Luis Almeida
email: luis.almeida@sorbonne-universite.fr

Jesús Bellver Arnau

Michel Duprez
email: duprez@ceremade.dauphine.fr

Yannick Privat
email: yannick.privat@unistra.fr

Jesús Bellver
email: bellver@ljll.math.upmc.fr

Minimal cost-time strategies for mosquito population replacement

Keywords: minimal time, optimal control, Wolbachia, ordinary differential systems, epidemic vector control

come

Introduction

Arboviruses are a major threat for human health throughout the world, being responsible for diseases such as Dengue, Zika, Chikungunya or Yellow fever [START_REF] Guzman | Dengue: a continuing global threat[END_REF][START_REF] Organization | Dengue: guidelines for diagnosis, treatment, prevention and control[END_REF]. This has led to the development of increasingly sophisticated techniques to fight against these viruses, specially, techniques targeting the vector transmitting the diseases, i.e. the mosquito [START_REF] Alphey | Sterile-insect methods for control of mosquito-borne diseases: an analysis[END_REF][START_REF] Alphey | Genetic control of mosquitoes[END_REF][START_REF] Hoffmann | Successful establishment of Wolbachia in Aedes populations to suppress dengue transmission[END_REF].

Recently, there has been increasing interest in using the well-known bacterium Wolbachia which lives inside insect cells [START_REF] Hertig | Studies on rickettsia-like micro-organisms in insects[END_REF], as a tool for carrying out this vector-targeted control [START_REF] Bliman | Ensuring successful introduction of wolbachia in natural populations of aedes aegypti by means of feedback control[END_REF][START_REF] Hughes | Modelling the use of wolbachia to control dengue fever transmission[END_REF][START_REF] Schraiber | Constraints on the use of lifespanshortening wolbachia to control dengue fever[END_REF][START_REF] Ndii | Modelling the transmission dynamics of dengue in the presence of wolbachia[END_REF][START_REF] Hoffmann | Successful establishment of Wolbachia in Aedes populations to suppress dengue transmission[END_REF][START_REF] Sallet | Monotone dynamical systems and some models of wolbachia in aedes aegypti populations[END_REF].

There is a strategy named "Incompatible Insect Technique" [START_REF] Strugarek | On the use of the sterile insect release technique to reduce or eliminate mosquito populations[END_REF] for using Wolbachia-infected male mosquito releases to control mosquito populations. It is quite similar to the more widespread sterile insect technique (they are sometimes used together). It is based on a a phenomenon called cytoplasmic incompatibility (CI) [START_REF] Sinkins | Wolbachia and cytoplasmic incompatibility in mosquitoes[END_REF][START_REF] Kambhampati | Unidirectional cytoplasmic incompatibility in the mosquito, aedes albopictus[END_REF], which produces cross sterility between Wolbachia-infected males and uninfected females.

This paper is concerned with another way of using Wolbachia-infected mosquitoes to control vector populations. It takes advantage not only of CI, but also of the vertical transmission of Wolbachia in mosquitoes (it is transmitted from the mother to its offspring) and of the fact that mosquitoes carrying this bacterium show a significant reduction in their vectorial capacity for several arboviroses [START_REF] Walker | The w mel wolbachia strain blocks dengue and invades caged aedes aegypti populations[END_REF][START_REF] Moreira | A wolbachia symbiont in aedes aegypti limits infection with dengue, chikungunya, and plasmodium[END_REF][START_REF] Turley | Wolbachia infection reduces blood-feeding success in the dengue fever mosquito, aedes aegypti[END_REF][START_REF] Mousson | The native wolbachia symbionts limit transmission of dengue virus in aedes albopictus[END_REF]. Together, these properties make population replacement an interesting tool to fight against these diseases. Thus, this second technique consists in releasing both males and females to achieve a (at least partial) replacement of the original mosquito population (which were efficient vectors for the disease) by one of Wolbachia-infected mosquitoes (which will not be good vectors for it).

In this work, we focus on taking advantage of mathematical modeling to find ways to minimize the amount of mosquitoes needed to ensure an effective population replacement. We will first fix the time horizon over which we will be able to act and consider a cost functional modeling the population replacement strategy. Next, a convex combination of the time horizon and the previous cost functional will be considered to get an optimized cost-time strategy. To investigate this issue, let us introduce a model for two interacting mosquito populations: a Wolbachia-free population n 1 , and a Wolbachia carrying one, n 2 . The resulting system, introduced in [START_REF] Almeida | Optimal releases for population replacement strategies: application to wolbachia[END_REF], reads

       dn1(t) dt = b 1 n 1 (t) 1 -s h n2(t) n1(t)+n2(t) 1 -n1(t)+n2(t) K -d 1 n 1 (t), dn2(t) dt = b 2 n 2 (t) 1 -n1(t)+n2(t) K -d 2 n 2 (t) + u(t) , t > 0, n 1 (0) = n 0 1 , n 2 (0) = n 0 2 , (1)
where u(•) plays the role of a control function that will be made precise in the sequel.

The aim is to achieve a population replacement meaning that, starting from the equilibrium (n * 1 , 0) = K 1 -d1 b1 , 0 , one wants to reach the equilibrium (0, n * 2 ) = 0, K 1 -d2 b2 . In the system, the positive constant K represents the carrying capacity of the environment (concerning the sum of the two populations n 1 (t) + n 2 (t)), d i and b i , i = 1, 2 denote respectively the death and birth rates of the mosquitoes and s h is the CI rate.

To act on the system, we will use the control function appearing in the second equation, representing the rate at which Wolbachia-infected mosquitoes are introduced into the population.

We will impose the following biological constraint on the control function u: the rate at which we can instantaneously release the mosquitoes will be bounded above by M . Another natural biological constraint is to limit the total amount of mosquitoes we release up to time T , as done in [START_REF] Almeida | Mosquito population control strategies for fighting against arboviruses[END_REF] and [START_REF] Almeida | Optimal releases for population replacement strategies: application to wolbachia[END_REF]. In this article, we take a different approach, looking actually at minimizing the total number of mosquitoes used. We thus introduce the space of admissible controls

U T,M := {u ∈ L ∞ ([0, T ]) , 0 u M a.e. in (0, T )} . (2)
Before going further, in order to simplify the analytical study of the system, we will work on a reduction of the problem already used in [START_REF] Almeida | Optimal releases for population replacement strategies: application to wolbachia[END_REF]. It is shown there that, under the hypothesis of having a high birth rate, i.e. considering b

1 = b 0 1 ε , b 2 = b 0 2
ε and taking the limit ε → 0, the proportion n2 n1+n2 of Wolbachia-infected mosquitoes in the population, uniformly converges to p, the solution of a simple scalar ODE, namely

dp dt (t) = f (p(t)) + u(t)g(p(t)), t > 0 p(0) = 0, (3)
where

f (p) = p(1 -p) d 1 b 0 2 -d 2 b 0 1 (1 -s h p) b 0 1 (1 -p)(1 -s h p) + b 0 2 p and g(p) = 1 K b 0 1 (1 -p)(1 -s h p) b 0 1 (1 -p)(1 -s h p) + b 0 2 p .
We remark that in the absence of a control function, the mosquito proportion equation simplifies into dp dt = f (p). This system is bistable, with two stable equilibria at p = 0 and p = 1 and one unstable equilibrium at p = θ = 1 s h 1 -

d1b 0 2 d2b 0 1
, the only root of f strictly between 0 and 1 assuming that

1 -s h < d 1 b 0 2 d 2 b 0 1 < 1. (4)
In what follows, it will be useful to notice that the derivative of the function f /g has a unique zero p * in (0, θ) defined by

p * = 1 s h 1 - d 1 b 0 2 d 2 b 0 1 . (5)
In [START_REF] Almeida | Optimal releases for population replacement strategies: application to wolbachia[END_REF], for the aforementioned system of two equations, the problem inf

u∈U T ,C,M J(u), with J(u) = 1 2 n 1 (T ) 2 + 1 2 [(n * 2 -n 2 (T )) + ] 2 (6)
and

U T,C,M = u ∈ U T,M , T 0 u(t)dt C is considered.
Then, denoting by J ε (u) the criterion J(u) where the birth rates b 1 and b 2 have been respectively replaced by ε , with ε > 0, a Γ-convergence type result is proven. More precisely, any solution of the reduced problem is close to the solutions of the original problem for the weak-star topology of L ∞ (0, T ) and moreover

lim ε→0 inf u∈U T ,C,M J ε (u) = inf u∈U T ,C,M J 0 (u), where J 0 (u) = lim ε→0 J ε (u) = K(1 -p(T )) 2 (7)
and p is the solution of (3) associated to the control function choice u(•). The arguments exposed in [START_REF] Almeida | Optimal releases for population replacement strategies: application to wolbachia[END_REF] can be adapted to our problem, allowing us to investigate the minimization of J 0 given by [START_REF] Guzman | Dengue: a continuing global threat[END_REF], which is easier to study both analytically and numerically, instead of the full problem [START_REF] Bliman | Ensuring successful introduction of wolbachia in natural populations of aedes aegypti by means of feedback control[END_REF], since the solutions of both problems will be close in the sense of the Γ-convergence.

In accordance with the stability considerations above concerning system (3) without control, we will impose as final state constraint p(T ) = θ since once we are above this state, the system will evolve by itself (with no need to control it any longer) to p = 1, the state of total invasion. By analogy with the two-equation system, θ represents the threshold of the basin of attraction of the equilibrium (0, n * 2 ). Since our goal in this work is to minimize the cost of our action on the system, we will address the issue of minimizing the total number mosquitoes released, in other words, the integral over time of the rate at which mosquitoes are being released, namely

J(u) = T 0 u(s) ds. (8)
2 Optimal control with a finite horizon of time

We first consider an optimal control problem where the time window [0, T ] in which we are going to act on the system is fixed. This leads us to deal with the optimal control problem

   inf u∈U T ,M J(u), p = f (p) + ug(p), p(0) = 0 , p(T ) = θ , (P T,M )
where J(u) is defined by ( 8) and U T,M is given by (2).

Theorem 1. Let us introduce

m * := max p∈[0,θ] - f (p) g(p) and T * = θ 0 dν f (ν) + M g(ν)
for M > 0.

Let us assume that M > m * , T T * and that (4) is true. Then there exists a bang-bang control u * ∈ U T,M solving problem (P T,M ). Furthermore, every function u * defined by u

* ξ = M 1 (0+ξ,T * +ξ) , with ξ ∈ [0, T -T * ] solves the problem, and J(u * ξ ) = M T * .
Proof. Observe first that T * is constructed to be the exact time such that p(T ) = θ whenever one takes the "maximal" control equal to M on (0, T * ). For this reason, if T = T * the set of admissible controls reduces to a singleton and we will assume from now on that T > T * . For the sake of readability, the proof of the existence of solutions is going to be treated separately in Appendix A. We focus here on deriving and exploiting the necessary optimality conditions. To this aim, let u * be a solution of Problem (P T,M ). To apply the Pontryagin's Maximum Principle (PMP), let us introduce U = [0, M ], as well as the Hamiltonian H of the system, given by

H : IR + × IR × IR × {0, -1} × U (t, p, q, q 0 , u) → q(f (p) + ug(p)) + q 0 u.
Here, q satisfies

q (t) = - ∂H ∂p = -q(f (p) + ug (p)), t ∈ (0, T )
so that q(t) = q(0)e -t 0 f (p(s))+u(s)g (p(s))ds , and thus q(t) has a constant sign. The instantaneous maximization condition reads

u * (t) ∈ arg max v∈U H(t, p, q, q 0 , v) = arg max v∈U qg(p) + q 0 v. (10)
This condition implies that q is positive in (0, T ). Indeed, assume by contradiction that q 0 in (0, T ). Since q has a constant sign, since q 0 ∈ {0, 1}, and since the pair (q, q 0 ) is nontrivial according to the Pontryagin Maximum Principle, there are two possibilities: either q 0 = -1 and q 0 in (0, T ) or q 0 = 0 and q < 0 in (0, T ). In both cases, the optimality condition yields that u * (t) = 0 for all t ∈ [0, T ], which is in contradiction with the condition p(T ) = θ. We thus get that q(0) > 0 and q(t) > 0 in (0, T ). Now, let us show that q 0 = -1. To this aim, let us assume by contradiction that q 0 = 0. Hence, the optimality condition reads u * (t) ∈ arg max v∈U qg(p)v, and since both q and g • p are positive in [0, T ], one has u * (t) = M 1 [0,T ] . But this control function verifies the final state constraint p(T ) = θ if and only if T = T * . We have thus reached a contradiction if T > T * . Therefore, it follows that q 0 = -1.

Let us introduce the function w given by w(t) = q(t)g(p(t)). The maximization condition (10

) yields      w(t) 1 on {u * = 0}, w(t) = 1 on {0 < u * < M }, w(t) 1 on {u * = M }.
Let us finally prove that any solution is bang-bang. The following approach is mainly inspired from [2, Proof of Lemma 7]. For the sake of readability, we recall the main steps and refer to this reference for further details.

Note that

w (t) = q (t)g(p(t)) + q(t)g (p(t))p (t) = -q(t) (f (p(t)) + u(t)g (p(t))) g(p(t)) +q(t)g (p(t)) (f (p(t)) + u(t)g(p(t))) = q(t) (-f (p(t))g(p(t)) + f (p(t))g (p(t))) = q(t)g(p(t)) 2 - f g (p(t)).
Looking at the monotonicity of p → -f (p)/g(p), we deduce that w is increasing if 0 < p(t) < p * according to (4), and decreasing if p * < p(t) < θ, where p * is defined by [START_REF] Beal | Gekko optimization suite[END_REF]. Recall that one has 0 < p * < θ according to (4).

To prove that u * is bang-bang, we show that w cannot be constant on a measurable set of positive measure. By contradiction, assume that w is constant on a measurable set I of positive measure. Necessarily, we also have (-f /g) (p(t)) = 0 on I which implies that p(t) = p * on I. This implies that for a.e. t ∈ I, p(t) is constant. In order to get that, we must have, p = 0 on that set1 . At this step, one has

{0 < u * < M } ⊂ {t ∈ (0, T ) | p(t) = p * } ⊂ {t ∈ (0, T ) | u * (t) = -f (p * )/g(p * )}.
Since M > m * , one has -f (p * )/g(p * ) ∈ (0, M ) which shows that the converse inclusion is true, and therefore,

{0 < u * < M } = {t ∈ (0, T ) | u * (t) = -f (p * )/g(p * )}.
Using that {w = 1} ⊂ {p = p * }, one gets

{0 < u * < M } = {w = 1}, {u * = M } = {w > 1}, {u * = 0} = {w < 1}
and in particular, {u * = M } and {u * = 0} are open sets.

Let I be a maximal interval on which p is equal to p * . On I, we have u

* (t) = -f (p * )/g(p *
). If at the end of the interval we have u = 0, then p must decrease since p * < θ. Therefore, (-f /g) (p(t)) > 0 so that w increases. But this is in contradiction with the necessary optimality conditions (on {u * = 0}, one has w 1). If at the end of the interval we have u = M then p increases and w decreases, leading again to a contradiction. Hence, we must have |I| = 0 or I = [0, T ]. Since p(t) = p * on I, p(0) = 0 and p(T ) = θ, this is impossible.

Therefore, u * is equal to 0 or M almost everywhere, meaning that it is bang-bang.

Finally, let us prove that the set {u * = M } is a single interval. We argue by contradiction. Using the fact that the solution is bang-bang, M > m * and that {u * = M } is open, if {u * = M } is not one open interval, there exists (t 1 , t 2 ) on which u * = 0 and p < 0. Since the final state is fixed, p(T ) = θ, then there necessarily exists a time t 3 > t 2 such that p(t 3 ) = p(t 1 ). Let us define ũ as

ũ(t) =      0 t ∈ (0, t 3 -t 1 ), u * (t -t 3 + t 1 ) t ∈ (t 3 -t 1 , t 3 ), u * (t) t ∈ (t 3 , T ).
We can easily check that since u * ∈ U T,M , then ũ ∈ U T,M and if p defined as the solution to

p (t) = f (p(t)) + u * (t)g(p(t)), t ∈ [0, T ] p(0) = 0
satisfies p(T ) = θ, then the function p defined as the solution to p (t) = f (p(t)) + ũ(t)g(p(t)), t ∈ [0, T ] p(0) = 0 satisfies p(T ) = θ and is moreover non-decreasing. Now performing a direct comparison between the cost of both controls we obtain 3 Optimal control with a free horizon of time

J(u * ) -J(ũ) = T 0 u * (t)dt - T 0 ũ(t)dt =
In the previous section, we obtained infinitely many solutions for problem (P T,M ), showing that the system presents a natural time during which one should act upon it, i.e. T * . This is supported by the fact that we need to assume T T * in order to have existence of solutions and by the fact that in case T > T * , there exists one or two time interval of size T -T * > 0 in which u * (t) = 0. This motivates the introduction of a new problem, in which the final time T is free. The functional we are interested in minimizing in this section is a convex combination of the cost J used in the previous section, and the final time:

     inf u∈U T ,M T >0
J α (T, u), p = f (p) + ug(p), p(0) = 0 , p(T ) = θ,

(P α M )
where U T,M is given by (2), α ∈ [0, 1] and

J α (T, u) = (1 -α) T 0 u(s)ds + αT.
We expect to see the intervals where u = 0 disappear. Let us state the main result of this section.

Theorem 2. Let M > m * and α > 0. Then, the unique solution of Problem

(P α M ) is u * (t) =
M 1 [0,T * ] and T * given by [START_REF] Hertig | Studies on rickettsia-like micro-organisms in insects[END_REF]. Thus the optimal value is J(u

* ) = T * ((1 -α)M + α).
Proof. Existence is proved separately in Appendix A. Here, we focus on the characterization of the solution using first-order optimality conditions. We begin by excluding the case α = 1, because the solution is trivial in this case. The problem reads

     inf u∈U T ,C,M T >0 T, p = f (p) + ug(p), p(0) = 0 , p(T ) = θ (P 1 T,M )
and the solution is clearly the constant function u * = M . Indeed, it is easy to prove that, if the set {u * < M } has a positive measure, then one decreases the minimal time to reach θ by increasing u * . Moreover, by the reasoning carried out in Theorem 1, we know that T * is the time that it takes for the system p M = f (p M ) + M g(p M ), p M (0) = 0, to reach the point p M (T * ) = θ. The conclusion follows.

Next, let us assume that α < 1 and let (T, u * ) denote an optimal pair. To apply the Pontryagin Maximum Principle (PMP), we need to define the Hamiltonian of the system H : IR + × IR × IR × {0, -1} × U (t, p, q, q 0 , u) = q(f (p) + ug(p)) + q 0 (1 -α)u.

Here, q satisfies the equation

q = - ∂H ∂p (t, p, q, q 0 , u * ) = -q(f (p) + u * g (p))
and therefore, q has a constant sign. The instantaneous maximization condition reads

u * (t) ∈ arg max v∈U H(t, p, q, q 0 , v) = arg max v∈U w(t) + q 0 (1 -α) v,
where w(t) = q(t)g(p(t)).

Thanks to a reasoning analogous to the one done in Theorem 1, we also get that q is positive. Since the final time is free, we have the extra condition u * (T ) = v T , where v T solves the onedimensional optimization problem

max v T ∈U H(T, θ, q(T ), q 0 , v T ) = max v T ∈U (w(T ) + q 0 (1 -α))v T = -q 0 α.
This condition rules out the case q 0 = 0 because, if we assumed that q 0 = 0, then we would have max

0 v T M
w(T )v T = 0. Since w(t) > 0, this would imply that we should have at the same time v T = M and w(T )M = 0, leading to a contradiction. Therefore, q 0 = -1 and we infer that the first order optimality conditions imply that

           w(t) 1 -α on {u * = 0}, w(t) = 1 -α on {0 < u * < M }, w(t) 1 -α on {u * = M }, max 0 v T M (w(T ) -(1 -α))v T = α. (11)
Recall that w is increasing if p(t) ∈ (0, p * ) and decreasing if p(t) ∈ (p * , θ), where p * is given by ( 5).

Mimicking the reasoning done in the proof of Theorem 1 (in the case where T is fixed), one can assert that u * is bang-bang and that the set {u * = M } is one single open interval. Since α > 0, the last condition of ( 11) yields u * (T ) > 0, so u * (T ) = M . This, together with the fact that the time it takes the system to reach p = θ at speed M is T * , allows us to conclude that solutions must be of the form u * (t) = M 1 [ξ,ξ+T * ] , with ξ 0. Indeed, once the function has switched to u * = M it can not switch back to u * = 0. Then, looking at the functional we want to minimize, we conclude that ξ = 0, since the cost term is independent of ξ and the term αT is increasing with respect to ξ.

As a result, the (unique) solution is Cytoplasmatic incompatibility level 0.9

u * = M 1 [0,T * ] .
Remark: if we set α = 0 in problem (P α M ), we recover problem (P T,M ) but without a restriction on the final time. This allows T to go to infinity and explains that all the pairs of the form (u T , T ) where

T > T * and u T (t) = M 1 [ξ,ξ+T * ] ,
and ξ ∈ [0, T -T * ), solve this problem. Once the system has reached the final state p(T ) = θ it can stay there indefinitely without using any mosquitoes (i.e. u = 0). It is not so realistic from a practical point of view, and justifies fixing T to avoid the emergence of such non-compact families of solutions.

Numerical simulations

This section is devoted to some numerical simulations. We will use the Python package GEKKO (see [START_REF] Beal | Gekko optimization suite[END_REF]) which solves among other things optimal control problems under constraints given by differential equations thanks to nonlinear programming solvers.

A particular attention has been paid to the development of a user-friendly source code, which is available online at https://github.com/jesusbellver/ Minimal-cost-time-strategies-for-mosquito-population-replacment with the hope that it may serve as a useful basis for other research.

1D Case

Hereafter, we provide some simulations for the reduced problems (P T,M ) and (P α M ), that can be seen as numerical confirmations of the theoretical results stated in Theorems 1 and 2. The parameters considered for these simulations are given in Table 1, using the biological parameters considered in [START_REF] Almeida | Optimal releases for population replacement strategies: application to wolbachia[END_REF].

Simulations for Problem (P T,M ) are shown in Fig. 1. To deal with the constraint p(T ) = θ, we added a penalization term in the definition of the functional. As expected, we recover in Fig. 1 that the optimal control is bang-bang and that all mosquitoes are only released for an interval of time. Although other optimal solutions may exist, we obtain a particular one, where the action is concentrated at the beginning of the time interval [0, T ].

Simulations for Problem (P α M ) are provided in Fig. 2. As proven in Theorem 2, the effect of letting the final time T free and adding it with a weight to the functional we want to minimize, Figure 1: Simulation of the reduced problem (P T,M ). The penalization parameter is ε = 0.01 and the number of elements in the ODEs discretization is equal to 300. leads to the absence of an interval during which no action is taken, for every α ∈ (0, 1]. In Fig. 2, numerical solutions are plotted for α = 0.01. The final time obtained is T * num ≈ 0.0238519, which is very close to the expected theoretical one (which does not depend on α)

T * = θ 0 dν f (ν) + M g(ν) ≈ 0.0238122.
Results for other values of α are similar.

2D Case

In this section, we provide simulations for optimal control problems involving the full system (1). We will use the parameters of Table 2, where the biological parameters have been extracted from Cytoplasmatic incompatibility level 0.9 [START_REF] Strugarek | On the use of the sterile insect release technique to reduce or eliminate mosquito populations[END_REF]. We choose M to be ten times higher than the birth rate of wild mosquitoes, in analogy with the simulations for the reduced problem.

To compute the carrying capacity K, we used the same procedure as in [START_REF] Strugarek | On the use of the sterile insect release technique to reduce or eliminate mosquito populations[END_REF], but adapting it to our model. We will make our results relevant for an island of 74 ha with a mosquito density of 69 ha -1 , so the amount of wild mosquitoes at equilibrium is n * 1 = 74 × 69 = 5106. Then, since n * 1 = K 1 -d1 b1 , we obtain the following carrying capacity of the environment

K = n * 1 1 -d1 b1 ≈ 5124.3011.
We first deal with the case where T is fixed, in other words, we solve the optimal control problem inf

u∈U T ,M T 0 u(s) ds + 1 ε max {n 1 (T ) -10, n * 2 -10 -n 2 (T ), 0} , (12)
where • IR 2 stands for the Euclidean norm in IR 2 . The minimized criterion is a combination of the total amount of mosquitoes used and a penalization term standing for the final distance to the region [0, 10] × [n * 2 -10, n * 2 ], with ε = 0.0001. It is easy to show that, because of the pointwise constraint on the control function u(•), the steady-state (0, n * 2 ) of System (1) cannot be reached in time T . This is why we chose to penalize the final distance to an arbitrary region that is clearly included in the basin of attraction of the steady-state (0, n * 2 ) but is reachable. The simulations are performed for different final times, and results are given on Fig. 3. We observe that if T is not large enough to get sufficiently close to the point (0, n * 2 ), then the control is the function u equal to M almost everywhere. When T increases, the action is carried out in two stages: first, one has u = M at least until the system enters the basin of attraction of the equilibrium point (0, n * 2 ), then u = 0 to let the system evolve without using mosquitoes. The larger T is, the less it seems necessary to act. A possible explanation is that with only a little action, it is possible to enter the basin of attraction of (0, n * 2 ). Therefore, if T is big, we can stop acting early in order to decrease the amount of mosquitoes used. Instead, if T is small, we need to release a lot of mosquitoes because otherwise, the system would not get close enough to (0, n * 2 ) by itself. Finally, on Fig. 4, simulations are carried out for the full system (1), by letting T free and replacing the previous cost functional by A Existence of solutions for problem (P T,M )

u → (1 -α) T 0 u(t)dt + αT + 1 ε max {n 1 (T ) -10, n * 2 -10 -n 2 (T ), 0} ,
The set of admissible controls for Problem (P T,M ) is

D = {u ∈ U T,M , p(T ) θ} .
Let us first prove that D is non-empty. This can be done by seeking under which assumptions one can ensure that constant controls u(t) = ū1 [0,T ] belong to the set D. Let us introduce p ū solving p ū = f (p ū) + ūg(p ū) in (0,T), p ū(0) = 0.

By integrating both sides of the differential equation, we get that the time it takes for p ū to reach the point θ, called T ū, is

T ū = θ 0 dν f (ν) + ūg(ν)
.

Note that T ū is finite if one imposes ū > m * . Moreover, since we want ū ∈ U T,M , we have that ū M , so ū ∈]m * , M ]. Finally, since the final time T is fixed, we need T ū T , and using that θ 0 dν f (ν)+ūg(ν) is decreasing with respect to ū, we deduce that T * T ū. Thus, we can conclude that D contains at least one constant control, and therefore is non-empty if, and only if, T * T , as we assumed.

Using now the fact that D is non-empty, we consider a minimizing sequence (u n ) n∈N ∈ D N for the problem (P T,M ). We have 0 u n M a.e. for all n ∈ N. Hence, the sequence (u n ) n∈N is uniformly bounded. Also, since L 1 (0, T ) = L ∞ (0, T ), using the Banach-Alaouglu theorem we conclude that D is weakly-* compact and therefore, up to a subsequence, u n * n→∞ u * , i.e. (u n ) n∈N converges for the weak-* topology of L ∞ (0, T ), and 0 u * M a.e., so that u * ∈ D.

We now consider (p n ) n∈N where p n solves p n = f (p n ) + u n g(p n ) with p n (0) = 0. Using the fact that f, g ∈ C ∞ ([0, 1]) and 0

p n 1, we deduce that (p n ) n∈IN is bounded in L ∞ (0, T ). Hence p n ∈ C 0 ([0, 1]) and therefore, using the Ascoli-Arzelá theorem, we conclude that, up to a subsequence, p n C 0 -→ p * where p * ∈ W 1,∞ (0, T ). To conclude, since u n * n→∞ u * , then T 0 ϕu n → T 0 ϕu * for all ϕ ∈ L 1 (0, T ). In particular, for ϕ : t → 1, we have Since the solutions of this system are the same as those of the system we are interested in, we will study the existence of solutions for the new one. Let us define the set of (T, ũ) verifying the constraints of System ( Pα M ), i.e. Each term of the sum being bounded from below by 0, it :follows that both of them are also bounded from above. Since α > 0, (T n ) n∈IN is bounded, therefore, up to a subsequence, T n → T < ∞. By mimicking the arguments used in Section A, one shows that, up to a subsequence, (ũ n ) n∈IN converges to ũ * ∈ U T ,M weakly-star in L ∞ (0, 1; [0, M ]). Moreover,(p n ) n∈IN converges to p * in C 0 ([0, T ]), where p * solves the equation (p * ) = f (p * ) + ũ * g(p * ), in (0, T ) and p * (0) = 0. As a consequence, ( J(T n , ũn )) n∈IN converges to J( T , ũ * ), which concludes the proof.

u

  * (t) dt > 0, which contradicts the optimality of u * . Therefore, {u * = M } is a single interval and it follows that{u * = 0} = {p ∈ {0, θ}} = {p = 0} ∪ {p = θ}.We conclude that all the mosquitoes are released in a single interval such that |{u * = M }| = T * , and thus the set {u * = 0} splits into two intervals verifying|{u * = 0}| = |{p = 0}| + |{p = θ}| = T -T * .Using that for all ξ ∈ [0, T -T * ], u * ξ = M 1 (0+ξ,T * +ξ) satisfies this property, we get that J(u * ξ ) = M T * and there are infinitely many solutions to Problem (P T,M ).

Figure 2 :

 2 Figure 2: Simulation of the reduced problem (P α M ) with α = 0.01. The number of elements in the ODEs discretization is equal to 300.

Figure 3 :

 3 Figure 3: Simulation of the full problem (12) with T fixed for T = 195 (first row), T = 210 (second row) and T = 250 (third row). The time step in the ODE discretization is ∆t = T /300.

Figure 4 :

 4 Figure 4: Simulation of the full problem with T free for α = 0.1 (first row), α = 0.5 (second row) and α = 0.9 (third row).The number of points in the ODE discretization is 101.

T 0 1 •

 1 u n → T 0 1 • u * . Hence J(u * ) = lim n→∞ J(u n ) = infu∈D J(u) and, therefore, problem (P T,M ) admits a solution. B Existence of solutions for problem (P α M ) In order to simplify the study of the existence of solutions, and to avoid working on a variable domain, we make the following change of variables. We define p(s) := p(T s) and ũ(s) := u(T s), s ∈ [0, 1]. Then, we are led to consider the problem    p (s) = T (f (p(s)) + ũ(s)g(p(s))) , p(0) = 0 , p(1) = θ, inf ũ∈L ∞ (0,1;[0,M ])

D 1 0

 1 := (T, ũ) ∈ R + × U 1,M × [0, 1] | p(1) θ .This set is clearly non-empty (consider for instance T = T * and u(•) = M 1 [0,T * ] (T * •)).Consider a minimizing sequence (T n , ũn ) n∈N ∈ D N and let pn be the associated solution of the ODE definig Problem ( Pα M ). By minimality, one has lim n→∞ J(T n , ũn ) < ∞, i.e. lim n→∞ (1 -α) T n ũn (s)ds + αT n < ∞.

Table 1 :

 1 Simulation parameter values considered in (P T,M ) and (P α M )

	Category	Parameter	Name	Value
	Optimization	T M	Final time Maximal instantaneous release rate	0.5 10
	Biology	b 0 1 b 0 2 d 0 1 d 0 2	Normalized wild birth rate Normalized infected birth rate Wild death rate Infected death rate	1 0.9 0.27 0.3
		K	Normalized carrying capacity	1
		s h

Table 2 :

 2 Simulation parameters value for the optimal control problems involving the full system[START_REF] Almeida | Mosquito population control strategies for fighting against arboviruses[END_REF]

	Category	Parameter	Name	Value
	Optimization	M	Maximal instantaneous release rate	112
		b 1	Wild birth rate	11.2
		b 2	Infected birth rate	10.1
	Biology	d 1	Wild death rate	0.04
		d 2	Infected death rate	0.044
		K	Carrying capacity	5124
		s h

Recall that if a function F in H 1 (0, T ) is constant on a measurable subset of positive Lebesgue measure, then its derivative is equal to 0, a.e. on I, see e.g.[START_REF] Henrot | Shape Variation and Optimization[END_REF] Lemma 3.1.8]

Acknowledgement

This research was supported by the Project "Analysis and simulation of optimal shapes -application to life science" of the Paris City Hall. This program has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sk lodowska-Curie grant agreement No 754362


M Merhej

S Ecoffey

B Sadani

B Lee-Sang

Thierry Baron

S David

D Drouin

B Salem

B Lee -Sang

A fabrication

Keywords: Vapor-liquid-solid (VLS) growth, Silicon germanium nanowire (SiGe NW), Horizontal growth, Nanotrenches

HAL is

Introduction

Three-dimensional (3D) integration is a promising complementary approach of the conventional planar technologies. It allows leveraging interconnect delay latency, increases overall performance and energy efficiency and decreases the footprint of an Integrated Circuit (IC) [START_REF] Shulaker | Monolithic 3D Integration : A Path From Concept To Reality[END_REF]. Today, 3D ICs in which active devices are vertically stacked upon each other, have shown significant power gains for different applications like memory-on-logic [START_REF] Dutoit | A 0.9 pJ/bit, 12.8 GByte/s Wide IO memory interface in a 3D-IC NoCbased MPSoC[END_REF]. This 3D technology relies on vertical interconnects between circuit layers known as Through Silicon Vias (TSVs). Different strategies are adopted in the 3D-ICs integration: chip level [START_REF] Al-Sarawi | A review of 3-D packaging technology[END_REF] and wafer level [START_REF] Koyanagi | Future system-on-silicon LSI chips[END_REF]. The monolithic 3D integration [START_REF] Shen | Monolithic 3D chip integrated with 500ns NVM, 3ps logic circuits and SRAM[END_REF] [START_REF] Batude | 3DVLSI with CoolCube process: An alternative path to scaling[END_REF] is another lower-cost 3D approach, that offers the possibility to elaborate new circuit layers sequentially over the previously fabricated layers, and on the same substrate. A major processing difficulty of this attractive technological approach is that the fabrication process in the upper layers circuitry must be below 550 °C [START_REF] Batude | 3DVLSI with CoolCube process: An alternative path to scaling[END_REF] in order to preserve the performances of the devices in the front-end of the line (FEOL).

Recently, self-assembled horizontal semiconductor nanowires, prepared by bottom-up techniques has spurred a new interest in the fabrication of nano-devices. This bottom-up technique provides cost-effective nanowires compared to the top-down technique [START_REF] Alaca | Guided self-assembly of metallic nanowires and channels Guided self-assembly of metallic nanowires and channels[END_REF], with respect to the thermal budget for 3D integration processes. It allows for an extremely reduced nanowires dimension and a wide choice of materials and compositions for integration with planar devices.

Currently, there are mainly two general strategies to address the challenging alignment and to fabricate self-assembled horizontal NW based FETs: an indirect post-growth assembly or a direct assembly. Post-growth assembly in liquid, such as dielectrophoresis is an indirect approach that was used in order to create arrays out of randomly dispersed nanowires [START_REF] Raychaudhuri | Precise semiconductor nanowire placement through dielectrophoresis[END_REF]. This strategy requires specific fabrication processes to create a contact pad and metal wires, for applying non-uniform AC signals. Connecting NWs with a precise alignment over a large area is considered as the main obstacle for post-growth assembly. Recent studies have shown a well ordered and precise NWs directly assembled during their CVD-VLS growth [START_REF] Oh | 3D-transistor array based on horizontally suspended silicon Nano-bridges grown via a bottom-up technique[END_REF] [10]. Device fabrication using planar NW growth without any crystalline seed is still facing difficulties [START_REF] Lecestre | Confined and guided catalytic growth of crystalline silicon films on a dielectric substrate[END_REF], therefore direct growth between pre-defined electrodes could be an effective alternative [START_REF] Salem | Self-connected horizontal silicon nanowire field effect transistor[END_REF].

We propose in this work a low thermal budget process to electrically connect self-assembled horizontal silicon-germanium nanowires (SiGe NWs). We first present the horizontal growth parameter optimization at a low temperature between predefined electrodes, on a SiO2/Si substrate. Afterward, we propose a technological process to control and eliminate the random growth of horizontal NWs. Finally, we present this guided growth according to different parameters including the wet etching, and the width of the trenches. We show the first electrical results of a connected nanowire between metal electrodes.

II. Experiment and methodology:

A preliminary study on a SiO2/Si substrate with two predefined metallic electrodes has been established to optimize the growth parameters of the horizontal SiGe NW. The NWs growth have been conducted into a Chemical Vapor Deposition (CVD) furnace, using patterned electrodes of gold seed layer covered with 120 nm titanium which is used as a protected layer (see Figure 1). A 90 sccm of silane and 45 sccm of germane (~10% in H2) gas precursors were used as a source of silicon and germanium respectively, 40 sccm of HCl as reactive gases [START_REF] Dubuc | A nanodamascene process for advanced singleelectron transistor fabrication[END_REF],

and H2 as a carrier gas. For temperature below 400°C, no growth has been detected, while SiGe NWs horizontal growth was observed over the range of 400-450°C. It was determined that a 40 min NW growth at 450°C with a pressure of 4.5 Torr produces a 2.5-4.5 µm nanowire length to connect the electrodes as shown in figure 2. This study also shows the importance of a well-guided growth to fabricate NW based FET since neither number nor connectivity control has been obtained.

We note that the nanowires growth has taken place between the pre-defined electrodes, with nucleation on the pad trenches (gold layer). Based on the optimized growth parameters, a damascene process was developed on 150 nm thick silicon dioxide thermally grown on a silicon substrate. The damascene process with nanotrenches in which the NWs will be guided is detailed in Figure 3. This process combines photolithography, dry/wet etching, electron beam lithography, and chemical-mechanical planarization techniques. The damascene method proposed in this work benefited from the previous development related to the fabrication of single electron transistors [START_REF] Dubuc | A nanodamascene process for advanced singleelectron transistor fabrication[END_REF], and top-down ultra-thin nanowires [START_REF] Guilmain | A damascene platform for controlled ultra-thin nanowire fabrication[END_REF].

First dummies structures and contact electrodes are patterned in the dielectric using photolithography [step (a) of figure 3]. Then these structures are etched using an ICP reactive ion etching [START_REF] Guilmain | SiO2 shallow nanostructures ICP etching using ZEP electroresist[END_REF] to create 100 nm deep microtrenches [step b of figure 3]. The dummies are essential for planarization uniformity [START_REF] Oliver | Chemical-Mechanical Planarization of Semiconductor Materials[END_REF]. A Titanium film (5 nm) and gold film (50 nm) were deposited inside the trenches using e-beam evaporation [step c of figure 3], followed by a liftoff [step d of figure 3].

Another 150 nm titanium film deposited by sputtering over the entire sample, in a way to fill the trenches on the top of the gold catalyst [step e of figure 3]. This titanium layer has to be thicker than the deepest trenches of the sample, to enhance CMP quality, and ensure uniformity of the planarization [START_REF] Ecoffey | Technology platform for the fabrication of titanium nanostructures[END_REF]. The CMP is used to eliminate all excess titanium material above the original oxide surface, and to planarize trenches down to the SiO2 surface [step f of figure 3].

These CMP experiments have been carried out on 10×10 mm 2 Si samples. A commercial slurry with 50 nm silica nanoparticles diluted in isopropyl alcohol (IPA) in 1:1 volume ratio was chosen. The selectivity studies on both materials have been used to control planarization at a nanometer scale [START_REF] Merhej | Fabrication of top-down gold nanostructures using a damascene process[END_REF]. Finally, 100 nm deep perpendicular nanotrenches were patterned using an e-beam lithography (EBL) on a ZEP resist [step g of figure 3], followed by an ICP plasma etching in the oxide layer [step h of figure 3]. The fabricated nanotrenches are 2 µm long, while the width varies from 70 nm to 1µm, depending on the EBL patterns.

Figure 4 depicts a SEM top view image of a 100 nm deep nanotrenches in the dielectric layer, after the final etching step at the end of the pre-described process. The darker lines represent 2 µm wide predefined titanium electrodes, while the brighter color reflects the oxide material. To eliminate the remaining SiO2 layer on the nanotrenches sidewall, an extra wet etching step is added to the process before removing the resist. This etching must be isotropic and very selective in the thin films of both Ti and resist. For that different wet etching solutions have been evaluated: hydrofluoric acid HF (10%) based-solution a commercial solution known as "Pad-etch" [START_REF] Knotter | Etching mechanism of vitreous silicon dioxide in HF-based solutions[END_REF]. Table 1 shows the etching rates of HF (10%) and "Pad-etch" solutions versus deposited titanium and e-beam resist films. While HF is the preferred solution to wet etch the SiO2 , its etch rate versus Ti and resist is too fast to allow a good control on thin films. Figure 6 shows etched Ti and SiO2 thicknesses versus time for HF and Pad-etch solutions. The Pad-etch [START_REF] Williams | Etch Rates for Micromachining Processing__ Part II[END_REF] solution has similar etch rates as HF, however its selectivity versus Ti is far better.

HF solution etches Ti almost as fast as SiO2 under certain conditions.

The surfactant used in the Pad-etch [START_REF] Williams | Etch Rates for Micromachining Processing__ Part II[END_REF] allows to obtain low etch rates on titanium because it acts as a corrosion inhibitor. After a certain amount of time (between 30 and 40 s on Fig. 6), the preferential adsorption of the corrosion inhibitor on the TiO2 is decreased due to the etching / removal of the TiO2 layer and thus appearance of the Ti surface. The Ti is then etched rapidly in the pad etch solution.

The "pad-etch" solution has been used on the samples after nanotrenches plasma etching process and before resist removal. A final acetone/IPA cleaning removes all the remaining resist film before loading the samples into the CVD machine. A 40 min VLS growth is carried out at 450° C with a pressure of 4.5 Torr immediately after the cleaning to avoid further oxidation.

III. Results and Discussion

Figure 7 depicts nanowire growth in different nanotrenches cavities dimensions with the optimized fabrication process described above. The width of these cavities varies between 500 nm (figure 7-A), 200 nm (figure 7-B), and 100 nm (figure 7-C). The NW growth took place in different trenches regardless of their widths. Among 10 samples tested with 100 different nanotrenches each, the yield of the nanowires growing inside the cavities is about 5 to 10 %.

This limited yield is attributed to a poor control of the pad-etch process. The presence of many empty trenches is probably due to the wet etching time used. Indeed, this time is not long enough to eliminate the entire oxide layer on most of the trenches. Nevertheless, and on other nanotrenches, this time was enough to remove the oxide and expose the catalyst. Figure 7-C is an example that shows a non-homogeneity in growing NWs inside the trenches. This reflects the optimization needed to have a better control of the wet etching process mainly by controlling the etching and over etching time to completely remove the SiO2 on the sidewalls. Further optimization of the nanowire nucleation is also needed.

Several nanowires could nucleate from a rectangular gold region. The trench width impacts the growth following the initial nucleation. The transport limitations influence local precursor pressures and thus induce kinking. The contact between the sidewall trenches and the liquid catalyst play an important role to maintain a straight nanowire inside the cavity, thus to connect the electrodes. Indeed, different physical effects such as surface chemistry and roughness, adhesion, temperature, and pressure can affect the frictional forces that exist between the sidewalls and the liquid catalyst [START_REF] Lecestre | Confined VLS growth and structural characterization of silicon nanoribbons[END_REF]. If a guided assembly is to be achieved, efficient nucleation will also be required.

Our fabrication scheme allowed a preliminary demonstration for the elaboration of different single nanowires, where only one nanowire connected a pair of electrodes to fabricate S/D contacts. Optimization of the plasma and the wet etching processes is ongoing to increase the yield of NWs growth, and get a deeper understanding of the guided NW growth to control it. a typical Schottky behavior. However, the overall resistance differs according to the sign of the applied drain voltage, thus revealing an electrical dissymmetry of the source and drain contacts (Figure 8-c). This phenomenon can be referred to a non-uniform cover of the titanium electrode that connects the nanowire. An additional silicidation process is considered as a solution to reduce the contact resistance between the nanowire and the metal, and thus enhance the electrical properties [START_REF] Rosaz | Highperformance silicon nanowire field-effect transistor with silicided contacts[END_REF].

IV. Conclusion

In conclusion, a preliminary demonstration of planar SiGe NWs grown via a bottom-up (VLS) technique over a dielectric surface has been demonstrated. This process can be implemented in the back end of the line since all fabrication steps respect the thermal budget. It may offer a direct contact between the SiGe NWs and metallic source/drain electrodes without any additional treatments for aligning. Connected NWs between predefined electrodes can be exploited as channels to fabricate nanowire-based devices in the BEOL for further device development.

V.

Figure 1 :

 1 Figure 1: Schematic illustration of a direct growth SiGe NW between two predefined deposited and patterned electrodes.

Figure 2 :

 2 Figure 2: SEM image of SiGe NWs grown by Vapor-Liquid-Solid mechanism between two pre-defined electrodes on an oxide plane surface.

Figure 3 :

 3 Figure3: The damascene process for guided horizontal nanowire growth. The left side of each part of the figure is the cross section along the AA' axis. a) A UV lithography is realized to pattern the titanium electrodes over the oxide layer grown on a silicon substrate. b) 100 nm trenches are formed into the oxide layer. c) A 50nm gold film is deposited on the surface using e-beam evaporation and d) then removed from the surface by a lift-off technique. e) A 180nm titanium film is deposited by sputtering over the entire device area and f) then planarized with CMP down to the SiO2 surface with the Au/Ti metal stack is embedded in the oxide layer. g) E-beam lithography is used to pattern different sizes of nanotrenches into the oxide. h) A second 100 nm oxide etching step is made to create nanotrenches between the electrodes.

Figure 4 :

 4 Figure 4: SEM image of the oxide nanotrenches created between titanium electrodes. The bright color represents the oxide while the dark color depicts the titanium.

Figure 5 :

 5 Figure 5: FIB-STEM images revealing the remaining silicon oxide layer the Au catalyst at the end of the UV trench sidewalls after the final etching step of the nanotrenches.

Figure 6 :

 6 Figure 6: Average oxide and titanium removal thickness as a function of the time, for different etchant solutions.

Figure 7 :

 7 Figure 7: Nanowire growth in several nanotrenches cavities dimensions a): 500 nm width, b): 200 nm width, c) 100 nm width.

Figure 8 :

 8 Figure 8: SEM micrographs of a) the contact pad area (in square) for electrical measurement and b) SEM image of a SiGe NW grown by VLS inside a nanotrenche (100 nm width) between two titanium/gold pre-defined electrodes. c) Output characteristics of the connected intrinsic SiGe NW with voltage varying from -1 to 1 V.

Table 1 :

 1 Etch rate comparison on titanium and e-beam resist films between HF (10 %) and Pad-etch based etchant solutions.

		Titanium	E-beam resist
	HF 10%	> 900 nm/min	> 800 nm/min
	Pad etch	< 2 nm/min	< 1 nm/min


Pascal Nguyen

Thanh Binh Nguyen
email: thanh-binh.nguyen@cnrs.fr

Pascal Retailleau

Umpolung Strategy with 2-Aminothiophenols: Access to 2-Arylbenzothiazine Derivatives from Alkyl Aryl Ketones

Keywords: sulfur, 1,4-benzothiazines, DMSO, redox condensation

Umpolung Strategy with 2-Aminothiophenols: Access to 2-Arylbenzothiazine Derivatives from Alkyl Aryl Ketones

T .B.

Among nitrogen and sulfur heterocycles, 1,4benzothiazines [1] constitute an important motif that is widely present in natural products and bioactive molecules as dimeric and other oligomeric forms. [2] For example, pheomelanins are polymeric structures of melanin pigments in mammalian hair and chicken feathers. [3] Trichochromes are produced from the same sources, of lower molecular weight and usually yellow, red or violet. All these structures contain multiple 1,4benzothiazines motifs. The synthesis of 1,4-benzothiazine derivatives was based mainly on non-redox condensation reactions of o-aminothiophenols, which act as a bis-nucleophile, with a 1,2-bis-electrophile. [4] Since both thiol SH and amino NH2 groups could act as a nucleophile, a frequently encountered issue of this approach is difficulty in controlling regiochemistry of the reactions. To overcome this drawback, the transformation could be performed in a stepwise manner in which the nitrogen or sulfur terminus is temporarily masked or protected and then released later for cyclization. Such a process requires however additional operations of protection/deprotection and thus lower the efficiency of the global transformation.

Scheme 1. Umpolung Approach to 1, 4-Benzothiazine In this context, the development of new conditions allowing direct access to original 2-aryl-1,4benzothiazine motif from 2-aminothiophenols and aryl alkyl ketones would be highly desirable. [START_REF] Lin | For a recent example of oxidative condensation of 2aminothiophenols with enolizable ketones at 110 ºC[END_REF] Taking advantage of the facile oxidation of the thiols into the corresponding disulfide and the acid-catalyzed condensation of an amino group with a ketone group, we suggest here the use of 2-aminothiophenols 1 for direct oxidative functionalization of readily available enolizable ketones via an in-situ and easily reversal of polarity of 1 from a 1,2-bis-nucleophile to 1,2nucleophile/electrophile.

We recently reported the reaction of 2aminothiophenol with a,a'-enolizable ketones such as cycloalkanones or dialkyl ketones to provide spirobis(1,4-benzothiazines) A (Scheme 2). The reaction was efficiently catalyzed by a strong Brønsted acid such as TFA and required only DMSO as an oxidant in near stoichiometric amounts. [6] We were eager to investigate the behavior of a-enolizable

S H N N S O NH 2 HO 2 C R 2 OH HO 2 C OH R 1 Trichochrome B R 1 = CH 2 CH(NH 2 )CO 2 H, R 2 = H Trichochrome C R 1 = H, R 2 = CH 2 CH(NH 2 )CO 2 H S H N S N N S N OH O NH 2 HO 2 C HO 2 C OH H 2 N CO 2 H CO 2 H
Pheomelanins the arrows denote where the polymer continues Previously, access to such dimeric 1,4benzothiazines was based mainly on the condensation of o-aminothiophenols with 1,2-biselectrophiles. [7] Recently, 3aa analogues were obtained via a reaction of 2,2′-dithiodianilines and acetophenones in HOAc under aerobic conditions. One of the major disadvantages of this method is the use of AIBN. [8] This unstable initiator should be stored at low temperature in the dark to avoid thermal and photochemical decomposition and accidental explosion. Moreover, the reactions were reported only with 2,2′-dithiodianilines, which should be prepared from their monomeric precursors 2-aminothiophenols by an additional oxidation.

+ SH NH 2 O X + S N X 1 X 2
Based on this serendipitous observation and the above-mentioned literature precedent, we focused our investigation on such a new reactivity and report here interesting aspects of the reaction of 2aminothiophenols with alkyl aryl ketones in DMSO catalyzed by a strong Brønsted acid. The initial reaction conditions were applied to an equimolar mixture of both starting materials and resulted in an better yield (Table 1, entry 1). The important role of TFA catalyst was further demonstrated. Indeed, when the reaction was performed in the absence of TFA, the reaction proceeded in an uncontrolled way, leading to complex mixtures from which we could isolate 2phenylbenzothiazole 8 in low yield (22%). In this case, the expected product 3aa was formed only in a trace amount that could be detected by the characteristic 1 H NMR signal of the aliphatic C-H proton (entry 2). On the other hand, the background reaction conditions could result in higher yield (21%) of 3aa at a higher temperature (100 ºC) (entry 3). When a base such as DBU [START_REF] Lin | For a recent example of oxidative condensation of 2aminothiophenols with enolizable ketones at 110 ºC[END_REF] or cyclohexylamine was used as a catalyst (entry 4) or additive (entry 5), the reactions in both cases stopped at the formation of monomer 3-phenyl-2H-benzo [b][1,4]thiazine 7a.

A Lewis acid such as FeCl3•6H2O or another Brønsted acid PTSA showed also good catalytic effect (entries 7 and 8). While changing reaction temperatures reduced the yield of 3aa (entries 8 and 10), lower the DMSO amount to only 2 equiv (entry 9) enhanced significantly the reaction. This improvement could be explained by an increase in the concentrations of both starting materials and catalyst and favored the dimerization of the monomeric intermediate 7a to 3aa. We found that the reaction the reaction conducted under an inert atmosphere led to practically the same result (entry 11). The reaction could be performed with 2,2′-dithiodianiline B instead of 1a (entry 12). Finally, we found that a weak acid such as HOAc was not suitable as a catalyst for this reaction (entry 13).

Table 1. Optimization of the Reaction Conditions Entry [a] Catalyst (x mol %) y T (ºC) Yield (%) [b] 1 TFA ( 10 [d] TFA (10) 2 80 3aa, 76 12 [b]e TFA (10) 2 80 3aa, 73 13 HOAc (10) 2 80 3aa, 25 [a] Conditions: 1a (1 mmol, 125 mg), acetophenone (1 mmol, 120 mg), additive or catalyst (x mol %), DMSO (y mmol) under air. [b] Isolated yield unless otherwise noted. [c] Determined by 1 H NMR. [d] Reaction performed under an Ar atmosphere. [e] 2,2′-Dithiodianiline B (0.5 mmol) were used in place of 1a.

The optimized reaction conditions (Table 1, entry 9) were next applied to a range of aryl methyl ketones (Scheme 3). Functional groups such as alkyl, alkoxy, halogen and cyano were compatible with the reaction conditions, leading to the expected products 3ab-3am in moderate to good yields. In most cases, the products could be isolated by a simple trituration of the reaction mixture with methanol. While good to excellent yields were obtained with alkyl, alkoxy, fluoro, chloro and cyano substituted acetophenones, the reactions were less efficient with m-and p-bromo-acetophenones.

In some cases, while the products 3 are sparingly soluble in common organic solvents such as chloroform or DMSO, their solubility is significantly enhanced by addition of TFA as exemplified by the case of dimer product 3ad issued from pmethoxyacetophenone 2d. The 1 H and 13 C NMR spectra of 3ad were recorded in CDCl3 and TFA and its crystals for X-ray crystallographic study were obtained as 3ad•2TFA salt from slow evaporation of this salt in chloroform. Acetyl hetarenes derived from thiophene, furan and benzofuran were shown to be competent substrates, providing the expected product 3an, 3ao, 3bo and 3ap in excellent yields. As an a-enolizable aliphatic methyl ketone, pinacolone 2q reacted sluggishly, leading to the desired product 3aq in a low yield, which could be ascribed to the important steric hindrance of the t-butyl group. The reactivity is ameliorated with a less sterically demanding ketone such as methyl neopentyl ketone 2r. In this case, although the ketone substrate 2r is a,a'-enolizable, the isolated product 3ar was issued from the oxidation that exclusively occurred at the methyl ketone group.

The reaction was next extended to propiophenone 4a (Scheme 4a). To our surprise, while the corresponding bis-1,4-benzothiazine was not formed, we could isolate from the reaction mixture a white solid which was identified as an original heteropropellane 5a (22%). Based on this observation, the molar ratio 1a:4a was adjusted to 2:1. Gratifyingly, such a simple modification allowed us to increase the yield of 5a up to 79%. This set of reaction conditions were applied to p-substituted propiophenones 4b-d and resulted in the expected heteropropellanes 5b-5d, which were isolated in high yields by simple filtration in all cases.

With butyrophenone 6a, valerophenone 6b and hexyl phenyl ketone 6c, neither bis-1,4-benzothiazine nor heteropropellane was formed. the reaction stopped at the formation of the corresponding benzothiazines 7a-c in high yields by 1 H NMR of the crude mixtures (Scheme 4b) but their isolation by column chromatography was shown to be problematic due to reactions with oxygen. Indeed, freshly purified samples upon leaving under air turned from bright yellow of benzothiazine 7a-c to dark brown mixtures from which large and single crystals were separated and determined as 2-phenylbenzothiazole 8 for all these three cases. On the other hand, a-substituted alkyl phenyl ketones such as isobutyrophenone and benzoylcyclohexane were found to yield 3,3-dialkyl-1,4-benzothiazines 7d,e, which were bench-stable and could be isolated in high yields (Scheme 5).

Scheme 5. Reactions with a-Substituted Propiophenones

Based on these observations, we tentatively suggest a mechanism (Scheme 6) for the formation of benzothiazines 3, 5a-d and 7d-e as well as of benzothiazole 8 from autoxidation of 7a-c. The first step was the formation of disulfide B by oxidation of 1a with DMSO, confirmed by a control experiment. [11] Disulfide B condensed next with ketone C to provide imine-enamine D-E. Cyclization of E led to benzothiazine 7 along with the extrusion of 1a, which was recycled to disulfide B by DMSO (Scheme 6a). Ketones 7 bear a small substituent (R = H or Me, 7H or 7Me) were oxidized easily to sulfoxide by DMSO (Scheme 6b). The rearrangement of sulfoxide F to sulfonium J was promoted by acid via intermediates G-I. Dimer 3aa was obtained by the condensation of J (R = H) with 7a, which is the enamine form of 7 (R = H). When R = Me, the similar condensation is hindered. As a result, a reaction of J with an additional molecule of 1a would provide sulfonium M, which underwent a cyclization by the mercapto group from the less hindered face to give 5a as a single diastereomer. Scheme 6. Proposed reaction pathways One of the possible reason for the formation of 3aa without any added acid catalyst is the reaction is autocatatalyzed by the acidic thiol function of 2aminothiophenol 1a (pKa = 6.59) [12] as well as by other oxygenated acidic species issued from oxidation of the thiol function of 1a by DMSO such as sulfinic and sulfonic acids. To confirm this hypothesis, we performed control experiments in the presence of an organic base (1 equiv) such as pyridine or Nmethylpiperidine. Under the same conditions of heating (DMSO (3 equiv), 80 ºC, 16 h), such base prevented the formation of 3aa from occurring. While acetophenone remained unchanged, 2aminothiophenol 1a was oxidized mainly into B.

Benzothiazines bearing larger R groups (Et, n-Pr, n-Bu) were more stable against oxidation by DMSO but not air-stable, being oxidized by O2 to ketone H. In this case, the transformation of H to sulfonium K was hindered by the larger R group. Instead, cyclization of H to benzothiazoline L occurred easily. Elimination of RCHO from L yielded 8. [9] The autoxidation of benzothiazines 7d,e is totally inhibited by the two R groups.

The presence of sulfonium J (R = H) could be confirmed by trapping experiments with anilines 9a,b, providing imine 10a,b in moderate yields (Scheme 7). Previously, similar reactions were reported to involve both O2/DMSO oxidant and KI catalyst at 120 °C. [10] In conclusion, we have developed a simple method for functionalization of aryl alkyl ketones with oaminothiophenols to provide a wide range of 2arylbenzothiazines. The method is highlighted by the use of TFA as an acid catalyst and DMSO in near stoichiometric amount as an oxidant. Although 1,4benzothiazines 7 are common reaction intermediates, the final reaction outcomes were determined by the size and electronic effect of the alkyl group of the ketones. The results presented here confirm that complex structures could be built from simple starting materials under simple conditions. Further study on the mechanism as well as the reaction scopes are ongoing in our laboratory.

Experimental Section

General procedure for the synthesis of 3 from 2aminothiophenols 1 and methyl ketone 2 (Table 1 andScheme 3) A mixture of 2-aminothiophenol 1 (1 mmol), acetophenone 2 (1 mmol), TFA (0.1 mmol, 11 mg) in DMSO (2 mmol, 0.15 mL) was stirred in a 7-mL tube closed with a rubber septum at 80 ºC for 16 h. After being cooled to room temperature, the crude reaction mixture was purified as indicated for each compound, by trituration or chromatography on silica gel column.

Products 7d,e and 12a,b were synthesized in the same manner with DMSO (4.5 mmol, 0.32 mL). In these cases, the products were purified by chromatography on silica gel column.

Representative procedure for the synthesis of heteropropellanes 5 from 2-aminothiophenol 1a and propiophenone 4

A mixture of 2-aminothiophenol 1a (2 mmol, 250 mg), propiophenone 4a (1 mmol, 134 mg), TFA (0.1 mmol, 11 mg) in DMSO (4.5 mmol, 0.32 mL) was stirred in a 7-mL tube closed with a rubber septum at 80 ºC for 16 h. After being cooled to room temperature, the crude reaction mixture was purified as by trituration and filtration with cold methanol (0-4 ºC) to provide heteropropellane 5a as a white solid.

CCDC 2016315-2016319; 2016321-2016322 and 2031498-2031499 contain the supplementary crystallographic data for 3ad•2TFA, 3ah, 3ai, 3aq, 3ar, 7d, 7e, 5a and 12a respectively in this paper. These data can be obtained free of charge from The Cambridge Crystallographic Data Centre via www.ccdc.cam.ac.uk/data_request/cif.

COMMUNICATION

Figure 1 .

 1 Figure 1. Selected Natural 1,4-Benzothiazines

  Our previous work with α,α'-enolizable ketones: Adv. Synth. Catal. 2020, DOI: 10.1002/adsc.202000725 α' α Our initial observation with α-enolizable ketone:

Scheme 3 .

 3 Scheme 3. Reactions of 1a,b with Methyl Ketones 2a-q

Scheme 4 .

 4 Scheme 4. Reactions with Propiophenones and Homologues

Scheme 7 .

 7 Scheme 7. Three-component Access to 10a,b via J

  Et, n-Pr, n-Bu, R 2 = H R 1 = CN, CO2Et, R 2 = H


Florent Dubray

Simona Moldovan

Cassandre Kouvatas

Julien Grand

Cindy Aquino

Nicolas Barrier

Jean-Pierre Gilson

Nikolai Nesterenko

Delphine Minoux

Svetlana Mintova

Direct Evidences for Single Molybdenum Atoms Incorporated in the Framework of MFI Zeolite Nanocrystals

This article is

ABSTRACT: Direct evidence of the successful incorporation of atomically dispersed molybdenum (Mo) atoms into the framework of nanosized highly crystalline MFI zeolite is demonstrated for the first time. Homogeneous distribution with a size of 0.05 nm is observed by STEM-HAADF. 31 P MAS NMR and FTIR spectroscopy, using trimethylphosphine oxide (TMPO) and deuterated acetonitrile as probe molecules, reveal a homogeneous distribution of Mo in the framework of MFI nanozeolites, and the presence of Lewis acidity. 31 P MAS NMR of TMPO shows probe molecules interacting with isolated Mo atoms in the framework, and physisorbed probe molecules in the zeolite channels. Moreover, 2D 31 P-31 P MAS radio frequency-driven recoupling (RFDR) NMR indicates the presence of one type of Mo species in different crystallographic positions in the MFI framework. The substitution of framework Si by Mo significantly reduces the silanol defect content, making the resulting zeolite highly hydrophobic. In addition, the insertion of Mo into the MFI structure induces a symmetry lowering, from orthorhombic (Pnma), typical of high silica MFI, to monoclinic (P21/n), as well as an expansion of unit cell volume. The novel material opens many opportunities of catalysts design for application in mature and emerging fields.

The introduction of metal ions into zeolites is known to modify their acid and/or redox properties, and can be achieved by different routes as recently reviewed. [START_REF] Kosinov | Engineering of Transition Metal Catalysts Confined in Zeolites[END_REF] One such route is framework isomorphous substitution where framework silicon ions are replaced by transition metal ions. The classic zeolite literature states that elements such as Ga, Ce, Be, B, Fe, Cr, P, and Mg can isomorphously substitute Si and Al in zeolite framework. [START_REF] Barrer | Hydrothermal Chemistry of Zeolites[END_REF][START_REF] Jacobs | Acid Zeolites: an Attempt to Develop Unifying Concepts[END_REF] In the 80s the introduction of Ti in the MFI zeolites was reported by ENI researchers. [START_REF] Moliner | State of the Art of Lewis Acid-Containing Zeolites: Lessons from Fine Chemistry to New Biomass Transformation Processes[END_REF] Two years ago the introduction of W into the framework of MFI zeolite was reported by our group. [START_REF] Grand | One-Pot Synthesis of Silanol-Free Nanosized MFI Zeolite[END_REF] In general, this substitution greatly influences the physicochemical properties of the materials and opens many opportunities for new application in catalysis and adsorption. [START_REF] Price | Ga2O3/HZSM-5 Propane Aromatization Catalysts: Formation of Active Centers via Solid-State Reaction[END_REF][START_REF] Kharitonov | Selective Oxidation of Benzene to Phenol over FeAlPO Catalysts Using Nitrous Oxide as Oxidant USSR Pat[END_REF][START_REF] Panov | Oxidation of Benzene to Phenol by Nitrous Oxide over Fe-ZSM-5 Zeolites[END_REF][START_REF] Bellussi | Zeolite Science and Technology at Eni[END_REF][START_REF] Taramasso | Preparation of Porous Crystalline Synthetic Material Comprised of Silicon and Titanium Oxides US pat[END_REF] Ti-containing materials (TS-1) are very active and selective in partial catalytic oxidation of fine chemicals, in particular in the presence of H2O2. The best known examples of such transformations are propylene epoxidation (Hydrox. Process, HPPO), hydroxylation of phenol, and cyclohexanone ammoximation. [START_REF] Bellussi | Zeolite Science and Technology at Eni[END_REF] Properties of the metal-containing zeolites, such as Lewis acidity and higher resilience to water, are very valuable for future application in biomass upgrading. [START_REF] Moliner | State of the Art of Lewis Acid-Containing Zeolites: Lessons from Fine Chemistry to New Biomass Transformation Processes[END_REF] Our group described the preparation of a nanosized MFI zeolite containing atomically dispersed tungsten. [START_REF] Grand | One-Pot Synthesis of Silanol-Free Nanosized MFI Zeolite[END_REF] The resulting W-containing MFI zeolite possessed exceptional stability, high hydrophobicity, and no silanol defects. These properties give them a great potential in catalysis as silanols (often referred to as framework and external defects) display unwanted hydrophilic properties related to water adsorption on silanol sites. [START_REF] Mintova | Nanosized Microporous Crystals: Emerging Applications[END_REF][START_REF] Chezeau | Influence of Synthesis Conditions and Postsynthesis Treatments on the Nature and Quantity of Structural Defects in Highly Siliceous MFI Zeolites: A High-Resolution Solid-State 29 Si n.m.r. Study[END_REF] However, the proven incorporation of the molybdenum atoms in the MFI was not described in literature so far. Attempts were reported by Raghavan et al. [START_REF] Raghavan | Synthesis and Characterization of Molybdenum Containing Silicalite-2 (MoS-2) Molecular Sieve[END_REF] who prepared a molybdenum containing silicalite-2 (MEL structure) by hydrothermal synthesis in aluminum-free alkaline mixtures that was active for the oxidative dehydrogenation of ethanol to acetaldehyde. Then, the synthetic effort to introduce Mo in silicalite-1 (MFI structure) was also reported by the same group. [START_REF] Raghavan | Selective Catalytic Oxidation of Thioethers to Sulfoxides over Mo-Silicalite-1 (MoS-1) Molecular Sieves[END_REF] Interestingly, the samples were more hydrophilic than silicalite-1 (more defects) and thermally stable; they displayed a high activity in the catalytic oxidation of thioethers to the corresponding sulfoxides. A non-linear unit cell volume expansion with the introduction of molybdenum was observed, indicating a significant co-formation of non-framework metallic species. The partial incorporation of molybdenum in the silicalite-1 framework synthesized in fluoride media was reported by Tavalaro. [START_REF] Tavolaro | Thermal Characterization of the Metal-Silicalites Obtained from Aqueous Nonalkaline Fluoride Gels[END_REF] The framework insertion of a portion of the molybdenum was claimed to occur based on the thermal decomposition behavior of TPA + cations blocked in the zeolite micropores without any further evidences.

Here we present for the first time direct evidence for the hydrothermal synthesis of a nanosized MFI zeolite containing atomically dispersed Mo in the zeolite framework, hereinafter referred to as Mo-MFI-D. A reference sample, prepared by molybdenum wet-impregnation of silicalite-1, hereinafter referred to as Mo- The distribution of molybdenum in the Mo-MFI-P and Mo-MFI-D samples was investigated by STEM-EDS (Figure 1). Both Mocontaining samples show a homogeneous distribution of Mo with a mean content of 0.8 to 1.2 wt. %. STEM-HAADF results reveal structural information of the materials by providing Z-contrast images. The white dot-like features identified by the high-resolution STEM-HAADF micrographs disclose the homogeneous distribution of Mo in Mo-MFI-D. The size of the white dots is 0.05 nm, which is approximately 10 times smaller than the crystalline fringes. This indicates that atomically dispersed Mo atoms are part of the MFI framework (Figure 2a). Similar dots that are of a larger size (1.5 nm) are observed in Mo-MFI-P (Figure 2b). For comparison, the STEM-HAADF micrograph of the pure siliceous calcined MFI zeolite used for the preparation of sample Mo-MFI-P prior ion-exchange is presented in Figure S1. The larger white dots observed in sample Mo-MFI-P correspond to small molybdenum oxide clusters homogeneously distributed in the MFI sample as indicated by Raman spectroscopy (Figure S2). The bands at 247, 621, 890, and 980 cm -1 in the Raman spectrum of Mo-MFI-P are assigned to molybdenum oxide. [START_REF] Dieterle | Raman Spectroscopy of Molybdenum Oxides -Part II. Resonance Raman Spectroscopic Characterization of the Molybdenum Oxides Mo4O11 and MoO2[END_REF] These bands are not observed in Mo-MFI-D, indicating the absence of any oxide phases in this sample. A small peak at 980 cm -1 in Mo-MFI-D is not attributed to an oxide phase (narrow signal), but to a small amount of silanols (wide band). Furthermore, in Mo-MFI-D, two new bands (803 and 820 cm -1 ) possibly originate from single Mo atoms in the MFI framework positions. Two shoulders (332 and 416 cm -1 ) are attributed to the perturbation of Si-O-Si vibrations linked to the presence of neighboring Mo-O-Si species, or possibly Mo-O-Si vibrations. [START_REF] Xia | Confirmation of the Isomorphous Substitution by Sn Atoms in the Framework Positions of MFI-Typed Zeolite[END_REF] This result is further supported by the presence of a broad band at 1105 cm -1 in the FTIR spectrum of sample Mo-MFI-D, corresponding to Si-O-Mo framework vibration (Figure S3). The two samples differ substantially: (i) sample Mo-MFI-D contains Mo atoms in the MFI framework and no oxide phase, with a negligible amount of isolated silanols (3745 cm -1 ), and (ii) sample Mo-MFI-P contains molybdenum oxide and silanols as shown by FTIR (Figure S4).

Upon direct introduction of molybdenum in the MFI framework (Mo-MFI-D), the diffraction peaks at 23.30, 23.75, and 24.50 ° 2θ split (Figure S5). This is a direct consequence of the symmetry lowering from orthorhombic (Pnma) to monoclinic (P21/n11), confirmed by Le Bail profile refinements of the respective diffraction patterns (see Table S1 and Figure S5). The unit cell volume of Mo-MFI-D expands also from 5337.4 to 5360.6 Å 3 compared to Mo-MFI-P. Such a polymorphic transition is observed in MFI zeolite, and its transition temperature depends on the incorporation of Telements such as aluminum and titanium into the MFI framework. [START_REF] Wu | ZSM-5-Type Materials. Factors Affecting Crystal Symmetry[END_REF][START_REF] Van Kningsveld | The Monoclinic Framework Structure of Zeolite H-ZSM-5. Comparison with the Orthorhombic Framework of As-Synthesized ZSM-5[END_REF][START_REF] Marra | Orthorombic to Monoclinic Phase Transition in High-Ti-Loaded TS-1: an Attempt to Locate Ti in the MFI Framework by Low Temperature XRD[END_REF] The Mo-MFI-D sample exhibits monoclinic symmetry, although the incorporation of Mo in the MFI framework is expected to lower the transition temperature of the monoclinic-orthorhombic phase transition instead. [START_REF] Ardit | Monoclinic-Orthorhombic Phase Transition in ZSM-5 Zeolite : Spontaneous Strain Variation and Thermodynamic Properties[END_REF] This result is probably a consequence of the hydrophobic and silanol defect-free properties of the Mo-MFI-D material. An increase in the unit cell volume proportional to the amount of metals introduced into the zeolite framework has been previously reported. [START_REF] Borin | Synthesis of TS-1 Molecular Sieves Using a New Ti Source[END_REF][START_REF] Tuel | Influence of the Nature of Silicon and Titanium Alkoxides on the Incorporation of Titanium in TS-1 App[END_REF] The unit cell volume expansion and symmetry change of the MFI type zeolite are due to distortion of the framework to accommodate Mo atoms with an ionic radius larger than Si. [START_REF] Ch | Comparison of All Sites for Ti Substitution in Zeolite TS-1 by an Accurate Embedded-Cluster Method[END_REF] The space group transition and volume expansion observed for Mo-MFI-D provide additional evidence for the incorporation of molybdenum atoms into the framework of the MFI type zeolite. In contrast, Mo-MFI-P does not show any evidence for the presence of framework Mo. This is based on its orthorhombic symmetry and its unit cell volume that are both similar to purely siliceous MFI type zeolite (orthorhombic symmetry, measured volume = 5333.1 Å 3 ).

The nature of the silicon environment in the MFI type zeolite framework was investigated by 29 Si magic-angle spinning (MAS) NMR and [ 1 H] cross-polarization (CP) 29 Si MAS NMR spectroscopy (Figure S6). The absence of Q3 species in Mo-MFI-D is highlighted by a one-pulse experiment (absence of a -104 ppm peak). Few silanols are observed by 29 Si CP MAS NMR for Mo-MFI-D, while for Mo-MFI-P a higher concentration of Q3 species is detected. Moreover, well resolved Q4 species in Mo-MFI-D are attributed to the local homogeneity of the sample. Mo-MFI-D can be described as a silanol-free material with a high degree of local homogeneity.

The nature of the Mo and the acidity of the zeolites is studied using trimethylphosphine oxide (TMPO) adsorption monitored by 31 P MAS NMR spectroscopy (Figure 3). The spectra of Mo-MFI-D exhibits contributions from TMPO interacting with framework Mo atoms (45 ppm) as well as physisorbed TMPO (29 ppm). [START_REF] Zheng | 31 P NMR Chemical Shifts of Phosphorous Probes as Reliable and Practical Acidity Scales for Solid and Liquid Catalysts[END_REF] The absence of a peak at 50 ppm indicates that no or few silanols are present on Mo-MFI-D. This is further supported by 2D 31 P-31 P MAS RFDR NMR (Figure 3). The symmetry and shape of the 45 ppm peak related to framework molybdenum Lewis acid sites indicates the presence of a single chemical species, with some small variations in its environment. This is potentially indicative of Mo atoms in different crystalline positions of the framework. In contrast, the spectra of Mo-MFI-P exhibits contributions from silanols (50 ppm), crystalline TMPO (42 ppm), and physisorbed TMPO (29 ppm). The 50 ppm peak in Mo-MFI-P is attributed to the presence of different species such as Q2, Q3, silanol nests, and isolated silanols. The absence of correlation in the 2D experiment indicates that the amount of TMPO is too low to observe interactions between probe molecules and high enough to observe physisorbed species. Moreover, the peak at 29 ppm attributed to physisorbed TMPO is narrower on Mo-MFI-D than on Mo-MFI-P. This indicates that the local geometry for the physisorbed TMPO is different in the two samples, possibly due to a lower anisotropy of the local environment in Mo-MFI-D compared to Mo-MFI-P. The higher anisotropy of the TMPO would then be attributed to the presence of different species in the zeolite pores, such as Mo oxides and silanols in Mo-MFI-P. In Mo-MFI-D, the narrow peak at 29 ppm may be attributed to the local homogeneity in the micropores, i.e., a single type of Lewis acid site as discussed above. Additionally, the peak corresponding to physisorbed TMPO on Mo-MFI-D is shifted by -0.5 ppm compared to Mo-MFI-P. A slightly lower chemical shift in Mo-MFI-D is linked to the absence of silanol groups and thus to a higher hydrophobicity compared to Mo-MFI-P. It was reported that the higher the chemical shift of TMPO, the stronger the acidity of the zeolite. [START_REF] Zheng | 31 P NMR Chemical Shifts of Phosphorous Probes as Reliable and Practical Acidity Scales for Solid and Liquid Catalysts[END_REF] The presence of a significant number of silanol groups in Mo-MFI-P, compared to Mo-MFI-D, makes the material more acidic, as observed by a shifting of the peak attributed to physisorbed TMPO to higher ppm values compared to Mo-MFI-D. The acidity of these two molybdenum containing zeolites is also investigated by the adsorption of deuterated acetonitrile (CD3CN) monitored by FTIR spectroscopy (Figure 4). The spectra of Mo-MFI-D exhibits one band at 2308 cm -1 corresponding to CD3CN adsorbed on the Lewis acid sites brought by the atomically dispersed framework molybdenum and low intensity bands at 2283 and 2274 cm -1 indicative of trace amounts of alkali metal and silanols. The 2308 cm -1 band is attributed to closed Lewis acid sites, i.e. non-hydrolysed framework Mo, that may also correspond to Mo VI in the MFI framework (O=Mo(-OSi)4), based on previous reports on Me-containing zeolites. [START_REF] Bermejo-Deval | Active Sites in Sn-Beta for Glucose Isomerization to Fructose and Epimerization to Mannose[END_REF][START_REF] Boronat | Determination of the Catalytically Active Oxidation Lewis Sites in Sn-Beta Zeolites, and Their Optimisation by the Combination of Theoretical and Experimental Studies[END_REF][START_REF] Otomo | Modification of Sn-Beta Zeolite: Characterization of Acidic/Basic Properties and Catalytic Performances in Baeyer-Villiger Oxidation[END_REF] In contrast, the spectra of Mo-MFI-P exhibits two bands at 2317 and 2303 cm -1 indicating the presence of molybdenum oxide species. In addition, low intensity band at 2283 cm -1 is due to trace amounts of alkali metals, and peak at 2274 cm -1 corresponds to silanol groups. The bands at 2303 and 2317 cm -1 highlight the presence in Mo-MFI-P of two different molybdenum oxide species with slightly different acidity.

In summary, a nanosized MFI type zeolite with atomically dispersed molybdenum in framework is prepared. Mo-MFI zeolite displays high local homogeneity of the atomically dispersed Mo atoms, low silanol content, and high hydrophobicity. These properties originate from the incorporation of Mo atoms into the MFI framework as closed Lewis acid sites (O=Mo(-OSi)4). The insertion of Mo in the MFI structure induces a space group transition and increase of the unit cell volume. The uniform nature of the Mo incorporated into the MFI framework is confirmed by IR spectroscopy (deuterated acetonitrile probe) and 31 P MAS NMR spectroscopy (trimethylphosphine oxide probe). The homogeneous distribution of the atomically distributed Mo in the MFI framework is proven by STEM-HAADF. The materials open many opportunities of catalysts and adsorbents design for applications in mature and emerging fields.

ASSOCIATED CONTENT Supporting Information

Synthesis description, Characterization details, STEM-HAADF micrograph of pure siliceous MFI zeolite (Figure S1), Raman spectra (Figure S2), FTIR spectra using KBr (Figure S3), FTIR spectra using self-supported zeolite wafers (Figure S4), XRD patterns (Figure S5), 29 Si MAS NMR and CP{ 1 H} 29 Si MAS NMR spectra (Figure S6), Le Bail profile refinements (Table S1).

Figure 1 .Figure 2 .

 12 Figure 1. STEM-EDS elemental maps of (a,b,c,d) Mo-MFI-D and (e,f,g,h) Mo-MFI-P: Si (orange), O (blue), and Mo (green)

Figure 3 .

 3 Figure 3. 31 P-31 P MAS RFDR NMR spectra (top: 1D 31 P MAS NMR spectra) of (a) Mo-MFI-P and (b) Mo-MFI-D under 1 H spinal-64 decoupling (magnetic field of 500 MHz, MAS at 12 kHz and 14 kHz for 1D and RFDR spectra, respectively)

Figure 4 .

 4 Figure 4. FTIR spectra of CD3CN adsorbed on (a) Mo-MFI-D (3.7, 11.3, 26.2, 41.4 x10 -3 mmol/g), and (b) Mo-MFI-P (2.6, 7.8, 18.1, 28.0 x10 -3 mmol/g) at room temperature.

AUTHOR INFORMATION Corresponding Author

*mintova@ensicaen.fr

Notes

The authors declare no competing financial interests.

ACKNOWLEDGMENT

This research was supported by TOTAL and Industrial Chair ANR-TOTAL "Nanoclean Energy". GENESIS is supported by the Région Haute-Normandie, the métropole Rouen Normandie, the CNRS via LABEX EMC and the French National Research Agency as a part of the program "Investissements d'avenir" with the reference ANR-11-EQPX-0020.