Keywords: Output feedback, High-gain observers, Cascade ODE-PDE systems

In this paper it is proposed to extend the result described in Khalil and Praly (2014) and the references therein, regarding the high-gain observer-based nonlinear control to the case of systems with diffusion sensor dynamic. Based on some usual hypotheses, we provide sufficient conditions involving the high-gain parameter and the length on the PDE sensor. In fact it is brought into light an explicit trade off between them: the larger the observer gain, the smaller the length of the PDE sensor needs to be. The stability analysis of the closed loop system is based on a Lyapunov functional.

High-gain Observer-based Output Feedback Control with Sensor Dynamic Governed by Parabolic PDE

Tarek Ahmed-Ali * Francoise Lamnabhi-Lagarrigue * * Hassan K. Khalil * * *

INTRODUCTION

The present paper deals with the design of output feedback control for a class of nonlinear cascade ODE-PDE systems. Throughout the past decades, the high-gain observers have been used extensively for the design of output feedback control of nonlinear systems, see [START_REF] Khalil | High-gain observers in nonlinear feedback control[END_REF] and the references therein. An important advantage of using the high-gain observers is that they can recover the performances of state feedback control in the sense that, for instance, the trajectories of the system under output feedback approach those under state feedback as the observer gain increases.

Moreover, from the work [START_REF] Atassi | A separation principle for the stabilization of a class of nonlinear systems[END_REF], it is well known that the separation principle holds not only because the observer gain is made high, but also because, by designing the feedback control as a globally bounded function, the state of the plant is protected from the peaking phenomenon [START_REF] Khalil | Nonlinear Systems[END_REF] when the high-gain observer estimates are used instead of the true states.

In fact, the design of output feedback control for cascade ODE-PDE systems has not been yet very much studied, one can cite for instance the work [START_REF] Krstic | Compensating actuator and sensor dynamics governed by diffusion PDEs[END_REF] for a linear ODE or the work [START_REF] Wu | Observer design and output feedback stabilization for nonlinear multivariable systems with diffusion PDE-governed sensor dynamics[END_REF] where a nonlinear ODE is considered. However in this later work, the nonlinear ODE is restrictive and the method, based on LMIs, does not provide explicit conditions regarding the length that the PDE must satisfy in order to ensure global exponential stability of the overall system.

In the present paper we propose to extend the design of high-gain observer-based output feedback control for nonlinear systems with sensors described by heat PDEs. More precisely, we will derive explicit sufficient conditions, involving both the high-gain and the length of the PDE, ensuring exponential convergence of the overall closed cascade ODE-PDE. It has also to be noticed that the observer designed here is more simple than those designed in [START_REF] Ahmed-Ali | Observer design for a class of nonlinear ODE-PDE cascade systems[END_REF] and Ahmed-Ali et al. (2019a) for the same cascade ODE-PDE systems which used backstepping technics. A sampled-data version of our algorithm is also presented with its convergence conditions. A conference version of the present work has been presented at [START_REF] Ahmed-Ali | High-gain observer-based output feedback control with sensor dynamic governed by parabolic pde[END_REF]. The main difference between the two versions is in the the fact that this version contains supplementary results by treating the sampled-data case.

Notations and preliminaries

Throughout the paper the superscript T stands for matrix transposition, R n denotes the n-dimensional Euclidean space with vector norm |.|, I n is the n × n identity matrix, R n×m is the set of all n×m real matrices, and the notation P > 0, for P ∈ R n×n , means that P is symmetric and positive definite. In matrices, symmetric terms are denoted * ; λ min (P ) (resp.λ max (P )) denotes the smallest (resp. largest) eigenvalue. L 2 (0, D) is the Hilbert space of square integrable functions z :→ R with the corresponding norm

z L 2 = D 0 z 2 (x)dx. H k (0, D)
is the Sobolev space of functions z : [0, D] → R having k square integrable weak derivatives. Given a two-argument function u(x, t), then its partial derivatives are denoted u t = ∂u ∂t , u xx = ∂ 2 u ∂x 2 . u[t] and u x [t] refer to the functions defined on 0 ≤ x ≤ D by (u[t])(x) = u(x, t) and (u x [t])(x) = ∂u(x,t) ∂x . In this paper the Halany's type Inequalities, [START_REF] Fridman | Robust sampled-data control of a class of semilinear parabolic systems[END_REF] summarized in the following lemma will be used. Lemma 1. Let 0 < δ 1 < 2δ and let V : [t 0 -h, ∞) → [0, ∞) be an absolutely continuous function which satisfies 2α(t-t0) sup -h≤s≤0 V (s) where α is the unique positive solution of equation

V (t) ≤ -2δV (t) + δ 1 sup -h≤s≤0 V (t + s) Then V (t) ≤ e -
α = δ - δ 1 e 2αh 2
2. PROBLEM STATEMENT Let us consider the following class of systems in the state space R n × L 2 (0, D).

             Ẋi = X i+1 , i = 1, . . . , n -1 Ẋn = f (X, v) u(D, t) = X 1 u x (0, t) = 0 u t = u xx , x ∈ [0, D] y(t) = u(0, t) (1)
where v and y represent respectively the input and the output of the above system. Throughout the paper, we assume the following hypotheses: H1: The function f is continuous globally Lipschitz in both X and v with a Lipschitz constant K 0 . H2: : There exists a continuous globally Lipschitz function α(X), such that the following dynamical system Ẋi = X i+1 Ẋn = f (X, α(X))

is globally exponentially stable.

Using the converse Lyapunov Theorem [START_REF] Khalil | Nonlinear Systems[END_REF] we can say that there exists a function V 0 (X) > 0 and positive parameters c i , i = 1, . . . , 4 such that:

           c 1 |X| 2 ≤ V 0 (X) ≤ c 2 |X| 2 | ∂V0 ∂Xi | ≤ c 3 |X| n-1 i=1 ∂V0 ∂Xi X i+1 + ∂V0 ∂Xn f (X, α(X)) ≤ -c 4 |X| 2 (2)
for all X.

If we consider the changes of coordinates p(x, t) = u(x, t)-X 1 then the above system can be written as follows

             Ẋi = X i+1 , i = 1, . . . , n -1 Ẋn = f (X, v) p(D, t) = 0 p x (0, t) = 0 p t = p xx -X 2 , x ∈ [0, D] y(t) = p(0, t) + X 1 (3)

OUTPUT FEEDBACK DESIGN

Based on the above hypotheses, we propose the following high-gain observer-based output feedback control:

             Żi = Z i+1 -l i θ i (û(0, t) -y), i = 1, . . . , n -1 Żn = f (Z, v) -l n θ n (û(0, t) -y) v = α(Z) û(D, t) = Z 1 ûx (0, t) = 0 ût = ûxx -l 1 θ(û(0, t) -y) x ∈ [0, D] (4) The vector gains L = (l 1 , • • • , l n ) ∈ R n×1 is chosen such that the matrix (A -LC) is Hurwitz where A =           0 1 0 . . . . . . 0 0 0 0 1 0 . . . 0 0 0 0 0 1 0 . . . 0 . . . . . . . . . 0 1 0 . . . 0 0 . . . 0 0 1 0 0 0 . . . . . . 0 0 1 0 0 . . . 0 0 0 0           ∈ R n×2 and C = [1, 0, • • • , 0] ∈ R 1×n . Let us use the change of coordinates p(x, t) = û(x, t) -Z 1
then the observer (4) can be expressed as follows

             Żi = Z i+1 -l i θ i (û(0, t) -y), i = 1, . . . , n -1 Żn = f (Z, v) -l n θ n (û(0, t) -y) v = α(Z) p(D, t) = 0 px (0, t) = 0 pt = pxx -Z 2 x ∈ [0, D] (5)
Remark 2. The existence and uniqueness of solutions of system (5) can be analyzed by using Theorem 2.1 of [START_REF] Karafyllis | Iss in different norms forel 1-d parabolic pdes with boundary disturbances. siam journal on control and optimization[END_REF]. As in Remark 4 of [START_REF] Ahmed-Ali | Sampled boundary observer for strict-feedback nonlinear ode systems with parabolic pde sensor[END_REF] where the considered system is very similar to the one considered here, we can say that for every p

[0] ∈ C 2 (0, D) with (p[0])(D) = (p[0]) (0) = 0, there exists an unique solution p ∈ ([0, D] × C 0 [0, ∞)) ∩ ([0, D] × C 1 (0, ∞)) with p[t] ∈ C 2 (0, D)
for all t ≥ 0. We can also use the same arguments for system (3) to say that for every p

[0] ∈ C 2 (0, D) with (p[0])(D) = (p[0]) (0) = 0, there exists an unique solution p ∈ ([0, D] × C 0 [0, ∞)) ∩ ([0, D] × C 1 (0, ∞)) with p[t] ∈ C 2 (0, D) for all t ≥ 0 and η ∈ (C 1 (0, ∞); R n × R n ). From this we can also claim that both u ∈ ([0, D] × C 0 [0, ∞)) ∩ ([0, D] × C 1 (0, ∞)), u[t] ∈ C 2 (0, D), û ∈ ([0, D] × C 0 [0, ∞)) ∩ ([0, D] × C 1 (0, ∞)) and û[t] ∈ C 2 (0, D).
Let us consider the dynamical error system e = Z -X and ũ = û -u, then for any x ∈ [0, D], we obtain

                 ėi = e i+1 -l i θ i ũ(0, t), i = 1, . . . , n -1 ėn = f (Z, α(Z)) -f (X, α(Z)) -l n θ n ũ(0, t) Ẋi = X i+1 , i = 1, . . . , n -1 Ẋn = f (X, α(Z)) ũ(D, t) = e 1 ũx (0, t) = 0 ũt = ũxx -l 1 θũ(0, t), x ∈ [0, D] (6) Now if we introduce the change of coordinate w(x, t) = ũ(x, t) -e 1 , then we have                  ėi = e i+1 -l i θ i e 1 -l i θ i w(0, t) i = 1, . . . , n -1 ėn = f (Z, α(Z)) -f (X, α(Z)) -l n θ n e 1 -l n θ n w(0, t) Ẋi = X i+1 , i = 1, . . . , n -1 Ẋn = f (X, α(Z)) w(D, t) = 0 w x (0, t) = 0. w t = w xx -e 2 , , x ∈ [0, D]
(7) By using the classical change of coordinates

ξ i = θ 1-i e i , we derive                    ξi = θξ i+1 -θl i ξ 1 -l i θw(0, t), i = 1, . . . , n -1 ξn = θ 1-n [f (Z, α(Z)) -f (X, α(Z))] -θl n ξ 1 -θl n w(0, t) Ẋi = X i+1 , i = 1, . . . , n -1 Ẋn = f (X, α(Z)) 'w(D, t) = 0 w x (0, t) = 0 w t = w xx -θξ 2
Let us now introduce the following augmented vector state η = [ξ, X] T , where ξ = (ξ 1 , . . . , ξ n ) T . Then the above system can be written as

     η = F 0 (η, w(0, t)) w(D, t) = 0 w x (0, t) = 0 w t = w xx -θξ 2 (8)
where F 0 (η, w(0, t)) is given by

       θξ i+1 -θl i ξ 1 -l i θw(0, t), i = 1, . . . , n -1 θ 1-n [f (∆ξ + X, α(Z)) -f (X, α(Z))] -θlnξ 1 -θlnw(0, t) X i+1 , i = 1, . . . , n -1 f (X, α(∆ξ + X))        .
where ∆ = diag(1, . . . , θ n-1 ). Notice that since w(D, t) = 0, then w(0, t) = -D 0 w x (x, t)dx.

Theorem 3. Consider the system (1). Then for all (X(0),

Z(0)) ∈ (R n × R n ) and (u[0], û[0]) ∈ (C 2 (0, D) × C 2 (0, D)) with (u[0])(D) = X 1 (0), (u[0]) (0) = 0, (û[0])(D) = Z 1 (0), (û[0]
) (0) = 0, there exist θ 0 and D (θ) such that ∀θ > θ 0 and ∀D ∈ (0, D (θ)) the following inequality holds:

|X| + |e| + D 0 ũ2 (x, t)dx ≤ M 1 e -σ 1 t
for some positive constant M 1 > 0 and σ 1 > 0 where M 1 depends on the initial conditions.

Proof. In order to prove the exponential stability of the system (8), we will divide the proof in three parts: for the infinite dimensional sub-system, for the finite dimensional one and finally for the overall error system.

Infinite dimensional sub-system

In this first part we will analyse the sub-system

w t = w xx -θξ 2 w(D, t) = 0 w x (0, t) = 0 (9)
In order to do this, we consider the following functional :

W = 1 2 D 0 w 2 (x, t)dx + 1 2 D 0 w 2 x (x, t)dx
Then, the time derivative of W(t) along the trajectory of the system (5) is

Ẇ (t) = D 0 w(x, t)w t (x, t)dx + D 0 w x (x, t)w xt (x, t)dx = D 0 w(x, t)(w xx (x, t) -θξ 2 )dx + D 0 w x (x, t)w tx (x, t)dx
From the fact that w(D, t) = 0, then we have w t (D, t) = 0. From this and by using the integration by parts on [0, D], we can easily derive that

Ẇ = - D 0 w 2 x (x, t)dx -θ D 0 w(x, t)ξ 2 dx - D 0 w 2 xx (x, t)dx +θ D 0 w xx ξ 2 dx Using Young's inequality leads to Ẇ ≤ - D 0 w 2 x (x, t)dx - 1 2 D 0 w 2 xx (x, t)dx + 1 2 D 0 w 2 (x, t)dx + Dθ 2 |ξ 2 | 2
Using Wirtinger's inequality Fridman and Blighovsky (2012), we have

Ẇ ≤ -( π 2 4D 2 - 1 2 ) D 0 w 2 (x, t)dx - π 2 8D 2 D 0 w 2 x (x, t)dx + Dθ 2 |ξ 2 | 2 . Since |ξ 2 | ≤ |η|, then Ẇ ≤ -( π 2 4D 2 - 1 2 ) D 0 w 2 (x, t)dx - π 2 8D 2 D 0 w 2 x (x, t)dx + Dθ 2 |η| 2

Finite-dimensional system

Let us now analyse the unperturbed sub-system η = F 0 (η, 0).

By considering the following Lyapunov function,

V (η) = 1 θ 2(n-1) V 0 (X) + ξ T P ξ
where P is a positive definite symmetric matrix which satisfies P (A -LC) + (A -LC) T P = -I n then, we can derive the following inequalities satisfied by V :

   c 1 |η| 2 ≤ V (η) ≤ c 2 |η| 2 | ∂V ∂η | ≤ c 3 |η| where              c 1 = min c1 θ 2(n-1) , λ min (P ) c 2 = max c2 θ 2(n-1) , λ max (P ) c 3 = max c3 θ 2(n-1) , 2λ max (P )
with c 1 , c 2 and c 3 defined in (2). Now let us compute the derivative of V along the solution of the unperturbed system, we obtain, V = ∂V ∂η F 0 (η, 0).

After some computations as in [START_REF] Khalil | High-gain observers in nonlinear feedback control[END_REF], we can easily derive the following inequality:

V ≤ - c 4 2θ 2(n-1) |X| 2 -θ -2λ max (P )K 0 - 2c 2 3 c 4 K 2 0 |ξ| 2 .
for some positive constant K 0 independent of θ, and c 4 defined in (2).

Choosing θ > θ 0 such that

θ 0 = max 2λ max (P )K 0 + 2c 2 3 c 4 K 2 0 + c 4 2 , 1 (10) we deduce that V ≤ -c 4 |η| 2 with c 4 = c 4 2θ 2(n-1) .
From this, we deduce that the unperturbed system is globally uniformly exponentially stable. Now let us compute the derivative V along the sub-system η = F 0 (η, w(0, t)), we obtain,

V = ∂V ∂η F 0 (η, w(0, t))
which can be rewritten as follows:

V = ∂V ∂η F 0 (η, 0) + ∂V ∂η (F 0 (η, w(0, t)) -F 0 (η, 0))
Using the above inequalities, we easily derive that

V ≤ -c 4 |η| 2 + | ∂V ∂η || (F 0 (η, w(0, t)) -F 0 (η, 0)) |
Using again Young's inequality, this leads to:

V ≤ -c 4 |η| 2 + β| ∂V ∂η | 2 + 1 β | (F 0 (η, w(0, t)) -F 0 (η, 0)) | 2
for β > 0.

On the other hand, we can easily see that, from the definition of F 0 ,

| (F 0 (η, w(0, t)) -F 0 (η, 0)) | ≤ θ|L||w(0, t)|
which also gives

| (F 0 (η, w(0, t)) -F 0 (η, 0)) | ≤ θ|L| D 0 |w x (x, t)|dx
Now using the inequality of Cauchy-Schwarz, we obtain

| (F 0 (η, w(0, t)) -F 0 (η, 0)) | 2 ≤ θ 2 |L| 2 D D 0 w 2 x (x, t)dx
and from this we derive that

V ≤ -(c 4 -βc 2 3 )|η| 2 + 1 β |θ 2 |L| 2 D D 0 w 2 x (x, t)dx

Stability of the overall error system

At this stage, we now consider the Lyapunov functionnal W 1 = W + V . Then from the two previous parts, its time derivative satisfies the following inequality

Ẇ1 ≤ -c 4 -βc 3 2 -Dθ 2 |η| 2 - π 2 4D 2 - 1 2 D 0 w 2 (x, t)dx - π 2 8D 2 - 1 β θ 2 |L| 2 D D 0 w 2 x (x, t)dx (11)
For any > 0, we can also write :

Ẇ1 + 2 W 1 ≤ -c 4 -βc 3 2 -Dθ 2 -2 c 2 |η| 2 - π 2 4D 2 - 1 2 - D 0 w 2 (x, t)dx - π 2 8D 2 - 1 β θ 2 |L| 2 D - D 0 w 2 x (x, t)dx (12)
In order to ensure the exponential stability, it is sufficient to set → 0, and it is not difficult to see that we need to choose β sufficiently small so that c 4 -βc 3 2 > 0 and we have to choose D such that

D < D (θ) = min 1 θ 2 (c 4 -βc 3 2 ), π √ 2 , π 2 β 8θ 2 |L| 2 1/3 (13)
This ends the proof.

Remark:

The expression for D * (θ) shows that as θ increases, D * (θ) decreases. Hence there is a trade off between θ and D; the larger the observer gain θ, the smaller the length of the PDE sensor.

EXTENSION TO SAMPLED-DATA CASE

In this section, we propose an extension of the above result to the case where the output is available only at sampling instants t k :

0 = t 0 < t 1 < . . . , < t k < . . . , lim i→∞ t k = ∞ with t k+1 -t k ≤ h.
For this case, we propose the following high-gain observer-based output feedback control:

for t ∈ [t k , t k+1 ),              Żi = Z i+1 -l i θ i (û(0, t k ) -y t k ), i = 1, . . . , n -1 Żn = f (Z, v) -l n θ n (û(0, t k ) -y t k ) v = α(Z) û(D, t) = Z 1 ûx (0, t) = 0 ût = ûxx -l 1 θ(û(0, t k ) -y t k ) x ∈ [0, D]
(14) Using the same computations and change of coordinates as the continuous time case, we easily derive the following error system :

for t ∈ [t k , t k+1 )            ξi = θξ i+1 -θl i ξ 1 (t k ) -l i θw(0, t k ), i = 1, . . . , n -1 ξn = θ 1-n [f (Z, α(Z)) -f (X, α(Z))] -θl n ξ 1 (t k ) -θl n w(0, t k ) Ẋi = X i+1 , i = 1, . . . , n -1 Ẋn = f (X, α(Z))
and

w t = w xx -θξ 2 w(D, t) = 0 w x (0, t) = 0
Remark 4. The analysis of well posedness done in the above section (continuous-time case) remains valid for each interval [t k , t k+1 ). This allows us to also ensure the existence of solutions with the same properties than the continuous-time case.

Since w(D, t k ) = 0 then w(0, t k ) = -D 0 w x (x, t k )dx Then the above system can be obviously re-written as follows :

ξ = θ(A -LC)ξ -θLC(ξ(t k ) -ξ(t)) +bθ 1-n [f (∆ξ + X, α(Z)) -f (X, α(Z))] + θL D 0 wx(x, t k )dx, Ẋi = X i+1 , i = 1, . . . , n -1 Ẋn = f (X, α(Z)) wt = wxx -θξ 2 w(D, t) = 0 wx(0, t) = 0 ( 15
)
Where b T = (0, 0, . . . , 0, 1) n . Now, in order to study the convergence of the above error system, we consider the following Lyapunov functional inspired from the work of [START_REF] Selivanov | Observer-based input-to-state stabilization of networked control systems[END_REF] :

W 2 = W 1 + W 3 W 3 = W h 2 e 2δh t t i e 2δ(s-t) | ξ(s)| 2 ds - π 2 4 W t t i |ξ(s) -ξ(t i )| 2 ds (16)
where δ and W are two positive constants which will be defined latter. By using generalized Wirtinger's inequality, we easily deduce that both W 3 is nonnegative and does not grow at the jumps occurring at instants t i [START_REF] Selivanov | Observer-based input-to-state stabilization of networked control systems[END_REF]. Moreover we have

Ẇ3 + 2δW 3 ≤ W h 2 e 2δh | ξ(t)| 2 - π 2 4 W |ξ(t) -ξ(t i )| 2 (17)
Using Young's inequality, we have

| ξ(t)| 2 ≤ 4θ 2 [|A -LC| + K 0 ] 2 |ξ(t)| 2 + 4θ 2 |L| 2 |ξ(t) -ξ(t k )| 2 + 4θ 2 |L| 2 D D 0 w 2 x (x, t k )dx then Ẇ3 + 2δW 3 ≤ W h 2 e 2δh [|A -LC| + K 0 ] 2 |ξ(t)| 2 + W h 2 e 2δh 4θ 2 |L| 2 D D 0 w 2 x (x, t k )dx - π 2 4 W -W h 2 e 2δh 4θ 2 |L| 2 |ξ(t) -ξ(t i )| 2
Now, let us compute the time derivative of W 2 along systems (15), then after some simple computations, we deduce the following inequality

Ẇ2 ≤ -c 4 -δc 3 2 -Dθ 2 |η| 2 - π 2 4D 2 - 1 2 D 0 w 2 (x, t)dx - π 2 8D 2 D 0 w 2 x (x, t)dx + 2 δ θ 2 |L| 2 D D 0 w 2 x (x, t k )dx -2δW 3 + W h 2 e 2δh [|A -LC| + K 0 ] 2 |ξ(t)| 2 + W h 2 e 2δh 4θ 2 |L| 2 D D 0 w 2 x (x, t k )dx - π 2 4 W -W h 2 e 2δh 4θ 2 |L| 2 - 2 δ θ 2 |L| 2 |ξ(t) -ξ(t i )| 2 (18)
which also gives

Ẇ2 + 2δW 2 ≤ -c 4 -δc 3 2 -Dθ 2 -W h 2 e 2δh [|A -LC| + K 0 ] 2 -2δc 2 |η| 2 - π 2 4D 2 - 1 2 -δ D 0 w 2 (x, t)dx - π 2 8D 2 -δ D 0 w 2 x (x, t)dx + 2 δ θ 2 |L| 2 D + W h 2 e 2δh 4θ 2 |L| 2 D D 0 w 2 x (x, t k )dx
Combining this, with the above conditions, we deduce that for all θ, satisfying conditions (10), ( 21), and if

               δ = 1 θ 2n , W = 4 π 2 δ|L| 2 + 2 δ θ 2 |L| 2 D < D * (θ) = min δ 2 δ |L| 2 θ 2 +δ|L| 2 , δ θ 2 , π 2 √ 2δ , π 2 1 δ+ 1 2 W h 2 e 2δh 4θ 2 ≤ δ (24)
then system (15) is exponentially convergent. It has to be noticed that since the function W h 2 e 2δh 4θ 2 is an increasing function then, ∀θ, there exists h(θ) such that ∀h ∈ (0, h(θ)) the inequality W h 2 e 2δh 4θ 2 ≤ δ holds. From this we can state the following Theorem : Theorem 5. Consider system (1). Then, there exist θ 1 , D (θ), h(θ) such that ∀θ > θ 1 , ∀D ∈ (0, D (θ)) and ∀h ∈ (0, h(θ)), the observation error system (15) converges exponentially to zero.

CONCLUSION

An extension of the design of high-gain observer-based output feedback control for nonlinear systems with sensors described by heat PDEs has been proposed by bringing into light explicitly the expected trade-off between the gain of the observer and the length of the PDE. Further works will be undertaken in the same vein by i) considering actuators PDEs and ii) by relaxing the global exponential assumption. This later extension will require careful analysis of the peaking phenomenon. On the other hand it's well known that a heat equation can be decoupled into a finite dimensional unstable part and an infinite dimensional stable one. We are currently investigating how we can use this nice property in order to design a new kind of high observers for the system considered here and other classes of systems [START_REF] Orlov | Robust stabilization of infinite-dimensional systems using sliding-mode output feedback control[END_REF]. We will also investigate a sliding modes observer design for the above class of systems in presence of disturbances [START_REF] Dimassi | A sliding mode observer for a linear reactionconvectiondiffusion equation with disturbances[END_REF].


Types of molecules reported in Gilman et al.

The potential to detect wildfire-related species in the infrared

Introduction:

Wildfires are a cause for growing concern for air quality and the climate. They represent a large source of emissions of gas-phase and particulate species. As a consequence of global warming, the spatial extent, severity and duration of wildfires is likely to increase, with current indications suggesting that this may already be happening. Wildfires are a truly global phenomenon, with largescale occurrences on each continent besides Antarctica.

Wildfire plume composition:

Like all pyrolysis sources, as an emission source, wildfires are highly complex, with a large variety of primary and secondary particulates and gases. Particulates are not expected to be long-lived, and so have an important, yet mostly short-term impact on climate and air quality. Conversely the gas-phase species, which are composed of volatilized, pyrolyzed or partially oxidized molecules will exhibit a broad range of physical and chemical behaviours in the atmosphere.

Overlap between HITRAN and wildfire measurements:

Remote sensing of wildfire components will depend upon the availability of spectroscopic data. If the current contents of the HITRAN database of cross sections is compared with measurements of wildfire plumes and fire lab experiments, major differences are observed, including much larger numbers of unsaturated, benzenoid and furan compounds in wildfire scenarios.

Lifetimes in the atmosphere: Not all wildfire chemicals will survive transport, many of these components will be very shortlived such as furans, polyenes and other unsaturated species, which possess lifetimes of less than a day. Others could be long-lived, such as hydrogen cyanide (HCN), isocyanic acid (HNCO) and acetonitrile (CH 3 CN), which can persist for several years. Some species, such as the peroxyacyl nitrates, possess short lifetimes in the boundary layer, but may become long-lived as they are lofted. The main factors that govern the persistence of these chemicals are their reaction rates with atmospheric oxidants, their photolysis rates and their wet deposition rates.

Limb-sounding detection?

Making new measurements:

New measurements will be necessary in many cases, to improve our understanding of the long-range transport and possible transformation products of important wildfire chemical components. For this purpose, the laboratory facilities at CNRS-Orléans together with ACTRIS national platform, HELIOS, will be available to understand these processes further.

Potential screening approach and workflow for assessing candidate molecules for remote sensing:

With existing measurement databases and estimation techniques, it is possible to assess which types of chemical components are likely to survive transport and have the potential for long-term climatic effects as well as being amenable to limb-sounding detection. Such an approach would also identify which types of species are poorly understood and require new measurements. (Above): There are many possible loss processes in the atmosphere, but also 1000s of different wildfire emissions, we can therefore expect that many chemicals survive long-range transport. Chemicals containing nitrogen, sulphur and phosphorus are of particular interest since these are likely to have large impacts on the environment.

  FIGAERO-ToF-CIMS

  Institut de Combustion Aérothermique Réactivité Environnement (ICARE), Orléans, France HELIOS platform: Situated in Central France, HELIOS is the largest outdoor chamber in France, and the third largest in Europe. It is equipped with a broad range of instrumentation, including many online and off-line mass spectrometric methods, open-path spectroscopic methods and various wet-chemical approaches. This allows us to cover much of the complex parameter-space encountered in real-world emissions such as wildfire plumes. This includes rate coefficients, reaction mechanisms and absorption cross sections. (Left): Chemical lifetimes in the atmosphere are controlled by fundamental kinetic parameters, such as reaction rate coefficients (k), photolysis rates (J) and heterogeneous deposition rates. Parameters such as these can be determined in laboratory experiments, or through estimation.


Simon Chouteau

Maria Mitronika

Antoine Goullet

Mireille Richard-Plouet

Luc Stafford

Agnès Granier

Kinetics driving nanocomposite thin-film deposition in low-pressure misty plasma processes

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

After decades of surface engineering research by plasma deposition of functional thin films, attention has now shifted towards multifunctional coatings combining several properties in a single-layer system [START_REF] Tiginyanu | Nanostructures and Thin Films for Multifunctional Applications: Technology, Properties and Devices[END_REF][START_REF] Faustini | [END_REF][3] . One of the approaches to achieve multifunctionality consists in embedding nano-objects in a thin film matrix, creating a multiphase material commonly called nanocomposite [4][5][6] . The high surface-to-volume ratio and extended interface area in nanocomposite thin films allow for a wider range of material properties compared to macroscopically combined materials. Numerous ways were implemented to introduce nano-objects in plasma-deposited thin films, including: plasma-enhanced chemical vapor deposition (PECVD) from metal-organic precursors resulting in nanometer-scale metal inclusions in organic matrices 7,8 , magnetron cosputtering of two targets corresponding to the matrix and reinforcement material [9][10][11] , combinations of PECVD for matrix deposition and sputtering for nanoparticles deposition [12][13][14][15] , or direct injection of solid nanoparticles in plasma deposition systems through gas aggregation sources [16][17][18] . In all cases, the presence of solid nano-objects in the so-called "dusty" plasma results in additional processing challenges. For instance, the plasma undergoes a decrease in electron density due to electron collection by particles, which is accompanied by a rise in electron temperature to generate more electrons per each ionization collision [19][20][21] . The nano-objects can also be subjected to various plasma interactions, such as charging and agglomeration [21][START_REF] Lieberman | Principles of Plasma Discharges and Materials Processing: Lieberman/Plasma 2e[END_REF][START_REF] Lopez | [END_REF][24][25][26][27] . This last point is critical as it affects the final size and distribution of nano-objects. Since nanoscale objects tend to be more reactive as they get smaller, agglomeration remains a major obstacle to the synthesis of nanocomposite thin films with well-distributed small-scale nanoparticles (< 10 nm).

Recently, the use of liquid colloidal solutions sprayed as mists in PECVD systems proved to be an adequate strategy for the growth of high-quality nanocomposite thin films [28][29][30][31][32][33][34][35][36][37] . More specifically, the early works of Ross et al. 38 and Ogawa et al. 39,40 opened a new area of research with the direct injection of liquid precursors and colloidal solutions in low-pressure PECVD systems. The corpus on these particular experiments is still remarkably slight: only a handful of published articles exists to this day [38][39][40][41][42][43][44] . In these hybrid, aerosol-assisted plasma processes (or "misty plasma" processes, as first coined by Coppins 45,46 ), the colloidal droplets are used as a vector to deliver the nanoparticles from the injection system to the substrate. Ideally, the liquid solvent evaporates during transport, leaving nothing but the nanoparticles on the surface of the sample, which will be quickly covered by the continuous deposition of the matrix. In other words, the droplets "feed" nanoparticles to the nanocomposite, and the plasma rids the sample of solvent residue and provides the matrix material by fragmentation of a gaseous precursor. Since agglomeration in the gas phase is prevented by the dispersion of the nanoparticles in liquid droplets, this strategy is well suited for the growth of nanocomposite thin films containing small, nonaggregated nanoparticles, as shown in a recent publication 41 . However, the interactions with mists expectedly bring additional processing challenges, which can affect thin film growth. For example, a rather predictable consequence for low-pressure systems would be the pressure variations induced by liquid droplet evaporation. So far, most research involving misty plasmas for the growth of nanocomposite thin films focused on process capabilities and/or film properties 42,[47][48][49] . In this work, the mist-assisted plasma deposition of SiO2/TiO2 nanocomposite thin films is investigated using in situ optical emission spectroscopy and in situ spectroscopic ellipsometry. It is shown that a deeper understanding of the plasma deposition kinetics is critical to gain a fine control on the plasmasynthesized nanocomposites' properties.

The reactor used in this study has been described in a previous publication 41 . In short, an inductively coupled plasma is generated in an oxygen-argon gas mixture (95% O2, 5% Ar) using a helicon antenna with an applied RF power of 400 W. The plasma then diffuses from the source region to a 21 L processing chamber, wherein a 0.11 standard cubic centimeters per minute (SCCM) flow of hexamethyldisiloxane (HMDSO, Sigma Aldrich) is injected as a precursor for plasma deposition of SiO2 on a silicon substrate. In addition to the gaseous organosilicon precursor, the processing chamber also hosts a Kemstream direct-liquid injector (DLI). A lab-made colloidal solution containing monodisperse TiO2 nanoparticles (≈ 3 nm diameter) is injected through the DLI in a pulsed fashion, creating sequences of 1-ms pulses with a frequency of 0.5 Hz. The parameter N defines the number of pulses per minute cycle. For instance, N = 10 results in sequence of 10 pulses at 0.5 Hz (namely 20 seconds total) followed by a 40-seconds period without injection. The solvent used for the colloidal solution is a mixture of propylene carbonate (PC), propylene glycol (PG) and methanol with a volume fraction of 20.25/54.75/25, respectively. The solution is very stable over time; detailed synthesis protocol has been the subject of earlier publications 50,51 .

The residual pressure before plasma deposition was in the order of 10 -3 mTorr, and the working pressure was 3 mTorr with a total gas flow of 24 SCCM. Pressure variations due to the pulsed liquid injection were recorded using an MKS capacitance gauge. Time-resolved optical emission spectroscopy was carried out using an Avantes AvaSpec-3648-2-USB2 dual-channel spectrometer, scanning between 500 and 900 nm with an integration time of 100 ms. Film growth was monitored by in situ spectroscopic ellipsometry using a J.A. Woollam M-2000 ellipsometer.

Samples were modeled as heterogeneous structures with a Bruggeman effective medium approximation, comprised of data from Herzinger et al. 52 for the SiO2 matrix and a custom Tauc-Lorentz dispersion law for the TiO2 nanoparticles 41,53 .

Figure 1 shows the pressure variations over time in the processing chamber. Starting at t = 0 s, a sequence of N 1-ms-long pulses is injected in the reactor. A sharp pressure rise is observed after each pulse due the injection and evaporation of the liquid droplets. For a single pulse, the pressure doubles its initial value, yielding Δp ≈ 3 mTorr. For repeated injections, a cumulative trend appears: the system does not have enough time between two pulses to return to the nominal 3 mTorr conditions. In other words, evaporated solvent molecules from a given pulse remain in the gas phase when the next pulse occurs. This leads to alternating pressure conditions during deposition, with a "high-pressure" period immediately after the injection(s), and a "low-pressure" 3-mTorr period the rest of the time. This result hints at two distinct kinetics driving plasma deposition, one for each period. Figure 1 further shows that the post-injection pressure deviates from a simple first-order decay. A good fit was reached with a sum of two exponential decays: for any number of pulses and many pressure datasets, the time constants were τ1 = 1.6 ± 0.3 s and τ2 = 8.6 ± 1.0 s. Although assigning a definite physical meaning to these values is still delicate at this stage, it can be noted that the order of magnitude of a few seconds is consistent with the droplet evaporation times calculated by Ogawa et al. 39 in low-pressure environments. Since the sole evacuation of excess pressure at maximum pumping capacity would have led to first-order kinetics with much shorter timescales, the pressure decrease seems evaporation-limited (as opposed to pumping-limited). In addition, with the amount of liquid injected per pulse (~1 µL/pulse), the expected change in pressure for a prompt and total solvent evaporation would be Δp ≈ 15 mTorr. This value is higher than the Δp ≈ 3 mTorr observed in the experiments, which confirms that droplet evaporation is not instantaneous but rather occurs over a few-seconds timescale.

𝑝(𝑡) = 𝑝 0 + 𝐴𝑒 -𝑡 𝜏 1 ⁄ + 𝐵𝑒 -𝑡 𝜏 2 ⁄
, where p0 = 3 mTorr, τ1 = 1.6 ± 0.3 s and τ2 = 8.6 ± 1.0 s.

Selected results obtained from time-resolved optical emission spectroscopy are presented in Figure 2 for a single liquid pulse starting at t = 0 s. While Fig. 2a displays an Ar line intensity (750 nm), Figs. 2b and2c show oxygen-to-argon (844 nm/750 nm) and hydrogen-to-argon (656 nm/750 nm) line intensity ratios. In the last two cases, Ar is used as an actinometer gas to probe the populations of O and H atoms 54,55 . As shown in Fig. 2, similar trends are observed with or without TiO2 nanoparticles in the injected solution (black curves vs. red curves). This result reveals that misty plasma effects are dominated by plasma-droplets interactions rather than plasmananoparticles interactions. Indeed, having solid nanoparticles in low-pressure plasma is known to cause significant changes in the fundamental plasma parameters, and therefore on the optical emission spectrum, as the nanoparticles behave as "electron wells" 20 . The non-observation of these changes thus indicates that the droplets reach the substrate while still in the liquid state, which is a key point to avoid in-plasma nanoparticles agglomeration and to ensure a good distribution of nanoparticles in the plasma-deposited nanocomposite films 41 . (Paschen notation) giving rise to the emission at 750 nm is mostly populated by electron impact on ground-state Ar atoms, this suggests a time variation of electron temperature Te (assuming Maxwellian electron energy distribution function) and/or electron density ne 20,56 . Over the range of experimental conditions investigated, previous Langmuir probe measurements recorded near the substrate in oxygen plasma indicated that both Te and ne decreased as the pressure rose from 1 to 25 mTorr 57 . This behavior is typical of large-volume diffusion plasmas, where the spatial profile of the electron density changes with increasing pressure 58,59 . Hence, the trend displayed in Fig. 2a most likely accounts for Te and ne variations. Here, a single first-order decay was sufficient to fit the data, with a time constant τ Ar = 2.0 ± 0.3 s. This result reveals that Te and ne are more strongly influenced by the more prominent pressure variations described by τ1 = 1.6 ± 0.3 s than the smoother one linked to τ2 = 8.6 ± 1.0 s.

In Fig. 2b, the behavior of the oxygen-to-argon line ratio follows closely the pressure rise, with the difference that a quick dip is observed before the prominent rise and subsequent decay.

This decrease of O atoms can be linked to either an oxygen depletion by chemical reactions with the solvent or a lower dissociation rate of O2 by electron-impact due to lower Te and ne values early in the pulse cycle. After the dip, the rise in the oxygen population with respect to the continuous O2/HMDSO plasma is unlikely to originate from the fragmentation of oxidation products, since the latter are rather stable molecules (H2O, CO, CO2) 54 . On the other hand, previous studies have shown that oxidation products can prevent O atoms loss at the walls by occupying surface recombination sites 54 . This effect was also demonstrated for Cl atoms loss with Cl2 as a recombination site obstructer 60,61 . As fewer O atoms are lost at the walls, the O atoms population in the plasma must increase. Hence, such a rise in the oxygen population most likely results from a reduction of the loss coefficient of O atoms at the walls due to a competition for surface recombination sites between O atoms and oxidation products. Here, the first-order decay time for O atoms, τ O = 8.1 ± 1.7 s, becomes comparable to the second time constant τ2 = 8.6 ± 1.0 s observed on Figure 1. Such slower dynamics are to be linked to heterogeneous surface reactions (recombination and desorption) at the reactor walls.

In the case of H atoms (Fig. 2c), the dramatic increase in hydrogen-to-argon line intensity ratio can quite clearly be ascribed to the high hydrogen content in the molecules constituting the misty solvent. However, this surge is much more prominent than what was previously observed with pressure: while the hydrogen-to-argon line ratio linked to atomic hydrogen increased by more than two orders of magnitude, the pressure linked to all solvent atoms and molecules rose by only a factor of two. This difference highlights that additional hydrogen atoms produced by plasmadroplet interactions represent only a small fraction of the total hydrogen content available in the mist, but remains significant with respect to the small amount of hydrogen atoms coming from continuous HMDSO fragmentation. A clear change in trend can also be seen in Fig. 2c around t = 10 s, which was not present when injecting methanol alone (no PC nor PG in the solution, not shown in Fig. 2). Since the presence of nanoparticles in the dispersions does not affect the spectral behavior of the plasma, this result suggests that the two kinetics can be attributed to successive evaporation/oxidation of methanol and PC/PG. In line with the pressure variations depicted in Fig. 1, H-to-Ar line ratios were also fitted with a sum of two first-order decays. Here, the corresponding time constants were τ1 H = 0.6 ± 0.2 s and τ2 H = 4.5 ± 1.2 s. These values are lower than the pressure time constants τ1 and τ2 and the time constant for O atoms τ O . This indicates that solvent injection gives rise to additional hydrogen atoms loss mechanisms, for example through hydrogen abstraction by impinging H atoms to form volatile H2 molecules 62,63 .

Figure 3 shows the evolution of film thickness and nanoparticles volume fraction over a given time interval during a 60-minutes plasma deposition, as determined by in situ spectroscopic ellipsometry 41 . Matrix growth appears to slow down periodically, with the plateaus corresponding to the injection periods (light blue sections in Fig. 3). This observation is consistent with the pressure increase and Ar line emission intensity decrease coming after the injection: at higher pressures, Te and ne are reduced and the excitation kinetics of Ar 2p states by electron impact is lower. Accordingly, the dissociation kinetics of HMDSO by electron impact is also slower, which slows down plasma deposition 64 . Interestingly, because the reduction in deposition rate occurs during the mist injection periods, the nanoparticles are supplied to the sample at the same time, and their fraction in the film increases (orange curve in Fig. 3). This percentage decreases again between the injection periods, as no additional nanoparticles are injected while SiO2 deposition continues. However, the deposition rate of SiO2 is low enough to keep the TiO2 fraction within a relatively narrow range of values (see right vertical scale in Fig. 3). Hence, the growth of a nanocomposite thin film in the misty plasma consists of two alternating steps (either matrix-or nanoparticles-predominant), occurring at regular intervals determined by the number of pulses N.

From the spectroscopic ellipsometry results presented in Fig. 3, it can also be noticed that during each "no injection" step, the slope of film thickness versus time is the same as the one of a standard SiO2 PECVD in the same experimental conditions (dashed line in Fig. 3). This observation is more appreciable in Fig. 4, presenting the deposition rate as a function of time for depositions with and without liquid injection. Naturally, for a sample deposited without any mist injection, the deposition rate remains constant throughout the whole process and lies around 2 nm/min. At N = 30 pulses per minute, which is the maximum value at a frequency of 0.5 Hz, the deposition rate is also relatively stable, at a lower value of around 0.4 nm/min. At N = 30 the pressure stays relatively high (up to 20 mTorr) and the deposition rate is therefore continuously hampered. From the N = 10 curve, the deposition oscillates between a slow growth step (corresponding to the liquid injection, as identified before) and a "fast" growth step. An axis break was added to the graph to illustrate that this trend is preserved for longer deposition times. Adjusting the number, duration or frequency of the pulses will therefore modify the span of the slow deposition step, which enables to control the fraction of nanoparticles introduced in the film and ultimately its final properties. In summary, the present work showed that low-pressure misty plasma deposition can exhibit different process kinetics that can play a crucial role on nanocomposite film properties.

Pressure variations in the mist-assisted deposition displayed a cumulative trend for series of liquid injections, dividing deposition in two steps of "high-" and "low-pressure" conditions. Such changes readily modified the plasma parameters, including the electron temperature and density. Overall, pressure and spectroscopy measurements revealed two distinct decay timescales, a faster one linked to plasma-droplet interactions and a slower one linked to heterogeneous surface reactions such as recombination, desorption and abstraction at the plasma reactor walls. In situ spectroscopic ellipsometry exposed how these changes in the gas phase directly affected film growth. At higher pressures, lower Te and ne yielded limited precursor fragmentation and consequently slower matrix deposition. Thus, matrix growth was slowed down synchronously with the nanoparticles input.

This new insight on plasma-mist deposition kinetics allows for a better connection between injection parameters and film properties.

Figure 1 -

 1 Figure 1 -Pressure variations over time upon liquid injection. Each pulse sequence starts at t = 0. Decreasing parts of the curves were fitted with a sum of two exponential decays:

Figure 2 -

 2 Figure 2 -Time-resolved optical emission spectroscopy for argon line intensity (a), oxygen-toargon line ratio (b), and hydrogen-to-argon line ratio (c) after a single liquid injection at t = 0.

Figure 3 -

 3 Figure 3 -Film thickness and nanoparticles content evolution of TiO2-SiO2 nanocomposite thin films over a given time interval, as determined by in situ spectroscopic ellipsometry. Mist injection periods of ten pulses at 0.5 Hz are marked in light blue. Evolution of SiO2 thickness during growth without mist injection is added as a dashed line.

Figure 4 -

 4 Figure 4 -Deposition rate of SiO2 thin films (black) and TiO2-SiO2 nanocomposite thin films as a function of time for N=10 (blue) and N=30 (green).

All authors would like to acknowledge the financial support of the CNRS, Université de Montréal and Nantes Université through their contributions to the International Research Network on Controlled Multifunctional Nanomaterials.


E W Awin

K C H Kumar

S Bernard

R Kumar

Mechanical characterization of spark plasma sintered titania-silicon oxycarbide (TiO 2 /SiOC) nanocomposites Mechanische Charakterisierung von sparkplasmagesinterten Titan-Silcium-Oxidkarbid-Verbundwerkstoffen (TiO 2 /SiOC)

Keywords: Spark plasma sintering, Nanoindentation, Fracture toughness, Ball-onthree ball method

In this work, titania-silicon oxycarbide nanocomposites synthesized via the polymer derived ceramics route have been sintered into crack-free monoliths using spark plasma sintering and their mechanical properties as well as their apparent density have been characterized. The x-ray diffractogram clearly showed the stabilization of anatase phase of size less than 10 nm size in an amorphous silicon oxycarbide matrix at 1200 °C. The hardness and elastic modulus determined using nanoindentation were found to be 9 GPa and 82 GPa, respectively, and the fracture toughness calculated using indentation crack length method was found to be 2.34 MPa m 1/2 . In addition to this, ball-on-three ball technique was used to evaluate the biaxial flexural strength and fractographic studies were carried out to understand the fracture mechanisms.

Introduction

In recent years, silicon oxycarbide based ceramics have gained a lot of attention in the research community due to their excellent mechanical properties as well as their high chemical and thermal stability [1]. Silicon oxycarbide based ceramics have been synthesized through a wide variety of processing routes, out of which commonly used techniques are the sol-gel and the polymer-derived ceramics route [2,3]. The main advantage of implementing the polymer derived ceramics route is in the flexibility to control and 1 Laboratory for High Performance Ceramics, Department of Metallurgic and Materials Engineering, Indian Institute of Technology-Madras (IIT Madras), Chennai, India 2 Université de Limoges, CNRS, IRCER, UMR 7315, Limoges, France adapt the structure and the composition of the ceramics for the targeted application based on the chemistry of the selected preceramic polymer as precursor [4]. The development of nanocomposites in which the structure of at least one phase is designed at nanoscale allows improved performance in applications that can reach far beyond those of conventional polymer derived ceramics [5][6][7][8]. For example, nano-titania (anatase phase) based silicon oxycarbide ceramics, i. e., titania-silicon oxycarbide nanocomposites, as a material for photocatalytic application has the advantage compared to conventional materials due E P to the combinatorial effect: the silicon oxycarbide phase providing mechanical support whereas titania promoting photocatalysis [START_REF] Awin | Materials[END_REF]. However, the mechanical properties of polymer derived ceramics-based nanocomposites have received limited attention so far compared to the photocatalytic aspects.

There have been only very few studies carried out on the mechanical stability of silicon oxycarbide based ceramics. Recently, the dependence of fracture toughness and elastic modulus on the presence of free carbon, crystalline phases of silicon carbide, zirconia and hafnia in silicon oxycarbide based glass-ceramics were reported [START_REF] Sorarù | [END_REF]11]. The flexural strength data of silicon oxycarbide ceramics was reported to be in the range of 350 MPa-390 MPa [12,13]. The biaxial flexural strength of a crack-free dense monolith produced by field-assisted sintering was found to be 173 MPa-222 MPa [14]. Recently, studies carried out on the strontium and barium doped silicon oxycarbide ceramics revealed that the initial particle size, as well as the porosity, affects the flexural strength [15,[START_REF] Eom | Nippon Seramikkusu Kyokai[END_REF]. The biaxial flexural strength for sub-micron sized strontium and barium doped silicon oxycarbide ceramics were reported to be 220 MPa and 209 MPa, respectively.

Lifetime estimation of a material requires the knowledge of its mechanical strength. The conventional test methods such as three and fourpoint bend tests, ball-on-ring, ring-on-ring test method and pin-on-three balls for determining the mechanical strength of a material have many disadvantages [17]. In the case of three and fourpoint bending test, the requirement of a large number of samples and its preparation alter the testing surface. On the contrary, in the case of ball-on-ring and pin-on-three ball method, a slight error in the sample geometry and uncertainty in loading could lead to unevenness in load transfer during the test [17][18][19]. In the present work, a comprehensive mechanical characterization of spark plasma sintered titania-silicon oxycarbide nanocomposites using indentation techniques and ball-on-three ball test was carried out for the first time. In ball-on-three ball tests, the problem associated with the above-mentioned methods is lessened to a large extent.

© 2022 Wiley-VCH GmbH

Experimental details

Commercially available polyhydrido methylsiloxane and titanium n-tetra butoxide were taken in equal amount and mixed continuously for 60 min using a magnetic stirrer. The solution was transferred to a horizontal tubular furnace in an alumina boat. The crosslinking was carried out at 300 °C for 2 h and subsequently pyrolyzed at 1200 °C maintaining a heating rate of 5 °C/min and holding time of 5 h in an ambient atmosphere. A foamy structure was obtained, which was crushed using a mortar and pestle and then subjected to spark plasma sintering (Sumitomo Coal Mining Co. Ltd, Japan). A 13 mm diameter die was used for the compaction of powder wherein graphite foils were placed between the die, punch and the powder for the easy passage of electric current. The sintering was performed in vacuum under a constant pressure of 30 MPa. The desired temperature (1200 °C) was attained at a heating rate of 100 °C/min, and dwell duration was 10 min.

The density of the sintered samples was measured using the water displacement method. The thermogravimetric analysis (NETZSCH STA 409, Selb, Germany) was performed in static air flow maintaining a heating rate of 5 °C/min to 1400 °C. The phase evolution and microstructural characterization were done using x-ray diffractometer (Bruker Discover D8, USA) and scanning electron microscope (FEI Quanta 200, USA), respectively. The Raman spectrum (Horiba, Japan) was recorded using a He Ne laser source operating at 488 nm in the range 100 cm 1 -4000 cm 1 . The fracture toughness of the material was determined using indentation crack length method using Anstis equation (1). A load of 10 kg was applied, and crack length was measured using scanning electron microscope.

� � 0:5 � � K IC  0:016 H c 1:5 (1)
where K IC is the fracture toughness (MPa m 1/2 ), P is the applied load (N), E is the elastic modulus (GPa), H is the hardness (GPa), and c is the radius of the critical crack (μm). The Poisson's ratio of the samples was determined by a non-destructive ultrasonic pulse echo V s t ff i  method. The longitudinal wave (10 MHz) and shear wave (5 MHz) transducers were used to measure the ultrasonic velocity.

The biaxial flexural strength test was carried out by ball-on-three ball tests with the aid of a universal testing machine (Zwick Z010, Zwick/ Roell, Ulm, Germany), Figure 1. Three balls were kept at the base on which the sintered sample was placed, and the fourth ball was symmetrically loaded above the specimen. A preload of 5 N was applied before the start of the experiment to retain the balls in position, maintaining crosshead speed and strain rate constant (1 mm/ min and 0.001 s 1 , respectively). The load was subsequently increased till failure, and this load was used to calculate the biaxial fracture strength.

The ball-on-three ball test was restricted to five samples, and Weibull analysis was performed with these data points (relative humidity = 4 0 % and temperature = 2 7 °C). The sample thickness-to-radius ratio, t/R ratio was maintained at 0.25 for all the samples as ball-on-three ball test was proven to be valid in the range (0.05 < t/R < 0.6) for ceramics [17]. The sample being too thick or thin, could induce Hertzian contact stress or buckling issues, respectively.

The failure of a brittle material is usually elicited by the largest flaw in the sample volume. The probability of failure, F for a given sample, as por- trayed by Weibull analysis is given by the expression

� � � m � F s; V  1-exp 1 V 0 s 0 (2)
where s is the applied stress (MPa), V is the sample volume (m 3 ), V 0 is the reference volume, s 0 is the stress at which probability of failure is 63 % (characteristic strength) and m is Weibull modulus which depicts the scatter of stress distribution.

The biaxial flexure strength of the sample was calculated from the fracture load using the equation

s m ax  A P ( 3
)
where A is a dimensionless factor which depends on the specimen geometry and Poisson's ratio (#), P is the applied load (N) at failure and t is the thickness (mm) of the sample. The specimen is defined by its radius R, whereas R a is the radius of the support circle formed by the three balls and R b is the radius i of the balls (8 mm), which is related by R a  R b . The biaxial flexural strength was calculated using a Web-Mathematica tool developed for discs (www.isfk.at/en/960).

Results and discussion

The titania-silicon oxycarbide nanocomposite was synthesized through the polymer derived ceramics route as discussed elsewhere [START_REF] Awin | Materials[END_REF]. The thermogravimetric curve exhibits three stages of mass loss at 25 °C-250 °C (loss of alcohols and water), 250 °C-450 °C (polymer decomposition) and 450-800 °C (release of hydrogen and methane gases) [START_REF] Awin | Materials[END_REF], Figure 2a. The titania-silicon oxycarbide nanocomposites were found to amorphous till 1000 °C and further increase in pyrolysis temperature led to the crystallization of anatase phase at 1200 °C; distributed in an amorphous silicon oxycarbide matrix, Figure 2b [9]. The Rietveld refinement (MAUD) performed on the powder sample indicated that the nanocomposite contains 34 wt. % of the anatase phase [START_REF] Awin | Materials[END_REF]. The density of the spark plasma sintered sample was determined using the water displace- ment technique (Archimedes principle) and was found to be 2.33 g/cm 3 . As inferred from the xray diffractogram of the spark plasma sintered sample, the first three dominant peaks at 2θ = 25.2°, 47.9° and 37.7° confirm the presence of anatase phase in an amorphous silicon oxycarbide, Figure 2c. A broad hump at 22° indicates the presence of amorphous silica. The crystallite size of anatase was determined using Scherrer's formula and was found to be 9 nm.

The peaks at 145 cm 1 , 198 cm 1 and 636 cm 1 corresponding to E band whereas 394 cm 1 and 510 cm 1 attributing to B 1g and A 1g + B 1g bands, respectively confirms the presence of anatase phase from the Raman spectra, Figure 2d. The inset of Figure 2b clearly presents the D (1343 cm 1 ) and G (1594 cm 1 ) peaks of graphitic carbon which relates to the defects and sp 2 hybridized carbon, respectively. The additional peaks observed at 2691 cm 1 and 2919 cm 1 corresponds to the 2D and D + G bands, respectively.

The scanning electron micrograph of the sintered sample clearly indicated the presence of pores, Figure 3. The presence of pores could be attributed to low sintering temperature used in this study.

The elastic modulus and hardness calculated using Oliver and Pharr method were found to be 82 GPa  2 GPa and 8.6 GPa  0.9 GPa, respectively [20], Figure 4a. The hardness values determined using nanoindentation was found to be © 2022 Wiley-VCH GmbH higher when compared to bulk Vickers hardness (5.7 GPa). This could be attributed to the indentation size effect, and the difference in the load applied. The power-law relation as given by the equation ( 4) was used to fit the unloading segment of the load-displacement curve.

P  B h h f  m ( 4
)
where P is the applied load (mN), B&m are fitting parameters, h is the displacement (nm) and h f is the final indentation depth. The power-law index, m was determined to be 1.5 and was analogous to the reported values of polymer derived ceramics [21]. The nanocomposite exhibited considerable elastic recovery (~64 %) during unloading as per the equation ( 5), which was analogous to the reported values of fused silica [22].

% R  h max h f (5) max
The presence of turbostratic carbon at the interphase boundaries between the nanocrystalline titania phases could be attributed to the high elastic recovery in the nanocomposite. The turbostratic carbon is expected to relax upon unloading, causing high elastic recovery. Moreover, in the case of the nanocomposites, the calculated E/H ratio of 9 was comparable to that of fused quartz and silicon oxycarbide implying the ceramic to possess a cracking pattern similar to that of the aforementioned material systems [23]. This also suggests the fact that the uniformly distributed titania nanocrystals in the amorphous silicon oxycarbide matrix have a minimal influence in governing the mechanical properties.

The fracture toughness of the material determined using Anstis equation was found to be 2.34 MPa m 1/2 which was analogous to silicon oxycarbide ceramics [11,24]. The crack evolution pattern in titania-silicon oxycarbide nanocomposite was comparable to that observed in polymer derived silicon oxycarbide ceramics. The crack pat-© 2022 Wiley-VCH GmbH tern observed was a combination of conical and radial crack, Figure . 4b. The investigation performed on the silicon oxycarbide glasses (pyrolyzed at temperatures less than 1000 °C) suggests that the densification of Si O C networks beneath the indenter is responsible for the development of conical cracks. The formation of radial cracks was observed at higher temperatures (> 1000 °C) which were attributed to the reduction in the mobility of Si O C network by the covalently bonded carbon atoms. At higher temperatures, the deformation occurs by shear flow rather than by densification under the indenter. The similarity in the structure of titania-silicon oxycarbide nanocomposite to that of silicon oxycarbide, with Si O and Si C could be a reason for the exhibition of mixed conical and radial cracks. In fact, it was also shown that the glassceramics possessing high Poisson's ratio (#), # > 0.25 resulted in the formation of radial crack upon indentation. On the contrary, for lower # values ( # < 0.20), cone cracks were observed [25]. In the current study, the Poisson's ratio of titania-silicon oxycarbide nanocomposite was found to be 0.23. Hence, it could be assumed that the intermediate value of Poisson's ratio might have led to the formation of a combined radial and conical crack.

The Poisson's ratio and elastic modulus were calculated from the following expression, Table 1

V l 2 V t 2 V 2 2 V 2
V l 11  #1 2# 1 #
where V l and V t are the calculated values of longitudinal and transverse velocities (m/s). The biaxial flexure strength results carried out on the sintered samples as a Weibull diagram. The results are plotted as a function of fracture stress (rearranging) vs. probability of failure (taking the double logarithm of equation 2), Figure 5. The Weibull modulus, m = 18.1  3 as well as characteristic biaxial flexural strength i.e, probability of failure of 63.21 %, σ 0 = 92.7 MPa were evaluated according to EN-843-5 [26].

Most of the specimens broke into three pieces whereas in one case, it was observed to fracture into two pieces, Figure 6. It is believed that as the strength (stored elastic energy) increases, the number of fracture pieces increases.

The stresses at which sintered samples fractured were in a narrow range, and it can be therefore inferred that this material exhibits high mechanical reliability [25]. The low flexural strength (92.7 MPa) of titania-silicon oxycarbide nanocomposite obtained in this study when compared to its counterpart silicon oxycarbide (153 MPa) could be attributed to the low sintering temperature. The presence of pores would have degraded the flexural strength of the nanocomposite [15]. The influence of free carbon on mechanical properties has always been a subject to controversy. Recently, biaxial flexural strength of 922 MPa was reported for an amorphous silicon oxycarbide ceramic sintered at 1000 °C [27]. The considerable increase in biaxial characteristic stress value has been attributed to the presence of a large amount of free carbon content. On the contrary, a diminution in mechanical properties with an increase in free carbon content was also reported [START_REF] Sorarù | [END_REF]. The density of the sintered sam- ple was also found to be lower with an increase in free carbon content. Even though, the biaxial stress has no direct repercussion on the values of hardness and elastic modulus, the fact that the free carbon plays a significant role in the densification mechanism; it is believed to lower the biaxial flexural strength of the components. In the present study, the I D /I G ratio of 1.008 implies the existence of disordered carbon in the nanocomposite. Hence, it could be inferred that the presence of free carbon could have resulted in the drop of biaxial stress values. It is inferred from the plot of characteristic stress (σ 0 ) versus Weibull modulus (m) of different ceramic materials and bulk metallic glasses that the titania-silicon oxycarbide nanocomposite exhibits a Weibull modulus value of m = 18 are comparable to the values of glass ceramics, alumina/silicon carbide/silicon nitride and zirconia-based ceramics, Figure 7 [28][29][30][31]. Taking into consideration of the characteristic stress value, the titania-silicon oxycarbide nanocomposite was in par with the glassceramics. However, the σ 0 was found to be lower than other ceramic-based materials. The high value of m implies that the nanocomposite is reliable for engineering applications.

The polymer derived ceramic approach used for the processing of nanocomposite could result in the formation of pores. The polymer to ceramic conversion happens with the formation of cracks and pores due to the volatilization of low molecular components in the pre-ceramic polymer. The frac- tographic analysis performed on the ball-on-three ball fractured samples enabled us to identify the origin of fracture, Figure 8. The breakage of specimens into fragments and small effective volume to surface ratio makes it difficult to identify the origin of fracture. However, in ball-on-three ball tests, failure is expected to happen at the center of the disc opposite to the loading where the tensile stress is high [32]. From the fractographs, it was clear that the defect present on the surface of the specimen was the reason for failure of many of the samples. The hackle lines (dotted lines) which appear on the fractured surface indicate that fracture has initiated at the surface. The region subjected to the highest tensile stress is shown as dotted box and is expected to be the site of fracture origin, Figure . 8. In all the cases, Type 2 cracks were observed [33]. This could be due to the scratches or pores present on the surface of the specimen, which is substantiated from scanning electron micrographs. When compared to the biaxial flexural strength of the sintered samples, the current material exhibits low values, which could be attributed to the porosity present in the sample. The brittle nature of the sample as well as the porosity, was well understood from the fractographs.

Conclusions

The spark plasma sintered titania-silicon oxycarbide nanocomposites synthesized through precursor derived ceramics route resulted in the stabilization of nanocrystalline anatase phase in an amorphous silicon oxycarbide matrix at a temperature as high as 1200 °C. The presence of free carbon was revealed by Raman spectra, and the microstructural characterization confirmed the presence of pores. The hardness and elastic modulus determined using nanoindentation exhibited lower values than expected, which could be attributed to the lower sintering temperature, in turn affecting the densification process. The fracture toughness was calculated using indentation crack length method and the cracking pattern observed was a combination of conical and radial cracks suggesting the deformation was prominently due to shear flow under the indenter. The low value of biaxial flexural strength determined using ball-on-three ball test clearly indicates the effect of porosity and the high Weibull modulus value reciprocates the mechanical stability in the design point of view.

Figure 1 .

 1 Figure 1. Photograph of Zwick Z010 instrument and the schematic of ball-on-three ball set up.

Figure 2 .

 2 Figure 2. (a) TGA curve indicating the mass loss behavior with respect to temperature (b) X-ray diffractogram indicating the phase evolution of the nanocomposite as a function of pyrolysis temperature (c) X-ray diffractogram of spark plasma sintered nanocomposite revealing the formation of titania and (d) Raman spectra confirming the anatase phase as well as the existence of free carbon.

Figure 3 .

 3 Figure 3. Scanning electron micrograph of the sintered pellet. Inset image shows nanocrystalline grains of titania.

Figure 4 .

 4 Figure 4. (a) Load vs. displacement curve of titania-silicon oxycarbide nanocomposite (inset image displays the nanoindent impression) and (b) Scanning electron micrograph indicating the evolution of conical and radial cracks.

Figure 5 .

 5 Figure 5. Weibull plot: the probability of failure as a function of failure stress.

Figure 6 .

 6 Figure 6. Ball-on-three ball fractured specimens (a) three-fold and (b) two-fold symmetry.

Figure 7 .

 7 Figure 7. Characteristic stress of different ceramic based materials as a function of Weibull modulus.

Figure 8 .

 8 Figure 8. Fractograph of the ball-on-three ball samples indicating the area subjected to high tensile stresses (dotted box) and hackle lines (dotted lines) indicating the origin of failure.

Table 1 .

 1 Poisson's ratio and elastic constant of titania-silicon oxycarbide nanocomposite acquired using pulseecho resonance method.

	Sample	Thickness V l	V t	#	E
			(m/s) (m/s)		(GPa)
	titania-	3.75	5161 3047 0.23 53.6
	silicon
	oxycarbide

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Acknowledgements

We would like to acknowledge the funding received from Indo-French Centre for the Promotion of Advanced Research (IFCPAR/CEFIPRA), a bilateral Indo-French project (Project No. 5108-1) to carry out this work.