer design with linearization error dynamics[END_REF].

Remarks 1: i) The R n -valued form ω can be viewed as an isomorphism T U n → U × R n which brings each τ i to the canonical vector basis ∂ ∂z I . Moreover, dω = 0 means that there is a local diffeomorphism φ : U → U such that ω is the tangent map of φ.

ii) The diffeomorphism φ(x) = z is determined by ω = φ * (x), which can be given locally as follows

z i = φ i (x) = γ ω i + φ i (0) f or 1 ≤ i ≤ n,
where γ is a smooth path from 0 to x lying in a neighborhood V 0 ⊆ U of 0.

The following simple example is studied in order to illustrate Theorem 3.

Example 1: Let us consider the following system

             ẋ1 = γ(y) 1+x4 x 1 x 3 , ẋ2 = β(y) 1+x4 x 1 , ẋ3 = µ(y)x 2 , ẋ4 = γ(y)x 3 , y = x 4 . (11) which gives 8 > > > > > > < > > > > > > : θ1 = dx4, θ2 = γdx3 + γ ′ x3dx4, θ3 = γµdx2 + 2γ ′ γx3dx3 + `(γµ) ′ x2 + (γ ′ γ) ′ x 2 3 ´dx4, θ4 = γµ β 1+x 4 dx1 + `2γ ′ µ + (γµ) ′ ´γx3dx2 + `2γ ′ γµx2 + γ (γµ) ′ x2 + 3γ (γ ′ γ) ′ x 2 3
´dx3 +O [2] (x1, x2, x3) θ1.

Then we have τ

1 = 1+x4 γµβ ∂ ∂x1 .
According to Proposition 2.3 in [START_REF] Zheng | Single output-dependent observability normal form[END_REF], α 1 = c1 c2 β, α 2 = c2 c3 µ and α 3 = c 3 γ, so the new vector fields are

e τ1 = c1 (1 + x4) ∂ ∂x1 , e τ2 = c2 ∂ ∂x2 , e τ3 = c3 ∂ ∂x3 , e τ4 = ∂ ∂x4 + x1 1 + x4 ∂ ∂x1 .
It is clear that [ τ i , τ j ] = 0 for all 1 ≤ i, j ≤ 4. Therefore, according to Theorem 3, system [START_REF] Kazantzis | Nonlinear observer design using Lyapunov's auxiliary theorem[END_REF] can be transformed into SODO normal form [START_REF] Busawon | A simple observer for a class of nonlinear systems[END_REF].

Moreover as

Λ = θ( τ 1 , • • • , τ n ), a straightforward com- putation gives ω = Λ -1 θ = d x1 c1(1+x4) , d x2 c2 , d x3 c3 , dx 4 T .
As ω = dφ, thus the diffeomorphism which transforms system [START_REF] Kazantzis | Nonlinear observer design using Lyapunov's auxiliary theorem[END_REF] into SODO normal form (2) is

φ(x) = z = x 1 c 1 (1 + x 4 ) , x 2 c 2 , x 3 c 3 , x 4 T .
with which system (11) could be transformed into

       ż1 = 0, ż2 = c1 c2 β(y)z 1 , ż3 = c2 c3 µ(y)z 2 , ż4 = c 3 γ(y)z 3 .
So far, in this paper, we have only considered systems without inputs. The next section is devoted to systems that are also driven by an input term.

IV. EXTENSION TO SYSTEMS WITH INPUTS

Consider a system with inputs in the following form

ẋ = f (x) + g(x, u), y = h(x), (12)
where

x ∈ U ⊂ R n , f : U ⊂ R n → R n , g : U × R m → R n , h : U ⊂ R n → R
are analytic functions and for x ∈ U, g(x, 0) = 0.

For system [START_REF] Krener | Linearization by output injection and nonlinear observer[END_REF], the SODO normal form along its output trajectory y(t) is as follows

ż = A(y)z + β(y) + η(y, u), y = z n = Cz, (13)
where A(y) and β(y) are given in (2) and η(y, u) = η 1 (y, u), η 2 (y, u),

• • • , η n (y, u) T .
Theorem 4: System (12) can be transformed into SODO normal form (13) by a diffeomorphism if and only if i) one of conditions in Theorem 3 is fulfilled.

ii) [g, τ i ] = 0 for 1 ≤ i ≤ n -1.

Proof: From Theorem 3, we can state that there exists a diffeomorphism φ such that φ * (f ) = A(y)z + β(y).

For 1 ≤ i ≤ n -1, because φ * ( τ i ) = ∂ ∂zi is constant, hence we have ∂ ∂z i φ * (g) = φ * ([g, τ i ]) = 0.
Therefore φ * (g) = η(y, u). Thus, we obtain the form [START_REF] Krener | Nonlinear observer with linearizable error dynamics[END_REF].

Remark 1: If g(x, u) = g 1 (x)u 1 + • • • + g m (x
)u m , and also both conditions i) and ii) of Theorem 4 are fulfilled, then η(y, u)

= B 1 (y)u 1 + • • • + B m (y)u m .
Let us now study some special cases of the term η(y, u). Corollary 1: Assume that conditions i) and ii) of Theorem 4 are fulfilled,

a) if [g, τ n ] = 0, then η(y, u) = η(u). b) if g(x, u) = g 1 (x)u 1 + • • • + g m (x)u m and [g k , τ i ] = 0, f or 1 ≤ i ≤ n and 1 ≤ k ≤ m, then η(y, u) = B 1 u 1 + • • • + B m u m ,
where B i are constant vector fields.

Example 2: Let us consider the following system

         ẋ1 = γ(y) 1+x3 x 1 x 2 + x1 1+x3 u, ẋ2 = µ(y) 1+x3 x 1 , ẋ3 = γ(y)x 2 + u, y = x 3 . (14)
A similar computation gives:α 1 (y) = c1 c2 µ (y) and α 2 (y) = c 2 γ (y) . Therefore, we obtain

τ 1 = c 1 (1 + x 3 ) ∂ ∂x1 , τ 2 = c 2 ∂ ∂x2 and τ 3 = ∂ ∂x3 + x1 1+x3 ∂ ∂x1 . As g = x1 1+x3 ∂ ∂x1 + ∂ ∂x3 = τ 3 then [g, τ 1 ] = [g, τ 2 ] = 0 and system (14) is transformed into        ż1 = 0, ż2 = c1 c2 µ (y) z 1 , ż3 = c 2 γ (y) z 2 + u, y = z 3 . ( 15
)
by the following diffeomorphism

φ(x) = z = x 1 c 1 (1 + x 3 ) , x 2 c 2 , x 3 T .
Following the proposed high gain observer in the form (3), the corresponding observer for the system (15) can be designed as follows: where ρ is the tunable gain. For a more specific but simple simulation, choose c 1 = c 2 = 1, u (t) = 1, µ (y) = 1 + y 2 , and γ (y) = 2 + cos(y). Its simulation results are presented in Fig. 1 which presents the convergence of system's states and their estimations.

     . ẑ1 = -ρ 3 γµ (ẑ 3 -z 3 ) , . ẑ2 = c1 c2 µ (y) ẑ1 -3 ρ 2 γ (ẑ 3 -z 3 ) , . ẑ3 = c 2 γ (y) ẑ2 -3ρ (ẑ 3 -z 3 ) + u,
In addition, in order to solve the left invertibility problem, the Observability Matching Condition (OMC) for system [START_REF] Krener | Linearization by output injection and nonlinear observer[END_REF] with m = 1 is as follows

L g L i-1 f h = 0, ∀x ∈ U, 1 ≤ i ≤ n -1, L g L n-1 f h = 0.
Corollary 2: Assume conditions i) and ii) of Theorem 4 are fulfilled and the OMC is verified then

η(y, u) = η 1 (y, u), 0, • • • , 0 T .
Remark 2: The OMC for system [START_REF] Krener | Linearization by output injection and nonlinear observer[END_REF] with m = 1 is equivalent to g ∈ span{ τ 1 }.

We give another example in order to highlight Corollary 2. And it is obvious that the input u can be represented as an algebraic function of the output and its derivative as follows:

u = γ 2 µ ... y -γµγ ′ ÿ -γ 2 µ ′ ÿ -γµγ ′′ ẏ + γµ (γ ′ ) 2 ẏ + γγ ′ µ ′ ẏ γ 3 µ 2 V. CONCLUSION
This paper studied the SODO normal form. Geometrical conditions, which allow us to determine whether a nonlinear system can be transformed locally into the studied normal form by means of a diffeomorphism and of an output injection, are deduced. Moreover, an extension of our results is stated for a class of nonlinear systems with inputs.

of z 1 Observation error of z 2 Observation error of z 3 Fig. 1 .

 1231 Fig. 1. Observation errors of z 1 , z 2 and z 3

Example 3 : 1 ∂τ 2 =

 312 Consider the following system       ẋ1 = u, ẋ2 = µ(y)x 1 + µ(y)x 2 1 + x2 1+x1 u, ẋ3 = γ(y) x2 1+x1 , y = x 3 .(16)A straightforward computation gives α 1 (y) = c1 c2 µ (y) and α 2 (y) = c 2 γ (y). Thus, we have τ 1 = c c 2 (1 + x 1 ) ∂ ∂x2 and τ 3 = ∂ ∂x3 . As g ∈ span{ τ 1 }, then the OMC condition is fulfilled, therefore system (16) could be transformed by the diffeo-morphism: φ(x) = z = x1 c1 , x2 c2(1+x1) , x 3 c1 , ż2 = c1 c2 µ (y) z 1 , ż3 = c 2 γ (y) z 2 , y = z 3 .


Tamara Kujawska

Wojciech Secomski

Michal Byra

Michiel Postema

Andrzej Nowicki

Michał Byra

Annular phased array transducer for preclinical testing of anti-cancer drug efficacy on small animals

Keywords: spherical annular phased array transducer, pulsed HIFU beam, electronically adjustable focal length, local tissue heating, thermal ablation, necrotic lesion

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

Introduction.

Pulsed High Intensity Focused Ultrasound (HIFU) is a promising non-invasive technique, i. a.

for the thermo-ablative treatment of solid tumors located deep under the skin [START_REF] Ter Haar | Therapeutic applications of ultrasound[END_REF][START_REF] Miller | Overview of therapeutic ultrasound applications and safety consideratios[END_REF]. The depth at which the thermal lesion is desired to form depends on the geometry and acoustic parameters of the transducer as well as on geometry and acoustic and thermal properties of propagation media used.

An advantage of this technique is its ability to provide non-invasive damage of a small volume within tissue without damaging adjacent structures. This volume is determined by the dimensions of the ellipsoidal focal spot of the heating beam used and typically is of the order of 1•λ x 1•λ x K•λ [START_REF] Ebbini | Ultrasound-guided therapeutic focused ultrasound: Current status and future directions[END_REF],

where λ is the wavelength, and K denotes a constant determined by the transducer geometry (typical values are 5-7). The three-dimensional scanning of the tumor by the focal spot leads to thermal ablation of the entire tumor volume leaving surrounding healthy tissues intact. Acceptance of any promising new therapy and its widespread use in clinical practice depends on the accuracy and repeatability of treatments. An accuracy of treatment depends on 1) the accuracy of guidance of the focal spot of the heating beam [START_REF] Yang | Extracorporeal liver ablation using sonography-guided high intensity focused ultrasound[END_REF][START_REF] Hutchinson | The feasibility of MRI feedback control for intracavitary phased array hyperthermia treatments[END_REF][START_REF] Mcdannold | Usefulness of MR imaging-derived thermometry and dosimetry in determining the threshold for tissue damage induced by thermal surgery in rabbits[END_REF][START_REF] Kaye | Rapid MR-ARFI method for focal spot localization during focused ultrasound therapy[END_REF], 2) the accuracy of characterization of the acoustic field applied [START_REF] Ter Haar | Guidance on reporting ultrasound exposure conditions for bio-effects studies[END_REF], and 3) the accuracy of controlling of the ultrasound energy delivered [START_REF] Fry | Thereshold ultrasonic dosages for structural changes in the mammalian brain[END_REF][START_REF] Frizzel | Threshold dosages for damage to mammalian liver by high intensity focused ultrasound[END_REF][START_REF] Sapareto | Thermal dose determination in cancer therapy[END_REF]. Biological effects induced in tissues exposed to the high intensity focused ultrasound beams are typically the result of heating and mechanical effects caused by inertial cavitation.

The main objective of this study was to design, manufacture and test a spherical ultrasound annular array transducer suitable for preclinical studies on efficacy of thermal ablation and of tested anti-cancer drugs activated by HIFU beams in the treatment of a variety of solid tumors implanted to various organs of small animals at the depth of the order of 1-2 cm under the skin. The transducer testing consisted of examining the impact of geometry and the acoustic parameters of each beam generated on location and extent of a local necrotic lesion induced within a tested tissue: 1) average acoustic power, 2) focal length, and 3) exposure time. The results of these studies were necessary to assess the feasibility of application of such beams for preclinical studies on small animals as well as to explain the characteristics of thermal lesions formed in the tested tissue ex vivo.

Materials and methods

To generate the heating beams of the selected acoustic power and different focal lengths we designed and built the spherical 7-element annular phased array transducer. To measure temperature rises induced locally in the tested tissue sample by the heating beams the custom-made experimental setup was designed and constructed. To visualize the thermal lesions formed within the tissue an ultrasound scanner equipped with a linear imaging probe as well as a magnetic resonance scanner were used. Each tissue sample was exposed to three parallel HIFU beams of the same: 6W, or 12W, or 18W, average acoustic power and 25mm, 30mm, and 35mm focal lengths.

HIFU transducer

Both, the pulsed nonlinear acoustic pressure and temperature fields were produced by the custom-made, spherical, 7-element annular phased array transducer having an external diameter of 29 mm, resonance frequency of 2 MHz and f-number of 2, as shown in Fig. 1.

Fig. 1

The transducer was made of Pz26 piezo-ceramics (Meggitt, Kvistgaard, Denmark) exhibiting very low aging rate (˂ 1% per time decade) and extremely stable performance (over several years).

The transducer was air-backed and had one quarter-wavelength matching layer. Each element of the HIFU transducer had the same radiating surface. All elements generated tone bursts with 20-cycle duration, 0.2 duty-cycle and programmable time delay providing the beam focusing in water at three axial distances 25 mm, 30 mm, and 35 mm from the center of the transducer.

Experimental setup

The experimental setup along with its scheme and cross-section are shown, respectively in Fig. 2 and Fig. 1. All measurements were carried out at 36 o C.

Fig. 2

As shown in Fig. 1 and Fig. 2 a custom-made transmitting electronics (IPPT PAN, Warsaw, Poland) contained seven broadband power amplifiers driven by seven 14-bit 80-MHz AD9717 converters (Analog Devices, Norwood, MA, USA) controlled by a programmable Altera Arria EP1AGX FPGA logic system (Altera Corporation, San Jose, CA, USA). The amplifiers excited the elements of the annular phased array transducer to produce the heating beam. The voltage applied to each individual element was adjusted between 89 V pp and 150V pp to generate focused beams with the same average acoustic power and increasing focal lengths. The acoustic power for each beam was measured using an UPM-DT-1E ultrasound power meter (Ohmic Instruments, St. Charles, MO, USA). Each set of time delays, required to focus the beam energy within the tissue sample at the selected depth, was processed using an Altera Quartus software (Altera Corporation, San Jose, CA, USA). The annular array transducer was mounted rigidly in a wall of a container filled with distilled water of a temperature controlled by an ME AC 190 thermostat (Julabo Laborthechnik GmbH, Seelbach, Germany). The waveform of tone bursts generated by transducer elements were recorded using a TDS3012B digital oscilloscope (Tektronix, Beaverton, USA).

Pulsed HIFU beams were generated by the annular phased array transducer designed, as shown in Fig. 2. The tone bursts propagated through a two-layer media system consisting of a 15-mm layer of water and a 40-mm layer of pork loin. The thickness of the water layer was selected using our nonlinear propagation model [START_REF] Wójcik | Wave envelopes method for description of nonlinear acoustic wave propagation[END_REF] in water as the axial distance from the transducer used at which the amplitude of the second harmonic component starts to be discernible. As shown in previous publication [START_REF] Kujawska | Determination of nonlinear medium parameter B/A using model assisted variable-length measurement approach[END_REF] this distance is constant and independent on the source pressure level providing formation of weak or moderate nonlinear fields. To provide an uniform pressure distribution on the surface of the annular array transducer its rings had the same radiating aperture area as the central disk. The focal length of each HIFU beam was controlled electronically by using programmable time delays to the pulsed excitation of the transducer elements. A set of time delays for each focal length was determined by measurements of pressure waveforms generated in water by both, the central disc and consecutive rings using a calibrated S/N 1661 needle hydrophone (Precision Acoustics, Dorchester, UK). The axial position of the maximum heating spot and exposure time needed to locally heat the tissue to induce necrosis, corresponding to a 56 °C temperature, were determined from measurements of temperature by TP-201 thermocouples (Czaki Termo-Product, Raszyn, Poland) placed along the acoustic axis of each HIFU beam. After exposure of each tissue sample to three HIFU beams with the same acoustic power and different focal lengths the necrotic lesions within the tissue at three depths were visualized using both, a Sonix Touch ultrasound scanner (Ultrasonix, British Columbia, Canada) and a Biospec 70/30USR magnetic resonance imaging system (Bruker BioSpin Corporation, Billerica, MA, USA). Consiquently, the location and extent of the necrotic lesions was estimated and compared with those predicted using numerical model. The experimental verification of the obtained results was carried out optically with a digital caliper after cutting the tissue sample along the plane containing the three axes of the HIFU beams used.

Acoustic pressure/intensity field characterization

The beam patterns for the transducer used were characterized numerically and measured in water at room temperature in free-field conditions using a S/N 1661 calibrated 0.2 mm needle hydrophone (Precision Acoustics, Dorchester, Dorset, UK). The initial (source) and focal (in the focal plane) pressure amplitudes were determined from predicted spatial acoustic pressure distributions using our nonlinear propagation model [START_REF] Wójcik | Wave envelopes method for description of nonlinear acoustic wave propagation[END_REF] in water and then their fitting to those measured for each beam used. For the beams with the same 6W, or 12W, or 18W acoustic power and 25mm, 30mm, and 35 mm focal lengths the source pressure was found to be equal to 0.369 MPa, 0.522 MPa and 0.639 MPa with an uncertainty of about ± 4.4 % and the initial intensity I SATA of the beam was found to be equal to 0.9 W/cm 2 , 1.8 W/cm 2 , and 2.7 W/cm 2 with an uncertainty of about ± 9 %, respectively. The focal pressure was found to be equal to 8.4 MPa, 6.8 MPa and 5.5

MPa for 6W beams, 12.8 MPa, 10.5 MPa and 8.5 MPa for 12W beams and 17.2 MPa, 14.2 MPa and 11.7 MPa for 18W beams, respectively.

The focal intensity I SATA for each beam produced within the tested tissue was determined from numerical simulations of spatial pressure distributions in the beams propagating in the two-layer media system: water/tissue (15mm/40mm) using our nonlinear propagation model [START_REF] Wójcik | Wave envelopes method for description of nonlinear acoustic wave propagation[END_REF] required the following input parameters whose values were taken from literature: the pork loin density ρ 0 = 1060 kg/m 3 [START_REF] Duck | Physical properties of tissue: a comprehensive reference book[END_REF], sound velocity c = 1620 m/s [START_REF] Koch | Ultrasound velocity and attenuation of porcine soft tissues with respect to structure and composition: I[END_REF], attenuation coefficient α = 11•10 -6 Np/(m•Hz b ) [START_REF] Koch | Ultrasound velocity and attenuation of porcine soft tissues with respect to structure and composition: I[END_REF], power index of attenuation dependence on frequency b = 1 [START_REF] Nassiri | Attenuation of ultrasound in skeletal muscle[END_REF], nonlinearity parameter B/A = 8 [START_REF] Law | Determination of the nonlinearity parameter B/A of biological media[END_REF] and thermal conductivity K t = 0.52 W/(m• o C) [START_REF] Kujawska | Determination of tissue thermal conductivity by measuring and modeling temperature rise induced in tissue by pulsed focused ultrasound[END_REF]. As mentioned above, each tissue sample was exposed to three parallel HIFU beams with the same average acoustic power and different focal lengths. The distance between axes of the beams was chosen to be 10 mm (in some cases 3mm).

The beams were focused within each sample at 10mm, 15mm, and 20mm depths under the water/tissue interface. The calculated values of the focal intensity I SATA in three beams used were found to be equal to 410 W/cm 2 , 260 W/cm 2 , and 170 W/cm 2 for 6W beams; 940 W/cm 2 , 630 W/cm 2 , and 410 W/cm 2 for 12W beams, and 1690 W/cm 2 , 1150 W/cm 2 , and 780 W/cm 2 for 18W beams, respectively.

Temperature measurement protocol

Measurements of temperature induced locally in each pork loin sample by three parallel HIFU beams with the same acoustic power and increasing focal length were carried out using the experimental setup shown in Fig. 1. Both, the ultrasound and magnetic resonance imaging of the three necrotic lesions formed within each tissue sample as well as its cutting were performed after exposure to the three heating beams as is shown in Fig. 3. As already mentioned, the path-length of the pulse propagation in water before penetrating into the tissue was selected from axial harmonic pressure distributions calculated for nonlinear HIFU beams produced in water. For the heating beams used here this distance was found to be 15 mm. Selection of thickness of the water layer in two-layer media system used was done to maximize the effect of the harmonic waves generation in the focal region. Their attenuation increase nonlinearly with frequency being one of the reasons of thermal effects induced within tissue.

Temperature rises induced in each tested tissue sample by three HIFU beams used were measured by TP-201 thermocouples (Czaki Termo-Product, Raszyn, Poland) with a diameter of 0.2 mm. The thermocouples were placed inside the tissue sample along the heating beam axis using thin 0.5 mm, hypodermic needles inserted through a cover of the water container in steps of 5 mm, providing their exact position on the axis of each beam used. The uncertainty of position of the thermocouples was ± 0.5 mm. Due to a small diameter their influence on measurement results were considered negligible. The temperature rise detected by each thermocouple was recorded in steps of 1 second by USB-TEMP unit (Measurement Computing, Norton, USA) and transferred to the PC memory. To process data obtained and to visualize the temperature-time plots a TracerDAQ software (MicroDAQ.com Ltd, Contoocook, NH, USA) was used. The maximum temperature rise determined the position of the maximum heating spot, which, in turn, determined the depth of location of the thermal lesion. Example of the temperature rises ΔT(t) induced in the tested tissue during its exposure to three HIFU beams and measured by thermocouples placed along the acoustic axis of each beam is shown in Fig. 4.

Fig. 4

After three exposures the necrotic lesions induced in each tissue sample were visualized using both, the ultrasound and magnetic resonance imaging techniques. Then, the tissue sample, being inside the holder, was sliced carefully along the plane containing three beam axes to make the lesions optically visible and to verify manually their location and dimensions with accuracy within ± 0.1 mm using digital caliper.

Ultrasound and magnetic resonance imaging of necrotic lesions

For ultrasound imaging of the necrotic lesions a Sonix Touch scanner (Ultrasonix, British Columbia, Canada) equipped with a L14-5/38 linear phased array probe operating at a 10 MHz frequency was used. Ultrasound imaging was done immediately after three exposures of each sample. Magnetic resonance imaging was done 30 ± 5 minutes after three exposures using Biospec 70/30USR high-field (7T) magnetic resonance scanner for small animals (Bruker BioSpin Corporation, Billerica, MA, USA).

Results and discussion

Quantitative analysis of spatial acoustic pressure/intensity distributions in the pulsed nonlinear beams produced in the two-layer media system: water/tissue and predicted using our nonlinear propagation model allowed to show that for the beams with the same acoustic power and increasing focal length the maximum pressure decreases, while the full width half maximum (FWHM) dimensions of the ellipsoidal focal volume increases. For example, for 6W beams the maximum pressure decreased from 8.5 MPa to 5.5 MPa, while dimensions of the ellipsoidal focal volume increased from 1.1mm x 1.1mm x 6.8mm for the beam of a 25mm focal length to 1.4mm x 1.4mm x 11.2mm for the beam of a 35mm focal length. It means that focal volume increased about 3 times.

Meanwhile, for 3 times more powerful (18W) beams the maximum pressure decreased from 17.2MPa to 11.7MPa, while ellipsoidal focal dimensions increased from 0.9mm x 0.9mm x 6.4mm for the beam of 25-mm focal length to 1.1 mm x 1.1 mm x 10 mm for the beam of the 35-mm focal length. So, focal volume increased about 2 times. Comparing the dimensions of the focal volume for the beams with the same focal lengths and increasing power we have shown that the higher the acoustic power, the smaller sizes of the focal volume. This is probably due to the shorter exposure time necessary for beams of the greater power to heat tissue locally to 56 °C, which in turn leads to a reduced heat diffusion.

Quantitative analysis of the temperature rises, induced locally in each pork loin sample during its exposure to the three HIFU beams and measured by thermocouples, revealed that for 6W beams the axial positions of the maximum heating within tissue were approximately 2.5 mm shallower under water/tissue interface than those resulting from the geometry, i.e. 10mm, 15mm, and 20mm, respectively. In the case of 18W beams the axial positions of the maximum heating corresponded to those resulting from the geometry. For the beams of 12W the axial position of the maximum heating for the beam with 25-mm focal length was found at the same depth as resulting from the geometry, meanwhile, as for the beam with 30-mm focal length it was about 2.5 mm shallower under the tissue penetration surface (between 10mm and 15mm), similar as for the beam with 35mm focal length (between 15mm and 20mm). This is illustrated in Fig. 4. The exposure time for each beam used was determined from the temperature-time ΔT(t) relationship as the duration required to heat locally the tested tissue to 56 °C leading to the formation of the necrotic lesion.

The measurement results allowed to show that the larger the focal length of the beam, the longer the exposure time is required to form the necrotic lesion (to heat locally the tested tissue to 56 °C) as shown in Fig. 4. This results from the lower focal intensity for the beams with the larger focal length and from increased heat diffusion during longer exposure needed to induce necrosis. This, in turn, leads to expansion of the heated tissue volume and, thus, to increased extension of the necrotic lesion. Considering that the essence of thermo-ablative anti-tumor HIFU therapy involves the rapid (less than 3 seconds) heating of a small volume within a tumor to a temperature of 56 °C and scanning the beam focus throughout the entire tumor we showed that only the beams with an output exceeding 12W were able to meet this condition. As can be seen from Fig. 4 the exposure time needed to heat locally the tested tissue to 56 o C varied between 2 second for the beam of 25-mm focal length and 3.8 seconds for the beam of 35-mm focal length.

On the ultrasound images of the axial cross-section (xz plane) of each sample after its exposure to the three HIFU beams with the same acoustic power and increasing focal lengths, the necrotic lesions were visible, although their contours were not clearly defined because of blurry edges, as shown in Fig. 5. The sizes of thermal lesions visible on the ultrasound images were larger than those measured by caliper. Probable cause of this is insufficient rate of temperature increase to form a clear boundary between necrotic lesion and undamaged tissue structures as well as to induce in tissue gas bubbles improving the contrast of ultrasound imaging due to the induction of inertial cavitation.

Fig. 5

However, after cutting each sample along the xz plane, the necrotic lesions with clearly outlined edges were visible to the naked eye, as shown in Fig. 6.

Fig. 6

The magnetic resonance images of axial cross-sections of each tissue sample in planes parallel to xz plane and spaced apart by 0.5 mm showed much more clearly defined boundaries of the necrotic lesions, as can be seen Fig. 7.

Fig. 7

Optical visualization and measuring of three thermal lesions allowed to verify experimentally their extent and location under the water/tissue interface. This was done by comparison the location and dimensions of the necrotic lesions measured by digital caliper after cutting each sample with those predicted numerically as well as with those imaged using ultrasound and magnetic resonance imaging techniques. Results of comparison of the calculated sizes of thermal lesions with those measured by the caliper have shown that both, the nonlinear propagation and bio-heat transfer models predict well the acoustic pressure and temperature fields produced in the two-layer media system used. The agreement between the calculation results and measurement data was within ± 9 %. The sizes of thermal lesions visible on the ultrasound images were not in good correlation with those measured by caliper, although their location corresponded to measured ones, as shown in Fig. 6. Meanwhile, the location and extent of thermal lesions visible on the magnetic resonance images fully corresponded to those measured by caliper, as demonstrated in Fig. 7.

Conclusions

In conclusion, we have shown, that it is feasible to manufacture spherical annular phased array transducer with electronically adjustable focal length suitable for preclinical studies of efficacy of tested, anti-cancer, ultrasound-activated drugs in the treatment of a variety of primary solid tumors localized in various organs of small animals at the depth of the order of 1-2 cm under the skin. Our experiments revealed the encouraging results showing the feasibility of inducing within tissues local necrotic lesions using the designed, custom-made, spherical, 7-element annular array transducer generating HIFU beams with electronically adjustable focal length. The necrotic lesions with clearly outlined edges were formed within pork loin tissue ex vivo at desired depths starting from the beams of 12W acoustic power. Optical inspection of the axial cross-section of each sample exposed to the three HIFU beams with the same acoustic power and increasing focal lengths confirmed that the location and extent of the local necrotic lesions induced within the tested tissue are in good correlation with those predicted by numerical modeling and verified using both, the ultrasound and magnetic resonance imaging techniques as well as optical revision using digital caliper after slicing the tissue sample.

The necrotic lesions had clearly outlined edges. On the basis of measurement data we showed that for exposure parameters used, the location and extent of the necrotic lesions induced in the tested tissue ex vivo strongly depend on the acoustic power, focal length and exposure time of the heating beam used. This is important for preclinical applications proposed.

The theoretical and experimental studies described here indicated that pulsed HIFU beams with electronically adjustable focal length, generated by the annular array transducer designed can be used as an effective, non-invasive tool useful for both, 1) the thermo-ablative treatment of primary solid tumors localized within various organs of small animals at the depths of 1-2cm under the skin and 2) their treatment based on delivery of anti-cancer drugs activated by HIFU beams. However, further studies are needed to test the developed transducer on solid tumors in vivo implanted to small animals.

List of figure captions

Fig. 3

 3 Fig.3

Fig. 1 .

 1 Fig. 1. Scheme and cross-section of the experimental setup. The cylindrical chamber with the tissue

Fig. 2 .

 2 Fig. 2. Photo of the experimental setup for measuring temperature rises induced in the tested tissue

Fig. 3 .

 3 Fig. 3. Scheme of axial cross-section of a tissue sample exposed to three parallel HIFU beams of the same

Fig. 4 .

 4 Fig. 4. Tissue temperature measured at penetration depths of 5mm (red), 10mm (blue), 15mm (black),

Fig. 5 .

 5 Fig. 5. Ultrasound images of the tested tissue sample before (A) and after (B) exposure to the three

Fig. 6 .

 6 Fig. 6. The axial cross-sections of tested tissue samples exposed to the three parallel HIFU beams of

Fig. 7 .

 7 Fig. 7. Magnetic resonance images (T1) of axial cross-sections for the tested tissue exposed to three

Acknowledgments

The financial support of National Science Centre of Poland (Project 2011/01/B/ST7/06735) is gratefully acknowledged.


Damien Texier
email: damien.texier@mines-albi.fr

Maxime Ecochard

Thomas Gheno

Daniel Monceau

Philippe Mehdi Salem

Philippe Lours

Screening for Al2O3 failure in MCrAlY APS coatings using short-term oxidation at high temperature

Keywords:

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

Introduction

Nickel-based superalloys are widely used in the hot sections of turbomachines for aerospace and power plant applications. Their excellent mechanical properties at high temperatures and their good resistance to oxidizing and corrosive environments make them an ideal candidate for high temperature structural applications [1][START_REF] Reed | The Superalloys: Fundamentals and Applications[END_REF][START_REF] Young | High Temperature Oxidation and Corrosion of Metals[END_REF]. However, their surface reactivity at high temperature can be a limiting factor, especially with the increase in turbine inlet temperature and the severity of thermal cycling. MCrAlY (where M = Ni, Co or Ni/Co) or aluminide coatings are commonly used in the hottest parts of the turbines to protect the superalloy [START_REF] Mévrel | [END_REF][START_REF] Rhys-Jones | Coatings for Blade and Vane Applications in Gas Turbines[END_REF][START_REF] Bose | High Temperature Coatings[END_REF]. At temperatures above ~ 1000 • C, MCrAlY coatings develop a slow-growing, compact and adherent α-Al 2 O 3 scale, which provides the required protection [START_REF] Evans | [END_REF]8]. However, their oxidation resistance can be a limiting factor due to the limited thickness of coatings, i. e., the "reservoir" of Al to form the protective α-Al 2 O 3 scale.

MCrAlY overlay coatings can be deposited using various line-of-sight or non-line-of-sight deposition processes depending on the geometrical complexity of the component to be coated. Extensive usage of projection processes makes some of these deposition techniques affordable and MCrAlY coatings have thus demonstrated a very interesting trade-off between a low-cost process and excellent protection for various industrial applications [9]. MCrAlY coatings are widely applied to first and second stage turbine blades and nozzle guide vanes, compressors, turbine shrouds, and turbine blades, i.e., for components exceeding 850 • C. Ever-more severe operating conditions are increasingly critical for the materials that are used, as cyclic loading promotes mechano-chemical coupling and the onset of early strain-and oxidation-assisted damage and cracking at the microstructure scale [10][11][12][13][14]. Repeated through-thickness cracking and/or spallation of the protective Al 2 O 3 scale can bring the local Al concentration down to a critical-healing level for which reforming Al 2 O 3 would no longer be possible [15,16]. This mechanism associated with change in oxidation products due to thermal and/or mechanical cracking of the protective oxide scale is termed Mechanically Induced Chemical Failure (MICF). Below the critical Al concentration, Al 2 O 3 failure results in the rapid growth of oxides of the base elements (Ni, Co) [17,18,8]. This local change in oxidation behavior is highly detrimental for the integrity of the bond-coat itself. The occurrence of this localized breakaway can concomitantly be accelerated by coating-thick Al depletion due to interdiffusion with the underlying alloy [19,20,8].

In components for which weight is of concern, MCrAlY coatings can be as thin as tens of micrometers. During prolonged exposure, oxidationinduced depletion can then extend to the whole coating section, to a point where the Al reservoir is fully consumed and Al 2 O 3 is no longer the most stable oxide in contact with the metal. Failure of the protective oxide in this case has been called "intrinsic chemical failure (InCF)" by Evans [16]. It is usually accompanied by an abrupt acceleration of the mass gain, associated with the growth of base metal oxides [21,22,18,12,16,[23][24][25], in a process termed generalized breakaway. In some cases, however, if the protective scale is mechanically intact, its chemical failure can lead to the slow growth of the next most stable oxide (Cr 2 O 3 in the case of (Al,Cr)-rich materials) at the alloy/scale interface, with kinetics limited by diffusion across the initial scale [16,23,24]. This can extend the coating lifetime before breakaway eventually occurs.

Localized and generalized breakaway are highly detrimental to the integrity of the bond-coat and underlying alloy, and even more damaging in thermal barrier coating (TBC) systems, where the bondcoat is overlaid with a thermal-insulating ceramic topcoat [26,27]. In TBCs, fast oxide growth at the bond-coat/top-coat interface may cause the top-coat to spall, thereby exposing the metal parts to excessive temperatures. The investigation of breakaway events, especially generalized breakaway, normally requires very-long-term exposure at high temperature (thousands of hours). A method for fast screening of breakaway events would be useful to accelerate materials development in association with numerical simulations [28]. Experimental testing can be accelerated using ultrathin specimens, with thicknesses comparable with the diffusion length of the alloying elements, i.e., in the order of tens of micrometers.

In the present study, oxidation experiments were performed at 1150 • C on ultrathin freestanding NiCoCrAlY coating specimens prepared with a dedicated precision jig [29]. InCF, i.e., the necessary condition for further generalized breakaway, was observed in hours in the thinnest specimens due to through-thickness Al depletion. Interrupted oxidation tests were performed to document the onset and progression of the generalized breakaway as a function of the exposure time. Thermogravimetric analyses were conducted to study the mass variation associated with the formation of the oxidation products. X-ray diffraction, chemical and metallographic analyses were carried out to identify the oxidation products in relation with the composition of the metal substrate.

Materials and experimental procedures

Material and specimen preparation

MCrAlY coatings used in the present study were prepared by air plasma spray (APS) deposition process by PRAXAIR© research laboratory. The NiCoCrAlY powder was the commercial NI-191-4 powder from PRAXAIR© [30]. The size distribution of the NiCoCrAlY powder is centered at 45 ± 16 μm (spherical particles). The powder composition is reported in Table 1. The coating was initially deposited on a 304 stainless steel plate with dimensions of 50 × 250 × 2 mm 3 . No surface treatment was applied on the 304 stainless steel plate to favor natural debonding of the NiCoCrAlY coating from this substrate plate during the deposition process. Indeed, substantial strains generated by the APS process of thick coating resulted in the coating delamination during the deposition process, leading to 550-650 μm thick freestanding NiC-oCrAlY plates. Then, 50 × 25 mm 2 samples were cut from the freestanding NiCoCrAlY coating and subjected to a standard heat treatment under air: 8 h at 1080 • C followed by 20 h at 870 • C. The metallurgical state of the APS coating after this standard heat treatment is hereafter denoted "as-received" ("AR") microstructure in the present paper. This standard heat treatment is generally applied for improving the coating adhesion onto Ni-based superalloy substrates due to chemical interdiffusion and for optimization of the mechanical properties of the Ni-based superalloy substrates [31]. Despite the use of freestanding coating samples in the present paper, this heat-treatment was performed to investigate the oxidation behavior of the industrially used metallurgical state and microstructure of the NiCoCrAlY APS coating. The "AR" microstructure of the APS processed NiCoCrAlY coating consists in a fine γ/β microstructure with a 41% γ-59% β surface fraction proportion (obtained with image analyses on backscattered electron (BSE) images). Typical features of sprayed materials are observed, i.e., melted and resolidified powder particles (MRPP), un-melted powder particle (UMPP), pores, Al 2 O 3 dispersoids and AlN precipitates located in the intersplat regions. AlN precipitates were found in a larger proportion compared to Al 2 O 3 dispersoids. Samples for oxidation testing were cut with a precision cutting machine to the final dimensions 10 × 15 mm 2 . They were then thinned using a custom precision jig and SiC papers down to a P4000 grit, to a final thickness of either 21 ± 3, 31 ± 3, 59 ± 3, 90 ± 3 or 238 ± 3 μm. Five samples were prepared for each thickness range (that is, 25 samples in total). The sample thicknesses will be referred to as "~ XX μm" for samples with a thickness of XX ± 3 μm. The thickness variation along each specimen was about ±2-3 μm. The polishing procedure is described in detail in Ref. [29]. The sample thicknesses are comparable with the diffusion length of the alloying elements in the metal after relatively short exposure times (≤300 h) at 1150 • C. The samples were cleaned with acetone and ethanol. The thicknesses were measured using a micrometer with an uncertainty of ± 1 μm, and weights were measured with a precision balance (Sartorius MC5), with an uncertainty of ± 1.0 × 10 -6 g. Each specimen was photographed using a calibrated binocular magnifying glass (Leica DMS 300) to measure the sample surface, S, necessary to determine the evolution of the specific mass Δm(t)/S.

Oxidation testing

Two types of oxidation experiments were performed, both at 1150 • C: (i) thermogravimetric analysis (TGA) was used to continuously record mass variations, and (ii) an interrupted furnace test was used to observe local changes in the oxidation behavior associated with mass variation. TGA tests were performed for 300 hours using a Setaram Setsys balance. The sample was suspended using a platinum wire. These tests were interrupted once at 150 h to observe the sample surface. They were conducted under either static or flowing air, in order to evaluate the effect of air and oxygen replenishment on the oxidation kinetics.

The interrupted furnace test was conducted in a Nabertherm N 11/H furnace in laboratory air (static air) for a cumulative time of up to 500 hours. The specimens were hung on alumina supports using holders made of platinum wire. The specimens were weighed and their surface observed by optical and scanning electron microscopy periodically (after 5, 10, 15, 20, 50, 100, 150, 250, 350, and 500 h). Some were removed from the test at an interruption and used for X-ray diffraction (XRD) and cross-sectional observations using scanning electron microscopy (SEM).

Microstructural and oxidation product analyses

Surface and cross-sectional observations of the oxidized samples were conducted using optical microscopy (OM) and scanning electron microscopy (SEM) in backscattered electron mode. Based on the optical semi-transparency of the α-Al 2 O 3 scale in the visible spectrum, optical microscopy aimed to investigate potential change in oxidation products at the surface and/or at the α-Al 2 O 3 /metal interface. In contrast, α-Al 2 O 3 being opaque to electrons, solely surface observations could be conducted using scanning electron microscopy (SEM). In this study, SEM observations were conducted with a tabletop SEM Hitachi TM 3030 Plus for routine analyses of the external surface between interrupted oxidation tests, a Nova NanoSEM 450 field-emission gun SEM (FEG-SEM) from Thermo Fischer Scientific and a JSM 7800F Prime FEG-SEM from JEOL for cross-sectional observations. Ten cross-sectional micrographs were taken for each thickness/exposure duration variants with the same SEM operating conditions (tension = 10 keV, probe current = 1.5 nA, working distance = 6 mm) for image analyses using Fiji software [32] to segment the different oxides in presence and their thickness evolution. The Nova NanoSEM 450 FEG-SEM was equipped with a GENESIS APEX 2i EDS detector from EDAX for chemical profiles and mapping. Focused ion beam (FIB) cross-sectional preparations and observations were performed using a Dual-Beam FEG-SEM HELIOS 600i from Thermo Fischer Scientific equipped with a SDD X-Max 80 mm 2 detector and EDX Aztec Advanced (Oxford Instruments).

X-Ray diffraction analyses

X-Ray Diffraction (XRD) analyses of the oxidized samples were performed with a diffractometer X'Pert PANALYTICAL from Philips using a Cu-Kα radiation (λ = 1.54 Å) for the identification of crystallographic structure of the oxides. X-ray scans were carried out in a range of angles (2θ) from 15 • to 100 • with a step size of 0.033 • and an acquisition time per step of 200 s (linear detector with a range of 2θ = 2.1 • ).

Results

High temperature oxidation kinetics

Furnace tests and TGA were performed on NiCoCrAlY APS freestanding coating samples with different thicknesses ranging from 24 to 238 μm at 1150 • C during 300 h. The mass gain is plotted as a function of the time for both types of oxidation tests in Fig. 1(a). The 227 μm-thick sample oxidized under air flow is considered as a reference sample since it was not subject to InCF in the investigated time range. The mass variation, Δm(t)/S, of the reference sample progressively increased up to 1.3 mg⋅cm -2 after 300 h. A log-log representation of the mass gain evolution is given in Fig. 1(b) to analyze the oxidation kinetics based on a n th root rate law, i.e., a power rate law with exponent 1/n. During the first 2.5 h of oxidation, the 227 μm-thick sample followed nearly parabolic kinetics (n ≈ 2). The mass variation then deviated from a parabolic behavior and the exponent stabilized at n = 3.9. The reasons underlying this change in oxidation kinetics will be discussed in Section 4. As will be shown, both short-term and long-term oxidation kinetics can be described using a single logarithmic rate law. The 238 μm-thick sample oxidized using furnace tests follows a n th root rate law with n = 2.4. Some variability in oxidation kinetics was noticed for furnace tests.

TGA and furnace tests were also conducted on ultrathin samples, i.e., on volume-limited samples, to trigger InCF. For the 29 μm-thick sample oxidized under air flow (light blue curve in Fig. 1), the mass gain was very similar to the reference behavior during the first [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30] h. After this period, the mass gain progressively slowed down up to 100 h to reach a maximal value of 0.85 mg⋅cm -2 . Assuming that only Al 2 O 3 formed in the first stage of oxidation, the mass variation of Al, Δm Al (t), can be expressed as:

2Al + 3 2 O 2 ⟶Al 2 O 3 (1)
Δm Al (t) S = 2 3 M Al M O Δm(t) S ( 2
)
with Δm(t), the recorded mass gain corresponding to the oxygen uptake, M Al , the molar mass of aluminum, M O , the molar mass of oxygen, and S, the sample surface. The maximal value of the mass gain (0.85 mg⋅cm -2 ), if only attributed to the addition of oxygen to form the Al 2 O 3 scale, corresponds to the consumption of 106% of the Al content initially present in the 29 μm-thick sample. Furthermore, a decrease in mass was observed for exposure times longer than 100 h (0.60 mg⋅cm -2 at 290 h, i.e., a decrease of 29.5% of the maximal mass gain for this sample). Similar behaviors were observed for thin and thick samples using static air furnace tests. However, no apparent mass loss was measured using discontinuous weight measurements under static air. A stabilization of the mass gain was observed for samples with a thickness of 24 μm and 33 μm. Similar to the ultrathin TGA sample, the maximum mass gain corresponds to the whole consumption of Al to form Al 2 O 3 . A TGA test using static air was conducted on a 16 μm-thick sample for comparison with the furnace tests (dark blue curve in Fig. 1). During the first tens of hours, the oxidation behavior of the sample was similar to the one of samples tested under air flow condition. However, a stabilization of the mass gain (or even a slight increase) is observed when 100% of the Al of the sample is oxidized. It is worth reminding that mass loss was observed for the ultrathin sample tested under air flow (light blue curve in Fig. 1) once 100% of the Al content of the sample was oxidized.

Surface observations at different exposure times

Optical macrographs of the sample surface are presented in Fig. 2.

Samples thicker than 61 μm present a homogeneous light gray surface, typical of an external Al 2 O 3 scale. Some oxide spallation (bright spots on the sample surface) is observed due to cooling at each interruption. In contrast, darker gray regions are observed for 24 to 33 μm-thick samples in this time window. This variation in optical properties is associated

Surface observations of the oxidized samples before and after InCF

The sample surfaces were examined using SEM in order to identify the oxides present before and after InCF.

X-ray diffraction analyses of the oxidized samples

X-ray diffraction analyses were performed on oxidized samples with a thickness of 21 ± 3, 31 ± 3 and 238 ± 3 μm after cumulative oxidation exposures of 20, 50, 100 and 250 h (Fig. 6). The ratio of intensity of the {104} Al 2 O 3 peak (red curves) and {104} Cr 2 O 3 peak (blue curves) related to the {111} γ-Ni peak was calculated in order to qualitatively document the thickness evolution of the oxide scale (Fig. 7).

For the reference sample exclusively forming an Al 2 O 3 scale, the relative intensity of the {104} Al 2 O 3 peak increases with the exposure time with a trend very similar to the one of the mass gain evolution. For

Cross-sectional analyses of the oxidized samples

Cross-sectional observations of the ~21 μm-thick samples oxidized during 20 and 50 h were conducted to document the evolution of the oxide scale and the underlying alloy microstructure before and after InCF. It is worth reminding that the "AR" sprayed NiCoCrAlY coating consisted in a γ/β microstructure with AlN precipitates and Al 2 O 3 dispersoids located in the bulk in intersplat regions. After 20 h, most of the sample surface is not affected by InCF, as illustrated in Figs. 2 and3 alloy. The Cr 2 O 3 contains small amounts of Al detected using spot analyses in coarse Cr 2 O 3 regions during EDS analyses (see EDS analysis in supplementary materials in "Supplementary note 4/Supplementary Table 3/p. 10"). In addition, chromium nitride precipitates are found sparsely distributed at the Cr 2 O 3 /metal interface. The crystal structure of this nitride phase could not be identified, as it was not present in sufficient amounts to give visible signal in the diffractograms (Fig. 6). However, EDS analysis showed that its composition corresponds to the CrN compound (see supplementary materials). Cr 2 O 3 as well as Y-rich oxide dispersoids are also observed in the bulk of the sample. Therefore, the darker regions evidenced from surface observations when InCF occurs correspond to the formation of the Cr 2 O 3 scale plus CrN precipitates beneath the external Al 2 O 3 scale (Fig. 2). The readers are referred to supplementary materials for additional EDS analysis of the oxide scale.

The evolution of the microstructure beneath the oxide scale and the composition of the oxide scale for each sample thickness ranging from 18 to 241 μm was illustrated in Fig. 9. Micrographs of the ~238 μm-thick samples clearly show no Al 2 O 3 failure, i.e., the reference condition. The growth of the Al 2 O 3 scale is accompanied by an extension of the β-NiAldepleted region with the exposure time, not affecting the sample core. As Analysis of the BSE micrographs aimed to quantitatively measure the evolution of oxide thicknesses but also the metal recession for several sample thickness-oxidation duration couples (Fig. 10). For the reference sample, i.e., the ~238 μm-thick samples, the Al 2 O 3 scale grew with a power law exponent of n ≈ 2.5, which is lower than the one identified from mass gain data. For the ~21 μmand ~31 μm-thick samples, the growth of the Al 2 O 3 scale is similar to the one of the reference samples as long as no Cr 2 O 3 scale/CrN precipitate is observed. Once Cr 2 O 3 /CrN develops, the thickness of the Al 2 O 3 scale stabilizes and even slightly decreases with the oxidation time. Furthermore, the Cr 2 O 3 /CrN scale was measured and its thickness was found to slightly decrease with the time after InCF occurrence, leading to a decrease of the overall external oxide scale thickness. In addition, the extent of metal recession is smaller than the Al 2 O 3 thickness for all exposure times for the ~238 μm-thick samples. It is worth noting that the extension of the metal recession is much greater than the growth of the Al 2 O 3 scale after InCF for ultrathin samples.

Chemical composition profiles before and after InCF

Ten EDS profiles were performed for each oxidation condition to assess the evolution of the composition within the metallic material before and after InCF for the ~21 μm-thick samples (Fig. 11). For information and comparison, the elemental composition of the γ phase of the 20 h oxidation condition has been reported as dashed lines in Fig. 11, with colors corresponding to the different elements. After 20 h, residual β-NiAl (Al-rich/Cr-poor regions) are observed in the bulk of the specimen, and InCF has not yet started. A slight gradient of composition is observed in the 2-3 μm-thick region near the Al 2 O 3 -metal interface. For information, InCF was seen to occur between 20 and 50 h for the ~21 μm-thick samples (see e.g. Fig. 3). After 50 h and longer, no β is left and Al has been fully consumed. The Cr content is found to be minimal (≈11 % (at.) at 50 h while the reference concentration is about two times higher). This modification in Cr content is partly attributed to Cr 2 O 3 and CrN when InCF occurs.

FIB cross-section observation of local InCF: (Ni,Co)-rich oxides

Focused ion beam (FIB)-cross sections of the (Ni,Co)-rich oxides located in islets of (Ni,Co)Al 2 O 4 spinels were performed (Fig. 12). The contour of the external oxide has been marked with white dashed lines to differentiate FIB cross-sectional observations from surface observations on the tilted view. The morphology of the external oxide scale beneath the (Ni,Co)-rich oxide observed on the FIB-cross section is quite complex, consisting of different layers: (1) the original and protective external Al 2 O 3 scale, (2) intrusions of Al 2 O 3 within the bulk due to the splat microstructure of the APS coating (see e.g. Ref. [33]), (3) the Cr 2 O 3 layer growing after InCF, (4) (Ni,Co)Al 2 O 4 spinels corresponding to the light gray oxide islets shown on the BSE surface observations, (5) (Ni, Co)-rich oxides being the last stage of InCF observed in the present study, and (6) some remaining metallic regions. Such (Ni,Co)-rich oxides were only observed in regions presenting Al 2 O 3 intrusion, i.e., severe intersplat oxidation.

Discussion

The motivation of the present paper was to predict general InCF of ultrathin samples using oxidation properties obtained on bulk samples of an Al 2 O 3 -forming MCrAlY alloy. Ultrathin samples with different thicknesses, i.e., with different aluminum reservoirs, were thus investigated in order to document the incubation time for the onset of the InCF.

Testing ultrathin samples, i.e., thinner than 60 μm, was demonstrated to activate InCF in a reasonable time at operating temperatures (tens of hours). The identification of the oxidation kinetics and related aluminum consumption is highly important both for short-and longterm oxidation for further prediction of the InCF event. Material, technical and mechanistic issues are addressed in the following sections.

Identification of the oxidation behavior and time to InCF

The oxidation kinetics of the MCrAlY samples were evaluated using both mass gain (Fig. 1) and oxide thickness data (Fig. 10). A direct comparison between the two types of data can be made by converting oxide thicknesses into equivalent mass. Assuming that the Al 2 O 3 is compact and exempt of pores, the mass gain that corresponds to the growth of a thickness ξ Al2O3 of Al 2 O 3 is given by:

Δm(t) S = 3⋅ M O M Al2O3 ⋅ρ Al2O3 ⋅ξ Al2O3 (3)
with M Al2O3 and ρ Al2O3 the molar mass and density of Al 2 O 3 . The mass gains obtained from oxide scale thicknesses measured on the ~238 μmthick samples are plotted along with the TGA data from the 227 μm-thick sample in Fig. 13(a). The mass gains estimated from thickness measurements are lower than those directly measured by TGA. This difference presumably arises from oxidation within the bulk of the MCrAlY material, in intersplat regions (see Figs. 7 and 8 from Ref. [33]). While not included in external scale thicknesses, intersplat oxidation contributes to the mass gain, and in turn to Al consumption [34,26,[35][36][37][38][39]33]. As a first approximation, its contribution can be estimated as the difference between both sets of results. It is seen to gradually decrease over the course of oxidation, as it represents about 46%, 37%, 27%, and 12% of the mass gain for oxidation times of 20, 50, 100 and 250 h, respectively.

Intersplat oxidation causes the oxidation behavior of the APSprocessed MCrAlY material to deviate from purely parabolic kinetics. A defect-free material would tend toward the kinetics obtained from thickness measurements. It is noted that the latter also exhibits a subparabolic behavior (n = 2.5 in a power law, i.e., between parabolic and cubic kinetics). This deviation likely finds its source in microstructural evolutions of the Al 2 O 3 [40]: at 1150 • C, Al 2 O 3 grain growth becomes significant, while scale growth still primarily occurs by grain boundary diffusion. This tends to accelerate the decrease of the oxidation rate, i.e., to produce sub-parabolic kinetics. The oxidation kinetics may also be affected by the initial formation of metastable Al 2 O 3 (θ-and γ-Al 2 O 3 ), which grow orders of magnitude faster than α-Al 2 O 3 [41][42][43][44][45],

although their presence at 1150 • C would be short-lived [46]. The size of the phase domains in the β/γ microstructure is also reported to affect the formation of transition oxides in the early stages [47,48].

The time at which InCF occurs can be predicted provided that a model describing the consumption kinetics of Al is available. A good fit of the TGA data recorded with the 227 μm-thick sample oxidized under flowing air can be obtained using either a power law,

Δm(t) S = ̅̅̅̅̅̅̅ k n ⋅t n √ , (4)
with n = 3.9 for t > 2.5 h, or a logarithmic law,

Δm(t) S = k l1 ⋅ln(k l2 ⋅t + 1), (5)
with a single set of constants k l1 and k l2 for the whole duration of the experiment (Fig. 13(a)). On the other hand, scale thickening kinetics measured on the ~238 μm-thick samples are best described by a power law (Eq. ( 4)) with n = 2.5. The adjusted values of the parameters for both models are reported in Table 2. Alumina fails when the sample's Al is fully consumed (Figs. 3 and11). Using Eq. ( 2), the corresponding mass gain, Δm(t InCF ), can be expressed as a function of the initial Al mass:

1 M O Δm(t InCF ) S = 3 2⋅M Al m 0 Al S (6)
Given the dimensions of ultrathin samples, the base area is about half the surface area: A ≈ S/2. For a given oxidation rate law, the time to InCF can thus be expressed as a function of the initial sample thickness. One obtains:

t InCF = 1 k n ( 3⋅M O 4⋅M Al ⋅ρ MCrAlY ⋅ξ MCrAlY ⋅w Al
) n [START_REF] Evans | [END_REF] for a power rate law, and:

t InCF = 1 k lt ( exp ( 3⋅M O 4⋅M Al ⋅ ρ MCrAlY ⋅ξ MCrAlY ⋅w Al k lm ) - 1
) ( 8
)
for a logarithmic rate law, where w Al is the initial Al mass fraction. The evolution of t InCF as a function of the initial thickness of the MCrAlY sample is plotted in Fig. 13(b) for the different models. Improper identification of the oxidation kinetics for short-and long-term exposures can lead to important difference in the estimated t InCF for ultrathin products. For instance, t InCF is about 3.5 times greater when estimated from a model adjusted on the scale thickness data, versus TGA data, for a The difference progressively decreases with the sample thickness for samples thinner than 45 μm. For thicker samples, on the contrary, t InCF estimated from thickness data is lower than the one obtained from TGA data, and the difference gradually increases with the sample thickness. Estimations of t InCF based on external scale thickness measurements miss the contribution of intersplat oxidation in the Al mass balance, and are therefore less suitable than estimations based on mass gain data. In terms of kinetics modeling, the logarithmic rate law has no physical justification here, but offers a good description of the TGA data over the whole course of the 300 h experiment with a single set of parameters. On the other hand, a power rate law only gives a good fit to the TGA data after a transition regime (t = 2.5 h). The prediction of t InCF is based on the consumption of Al integrated from the start of oxidation. Therefore, the logarithmic model is found to be better suited to the purpose of estimating t InCF .

In the present study, samples thinner than 60 μm exhibited InCF in the investigated t < 300 h window, as reported in Figs. 2 and3 . Experimental and calculated values of t InCF are compared for the ~21 μmand ~31 μm-thick samples in Fig. 14. Tens of hours were needed for both sample sizes for InCF to spread over the whole surface. Therefore, different categories have been defined in Fig. 14 depending on the progress of InCF: no InCF, less and more than 10 % of the sample surface affected by InCF. Experimental results are in good agreement with the logarithmic model adjusted on TGA data.

It is reminded that the time to InCF estimated from a model adjusted on oxide thickness data corresponds to a defect-free sample, without oxide intrusions. A defect-free sample would be estimated to withstand 3.6 times, 2.5 times, and 1.4 times longer than this APS-processed sample for an initial thickness of 20 μm, 30 μm, and 40 μm, respectively. Intersplat oxidation is highly process-dependent and can be significantly avoided using additional treatments to close intersplat pores and channels [49] or by improving parameters of the deposition/projection processes and post-heat treatment [36,39]. Based on the present estimations, i.e., considering only oxidation-induced InCF, improving process parameters so as to reduce intersplat oxidation could potentially offer significant gains in coating lifetime.

Ultrathin sample preparation and time to InCF

InCF was found to take hours and even tens of hours to fully affect ultrathin samples (Figs. 2 and3). As noticed in Fig. 14, slight changes in sample thickness could highly affect the time at which this change in oxidation products occurs. In the present paper, ultrathin samples were polished using a precision jig to control geometrical tolerances of the samples and more precisely the thickness variation. Centimeter-size but micrometer-thick samples are needed to accurately measure the evolution of the mass gain due to oxidation. However, the preparation of such samples is challenging and leads to local variation in thickness. This local variation in thickness were measured to be ± 2-3 μm over the whole surface sample, as reported in Ref. [29,[START_REF] Texier | Mesure et évolution des gradients de propriétés mécaniques dans le système superalliages à base de nickel MC2 revêtu MCrAlY[END_REF]. The variation in thickness could arise from a mismounting of the samples on the glass support (parallelism and curvature), a defect when grinding of the precision jig, edge effects leading to rounded edges. The thickness distribution function of a sample can thus vary from one sample to another. This distribution function was not assessed prior to oxidation for all the samples due to technical reasons and only the average thickness was taken into account. For the sake of simplicity, the thickness of the samples is considered to follow a normal distribution with a standard deviation of 1 μm. Therefore, the probability density function can be expressed as:

f (ξ MCrAlY ) = 1 σ ̅̅̅̅̅ 2π √ exp ( - (ξ MCrAlY -ξ MCrAlY ) 2 2σ 2 ) ( 9
)
with ξ MCrAlY the local thickness of the sample, ξ MCrAlY the average thickness, and σ the standard deviation in thickness (1 μm). The cumulative density function can thus be expressed as follows:

F(ξ MCrAlY ) = 1 2 [ 1 + erf ( ξ MCrAlY -ξ MCrAlY σ ̅̅ ̅ 2 √ ) ] (10)
Assuming that the lateral extension of the InCF on the sample surface is not driven by a nucleation/growth phenomenon and only depends on the local thickness of the sample, ξ MCrAlY can be expressed as a function of t InCF using Eq. ( 8):

ξ MCrAlY = 4⋅M Al ⋅k lm 3⋅M O ⋅ρ MCrAlY ⋅w Al ⋅ln(k lt ⋅t InCF + 1) (11)
A cumulative density function of InCF time, F(t InCF ), can then be obtained by substituting Eq. ( 11) in Eq. ( 10). This function F(t InCF ) is a model of the surface fraction of the sample affected by InCF. The experimental and modeled fractions of InCF are plotted as a function of An additional source of variability in the evolution of the surface coverage of post-InCF material is found in the coating microstructure. As mentioned earlier, intersplat oxide intrusions are present in the specimens due to oxygen ingress in intersplat channels during high temperature exposure (e.g. Fig. 8 from Ref. [33]). Intersplat oxidation consumes Al locally; this affects the onset of InCF locally, and therefore contributes to variability in the overall t InCF distribution. Investigations of Al 2 O 3 failure in thin foils of commercial alloys have been reported in the literature, e.g. in Refs. [21,23,24,[START_REF] Al-Badairy | [END_REF]18,[52][53][54][55]. In these studies, the samples used for oxidation testing were primarily produced by rolling, as they would be in a number of practical applications. Rolling provides good control over the sample dimensions, but also produces particular microstructures (with strong morphological and crystallographic texture), which may not necessarily be wanted, depending on the application. In the present study, the samples were prepared by gentle mechanical polishing. This technique has a lesser effect on the alloy microstructure, and can therefore be used to study a wide variety of materials. The samples used here were centimeter-size with thicknesses down to 15 ± 2-3 μm. A yet better control of the thickness distribution would help future work on protective oxide failure and its numerical simulation.

InCF: Mechanisms and kinetics

The criterion to activate InCF was found to correspond to the full consumption of the aluminum present in the MCrAlY material (Fig. 11). At this stage, the microstructure of the MCrAlY material is fully γ. No remaining β phase was observed in the bulk of the sample (Figs. 8 and9). As aforementioned, both external oxidation and oxidation in the volume of the material were observed to participate in the aluminum consumption. Prior to InCF, the external oxide scale was constituted of α-Al 2 O 3 with few islets of (Ni,Co)Al 2 O 4 spinels sparsely distributed at the external surface (Fig. 4). The external Al 2 O 3 layer was found to follow a sub-parabolic oxidation behavior, i.e., lying between a parabolic and cubic oxidation behavior, as depicted in Figs. 10 and [54]. They also reported the formation of (Al,Cr) 2 O 3 beneath the external Al 2 O 3 scale during thermal cycling for a shorter exposure time (500 h) in regions subjected to oxide cracking, an illustration of the accelerating role of mechanical damage on Al 2 O 3 failure (MICF). While the formation of a Cr 2 O 3 layer beneath the Al 2 O 3 scale was already reported in the literature, the formation of CrN precipitates at the oxide/metal interface due to InCF under isothermal oxidation at high temperature was not reported yet, to the authors' knowledge. Aluminum consumption in the alloy eventually leads to the dissolution of the AlN precipitates. This releases N, which can then diffuse toward the alloy/scale interface to react with Cr and form CrN. In addition, N may also come from the atmosphere, and reach the alloy/scale interface by diffusion/transport through the Al 2 O 3 scale after InCF. Indeed, apparently compact Al 2 O 3 scales have been observed to lose their protective properties and become permeable to N after chemical failure [58,59]. In addition to the formation of sub-surface Cr 2 O 3 plus CrN precipitates, the external oxide scale slightly differs after InCF. The network of (Ni,Co)Al 2 O 4 spinels at the external surface intensified, as illustrated in Fig. 4. Spinels were also reported for Al In the present work, the mass gain was found to progressively slow down for experiments conducted under static air, and a mass loss was recorded in experiments under air flow, after InCF and Cr 2 O 3 formation. This mass loss is likely to be due to Cr diffusion across the Al 2 O 3 and Cr 2 O 3 reactive volatilization [61] at the scale surface, which would be insignificant in a static atmosphere. Cr-depletion was also observed after the onset of Cr 2 O 3 formation using EDS profile measurements within the MCrAlY material (Fig. 11). Ni and Co gradually decreased to participate in the spinel formation. Similar mass gain behavior were reported for other Al 2 O 3 -or Cr 2 O 3 -forming materials, as long as the protective oxide scale remains undamaged, compact, dense and adherent [23,24,52,53].

Oxide formation is also accompanied by the metal recession due to consumption of elements participating in the growth of the oxide scale. In the present paper, the metal recession prior to InCF was found to extend less than the thickness of the oxide scale (Fig. 10). This statement is in good agreement with the Pilling-Bedworth ratio of the Al 2 O 3 for NiCoCrAlY alloys. After InCF, the metal recession significantly increased and is even more important than the growth of the oxide scale. This finding is not in good agreement with a Pilling-Bedworth ratio approach since R Cr2O3 PB is greater than R Al2O3 PB . Strehl et al. also reported a significant metal recession of thin FeCrAl foils [24]. They tried to relate this intense metal recession with the potential elongation of the sample due to creep during isothermal oxidation, stress being inherent to the growth of the oxide willing to expand more than the consumed metal. The to oxide spallation. Samples thinner than 60 μm did not show any traces of oxide spallation while thicker samples did under interrupted oxidation tests (Fig. 2). Such resistance of ultrathin samples to oxide spallation due to strain accommodation (sample curvature for instance) was also reported by Orosz et al. [62] and Duan et al. [55]. Despite this point is in favor of ultrathin samples, i.e., thin foils, the same material used as a thin coating could yet be subjected to spallation due to the substrate stiffness. Therefore, both assessment of oxide spallation kinetics and progressive consumption of aluminum/chromium are required to better predict InCF/MICF under real service conditions. In the present study, APS-processed MCrAlY were polished to investigate the oxidation resistance of ultrathin products at high temperature. However, APS-processed MCrAlY generally presents a rough surface inherent to the deposition process. Oxidation of a rough surface can lead to different oxidation products and rates depending on the curvature of the outer surface. Onset of breakaway associated with oxide micro-cracking in rough regions was also reported [63,64]. This could be attributed to non-uniform consumption of Y and Al beneath the rough surfaces and repeated-cracking/re-growth of the Al 2 O 3 scale during temperature cycling.

Conclusion

The oxidation resistance of an APS-processed MCrAlY material was investigated at 1150 • C using freestanding coating samples of various thicknesses. This screening technique was proposed to assess intrinsic chemical failure (InCF) of ultrathin samples for short-term oxidation testing. The following conclusions can be drawn from the present work:

• InCF occurred after the alloy was fully depleted of its Al, and led to the formation of a Cr

Table 2

Parameters of the oxidation models adjusted on TGA and oxide thickness data obtained from the reference samples (227 μmand ~238 μm-thick samples).

Fig. 1 .

 1 Fig. 1. (a) Mass gain evolution of a NiCoCrAlY APS freestanding coating samples with different thicknesses ranging from 24 to 238 μm. Solid lines correspond to TGA tests conducted on the thinnest and thickest samples. Markers correspond to furnace tests and associated dashed lines are drawn for reading convenience, (b) Log-log representation of the mass gain evolution as a function of the time.

Fig. 2 .

 2 Fig. 2. Optical macrographs of oxidized samples with various thicknesses at different oxidation times showing InCF for samples thinner than 61 μm (dark gray regions) and Al 2 O 3 spallation for thicker samples (nearly white spots). The onset of InCF is highlighted with green arrows.

Fig. 3 .

 3 Fig. 3. (a) Evolution of the fraction surface of the InCF affected region as a function of the time for various thin samples. (b) Evolution of the fraction surface of the InCF affected region as a function of the relative aluminum consumption considering that the mass gain is attributed to the formation of exclusively Al 2 O 3 .

  the ~21 μmand ~31 μm-thick samples, the relative intensity of the Al 2 O 3 peak also increases with the exposure time with a different behavior compared to that of the reference sample. Prior to InCF, the relative intensity of the Al 2 O 3 peak is similar to the one of the reference sample. However, the relative intensity of the Al 2 O 3 peak progressively increases when Cr 2 O 3 is formed. The increase of the Al 2 O 3 peak after InCF does not necessarily signify a growth of the Al 2 O 3 scale since the intensity of the Ni peak is weakened by the growth of the underlying Cr 2 O 3 scale. The slope of the Al 2 O 3 signal as a function of the exposure time is similar for both the ~21 μmand ~31 μm-thick samples when Cr 2 O 3 is formed. The relative intensity of the Cr 2 O 3 peak also increases with the time in a linear manner when InCF is observed.

  . In regions not affected by InCF, an external Al 2 O 3 scale exclusively forms, as shown on EDS maps in Fig. 8. Residual β-NiAl regions (Al-rich/Crpoor regions on EDS micrographs and medium gray regions on BSE micrographs) and AlN precipitates are observed in the bulk of the specimen, showing that the Al reservoir is not fully exhausted yet. After 50 h, a continuous Cr 2 O 3 layer (medium gray in BSE micrograph) has developed beneath the Al 2 O 3 scale and no β-NiAl or AlN is left in the

Fig. 4 .

 4 Fig. 4. SEM observations in a backscattered electron mode of the external surface of the oxidized samples before (~61 μm-thick sample) and after (~21 μm-thick sample) InCF showing some islets of (Ni,Co)Al 2 O 4 spinels and changes in the external oxide scale.

Fig. 5 .

 5 Fig. 5. Observation of (Ni,Co)-rich oxides as the final stage of the InCF growing on (Ni,Co)Al 2 O 4 islets after 100 h at 1150 • C for a ~21 μm-thick sample. (a) Surface observation in backscattered electron mode, (b) Element maps on surface observations.

Fig. 6 .

 6 Fig. 6. XRD diagrams for samples with a thickness of (a) ~21 μm, (b) ~31 μm, and (c) ~238 μm oxidized during 20, 50, 100 and 250 h.

Fig. 7 .

 7 Fig. 7. Relative peak intensity of the α-Al 2 O 3 {104} and Cr 2 O 3 {104} peaks compared to the γ-Ni {111} peak for above-mentioned oxidized samples.

Fig. 8 .

 8 Fig. 8. Cross-sectional observations of the ~21 μm-thick samples oxidized during 20 and 50 h before and after InCF using SEM micrographs in BSE mode and Al and Cr element maps from EDS analyses. Cr 2 O 3 and CrN precipitates developed beneath the Al 2 O 3 scale when the remaining metallic material is depleted from β-NiAl phases and AlN precipitates. White arrows on the right of the Cr element maps show the initial thickness of the sample before oxidation.

Fig. 9 .

 9 Fig. 9. Evolution of the oxide scale and the microstructure beneath the oxide scale after oxidation during 20 and 250 h for three different sample thicknesses (SEM observations in a backscaterred electron mode). Blue boxes on the right of the micrographs show the initial thickness of the sample before oxidation.

  13 (a). Internal Al 2 O 3 dispersoids and AlN precipitates were also found in intersplat regions. After InCF, the reaction microstructure evolves, as Cr 2 O 3 develops beneath the Al 2 O 3 scale and CrN forms beneath the Cr 2 O 3 (Figs.8 and 9), while AlN is no longer present in the coating. Some Al is found in the newly formed Cr 2 O 3 (Fig.11and supplementary materials). Indeed, Cr 2 O 3 and Al 2 O 3 show substantial mutual miscibility above 1000 • C [56,57]. Chyrkin et al. reported the formation of an (Al, Cr) 2 O 3 oxide beneath the external Al 2 O 3 near the specimen edge and above the external Al 2 O 3 in the central region of the specimen after isothermal oxidation of thin Haynes 214 foils at 1200 • C for 720 h

Fig. 10 .

 10 Fig. 10. Evolution of the thickness of the oxide scale and the extension of the metal recession zone as a function of the time for the ~21 μm-thick samples, ~31 μm-thick samples and ~238 μm-thick samples.

Fig. 11 .

 11 Fig. 11. EDS profiles showing the relative composition of Al, Cr, Co, Ni, and O in the γ-phase. Profiles were plotted for ~21 μm-thick samples for exposure times of 20, 50, 100, and 250 h at 1150 • C. The chemical composition of the γ phase in the 20 h oxidized sample was depicted as horizontal dashed lines for comparison.

Fig. 12 .

 12 Fig. 12. SEM observation of (Ni,Co)-rich oxides as the final stage of the InCF growing on (Ni,Co)Al 2 O 4 islets for this oxidation conditions and material. (a) Surface observation in backscattered electron mode (the green dashed line corresponds to the FIB section trace), (b) Micrographs in secondary electron mode and element maps of the FIB-cross-section view.

Fig. 13 .

 13 Fig. 13. (a) Comparison of oxidation kinetics based on TGA data and oxide thickness measurements, and (b) Estimation of the time to InCF based on the indicated oxidation models.

  2 O 3 layer at the Al 2 O 3 -alloy interface, of subsurface CrN precipitates, and of (Ni,Co)(Al,Cr) 2 O 4 islands at the surface of the Al 2 O 3 . • The mass gain did not significantly accelerate following InCF, due to the fact that the Al 2 O 3 layer remained mechanically intact. The Al 2 O 3 lost its protective character in that it allowed the oxidation of Cr, Ni and Co, but it retained a beneficial role in limiting their diffusion rate; • The lamellar microstructure of the APS-processed material resulted in intersplat oxidation, which locally accelerated Al consumption and thus the occurrence of InCF. Therefore, improvements in the material microstructure could significantly delay InCF in sprayed materials; • Intense metal recession was found after InCF without significant size variation of the external surface area; • Ultrathin samples were less prone to oxide spallation than thicker samples, due to their higher compliance; • The identification of the oxidation kinetics on bulk samples is necessary to well predict the time to InCF on ultrathin samples; • The present study demonstrated the interest of using ultrathin samples to rapidly screen onset of InCF for lifetime evaluations of materials under isothermal or thermal cycling conditions.

Table 1

 1

	Nominal composition of the NI-191-4 powder used to deposit the NiCoCrAlY
	coating [30] (in %(at) and %(wt))
	Element	Ni	Co	Cr	Al	Y
	x i (at.%)	Balance	18.8	16.5	23.0	0.3
	wi (wt.%)	Balance	22.0	17.0	12.3	0.5

  The intensity of the XRD diagrams was normalized by the intensity of the γ-Ni peak correspond-

	ing to {111} planes. The ~238 μm-thick samples exclusively formed
	α-Al 2 O 3 at all time steps examined, with the relative intensity of the
	{104} Al 2 O 3 peak gradually increasing with time. For the ~21 and ~31
	μm-thick samples, Al 2 O 3 is observed for all the exposure durations and
	Cr 2 O 3 is identified at 50 h and 100 h, respectively. Neither (Ni,Co)-rich
	oxides nor (Ni,Co)Al 2 O 4 spinels are identified after InCF on XRD dia-
	grams owing to their sparse occurrence.

  2 O 3 -forming alloys even without the growth of the sub-surface Cr 2 O 3 , such as Ni(Al,Cr) 2 O 4 , (Ni,Co)Al 2 O 4 and (Ni,Co)(Al, Cr) 2 O 4 spinels [47,35,53,54,39]. Furthermore, NiO was found to form in

Acknowledgements

This work was supported by the Agence Nationale de la Recherche (ANR) [ANR-18-CE08-0003; ANR-JCJC-COMPAACT project funded from the AAPG2018]. The authors are grateful to Stephane Le Blond Du Plouy and Claudie Josse from the Raymond Castaing Microanalysis Centre (Mixed Services Unit 3623) for access to microscopy facilities and assistance. The authors are grateful to Karim CHOQUET and Geoffrey SALVI (ICA-IMT Mines Albi) and Remi ROUMIGUIER (MIDIVAL) for technical assistance.

Data

Data availability

The raw/processed data required to reproduce these findings cannot be shared at this time due to technical or time limitations.

Declaration of Competing Interest

The authors report no declarations of interest.

Appendix A. Supplementary data

Supplementary data associated with this article can be found, in the online version, at https://doi.org/10.1016/j.corsci.2021.109334.