Philipp Braun
email: philipp.braun@anu.edu.au

Giulia Giordano
email: giulia.giordano@unitn.it

Christopher M Kellett
email: chris.kellett@anu.edu.au

Luca Zaccarian
email: luca.zaccarian@laas.fr

An Asymmetric Stabilizer Based on Scheduling Shifted Coordinates for Single-Input Linear Systems With Asymmetric Saturation

HAL is

I. INTRODUCTION

While historically input saturation has been mostly studied using symmetric limits (see, e.g., [START_REF] Hu | Control systems with actuator saturation: analysis and design[END_REF], [START_REF] Tarbouriech | Stability and stabilization of linear systems with saturating actuators[END_REF]), recent technological challenges show that asymmetric limits often arise in practice, thereby making the symmetric solutions (typically based on focusing at the smallest limit) quite conservative.

As a consequence, the community started looking into nonsymmetric Lyapunov certificates in the presence of symmetric stabilizers (see [START_REF] Benzaouia | Constrained stabilization: an enlargement technique of positively invariant sets[END_REF] and references therein, and also [START_REF] Hu | Null controllability and stabilization of linear systems subject to asymmetric actuator saturation[END_REF], [START_REF] Hu | Null controllability and stabilization of linear systems subject to asymmetric actuator saturation[END_REF]). Additionally, [START_REF] Li | An asymmetric Lyapunov function for linear systems with asymmetric actuator saturation[END_REF] and [START_REF] Groff | Regional stability of discrete-time linear systems subject to asymmetric input saturation[END_REF] use a piecewise quadratic Lyapunov function with symmetric stabilizing saturated linear feedbacks, in the continuous-time and discrete-time cases, respectively. Alternative recent techniques include symmetrically stabilizing a shifted equilibrium (see [2, ch. 8] and references therein), but this comes at the cost of not stabilizing the origin any longer. Finally, more sophisticad stabilizers have been proposed in [START_REF] Yuan | Switching control of linear systems subject to asymmetric actuator saturation[END_REF], where a switching dynamical controller is designed, capable of exploiting the available range of the control action on both sides of the saturation levels. In an attempt to provide enlarged regions of attraction, we recently proposed in [START_REF] Mariano | Asymmetric state feedback for linear plants with asymmetric input saturation[END_REF] the design of an asymmetric stabilizer, based on the convex scaling proposed in [START_REF] Blanchini | Any domain of attraction for a linear constrained system is a tracking domain of attraction[END_REF] for the shifted stabilizer. That solution provides significantly larger guarantees but has the drawback of 1) restricting the achievable region of attraction with a conservative point inclusion condition required to apply the technique in [START_REF] Blanchini | Any domain of attraction for a linear constrained system is a tracking domain of attraction[END_REF] and 2) reducing the local performance of the symmetric solution.

In this work we propose a novel asymmetric scheme based on focusing on the shifted equilibrium, as in [2, ch. 8], and continuously driving that equilibrium back to the origin by relying on the solution of a parametric optimization problem. As compared to [START_REF] Mariano | Asymmetric state feedback for linear plants with asymmetric input saturation[END_REF], this solution is not based on convex scalings and therefore preserves the convergence rate given by the symmetric stabilizer in its guaranteed ellipsoidal set. Moreover, it does not require any point inclusion conditions, therefore obtaining an estimate of the basin of attraction containing all the contractive ellipsoids that can be constructed in shifted coordinates. While the proposed framework applies to multi-input plants, we focus most of the paper on the singleinput case, for which we can compute explicit expressions of the control law solving the optimization. In the rest of the paper we discuss symmetric and shifted asymmetric stabilizers in Section II, then for the single-input case we propose our optimization-based solution in Section III and its explicit version in Section IV. Numerical examples in Section V show the ability of our controller to provide greatly enlarged certified stability regions.

Notation. For u -, u + ∈ R m ≥0 , m ∈ N, sat [u -,u + ] (u) = max{min{u + ,
u}, -u -} defines the saturation, where the maximum/minimum are to be understood componentwise. The deadzone is defined through dz [u -,u + ] (u) = usat [u -,u + ] (u). For Z ∈ R n×n , He(•) denotes the function He(Z) = Z + Z . For Z ∈ R n×m and z ∈ R n , Z [k] and z k denote the k-th row and the k-th entry, respectively. A vector

v ∈ R n satisfies v ≤ min{u -, u + } if v k ≤ u - k and v k ≤ u +
k for all k ∈ N. A symmetric positive definite matrix P ∈ R n×n can be uniquely decomposed as P = P

1 2 P 1 2
where P 1 2 ∈ R n×n is symmetric and positive definite. We use the norms |x| =

√

x T x, x ∈ R n , and |x| P = √ x T P x, P ∈ R n×n positive definite. Finally, I ∈ R n×n denotes the identity matrix, 1 ∈ R n satisfies 1 k = 1, k ∈ {1, . . . , n} and int(A) denotes the interior of a set A ⊂ R n .

II. SYMMETRIC AND SHIFTED STABILIZERS

We consider linear saturated continuous time systems

ẋ = Ax + B sat [u -,u + ] (u) (1)
with state x ∈ R n , input u ∈ R m , A ∈ R n×n , B ∈ R n×m
and saturation limits u -, u + ∈ R m >0 . We define the average saturation range ū ∈ R m and the average saturation center

u • ∈ R m as ū = 1 2 (u + + u -), u • = 1 2 (u + -u -). (2)
For simplicity, we assume that the average saturation range ūk satisfies ūk = 1 for all k ∈ {1, . . . , m}; this is not restrictive and can always be assumed without loss of generality for u -, u + ∈ R m >0 by scaling the columns of B.

Assumption 1: The average saturation range satisfies ū = 1 ∈ R m .

To convey the idea of the controller design and to obtain explicit formulations, we restrict our presentation in most parts of the paper to the single input case and we assume that the matrix A is non-singular. However, by discussing different cases and by using a more convoluted notation, the ideas seem to extend to the multi input case and to dynamics (1) with singular matrix A.

Assumption 2: The pair (A, B) is stabilizable and A is non-singular.

Of particular interest in this paper is the subspace of induced equilibria

Γ = {x e ∈ R n : Ax e + Bu e = 0, u e ∈ R m }. (3)
Under Assumption 2, there is a continuous mapping u e → x e (u e ) defined as

x e (u e ) = -A -1 Bu e , (4)
characterizing pairs of induced equilibria (x e , u e ) ∈ Γ × R m through the input u e ∈ R m . Note that, in the definition of Γ, the saturation levels are not present. System (1) with u ∈ [-u -, u + ] can only be stabilized at x e if a corresponding input satisfies u e ∈ (-u -, u + ).

In a neighborhood of the origin, we are looking for a feedback law

u = Kx + L dz [u -,u + ] (u), (5)
with K ∈ R m×n , L ∈ R m×m , asymptotically stabilizing the origin. Combining (1) and ( 5), the closed loop dynamics can be written as

ẋ = (A + BK)x -(B -BL) dz [u -,u + ] (u) u = Kx + L dz [u -,u + ] (u). (6)
To characterize regions of attraction of asymptotically stable (induced) equilibria x e , we consider sublevel sets of quadratic functions. In particular, for κ ∈ R ≥0 , x e ∈ R n and P ∈ R n×n positive definite we define the set

E κ xe (P ) = {x ∈ R n : |x -x e | P ≤ κ}. (7)
Proposition 1 (Symmetric Stabilizer, [10, Theorem 1]):

Given the plant (1), let v ∈ R m ≥0 with v ≤ min{u -, u + } and let α ∈ R ≥0 . Moreover, let Q v ∈ R n×n , W v , Y v ∈ R m×n , U v , X v ∈ R m×m be a solution of the optimization problem max Q,W,Y,U,X log det(Q) (8) subject to U > 0 diagonal, Q = Q > 0 He AQ + BW + αQ -BU + BX W + Y X -U < 0 v 2 k Y [k] Y [k] Q ≥ 0, k = 1, . . . , m.
Then, for

K = W v Q -1 v , L = X v U -1 v , P v = Q -1 v ( 9
)
the nonlinear algebraic loop in (5) is well posed (i.e., its solution is unique and Lipschitz) and function x P v x exponentially decreases with rate larger than 2α within the sublevel set E 1 0 (P v ). Consequently, the origin of ( 6) is locally exponentially stable, with basin of attraction containing the set E 1 0 (P v ). The subscript v is used to indicate the dependence of P on the selection of vector v ∈ R m ≥0 . In particular, we observe that LMIs [START_REF] Hu | Null controllability and stabilization of linear systems subject to asymmetric actuator saturation[END_REF] are homogeneous in the decision variables, except for v 2 k in the constraints. Therefore, scaling vector v in Proposition 1 by a positive scalar κ leads to scaling the corresponding optimal solution (8) by κ 2 . More specifically, according to [START_REF] Li | An asymmetric Lyapunov function for linear systems with asymmetric actuator saturation[END_REF], this corresponds to a scaled P κv = κ -2 P v , whereas gains K and L remain unchanged because the scaling cancels out (this is the reason why no subscript is used in K and L). Finally, the certified ellipsoidal set scales from

E 1 0 (P v ) to E 1 0 (P κv ) = E κ 0 (P v ).
This fact is stated in the next corollary.

Corollary 1 (Homogeneity): Let K, L, P v defined in (9) correspond to an optimal solution of ( 8) for v ∈ R m ≥0 . Then for all κ ∈ R ≥0 with κv ≤ min{u -, u + }, and with the same gains K and L, function x P v x exponentially decreases with rate larger than 2α within the set E κ 0 (P v ). Proposition 1 can be used to define a control law stabilizing an induced equilibrium, instead of the origin. Consider an equilibrium pair (x e , u e ) satisfying ( 4) and assume that u -+ u e , u + -u e ∈ R m >0 . Additionally, consider the coordinate transformation

x = x -x e and ũ = u -u e .
Then, it holds that

ẋ = ẋ = Ax + B sat [u -,u + ] (u) = Ax + Ax e + B sat [u -,u + ] (ũ + u e ) = Ax + Ax e + B(u e + sat [u -+ue,u + -ue] (ũ)) = Ax + B sat [u -+ue,u + -ue] (ũ). (10a)
with the shifted input ũ selected as follows

ũ = K x + L dz [u -+ue,u + -ue] (ũ). (10b)
For the shifted dynamics [START_REF] Mariano | Asymmetric state feedback for linear plants with asymmetric input saturation[END_REF], it is evident that the same result as that of Corollary 1 applies. This fact is stated in the following corollary, where a more convenient expression of u is deduced from (10b) exploiting the identities (4) and dz [u -+ue,u + -ue] (ũ) = dz [u -,u + ] (u), which follows straightforwardly from (10a).

Corollary 2: Let Assumption 2 be satisfied. Consider an equilibrium pair (x e , u e ) defined through [START_REF] Grant | CVX: Matlab software for disciplined convex programming[END_REF]. Given κ ∈ R >0 and v ∈ R m ≥0 assume that κv ≤ min{u -+ u e , u + -u e }, and let K, L, P v defined in [START_REF] Hu | Null controllability and stabilization of linear systems subject to asymmetric actuator saturation[END_REF] correspond to an optimal solution of (9). Then the following selection of the input u

u = u e + K(x + A -1 Bu e ) + L dz [u -,u + ] (u) (11)
ensures that function

V xe : R n → R ≥0 V xe (x) = (x -x e ) P v (x -x e ) = |x -x e | 2 Pv = |x| 2 Pv (12)
exponentially decreases with rate larger than 2α within the sublevel set x ∈ E κ 0 (P v ). Consequently, selection (11) locally exponentially stabilizes the (induced) equilibrium x e (u e ) of (1) with basin of attraction containing the shifted ellipsoid E κ xe (P v ) defined in [START_REF] Hu | Null controllability and stabilization of linear systems subject to asymmetric actuator saturation[END_REF]. Remark 1: Expression (11) specifies the control input u only implicitly, even though Proposition 1 ensures that the corresponding solution is Lipschitz. Proceeding as in [10, Lemma 3], for the single-input case m = 1, the selection

u = u e + K(x + A -1 Bu e ) (13) + L(I -L) -1 dz [u -,u + ] (u e + K(x + A -1 Bu e ))
can be proven to be the explicit solution to [START_REF] Still | Lectures on parametric optimization: An introduction[END_REF].

• Before we conclude this section with an example putting these results into context, we point out that an appropriate coordinate transformation x → P

1 2 v x, i.e., χ = P 1 2 v x ⇐⇒ x = (P 1 2 v ) -1 χ, (14)
with

P 1 2
v ∈ R n×n symmetric and positive definite, allows us to consider, instead of the Lyapunov function with ellipsoidal level sets, a Lyapunov function with circular level sets, without loss of generality. In particular, using the notation introduced in the results discussed in this section, the function [START_REF] Tarbouriech | Stability and stabilization of linear systems with saturating actuators[END_REF] reduces to the Euclidean norm in the χ-coordinates,

V χe : R n → R ≥0 , V χe (χ) = |χ-χ e | 2 . Introducing the vector C := P 1 2 v A -1 B, the set of induced equilibria (4) is given by χ(u e ) = P 1 2 v x e (u e ) = -P 1 2 v A -1 Bu e (15)
in the χ-coordinates. While using coordinates χ instead of x is not necessary in the following sections, the interpretations in the χ-coordinates are more illustrative in some places.

Example 1: Consider the dynamical system (1) defined through the matrices A = 0.6 -0.5 0.3 1.0 and B = [ 1 3 ]. From (8)1 the positive definite matrix

P 1 = Q -1 1 = 0.
v = 1, if κ ≤ min{u -, u + } for κ ∈ R >0
, then the origin of the closed-loop system ( 6) is asymptotically stable and the domain of attraction contains the set E κ 0 (P 1 ) according to Proposition 1 and Corollary 1.

Assume now that the saturation levels are chosen as u -= 1.5 and u + = 0.5. Then Proposition 1 and Corollary 1 can be applied with v = 1 and κ = 0.5 guaranteeing that the blue sublevel set in Fig. 2 is contained in the basin of attraction of the closed-loop system [START_REF] Hu | Control systems with actuator saturation: analysis and design[END_REF]. Alternatively, the sublevel sets guaranteeing asymptotic stability of the induced equilibria x e = -A -1 Bu e , u e ∈ (-1.5, 0.5), using the feedback law (11) can be derived through Corollary 2. For u e = -0.5, the assumptions of Corollary 2 are satisfied for κ = 1, leading to a larger sublevel set E 1 xe(ue) (P 1 ), compared with E 0.5 0 (P 1 ), for which convergence to x e is guaranteed (see Fig. 2).

III. OPTIMIZATION-BASED SHIFTED STABILIZER

Fig. 2 and the discussions above clearly indicate that the size of the estimate of the region of attraction varies significantly with the induced equilibrium to be stabilized. We exploit here this potential by deriving a modified controller with a larger estimate of the basin of attraction. In particular, the basin of attraction of our modified feedback law includes all sublevel sets E κ xe(ue) (P 1 ), u e ∈ (-u -, u + ), for which asymptotic stability of x e is guaranteed through Proposition 1, Corollaries 1 and 2 with the shifted feedback law [START_REF] Still | Lectures on parametric optimization: An introduction[END_REF]. From this point onwards, we restrict our attention to the single input case, i.e., we assume that m = 1.

A. Properties of the shifted stabilizer

Let P ū be the solution of (8) with ū = 1 defined in [START_REF] Benzaouia | Saturated Control of Linear Systems[END_REF]. Then, the function

β : [-u -, u + ] → [0, 1] defined as β(u e ) = min{u -+ u e , u + -u e } (16) = u -+ u e , for u e ∈ [-u -, u • ] u + -u e , for u e ∈ [u • , u + ]
defines the maximal value β(u e ) = κ in Corollary 2 (with v = ū) such that the assumptions of Corollary 2 are satisfied. Moreover, according to Corollary 1, the feedback law [START_REF] Still | Lectures on parametric optimization: An introduction[END_REF] guarantees the asymptotic stability of x e (u e ) with region of attraction containing the set

E β(ue) xe(ue) (P ū) = {x ∈ R n : |x -x e (u e )| Pū ≤ β(u e )}. (17)
In the χ-coordinates, introducing χ e (u e ) := P

1 2
ū x e (u e ) (see ( 14)), the set E β(ue) xe(ue) (P ū) simplifies to E β(ue) χe(ue) (I) = {χ ∈ R n : |χ -χ e (u e )| ≤ β(u e )}, (18) which we will work with in the following. In this section we will prove that the proposed scheduled law is stabilizing with basin of attraction containing the union of the sublevel sets generated by all possible values of u e ∈ (-u -, u + ), (I) holds. Moreover, the objective function is defined so that, from the set of feasible solutions, the one with the shortest distance to the origin is selected.

R x = ue∈(-u -,u + ) E β(ue) xe(ue) (P ū), R χ = ue∈(-u -,u + ) E β(ue) χe(ue) (I), (19)
Lemma 1: Let m = 1 and assume that Assumptions 1 and 2 are satisfied. Consider the optimization problem (20) where β and R χ are defined in ( 16) and (19), respectively. Then the following properties are satisfied:

1) For all χ ∈ int(R χ ), (20) is feasible, the feasible set is convex and its interior is nonempty; 2) u e (χ) = 0 for all χ ∈ E β(0) 0 (I); 3) u e (χ) satisfies |χ -χ e (u e )| = β(u e ) for all χ ∈ int(R χ )\E β(0) 0 (I); 4) u e (χ) ∈ (u -, u + ) is unique for all χ ∈ int(R χ ); and 5) u e (•) : int(R χ ) → (-u -, u + ) is continuous.

Proof: Item 1. Feasibility follows immediately from the definition of the function β and the set R χ . Moreover, since χ is in the interior of R χ , a feasible point can be infinitesimally increased/decreased while remaining feasible. Let χ ∈ R χ and take u e1 , u e2 ∈ (-u -, u + ) such that |χ -

χ e (u e1 )| ≤ β(u e1 ) and |χ -χ e (u e2 )| ≤ β(u e2 ). Then for all λ ∈ [0, 1] it holds that |χ-χ e (λ 1 u e1 + (1 -λ)u e2 )| ≤ λ|χ -χ e (u e1 )| + (1 -λ)|χ -χ e (u e2 )| ≤ λβ(u e1 ) + (1 -λ)β(u e2 ) ≤ β(λu e1 + (1 -λ)u e2 ).
The last step follows from the concavity of β in (16).

Item 2. The second item follows immediately from (18) together with the objective function. ū A -1 Bu e , it holds that u 1 e = -u 2 e and the origin can be defined as the average u # e = 0 = 1 2 (u 1 e + u 2 e ). Since β in ( 16) is a concave function, then it holds that min{β(u 1 e ), β(u 2 e )} ≤ β(u # e ) = β(0), i.e., χ ∈ E β(0) 0 (I), which implies that the optimal solution is χ e = -P •

1 2 ū A -1 Bu # e = 0,

B. Stabilization with scheduled shifted coordinates

With Lemma 1 we are able to construct a control law stabilizing the origin from any initial condition satisfying x ∈ int(R x ). Recall that, through the coordinate transformation (14), there is a one to one mapping between x and χ, therefore we can equivalently express the control law as a function of χ or x. Opting for the formulation as a function of x, we can now state the main feedback controller proposed in this paper, combining the results of Corollary 2 and Lemma 1 (together with Remark 1 for the case m = 1 addressed here). The controller is given by the state feedback law

u = u e (x) + K(x + A -1 Bu e (x))) (22) + L(I -L) -1 dz [u -,u + ] u e (x) + K(x + A -1 Bu e (x)) ,
defined through the optimal solution of (20). Based on Lemma 1, our main theorem proves its properties.

Theorem 1: Consider system (1) with m = 1, satisfying Assumptions 1 and 2. Let u -, u + ∈ R >0 , let P ū denote the solution of (8) for v = ū and α > 0 arbitrary. Then the feedback control law (22) 1) is well defined and continuous for all x ∈ int(R x ); 2) asymptotically stabilizes the origin of (1), and the basin of attraction contains the set int(R x ).

Proof: The proof exploits Lemma 1. The equivalence between the coordinate representations χ and x allows us to apply the results of Lemma 1 also in the x coordinates.

Item 1. The property that ( 22) is well defined in x ∈ int(R x ) and continuity of the feedback law follow immediately from the corresponding properties in Lemma 1.

Item 2. First note that, through the second item in Lemma 1, in the set E β(0) 0 (P ū), the modified control law (22) and the original control law (5) coincide. Thus the origin is Lyapunov stable and E β(0) 0 (P ū) is contained in the region of attraction of the origin. Moreover, for x ∈ int(R x ) the control law (22) coincides with [START_REF] Still | Lectures on parametric optimization: An introduction[END_REF], and thus it stabilizes the reference point x e (u e ) = -A -1 Bu e . Note that by construction x ∈ E

β(u e ) xe(u e ) (P ū). Since E β(u e )
xe(u e ) (P ū) denotes the sublevel set of V xe(u e ) (•) defined in [START_REF] Tarbouriech | Stability and stabilization of linear systems with saturating actuators[END_REF], given any closed-loop solution x(•) and for all t, the distance to x(t) -x e (u e (x(t))) in terms of the norm t → |x(t)x e (u e (x(t)))| Pū is contractive with rate at least 2α > 0. Contractivity implies that any solution starting in int(R x ) satisfies x(t) ∈ int(R x ) for all t ≥ 0. Furthermore, the contractivity of the boundary of E

β(u e )
xe(u e ) (P ū), together with the selection of the objective function in (20) implies that t → |x e (x(t))| is a decreasing function.

It is left to show that |x e (u e (x(t)))| = 0 is satisfied for t ∈ R ≥0 sufficiently large. Assume, for the sake of a contradiction, that x e (u e (x(t))) → x # e = 0 for t → ∞. In this case x(t) → x # e for t → ∞ according to the definition and the properties of the control law [START_REF] Still | Lectures on parametric optimization: An introduction[END_REF]. However, the fact that β(x # e ) > 0 together with the selection of the objective function in (20) implies that |x e (u e (x(t)))| < |x # e | for t ∈ R ≥0 sufficiently large, leading to a contradiction.

Remark 3: Based on Theorem 1, it is possible to show that the continuous function W : R x → R ≥0 defined as

W (x) = |x -x e (u e (x))| 2 Pū β(u e (x)) 2 + (u e (x)) 2 (23)
is a Lyapunov-like function for the closed-loop system. Indeed W satisfies quadratic upper and lower bounds in R x and decreases along the solutions of (1), (22). However, function W would certify local asymptotic stability only if u e is Lipschitz, which does not seem to be true in general. • In the proof of Theorem 1, we have exploited the fact that the original feedback law [START_REF] Groff | Regional stability of discrete-time linear systems subject to asymmetric input saturation[END_REF] and the modified feedback law (22) are the same in a neighborhood around the origin. This implies that performance is locally preserved, which we state for completeness in the following corollary. This result could not be attained by the asymmetric solution proposed in [START_REF] Mariano | Asymmetric state feedback for linear plants with asymmetric input saturation[END_REF], due to the convex scaling performed therein, which reduced by one half the convergence rate (see [START_REF] Blanchini | Any domain of attraction for a linear constrained system is a tracking domain of attraction[END_REF]Thm 3.3]).

Corollary 3: Let the assumptions of Theorem 1 be satisfied. For x ∈ E β(0) 0 (P ū), controller (22) coincides with controller (5), thus locally preserving the performance.

IV. EXPLICIT EXPRESSION OF THE STABILIZER So far the feedback law ( 22) is only implicitly defined through the optimal solution of (20). However, due to the simplicity of the optimization problem in the single input case, a finite set of possible optimal solutions can be derived offline, from which the selection of the optimal solution online is straightforward. To present the corresponding result we use the quantity C defined in (21) and define u e in terms of χ instead of x. Additionally, we define u #i e ∈ C ∪ {±∞}, i ∈ {1, . . . , 6} with

u #1 e (χ) = -|χ| 2 + (u -) 2 2(χ C -u -) , u #2 e (χ) = -|χ| 2 + (u + ) 2 2(χ C + u + ) , u #3,4 e (χ) = χ C -u -± (χ C) 2 -|C| 2 |χ| 2 +|Cu --χ| 2 1 -|C| 2 , u #5,6 e (χ) = χ C +u + ± (χ C) 2 -|C| 2 |χ| 2 +|Cu + +χ| 2 1 -|C| 2 .
Theorem 2: Let the assumptions of Theorem 1 be satisfied and consider the optimization problem (20) and C defined in (21

). If |C| = 1 define S = ({u #1 e } ∩ [-u -, u • ]) ∪ ({u #2 e } ∩ [u • , u + ])
and for |C| = 1 consider

S=({u #3 e , u #4 e }∩[-u -, u • ])∪({u #5 e , u #6 e }∩[u • , u + ]).
Then, for χ ∈ int(R χ )\E β(0) 0 (I), the optimal solution of (20) satisfies

u e (χ) ∈ argmin ue∈S |χ e (u e )| 2 . (24)
Moreover, for χ ∈ E β(0) 0 (I) the optimal solution of (20) is given by u e = 0.

Note that the set S, which can be easily constructed, contains at most 4 points. Thus, the optimal solution of (20) is obtained by calculating and comparing the objective values of u e ∈ S in (24).

Proof: The fact that u e (χ) = 0 for χ ∈ E β 0 (0)(I) follows from Lemma 1, item 2. The six values u #i e (χ), i ∈ {1, . . . , 6} are obtained by solving the constraint

|χ -χ e (u e )| 2 = (β(u e )) 2 (25)
of the optimization problem (20) for β(u e ) = u -+ u e and β(u e ) = u + -u e according to the definition of β in (16). The consideration of equality instead of inequality in (25) is justified through Lemma 1, item 3. It follows from standard calculations that u #i e (χ), i ∈ {1, 2}, represent the solutions of (25) in the case |C| = 1 while u #i e (χ), i ∈ {3, . . . , 6} are the solutions of (25) in the case |C| = 1. Since Lemma 1, item 1, guarantees that S = ∅, the assertion follows.

V. NUMERICAL ILLUSTRATION

VI. CONCLUSIONS

Through a suitably scheduled shift of the equilibrium, an enlarged estimate of the basin of attraction for bounded control laws with asymmetric saturations has been obtained. The control law is implicitly defined through an optimization problem. The focus on the single input setting enables an illustrative interpretation of the controller design and allows for an explicit representation of the control law.

Future work will address the multi input case and remove the assumption that A be non-singular, by directly representing Γ through the kernel of [A, B]. To be able to derive explicit solutions of (20) in the multi input setting, the 2 As in Example 1, the absolute values of the unknowns are constrained to be less than or equal to 10. optimization problem may be reduced to a one dimensional subspace in the kernel of [A, B], depending on the state position. Alternatively, sample-and-hold approaches may be investigated, where equilibrium pairs (x e , u e ) are only updated at discrete time steps.

Fig. 1 . 2 1 2 1

 122 Fig. 1.Level sets of the function V (x) = x P 1 x in the original coordinates (left) and in the rotated coordinates χ = P 1 21 x (right). Additionally, the subspace of induced equilibria (3) is shown in red.

Fig. 2 .

 2 Fig. 2. Sublevel sets κ xe (P ) for different (induced) equilibria for which the combination of Proposition 1 and Corollaries 1 and 2 guarantee asymptotic stability of the (induced) equilibrium xe of the closed loop system.

  expressed in the xand χ-coordinates, respectively. For χ ∈ int(R χ ) we consider the following optimization problem u e (χ) ∈ argmin ue∈(-u -,u + ) χ e (u e ) χ e (u e ) subject to |χ -χ e (u e )| ≤ β(u e ).

2 ū 2 ū A - 1

 221 on a state χ ∈ int(R χ ), optimization problem (20) implicitly defines an induced equilibrium pair (χ e , u e ), with χ e = χ e (u e ) := P 1 x e (u e ) = -Cu e := -P 1 Bu e (21) such that χ ∈ E β(u e ) χ e

Item 3 ., u 2 e with u 1 e = u 2 e

 312 Assume that |χ -χ e (u e )| < β(u e ). Since χ / ∈ E β(0) 0 (I) it follows that χ e (u e ) = 0, i.e., u e = 0. Since β(•), χ e (•) and | • | are continuous functions, there exists u # e ∈ (-u -, u + ) with |u # e | < |u e | and |χ -χ e (u # e )| < β(u # e ). Moreover, the condition |u # e | < |u e | implies that χ e (u # e ) χ e (u # e ) < χ e (u e ) χ e (u e ), which contradicts the optimality of u e and thus completes the proof. Item 4. For χ ∈ E β 0 (0)(I) it holds that u e = 0 and χ(u e ) = 0 and thus uniqueness follows since the objective function satisfies |χ e (u e )| 2 = 0 if and only if u e = 0. Consider now χ ∈ int(R χ )\E β(0) 0 (I) and let u 1 e denote two optimal solutions. Due to the objective function and the fact that χ e = -P 1 2

thus contradicting the optimality of u 1 e , u 2 e . Item 5 . 2 :

 152 Note that in the domain of interest χ ∈ int(R χ ), u e ∈ (-u -, u + ), the constraints in (20) coincide with nonpositivity of the two continuously differentiable functionsg 1 (χ, u e ) = |χ + Cu e | 2 -(u -+ u e )2 and g 2 (χ, u e ) = |χ + Cu e | 2 -(u + -u e ) 2 . Continuity of u e (•) then follows from [11, Lemma 5.3(b)] and [11, Lemma 5.4(b)] which establish lower and upper semicontinuity of u e (•), respectively. The assumptions of [11, Lemma 5.3(b)] are satisfied since u e (•) is restricted to the bounded domain u e (•) ∈ [-u -, u + ] (see property LC on [11, Page 52]). The assumptions of [11, Lemma 5.4(b)] are satisfied since the constraint qualification (CQ) (see [11, Definition 5.3]) holds because the feasible set is convex with nonempty interior (see item 1).Remark Lemma 1 establishes continuity of u e . Establishing stronger Lipschitz continuity is an open problem, in general, even though geometrical considerations suggest that Lipschitz continuity does not hold when |C| = 1.

Example 2 :Fig. 3 .

 23 Fig. 3. Closed loop solutions using the feedback law (22) converging to the origin (black). Additionally, Rx and Rχ (cyan), E β(0) 0 (P 1 ) and E β(0) 0 (I) (blue) and the subspace of induced equilibria Γ (red) are shown.

Fig. 4 . 3 :

 43 Fig. 4. Evolution of input u, state x and points ue and xe for the initial conditions x 0 = [4.8, 3] (left) and x 0 = [-1.6, -0.96] (right). The kink occurs when the slope of β(ue) changes from increasing to decreasing (left) and when the state x enters the set E β(0) 0 (P 1 ) (right).

Fig. 5 .

 5 Fig. 5. Example 2: Level sets of the function W in (23) for the dynamics defined in Example 1 and decrease of W along the solutions of (1), (22).

Fig. 6 .

 6 Fig. 6. Example 3: Estimates of the region of attraction and a closed-loop solution using the feedback law (22) for dynamics (26).

To avoid numerical problems, the absolute values of unknowns in[START_REF] Hu | Null controllability and stabilization of linear systems subject to asymmetric actuator saturation[END_REF] are additionally constrained to be less or equal to 10. Here, (8) is solved through CVX[START_REF] Grant | CVX: Matlab software for disciplined convex programming[END_REF] in Matlab.

* P. Braun and L. Zaccarian are supported in part by the Agence Nationale de la Recherche (ANR) via grant "Hybrid And Networked Dynamical sYstems" (HANDY), number ANR-18-CE40-0010.


Zeinab Mousavi

Marie-Claude Heuzey

Musa R Kamal

Emmanuel Flahaut

Pierre J Carreau
email: pcarreau@polymtl.ca

Rheological, electrical and dynamic thermomechanical properties: comparison between multiwall and double-wall carbon nanotubes in polylactide and polyamide 11

Keywords: Nanocomposites, PLA, PA11, Carbon nanotubes, MWCNT, DWCNT, Rheology

In this study multiwall carbon nanotubes (MWCNTs) and double-wall carbon nanotubes (DWCNTs) have been dispersed in polylactide (PLA) and polyamide 11 (PA11) using an internal mixer. Rheological characterization confirmed the formation of carbon nanotube (CNT) networks in PLA and PA11 attributed to well dispersed CNTs in the respective matrices. A lower rheological percolation threshold of PLA/MWCNT nanocomposites (less than 0.5 wt. %) compared to PA11/MWCNT nanocomposites (about 2 wt. %) confirmed the greater affinity of CNTs with PLA. The threshold for DWCNTs was below 0.5 wt. % in PLA and between 0.5 and 1 wt. % in PA11. PLA-based nanocomposites also showed higher electrical conductivity values compared to PA11-based nanocomposites. Nanocomposites containing DWCNTs exhibited higher electrical conductivities compared to those containing MWCNTs due to the higher aspect ratio of DWCNTs. Dynamic mechanical thermal analysis (DMTA) showed enhanced storage modulus values and reduced damping behavior with increasing content of CNTs for both polymers.

Introduction

Environmental concerns regarding the production of petroleum-based polymers and accumulation of their waste in the natural ecosystem have resulted in increasing interest in using biobased and biodegradable polymers. Polylactide (PLA) and polyamide 11 (PA11) are both derived from renewable resources. PLA is manufactured by polymerization of lactic acid or lactide, which is obtained by fermentation of starch/sugar extracted from agricultural products. [START_REF] Murariu | PLA composites: From production to properties[END_REF] In addition, it is also biodegradable. PA11 is based on castor oil. It is not biodegradable, but, due to low generation of CO 2 emission during its production, it has less negative impacts on the environment than its petroleum based counterparts. [START_REF] Venkatraman | Developing high performance PA 11/cellulose nanocomposites for industrial-scale melt processing[END_REF] PLA has high stiffness and strength. However, its low ductility, slow crystallization and low heat-resistance limit its applications. On the other hand, PA11 has higher ductility and better thermomechanical performance, compared to PLA. Several techniques have been used for improving the ductility and thermal properties of PLA. These techniques include blending with another polymer, which possesses the complementary characteristics [START_REF] Rasal | Poly(lactic acid) modifications[END_REF][START_REF] Nofar | Poly (lactic acid) blends: Processing, properties and applications[END_REF] and incorporating nanoparticles. [START_REF] Banerjee | An overview of the recent advances in polylactidebased sustainable nanocomposites[END_REF][START_REF] Raquez | Polylactide (PLA)-based nanocomposites[END_REF] Adding nanoparticles such as CNTs improves the thermal and mechanical properties of PLA. They would also impart added functionality to the material, i.e., electrical and thermal conductivity. [START_REF] Kaseem | A review on recent researches on polylactic acid/carbon nanotube composites[END_REF] Incorporating CNTs into these bio-based polymers enables the production of materials that can be used for electrostatic dissipation and electromagnetic interference shielding. Their application as liquid sensors has also been reported where the sensing properties are associated with a change in the electrical conductivity of these nanocomposites when exposed to a specific liquid. [START_REF] Kobashi | Liquid sensing of melt-processed poly(lactic acid)/multi-walled carbon nanotube composite films[END_REF] While single wall carbon nanotubes (SWCNTs) can be described by a single graphene layer rolled over itself, MWCNTs are composed of concentric carbon nanotubes arranged like Russian dolls, with a spacing of 0.34 nm between each other. [START_REF] Monthioux | Introduction to Carbon Nanotubes[END_REF] DWCNTs, with only one inner and one outer wall, are at the interface between these two categories of carbon nanotubes, combining advantages of both such as the high flexibility and mechanical strength of SWCNTs together with the protection of the inner wall and the much larger ratio of electrically conducting nanotubes. [START_REF] Seichepine | A simple and versatile method for statistical analysis of the electrical properties of individual double walled carbon nanotubes[END_REF] DWCNTs are expected to possess higher flexural modulus than SWCNTs due to the two walls and higher toughness than MWCNTs. [START_REF] Moniruzzaman | Polymer nanocomposites containing carbon nanotubes[END_REF] Dispersing nanoparticles in polymers is hindered by the inherent thermodynamics drive of nanoparticles/nanotubes to form agglomerates/bundles. The efficiency of nanoparticles in improving the properties of a polymer matrix depends on the quality of the dispersion in the matrix, which is a function of the level of interaction between the polymer chains and nanoparticles or their affinity. Melt mixing is the industrially preferred method for preparation of nanocomposites and it is more environmentally friendly (solvent free) compared to other methods such as solution casting and in-situ polymerization. It utilizes high shear/elongational forces and high temperatures to disperse the nanoparticles in a polymer. It is generally less effective than solution casting and is limited to lower contents of nanoparticles due to high viscosities at higher nanoparticle contents. The main steps involved in dispersing CNTs in a polymer matrix include wetting of primary agglomerates by the polymer and infiltration of polymer chains into the primary agglomerates followed by rupture and erosion and the distribution of individualized nanotubes into the matrix. [START_REF] Alig | Establishment, morphology and properties of carbon nanotube networks in polymer melts[END_REF] The surface energies between the nanotubes and polymer chains determines the level of interactions between these components, their chemical affinity and interfacial adhesion. Better affinity between the carbon nanotubes and the polymer matrix enhances the quality of dispersion of the nanotubes in the polymer. Improved dispersion yields a more significant effect on the rheological, mechanical and electrical properties of the polymer/CNT nanocomposites.

Morphological characterization for evaluating the nanotube dispersion in the polymer is required over a range of length scales including optical microscopy and scanning and transmission electron microscopy. Rheological characterization of nanocomposites also provides insight into the microstructure of the material.

Several publications on the dispersion of CNTs in PLA exist in the literature. Surface modification of MWCNTs for improved CNT dispersion in PLA was reported. [START_REF] Sahoo | Polymer nanocomposites based on functionalized carbon nanotubes[END_REF][START_REF] Ramontja | High-performance carbon nanotubereinforced bioplastic[END_REF][START_REF] Lin | The preparation of multiwalled carbon nanotube/poly(lactic acid) composites with excellent conductivity[END_REF] Carboxylic and hydroxyl functionalized CNTs and purified CNTs were dispersed in PLA via melt blending.

Carboxylic functionalized CNTs allowed for the best dispersion state according to transmission electron microscopy (TEM) analysis and rheological characterization, while still some bundles were present. [START_REF] Wu | Viscoelasticity and thermal stability of polylactide composites with various functionalized carbon nanotubes[END_REF] Wu et al. [START_REF] Wu | Relations between the aspect ratio of carbon nanotubes and the formation of percolation networks in biodegradable polylactide/carbon nanotube composites[END_REF] studied the effect of CNT aspect ratio on the formation of percolation networks in PLA/CNT nanocomposites. High and low aspect ratio carboxylic functionalized CNTs were added to PLA. The nanocomposites containing the higher aspect ratio CNTs yielded a lower rheological percolation threshold. In another study, PLA-grafted CNTs were obtained by reaction between acrylic acid grafted PLA and hydroxyl functionalized CNTs 18 ; these authors compared the dispersion of unmodified and hydroxyl functionalized CNTs in the modified PLA.

The latter showed a better dispersion, but the presence of some agglomerates was reported above 3 wt. % CNTs. [START_REF] Ramontja | High-performance carbon nanotubereinforced bioplastic[END_REF] In another study, PLA and CNTs were mixed without chemical modification by generating hybrid crystalline fibrils (shish) in which the extended polymer chains cover the CNTs using multi stage stretching extrusion. [START_REF] Li | Extensional flowinduced hybrid crystalline fibrils (shish) in CNT/PLA nanocomposite[END_REF] Significant enhancements in elongation at break, strength and modulus were observed compared to un-stretched PLA/CNT nanocomposites. Villmow et al. [START_REF] Villmow | Influence of twin-screw extrusion conditions on the dispersion of multi-walled carbon nanotubes in a poly(lactic acid) matrix[END_REF] prepared PLA/CNT nanocomposites through melt mixing and diluting a masterbatch via a twinscrew extruder. Higher screw rotation speeds and a screw profile containing mainly mixing elements improved the dispersion of CNTs in the PLA. A series of diluted nanocomposites from the masterbatch prepared under optimized extrusion conditions showed no agglomerates of diameter larger than 5 µm by light microscopy, an electrical percolation threshold below 0.5 wt. % CNTs and electrical conductivity (σ) of 3 × 10 -3 S.cm -1 at 0.5 wt. % CNTs.

Huang et al. [START_REF] Huang | Enhanced dielectric properties of polyamide 11/multi-walled carbon nanotubes composites[END_REF] prepared PA11/MWCNT nanocomposites by melt mixing in an internal mixer. An acceptable dispersion of MWCNTs was observed via scanning electron micrographs. The electrical conductivity of the PA11 containing 2.5 vol. % MWCNTs was about 10 -5 S.cm -1 .

Ha et al. [START_REF] Ha | Morphology and properties of polyamide/multi-walled carbon nanotube composites[END_REF] compared the dispersion of MWCNTs in various polyamides including PA6, PA11 and PA12. Polyamides with longer hydrophobic methylene groups in the repeat unit chains showed lower surface resistivities. However, the molecular weight of these polyamides were not comparable.

Agglomerates of longer or thinner nanotubes are more challenging to disperse in a polymer. This is explained by higher strength of their initial agglomerates, due to much higher surface area and stronger interactions between the tubes compared to shorter nanotubes of larger diameter. Longer tubes also tend to have more entanglements between them. [START_REF] Kasaliwal | Influence of material and processing parameters on carbon nanotube dispersion in polymer melts[END_REF] On the one hand, the dispersion of DWCNTs is more challenging than MWCNTs because the former have a much larger specific surface area, and thus they have a stronger tendency to form bundles and agglomerates in polymer matrices. On the other hand, based on their higher aspect ratios, a more significant effect on the rheological and electrical properties is expected. Due to a larger specific surface area for stress transfer at the interface when a good dispersion and adhesion are achieved, hence, DWCNTs are better candidates for reinforcing polymers compared to MWCNTs. [START_REF] Gojny | Influence of different carbon nanotubes on the mechanical properties of epoxy matrix composites -A comparative study[END_REF] The literature on DWCNT/polymer nanocomposites is scarce. Some publications on DWCNTs (or SWCNTs) in various matrices other than PLA and PA11 and for different preparation techniques are briefly discussed here.

Pristine and modified SWCNTs by styrene maleic anhydride copolymer were melt mixed with PA12. The improved dispersion for modified SWCNTs was confirmed by DMTA and mechanical characterization, but the electrical conductivity was compromised due to coverage of SWCNTs by a polymer layer. No comparison with MWCNTs was made. 25 Gojny et al., comprehensively studied the effect of MWCNTs, DWCNTs and SWCNTs and their modification on the dispersion of CNTs and on the mechanical properties of epoxy/CNT nanocomposites prepared by calendaring. [START_REF] Gojny | Influence of different carbon nanotubes on the mechanical properties of epoxy matrix composites -A comparative study[END_REF] Nanocomposites containing amino-functionalized DWCNTs at 0.5 wt. % demonstrated the most significant improvements in strength (+10%), stiffness (+15%) and fracture toughness (+43%). In another study, DWCNTs were dispersed in an epoxy using sonication and mechanical stirring. The electrical percolation threshold was between 0.2 and 0.4 wt. %. The electrical conductivity of nanocomposites containing 2.5 wt. % DWCNTs was 2 × 10 -4 S.cm -1 . [START_REF] Barrau | DC and AC Conductivity of Carbon Nanotubes-Polyepoxy Composites[END_REF] Epoxy/DWCNTs nanocomposites were prepared using palmitic acid as a surfactant to improve the dispersion of the DWCNTs. Spectroscopic Raman maps were recorded to evaluate the uniformity of the dispersed DWCNTs in the epoxy. [START_REF] Bassil | Spectroscopic detection of carbon nanotube interaction with amphiphilic molecules in epoxy resin composites[END_REF] Tishkova et al., studied the electrical conductivity of DWCNTs dispersed in a polyether ether ketone (PEEK) matrix. [START_REF] Tishkova | Dispersing carbon nanotubes in polymer nanocomposites: detecting individually dispersed tubes[END_REF] Nanocomposites were prepared by dispersing the CNTs and PEEK powder in acetone followed by probe sonication and evaporation of the acetone. The electrical percolation threshold was in the range of 0.2-0.3 wt. % and, at 2 wt. %, the electrical conductivity was 3 × 10 -2 S.cm -1 .

In summary, there are several publications on the properties of PLA containing surface-modified and unmodified MWCNTs prepared through various methods. However, there are few publications on evaluating the properties of PA11/MWCNT nanocomposites prepared by similar methods and the literature is scarce on the dispersion of DWCNTs in polymers through melt mixing. In this work, we evaluate and compare the dispersion of MWCNTs and DWCNTs in PLA and PA11 prepared under identical processing conditions. We focus on the rheological, electrical and dynamic thermomechanical properties to obtain a better understanding of the relative affinity of CNTs to these polymers and of the effect of the CNT aspect ratio on nanocomposite properties.

Experimental

Materials

PLA 3001D with a melt flow rate (MFR) of 22 g/10 min (210 °C, 2.16 kg), melting temperature of 165 °C and density of 1.24 g/cm 3 was provided by NatureWorks LLC (Minnetonka, MN, USA). PA11 Rilsan LMFO with a melting temperature of 190 °C and density of 1.02 g/cm 3 was provided by Arkema (King of Prussia, PA, USA). MWCNTs were purchased from Cheap Tubes Inc. (Grafton, VT, USA). These nanotubes were prepared by a catalytic chemical vapor deposition process (CCVD). Their specifications according to the manufacturer are: 10-20 nm diameter, 10-30 µm length, 233 m 2 /g specific surface area and purity greater than 95 wt. %.

DWCNTs were prepared by a CCVD process at CIRIMAT in Paul Sabatier University, France. [START_REF] Flahaut | Gram-scale CCVD synthesis of doublewalled carbon nanotubes[END_REF] DWCNTs have outer diameters in the 1-3 nm range, 900-1000 m 2 /g specific surface area and purity greater than 90 wt. % (typically between 90 and 92-93 wt.%, with ca. 4 wt.% of Co, 1 wt.% of Mo and the rest of oxygen). Measuring the length of the DWCNTs is challenging because these nanotubes are largely entangled and have a strong tendency to form long bundles of 3-30 nm in diameter that could reach up to 100 µm.

Nanocomposite preparation

PLA was initially dried in a vacuum oven at 60 °C overnight and PA11, MWCNTs and DWCNTs were dried at 80 °C overnight. Nanocomposites were prepared using an internal mixer, DDRV501 Brabender (C.W. Brabender Instruments Inc., South Hackensack, NJ, USA) at 200 °C, under a nitrogen atmosphere at a mixing speed of 90 RPM and a mixing time of 8 min.

Disks of 25 mm diameter and 1.5 mm thickness were compression molded at 200 °C under nitrogen for 11 min followed by 3 min at room temperature. These samples were used for rheological and morphological characterization. The compressive force was gradually increased to 29 kN. Samples were dried before each step to prevent the hydrolytic degradation of the polymers. Samples for DMTA and electrical resistivity measurements were prepared under similar conditions. The nomenclature indicates the acronym of the matrix followed by the type of carbon nanotube and its content in wt. %. For example, PLA-DWCNT2 indicates 2 wt. % of DWCNTs incorporated into PLA. CNT contents in PLA and PA11 were 0.2, 0.5, 1, 2, 3 and 5 wt. % and 0.2, 1, 2, 3 and 5 wt. %, respectively. Considering the density of CNTs, namely 2.1 g.cm -3 , these values correspond to 0.1, 0.3, 0.6, 1.1, 1.7 and 2.7 vol. % in PLA and 0.1, 0.5, 0.9, 1.4 and 2.3 vol. % in PA11. It should be noted that the content of CNTs in PA11 in vol. % are slightly lower than their content in PLA.

Characterization Morphology

Scanning electron microscopy (SEM) was carried out on nanocomposites using a field emission scanning electron microscope (FE-SEM JSM 7600F, JEOL, Tokyo, Japan) operating at 5 kV.

Samples for SEM were cryo-fractured in liquid nitrogen and a 20 nm chromium coating was deposited on the samples. Dispersion of CNTs in the polymer matrices at the nanoscale were evaluated using a JEM 2100F, JEOL, (Tokyo, Japan) transmission electron microscope (TEM).

Ultra-thin sections were cut for TEM using an ultra-microtome Ultracut FC microtome (Leica Microsystems, Wetzlar, Germany) with a diamond knife. For measuring the length of MWCNTs using TEM, a dilute suspension of MWCNTs (0.1 wt. %) in an aqueous solution of sodium dodecyl sulphate was prepared. [START_REF] Ham | An explanation of dispersion states of single-walled carbon nanotubes in solvents and aqueous surfactant solutions using solubility parameters[END_REF][START_REF] Borode | Surfactant-aided dispersion of carbon nanomaterials in aqueous solution[END_REF] The samples were sonicated using an ultrasonic bath (FS30, 100 Watts, Fischer Scientific, Pittsburg, PA, USA) for less than 5 minutes. Immediately after the sonication, the TEM grids were dipped in the solution then the excess solution was removed using a tissue paper. The observation was carried out the same day, after a few hours at the latest, so the samples can be considered as dry. ImageJ software 32 was used for quantitative analysis of the images.

Rheometry

The rheological properties of the neat polymers and nanocomposites in the linear viscoelastic range were measured using a stress controlled MCR302 rheometer (Anton Paar, Graz, Austria) at 200 °C under nitrogen. A parallel plate flow geometry was used. Time sweep tests were carried out to ensure the thermal stability of the neat polymers and nanocomposites during the small amplitude oscillatory shear (SAOS) experiments. The SAOS experiments were carried out between 0.1 and 100 rad/s.

Electrical resistivity measurements

The volume electrical resistivity of the samples was determined by measuring the DC resistance across the sample thickness using a Keithley 6517B electrometer/high resistance meter (Keithley instruments, Cleveland, OH, USA) or an Agilent 34401 A 6 ½ digital multimeter (Keysight Technologies, (formerly Agilent Technologies), Santa Rosa, CA, USA) connected with a twopoint probe. For samples with high resistance, measurements were carried out using the Keithley 6517B electrometer connected to a Keithley 8009 resistivity text fixture (Keithley instruments, Cleveland, OH, USA). 60 × 5 × 0.6 mm 3 stripes were used for the measurements. 65 × 65 × 0.6 mm 3 sheets were used for the measurements with the Keithley 8009 resistivity test fixture.

Samples were dried before the measurements and the surface of the specimens were wiped with ethanol before the tests. The voltage for the measurements was adjusted in the 1 -100 V range depending on the electrical resistivity of the samples. The electrical resistivity of a minimum of three replicates was measured for each sample. The measured volume resistances (R v ) were converted to volume resistivity (ρ v ) using the equation: ρ v = R v × A/t where A is the effective sample area and t is the thickness. The electrical conductivity (σ) is the inverse of the volume resistivity.

Dynamic mechanical thermal analysis (DMTA)

DMTA was carried out on the nanocomposite samples using DMA 2980 analyzer (TA Instruments, New Castle, DE, USA). Bars of 3.2 mm thick, 12.7 width and 60.5 mm length were tested in a dual cantilever bending mode at a strain amplitude of 25 µm and frequency of 1 Hz.

The temperature increased up to 140 °C at 5 °C/min rate. At least four replicates were tested for each sample.

Results and discussion

Morphology

Figures 1a andb show TEM images of the MWCNTs for a dilute suspension of MWCNTs (0.1 wt. %) in an aqueous solution of sodium dodecyl sulphate for two different scales. The length distribution of these MWCNTs, considering their bent morphology, is presented in Figure 2b. When exposed to ultrasonication in a bath for a short time, the length of a large population is between 0.5 µm and 3.5 µm (measured for 120 nanotubes). Figure 1b shows a single nanotube with an inner diameter of 10 nm and an outer diameter of 40 nm (about 40 concentric layers). A layer of SDS on the MWCNTs in some of the TEM images can be observed; however, it can clearly be distinguished from the MWCNTs due to a good contrast and, hence, it did not affect the diameter measurements. Figure 2a presents the distribution of the MWCNT diameter measured for 265 nanotubes. Their diameter is in the 10-45 nm range and a small percentage between 50 and 70 nm. The largest percentage is in the 20-25 nm range resulting in aspect ratios in the range of 11-350. The morphology of DWCNTs, which have a much smaller diameter and longer length compared to MWCNTs, is presented in Figures 1c andd. Single tubes and bundles of a few DWCNTs are observed in Figure 1d.

The quality of the dispersion of the carbon nanotubes in the polymer matrices was evaluated using SEM. A good dispersion is characterized by the good separation of the nanotubes into individual tubes, or at least individual small-diameter bundles. Agglomerates consist of single tubes, bundles, or both. [START_REF] Abbasi | Rheological properties and percolation in suspensions of multiwalled carbon nanotubes in polycarbonate[END_REF] A system is well distributed, but not well dispersed when the nanotube agglomerates or bundles are uniformly dispersed in the matrix. The SEM micrograph of PLA-MWCNT5 nanocomposites indicates well dispersed and well distributed MWCNTs in the PLA (Figure 3a). Figures 3b andc present micrographs of the PA11-MWCNT5 nanocomposites at two positions. The concentration in MWCNT appears higher in Figure 3c than in 3b, indicating less uniformly distributed MWCNTs in PA11 and thus a lower quality of the dispersion compared to PLA.

The dispersion of DWCNTs in a polymer matrix is much more difficult to achieve compared to MWCNTs, due to their larger aspect ratio and a strong tendency to form bundles. In the SEM micrographs of nanocomposites containing DWCNTs (Figures 4a-c), longer tubes compared to MWCNTs are observed and they appear to be present in more agglomerated form. Figures 4b and4c compare the PA11/DWCNT2 nanocomposite at two positions, showing an agglomerate in Figure 4c.

The TEM images of PLA-MWCNTs nanocomposites at 2 wt. % and 5 wt. % (Figures 5a andb, respectively) suggest a well dispersed state with the presence of many individual tubes (with diameters between 10 and 20 nm, a smaller percentage between 20 and 30 nm and very few between 30 and 40 nm) along with a few small agglomerates with diameters less than 0.3 µm as measured by the ImageJ software. Figures 5c and5d show image of PLA-DWCNT2 at two positions where the DWCNTs in the PLA have diameters between 2 and 6 nm, hence bundles of very few nanotubes.

Hansen solubility parameters (HSP)

The HSP theory defines the cohesive energy density of a chemical and its neighboring media based on the contributions from dispersive, polar and hydrogen-bonding interactions 34 and each chemical may be described by , and which are the dispersive, polar and hydrogen bonding components of the HSP, respectively. These parameters may be obtained for many solvents and polymers from a database available in HSPiP software package. [START_REF] Abbott | Hansen solubility parameters in practice software, ebook, datasets[END_REF] An indirect method for determining the unknown HSP values is through affinity tests between the chemical and solvents with known HSP values. Based on the idea of "like seeking like", a shorter distance defined below -between HSP of the material and solvent specifies higher affinity. The distance between the HSP of two materials is expressed by the following equation: 34

(1)

The HSP of nanoparticles can be used to predict their affinity and dispersion quality in solvents and polymers. A smaller distance between the HSP of components of a nanocomposite i.e.

matrix and nanoparticles means a better dispersion of the nanoparticle in that matrix and better interaction, thus a more thermodynamically favored state. [START_REF] Agrawal | Constrained nonlinear optimization for solubility parameters of poly(lactic acid) and poly(glycolic acid)-validation and comparison[END_REF] Here we estimated the HSP values of PA11 based on the HSP values of PA11 good solvents (Table Ⅰ). These solvents were mentioned in the safety data sheet of PA11 (Rilsan LMFO, Arkema). The HSP values of PA11 based on the solvents given in Table Ⅰ as good solvents and methanol as a bad solvent were estimated using the HSPiP software [START_REF] Abbott | Hansen solubility parameters in practice software, ebook, datasets[END_REF] to be: { : 19, : 6.5 and : 13.7} MPa 1/2 . Formic acid was not considered in the estimation of PA11 HSP values because its HSP values are farther from the other three solvents. We note that the limited number of solvents considered might result in inaccuracy of the estimated HSP values.

Rheology

The complex viscosity (η*) and storage modulus (G') of PLA-MWCNT and PA11-MWCNT nanocomposites obtained from SAOS measurements at 200 °C are presented in Figures 6a-d.

The neat PLA exhibits a Newtonian viscosity plateau at lower frequencies and a weak shearthinning behavior at higher frequencies. The storage modulus, G', is increasing as a function of frequency with a slope of 2 on the log-log plot of Figure 6b, as expected from the terminal behavior of a polymer. [START_REF] Ferry | Viscoelastic Properties of Polymers[END_REF] With increasing content of MWCNTs a significant increase in η* (500 to 10 6 Pa.s at 0.1 rad/s, hence 4 orders of magnitude, by adding 5 wt. % MWCNTs ) along with a reduction in the slope of G' at low frequencies is observed (Figures 6a andb, respectively).

These significant increases of η* and G' at low frequencies indicate a transition from a liquid to solid-like behavior due to formation of a percolated network of MWCNTs in the polymer attributed to well dispersed nanoparticles in the matrix. [START_REF] Abbasi | Rheological properties and percolation in suspensions of multiwalled carbon nanotubes in polycarbonate[END_REF] Increasing the content of nanoparticles results in increased interactions between nanoparticles (reduction of interparticle distance) and formation of a network. In preliminary experiments, the dispersion of commercially available surface modified MWCNTs, carboxylic and hydroxyl functionalized MWCNTs from Cheap Tubes in PLA were also examined. The untreated MWCNTs demonstrated a more significant increase of η*. Thus, no functionalized CNTs were used in the continuation of the work.

The neat PA11 also shows a Newtonian viscosity plateau at lower frequencies and a slight shearthinning behavior at high frequencies, along with a storage modulus increasing as a function of frequency with a slope of 2 on the log-log plot (Figures 6c andd In PLA an upturn in η* at low frequencies is observed by adding 0.5 wt. % MWCNTs while in PA11 at 1 wt. % only a small increase is observed. This indicates a better dispersion of MWCNTs in PLA compared to PA11 due to more interactions between the nanoparticles and polymer chains in PLA than in PA11 and the better affinity of the MWCNTs with PLA compared to PA11, confirming the HSP predictions.

The complex viscosity (η*) and storage modulus (G') of PLA-DWCNT and PA11-DWCNT nanocomposites are presented in Figures 7a-d. In PLA-DWCNT nanocomposites, adding DWCNTs at lower concentrations reduces η* (over the whole frequency range) and G' (at higher frequencies) of the nanocomposites below the values of the PLA matrix, which is explained by the degradation of the PLA matrix (Figures 7a andb, respectively). Determining the cause of this observation is out of the scope of this manuscript; however, one potential explanation could be the degradation of PLA caused by residual catalyst in the DWCNTs. [START_REF] Flahaut | Gram-scale CCVD synthesis of doublewalled carbon nanotubes[END_REF] Significant increases in η* and G' and the transition from liquid-to solid-like behavior are still observed for PLA-DWCNT nanocomposites at CNT contents above 1 wt. %. PA11-DWCNT nanocomposites also exhibit significant increases of η* and G' at low frequencies with increasing the DWCNTs content above 1 wt. % indicating the formation of nanoparticles network (Figures 7c andd). For PA11, at 2 wt. %, MWCNTs raise η* from 220 to 650 Pa.s, while DWCNTs raise it to 4500 Pa.s (at 0.17 rad/s).

At 2 wt. %, MWCNTs raise the PA11 G' to 33 vs. 650 Pa for DWCNTs. This is explained by the higher aspect ratio of DWCNTs vs. MWCNTs, which results in a lower rheological percolation threshold in nanocomposites containing DWCNTs.

An apparent yield stress also indicates the formation of an interconnected network of anisometric fillers in a polymer matrix. [START_REF] Utracki | Flow and flow orientation of composites containing anisometric particles[END_REF][START_REF] Shenoy | Rheology of Filled Polymer Systems[END_REF] The apparent yield stress for the nanocomposites is obtained by fitting a modified Herschel-Bulkley model (equation 2) to the data of the complex viscosity vs.

frequency.

(2)

= (3)
where in equation ( 2) is the magnitude of the complex modulus at the lowest frequency, is the maximum strain amplitude in the LVE range for each nanocomposite, k is a constant and n is the flow index. Table 3 reports the obtained apparent yield stresses and other parameters from fitting the modified Herschel-Bulkley model to the data. The fits of the model to the data of complex viscosity vs. frequency are also presented in Figures 6a, c and7a, c. Excellent fits of the modified Herschel-Bulkley model are observed.

Rheological percolation

The rheological percolation threshold is the value of the solid particle content above which the rheological properties increase exponentially. This value is usually determined by fitting an empirical power-law model (equation 4) to the G' vs. CNT content data: [START_REF] Moniruzzaman | Polymer nanocomposites containing carbon nanotubes[END_REF] for m > (4) where and n are power-law constants, m is the CNT content (wt. %) and is the percolation threshold (wt. %). The dependence of G' on the CNT content is demonstrated in

Electrical conductivity

The electrical resistivity measurements also provide information about the state of dispersion of electrically conductive nanoparticles in a polymer matrix. The electrical percolation threshold can also be obtained by fitting an empirical power-law model (similar to equation 4) to the data of the electric conductivity (σ) vs. CNT content. However, due to the lack of good fits of this empirical model, as for the rheological percolation approximate values of the electrical percolation threshold are presented. Figure 9 [START_REF] Abbasi | Rheological properties and percolation in suspensions of multiwalled carbon nanotubes in polycarbonate[END_REF] reported a σ value close to 10 -3 S. cm -1 at 3 wt. % MWCNTs in polycarbonate, which increased to about 10 -2 S. cm -1 at 16 wt. % MWCNTs. PA11 nanocomposites containing 2.5 vol% of MWCNTs had a σ value of 10 -5 S.cm -1 21 and 10 -1 S.cm -1 at 6 wt. % in PA12. [START_REF] Socher | Melt mixed nano composites of PA12 with MWNTs: Influence of MWNT and matrix properties on macrodispersion and electrical properties[END_REF] Comparing the rheological and electrical percolation thresholds, similar values below 0. (0.9-1.4 vol. %) and 3-5 wt. % (1.4-2.3 vol. %), respectively. A higher threshold for both the electrical and rheological percolations is observed, which could be a result of their worse dispersion compared to the other nanocomposites. Since no accurate values of the percolation thresholds could be determined, it is difficult to draw a definite conclusion on the differences between the rheological and electrical percolation thresholds. Higher threshold values for electrical vs. rheological percolation have been explained by the shorter distance between nanotubes required for electrical conductivity compared to that required for restricting polymer

Dynamic mechanical thermal analysis

In DMTA we observe increases in storage modulus of both types of nanocomposites with increasing content of CNTs (Figure 10). As expected, the storage modulus decreases with increasing temperature for all nanocomposites as they move from the glassy to the rubbery state. The storage modulus of PLA based nanocomposites increases again at higher temperatures due to the cold crystallization of PLA (Fig. 10a). The onset temperature of the PLA cold crystallization decreases with higher MWCNT content. Adding DWCNTs reduces the storage modulus of PLA, which is due to the degradation of the PLA as observed in rheology. reported a 30% increase in E' from 554 to 728 MPa (at 40 °C) and a 90% increase from 164 to 315 MPa (at 80 °C) by adding 6 wt. % SWCNTs to PA12. T g increased from 51.9 to 56.1 °C. was more difficult to achieve compared to MWCNTs, due to the larger aspect ratio and the presence of much stronger primary agglomerates and bundles. The SAOS data showed the transition from a liquid-to a solid-like behavior for both nanocomposites containing DWCNTs.

Conclusion

The percolation threshold of PLA based nanocomposites was lower compared to that of PA11, however a severe degradation of the PLA was observed when DWCNTs were used. In PA11,

DWCNTs formed networks at a lower percolation threshold compared to MWCNTs, due to their higher aspect ratio. The electrical conductivity of nanocomposites based on PLA were higher compared to those based on PA11, confirming the better affinity of CNTs with PLA. DWCNTs enhanced more significantly the electrical conductivity of PLA-based nanocomposites compared to MWCNTs at the same load, due to their higher aspect ratio resulting in lower electrical percolation threshold. Finally, enhanced storage modulus values were observed in DMTA for nanocomposites containing MWCNTs in both matrices, but the improvement was more significant in PLA than in PA11. A decrease in the peak of tan δ was also observed with increasing CNT content for both PLA and PA11 nanocomposites, showing the restrained mobility of polymer chains due to the dispersed nanoparticles. PA11 based nanocomposites containing DWCNTs exhibited a similar trend and at lower contents compared to MWCNTs.

Figure 1 .

 1 Figure 1. TEM images of (a-b) MWCNTs and (c-d) DWCNTs 29 .

Figure 2 .

 2 Figure 2. (a) Diameter and (b) length distribution of pristine MWCNTs.

Figure 3 .

 3 Figure 3. SEM micrographs of (a) PLA-MWCNT5 and (b-c) PA11-MWCNT5. Scale bars are 1 µm.

Figure 4 .

 4 Figure 4. SEM micrographs of (a) PLA-DWCNT2 and (b-c) PA11-DWCNT2. Scale bars are 100 nm.

Figure 5 .

 5 Figure 5. TEM images of (a) PLA-MWCNT2 (b) PLA-MWCNT5 and (c-d) PLA-DWCNT2.

  , respectively). Unlike many other grades of polyamide, this grade is thermally stable (less than 10% increase of η* at 200 °C during 40 min time sweep tests). The transition from liquid-to solid-like behavior in rheological properties is observed for the nanocomposites based on PA11 at higher MWCNT contents compared to those based on PLA.

Figure 6 .

 6 Figure 6. Complex viscosity versus angular frequency of (a) PLA-MWCNT and (c) PA11-MWCNT nanocomposites. Storage modulus versus angular frequency of (b) PLA-MWCNT and (d) PA11-MWCNT nanocomposites.

Figure 7 .

 7 Figure 7. Complex viscosity versus angular frequency of (a) PLA-DWCNT and (c) PA11-DWCNT nanocomposites. Storage modulus versus angular frequency of (b) PLA-DWCNT and (d) PA11-DWCNT nanocomposites.

Figure 8 .

 8 Figure 8. Due to the lack of good fits of this empirical model to the data, approximate values of the rheological percolation threshold are reported. A lower percolation threshold is observed for PLA-MWCNT compared to PA11-MWCNT (below 0.5 wt. % (0.3 vol. %) vs. between 2 and 3 wt. % (0.9 and 1.4 vol. %), respectively), confirming the better affinity of MWCNTs with PLA.For nanocomposites containing DWCNTs, G' values of PLA are affected by the degradation, but the percolation threshold is below 0.5 wt. % (0.3 vol.%). With PA11, at a specific nanoparticle content, the storage modulus is larger for PA11-DWCNT compared to PA11-MWCNT, due to the higher aspect ratio of the DWCNTs resulting in a lower percolation threshold between 0.2 and 1 wt. % (0.1-0.5 vol.%).

Figure 8 .

 8 Figure 8. Storage modulus as a function of CNT content (at 0.32 rad/s) for nanocomposites containing (a) MWCNTs and (b) DWCNTs.

  5 wt. % (0.3 vol. %) are obtained for the PLA based nanocomposites. The rheological and electrical percolation threshold values of the PA11 based nanocomposites containing DWCNTs are also in the range of 0.2-1 wt. % (0.1-0.5 vol. %) while for PA11 based nanocomposites containing MWCNTs, rheological and electrical percolation threshold values are in the range of 2-3 wt. %

Figure 9 .

 9 Figure 9. Electrical conductivity as a function of CNT content of nanocomposites containing (a) MWCNTs and (b) DWCNTs.

Figure 10 .

 10 Figure 10. (a and c) Dynamic storage modulus, E'. (b and d) tan δ as a function of temperature for nanocomposites based on PLA and PA11 containing MWCNTs or DWCNTs.

Figures 10b and d

  Figures 10b and d report the loss tangent, tan δ, where δ is the phase angle vs. temperature. Increasing the content of CNTs in both matrices reduces the area under the tan δ curve, indicating the more significant restrained polymer chain mobility due to presence of well dispersed carbon nanotubes. The tan δ peak is a characteristic of the glass transition temperature, T g , of the nanocomposites. The T g of nanocomposites based on PLA does not change

  PLA and PA11 nanocomposites containing various amounts of MWCNTs and DWCNTs were prepared using melt mixing. The dispersion quality of MWCNTs and DWCNTs in PLA and PA11 was evaluated through various characterization techniques. SEM and TEM micrographs revealed that MWCNTs were well dispersed and well distributed in PLA. Some agglomerates were observed in PA11/MWCNT nanocomposites. Rheological characterization confirmed the formation of a percolated network of MWCNTs in both PLA and PA11 by the non-terminal character of G' at low frequencies. The rheological percolation thresholds of MWCNTs in PLA were lower compared to those of PA11, indicative of the better affinity of MWCNTs for PLA than for PA11 as pointed out by the HSP predictions. The dispersion of DWCNTs in polymers

Table Ⅰ .

 Ⅰ HSP values of good solvents for PA11 (Rilsan, LMFO).[START_REF] Abbott | Hansen solubility parameters in practice software, ebook, datasets[END_REF]

	Solvent	(MPa 1/2 )	(MPa 1/2 )	(MPa 1/2 )
	m-cresol	18.5	6.5	13.7
	phenol	18.5	5.9	14.9
	Benzyl alcohol	18.4	6.3	13.7
	Formic acid	14.6	10.0	14.0
	(concentrated)

Table Ⅱ

 Ⅱ

		shows the HSP values of PLA, PA12 (close to PA11), PA11, MWCNTs and DWCNTs
	and the	values of PLA/MWCNT and PA11/MWCNT pairs. PLA 1 and PLA 2 correspond to

two sets of HSP values reported in the HSPiP software database for PLA. PA11 1 corresponds to the HSP values for PA11.

[START_REF] Abbott | Hansen solubility parameters in practice software, ebook, datasets[END_REF]

PA11 2 correspond to the values estimated above. As can be seen in

Table Ⅱ ,

 Ⅱ the distance between the HSP values are higher for PA11/CNT pairs than PLA/CNT pairs, indicating a better affinity of CNTs for PLA as compared to PA11.

Table Ⅱ .

 Ⅱ HSP values of PLA, PA12, PA11[START_REF] Abbott | Hansen solubility parameters in practice software, ebook, datasets[END_REF] and MWCNT and DWCNTs 37 .

					(Polymer/MWCNT)	(Polymer/DWCNT)
		(MPa 1/2 )	(MPa 1/2 )	(MPa 1/2 )	(MPa 1/2 )	(MPa 1/2 )
	PLA 1	18.6	9.7	6.0	3.5	2.9
	PLA 2	18.5	8.0	7.0	2.8	2.6
	PA12	18.5	8.1	9.1	4.4	4.3
	PA11 1	17.0	4.4	10.6	7.5	7.8
	PA11 2	19.0	6.5	13.7	8.3	8.5
	MWCNT	19.5	6.7	5.5	-	-
	DWCNTs	19.4	7.4	5.3	-	-

Table Ⅲ .

 Ⅲ Modified

	Herschel-Bulkley model parameter for (a) PLA-MWCNT (b) PA11-
	MWCNT and (c) PA11-DWCNT nanocomposites.
	CNT wt. %			n
	(a) PLA-MWCNT
	0.5	3	663	0.89
	1	64	787	0.83
	2	353	1160	0.77
	3	478	1810	0.68
	5	2420	2300	0.41
	(b) PA11-MWCNT
	1	0	256	0.96
	2	0	318	0.86
	3	42	589	0.67
	5	190	1080	0.43
	(c) PA11-DWCNT
	0.2	0	247	0.97
	1	0	355	0.89
	2	17	677	0.78

  reports the electrical conductivity of PLA and PA11 based nanocomposites. The nanocomposites based on PLA demonstrate higher σ values than those based on PA11 at the same loading of CNTs due to the better dispersion, confirming again S.cm -1 . Higher values of σ have been reported for nanocomposites containing MWCNTs (especially for PA11), probably due to different degrees of dispersion of nanoparticles. Villmow et al.[START_REF] Villmow | Influence of twin-screw extrusion conditions on the dispersion of multi-walled carbon nanotubes in a poly(lactic acid) matrix[END_REF] , obtained a σ value of 2 × 10 -2 S. cm -1 at 2 wt. % MWCNTs in PLA, but the electrical percolation threshold was not much lower than the values obtained in this study for PLA/MWCNT.Abbasi et al.,

	the better affinity of CNTs with PLA than with PA11. Nanocomposites containing DWCNTs
	impart higher electrical conductivity in both matrices compared to MWCNTs due to their larger
	aspect ratio and hence lower electrical percolation threshold. The electrical percolation threshold
	for PLA based nanocomposites is below 0.5 wt. % (0.3 vol. %) for both nanoparticles. The
	threshold for PA11 containing MWCNTs and DWCNTs is between 3 and 5 wt. % (1.4 and 2.3
	vol. %), and between 0.2 and 1 wt. % (0.1 and 0.5 vol. %), respectively. The highest electrical
	conductivity is observed for PLA-MWCNT10 at 2.6 ×10 -4 S.cm -1 . PLA-DWCNT2 shows 2.3 ×
	10 -5

  The increases of the storage modulus due to the addition of nanoparticles in the rubbery state are larger than that at the glassy state for all nanocomposites. For example, E' of PLA at 25 °C

	increases by 20% (from 1830 to 2206 MPa) by adding 5 wt. % MWCNTs, while at 80 °C it
	increases by 127% (from 33 to 76 MPa). In PA11-MWCNT5 nanocomposites, E' increases by
	12% (from 1020 to 1144 MPa) in the glassy state (25 °C) while in the rubbery state it increases
	by 46% (from 301 to 438 MPa) by adding 5 wt. % MWCNTs. Adding 2 wt. % DWCNTs to
	PA11 raises E' by 9% and 42% in the glassy and rubbery states, respectively.

Acknowledgments

Financial support from the Natural Science and Engineering Research Council (NSERC) of Canada is gratefully acknowledged. The authors also wish to acknowledge the help of Dr. Helia Sojoudiasli and Mr. Matthieu Gauthier for material preparation and processing. Finally, we wish to thank Mr. Jed Randall from NatureWorks for providing the PLA sample and Dr. Francois Bargain from Arkema for providing the PA11 sample.