Michel Cristofol
email: michel.cristofol@univ-amu.fr

Isma Kaddouri
email: isma.kaddouri@gmail.com

Grégoire Nadin
email: nadin@ann.jussieu.fr

Lionel Roques
email: lionel.roques@avignon.inra.fr

Coefficient determination via asymptotic spreading speeds

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

Introduction

This paper is devoted to the reconstruction of the linear part ∂ u f (x, 0) of the reaction term f (x, u) in the following reaction-diffusion problem:

(P ) ∂ t u -D∂ xx u = f (x, u), t > 0, x ∈ R, u(0, x) = u 0 (x) ≥ 0, x ∈ R.
This equation describes the space-time evolution of a concentration u(t, x) in a heterogeneous excitable environment [START_REF] Berestycki | Front propagation in periodic excitable media[END_REF][START_REF] Xin | Front propagation in heterogeneous media[END_REF]. It appears in several fields of applications such as physics, in combustion flame propagation models [START_REF] Berestycki | Quelques aspects mathématiques de la propagation des flammes prémélangées Nonlinear PDEs and their applications[END_REF], in chemistry [START_REF] Billingham | The development of traveling waves in quadratic and cubic autocatalysis with unequal diffusion rates I. permanent form of traveling waves Phil[END_REF], in ecology, to study the dynamics of a population [START_REF] Murray | Mathematical Biology Third Edition[END_REF][START_REF] Turchin | Quantitative analysis of movement: measuring and modeling population redistribution in animals and plants[END_REF] as well as in population genetics [START_REF] Aronson | Nonlinear diffusion in population genetics, combustion and nerve propagation Partial Differential Equations and Related Topics Lectures Notes in Mathematics[END_REF][START_REF] Aronson | Multidimensional nonlinear diffusion arising in population genetics[END_REF][START_REF] Roques | Allee effect promotes diversity in traveling waves of colonization[END_REF].

For Kolmogorov-Petrovsky-Piskunov type (KPP) nonlinearities (see definition below), it is well known that the solution u(t, x) of the problem (P ) mainly depends on the reaction term f (x, u) through its linearization ∂ u f (x, 0) around the steady state 0 (see for exemple [START_REF] Berestycki | Analysis of the periodically fragmented environment model: I-Species persistence[END_REF][START_REF] Cantrell | Spatial ecology via reaction-diffusion equations[END_REF]). In population dynamics, this term represents the intrinsic growth rate of a population. The persistence or the extinction of the population and its rate of range expansion only depend on f (x, u) through this term ∂ u f (x, 0) [START_REF] Berestycki | Front propagation in periodic excitable media[END_REF][START_REF] Berestycki | Analysis of the periodically fragmented environment model: I-Species persistence[END_REF][START_REF] Berestycki | Analysis of the periodically fragmented environment model: II -Biological invasions and pulsating travelling fronts[END_REF][START_REF] Freidlin | On the propagation of concentration waves in periodic and random media[END_REF][START_REF] Hamel | viscosity solution method for the spreading speed formula in slowly varying media Indiana[END_REF]. In practice, this term is often unknown or partially known.

In this work, we assume that D > 0, and that x → f (x, u) is a 1-periodic reaction term, that is:

∀x ∈ R, ∀s ∈ R, f (x, s) = f (x + 1, s). (1)
The reconstruction of coefficients for parabolic equations is a widely studied problem.

Several works are done in the case of parabolic operators defined on an open bounded set (see among others [START_REF] Alifanov | Extreme Methods for Solving Ill-Posed Problems with Application to Inverse Heat Transfer Problems[END_REF][START_REF] Chouli | Stable determination of semilinear term in a parabolic equation[END_REF][START_REF] Duchateau | Unicity in an inverse problem for an unknown reaction term in a reaction-diffusion equation[END_REF][START_REF] Egger | Global uniqueness and Hölder stability for recovering a nonlinear source term in a parabolic equation Inverse[END_REF][START_REF] Immanuvilov | Lipschitz stability in inverse parabolic problems by the Carleman estimate[END_REF][START_REF] Lorenzi | An inverse problem for a semilinear parabolic equation[END_REF][START_REF] Pilant | An inverse problem for a nonlinear parabolic equation[END_REF]). However in the unbounded case considered here, there are few references ( [START_REF] Kaddouri | Inverse problem for a parabolic equation with periodic and nonsmooth coefficients[END_REF][START_REF] Klibanov | Global uniqueness of a multidimensional inverse problem for a nonlinear parabolic equation Inverse[END_REF]) about coefficient identification.

In the case of bounded domains, several kinds of observations can be used for the determination of a coefficient. Many results are based on a method involving Carleman inequalities, which has been introduced in the seminal paper of Bukhgeim and Klibanov [START_REF] Bukhgeim | Uniqueness in the large of a class of multidimensional inverse problems Soviet Mathematics[END_REF]. This method involves, in addition to localized observations of the solution u(t, x) of the considered problem, an observation of the solution on the whole domain of study, at some fixed time. Following the approach by Imanuvilov and Yamamoto [START_REF] Immanuvilov | Lipschitz stability in inverse parabolic problems by the Carleman estimate[END_REF] for the reconstruction of a potential in the linear case, several results have been obtained in the nonlinear case [START_REF] Cristofol | Biological invasions: deriving the regions at risk from partial measurements[END_REF][START_REF] Cristofol | An inverse problem involving two coefficients in a nonlinear reaction-diffusion equation[END_REF][START_REF] Cristofol | Stable estimation of two coefficients in a nonlinear Fisher KPP equation Inverse Problems[END_REF][START_REF] Tort | Determination of the insolation function in the nonlinear Sellers climate model[END_REF]. More recently, other methods based on the maximum principle and the Hopf's lemma allowed to obtain uniqueness results using only pointwise observations in the one-dimensional case [START_REF] Cristofol | Uniqueness from pointwise observations in a multi-parameter inverse problem[END_REF][START_REF] Roques | On the determination of the nonlinearity from localized measurements in a reaction-diffusion equation[END_REF][START_REF] Roques | The inverse problem of determining several coefficients in a nonlinear Lotka-Volterra system Inverse[END_REF].

The study of the direct problem (P ) has been widely developed recently (see e.g. [START_REF] Berestycki | Front propagation in periodic excitable media[END_REF][START_REF] Berestycki | Elliptic eigenvalue problems with large drift and applications to nonlinear propagation phenomena[END_REF][START_REF] Berestycki | Asymptotic spreading in heterogeneous diffusive excitable media[END_REF][START_REF] Berestycki | Analysis of the periodically fragmented environment model: I-Species persistence[END_REF][START_REF] Berestycki | Spreading speeds for one-dimensional monostable reaction-diffusion equations[END_REF][START_REF] Weinberger | On spreading speeds and traveling waves for growth and migration in periodic habitat[END_REF][START_REF] Xin | Existence of planar flame fronts in convective-diffusive periodic media[END_REF]). The paper [START_REF] Berestycki | Elliptic eigenvalue problems with large drift and applications to nonlinear propagation phenomena[END_REF] is devoted to some nonlinear propagation phenomena in periodic and more general domains, for reaction-diffusion equations with KPP nonlinearities [START_REF] Kolmogorov | Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique Bull[END_REF]. In [START_REF] Berestycki | Spreading speeds for one-dimensional monostable reaction-diffusion equations[END_REF], the authors establish spreading properties for the solutions of equations of the form ∂ t u -a(x)∂ xx u -q(x)∂ x u = f (x, u). A common point of these works concerns the precise study of the speed of propagation of the solutions and of the asymptotic spreading speed.

On the other hand, the inverse problems associated with this class of equations have been little studied. In [START_REF] Choi | Inverse problem for a parabolic equation with space-periodic boundary conditions by a Carleman estimate Inverse[END_REF], the author treated the case of a linear parabolic operator in R n , associated with boundary and initial conditions which lead to a periodic solution.

Then the problem stated in R n can be easily written into a problem stated in a bounded cell. The observations required to solve this inverse problem have to be measured on a set which includes all the edges of the cell which induces a strong constraint. In a recent work [START_REF] Kaddouri | Inverse problem for a parabolic equation with periodic and nonsmooth coefficients[END_REF] the authors have improved this result by considering a nonlinear inverse parabolic problem with non-smooth coefficients and have obtained a more general reconstruction result without the above-mentioned constraint on the observation set.

Here, the initial condition u 0 is not necessarily periodic, and the solution u(t, x) is therefore non-periodic in general. In this context, we use a new type of observations, which are based on the spreading properties of the solutions of the Cauchy problem (P ). Before going further on, we recall the classical Kolmogorov-Petrovsky-Piskunov assumptions (denoted by KPP in reference to the seminal work [START_REF] Kolmogorov | Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique Bull[END_REF]):

       ∀x ∈ R, f (x, 0) = 0, ∃M > 0, such that for s ≥ M, and ∀x ∈ R, f (x, s) ≤ 0, ∀x ∈ R, s → f (x, s) s is decreasing for s > 0.
(

) 2
Under these hypotheses, the 1-periodicity assumption [START_REF] Alifanov | Extreme Methods for Solving Ill-Posed Problems with Application to Inverse Heat Transfer Problems[END_REF], and assuming that the steady state 0 is linearly unstable (see Proposition 2.1), it is known that the solution of the Cauchy problem (P ), starting with a compactly supported or Heaviside initial condition, propagates (to the right) with a finite asymptotic spreading speed w * [START_REF] Freidlin | On the propagation of concentration waves in periodic and random media[END_REF] in the following sense:

   lim t→+∞ u(t, x + ct) = 0, ∀c > w * , x ∈ R, lim inf t→+∞ u(t, x + ct) > 0, ∀ 0 ≤ c < w * , x ∈ R. (3)
This means that an observer moving to the right with a speed c larger than w * will see the solution converge to 0, whereas if he moves with a speed smaller than w * , he will see the solution remain above some positive constant. The existence of this constant asymptotic spreading speed, as well as the existence of traveling wave solutions which propagate with constant speeds c ≥ w * [START_REF] Berestycki | Front propagation in periodic excitable media[END_REF][START_REF] Berestycki | Analysis of the periodically fragmented environment model: II -Biological invasions and pulsating travelling fronts[END_REF][START_REF] Weinberger | On spreading speeds and traveling waves for growth and migration in periodic habitat[END_REF], are the key features which contributed to the success of the reaction-diffusion framework in applied sciences [START_REF] Murray | Mathematical Biology Third Edition[END_REF][START_REF] Shigesada | Biological Invasions: Theory and Practice Oxford Series in Ecology and Evolution[END_REF][START_REF] Turchin | Quantitative analysis of movement: measuring and modeling population redistribution in animals and plants[END_REF] since the pioneering work [START_REF] Kolmogorov | Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique Bull[END_REF]. In particular, whenever the parameters D, ∂ u f (x, 0) and u 0 in (P ) are properly fitted to experimental data, the predicted asymptotic spreading speed w * is often in accordance with observations of species range expansions [START_REF] Shigesada | Biological Invasions: Theory and Practice Oxford Series in Ecology and Evolution[END_REF]. It is therefore natural to consider the inverse problem of coefficient determination in equation (P ), based on observations of the spreading properties of its solution u(t, x).

The main objective of this paper is to determine the linear part

r(x) = ∂ u f (x, 0) (4)
of the reaction term f (x, u), using observations of the spreading speed of the solution of the Cauchy problem (P ). In that respect, instead of considering compactly supported or Heaviside initial conditions, we consider a 1-parameter family of initial conditions (u 0,λ ) λ∈I⊂(0,+∞) , which decay like e -λ x as x → +∞, and we use the associated asymptotic spreading speeds w λ as observations. The outline of the paper is as follows. In Section 2, we give our assumptions and we state our main result, in Sections 3-5 we detail the proofs of our results.

Assumptions and main results

Assumptions on f

In addition to the periodicity assumption (1) and the KPP assumption (2), we assume that the nonlinearity (x, u) → f (x, u) is of class C 0,α with respect to x locally uniformly in u ∈ R, and of class C 1 with respect to u. Besides, we assume that the linear part of the reaction term, r(x) = ∂ u f (x, 0) belongs to the function space M 1 defined by:

M 1 = {1-periodic function ρ ∈ C 0 (R), linear in [0, θ) and [θ, 1)}, (5)
for some (known) constant θ ∈ (0, 1). In other words, r(x) is a piecewise linear function.

Stationary states: existence, uniqueness and stability

Under our assumptions on the reaction term f and for continous and bounded initial conditions u 0 , it is known that the solution of the Cauchy problem (P ) converges when t → +∞ to a nonnegative stationary solution, that is, a solution p ≥ 0 of the problem:

-D ∂ xx p = f (x, p), x ∈ R. (6)
More precisely, in [START_REF] Berestycki | Analysis of the periodically fragmented environment model: I-Species persistence[END_REF] the authors have given a necessary and sufficient condition for the existence of a positive stationary state p. This condition is based on the description of the linear stability of the steady state 0, i.e., on the sign of the principal eigenvalue of the linear operator L 0 :

L 0 : ψ → -D∂ xx ψ -r(x)ψ, (7)
with periodicity assumptions on ψ. We recall that, from the Krein-Rutman Theorem [START_REF] Gilbarg | Elliptic Partial Differential Equations of Second Order[END_REF], there exists a unique k 0 ∈ R and a unique function

ψ 0 ∈ C 2 (R) such that          -D∂ xx ψ 0 -r(x)ψ 0 = k 0 ψ 0 , x ∈ R, ψ 0 is 1-periodic, ψ 0 (x) > 0, x ∈ R, ψ 0 (0) = 1. ( 8
)
The results of [START_REF] Berestycki | Analysis of the periodically fragmented environment model: I-Species persistence[END_REF] show that, under our assumptions on the function f, Proposition 2.1. 1) The equation ( 6) admits a positive and bounded solution p > 0 if and only if k 0 < 0.

2) If k 0 < 0, the bounded solution p > 0 of equation ( 6) is unique and 1-periodic in x.

3) Let u 0 ∈ C 0 (R) be bounded and such that u 0 ≥ 0, u 0 ≡ 0. If k 0 ≥ 0, then the solution u(t, x) of (P ) converges to 0 uniformly in x as t → +∞. If k 0 < 0, then u(t, x) converges to the solution p > 0 of equation ( 6) locally uniformly in x, as t → +∞.

The point 3) of the above proposition shows that, whatever the initial condition, the asymptotic spreading speed of the solution u(t, x) is equal to 0 if k 0 ≥ 0. In the remaining part of the manuscript, we need to observe positive spreading speeds. As a consequence, we have to assume that k 0 < 0. A sufficient condition on r(x) such that k 0 < 0 is (see proposition 2.9 in [START_REF] Berestycki | Analysis of the periodically fragmented environment model: I-Species persistence[END_REF]):

1 0 r(x) dx ≥ 0, with r ≡ 0 on [0, 1]. ( 9
)

Initial conditions and asymptotic spreading speed

In the case of Heaviside and compactly supported initial conditions, the asymptotic spreading speed w * exists, does not depend on this initial condition, is finite and satisfies (3). Now, we consider a family of bounded front-like initial conditions u 0,λ satisfying for some α ∈ (0, 1), x 0 > 0 and all λ > 0:

u 0,λ ∈ C 0,α (R), u 0,λ ≥ 0, lim inf x→-∞ u 0,λ (x) > 0, (10)
and

u 0,λ (x) = e -λx for x ≥ x 0 . (11)
We define the asymptotic spreading speed of the solution u λ (t, x) of the Cauchy problem (P ) starting with the initial condition u 0 (x) = u 0,λ (x) as the nonnegative real number w λ such that:

   lim t→+∞ u λ (t, x + ct) = 0, ∀c > w λ , x ∈ R, lim inf t→+∞ u λ (t, x + ct) > 0, ∀ 0 ≤ c < w λ , x ∈ R. (12)
Depending on the initial condition, the asymptotic spreading speed could be finite or not [START_REF] Hamel | Fast propagation for KPP equations with slowly decaying initial conditions[END_REF][START_REF] Roques | Recolonisation by diffusion can generate increasing rates of spread[END_REF]. In the case of Heaviside and compactly supported initial conditions, we have already mentioned that the asymptotic spreading speed w * is finite and (3) is satisfied. Besides, this asymptotic spreading speed is characterised by the following formula (see [START_REF] Freidlin | On the propagation of concentration waves in periodic and random media[END_REF]):

w * = min λ>0 -k λ λ , (13)
where k λ is the principal eigenvalue of the operator:

L λ : ψ λ → -Dψ ′′ λ + 2λDψ ′ λ -λ 2 Dψ λ -r(x)ψ λ , (14)
with periodicity conditions, where r(x) = ∂ u f (x, 0) . The next result describes the asymptotic spreading speed in terms of the initial condition. It is well known in the case of Heaviside and compactly supported initial conditions. To the best of our knowledge, it is not clearly stated in the existing literature for exponentially decaying initial conditions. In this proposition we show the existence of a finite asymptotic spreading speed w λ and we give a formula for w λ , under our assumptions on the initial conditions u 0,λ . Proposition 2.2. Assume that f satisfies (1), (2), the regularity assumptions of Section 2.1 and that k 0 < 0 (see Section 2.2). Let u 0,λ satisfy (10), [START_REF] Billingham | The development of traveling waves in quadratic and cubic autocatalysis with unequal diffusion rates I. permanent form of traveling waves Phil[END_REF] and u λ (t, x) be the solution of problem (P ) with initial condition u 0,λ (x). Then, we can associate to u λ (t, x) a finite asymptotic spreading speed w λ such that

   w λ = -k λ λ if 0 < λ < λ * , w λ = w * otherwise, (15)
where k λ is the principal eigenvalue of the operator defined by ( 14) and λ * = λ * (r) is the unique positive real number such that w * = -k λ * λ * . Proof. See the proof of this proposition in Section 4.

Uniqueness result

Finally we state our main uniqueness result.

Theorem 2.1. Let f (resp. f ) check ( 1), ( 2), the regularity assumptions of Section 2.1 and assume that k 0 < 0 (resp. k0 < 0 ) and r(x) ∈ M 1 (resp. ∂ u f (x, 0) = r(x) ∈ M 1 ). Let u λ (t, x) (resp. ũλ (t, x)), be the solution of the Cauchy problem (P ) (resp. ( P )) associated to f and u 0,λ (x) (resp. f and u 0,λ (x)). If we assume that the asymptotic spreading speeds w λ (resp. wλ ) associated to u λ (resp. ũλ ) coincide on a continuum of values of λ, that is to say

∃λ 1 > 0 such that ∀λ ∈ (0, λ 1 ), w λ = wλ , then r(x) = r(x) or r(x) = r(-x + θ).
This result shows that the linear part r(x) = ∂ u f (x, 0) of the reaction term f (x, u) in (P ) is uniquely determined (up to a symmetry) by the knowledge of the asymptotic spreading speeds w λ , for λ ∈ (0, λ 1 ). The proof is based on the equality of the first and the second moments of the functions r(x) and r(x) (see Lemma 3.2 and Lemma 3.3). The derivation of the equality of the higher-order moments would be more involved, but would enable one to extend the result of Theorem 2.1 to more general classes of functions than those in M 1 .

Proof of the main Theorem

Our goal is to reconstruct any potential r(x) element of M 1 , from observations of the asymptotic spreading speed which corresponds to a physical and measurable observation. Let w λ (r) and w λ (r) be the asymptotic spreading speed associated to r and r respectively. From the assumption of Theorem 2.1, we have w λ (r) = w λ (r) for all λ ∈ (0, λ 1 ). Thus, from Proposition 2.2, k λ (r) = k λ (r) for all λ ∈ (0, min(λ 1 , λ * (r), λ * (r))). This result follows from the analyticity of the coefficients in [START_REF] Choi | Inverse problem for a parabolic equation with space-periodic boundary conditions by a Carleman estimate Inverse[END_REF] and from the simplicity of k λ , the principal eigenvalue of L λ (see [START_REF] Kato | Perturbation theory for linear operators[END_REF]).

The analyticity of the functions λ → k λ (r) and λ → k λ (r) implies that (H) : k λ (r) = k λ (r), for all λ > 0.

We want to prove that the property (H) implies that: r(x) = r(x) or r(x) = r(-x + θ).

For this, we consider r(x), r(x) two elements of M 1 such that

r(x) = ax + b if x ∈ [0, θ), r(x) = cx + d if x ∈ [θ, 1), for a, b, c, d ∈ R, and
r(x) = ãx + b if x ∈ [0, θ), r(x) = cx + d if x ∈ [θ, 1), for ã, b, c, d ∈ R.
Using the regularity and periodicity hypothesis of r(x), we deduce

   r(x) = ax + b, for x ∈ [0, θ), r(x) = - aθ 1 -θ x + b + aθ 1 -θ , if x ∈ [θ, 1). ( 16
)
Thus, from ( 16), there exist

A, B ∈ R such that r(x) = Ar(x) + B, ∀x ∈ R. (17)
The proof will be developed in several parts. In a first step we prove the lemma: Lemma 3.2. For any r, r ∈ M 1 and verifying the hypothesis (H), we have:

1 0 r(x) dx = 1 0 r(x) dx = r.
Proof. We consider the following problem:

         -Dψ ′′ λ + 2λDψ ′ λ -(λ 2 D + r(x))ψ λ = k λ (r)ψ λ in R, ψ λ > 0, ψ λ 1-periodic , ψ λ (0) = 1. ( 18
)
According to [START_REF] Berestycki | Elliptic eigenvalue problems with large drift and applications to nonlinear propagation phenomena[END_REF] (Theorem 2.1 p.8), we have as λ → +∞

k λ (r) + λ 2 D → - 1 0 r(x) dx, and k λ (r) + λ 2 D → - 1 0 r(x) dx. (19)
Thus, using property (H) and [START_REF] Cristofol | Stable estimation of two coefficients in a nonlinear Fisher KPP equation Inverse Problems[END_REF] we deduce the result of Lemma 3.2 (see another proof of this result in Lemma 5.1). Lemma 3.3. For any r, r ∈ M 1 , and verifying the property (H), we have:

1 0 (r(x) -r) 2 dx = 1 0 (r(x) -r) 2 dx.
Proof. See the proof of this Lemma in Section 5.

From ( 17) we obtain

1 0 r2 (x)dx = A 2 1 0 r 2 (x)dx + 2AB 1 0 r(x)dx + B 2 ,
by using Lemma 3.2 and Lemma 3.3 we get

(1 -A 2 ) 1 0 r 2 (x)dx = B (1 + A) 1 0 r(x)dx , for A = -1 we obtain (1 -A) 1 0 r 2 (x)dx = B 1 0 r(x)dx,
by using the fact that B = (1 -A)

1 0 r(x)dx, for A = 1, we conclude that 1 0 r 2 (x)dx = 1 0 r(x)dx 2 . (20)
From the Cauchy inequality, the equality ( 20) is true if and only if r is constant. Thanks to [START_REF] Cristofol | Biological invasions: deriving the regions at risk from partial measurements[END_REF] we deduce that r is also constant, and according to Lemma 3.2 we obtain r = r. By symmetry we also deduce that if r is constant then r is constant so r = r. Now we consider the case A = 1. We have

1 0 r(x)dx = 1 0 r(x) dx + B.
By using the fact that 1 0 r(x) dx = 1 0 r(x) dx, we conclude that B = 0, so r(x) = r(x).

Finally, if A = -1, then r(x) = -r(x) + B, so r(x) = r(-x + θ).

Remark 1. In the homogeneous case, i.e, if f does not depend on x, the uniqueness result of Theorem 2.1 is obvious. In such case, it is known [START_REF] Aronson | Nonlinear diffusion in population genetics, combustion and nerve propagation Partial Differential Equations and Related Topics Lectures Notes in Mathematics[END_REF][START_REF] Kolmogorov | Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique Bull[END_REF] that case w * = 2 √ r D, thus w * = w * implies that r = r.

Proof of Proposition 2.2

Recall that for the Cauchy problem (P ) associated with compactly supported or Heaviside initial conditions, we can define an asymptotic speed of propagation (see formula (3)). In the case of exponentially decaying initial conditions of the type ( 10)- [START_REF] Billingham | The development of traveling waves in quadratic and cubic autocatalysis with unequal diffusion rates I. permanent form of traveling waves Phil[END_REF] we have to prove the existence of this speed. The proof is based on the construction of appropriate sub and super-solutions for the system (P ). We are going to use the notion of pulsating traveling wave for the equation:

∂ t u = D∂ xx u + g l (x, u), with g l (x, u) = 1 l f (x, lu) for some l > 0. Note that, ∂ u g l (x, 0) = ∂ u f (x, 0), ∀l > 0.
In view to carry out sub-solutions and super-solutions of problem (P ), we consider the following problem:

     -D∂ xx p l = g l (x, p l ), x ∈ R, p l is 1-periodic, p l > 0. (21)
Now, we consider the solution u λ (t, x) of (P ) with initial condition u 0,λ (x). We construct a super-solution for this problem. In that respect we first look for l such that the solution p l (x) of ( 21) satisfies:

min R p l > sup R u 0,λ . (22)
Let us show that such a value of l exists. Let p(x) = p 1 (x) be the unique positive solution of

-D∂ xx p = f (x, p), x ∈ R. (23)
It is known, from Proposition 2.1, that p(x) is strictly positive and periodic. Thus, it exists δ 1 , δ 2 > 0 such that 0 < δ 1 < p(x) < δ 2 on R.

Recall that the function p l satisfies: -D∂ xx p l = 1 l f (x, lp l ) for x ∈ R. Let us set q(x) = lp l (x). Then q verifies the following equation:

-D∂ xx q = f (x, q).
By uniqueness (Proposition 2.1), we deduce q(x) = p(x), and finally p l (x) = p(x) l . From [START_REF] Gilbarg | Elliptic Partial Differential Equations of Second Order[END_REF] we obtain:

δ 1 l < p l (x) < δ 2 l . ( 25
)
Let l 1 < 1 be such that

δ 1 l 1 > sup R u 0,λ . Then we can write min R p l 1 > sup R u 0,λ . (26)
In a first step we consider λ ′ ∈ (0, λ) such that λ ′ < λ * and let

c ′ = -k λ ′ λ ′ > c * = -k λ * λ * . We consider U l 1 c ′ (t,
x), a pulsating traveling wave solution of the problem:

         ∂ t U l 1 c ′ = D∂ xx U l 1 c ′ + g l 1 (x, U l 1 c ′ ), t ∈ R, x ∈ R, ∀z ∈ Z, ∀(t, x) ∈ R × R, U l 1 c ′ (t + z c ′ , x) = U l 1 c ′ (t, x -z), ∀(t, x) ∈ R × R, 0 ≤ U l 1 c ′ (t, x) ≤ p l 1 (x), lim x→-∞ |U l 1 c ′ (t, x) -p l 1 (x)| = 0 and lim x→+∞ U l 1 c ′ (t, x) = 0, ( 27
)
where the above limits hold locally in t. The existence of this solution follows from [START_REF] Berestycki | Analysis of the periodically fragmented environment model: II -Biological invasions and pulsating travelling fronts[END_REF] and its uniqueness (up to a translation) is proved in [START_REF] Hamel | Uniqueness and stability properties of monostable pulsating fronts[END_REF].

From [START_REF] Hamel | Qualitative properties of monostable pulsating fronts : exponential decay and monotonicity[END_REF], it is known that the asymptotic behavior of U l 1 c ′ (t, x) is:

U l 1 c ′ (t, x) ∼ Be -λ c ′ (x-c ′ t) ψ λ c ′ (x), as x → +∞, for B > 0, t ∈ R, (28)
where λ c ′ is such that

λ c ′ = inf{λ > 0 such that k λ + λc ′ = 0}, ( 29
)
and ψ λ c ′ (x) is the eigenfunction of the operator L λ c ′ (see ( 14)) associated to the principal eigenvalue k λ c ′ . From the convexity of the map λ → -k λ (see [START_REF] Berestycki | Analysis of the periodically fragmented environment model: I-Species persistence[END_REF]) and by definition of λ * (see Proposition 2.2), the equation k λ = -λc ′ admits at most two roots,

λ -≤ λ * ≤ λ + . Since λ ′ < λ * , we get that λ ′ = λ -= λ c ′ .
Moreover, since l 1 < 1 and f (x, s) s is decreasing we have

f (x, l 1 s) l 1 s > f (x, s) s .
Thus

g l 1 = 1 l 1 f (x, l 1 s) > f (x, s) for all s ≥ 0. Finally U l 1 c ′ (t, x) satisfies ∂ t U l 1 c ′ -D∂ xx U l 1 c ′ = g l 1 (x, U l 1 c ′ ) > f (x, U l 1 c ′ ), x ∈ R. Then U l 1 c ′ (t, x
) is a super-solution for the problem (P ). Now we want to prove that

U l 1 c ′ (0, x) ≥ u 0,λ (x), ∀x ∈ R. (30)
From ( 28), we get U l 1 c ′ (0, x) ∼ Be -λ ′ x ψ λ ′ (x), as x → +∞. Using that ψ λ ′ (x) is continuous, 1-periodic, ψ λ ′ (x) strictly positive on [0, 1] and λ ′ < λ, we obtain

Be -λ ′ x ψ λ ′ (x) > e -λx , for x large enough. ( 31
)
Then, there exists M > 0 such that for any x > M ,

U l 1 c ′ (0, x) ≥ e -λx = u 0,λ (x). ( 32
)
From ( 26) and ( 27), we deduce that there exists m < 0 such that

min x≤m U l 1 c ′ (0, x) ≥ sup x∈R u 0,λ (x). ( 33
)
It remains to prove that min

x∈[m,M ] U l 1 c ′ (0, x) ≥ sup x∈R u 0,λ (x). Let δ = min x∈[m,M ] U l 1 c ′ (0, x). (34)
From the strong parabolic maximum principle, we know that δ > 0. Let L ≥ m be large enough such that e -λx < δ, for all x > L.

Then U l 1 c ′ (0, x -L + m) verifies [START_REF] Immanuvilov | Lipschitz stability in inverse parabolic problems by the Carleman estimate[END_REF]. Indeed: -If x ≤ L, then x -L + m ≤ m. From [START_REF] Klibanov | Global uniqueness of a multidimensional inverse problem for a nonlinear parabolic equation Inverse[END_REF], it follows that

U l 1 c ′ (0, x -L + m) ≥ sup x∈R u 0,λ (x).
-

If x ∈ (L, L + M -m), then m ≤ x -L + m ≤ M and, from (34), U l 1 c ′ (0, x -L + m) > δ > e -λx = u 0,λ (x). -If x ≥ L + M -m, then x -L + m ≥ M
and, from [START_REF] Kato | Perturbation theory for linear operators[END_REF],

U l 1 c ′ (0, x -L + m) ≥ e -λ(x-L+m) ≥ e -λx e +λ(L-m) ≥ e -λx = u 0,λ (x).
Thus, even if it means translating U l 1 c ′ (0, x) to the right, we can assume that

U l 1 c ′ (0, x) > u 0,λ (x), ∀x ∈ R. Since U l 1 c ′ (0, x) > u 0,λ (x)
and since U l 1 c ′ is a super-solution of (P ), a comparison principle implies that for all λ ′ ∈ (0, min(λ, λ * )),

U l 1 c ′ (t, x) > u λ (t, x), t ≥ 0, x ∈ R, (35)
with c ′ = -k λ ′ λ ′ . In a second step we consider λ ′′ such that 0 < λ < λ ′′ < λ * and let

c ′′ = -k λ ′′ λ ′′ . Let l 2 > 1 be such that δ 2 l 2 < lim inf x→-∞ u 0,λ (x)
(see [START_REF] Hamel | Qualitative properties of monostable pulsating fronts : exponential decay and monotonicity[END_REF]). We consider U l 2 c ′′ (t, x) the pulsating traveling wave solution of the problem (27) (with l 2 instead of l 1 and c ′′ instead of c ′ ).

Using the same arguments as above, and the fact that max

R p l 2 < δ 2 l 2 < lim inf x→-∞ u 0,λ (x), it is easily seen that U l 2 c ′′ (t, x
) is a sub-solution of the problem (P ) satisfying

U l 2 c ′′ (0, x) < u 0,λ (x)
(even if it means translating U l 2 c ′′ (0, x) to the left). A comparison principle implies that for all

λ ′′ ∈ (λ, λ * ), U l 2 c ′′ (t, x) < u λ (x), t ≥ 0, x ∈ R. (36)
So we get from [START_REF] Lorenzi | An inverse problem for a semilinear parabolic equation[END_REF] and [START_REF] Murray | Mathematical Biology Third Edition[END_REF] an appropriate pair of sub and super-solutions for the problem (P ), and we can write:

U l 2 c ′′ (t, x) < u λ (t, x) < U l 1 c ′ (t, x), for all t ≥ 0, x ∈ R, λ ∈ (λ ′ , λ ′′ ).
Now we are going to complete the proof of the existence of an asymptotic spreading speed associated to problem (P ) with initial conditions of the type ( 10)- [START_REF] Billingham | The development of traveling waves in quadratic and cubic autocatalysis with unequal diffusion rates I. permanent form of traveling waves Phil[END_REF].

-First case: if λ < λ * , we prove that w λ = -k λ λ verifies [START_REF] Bukhgeim | Uniqueness in the large of a class of multidimensional inverse problems Soviet Mathematics[END_REF].

(i) Let c > w λ = -k λ λ , we want to prove that lim t→+∞ u λ (t, x + ct) = 0, for all x ∈ R. Let λ ′ ∈ (0, λ) be such that -k λ λ < -k λ ′ λ ′ < c; the existence of λ ′ follows from the continuity of s → k s s . We know from (35) that u λ (t, x + ct) < U l 1 c ′ (t, x + ct), with w λ < c ′ = -k λ ′ λ ′ < c. Since c ′ < c we obtain from (27) lim t→+∞ U l 1 c ′ (t, x+ct) = 0. We conclude that if c > w λ then lim t→+∞ u λ (t, x + ct) = 0, for all x ∈ R. ( 37
) (ii) Let c < w λ = -k λ λ . We want to prove that lim inf t→+∞ u λ (t, x + ct) > 0. Let λ ′′ ∈ (λ, λ * ) be such that c < -k λ ′′ λ ′′ < -k λ λ
. We know from [START_REF] Murray | Mathematical Biology Third Edition[END_REF], that

U l 2 c ′′ (t, x + ct) < u λ (t, x + ct), with c < c ′′ = -k λ ′′ λ ′′ < w λ . Since c < c ′′ we obtain from (27) that lim inf t→+∞ U l 2 c ′′ (t, x + ct) > 0.
We conclude that if c < w λ then lim inf t→+∞ u λ (t, x + ct) > 0, for all x ∈ R.

Using [START_REF] Pilant | An inverse problem for a nonlinear parabolic equation[END_REF] together with [START_REF] Roques | On the determination of the nonlinearity from localized measurements in a reaction-diffusion equation[END_REF] imply that w λ = -k λ λ is the spreading speed of u λ in the sense of ( 12) when λ ∈ (0, λ * ).

-Second case: if λ ≥ λ * , we prove that w λ = w * .

(i) Let c > w * , and consider λ ′ ∈ (0, λ * ) such that w

* < -k λ ′ λ ′ = c ′ < c. From (35), we know that 0 < u λ (t, x) < U l 1 c ′ (t, x), t ≥ 0, x ∈ R.
From (3) and since 0 < c ′ < c, it follows that lim t→+∞ u λ (t, x + ct) = 0, for all x ∈ R.

(ii) Let c < w * and V (t, x) be a solution of (P ) with compactly supported initial condition such that 0 ≤ V (0, x) < u 0,λ (x) for all x ∈ R. A comparison principle implies that 0 < V (t, x) < u λ (t, x) for all t > 0, x ∈ R.

Thus u λ (t, x + ct) > V (t, x + ct), for all t > 0, x ∈ R.

From [START_REF] Hamel | viscosity solution method for the spreading speed formula in slowly varying media Indiana[END_REF], we know that lim inf t→+∞ V (t, x + ct) > 0, for all x ∈ R.

Thus lim inf

t→+∞ u λ (t, x + ct) > 0, for all x ∈ R.

Using [START_REF] Roques | The inverse problem of determining several coefficients in a nonlinear Lotka-Volterra system Inverse[END_REF] together with [START_REF] Roques | Allee effect promotes diversity in traveling waves of colonization[END_REF] imply that w * is the spreading speed of u λ for all λ ≥ λ * .

Proof of Lemma 3.3

We consider the following problem

         Dψ ′′ λ -2λDψ ′ λ + r(x)ψ λ = -l λ (r)ψ λ in R, ψ λ > 0, ψ λ 1-periodic, ψ λ (0) = 1, (41)
where l λ (r) = k λ (r) + λ 2 D. We define the functions r λ and h λ by

l λ (r) = -r + r λ λ 2 ,
and

ψ λ (x) = 1 + f 1 (x) λ + h λ (x) λ 2 , where f 1 (x) := 1 2D
x 0 (r(s) -r)ds.

We derive from (41):

0 = (2Df ′ 1 -r(x) -l λ (r)) + 1 λ ( -Df ′′ 1 + 2Dh ′ λ -r(x)f 1 -l λ (r)f 1 ) + 1 λ 2 ( -Dh ′′ λ -r(x)h λ -l λ (r)h λ ) = ( -l λ (r) -r) + 1 λ ( -Df ′′ 1 + 2Dh ′ λ -r(x)f 1 + rf 1 ) + 1 λ 2 ( -Dh ′′ λ -r(x)h λ + rh λ ) - 1 λ 3 r λ f 1 - 1 λ 4 r λ h λ = 1 λ ( -Df ′′ 1 + 2Dh ′ λ -r(x)f 1 + rf 1 ) + 1 λ 2 ( -Dh ′′ λ -r(x)h λ + rh λ -r λ ) - 1 λ 3 r λ f 1 - 1 λ 4 r λ h λ . (42)
Now we prove that a 2 := lim Proof. We know (see [START_REF] Berestycki | Analysis of the periodically fragmented environment model: I-Species persistence[END_REF] for example) that

l λ (r) = sup{l ∈ R, ∃ψ ∈ W 2,∞ per (R), ψ > 0 in R, -L λ ψ ≥ lψ a.e. in R}, = inf{l ∈ R, ∃ψ ∈ W 2,∞ per (R), ψ > 0 in R, -L λ ψ ≤ lψ a.e. in R}.
Using ψ(x) = 1 + f 1 (x) λ as a test-function, taking λ large enough so that 1 + f 1 (x) λ > 0 for all x, we get almost everywhere:

-L λ ψ = - Df ′′ 1 (x) λ + 2λDf ′ 1 (x) λ -r(x) - r(x)f 1 (x) λ , = - r ′ (x) 2λ + r(x) -r -r(x) - r(x)f 1 (x) λ , ≥ - C 0 λ -r,
where C 0 is a constant which only depends on r ∞ and r ′ ∞ . Take C large enough so that

C + Cf 1 (x) λ + rf 1 (x) ≥ C 0 .
We derive from this inequality that

- C λ -r ψ ≤ - C 0 λ -r over R.
Hence -L λ ψ ≥ -C λ -r ψ over R and we derive from the definition of l λ (r) that

l λ (r) ≥ - C λ -r.
Similarly, one can prove that, taking C larger if necessary,

l λ (r) ≤ C λ -r. Define θ λ ∈ W 2,∞ per (R) so that ψ λ (x) = 1 + f 1 (x) λ + θ λ (x) λ ,
where f 1 (x) := 1 2D

x 0 (r(s) -r)ds.

Rewriting the identity -L λ ψ λ = l λ (r)ψ λ with this change of functions, we get:

-L λ θ λ + (r - r λ λ 2 )θ λ = Df ′′ 1 (x) + (r(x) -r + r λ λ 2 )f 1 (x) + r λ λ =: F λ . ( 43
)
Lemma 5.1 yields that r λ λ ≤ C for all λ. Thus, F λ ∈ L ∞ per (R) and there exists two constants λ > 0, C 0 > 0, so that for all λ ≥ λ, F λ ∞ ≤ C 0 . Multiplying (43) by θ ′′ λ and integrating, one gets, for all λ ≥ λ

1 0 (θ ′′ λ ) 2 dx = - 1 0 r(x) -r + r λ λ 2 θ λ θ ′′ λ dx - 1 0 F λ θ ′′ λ dx, = 1 0 r ′ (x)θ λ θ ′ λ dx + 1 0 r(x) -r + r λ λ 2 (θ ′ λ ) 2 dx - 1 0 F λ θ ′′ λ dx, ≤ C 1 ( θ ′ λ 2 L 2 (0,1) + θ ′′ λ L 2 (0,1) ), (44)
where C 1 is a constant which only depends on r ′ ∞ , r ∞ , the constant C given by Lemma 5.1 and C 0 , and where we have used the inequalities

θ λ L 2 (0,1) ≤ θ λ ∞ ≤ θ ′ λ L 1 (0,1) ≤ θ ′ λ L 2 (0,1) , (45)
since θ λ (0) = 0. Assume now that there exists a sequence (λ n ) n such that λ n ≥ λ and θ ′ λn L 2 (0,1) → +∞ as n → +∞. Let

ζ n := θ λn θ ′ λn L 2 (0,1) , so that ζ ′ n L 2 (0,1) = 1. Dividing (44) by θ ′ λn 2 L 2 (0,1) , one gets 1 0 (ζ ′′ n ) 2 dx ≤ C 1 1 + ζ ′′ n L 2 (0,1) θ ′ λn L 2 (0,1)
.

Thus, as lim n→+∞ θ ′ λn L 2 (0,1) = +∞, ( ζ ′′ n L 2 (0,1) ) n is bounded. As ζ ′ n L 2 (0,1) = 1 and ζ n L 2 (0,1) ≤ 1 for all n due to (45), we can extract a subsequence, that we still denote (ζ n ) n , which converges weakly in H 2 per (0, 1) as n → +∞. On the other hand, equation [START_REF] Tort | Determination of the insolation function in the nonlinear Sellers climate model[END_REF] gives

2λ n ζ ′ n = ζ ′′ n + (r(x) -r + r λn λ 2 n )ζ n + F λn θ ′ λn L 2 (0,1)
, and thus the sequence (λ n ζ ′ n ) n is bounded in L 2 (0, 1), which is a contradiction since ζ ′ n L 2 (0,1) = 1 for all n. We thus conclude that the family (θ ′ λ ) λ≥ λ is bounded in L 2 (0, 1). It follows that (θ λ ) λ≥ λ and (θ ′′ λ ) λ≥ λ are bounded in L 2 (0, 1) due to [START_REF] Turchin | Quantitative analysis of movement: measuring and modeling population redistribution in animals and plants[END_REF] and ( 45) respectively, and thus [START_REF] Tort | Determination of the insolation function in the nonlinear Sellers climate model[END_REF] implies that (λθ ′ λ ) λ≥ λ is bounded in L 2 (0, 1). The same chain of inequalities yields that (λθ λ ) λ≥ λ is bounded in H 2 (0, 1).

By replacing in [START_REF] Tort | Determination of the insolation function in the nonlinear Sellers climate model[END_REF]

We have shown previously that r λ and h λ H 2 (0,1) are bounded uniformly with respect to λ ≥ λ. Then r λ λ → 0 as λ → +∞. Passing to the limit in [START_REF] Xin | Existence of planar flame fronts in convective-diffusive periodic media[END_REF], we get that f 2 := lim λ→+∞ h λ is well-defined in H 1 (0, 1) and that

f ′ 2 = 1 2 (f ′′ 1 + (r(x) -r) D f 1 ).
Lastly, letting λ → +∞ in [START_REF] Xin | Front propagation in heterogeneous media[END_REF], we obtain lim λ→+∞ r λ = - By taking the limit in [START_REF] Xin | Existence of planar flame fronts in convective-diffusive periodic media[END_REF] we get:

-Df ′′ 1 + 2Df ′ 2 -r(x)f 1 + rf 1 = 0, (48)

Lemma 3 . 1 .

 31 For any r ∈ M 1 , the function : λ → k λ (r) is analytic.

λ→+∞ r λ and f 2

 2 (x) := lim λ→+∞ h λ (x) are well-defined by using Lemma 5.1. Lemma 5.1. There exists a constant C > 0 such that |l λ (r) + r| ≤ C/λ.

θ λ by h λ λ , we get 0 =r λ f 1 λ 2 - r λ h λ λ 3 .( 46 )and noticing that 1 0 1 0f 1 f ′ 1 dx = D 1 0 (f 2 1 )

 023461111 -Df ′′ 1 + 2Dh ′ λ + (r -r)f 1 -1 λ (Dh ′′ λ + (r -r)h λ +r λ )-Integrating (r -r(x))f 1 dx = 2D ′ dx = 0x) -r)h λ (x)dx -

1 0

 1 (r(x) -r)f 2 dx = a 2 . Now, we can claim that a 2 := lim λ→+∞ r λ and f 2 (x) := lim λ→+∞ h λ (x) are well-defined.

and a 2 = 1 0 2 = - 1 0 1 0

 21211 (r -r(x))f 2 (x)dx. Define M (x) := x 0 r(y)dy and integrate by parts:a (rx -M (x))f ′ 2 (x)dx(there is no boundary terms since M (1) = rdx = r). We compute f ′ 2 using the first equation in (48): x) -rx)(Df ′′ 1 + (r(x) -r)f 1 )dx, x) -r) 2 dx -x) -rx) 3 dx, x) -r) 2 dx + o( 1 λ 2 ),from which the conclusion follows from Lemma 3.2 and l λ (r) = l λ (r) for all λ > 0.

Acknowledgments

This work was partially supported by the franco-algerian agreement Tassili 11 MDU 834 (second author).


Introduction

This report is an appendix of a research work which will be published soon. A fault detection, isolation and adaptation (FDIA) formalism is introduced and this report demonstrates the two rules of non isolability.

The proposed FDIA framework assumes that one system observation produces two estimates of the same quantity: G and N . These estimates are stored in FDIA memory. K observations of the system are made.

Faults aecting G i and N i cause their value to be dierent from the true one P i . The index i stands for the i th observation. The state of G i (resp. N i ) of being faulty or not is denoted by the boolean variable f G i (resp. f N i ). For instance, f G i = 1 means that a fault aects G i then G i = P i . At the K th observation, the faulty states of every estimate is summarised by e called the set of faulty states. The purpose of FDIA is to isolate (i.e. determine) e which means to ascertain the faulty state of the estimates G i and N i for every observation i ≤ K.

FDIA is based on the use of the residual vector R(e). The terms of R are the results of boolean operations between the faulty states f G i and f N i . Some sets of faulty states produce unique residuals, isolation is then possible. However, some residuals are generated by several sets of faulty states, isolation is not possible.

The aim of this report is to prove the conditions on f G i and f N i for e to be not isolable. These are stated by two rules in Proposition 1: Proposition 1. A set of faulty states is not isolable if and only if, it complies with one of the following rules:

1. f N i = 1 , ∀i ∈ {1, . . . , K} and ∃!j ∈ {1, . . . , K} such as

f G j = 0 2. f G i = 1 , ∀i ∈ {1, . . . , K}
In other words it is not possible to isolate faults if:

1. Every estimates N is faulty and there is a unique true G.

Every G is faulty.

Section 2 introduces the notations required for the demonstration. Section 3 deduces mathematical properties of the proposed formalism. According to these properties, Section 4 reformulates the problem and demonstrate the proposition. Finally Section 5 concludes the demonstration.

Notations

A set of faulty states e is dened as e

= (f G i , f N i ) 1≤i≤K ∈ {0, 1} 2K for some K 1.
Let E be the set of sets of faulty states e.

e ∈ E

(2.1)

R is the function that associates a residual to a set of faulty states. R (e) is the residual of

e. R = r G i G j , r N i N j , r GpNq 1≤i,j,p,q≤K , i =j with r G i G j = f G i ∨ f G j , ∀i, j ∈ {1, . . . , K} , i > j (2.2) r G i N j = f G i ∨ f N j , ∀i, j ∈ {1, . . . , K} (2.3) r N i N j = f N i ⊕ f N j , ∀i, j ∈ {1, . . . , K} , i > j
(2.4) where ∨ and ⊕ are boolean or and exclusive or respectively. σ K is the set of permutations of {1, . . . , K} I stands for the set of isolable sets of faulty states.

I ⊂ E
(2.5) I c is the complement of I (i.e. the set of non-isolable sets of faulty states) C l,m is the set of sets of faulty states e = (f G i , f N i ) 1≤i≤K such as there are l f

G i = 1 and m f N i = 1.
Given a set of faulty states e ∈ E and two permutations σ, σ ∈ σ K , the set of faulty states obtained by permuting variables f G i (with σ) and variables

f N i (with σ ) is denoted by σ, σ • e = (f Gσ -1 i , f N σ -1 i ) 1≤i≤K
(2.6) where σ -1 (resp. σ -1 ) stands for the inverse permutation of σ (resp. σ ).

e l,m stands for canonical form of e ∈ C l,m .

e lm = (f G i , f N i ) 1≤i≤K with        f G i = 1, . . . , 1 l , 0, . . . , 0 K-l f M i = 1, . . . , 1 m , 0, . . . , 0 K-m
(2.7) Reciprocally, if σ -1 , σ -1 • e / ∈ I, then there exists σ -1 , σ -1 • e ∈ E such as σ -1 , σ -1 • e = σ -1 , σ -1 • e and R σ -1 , σ -1 • e = R σ -1 , σ -1 • e . One can apply the same permutation (σ, σ ) to those sets of faulty states.

C l,m C l,m , 1 ≤ l, m ≤ K is a partition of E. Let denote by Ẽ the disjoint union of C l,m where (l, m) = (K -1, K) and (l, m) = (K, K) Ẽ = 1≤l,m≤K C l,m | (l, m) = (K -1, K) and (l, m) = (K, K) (3
σ, σ • σ -1 , σ -1 • e = σ, σ • σ -1 , σ -1 • e and R σ, σ • σ -1 , σ -1 • e = R σ, σ • σ -1 , σ -1 • e =⇒ e =

New problem statement

According to the previous developments, studying isolability of a set of faulty states is equivalent to evaluating the isolability of the canonical set of faulty states of every class. The residuals are calculated using boolean operations between f G and f N variables. As stated in Section 2, the OR operator is used for f G f G and f G f N pairs combination and Exclusive OR is used for f N f N combination. The canonical sets of faulty states of class C l,m are represented in the Tables 5.2a, 5.2b and 5.2c. For f G (resp. f N ), the l (resp. m) ones are written rst and the K -l (resp. K -m) zeros are written then. The the result of the boolean operation is written in the table which forms the residual.

These tables oer the advantage of showing clearly the consequence of the parameters l and m on the residual of a set of faulty states. It has been shown previously that the isolability of a canonical set of faulty states is the same as the class it belongs to. The isolability study of a class is made by looking at the number of zeros in the tables with respect to l and m.

4.1 If l, m < K If l < K and m < K, there is a rectangle of zeros in Table 5.2c of size n c = (K -l) • (K -m) > 0.
Assume that e ∈ E is another set of faulty states with the same residual as e l,m . Looking at Table 5.2c we see that the K -l last variables f G i and the K -m last variables N i are necessarily 0. Then, because of the two sub-tables made of 1, we see that all the others variables are 1. In other words, e = e l,m . As a consequence, we have: 5.2b is the only one which can make a dierence in the residual. Moreover, N i ⊕ N j = N i ⊕ N j ∀i = j. Thus, for all m ∈ {1, ..., K} and e = (f G i , f N i ) ∈ C l,m , replacing N i by N i doesn't change the residual of e. Hence:

C K,m ⊂ I c , ∀m ∈ {1, . . . , K} (4.2)
If m = K and for all l, the Table 5.2b is full of zeros and the Table 5.2c is full of ones. Then the most signicant table is Table 5.2a.

It must be noticed rst that, l = K and l = K -1 makes Table 5.2a be full of ones. C K,K and C K-1,K have therefore the same residual. Then:

C K,K ⊂ I c (4.3) C K-1,K ⊂ I c (4.4) Secondly, if l ≤ K -2, there are (K -l) 2 -(K -l) > 0 zeros in Table 5.2a.
Assume that e is a set of faulty states with the same residual as e l,K . For the same reasons as in the case 4.1, this implies that the l rst variables f G i are 1 and the others are 0: the variables f G i of e are the same as f G i of e l,K . Moreover, in view of Table 5.2c, we see that all the variables f N i of e are 1 or else a 0 would appear in Table 5.2a corresponding to the residual of e. So, e = e l,K and we have shown that e l,K is isolable. This is sucient to obtain the following inclusion:

I c = C K-1,K ∪ K l=0 C K,l
(5.1)

In other words, a set of faulty states is not isolable if and only if, it complies to one of the following rules:

• N j = 1 , ∀j ∈ {1, . . . , K} and ∃!i ∈ {1, . . . , K} such as G i = 0

• G i = 1 , ∀i ∈ {1, . . . , K}

  .1) Proposition 3. If e ∈ E and σ, σ ∈ σ K then e ∈ I ⇐⇒ σ, σ • e ∈ I (3.2) Proof. e / ∈ I =⇒ ∃e = e | R (e ) = R (e). For a pair of permutations σ, σ ∈ σ K , then (σ, σ ) • e = (σ, σ ) • e and R ((σ, σ ) • e) = R ((σ, σ ) • e ) ; then (σ, σ ) • e / ∈ I.

  e and R e = R (e) =⇒ e / ∈ I If a set of faulty states e of E is isolable, then every permutation of e is isolable. More precisely, the following proposition holds : Corollary 1. For all e ∈ C l,m , e ∈ I ⇐⇒ C l,m ∈ I This is true in particular for the canonical set of faulty states e l,m of class C l,m . The canonical set of faulty states describes it entire class in terms of isolability.

C

  l,m ⊂ I , ∀l, m ∈ {1, . . . , K -1} (4.1) 4.2 If l = K If l = K, Tables 5.2a and 5.2c are full of ones for all m and Table

(

  C l,K ⊂ I , ∀l ∈ {1, . . . , K -2}

3

  PropertiesProposition 2. The class C l,m contains all the permutations of the canonical set of faulty states e l,m :C l,m = {(σ, σ ) • e l,m |σ, σ ∈ σ K } Proof. The permutation σ (resp. σ ) doesn't change the number of f G i (resp. f N i ) equals to one. Thus, if e ∈ C l,m and σ, σ ∈ σ K then (σ, σ ) • e ∈ C l,m .Reciprocally, every set of faulty states e ∈ C l,m can be obtained by permuting the variables f G i and f N i and the proposition follows.

	Remark 1. E =
	1≤l,m≤K


Braulio S Aranda

Pedro Aguirre

Sergio A Moya

Mickaële Bonneau

J A Gareth Williams

Loic Toupet

Muriel Escadeillas

Hubert Le Bozec

Véronique Guerchais
email: veronique.guerchais@univ-rennes1.fr

Luminescent bis-cyclometallated iridium(III) complexes containing phosphine-based ligands: influence of the P^N bridge

Keywords: Iridium complexes, Bis-cyclometallated complexes, Phosphine ligands, Luminescence, Blue phosphors

Three cationic bis-cyclometallated iridium complexes [Ir(C^N-ppyMe) 2 (P^N)][PF 6 ](1-3) with 4-methyl-2-phenylpyridine (ppyMe) as the C^Ncyclometallated ligands and 8-(diphenylphosphino)quinoline (Ph 2 Pqn, L 1 ), 2-(diphenylphosphinoamino)pyridine (Ph 2 PNHpy, L 2 ) or 2-(diphenylphosphinoamino)

pyrimidine (Ph 2 PNHpym, L 3 ) as P^N-ligands, were prepared, and their crystal structures and photoluminescence were investigated. The substitution of an "aromatic C Ar " (of a quinolyl group) by an "NH" bridge between the phosphine and imine ligating units of the P^N ligand dramatically changes the luminescence characteristics of the resulting complexes. The complexes 2-3 featuring the P-NH-N ligand emit in the blue region, whereas complex 1 incorporating the P-C Ar -N unit displays orange photoluminescence.

Introduction

Homoleptic and heteroleptic cyclometallated Ir(III) complexes are well known to emit phosphorescence at ambient temperature with high quantum yield. Numerous examples of cationic complexes incorporating a combination of two cyclometallated ligands and one diimine (e.g. bipyridine, phenanthroline or their derivatives) have been reported and the influence of the substitution of the diimine ligand on the electronic excited states described [1]. The emission energy of such heteroleptic complexes is readily fine-tuned by changing the nature of the cyclometallated ligand C^N and the diimine ligand. For example, blue and green-blue phosphorescent Ir(III) complexes have been reported. The commonly used approaches to widen the HOMO-LUMO gap and shift emission further to the blue are either the use of electron-withdrawing substituents such as F or CF 3 in the metallated aryl ring, which lowers the HOMO, or the replacement of the pyridine moiety of the C^N ligands by Nheterocyclic rings such as pyrazoles and N-heterocyclic carbenes to raise the LUMO [START_REF]To date, a large number of blue emissive iridium complexes has been reported[END_REF].

Another way to increase the energy gap is through the use of non-conjugated ancillary diarylphosphine chelates in place of the diimine [3]. Indeed, the π-accepting character of the PAr 2 group is expected to increase the metal-to-ligand charge-transfer (MLCT) energy level resulting in blue-shifted emission. Thus, blue-shifts of the emission maxima have been obtained using neutral [4,5] or cationic [6] heteroleptic Ir(III) complexes combining C^N ligands with either P^C or P^N chelates.

On the other hand, P^N ligands such as 8-(diphenylphosphino)quinoline (Ph 2 Pqn, L 1 ), 2-(diphenylphosphinoamino)pyridine (Ph 2 PNHpy, L 2 ) and 8-(diphenylphosphinoamino)pyrimidine (Ph 2 PNHpym, L 3 ) represent an interesting class of heterobidentate ligands that have been applied by some of us for the palladium catalyzed methoxy-carbonylation of olefins (Scheme 1) [7].

Ph 2 P N Ph 2 P N HN Ph 2 P N N HN L 1 L 2 L 3
Scheme 1. Chemical structures of the P^N ligands used in this study

Having in hand these chelating ligands, we sought to use them for the preparation of luminescent cationic Ir(III) complexes 1-3 (Scheme 2). Herein we report their synthesis and characterization, as well as their photophysical properties. We show that substitution of an "aromatic C Ar " (of a quinolyl group) by an "NH" bridge between the phosphine and imine ligands of the P^N ligand dramatically changes the luminescence characteristics of the resulting complexes. The complexes 2-3 featuring the P-NH-N ligands emit in the blue region, whereas the P-C Ar -N-incorporating complex 1 displays orange photoluminescence.

Experimental

Materials and general methods

All manipulations were performed using Schlenk techniques under an Ar atmosphere, but the workups were carried out in air. All solvents were dried and purified by standard procedures. All starting materials were used as received. Complex [Ir(C^N-Meppy) 2 (µ-Cl)] 2

(Meppy = 4-methyl-2-phenylpyridine) was prepared following the literature procedure [8].

The P^N ligands, Ph 2 Pqn, Ph 2 PNHpy and Ph 2 PNHpym, were synthesized according to reported methods [7,9]. with initial isotropic parameters [START_REF] Altomare | Sir97: a new tool for crystal structure determination and refinement[END_REF]. Atomic scattering factors were taken from international Tables for X-ray crystallography. ORTEP views were realized with PLATON [START_REF] Sheldrick | SHELX97-2 Program for Crystal Structure Refinement[END_REF]. The crystal structures of 1, 2 and 3 have been deposited at the Cambridge Crystallographic Data mmol), and the appropriate P^N-ligand L (2.1 equiv), and AgPF 6 (181 mg 0.716 mmol) were suspended in dichloromethane (15 mL). The reaction mixture was stirred under argon for 16 h at room temperature. The solution was filtered and the solvent was removed and the product was washed with diethyl ether (3 x 10 mL). Crystallization from a CH 2 Cl 2 /diethyl ether mixture gave the expected compound as a powder

NMR

Results and Discussion

Synthesis and characterization of Ir(III) complexes 1-3

The three P^N ligands were prepared following previously reported procedures [7,9]

X-Ray structures

The structures of the three complexes 1-3 were confirmed by X-ray crystal structure analyses, as shown in Figure 1. Selected bond length and bond angle data are listed in Tables 1 and2.

All complexes reveal a distorted octahedral geometry around iridium, consisting of two cyclometallated C^N ligands and one P^N ligand with the phosphorus atom being trans to a cyclometallated carbon atom. The ligand arrangement, where the two Ir-C bonds are in mutually cis positions, is similar to that of the parent phenylpyridine ligands in many heteroleptic Ir(III) complexes such as the diketonate complex Ir(C^N-ppy) 2 (O^O-acac) [13] and the cationic bipyridine derivatives [8]. The N-Ir and C-Ir bond distances involving the cyclometallated ligand are in the same range as those reported for related Ir complexes [START_REF]For examples of X-ray crystal structures of (C^N-ppy)Ir(III) complexes, see inter alia: (a) F. Neve, A. Crispini[END_REF].

The Ir-N3 bond distances (where N3 is the coordinated nitrogen atom of the P^N ligand in each case) are longer than those the coordinated Meppy ligand. In the three complexes, both the Ir-N3 and the Ir-P bond lengths of the P^N ligand are very similar.. The chelate bite angle of the phosphine ligand (∼ 80°) is the same for each complex. For the PPh 2 Pqn-containing complex, the deviations of atoms Ir1 and P1 from the least-square quinolyl ring plane are only 0.214 and 0.033 Å, respectively. These latter values are comparable to those found in other iridium(III) complexes [START_REF] Suzuki | For examples of X-ray crystal structures of Ir(III) complexes containing the Ph 2 Pqn ligand, see inter alia[END_REF]. They indicate that the chelate coordination of Ph 2 Pqn forms a strain-free planar five-membered ring.

Electronic absorption and luminescence properties

The UV-vis absorption data of complexes 1-3 in CH 2 Cl 2 are summarized in Table 3, and the corresponding spectra are shown in Figure 2. The absorption spectra of complexes 1-3

show intense absorption bands in the higher energy region (λ < 340 nm) which are dominated by intra-ligand 1 IL (π → π*) transitions of the C^N and P^N ligands, and a moderately intense lower-energy absorption band at longer wavelength (around 360 nm) assigned to the tail of the IL transitions overlapping with MLCT dπ(Ir) → π* (C^N and P^N) transitions.

The spectrum of complex 1 has a weak shoulder extending to longer wavelengths than for 2 and 3, possibly reflecting the more extended π-conjugation of the chelated quinolylphosphine ligand. The spectra of 2 and 3 are very similar to those reported for the related complex

Table 3

Photophysical data of the investigated iridium(III) complexes. [b,c] τ / τ / τ / τ /ns λ λ λ λ em at 77 K /nm [c,d] [%] [e] 1 293sh ( 22300 [a] Values refer to solutions in CH 2 Cl 2 at 298 K, except where indicated otherwise. [b] In degassed CH 2 Cl 2 solutions. [c] exc = 365 nm. [d] In EPA [e] Ru(bpy) 3 Cl 2 was used as the standard ( = 0.028 in air-equilibrated water).

Cpd λ λ λ λ abs / nm (ε ε ε ε [M -1 cm -1 ]) [a] λ λ λ λ em / nm
Upon irradiation, complexes 1-3 are photoluminescent in CH 2 Cl 2 solution at 298 K but they display very different emission characteristics depending on the nature of the P^N ligand.

The luminescence data are listed in Table 3 and the emission spectra are shown in Figure 3. ring in 3 has little influence on the emission maxima, but it is striking that the lifetime decreases by an order of magnitude. This could be due to a greater degree of distortion in the excited state of 3. Indeed, close inspection of the spectra of 2 and 3 reveals that, although the wavelengths of the emission maxima coincide, the relative intensity of the 0-0 vibrational band in the spectrum of 3 is substantially smaller than for 2, consistent with a larger degree of excited state distortion in the former. At 77 K, on the other hand, such distortion will be impeded, and indeed the spectra of the two complexes under these conditions are essentially identical.

The significant variation of emission wavelength between the use of "conjugated" and "non-conjugated" ligand P^N ligand has been previously reported. For instance, when diphenylnaphtylphosphine (dpn) is used as ancillary ligand in Ir(III) complexes [(C^N-Fppy) 2 Ir(dpn)] [16], a broad emission in the orange region is observed (∼580-630 nm for C^N (FppyH = 2-(2,4-difluorophenyl)pyridine) whereas the benzyl analogue [4a,b] gives rise to blue-emitting species.

Concluding remarks

Three cationic bis-cyclometallated iridium complexes with 4-methyl-2-phenylpyridine (Meppy) as the C^N-cyclometallated ligands and the P^N-ligands, 2-(diphenylphosphino) quinoline (Ph 2 Pqn, L 1 ), 2-(diphenylphosphinoamino)pyridine (Ph 2 PNHpy, L 2 ) and 2-(diphenylphosphinoamino) pyrimidine (Ph 2 PNHpym, L 3 ), were prepared, and their X-ray crystallography and photoluminescence were investigated. The above results demonstrate that phosphine derivatives, ubiquitous molecules used in catalysis, are also interesting ligands for the elaboration of luminescent metal complexes. Moreover, an efficient method to tune the color of the emission involves changing the degree of conjugation in these ancillary chelate ligands. Thus, we showed that the use of "non-conjugated" and π accepting phosphorus-based species as ancillary ligands allows the emission of cationic heteroleptic Ir(III) complexes to be shifted to the blue region.

Graphical abstract

Three cationic bis-cyclometallated iridium complexes with 4-methyl

  Scheme 2

Fig. 1 .

 1 Fig.1. ORTEP drawing displaying the molecular structure of complexes 1(a), 2 (b) and 3(c) with ellipsoids at the 50% probability level. Hydrogen atoms are omitted for clarity.

[

  Ir(C^N-ppy)(P^N-dppmp)][PF 6 ] having a P^N-2-diphenylphosphinomethylpyridine (dppmp) ligand, which contains a saturated "CH 2 " bridge instead of NH[6].

Fig 2 .

 2 Fig 2. UV-Vis absorption spectra (CH 2 Cl 2 , 298 K) for complexes 1-3.

Complex 1 Fig. 3

 13 Fig.3 Emission spectra of complexes 1-3 (λ ex = 365 nm): (a) in CH 2 Cl 2 at 298 K), (b) in EPA, at 77 K.

Fig. 3

 3 Fig.3Emission spectra of complexes 1-3 (λ ex = 365 nm): (a) in CH 2 Cl 2 at 298 K),(b) in EPA, at 77 K.

ray diffraction studies were grown by slow diffusion of pentane into a CH 2 Cl 2 solution of complexes 1-3 at 20°C. All measurements were made in the omega-scan technique on a CCD Sapphire 3 Xcalibur diffractometer (Agilent) with graphite monochromatized

	spectra were recorded on Bruker Ascend 400, AV 300 or AV 500 MHz
	spectrometers. UV-visible absorption spectra were recorded using an AnalitikJena Specord
	205 spectrophotometer. High resolution mass spectra (HRMS) were performed on a MS/MS
	MicrO-Tof-Q 2 at the CRMPO (Centre de Mesures Physiques de l'Ouest) in Rennes.
	Elemental analyses were performed on a FlashEA1112 at the CRMPO.
	Steady-state luminescence spectra were measured using a Jobin Yvon FluoroMax-2, fitted
	with a red-sensitive Hamamatsu R928 photomultiplier tube. The spectra shown are corrected
	for the wavelength dependence of the detector, and the quoted emission maxima refer to the
	values after correction. Phosphorescence lifetimes were measured by time-correlated single-
	photon counting following excitation at 374 nm with an EPL-375 pulsed diode laser. The
	emitted light was detected at 90˚ using a Peltier-cooled R928 PMT after passage through a
	monochromator. Spectra at 77 K were obtained with the sample loaded in a 4 mm o.d. quartz
	tube inside a quartz-walled Dewar filled with liquid nitrogen.
	Single crystals for X-

MoKα radiation. The data collection and refinement parameters are presented in Table

2

. The structures were solved with SIR-97

[START_REF] Binstead | Specfit/32 Global analysis system[END_REF]

which reveals the nonhydrogen atoms of the molecules. The whole structures were refined by full-matrix leastsquare techniques on F 2 , with hydrogens refined using the Riding mode. Structures were solved by Patterson or direct methods. The structures were completed by subsequent difference Fourier techniques and refined by full-matrix least squares on F 2 (SHELXL-97)

  45 H 36N 3 P 193 Ir : 842.22708, found : 842.2270. Anal. Calcd for C 45 H 36 N 3 F 6 P 2 Ir 1.5CH 2 Cl 2 : C, 50.[START_REF] Sheldrick | SHELX97-2 Program for Crystal Structure Refinement[END_REF]H, 3.53, N, 3.77. Found C, 50.25,H, 3.35, N, 3.76. Cl 2 ): 48.0 (s), -145.0 (q, PF 6 ). HRMS m/z calcd. for C 41 H 35 N 4 P 193 Ir : 807.22233, found: 807.2227. Anal. Calcd for C 41 H 35 N 4 F 6 P 2 Ir: C, 51.73, H, 3.71, N, 5.89. Cl 2 ): 39.1 (s), -145.0 (q, PF 6 ). HRMS m/z calcd. for C 40 H 34 N 5 P 193 Ir : 808.21758, found: 808.2177. Anal. Calcd for C 40 H 34 N 5 F 6 P 2 Ir: C, 50.42, H, 3.60, N, 7.35.

	Complex 2: yellow powder (75% yield). 1 H NMR (400 MHz, CD 2 Cl 2 ): 8.11 (d, 1H, 3 J = 6.0
	Hz), 7.92 (d, 1H, 3 J = 3.4 Hz), 7.82-7.66 (m, 6H), 7.54-7.44 (m, 5H), 7.30 (s, 1H), 7.28 (d,
	1H, 3 J = 6.0 Hz), 7.11-6.92 (m, 7H), 6.71 (t, 2H, 3 J = 9.2 Hz), 6.62-6.56 (m, 3H), 6.52 (dd,
	1H, 3 J = 5.8 Hz, 4 J = 1.3 Hz), 6.19 (t, 1H, 3 J = 5.9 Hz), 2.45 (s, 3H), 2.35 (s, 3H). 31 P{ 1 H}
	NMR (162 MHz, CD 2 Found C, 51.87, H, 3.78, N, 5.75.
	Complex 3: yellow green powder (70% yield). 1 H NMR (400 MHz, CD 2 Cl 2 ): 8.60 (s, 1H),
	8.11 (d, 1H, 3 J = 6.0 Hz), 7.78 (s, 1H), 7.75-7.67 (m, 5H), 7.59 (t, 1H, 3 J = 7.1 Hz), 7.54-7.49
	(m, 3H), 7.40 (d, 1H, 3 J = 6.0 Hz), 7.32 (s, 1H), 7.18-7.08 (m, 2H), 7.06-6.94 (m, 5H), 6.78-
	6.70 (m, 4H), 6.59-6.55 (m, 2H), 6.20 (t, 1H, 3 J = 6.1 Hz), 2.39 (s, 3H), 2.29 (s, 3H). 31 P{ 1 H}
	NMR (162 MHz, CD 2
	Complex 1: yellow-green powder (55% yield). 1 H NMR (400 MHz, CD 2 Cl 2 ): 8.60 (d, 1H,
	3 J = 8.4 Hz), 8.50-8.46 (m, 2H), 8.37 (d, 1H, 3 J = 8.0 Hz), 8.05 (t, 1H, 3 J = 7.6 Hz), 7.83-7.76
	(m, 5H), 7.59-7.55 (m, 1H), 7.50-7.41 (m, 4H), 7.31 (s, 1H), 7.15-7.10 (m, 2H), 7.05-6.95 (m,
	4H), 6.86 (td, 2H, 3 J = 6.5 Hz, 4 J = 2.3 Hz), 6.59 (d, 1H, 3 J = 7.2 Hz), 6.46 (d, 1H, 3 J = 5.8
	Hz), 6.36-6.32 (m, 3H), 6.22 (td, 1H, 3 J = 6.0 Hz, 4 J = 2.0 Hz), 2.32 (s, 3H), 2.27 (s, 3H).

. The cationic complexes [Ir(C^N-ppyMe) 2 (P^N)][PF 6 ](1-3) are readily accessible by treatment of the chloro-bridged dimer [Ir(C^N-Meppy) 2 (µ-Cl)] 2 with 2 equivalents of the appropriate P^N ligand in the presence of excess AgPF 6 to facilitate the removal of the chloride ligands. The

AgPF 6 also ensures that the complexes are isolated as their hexafluorophosphate salts, which offer better solubility over the chloride salts. All complexes are isolated in good yield after recrystallisation from CH 2 Cl 2 /diethyl ether. They were fully characterized by standard spectroscopic methods ( 1 H and 31 P NMR) and gave satisfactory analysis. In the 31 P{ 1 H} NMR spectrum of 1 (CD 2 Cl 2 ), the coordinated phosphine gives rise to a singlet at δ 8.20 ppm, which is shifted to higher field in comparison with that of the free ligand L 1 (δ (CHCl 3 ) = -14.20 ppm).

31 P{ 1 H} NMR (162 MHz, CD 2 Cl 2 ): 8.2 (s), -145.0 (q, PF 6 ). HRMS m/z calcd. for

C

Found C, 49.92,H, 3.46, N, 7.13.

Table 2

 2 Selected

	Bond	1	2	3
	C1-Ir-C13	86.70(12)	86.02(17)	88.55(17)
	C1-Ir-N1	79.49(12)	79.79(17)	80.38(17)
	C1-Ir-N2	91.76(12)	167.38(16)	80.38(17)
	C1-Ir-N3	95.96(11)	179.06(15)	174.88(16)
	C1-Ir-P1	177.40(9)	101.43(12)	95.10(13)
	C13-Ir-N1	96.68(11)	89.79(15)	93.11(18)
	C13-Ir-N2	79.97(11)	79.76(16)	78.91(18)
	C13-Ir-N3	174.61(11)	93.05(16)	95.87(15)
	C13-Ir-P1	95.56(9)	172.09(13)	176.29(12)
	N1-Ir-N2	170.84(11)	167.38(16)	171.22(15)
	N1-Ir-N3	88.42(10)	100.26(15)	96.76(15)
	N1-Ir-P1	101.49(8)	88.91(10)	86.87(11)
	N2-Ir-N3	95.24(10)	87.44(18)	87.77(15)
	N2-Ir-P1	87.37(7)	102.40(11)	101.34(11)
	N3-Ir-P1	81.69(8)	79.51(10)	80.45(10)
	NH-P1-CPh		104.6(2)	104.9(2)

bond angles (°) for complexes 1-3.

Acknowledgements

This work was supported by LIA CNRS 836 Chili (MIF) and Fondecyt (# 1120149). B.A gratefully acknowledges the French Embassy in Chile for a grant. M.B. thanks the CNRS and Durham University for a joint PhD studentship.


Viatcheslav Jouikov

Jacques Simonet

Graphene: Large Scale Chemical Functionalization By Cathodic Means

Keywords: graphene, modified graphene, redox electrodes, electrolyses

Large amounts of graphene were electrochemically functionalized in an adapted frit-separated two-compartment electrolysis cell. The employed technique corresponds to that of a fluidized bed with a dynamic suspension of graphene maintained by an efficient stirring system. A large collector electrode (e.g. a mercury pool or a glassy carbon plate) polarized at a potential adapted to the chemical modification is placed at the center of the cathodic compartment.

Modifications are based on two different types of electrochemical reactions: either i) generation of free radicals in situ adding to graphene or ii) the cathodic charge of graphene at the potentials < -1.8 V vs. Ag/AgCl leading to poly-nucleophilic graphene reactive toward electrophilic reagents present in the solution. Several selected examples underline the interest of this method for building modified graphene materials for further use as deposits on different substrates. The mass modification ratio of graphene could be estimated to be quite high, e.g. one COOH group per 100 graphene carbon atoms (ca 35 phenyl units).

Introduction

Glassy carbon (GC) is commonly used an electrode material for carrying out electrolyses, especially in the cathodic domain [1][START_REF] Peters | Organic Electrochemistry[END_REF]. Until now, a large panel of electro-organic reactions have been easily achieved at negatively polarized carbon. However, when using carbon as the electrode at quite negative potentials (E < -1.7 V), one has to take into account its cathodic

A C C E P T E D

M A N U S C R I P T charge. As a mater of fact, although considered for a long time as "inert", glassy carbon contains ribbons of graphite-like structures in discontinuity with "crystallite" boundaries representing all types of unsaturated forms of carbon. This provides extraordinary possibilities for exploiting the reactivity of glassy carbon, which is due to the presence of graphite, fullerene-like structures and graphene formed in this material during its preparation (high temperature pyrolysis of phenolic resins [START_REF] Jenkins | [END_REF][4]). Cathodic charging of these different species is often essential for inducing specific reactions with a large palette of electrophilic reagents (such as alkyl halides carrying redox groups) that are present in the vicinity of the surface [5][6].

It also appears essential to consider cathodically charged carbons as a material reactive towards a large number of electrophiles other than alkyl halides. Thus, highly oriented pyrolytic graphite (HOPG), natural graphite from Ceylon, and glassy carbon were reported to react with organic halides and also with carbon dioxide under quite simple and efficient conditions [7][8]. Recent preliminary experiments on the electrochemical modification of glassy carbon, made possible due to the presence of graphite, fullerenes, and possibly of graphene as electroactive impurities, demonstrated deep global modification of this material. Now, the growing interest for functionalization of graphene, quite obviously rich and promising for multiple applications [9], prompts us to consider that charged graphene species may be of high interest for fixing electrophilic compounds such as CO 2 [10] under the conditions previously described for other types of carbons.

Experimentally, cathodic polarization of graphene in non-aqueous solvents starts at E < -1.9 V vs. Ag/AgCl to give a series of cathodic steps described as a globally complex reversible charging rather comparable to that of natural graphite, when deposited onto solid conductor like gold or glassy carbon. Analytically, it was shown that these reduction steps (in close analogy to the chemical charge of graphite) yield a nucleophilic material capable to induce both internal and interfacial reactions. Thus, it is worth recalling that, when polar organic solvents are used, cathodic charge of highly oriented pyrolytic graphite (HOPG) [11] A C C E P T E D M A N U S C R I P T

ACCEPTED MANUSCRIPT

3 specifically leads to well defined insertion stages [C n -,TAA + ] via concomitant insertion of electrons and tetraalkylammonium salts (TAAX). Exfoliation of natural graphite by bulky ammonium salts has been reported [12] long before graphene was experimentally prepared.

Similarly, such "graphene salts", rather similar to tetraalkylammonium amalgams, could be considered both as reducing species and as poly-nucleophilic materials.

Therefore, modifications of graphene in macro-scale are nowadays important subjects since the development of new modes of addition of different groups (redox, hydrophilic, lipophilic, organo-metallic, etc.) to this material is of great importance. We focused this note on electrochemically induced chemical changes of graphene.

As underlined above and put forward in recent contributions [13], graphene can be seen both as a pro-nucleophilic material and as a radical scavenger, depending on the applied potential and on the nature of reagents present in the solution. We aim to propose here, for the first time, a simple and efficient method for achieving profound chemical functionalizations of graphene in quite large amounts by fixed-potential macro electrolyses using very simple electrolytic cells; the process can further be improved depending on reactions leading to the desired material(s). In this preliminary work, the experimental conditions are restrained to organic polar solvents such as N,N'-dimethylformamide (DMF) containing tetraalkylammonium salts like tetrabutylammonium tetrafluoroborate (TBABF 4 ). We point out here the preparation of graphene largely modified with redox entities like ferrocene and anthraquinone, or else with aromatic moieties (anthracene, pyrene, naphthalene) and also with further easily modifiable functions like allyl and propargyl groups. Obviously, this scope is far from being thoroughgoing. Moreover, for the first time we explore the capability of graphene, electrochemically charged in polar organic solvents, to act as a macro-nucleophilic substrate towards carbon dioxide to enable an easy building of [graphene-carboxylate] materials.

Experimental section

A C C E P T E D M A N U S C R I P T

and tetra-n-butylammonium tetrafluoroborates (TEABF 4 and TBABF 4 ), in reagent grade dimethylformamide (DMF). Experiments described in this work needed no special treatment of electrolytic solutions.

Potentials are referred to aqueous Ag/AgCl/KCl (sat) system. Voltammetric and fixed potential electrolyses were performed using three-electrode cells separated with a fritted glass.

The electrochemical instrumentation has been previously reported [START_REF] Jouikov | modes of graphene modification via radical (B1: addition of R produced by reduction of a soluble radical precursor) and nucleophilic (B2: transforming graphene into a poly-nucleophilic material) mechanisms. (C) Nyquist plots (inset: voltammetry) of the reduction of tetracyanobenzene (TCB) at modified graphene layers deposited on GC electrodes. (a) GC-GR-CH 2 -pyrene interface; R CT = 780 kohm. (b) GC-GR-COOH interface; R CT = 7.5 kohm. (c) Bare GC[END_REF].

Glassy carbon (GC) electrodes used as supports for graphene deposition had geometric area 7 mm 2 . All glassy carbon samples used as substrates were purchased from Tokai Carbon Co (code: GC Rod). All graphene modifications described in the present note concern a material purchased from XG Sciences as XGnP Graphene Nanoplatelets Grade C. This material typically consists of submicron platelets (particle diameter < 2 μm, with thickness of a few nm). The average surface is of the order of 750 m 2 /g and TEM images permit to detect almost transparent platelets. Oxygen content was reported to be < 2 wt % while that of carbon > 98.0 wt %.

Macro-modification of graphene were achieved by electrolyses in the presence either of pro-radicals such as {π-acceptor-CH 2 -Br}, often commercially available (Aldrich), or of electrophilic molecules such as substituted organic iodides (the synthesis of Fc-(CH 2 ) 6 -I was previously described [START_REF] Jouikov | modes of graphene modification via radical (B1: addition of R produced by reduction of a soluble radical precursor) and nucleophilic (B2: transforming graphene into a poly-nucleophilic material) mechanisms. (C) Nyquist plots (inset: voltammetry) of the reduction of tetracyanobenzene (TCB) at modified graphene layers deposited on GC electrodes. (a) GC-GR-CH 2 -pyrene interface; R CT = 780 kohm. (b) GC-GR-COOH interface; R CT = 7.5 kohm. (c) Bare GC[END_REF]) and CO 2 in saturated solutions in the polar organic solvent (DMF or acetonitrile).

Prior to deposition of the transformed graphene (for building modified electrodes and/or for analysis of the produced functionalized graphene), the electrodes were carefully polished with silicon carbide paper, first with {Struers 500} and then with {Struers 1200}. Then, graphene was mechanically deposited on the GC supports by rubbing the electrodes with pressure against a flat surface (glassy cardboard or polished agate) where graphene is disposed until shiny surfaces were obtained. After using these GC-graphene (GC-GR)

A C C E P T E D M A N U S C R I P T

electrodes, they were rinsed with water and then with acetone followed by final drying in a hot air stream (60 °C).

Results

Macro-electrolyses for graphene modification.

A typical procedure could be generalized as follows: about 150 mg of graphene are introduced to the cathodic compartment of an H-type electrolytic cell with the available volume of at least 10 mL (Figure 1 A) which was then filled with 6 mL of DMF containing TBABF 4 . An efficient stirring permits to form and to maintain graphene suspension similar to a fluidized bed system. The main current collector can be either a mercury pool (convenient when quite negative potentials, E -2 V, have to be reached for inducing electrochemical charge of graphene) or a GC plate (about 3-4 cm 2 ) placed in the center of the cathodic compartment. The anodic compartment simply contains an anode (carbon, graphite, or platinum grid) immersed in same solvent. Along with graphene, a suitable reagent has to be added to the catholyte (e.g. an electrophile, an RX compound capable to generate free R • radicals within a potential range > -1.5 V, or else bubbling CO 2 to saturation). As underlined in Figure 1 B, the applied potentials may differ according to the mode of modification. In addition, the formation of free radicals from RXs can be catalyzed by deposits of palladium or silver onto the main carbon electrode [13]. Lastly, the electrolyses have to be achieved under inert atmosphere (argon).

Basically, a potentiostatic system is required (to control the potential of the collector electrode). However, with small amounts of electricity injected in this process, a galvanostatic procedure could easily be used as well.

Graphene modification by attachment of carbon dioxide and ferrocene.

α. Grafting of CO 2 onto graphene was realized in a H-shaped cell using its saturated solution (P = 1 bar, room temperature) in DMF/0.1 M TBABF 4 (8 mL). The amount of graphene charged: 170 mg. The electric collector (GC plate, 3 cm 2 ) was polarized at -1.9 V. Under strong stirring, the initial current was 3 mA. After passing 11 C of electricity, the electrolysis

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was stopped; graphene material was filtered off and then rinsed several times with water, then with acetone. The obtained powder of carboxylated graphene was dried for 1 h (60 °C).

. Binding ferrocene to graphene was carried out using 1-iodohexyl-6-ferrocene (1.5 10 -2 mol L -1 ) in a similar solution (DMF/0.1 M TBABF 4 ). The graphene load was 120 mg and the applied potential (GC plate, 4 cm 2 ) was E = -1.95 V. Total amount of electricity consumed in the process was 4.5 C.

. Other modifications: several similar graphene modifications were as well achieved using π-acceptors bearing -CH 2 -X groups (X = I or Br). Upon one-electron reductive cleavage of the C-X end bond of the linker, these precursors lead to π-acceptor-CH 2 • radicals. For example, 2-bromomethyl-anthraquinone was successfully used to attach anthraquinone groups onto graphene. The same procedure was used for efficient grafting of other benzyl-type systems including pyrene, naphthalene, anthracene, and other substituted aromatics. Similarly, allylic and propargylic offshoots were successfully grafted to graphene, both groups being easily chemically modifiable to allow various ensuing functionalizations of this material.

Typically, mercury and GC macro-electrodes were used, and applied potentials were > -1 V.

Arbitrarily, we chose to limit the electricity consumption per mass of graphene present in the cell (actually about 5 coulombs per 100 mg of graphene).

What is the average level of grafting by means of the proposed procedure?

This is a question difficult to solve in the absence of a marker. In this preliminary approach, we opted for the use of redox markers such as anthraquinone (AQ), ferrocene (Fc), 4-nitrophenyl, 4-cyanophenyl, 4-iodophenyl and other groups providing a distinct redox response. In the case of carboxylated graphene, additional post-grafting transformations of the -CO 2 H group were achieved (e.g. easy addition of anthraquinone).

Main results (with CO 2 , AQ, and Fc) are displayed in Figure 2. The working principle is the same in all cases: the deposition of modified graphene (typically from 0.15 to 0.5 mg) in fine layers onto a GC plate (average area from 0.5 to 1 cm 2 ) followed by the integration of the global redox response of the GC coverage used as electrode. Several measurements (with

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different amounts of modified graphene) allowed us to check that electrochemical signals remain proportional to the deposited mass. Under the experimental conditions summarized above, the attachment of Fc corresponds to (5 ± 2) 10 -4 Fc per graphene carbon atom while that of AQ (using AQ-CH 2 -Br) similarly exhibits the levels of (5 ± 1) 10 -4 AQ/carbon.

Modification by CO 2 (quantified via its subsequent transformation in the AQ-tagged material)

appears quite efficient and corresponds to ≈10 -2 carboxylate groups per carbon. Experiments with much larger amounts of electricity remain to be done to produce more extensively modified graphenes.

Conclusion

Macro-electrolyses performed with graphene in fluidized bed mode in the solutions of DMF or ACN containing tetraalkylammonium salts lead to chemical modifications of graphene in a large amount. These functionalizations are exemplified with extensive carboxylation and with grafting of redox moieties like anthraquinone, ferrocene, nitrophenyl, etc. Experimental conditions of this process are very simple and could be applied, after deposition of modified graphenes onto conducting solid supports (glassy carbon, metals like gold, palladium, platinum, and silver) to a wide variety of modified electrodes usable in catalysis and analysis.

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Figure 2 RedoxFigure 1 A

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Figure 2 .

 2 Figure 2.

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Graphical abstract

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Highlights

Reduction of graphene nanoplatelets in polar aprotic solvents > Electrolysis to afford a cathodic charge of graphene in suspension > Generation of a polynucleophilic form of graphene reacting with organic nucleophiles > Alternative addition of electrogenerated radicals onto graphene depending on applied potential.