A Rodríguez

Del Nozal
email: arnozal@us.es

L Orihuela
email: dorihuela@uloyola.es

P Millan
email: pmillan@uloyola.es

Alexandre Seuret
email: aseuret@laas.fr

Luca

P Millán

L Zaccarian
email: zaccarian@laas.fr

Results on distributed state estimation

Keywords: Distributed estimation, Link failures, Multi-agent systems, Linear time-invariant systems, Switching topologies

come

INTRODUCTION

Traditional control systems consist of a centralized controller that collects all the measurements from a plant and carries out the necessary computations to assign the control input. To accomplish this goal, it is of crucial importance to place the controller node close to the plant, namely both near the sensors and the actuators. An alternative is to introduce point-to-point communication networks exclusively devoted to the control loop. In contrast to this centralize framework, the concept of Networked Control Systems (NCS) arose at the end of the last century. NCSs are spatially distributed systems for which the communication among the sensors, actuators and controllers is supported by a shared communication network. The use of a multipurpose shared network to connect spatially distributed elements results in flexible architectures and generally reduces both installation and maintenance costs. Despite of the apparent advantages of NCS, new challenges arise for the scientific community, some of them being summarized in [START_REF] Zhang | Survey on recent advances in networked control systems[END_REF]; [START_REF] Ge | Distributed networked control systems: A brief overview[END_REF].

One of the problems that received a great deal of attention in the last years is distributed estimation, as a first step towards achieving distributed control, or simply for monitoring purposes. The distributed estimation goal is to estimate the state of a plant by a network of agents that need to share (partial) information to accomplish a collective estimation goal. The reader is referred to [START_REF] Rego | Distributed state estimation for discrete-time linear time invariant systems: A survey[END_REF] for a recent survey of results in this area.

The above-mentioned survey papers emphasize that distributed estimators must deal, among several issues related to communications, with the occurrence of failures in the communication links/devices. These failures may suddenly isolate one agent or limit its ability to receive and send information. Therefore, proposing a distributed estimator capable of coping with link and communication failures, is the main purpose of this paper. The problem itself fits perfectly one of the challenges of the cyber-physical systems community, which is the operation continuum, see [START_REF] Engell | Core research and innovation areas in cyber-physical systems of systems[END_REF].

The literature on distributed estimation is huge and many approaches can be found. This revision will start by presenting the main active research lines in the last years from a broad point of view. Later, the scope will be narrowed to the end of framing the paper contributions against the latest advances in this area.

When considering perturbed systems, there are mainly three families of distributed observers, namely, distributed Kalman, H ∞ and set-membership filters, each one valid and optimized for specific models of disturbances and noises. Distributed Kalman filters (DKF) (first presented in [START_REF] Olfati-Saber | Distributed Kalman filtering for sensor networks[END_REF]) provide an optimal state estimation when the system model and the measurements of the agents are affected by Gaussian noises. However, an accurate model of these noise distributions is needed and sometimes this is difficult to obtain. The H ∞ filtering theory is used to develop distributed estimators providing state estimates with guaranteed performance. This strat-egy have been successfully applied in [START_REF] Ugrinovskii | Distributed robust filtering with H ∞ consensus of estimates[END_REF] and [START_REF] Shen | Distributed H ∞ -consensus filtering in sensor networks with multiple missing measurements: The finite-horizon case[END_REF]. Usually, distributed H ∞ filters rely on costly LMI design methods. Finally, set-membership approaches aim at finding a compact set where the plant state is certainly confined, see [START_REF] Orihuela | Negotiated distributed estimation with guaranteed performance for bandwidth-limited situations[END_REF]; [START_REF] Wang | Distributed zonotopic set-membership state estimation based on optimization methods with partial projection[END_REF]. They are conservative approaches that are adequate when the exogenous signals satisfy known bounds.

Regarding unperturbed and noiseless systems, different modifications of distributed Luenberger observers have been proposed. In pursuing of a decentralized design of the observers with minimum information, the authors of [START_REF] Park | Design of distributed LTI observers for state omniscience[END_REF] use state augmentations. With similar objectives, but using subspace decompositions (observable and unobservable modes), recent results have been presented in del [START_REF] Del Nozal | Distributed estimation based on multi-hop subspace decomposition[END_REF]; [START_REF] Kim | Distributed Luenberger observer design[END_REF]; [START_REF] Mitra | Distributed observers for LTI systems[END_REF]. The most interesting feature of these approaches is that they can provide necessary and sufficient design conditions, based on detectability properties accounting for the presence of the communication network, and to exploit this fact in the proposed observer structure.

The literature of distributed observers dealing with link failures and communication losses is less dense. The communication failures are modeled with different methods. For instance, in [START_REF] Ugrinovskii | Distributed robust estimation over randomly switching networks using H ∞ consensus[END_REF], Markov processes are employed to model random communication topologies. However, the local mode information of the entire network topology is non-Markovian, which complicates the problem solvability. To overcome this difficulty, Ugrinovskii (2013) emploies a two-step design procedure. The corresponding solution requires solving linear matrix inequalities subject to rank constraints, which are generally difficult. A different approach to model the communication failures can be found in [START_REF] Liu | On Kalman-consensus filtering with random link failures over sensor networks[END_REF] where Bernoulli variables are used. [START_REF] Liu | On Kalman-consensus filtering with random link failures over sensor networks[END_REF] introduces a weighted matrix in the consensus steps in order to implement distributed filtering. In addition, boundedness properties are thoroughly investigated, using statistic information of the random link failures. Regarding the strategy used to deal with the distributed estimation problem two main approaches can be found in the literature. On the one hand, the use of H ∞ strategies as discussed in [START_REF] Yan | Distributed H ∞ state estimation for a class of filtering networks with time-varying switching topologies and packet losses[END_REF] and [START_REF] Yu | Distributed H ∞ filtering in sensor networks with randomly occurred missing measurements and communication link failures[END_REF]. In [START_REF] Yan | Distributed H ∞ state estimation for a class of filtering networks with time-varying switching topologies and packet losses[END_REF], neural networks are used to estimate the system state using learning methods for the corresponding matrices. Instead, [START_REF] Yu | Distributed H ∞ filtering in sensor networks with randomly occurred missing measurements and communication link failures[END_REF] designs a filter on each node in the sensor network ensuring that the dynamics of the filtering error is meansquare stable and the prescribed average H ∞ performance constraint is met. On the other hand, the behaviour of the Kalman filter dealing with communication problems has been also studied. In [START_REF] Battilotti | Distributed Kalman filtering over sensor networks with unknown random link failures[END_REF] a failure detection device is introduced in every agent to detect link failures in the network at the receiving side. In addition, by using the maximum a posteriori probability decision rule, the authors propose a method to identify online the generally correlated multiple-valued stochastic output delay which guarantees (with some approximation) the minimum probability of error, given the available observations. Finally, the algorithm presented in Alonso-Román and Beferull-Lozano (2016) provides unbiased estimations when the steady-state value of the average consensus process becomes a random variable.

Within this setting, we focus here on distributed estimation in the presence of communication losses, and provide: • A distributed observer structure based on a multi-hop decomposition, which decomposes the state space in the observable subspace of each agent and the innovation introduced by the neighbors at each hop.

• A sufficient condition on the distributed observer gains ensuring uniform global exponential stability of the error dynamics over all possible communication losses satisfying a suitable persistence of excitation assumption.

• A set of assumptions (some necessary and some sufficient) under which it is always possible to design the gains of the distributed observers in order to meet the above mentioned sufficient conditions. This paper is organized as follows. Section 2 states the main problem and presents the necessary assumptions. Section 4 presents the proposed observation structure and the main results of the paper concerning stability and feasibility. Section 5 shows the observer performance in simulations. Finally, conclusions are drawn in Section 6.

Notation. A graph is a pair G = (V, E) comprising a set V = {1, 2, . . . , p} of vertices or agents, and a set E ⊂ V × V of edges or links. A directed graph is a graph in which edges have orientations, so that if (j, i) ∈ E, then agent i obtains information from agent j. A directed path from node i 1 to node i k is a sequence of edges such as (i 1 , i 2 ), (i 2 , i 3 ), . . ., (i k-1 , i k ) in a directed graph. The neighborhood of i, N i {j : (j, i) ∈ E}, is defined as the set of nodes with edges incoming to node i. Given ρ ∈ Z + , the ρ-hop reachable set of i, N i,ρ , is defined as the set of nodes with a direct path to i involving ρ edges. Note that the 1-hop reachable set of i corresponds to the neighborhood of i and the 0-hop reachable set of i matches with i.

Operators col(•, •), row(•, •) stacks subsequent matrices into a column/row vector, e.g. for A and B of appropriate dimensions, col(A, B) = [A B ] and row(A, B) = [A B]. |x| is the Euclidean norm of vector x. ||A|| stands for the induced matrix norm of matrix A.

PROBLEM STATEMENT

Consider a set of agents V = {1, 2, . . . , p} intending to distributedly estimate the state of the discrete-time LTI system

x(k + 1) = Ax(k),
(1)

y i (k) = C i x(k), ∀i ∈ V, ( 2
)
where x is the state vector, A is the system matrix, y i ∈ R mi is the output locally measured by agent i and C i ∈ R mi×n is its output matrix.

Since the agents are not able to reconstruct the whole state x based only on the local measurement y i (i.e. detectability of (C i , A) is not assumed), a communication network among them is required.

Thus, let G = (V, E) represents the directed graph modelling the communication network where no communication failures are allowed. For this directed graph, E ⊂ V ×V represents every communication channel between pairs of agents.

Collective detectability assumption

We introduce here some key concepts, useful for the developments of the rest of the paper. Definition 1. For the communication graph G, the ρ-hop output matrix of agent i, C i,ρ , is defined as:

C i,ρ := C i,ρ-1 col(C j,ρ-1 ) j∈Ni , ∀ρ ≥ 1, (3)
where C i,0 := C i .

Intuitively speaking, the ρ-hop output matrix C i,ρ of agent i, recursively defined in (3), comprises its output matrix C i and the output matrices of all the agents j with a direct path to i involving ρ or less edges. Based on this concept, we can formulate our first assumption. A similar assumption was introduced in del Nozal et al. ( 2019). Definition 2. System (1)-( 2) is collectively detectable if, for any i ∈ V, there exists a finite number of hops 2) is collectively detectable.

i ∈ Z >0 such that pair (C i, i , A) is detectable. Assumption 1. System (1)-(
As shown in del Nozal et al. ( 2019), Assumption 1 is necessary for the existence of a converging distributed state estimator. According to Definition 2, system (1)-( 2) is collectively detectable if, for each agent, the complete information provided by the network (that is, the ρ-hop output matrix with ρ sufficiently large) is sufficient to build a converging state observer. Recall that a strongly connected communication network is not required in contrast with other approaches as [START_REF] Wang | A distributed observer for a discrete-time linear system[END_REF]

Communication model and persistence assumption

In this paper, we consider that the topology of the network G can vary with time due to failures in the communication devices, jamming attacks (see [START_REF] Jin | Handbook of Research on Developments and Trends in Wireless Sensor Networks: From Principle to Practice[END_REF]) or packet dropouts.

To this end, a logic variable k → δ i (k) ∈ {0, 1} is associated to each node i to represent communication failures for agent i at time k. Using δ i , the set of links pointing to agent i, (i, j) : ∀j ∈ N i , are active at time k if and only if δ i (k) = 1. Otherwise, when δ i (k) = 0, these links are inactive due to a communication failure. As a result, E(k) and G(k) = (V, E(k)) represent, respectively, the set of active links at time k and characterize the timevarying graph at time k.

Whenever a loss occurs, agent i cannot receive information from its neighborhood. Consequently, it would be reasonable to operate based only on the system model and on the local measurement y i . However, if this situation is extended in time, the plant state cannot be detected in the directions that are not observable from that output. To rule out this scenario, we assume the following persistence of excitation property.

Definition 3. Graph k → G(k) = (V, E(k)
) enjoys a uniform local persistence property if for each i ∈ V there exist a finite time horizon τ i ∈ Z >0 and an integer

n τ i ∈ Z >0 such that τi-1 h=0 δ i (k + h) ≥ n τ i , ∀k ∈ Z ≥0 , (4)
namely, for each time interval {k, . . . , k

+ τ i -1}, k ∈ Z >0 , there exist at least n τ i distinct values of s ∈ {k, . . . , k + τ i -1} satisfying δ i (s) = 1. Assumption 2. The graph k → G(k) = (V, E(k)) enjoys a uniform local persistence property.
Assumption 2 requires that each agent i experiences no more than τ i -n τ i communication losses in each time window spanning τ i time instants. This implies that the agent receives enough information from the neighboring agents, which corresponds to some kind of persistence of excitation. While Assumption 2 is not necessary, in general, we emphasize that it does not hold true only in cases where the communication instants (when δ i = 1) become increasingly rare as time flows. Such a scenario is quite undesirable if one wants to achieve uniform convergence properties like those in our problem statement below.

Problem statement

Based on our standing Assumptions 1 and 2, we are ready to state our problem statement. Problem 1. Consider system (1)-( 2) and an interconnection graph G(k) = (V, E(k)). Under Assumptions 1 and 2, design a distributed observer providing, at each node i, an estimate xi of the state x of ( 1)-( 2), such that these estimates converge uniformly and exponentially to x. In particular, for each δ i , i = 1, . . . , p, satisfying Assumption 2, there must exist scalars M > 0 and λ ∈ (0, 1) such that, for any x(0) and any xi (0

), i ∈ V, |x(k) -xi (k)| 2 ≤ M λ k p j=1 |x(0) -xj (0)| 2 , ∀i ∈ V. (5)
The distributed observer that we design to solve the problem above generalizes the linear time-invariant solution in del [START_REF] Del Nozal | Distributed estimation based on multi-hop subspace decomposition[END_REF]. The novelty that we propose here is that we focus on linear dynamics subject to the "external" time-varying logical inputs δ i (k). Due to these extra inputs, the linear cascaded arguments of del Nozal et al. ( 2019) cannot be adopted, but we may resort to nonlinear Input to State Stability (ISS) results for timevarying systems.

MULTI-HOP SUBSPACE DECOMPOSITION

Before presenting the observer dynamics, we recall the basic concepts behind the multi-hop subspace decomposition presented in del Nozal et al.

(2019).

There always exists a coordinate transformation matrix Vi,ρ V i,ρ ∈ R n×n associated to pair (C i,ρ , A), such that the change of variable [ Vi,ρ V i,ρ ] x ∈ R n transforms the original state-space representation into the observability staircase form [START_REF] Hespanha | Linear systems theory[END_REF]. Note that Vi,ρ ∈ R n×n ō i,ρ is composed by n ō i,ρ column vectors in R n that form an orthogonal basis of the unobservable subspace of pair (C i,ρ , A). Correspondingly, V i,ρ ∈ R n×n o i,ρ is an orthogonal basis of its orthogonal complement.

Definition 4. The ρ-hop unobservable subspace from agent i, denoted Ōi,ρ , is composed of all system modes that cannot be observed from the output locally measured by agent i and those measured by all the agents belonging to the s-hop reachable nodes from i, ∀s ∈ {0, . . . , ρ}. Equivalently, the ρ-hop unobservable subspace from agent i is the unobservable subspace related to pair (C i,ρ , A) using the above coordinate transformation:

Ōi,ρ := Im( Vi,ρ ).

The orthogonal complement of Ōi,ρ , with some abuse of notation, is denoted ρ-hop observable subspace from agent i, O i,ρ := Im(V i,ρ ). We denote

n o i,ρ = dim(O i,ρ ).
According to Definition 4, it is clear that:

O i,ρ-1 ⊆ O i,ρ , ∀i ∈ V, ρ ≥ 0.
(6) where we consider O i,-1 = ∅. Then, the vectors of the "innovation" basis that generates

O i,ρ ∩ (O i,ρ-1 ) ⊥ can be stacked into a matrix W i,ρ ∈ R n×ni,ρ , where n i,ρ = n o i,ρ - n o i,ρ-1 , in such a way that: Im(W i,ρ ) := O i,ρ ∩ (O i,ρ-1 ) ⊥ , ρ ≥ 0, (7)
Let us define i ∈ Z >0 , to be selected later, as an arbitrary number of hops. From these definitions it is clear that for all ρ ∈ {0, . . . , i } and all i ∈ V, it holds that

Im(V i,ρ ) = Im ([W i,ρ V i,ρ-1 ]) , (8) Im( Vi,ρ-1 ) = Im W i,ρ Vi,ρ , (9)
with Vi,-1 := I n .

It is worth pointing out that Im(W i,ρ ) corresponds to the innovation introduced by the ρ-hop reachable set N i,ρ of agent i, that is, the observable modes for agent i at hop ρ that are not observable at hop ρ -1. Accordingly the transformation matrix T i ∈ R n×n , defined as

T i = [ Vi, i V i, i ]
, can be partitioned as follows, using the innovations at each hop:

T i := Vi,ρ Vi, i W i, i • • • W i,ρ+1
Vi,ρ

W i,ρ • • • W i,0 , (10)
for all ρ ∈ {0, . . . , i }, where it is easy to identify the observable and unobservable subspaces of the system by agent i at hop ρ. Note also that T i is orthogonal by construction, namely

T -1 i = T i .
The following lemma, proven in (del Nozal et al., 2019, Lemma 3), introduces some important properties that are central for the derivations of this paper. Lemma 1. (del Nozal et al., 2019, Lemma 3) For each agent i ∈ V, and any i > Z >0 , the next properties hold, ∀ρ, ρ ∈ {1, . . . , i } such that ρ = ρ :

(i) W i,ρ W i,ρ = 0, (ii) Im(W j,ρ-1 ) ⊆ Im(V i,ρ ), ∀j ∈ N i , (iii) Im(W i,ρ ) ⊆ j∈Ni Im(W j,ρ-1 ).

OBSERVER DESIGN FOR STABILITY DEALING WITH COMMUNICATION FAILURES

This section contains the main results of the paper. First we present the observer structure and then we derive the ensuing error dynamics. Finally, we provide design rules for the observer gains solving Problem 1 and we show that, under the prescribed assumptions, the design of these gains is feasible.

Observer structure and error dynamics

For any agent i, we propose the following observer structure:

xi (k + 1) = Ax i (k) + W i,0 L i (y i (k) -ŷi (k)) + δ i (k) i ρ=0 j∈Ni W i,ρ N i,j,ρ W j,ρ-1 (x j (k) -xi (k)), ( 11
)
where L i and N i,j,ρ are, respectively, a local gain and consensus gains to be selected later in such a way that Problem 1 is solved. The value of i is chosen so that collective detectability is fulfilled as per Assumption 1. This structure was presented in del Nozal et al. ( 2019) for δ i (k) = 1, ∀k. For a more detailed explanation of the proposed observer structure, the reader is referred to that paper.

For each agent i ∈ V, let us define the corresponding estimation error e i (k) := x(k) -xi (k). Similarly, it is possible to define the transformed estimation error as

ε i := col(ε i, i+1 , . . . , ε i,0 ) := T i e i , (12)
using the multi-hop subspace decomposition (10) introduced in Section 2. More specifically, the estimation error of agent i ∈ V, at hop ρ, is defined as: ε i,ρ (k) := W i,ρ e i (k), ∀ρ = 0, . . . , i + 1, where we denote W i, i+1 = Vi, i corresponding to the collectively unobservable but detectable system modes.

The following proposition clarifies the dynamics of these estimation errors. Its proof is a straightforward extension of the results in del Nozal et al. ( 2019) and is therefore omitted. Proposition 1. Consider the network of agents described by the graph G(k), where every agent i implements the observer structure (11) to estimate the state of system (1). Then the dynamics of the errors in (12) corresponds to

ε i,0 (k + 1) = (W i,0 AW i,0 -L i C i W i,0 )ε i,0 (k), ( 13
) ε i,ρ (k + 1) = ρ r=0 D i,(ρ,r) (δ i )ε i,r (k) (14) +δ i (k) j∈Ni N i,j,ρ ε j,ρ-1 (k), ρ ∈ {1, . . . , i }, with D i,(ρ,r) (δ i ) = W i,ρ AW i,r -δ i j∈Ni N i,j,ρ W j,ρ-1 W i,r .
The dynamics in Proposition 1 for ρ ∈ {1, . . . , i } can be compactly written as (we remove the dependence on k for simplicity):

ε + i,0 = (W i,0 AW i,0 -L i C i W i,0 )ε i,0 , (15a)
ε + i,ρ = D i,(ρ,ρ) (δ i )ε i,ρ + B i,ρ (δ i )u i,ρ , if ρ = 0 (15b)
where

B i,ρ (δ i ) = row D i,(ρ,r) (δ i ) r∈{0,...,ρ-1} | | row (δ i N i,j,ρ ) j∈Ni , u i,ρ (k) = col(ε i,r ) r∈{0,...,ρ-1} col(ε j,ρ-1 ) j∈Ni ,
which shows an interesting cascaded structure exploited in our main results of the next section.

Main result and tuning of the observer gains

By exploiting the cascaded dynamics (15), this section presents a design requirement that will be proven to be sufficient to guarantee the exponential estimation properties of Problem 1.

Note that, since the evolution of the transformed estimation error at hop ρ = 0 does not depend on the agents connectivity, the local gain L i can be easily tuned to ensure uniform exponential convergence to zero of the solutions to (15a). Instead, the connectivity properties in Assumption 2 are fundamental for the effectiveness of the design of the consensus gains N i,j,ρ , for which the cascade structure revealed with the multi-hop decomposition becomes crucial. Property 1. For each agent i, the local gain L i and consensus gains N i,j,ρ for hops ρ ∈ {1, . . . , i } are designed in such a way that the matrix 16) is Schur, and the following inequalities are met:

(W i,0 AW i,0 -L i C i W i,0 ) (
D i,(ρ,ρ) (δ i ) P i,ρ D i,(ρ,ρ) (δ i ) < µ i,ρ (δ i )P i,ρ , δ i ∈ {0, 1} (17) µ i,ρ := µ i,ρ (0) τi-nτ i µ i,ρ (1) nτ i < 1, ( 18
)
where µ i,ρ (δ i ), δ i = 0, 1 are two a scalar parameters depending on the switching signal δ i , satisfying µ i,ρ (1) ≤ µ i,ρ (0) and P i,ρ is a positive definite matrix with appropriate dimensions. Now, we are in position to introduce the main result of the paper in Theorem 1, which establishes that observer (11) solves Problem 1 whenever Property 1 is satisfied. Its proof is postponed to Section 4.3 to avoid breaking the flow of the exposition. Theorem 1. Consider plant (1) observed by a set of agents that can measure their local outputs (2), each of them implementing the observer structure (11). Under Assumptions 1 and 2, if the observer gains are designed according to Property 1, then Problem 1 is solved, namely the estimation errors satisfy (5).

The next theorem completes the statement of Theorem 1. Theorem 2. It is always possible, under Assumptions 1 and 2, to design matrices L i , N i,j,ρ , ∀i, ρ and j ∈ N i , that satisfy Property 1.

Proof. According to (del Nozal et al., 2019, Theorem 14), in the absence of communication failures, namely δ i (k) = 1 for all i and for all k, under Assumption 1 it is possible to design gain matrices L i , and N i,j,ρ to fix the convergence rate of the estimator arbitrarily fast (a detailed design method is presented there). In other words the value µ i,ρ (1) can be selected arbitrarily close to zero by appropriate choices of the gains. Due to the fact that the value of µ i,ρ (0) is determined by the open-loop system dynamics, and therefore independent of the observer gains, it is then possible to choose µ i,ρ (1) sufficiently small to ensure µ i,ρ (0) τi-nτ i µ i,ρ (1) nτ i < 1 for any given pair τ i , n τi from Assumption 2. 2

Proof of Theorem 1

Before proving the theorem, we introduce some preliminary results. Our proof is based on Input to State Stability (ISS) properties for systems of the form

ξ(k + 1) = f (ξ(k), u(k), k), (19
) which well represent dynamics (15b). In particular, we make use of the following property. Definition 5. System ( 19) is uniformly globally exponentially finite-gain ISS with respect to u if there exist scalars M > 0, λ ∈ (0, 1) and γ > 0, such that for any initial time k 0 , any initial condition ξ(k 0 ) and any uniformly bounded

input k → u(k), the corresponding solution k → ξ(k) satisfies 1 |ξ(k)| 2 ≤ M λ k-k0 |ξ(k 0 )| 2 + γ u 2 ∞ , ∀k ≥ 0, where u ∞ := sup k≥k0 |u(k)| denotes the l ∞ norm of the input u.
To prove the above ISS property for each one of the subsystems in (15b), for each i ∈ V and each ρ ∈ {1, . . . , i } we will use the following quadratic Lyapunov function

V i,ρ (ε i,ρ (k)) = ε i,ρ (k) P i,ρ ε i,ρ (k), (20)
where P i,ρ is a symmetric positive definite matrix. With some abuse of notation, we will use V i,ρ (k) in place of V i,ρ (ε i,ρ (k)) in the next derivations. Lemma 2. If Property 1 holds, then for each i ∈ V and each ρ ∈ {1, . . . , i } there exists a scalar γ i,ρ ∈ R + such that the Lyapunov function (20) satisfies

V i,ρ (k + 1) < µ i,ρ (δ i )V i,ρ (k) + γ i,ρ |u i,ρ (k)| 2 (21)
along the solutions to (15b).

Proof. Let us write the derivative of the Lyapunov function (20) along the error dynamics in (15). To this end, and to keep the notation more compact, let us denote V i,ρ (k) merely by V i,ρ and V i,ρ (k + 1) by V + i,ρ (and similarly for δ i and ε i,ρ and u i,ρ ). We then have

V + i,ρ = (ε + i,ρ ) P i,ρ ε + i,ρ = ε i,ρ D i,(ρ,ρ) (δ i ) P i,ρ D i,(ρ,ρ) (δ i )ε i,ρ + 2ε i,ρ D i,(ρ,ρ) (δ i ) P i,ρ B i,ρ (δ i )u i,ρ + u i,ρ B i,ρ (δ i ) P i,ρ B i,ρ (δ i )u i,ρ ≤ ε i,ρ D i,(ρ,ρ) (δ i ) P i,ρ D i,(ρ,ρ) (δ i )ε i,ρ + 2 D i,(ρ,ρ) (δ i ) P i,ρ B i,ρ (δ i ) • |ε i,ρ | • |u i,ρ | + B i,ρ (δ i ) P i,ρ B i,ρ (δ i ) • |u i,ρ | 2 .
By completing squares and using the Cauchy-Schwarz inequality, the following bound holds for any selection of η i,ρ ∈ R + :

2 |ε i,ρ | |u i,ρ | ≤ η i,ρ |ε i,ρ | 2 + 1 η i,ρ |u i,ρ | 2 (22)
Moreover, from Property 1, there exists a positive scalar

ν i,ρ ∈ R + such that D i,(ρ,ρ) (δ i ) P i,ρ D i,(ρ,ρ) (δ i ) < µ i,ρ (δ i )P i,ρ -ν i,ρ I.
Therefore, selecting η i,ρ in ( 22) small enough to satisfy D i,(ρ,ρ) (δ i ) P i,ρ B i,ρ (δ i ) η i,ρ < ν i,ρ , we may combine the previous bounds to prove (21) with the selection

γ i,ρ = max δi∈{0,1} η -1 i,ρ D i,(ρ,ρ) (δ i ) P i,ρ B i,ρ (δ i ) + B i,ρ (δ i ) P i,ρ B i,ρ (δ i ) .
2

Based on the previous lemma, we can now prove an ISS property, in the sense of Definition 5, for dynamics (15b) for each i ∈ V and each ρ ∈ {1, . . . , i }. This is established next. Lemma 3. If Property 1 and Assumption 2 hold, then for each i ∈ V and each ρ ∈ {1, . . . , i } the error system with dynamics (15b) is uniformly globally exponentially finitegain ISS with respect to u i,ρ .

Proof. The proof is based on demonstrating that the system error dynamics meets Definition 5 and consequently, it is exponentially finite-gain ISS with respect to u i,ρ .

First, exploiting µ i,ρ (1) ≤ µ i,ρ (0) in Property 1 and using Assumption 2, we may recursively apply expression (21) to τ i successive time instants to obtain

V i,ρ (k + τ i ) < µ i,ρ V i,ρ (k) + γ i,ρ ||u i,ρ || 2 ∞ , (23)
where ||u i,ρ || ∞ = sup k≥0 |u i,ρ (k)| the bound on u i,ρ (k), ∀k, µ i,ρ < 1 is defined in (18), and

γ i,ρ = τi-2 t=0 τi-1 s=t+1 µ i,ρ (δ i (k + s)) + 1 γ i,ρ ,
By recursively applying equation ( 23) evaluated at times of k = hτ i , we obtain (we focus on the case k 0 = 0 because the extension to the case k 0 = 0 is straightforward)

V i,ρ ((h + 1)τ i ) < µ i,ρ V i,ρ (hτ i ) + γ i,ρ ||u i,ρ || 2 ∞ . . . ( 24
) < µ h+1 i,ρ V i,ρ (0) + γ i,ρ ||u i,ρ || 2
∞ , where we used the fact that µ i,ρ < 1 implies that the following geometric series converges:

γ i,ρ := γ i,ρ 1 1 -µ i,ρ = γ i,ρ +∞ s=0 µ s i,ρ ≥ h s=0 µ s i,ρ γ i,ρ .
Consider now the intersample behavior of V i,ρ and note that linearity of the dynamics (15b) implies that there exists a large enough scalar σ such that

V i,ρ (hτ i + s) ≤ σV i,ρ (hτ i ) + σ||u i,ρ || 2 ∞ , ∀s ∈ {0, . . . , τ i -1}. ( 25
)
Combining bounds ( 24) and ( 25), we obtain the following bound for some suitable λ V ∈ (0, 1), M V > 0 and γ V > 0:

V i,ρ (k) ≤ M V λ k V V i,ρ (0) + γ V u 2 ∞ , ∀k ≥ 0. ( 26
)
Finally, from standard properties of positive definite matrices, we get λ min (P i,ρ )|ε i,ρ (k)| 2 ≤ V i,ρ (k) ≤ λ max (P i,ρ )|ε i,ρ (k)| 2 , which can be used twice in (26) to prove the desired ISS bound

|ε i,ρ (k)| 2 < λ M (P i,ρ ) λ m (P i,ρ ) M V λ k V |ε i,ρ (0)| 2 + γ V λ m (P i,ρ ) ||u i,ρ || 2 ∞ ,
as to be proven. 2

From Lemma 3, the proof of Theorem 1 can be presented.

Proof of Theorem 1. Since matrix ( 16) in Property 1 is Schur by assumption, the dynamics of the estimation error at hop ρ = 0 is exponentially stable for all agents, namely there exist M 0 > 0 and λ 0 ∈ (0, 1) such that |ε i,0 (k)| ≤ M 0 λ k 0 |ε i,0 (0)| for all k > 0 and all i ∈ V. Using this bound, and due to the cascaded-like expression of u i,ρ in (15b), we may concatenate the ISS bounds established in Lemma 3 to obtain that there exits λ ε ∈ (0, 1) and M ε > 0 such that vector ε = col(ε 1 , . . . , ε p ) satisfies the ISS bound

|ε(k)| 2 ≤ M ε λ k ε |ε(0)| 2 . ( 27
)
Since ε is equivalent (through linear transformation) to e = col(e 1 , . . . , e p ) (where we recall that e i = x -xi ), then the previous bound implies bound (5) in Problem 1, thus completing the proof.

2 Remark 1. We emphasize that the proof technique of this section, based on the time-varying dynamics (19), ensures that for each persistently exciting selection of δ i , i = 1, . . . , p, as per Definition 3, there exist M ε and λ ε satisfying (27) (equivalently (5) in Problem 1). However, we don't give here a guarantee that those scalars be uniform over the infinitely many persistently exciting selections of δ i . Nevertheless, we conjecture that a different proof technique may be used to prove a uniform exponential bound, valid for all such selections. Proving this uniform exponential convergence property is left as future work.

SIMULATION RESULTS

This section presents some simulation results that demonstrate the effectiveness of the proposed estimation algorithm. To this end, let us consider the following LTI autonomous system:

   x 1 x 2 x 3 x 4    + =    0.95 0 0 0 0 0.8606 -1.3368 0 0 0.1485 0.9315 0 0 0 0 1.015       x 1 x 2 x 3 x 4    .
The system is observed by a set of three agents that communicate according to the following diagram 1 ↔ 2 ↔ 3. Then, the interconnection graph is composed by V = {1, 2, 3} and the set of edges is E = {(1, 2), (2, 1), (2, 3), (3, 2)}, generating graph G = (V, E). In all of the examples discussed below, we consider the following output matrices for the three agents:

C 1 =    1 0 0 0 0 0 0 1    , C 2 =    0 1 0 0    , C 3 =    1 0 0 0 0 0 0 1    .
Example 1. Let us assume a scenario where the agents experience communication failures. We assume that at every time interval {k, . . . , k+τ i -1}, where τ i = 100, ∀i ∈ V, there exists at least n τi = 20 times when every agents i = 1, 2, 3, can communicate with their neighborhood.

According to (11), when the agent i cannot communicate, the observer dynamics only uses the local measurements y i . Thus, the error dynamics of the locally unobservable subspaces will evolve in open loop. In the case of unstable dynamics, the estimation error will accordingly grow. Figure 1 shows the evolution of the estimation error of agent 2, estimating states x 1 and x 4 . Recall that these states are measured by agents 1 and 3 and, according to the communication topology, these agents are one hop away from agent 2. When agent 2 can communicate, the estimation error decreases significantly. This is a consequence of the fast convergence rate fixed in the observer design. However, when agent 2 is not able to communicate with its neighbors the estimation error grows according to the unstable open-loop dynamics.

Example 2. This second example shows the unstable response of the state estimation error when Property 1 is not met. The consensus matrices designed in the previous example for τ i = 100, n τi = 20, do not satisfy Property 1 in the worsened scenario with τ i = 100, n τi = 2, for all i ∈ V, namely with increased communication losses.

By performing a parallel simulation to the one of the previous example, we now observe a diverging error response. In particular, Figure 2 shows the evolution of the estimation error of agent 2 estimating state x 4 . Note that the estimation error decreases when the communications are active. However, this is not enough to stabilize the estimation error.

Example 3. Let us present in this example an extension of the work proposed in this paper in which, instead of total communication losses, we consider that each link in the network can individually fail. In this case one may select the same observer structure (11) by setting δ i (k) = 0 when just one link incoming to agent i is failing. However this approach may be quite conservative.

Here we propose an alternative observer structure where the failure-related logical variables δ are associated to the links (that is, we call them δ i,j (k)) rather than to the nodes. In particular, we may modify (11) as follows xi (k + 1) = Ax i (k) + W i,0 L i (y i (k) -ŷi (k))

+ i ρ=0 j∈Ni δ i,j (k)W i,ρ N i,j,ρ W j,ρ-1 (x j (k) -xi (k)),
where δ i,j (k) is a binary variable indicating whether link (i, j) is active (δ i,j (k) = 1) or not (δ i,j (k) = 0) at time k.

Due to the topology considered in our example, only agent 2 can benefit from the proposed modification. Hence, Figure 3 shows the evolution of the estimation error of Agent 2 estimating state x 4 when using this revised structure. With the same gains N i,j,ρ used in the previous examples, the new structure successfully estimates the fourth state.

This last example encourages the authors to work towards a generalization of our approach to the case where the agents have partial communication or link failures, but still can make use of the information received from the rest their neighbors.

CONCLUSIONS

In this paper, the distributed state estimation problem of an autonomous LTI system by a lossy network of agents has been addressed. By using an observer structure based on a multi-hop subspace decomposition, each agent involved in the network can identify its observable subspace and the innovations introduced (whenever a communication loss does not occur) by its neighbors at each hop. Under some reasonable assumptions on the network connectivity, we have shown that it is always possible to find observer gains guaranteeing uniform exponential convergence to zero of all the estimation errors. Our architecture has been tested by means of simulations and the main results of the paper have been illustrated by a network of three agents. Future work includes proving a uniform version of our exponential bound.

Fig. 1 .

 1 Fig. 1. Estimation error of agent 2 in Example 1.

Fig. 3 .

 3 Fig. 2. Estimation error of agent 2 estimating state x 4 in Example 2.

Note that the standard definition of ISS does not include square powers in signals norms. Nevertheless, if this expression is fulfilled it is trivial to go back to the standard definition.

ACKNOWLEDGEMENTS

Research partially supported by grant TEC2016-80242-P founded by AEI/FEDER through the Laboratorio de Simulación Hardware-in-the-loop, and by ANR via grant HANDY, number ANR-18-CE40-0010.


L Breloy

C Negrell

A.-S Mora

W S J Li

V Brezová

S Caillol

D.-L Versace
email: versace@icmpe.cnrs.fr

Vanillin Derivative as Performing Type I Photoinitiator

Keywords: vanillin, natural oil acrylate monomer, free-radical polymerization, UV-light

A new efficient type I photoinitiator derived from vanillin was synthesized to initiate, according to a green photoinduced process, the free-radical polymerization (FRP) of acrylate monomers in a reduced time and under air. Interestingly, this unprecedented photoinitiator lead to high acrylate conversions even under air, with a higher efficiency than some reference photoinitiating systems commonly used in FRP.

INTRODUCTION

Over the past twenty years, free-radical photopolymerization (FRP) has gained considerable interests for the synthesis of new smart and multifunctional materials with multiple conventional applications [START_REF] Mendes-Felipe | Stateof-the-Art and Future Challenges of UV Curable Polymer-Based Smart Materials for Printing Technologies[END_REF][START_REF] Yagci | Photoinitiated Polymerization: Advances, Challenges, and Opportunities[END_REF] in microelectronics, coatings, adhesives, inks, dental fillings or the fabrication of 3D microstructures. The growing interest of FRP by industry and the increasing number of publications on the subject demonstrate that photopolymerization is becoming unavoidable. Many striking advantages of FRP can be pointed out over the thermal polymerization [START_REF] Chatani | The power of light in polymer science: photochemical processes to manipulate polymer formation, structure, and properties[END_REF] such as low temperature process at room temperature or below, fewer side reactions, reactions proceed in a few seconds or minutes without any solvent, thus avoiding volatile organic compounds to be released; this makes FRP an environmentally-friendly technique. Up to date, FRP has been extensively described and numerous interesting reviews [START_REF] Mendes-Felipe | Stateof-the-Art and Future Challenges of UV Curable Polymer-Based Smart Materials for Printing Technologies[END_REF][START_REF] Yagci | Photoinitiated Polymerization: Advances, Challenges, and Opportunities[END_REF] have been published. In this technology, polymerization occurs with the use of two types of photoinitiators [START_REF] Gruber | Photoinitiators for free radical polymerization[END_REF][START_REF] Fouassier | Photoinitiators for polymer synthesis : scope, reactivity and efficiency[END_REF][START_REF] Dietliker | A Compilation of Photoinitiators Commercially Available for UV Today[END_REF] which differ in the way radical species are generated. Type I photoinitiators are aromatic carbonyl organic compounds [START_REF] Fouassier | Photoinitiators for polymer synthesis : scope, reactivity and efficiency[END_REF] (benzil ketals, benzoin and derivatives, acetophenones, aminoalkyl phenones, O-acyl--oximino ketones, acylphosphine oxides, acylgermanes and -hydroxyalkyl ketones) and undergo upon light absorption an "cleavage" to produce free-radical species. Interestingly, the photochemically generated phosphinoyl radicals are widely exploited due to their high reactivity arising from the high electron density around the phosphorus atom and favorable steric conditions [START_REF] Sumiyoshi | On the reactivity of phosphonyl radicals towards olefinic compounds[END_REF][START_REF] Sumiyoshi | The photolysis of acylphosphine oxides: III: Laser flash photolysis studies with pivaloyl compounds[END_REF]. On the contrary, type II photoinitiating systems are capable of undergoing intermolecular H-abstraction or electron/proton transfer reactions when combined with co-initiators [START_REF] Dietliker | Chemistry and Technology of UV and EB Formulation for Coatings, Inks & Paints[END_REF][START_REF] Hoyle | Thiol-enes: Chemistry of the past with promise for the future[END_REF][START_REF] Hoyle | Thiol-click chemistry: a multifaceted toolbox for small molecule and polymer synthesis[END_REF][START_REF] Cramer | Kinetics of thiol-ene and thiol-acrylate photopolymerizations with real-time fourier transform infrared[END_REF][START_REF] El-Roz | Mechanistic Investigation of the Silane, Germane, and Stannane Behavior When Incorporated in Type I and Type II Photoinitiators of Polymerization in Aerated Media[END_REF] (i.e. H-donors like tertiary amines, alcohols, ethers, silanes, germanes, stannanes or thiols) or with the use of ketoesters [START_REF] Gauss | α-Ketoesters as Nonaromatic Photoinitiators for Radical Polymerization of (Meth)acrylates[END_REF]. Benzophenone, thioxanthone, benzil, and quinone derivatives are typical bimolecular type II photoinitiators which lead in presence of H-donors to the formation of both ketyl radical and another radical derived from the H-donor molecule [START_REF] Ledwith | Initiation of free-radical polymerization by photoinduced electron transfer processes[END_REF][START_REF] Davidson | The Chemistry of Excited Complexes: a Survey of Reactions[END_REF]. However, it is assumed that the initiation of the polymerization according to a bimolecular type II reaction is slower than that of a type I process, suffer from back electron transfer and the resulting ketyl radicals are usually unreactive toward vinyl monomers due to their high stability and steric hindrance [START_REF] Hutchison | Rôle of semi-pinacol radicals in the benzophenone-photoinitiated polymerization of methyl methacrylate[END_REF].

Another point raising is that the majority of the photoinitiators commonly used and described in literature are derived from petro-sourced molecules. Therefore, we focused our attention on the synthesis of a novel type I photoinitiator from a bio-sourced synthon. To the best of our knowledge, the synthesis and the photoreactivity of type I photoinitiators derived from biosourced molecules have never been described yet. Thus, the development of a type I photoinitiator from natural synthons remains a challenge to face. If oils and polysaccharides are already well exploited at industrial scale, there is a lag concerning aromatic compounds. Among these aromatic derivatives, vanillin can be considered as the most promising compound in this research area. Indeed, this molecule is among the only few phenolic compounds which can currently be produced industrially from lignin [START_REF] Fache | Vanillin Production from Lignin and Its Use as a Renewable Chemical[END_REF]. Its interest comes not only from its availability: this molecule bears two reactive groups strategic for organic modifications, a phenol and an aldehyde, and offers the great advantage to be safe. Main applications of vanillin nowadays concern perfumery of food industry, but vanillin derivatives have also a great potential as antibacterial agents [START_REF] Fitzgerald | Mode of antimicrobial action of vanillin against Escherichia coli, Lactobacillus plantarum and Listeria innocua[END_REF], for drug delivery [START_REF] Kamaraj | Curcumin drug delivery by vanillin-chitosan coated with calcium ferrite hybrid nanoparticles as carrier[END_REF], flame retardation [START_REF] Zhao | Renewable vanillin based flame retardant for poly(lactic acid): a way to enhance flame retardancy and toughness simultaneously[END_REF][START_REF] Xu | Vanillin-derived phosphoruscontaining compounds and ammonium polyphosphate as green fire-resistant systems for epoxy resins with balanced properties[END_REF], composites [START_REF] Stroescu | Chitosan-vanillin composites with antimicrobial properties[END_REF]. It was highlighted as a very promising building block for the preparation of bio-based monomers [START_REF] Fache | Vanillin, a promising biobased building-block for monomer synthesis[END_REF] and renewable polymers, more particularly high-performance thermosets [START_REF] Fache | Vanillin, a key-intermediate of biobased polymers[END_REF].

Here, we report for the first time, the synthesis of a new vanillin-derived type I photoinitiator and evaluate its capability to initiate in laminate and under air the FRP of two acrylate monomers: trimethylolpropane triacrylate (TMPTA), which is a reference monomer for FRP studies, and Soybean oil epoxidized acrylate (SOA), a cheap and natural oil acrylate monomer.

Its properties have been compared with those of classical type I and type II photoinitiating systems. The addition of a thiol cross-linker, trimethylolpropane tris(3-mercaptopropionate) (TT) was experimented to reduce oxygen sensitivity of the new vanillin-derived photoinitiator.

Mechanisms of photopolymerization process were described by electron paramagnetic resonance (EPR) and laser-flash Photolysis (LFP), and kinetic studies were carried out by realtime Fourier transform infrared spectroscopy (RT-FTIR).

EXPERIMENTAL SECTION

Materials

Synthesis of 4-allyloxy-3-methoxybenzaldehyde (A).

Vanillin (1 eq, 20.0 g, 131 mmol) was combined with dimethylformamid (DMF) in a round-bottom flask in an ice bath. K2CO3 (4 eq, 72.7 g, 52.8 mmol) was added and left to stir for 3 min. Allyl bromide (4 eq, 63.6 g, 52.8 mmol)

was added via syringe dropwise and stirred for 30 min. After, the reaction mixture was stirred for 48 h at 25 °C. 700 mL of deionized water was added to the reaction mixture and the aqueous phase was extracted with ethyl acetate (3 x 700 mL). Organic phase was washed with 700 mL of brine. Organic phase was dried with Na2SO4 and ethyl acetate was removed by rotary evaporator. Lastly, the product was purified by distillation under reduced pressure (10 -2 mBar and 60 °C). Isolated yield = 59%.

Synthesis of -hydroxy(4-allyloxy-3-methoxybenzyl)diphenylphosphine oxide (B). 4-

allyloxy-3-methoxybenzaldehyde (1 eq, 14.0 g, 72.8 mmol), diphenylphosphine oxide (1 eq, 14.7 g, 72.8 mmol), and dry tetrahydrofuran (THF; 280 mL) were combined in a round-bottom flask and sealed with a septum. Triethylamine (1 eq, 7.37 g, 72.8 mmol) was added dropwise into the reaction mixture at 25 °C and stirred for 24 h. The precipitated product was filtered and washed with THF several times and then dried. It used without further purification for next step.

Isolated yield = 85%. The thickness of the coatings was 12 μm for all experiments. The decrease of acrylate functions of TMPTA or SOA was continuously followed at 1636 cm -1 by real-time Fourier transform infrared spectroscopy (RT-FTIR, JASCO FTIR 4700) according to previously described method [START_REF] Breloy | β-Carotene/Limonene Derivatives/Eugenol: Green Synthesis of Antibacterial Coatings under Visible-Light Exposure[END_REF].

Laser-flash photolysis studies (LFP). LFP experiments were performed according to previously described procedures [START_REF] Breloy | Well-Defined Titanium Complex for Free-Radical and Cationic Photopolymerizations under Visible Light and Photoinduction of Ti-Based Nanoparticles[END_REF][START_REF] Breloy | Visible Light Anthraquinone Functional Phthalocyanine Photoinitiator for Free-Radical and Cationic Polymerizations[END_REF] and employed the pulses from a Spectra Physics GCR-150-30 Nd:YAG laser (355 nm, 7 ns pulse length). Experimental conditions for each measurement are detailed in the results section. For recording the transient absorption spectra and quenching studies with oxygen or TT, solutions of PM were prepared with an absorbance value ~0.6 at the excitation wavelength (355 nm) in an UV-cell with a path length of 1 cm.

EPR experiments. The X-band EPR spectra were measured by means of EMXplus spectrometer (Bruker) at 100 kHz modulation frequency with a high sensitivity probe-head (Bruker) using thin-walled quartz EPR tubes (Bruker). As the spin trapping agents 5,5dimethyl-1-pyrroline N-oxide (DMPO, Sigma-Aldrich, distilled before use) or N-tert-butyl-αphenylnitrone (PBN, Sigma-Aldrich) were applied. The samples prepared by the mixing of stock solutions in benzene were carefully saturated with argon. The prepared solutions were irradiated at 293 K directly in the EPR resonator using a LED@385 nm source (λmax = 385 nm;

Bluepoint LED, Hönle UV Technology) and the EPR spectra were recorded in situ during exposure. The g-values of spin-adducts were determined with an uncertainty of  0.0001 using a nuclear magnetic resonance teslameter and integrated frequency counter. The experimental EPR spectra were analyzed by Bruker software WinEPR and the simulated spectra were calculated with the EasySpin toolbox as previously described [START_REF] Breloy | Well-Defined Titanium Complex for Free-Radical and Cationic Photopolymerizations under Visible Light and Photoinduction of Ti-Based Nanoparticles[END_REF][START_REF] Breloy | Visible Light Anthraquinone Functional Phthalocyanine Photoinitiator for Free-Radical and Cationic Polymerizations[END_REF].

RESULTS and DISCUSSION

Both acyl-phosphine oxides are known to be used as classical photoinitiators. A methodology has been described by Nazir and co-workers [START_REF] Nazir | Push-Pull Acylo-Phosphine Oxides for Two-Photon-Induced Polymerization[END_REF]. First, the addition of aldehydes to diphenylphosphine oxide in dry solvent in the presence of base, gives α-hydroxyphosphine oxides, which are oxidized to desired acyl-phosphine oxides by MnO2. We used this synthesis on a derivative of vanillin to obtain a partially bio-based photoinitiator. To avoid the negative influence of the phenolic group of vanillin, which could lead for example to side reactions such as the addition of phenol group in basic media to the aldehyde or could play as a radical scavenger during free-radical photopolymerization, an allylation with allyl bromide was done in mild conditions with a soft base K2CO3 at room temperature, according to a previously described procedure [START_REF] Fache | Vanillin, a promising biobased building-block for monomer synthesis[END_REF]. The product (A) was characterized by 1 H and 13 Previous studies from Turro [START_REF] Sluggett | 2,4,6-Trimethylbenzoyl)diphenylphosphine Oxide Photochemistry. A Direct Time-Resolved Spectroscopic Study of Both Radical Fragments[END_REF][START_REF] Dursun | Free radical promoted cationic polymerization by using bisacylphosphine oxide photoinitiators: substituent effect on the reactivity of phosphinoyl radicals[END_REF] and Schnabel [START_REF] Sumiyoshi | On the reactivity of phosphonyl radicals towards olefinic compounds[END_REF][START_REF] Sumiyoshi | On the photolysis of acylphosphine oxides: 1. Laser flash photolysis studies with 2,4,6-trimethylbenzoyldiphenylphosphine oxide[END_REF] investigated the reactivity of phosphine oxides derivatives under UV-light irradiation (excitation = 355 nm) by LFP. They demonstrated that upon UV-light exposure, these photoinitiators underwent an homolytic photocleavage to produce benzoyl radicals and diphenylphosphinoyl radicals which are readily observable around 330 nm. No other active radical species were detected. To go further in the mechanism approach that we suggest during the photolysis of PM, EPR-ST technique is coupled with LFP experiments. EPR-ST is much more sensitive than LFP technique to observe short-lived and highly reactive free radicals in chemical systems at low concentration [START_REF] Swartz | Use of Electron Paramagnetic Resonance Spectroscopy to Evaluate the Redox State In Vivo[END_REF]; The generated spin-adducts possess considerably greater stability than that of the parent free radical species; for instance, phenyl radicals are not easily detected by LFP as max = 240 nm [START_REF] Hatton | The photochemistry of nitrosobenzene: direct observation of the phenyl radical-nitric oxide triplet radical pair in argon at 12 K[END_REF], LFP equipment lacks sensitivity for absorptions less 300 nm [START_REF] Hatton | The photochemistry of nitrosobenzene: direct observation of the phenyl radical-nitric oxide triplet radical pair in argon at 12 K[END_REF] and the signal of some phenyl derived radicals can be merged with diphenylphosphinoyl radicals between 320 and 380 nm [START_REF] Bartl | Photo-heterolysis and -homolysis of substituted diphenylmethyl halides, acetates, and phenyl ethers in acetonitrile: characterization of diphenylmethyl cations and radicals generated by 248-nm laser flash photolysis[END_REF]. Therefore, the generation of non-persistent radical species upon photoexcitation of PM was first monitored by EPR spin trapping technique. In situ irradiation of PM by LED@385 nm source in the presence of PBN spin trap results in the substantial intensity growth of complex EPR signal, corresponding to the simultaneous formation of  PBN-adducts of carbonand phosphorus-centered radicals (Figure 1). Simulation was calculated considering the presence of four  PBN-adducts with the spin-Hamiltonian parameters summarized in Table 2.

Scheme 2 describes the proposed homolytic cleavage of PM under light exposure. Interestingly, PM generates four radical species, i.e. two phosphorus and two carbon-centered radicals, which can be used as photoinitiating species for the FRP of acrylates (see Supporting Information).

One forces to note that the signals of PBN spin-adducts of substituted benzoyl radical (Table 2) are less intensive than those of phosphinoyl radicals because of their lower rates of formation and faster rate of relaxation compared to that of phosphinoyl radicals according to Turro and Khudyakov studies [START_REF] Turro | Single-phase primary electron spin polarization transfer in spin-trapping reactions[END_REF]. LFP experiments support the EPR results with the formation of (1) at 400 nm (see Figure S11 in Supporting Information). The transient of the diphenylphosphinoyl radical at ~ 325 nm is not observed because of the overlap of the ground state absorption/ground state bleaching of PM. Magnetic field (mT) Irradiation of a benzene solution of the PM/TT photoinitiating system in the presence of DMPO using LED@385 nm leads to an immediate formation of dominant signal characterized by the spin-Hamiltonian parameters fully compatible with the  DMPO-SR spin-adduct [START_REF] Breloy | Well-Defined Titanium Complex for Free-Radical and Cationic Photopolymerizations under Visible Light and Photoinduction of Ti-Based Nanoparticles[END_REF] (aN = 1.347 mT, aH  = 1.178 mT, aH = 0.091 mT, aH = 0.094 mT; g = 2.0061). This statement confirms the formation of thiyl radicals and the capability of radicals (1-4 in Scheme 2) to abstract H-atom from the thiol group of TT under light exposure. Simultaneously, upon in situ irradiation the generation of further signals corresponding to the DMPO-adducts of phosphorusand carbon-centered radicals is detectable, along with the low-intensity nine-line signal well compatible with the addition of hydrogen radical to DMPO (aN = 1.455 mT, aH  = 1.907 mT, aH  = 1.898 mT; g = 2.0059). As the stability of these DMPO-adducts is low under given experimental conditions, they disappeared from spectrum after switch-off the light source and  DMPO-SR signal dominates the EPR spectrum (Figure 2). Interestingly, the quenching rate constant of PM by TT and labelled kq TT (PM) was evaluated at 0.67  10 8 M -1 s -1 (see Figure S12 in Supporting Information for details). The high reactivity of ( 1)-( 4) (Scheme 2) toward Hdonor molecules is a consequence of the high degree of s-character and spin density localized on the carbonyl carbon, phenyl carbon and phosphorus atoms [START_REF] Sluggett | 2,4,6-Trimethylbenzoyl)diphenylphosphine Oxide Photochemistry. A Direct Time-Resolved Spectroscopic Study of Both Radical Fragments[END_REF]. To confirm the reactivity of PM toward acrylate monomers and its capability to initiate FRP under light irradiation, RT-FTIR experiments have been performed with TMPTA (Figure 3) and a bio-sourced acrylate monomer SOA (Figure 4). TMPTA was used as a reference. The final acrylate conversions for TMPTA and SOA according to PM/additives photoinitiating systems are summarized in Table 3. DMPA, BAPO, HCPK, QUIN/TT, PUR/TT AQ/TT and TX/TT were used as references for the FRP. DMPA photolyzes to produce both dimethoxybenzyl and carbonyl radicals [START_REF] Versace | The First Photochemical In Situ Production of Ti-Based Nanoparticles: A SH2 Strategy Using Bis(cyclopentadienyl)titanium Dichloride (Cp2TiCl2)[END_REF], BAPO generates both benzoyl and phosphinoyl radicals [START_REF] Copperwhite | The role of photoinitiator and chelating agent in the fabrication of optical waveguides from UVphotocurable organo-mineral sol-gel materials[END_REF], and AQ, QUIN, PUR and TX were used as type II photoinitiators when coupled with TT. TMPTA (Figure 3) or SOA (Figure 4) successfully polymerized in laminate when PM is used alone as a freeradical photoinitiator under light irradiation. Indeed, phosphinoyl and phenyl radicals are wellknown to be reactive toward acrylate functions [START_REF] Breloy | Well-Defined Titanium Complex for Free-Radical and Cationic Photopolymerizations under Visible Light and Photoinduction of Ti-Based Nanoparticles[END_REF][START_REF] Copperwhite | The role of photoinitiator and chelating agent in the fabrication of optical waveguides from UVphotocurable organo-mineral sol-gel materials[END_REF]. Not surprisingly, no polymerization occurs under air with TMPTA: the dissolved oxygen gets consumed by addition to initiator radicals and radicals of the growing polymer chain to form peroxyl radicals which are unreactive toward acrylate monomers [START_REF] O'brien | Impact of Oxygen on Photopolymerization Kinetics and Polymer Structure[END_REF]. As an example, PM is quenched by molecular oxygen efficiently with a quenching rate constant kq O2 (PM) which is evaluated to be 10 9 M -1 s - 1 (see Figure S13 in supporting information). On the opposite, PM promotes the FRP of SOA (Figure 4) under air with relatively high final conversion (40%) compared to that of TMPTA (0%): indeed, the high viscosity of SOA (1200 cps) compared to TMPTA (70-120 cps) reduces the diffusion of oxygen from the outside to the polymer coating, speeding up the FRP of SOA.

Other features can be expressed when PM is coupled with TT. As expected, the FRP of SOA successfully occurs in laminate or under air with PM/TT systems; indeed, previous investigations [START_REF] Hoyle | Thiol-enes: Chemistry of the past with promise for the future[END_REF][START_REF] Hoyle | Thiol-click chemistry: a multifaceted toolbox for small molecule and polymer synthesis[END_REF][START_REF] Breloy | β-Carotene/Limonene Derivatives/Eugenol: Green Synthesis of Antibacterial Coatings under Visible-Light Exposure[END_REF][START_REF] Breloy | Well-Defined Titanium Complex for Free-Radical and Cationic Photopolymerizations under Visible Light and Photoinduction of Ti-Based Nanoparticles[END_REF] supported the fact that thiyl radicals play a dual role i.e. they consumed dissolved oxygen, and the resulting peroxyl radicals can abstract hydrogen from TT to re-initiate the FRP (r1-r4). It is also noticeable that PM is more efficient than DMPA to initiate the FRP of TMPTA. This difference may be explained by the fact that the photolysis of PM leads to highly reactive phenyl radicals altogether with phosphinoyl and carbonyl radicals, unlike DMPA which generates only dimethoxybenzyl and carbonyl radicals [START_REF] Versace | Zinc complex based photoinitiating systems for acrylate polymerization under air; in situ formation of Zn-based fillers and composites[END_REF]. Finally, PM also shows initiating properties as effective as those of the well-known type I photoinitiator BAPO (Figure 5): indeed, BAPO like PM produces benzoyl and phosphinoyl radical species after its photolysis [START_REF] Copperwhite | The role of photoinitiator and chelating agent in the fabrication of optical waveguides from UVphotocurable organo-mineral sol-gel materials[END_REF].

PM + RSH + hv → phosphinoyl/carbon centered radicals + RS  (r1)

RS  + O2 → RSOO  (r2) RSOO  + RSH → RSOOH + RS  (r3)
RS  + acrylate → poly(acrylate) (r4)

1 H 6 (

 16 NMR (ppm, 400.13 MHz, CDCl3) : 3.96 (O-CH3, 3H, s), 4.73 (O-CH2-CH, 2H, d), 5.37 (CH=CHaHb, 1H, d, Jcis=10.5 Hz), 5.47 (CH=CHaHb, 1H, d, Jtrans=17.3 Hz), 6.11 (CH2-CH=CH2, 1H, m), 7.00 (Haromatic, 1H, d), 7.44 (Haromatic, 1H, s), 7.45 (Haromatic, 1H, d), 9.87 (HC=O, 1H, s). 13 C NMR (ppm, 100.61 MHz, CDCl3) : 56.1 (O-CH3), 69.8 (O-CH2-CH), 109.3 (CHaromatic), 111.9 (CHaromatic), 118.8 (CH=CH2), 126.CHaromatic), 130.0 (HC(O)-CHaromatic), 132.2 (CH=CH2), 149.8 (CH3-O-CHaromatic), 153.5 (CH2-O-CHaromatic), 191.0 (HC=O). HRMS (m/z, ES+, [M+H+]): C11H13O3; Calculated 193.0865; found 193.0867.

Scheme 1 .

 1 FigureS9, the UV-vis absorption spectrum of PM shows a higher molar extinction coefficient than DMPA, HCPK or BAPO used as references for type I photoinitiator. The overlap of the

Figure 1 .Scheme 2 .

 12 Figure 1. Experimental (1) and simulated (2) EPR spectra obtained after 900-s LED@385 nm exposure of PM/benzene solution in the presence of PBN spin trap under argon. EPR spectrometer settings: microwave frequency, ~ 9.436 GHz; microwave power, 1.114 mW; center field, 336.0 mT; sweep width, 7 mT; gain, 1.0010 5 ; modulation amplitude, 0.01 mT; scan, 90 s; time constant, 20.48 ms; number of scans, 10.

  a = photopolymerization of TMPTA, b = photopolymerization of SOA, np = no polymerization.

Figure 3 .

 3 Figure 3. Kinetic profiles of the free-radical photopolymerization of TMPTA 1) under air and 2) in laminate in the presence of (a) PM (1wt%), (b) PM/TT (1/1 wt%), (c) PM/TT (1/3 wt%), (d) AQ/TT (1/1 wt%), (e) TX/TT (1/1 wt%), (f) QUI/TT (1/1 wt%), (g) PUR/TT (1/1 wt%), (h) DMPA (1 wt%) and (i) HCPK (1 wt%) upon Hg-Xe lamp irradiation (60 mW/cm 2 ).

Figure 4 .

 4 Figure 4. Kinetic profiles for the free-radical photopolymerization of SOA 1) under air and 2) in laminate in the presence of (a) PM (1 wt%), (b) PM/TT (1/1 wt%), (c) PM/TT (1/3 wt%), (d) AQ/TT (1/1 wt%), (e) PUR/TT (1/1 wt%) and (f) QUI/TT (1/1 wt%) upon Hg-Xe lamp irradiation (60 mW/cm 2 ).

Figure 5 .

 5 Figure 5. Kinetic profiles for the free-radical photopolymerizations (FRP) of TMPTA in laminate with a) PM (1 wt%) and b) BAPO (1 wt%) and FRP of SOA in the presence of c) PM (1 wt%) and d) BAPO (1 wt%) upon Hg-Xe lamp irradiation (60 mW/cm 2 ).

Table 1 .

 1 Structure of the compounds used in this study.

	.	2,2-Dimethoxy-2-phenylacetophenone	(DMPA),	phenylbis	(2,4,6-
	trimethylbenzoyl)-phosphine oxide (BAPO) and trimethylolpropane triacrylate (TMPTA) were
	kindly provided by BASF and Sartomer, respectively. Soybean oil epoxidized acrylate (SOA),
	trimethylolpropane tris(3-mercaptopropionate) (TT, ≥ 95%), 1-hydroxy-cyclohexyl-phenyl-
	ketone (HCPK, 99%), Quinizarin (QUI, 96%), Purpurin (PUR, dye content 90%), allyl
	bromide, potassium carbonate (K2CO3), diphenylphosphine oxide, activated manganese oxide
	(MnO2) and celite were all provided by Sigma-Aldrich. 9,10-Anthraquinone (AQ, > 98%) and
	thioxanthone (TX, > 98%) were provided by Alfa Aesar and TCI respectively. Vanillin was also

= singlet, d = doublet, t = triplet, m = multiplet. MS measurements were performed on a Waters Synapt G2-S High Resolution Mass Spectrometer (HRMS) equipped with an ESI ionization source.

  UV-Visible absorption spectra of the photoinitiators were measured using a Perkin Elmer Lambda 2 UV-vis spectrophotometer in the 200-800 nm wavelength range.

	Photopolymerization kinetic studies. For each sample, photoinitiator (PI) and TT (PI/TT
	1%/1% w/w with respect to the monomer) were dissolved in acrylate monomer (TMPTA or
	SOA). Formulations were stirred and sonicated for 15 min before study. The resulting
	photosensitive formulations were laid down on a BaF2 pellet and irradiated with Hg-Xe lamp
	(Lightningcure LC8-02 lamp from Hamamatsu, 200 W, 60 mW.cm -2 ) under air or in laminate.
	93 (O-CH3, 3H, s), 4.72 (O-CH2-CH,
	2H, d), 5.35 (CH=CHaHb, 1H, d, Jcis=10.1 Hz), 5.43 (CH=CHaHb, 1H, d, Jtrans=17.1Hz), 6.08
	(CH2-CH=CH2, 1H, m), 6.95 (Haromatic, 1H, d), 7.5-7.9 (Haromatic, 10H, m). 13 C NMR (ppm,
	100.61 MHz, CDCl3) : 56.0 (O-CH3), 69.8 (O-CH2-CH), 111.0 (CHaromatic), 112.7 (CHaromatic),
	118.4 (CH=CH2), 118.9 (CHaromatic), 128.8 (CHaromatic in meta position of P), 132.1-132.4
	(CHaromatic), 149.5 (CH3-O-CHaromatic), 154.3 (CH2-O-CHaromatic). Traces of MnO2 in the sample

1 H NMR (ppm, 400.13 MHz, CDCl3) : 3.45 (O-CH3, 3H, s), 4.51 (O-CH2-CH, 2H, d), 5.25 (CH=CHaHb, 1H, d, Jcis=9.3 Hz), 5.34 (HO-CH, 1H, s), 5.38 (CH=CHaHb, 1H, d, Jtrans=17.2 Hz), 5.96 (OH, 1H, s), 6.06 (CH2-CH=CH2, 1H, m), 6.55 (Haromatic, 1H, d), 6.61 (Haromatic, 2H, s), 7.32 (Haromatic in meta position of P, 4H, m), 7.45 (Haromatic in para position of P, 2H, m), 7.68 (Haromatic in ortho position of P, 4H, m). 13 C NMR (ppm, 100.61 MHz, CDCl3) 55.5 (O-CH3), 69.7 (O-CH2-CH), 73.1-74.0 (CH-OH), 110.9 (CHaromatic), 112.5 (CHaromatic), 117.9 (CH=CH2), 120.2 (CHaromatic), 128.1 (CHaromatic in meta position of P), 129.8 (HO-CH-CHaromatic), 130.3 (P-CHaromatic), 131.8 (CHaromatic in para position of P), 132.1-132.5 (CHaromatic in ortho position of P), 133.3 (CH=CH2), 147.5 (CH3-O-CHaromatic), 148.7 (CH2-O-CHaromatic). 31 P NMR (ppm, 161.98 MHz, CDCl3) : 30.8. HRMS (m/z, ES+, [M+H+]): C23H24O4P; Calculated 395.1412; found 395.1415.

Synthesis of (4-allyloxy-3-methoxybenzoyl)diphenylphosphine oxide (PM). A solution of

-hydroxy(4-allyloxy-3-methoxybenzyl)diphenylphosphine oxide (1 eq, 11.0 g, 2.79 mmol) in anhydrous dichloromethane (DCM; 220 mL) and activated MnO2 (1 eq, 48.5 g, 558 mmol) in a round-bottom flask and purge for 30 min under N2. The mixture was stirred to react at 25°C for 18 h. Then it was filtered under celite and DCM was removed by rotary evaporator. Isolated yield = 95%. 1 H NMR (ppm, 400.13 MHz, CDCl3) : 3.of photoinitiator did not allow to obtain a correct

31

P NMR spectrum. HRMS (m/z, ES+, [M+H+]): C23H22O4P; Calculated 393.1256; found 393.1261.

Table 2 .

 2 Spin-Hamiltonian parameters of  PBN-adducts elucidated from the simulation analysis of experimental EPR spectrum shown in Figure1.

	Hyperfine coupling constants (mT) aN aH aP	g-value	Rel. concentration (%)		Radical trapped		References
	1.426 0.305	1.841	2.0060	41.8	P O •			[37, 38]
				O	•	C		OR	1
	1.441 0.412	1.714	2.0060	38.6	P	O	OR	[39]
								2
	1.442 0.452	-	2.0061	7.5	C •		OR	1	[37, 38]
					O
						OR
						2
	1.459 0.240	-	2.0061	12.1		•			[27, 28]

Table 3 .

 3 Final acrylate conversions of TMPTA and SOA determined by IR with different photoinitiating systems after 800 s of irradiation.

		TMPTA and SOA
	Photoinitiating	Under air	In laminate
	systems
	PM	np a (40%) b	72% a (86%) b
	BAPO	-	70% a (92%) b
	DMPA	np a	60% a
	HCPK	np a	70% a
	AQ/TT (1 wt%)	16% a (70%) b	71% a (87%) b
	QUI/TT (1 wt%)	np a (np) b	42% a (31%) b
	PUR/TT (1 wt%)	np a (15%) b	65% a (51%) b
	TX / TT (1 wt%)	np a	55% a
	PM / TT (1 wt%)	np a (50%) b	72% a (90%) b
	PM / TT (3 wt%)	25% a (60%) b	70% a (90%) b


Camille Chapelle

Ghislain David

Sylvain Caillol

Claire Negrell

Graziella Durand

Myriam Desroches Le Foll

Stéphane Trombotto

Water soluble 2,5-anhydro-D-mannofuranose end chain chitosan oligomers of very low molecular weight : synthesis and characterization

Keywords: Chitosan, depolymerization, variable resources, water soluble oligomers

Five chitosan with different acetylation degree (DA), molar masses and origin were depolymerized by nitrous acid treatment in acidic media leading to water soluble 2,5 anhydro-Dmannose end chain oligomers with DPn<20. The kinetics of the reaction was studied and the best work conditions were found to be 3h reaction at 50°C. It was shown that the DPn of oligomers only depends on the quantity of NaNO2 involved. Molar masses or DA do not have an impact on the depolymerization process when targeting oligomers with less than 20 units. This depolymerization process also leads to free AMF that might react with the free amines of obtained oligomers to form imines. This reaction is pH dependant and in acidic condition, leads to the formation of 5-hydromethyl-2-furfural (HMF). At the end, the oligomers were purified with dialysis to get rid of most free AMF (<5%) and other residual salts, and appeared to have no acute toxicity.

INTRODUCTION

Chitosan is a polysaccharide constituted of β-D-glucosamine and N-acetyl-β-D-glucosamine units linked by a glycosidic bond (1→4). It is the N-deacetylation of chitin, a polymer extracted from shrimp shells, squids, insects or mushrooms, poorly processable due to its insolubility in most solvent including water that leads to chitosan. Chitin and chitosan have the same chemical structure and differ only by the proportion of N-acetylated units along the chain, >60% for chitin and <60% for chitosan.

In fine, chitosan is characterized by its source (crustacean, insects, fungi…), its N-acetylation degree (DA) and its molar mass. A wide range of chitosan exists with various molar masses from 100,000 g/mol up to 500,000 g/mol with acetylation degree between 0 and 30 % 1 . Many interesting applications result from use of chitosan 2 3 4 5 . Nevertheless, chitosan use is limited due to its high viscosity, its insolubility in organic solutions and poor solubility in aqueous solution owing to the pH [START_REF] Kumar | A Review of Chitin and Chitosan Applications Q[END_REF] . Hence, a flourishing interest for lowering the molar masses to obtain chitosan oligomers appeared during the last decade 7 8 9 .

ChitoOligoSaccharides (COS) have indeed recently received considerable attention as functional biomolecules with a wide range of applications in food, agriculture, medicine, pharmaceutics and cosmetics. COS are often defined as chitosans with degree of polymerization DP less than 20 [START_REF] Naveed | Chitosan Oligosaccharide (COS): An Overview[END_REF] , i.e. with average molar masses lower than 4,000 g/mol. COS are either composed of β-D-glucosamine units or, can be a mixture of β-D-glucosamine and acetyl-β-D-glucosamine randomly organized. COS are defined by some unique properties such as water solubility, cell membrane penetrability, easy absorption and various biological activities. It is also worth to note that COS are easier to process, due to less chain entanglement and steric hindrance which confer them a direct solubility in water for DP<20 5 8 . COS are insoluble in acetone, butanol, ethanol, ethyl acetate, propanol and pyridine but as opposed to chitosan, partially soluble in methanol and in dimethyl sulfoxide and dimethyl formamide when heated [START_REF] Inamdar | Chitooligosaccharides : Synthesis , Characterization and Applications Chitooligosaccharides[END_REF] . Furthermore for these oligomers, the smaller are the molar masses, the better is the solubility. In addition, the utilization of COS allows to reduce the viscosity and to work with higher concentration. COS are reported to treat asthma, prevent tumor growth, malaria, control blood pressure and also have neuroprotective effects for Alzheimer disease treatment thanks to their biological activities, that include antibacterial, antifungal, antiviral, antitumor and antioxidant properties, which have been reported to overcome those of chitosan 9 12 13 14 15 .

Several ways to reduce chitosan molar masses including enzymatic, physical or chemical depolymerization have been studied over the last decades 8 16 17 18 19 . Enzymatic methods are really precise but very expensive and hardly applicable on large scale. Physical depolymerizations with microwave or sonication do not lead to oligomers and are mostly used to reduce molar masses and viscosity of chitosan solutions. They are often combined with chemicals like peroxide or concentrated acid to boost the depolymerization. Those methods are hardly controllable and reproducible. Allan and al. [START_REF] Allan | Molecular Weight Manipulation of Chitosan II: Prediction and Control of Extent of Depolymerization by Nitrous Acid[END_REF] developed a depolymerization method, based on the use of sodium nitrite (NaNO2). The principle is based on the action of nitrous acid, which cleaves the glycosidic linkages of chitosan. The reaction consists in the formation of a diazonium ion on a D-glucosamine unit by reaction between the free amine and the nitrous acid. A carbocation is obtained by nitrogen release. The nucleophilic addition of endocyclic oxygen then leads to a 2,5-anhydro-D-mannofuranose (AMF) structure as the new chain end (Scheme 1).

Scheme 1 Reaction of NaNO2 onto chitosan and side reaction leading to HMF Interestingly, they established an equation linking the obtained oligomer molar mass to the quantity of NaNO2. Later on, Tømmeraas et al. [START_REF] Tømmeraas | Preparation and Characterisation of Oligosaccharides Produced by Nitrous Acid Depolymerisation of Chitosans[END_REF] investigated on the reaction mechanism with detailed NMR analyses of fully N-acetylated and N-deacetylated oligomers. They especially underlined the formation of AMF chain end. They also showed the presence of secondary reactions: the resulting aldehyde of AMF chain end could react with the free amines of chitosan oligomers, leading to the formation of a Schiff base during lyophilisation. In acidic conditions, this Schiff base would then facilitate the elimination reactions leading to the formation of 5-hydroxymethyl-2-furfural (HMF), another byproduct. Shirui Mao and al [START_REF] Mao | The Depolymerization of Chitosan: Effects on Physicochemical and Biological Properties[END_REF] , studied some factors affecting the depolymerization process like chitosan concentration, reaction time and reproducibility. They studied high molecular weight (600 kg/mol and 400 kg/mol) and low molar masses (150 kg/mol) chitosans as well as cytotoxicity of the resulting COS with molar masses ranging from 2,800 to 49,000 g/mol but did not get further on the depolymerization process to reach lower molar mass oligomers. Although nitrous deamination has been studied and used many times to depolymerize chitosan 23 24 , this reaction was not investigated so far for reaching even smaller DPs , i.e. 5<DP<20. Only Moussa et al. synthesized oligomers of DP10 using this method but started from one single type of chitosan (DA 0%, Mn= 115,000 g/mol) [START_REF] Moussa | Reducing-End " Clickable " Functionalizations of Chitosan Oligomers for the Synthesis of Chitosan-Based Diblock Copolymers[END_REF] . Those oligomers could be of interest as they happen to be directly water soluble depending on the pH and even soluble in DMSO and exhibit better biological activities [START_REF] Liaqat | Chitooligosaccharides and Their Biological Activities : A Comprehensive Review[END_REF] . Besides, effect of both sourcing and DA for initial chitosan onto depolymerization process was also not highlighted.

The purpose of this paper is to obtain chitosan oligomers of different DP (average from 5 to 60) by nitrous deamination from different chitosans by varying the DA, the Mw and the origin (marine and insect). The kinetics of the reaction was studied as well as the ideal reaction parameters to lead to a robust method of chitosan depolymerization. The formation of free AMF and HMF was underlined and correlated to the different reaction parameters. The molar masses of the resulting oligomers were calculated using size exclusion chromatography (SEC). Structure identification was performed using infrared spectroscopy (IR), 1 H and 2D DOSY NMR spectroscopy, and headspace chromatography.

MATERIALS AND METHODS

Materials

Three different shrimp chitosan DA around 10-15% CH30 (Mw 30 kg/mol), CH150 (Mw 150 kg/mol), CH250 (Mw 250 kg/mol) were purchased from Glentham Science, chitosan with DA equal to 1% CHDA1 (150 kg/mol) was purchased from Mathani chitosan, insect chitosan with DA equal to 20% CHY (Mw 180 kg/mol) was purchased from Ynsect and used as received. Acetic acid (AcOH), sodium acetate (NaAc), sodium nitrite (NaNO2), lithium bromide (LiBr), sodium azide (NaN3), sodium hydroxide (NaOH), hydrochloric acid (HCl), acetone (>98%) were purchased from Sigma and used without purification, HPLC water (>99%) was purchased from VWR.

Depolymerization of chitosan

The depolymerization of high molecular weight chitosans was performed by nitrous deamination involving NaNO2 as previously described [START_REF] Tømmeraas | Preparation and Characterisation of Oligosaccharides Produced by Nitrous Acid Depolymerisation of Chitosans[END_REF] . Briefly, 1g (w/w) of chitosan was dissolved in 80 mL of AcOH (350µL in 90 mL of deionized water) solution overnight (pH 4.5). Once solubilized, a freshly prepared solution (10 mL) containing the desired equivalent ratio of NaNO2/NH2 (0.1, 0.2, 0.3, 0.4, 0.5, 0.6 eq) was added to the solution. The solution was stirred at definite temperature for a definite time. Oligomers of DP>20 (made with 0.1 eq of NaNO2/NH2) were precipitated by addition of 1M solution of NaOH and washed several times with deionized water until neutral pH. Oligomers with DP<20 did not precipitate with the addition of NaOH. The solutions were then freeze dried for 24h and the lyophilisates were dried under vacuum for 24h. The kineticss of the reaction was investigated for the five different chitosans. For that, chitosans were dissolved in deionized water with 1eq AcOH/NH2, 0.2 eq of NaNO2/NH2 were added and the reaction was led at room temperature (30°C). Samples were taken after 30 min, 1 h, 2 h, 3 h, 6 h and 24 h, freeze dried and analyzed using SEC with a PEO calibration. The same experiment was reproduced for CH150 at 10°C, 30°C, 50°C and 70°C in order to investigate influence of temperature on the reaction. Dialysis was performed using Spectra/Por dialysis membrane Biotech CE tubing MWCO: 100-500 D.

AMF synthesis

AMF was synthesized by action of nitrous acid on D-glucosamine. Briefly, 1 g (4.6 mmol, 1 eq) of D-(+)-Glucosamine hydrochloride was dissolved in 20 mL of deionized water during 3h.

Sodium nitrite (1 g, 14.5 mmol, 3.2 eq) was added and the reaction was led at room temperature for 6h. The final product was lyophilized for at least 24h. 1 H NMR for D-glucosamine (D2O, 400 MHz, 298 K, pH 5.5) : δ 5.42 (d 4Hz, 1H, H1α ), 4.9 (d 8Hz, 1H, H1β ), 3.4-3.9 (m, 1H H3, 1H H4, 1H H5, 2H H6), 3.3 (dd 10Hz, 1H, H2α ), 2.9 (dd 10Hz, 1H, H2β ), 4.8( HOD). 1 H NMR for AMF (D2O, 400 MHz, 298 K, pH 5.5) : δ 8.5 (s, 1H, H1), 5.1 (d, 5Hz, 1H, H1 gemdiol), 4.2 (t 6Hz, 1H, H2), 4,1 (t 6Hz, 1H, H3), 4,0 (m, 1H, H5), 3.8 (m, 1H, H4 2H, H6) as expected [START_REF] Claustre | An Easy Stereospecific Synthesis of 1-Amino-2,5-Anhydro-1-Deoxy-D-Mannitol and Arylamino Derivatives[END_REF] . In deuterated water, AMF is 70% in gem-diol and 30% of aldehyde form (SI 1 & 2). Characterization 1 H NMR, analyses were performed using a Bruker Avance 400 MHz NMR spectrometer at 298 K. 2D DOSY NMR was performed on a 600MHz Bruker Avance spectrometer. NMR samples were prepared by dissolving 15 mg of product in 0.5 mL of D2O. The chemical shifts were reported in part per million relative to tetra methylsilane. Spin multiplicity is shown by s=singlet, d=doublet, t=triplet, q=quadrat and m=multiple. DA of native chitosan and oligomers were determined using 1 H NMR. FTIR spectra were recorded on Nicolet 6700 FTIR spectrometer. GC-MS analysis were performed on GC 7890B AGILENT equipped with a capillary column Agilent 122-1364UI DB-624UI (60 m x 0.25 mm, 1.4 μm), with FID detector and mass spectrometer 5977B AGILENT. GC is directly linked to the headspace sampler 7697A Agilent, with and injection loop (1000 µL) and a transfer line. dn/dc were calculated using the differential refractive index (RI) detector Wyatt Optilab T-rEX. Molar masses were recorded using triple detection size exclusion chromatography (Tosoh TSK PW 2500, TSK PW 6000, TSK gel G Oligo PW) coupled with the same RI detector Wyatt Optilab T-rEX, multiangle light scattering Wyatt DAWN HELEOS II, and a Fused Silica cell, laser =664 nm. They were also recorded using size exclusion chromatography with RI detection (PL aquagel-OH 30 8µm 100-60.000 g/mol, PL aquagel-OH 40 8 µm 10.000-200.000 g/mol, TSK gel G Oligo PW 7µm 7.8 mm ID ×30 cm 0-6000 g/mol) calibrated using PEO/PEG standards. 10 mg/mL solutions of oligochitosans were prepared in acetate buffer (1,5% AcOH, 0.8% NaAc, 1 ppm NaN3, 5 ppm LiBr). Error were calculated by doing standard deviation on three different integration on each chromatogram and data plotted are the average value of those measurement. The pH (correction for isotopic effect: pD=pH+0.4) were measured by use of a Mettler Toledo pH electrode.

In order to validate the PEO equivalent method for molar masses measurement, triple detection SEC was performed. To do that, variation of refractive index according to concentration, dn/dc, were measured using a differential refractometer Wyatt Optilab T-rEX. 1mg/mL chitosan/oligochitosan solutions were prepared in degassed acetate buffer (AcOH 0.2M/0.15M ammonium acetate; pH=4.5) and diluted at 80%, 60%, 40%, 20%, 10% (w/w). Refractive indexes were measured according to solution concentration (658 nm, 0.5 mL/min). After that, 3mg/mL chitosan/oligochitosan solutions were prepared in the acetate buffer from above. 100µL of solution were filtrated on 0.45µm membrane CME filter and injected in the apparatus made of Tosoh TSK PW 2500 TSK PW 6000 coupled with a differential refractometer Wyatt Optilab T-rEX, multi angular light scattering Wyatt DAWN HELEOS II, and UV cellule Fused Silica, laser =664 nm. Molar masses (Mw, Mn and Ð) of chitosans and oligochitosans were determined taking in account the dn/dc Samples were prepared by dissolving oligomers in an ultra-pure water solution of anhydrous sodium hydrogen carbonate NaHCO3 (0.200 g/L), calcium chloride CaCl2 (0.260 g/L), magnesium chloride MgCl2, 6 H2O (0.148 g/L), anhydrous potassium sulfate K2SO4 (0.026 g/L). 100 mg/L solutions were prepared and diluted as followed: 35%, 10%, 3.5%, and 1% (w/w). Tests were performed at 20 °C.

Chitosan

RESULTS AND DISCUSSION

Depolymerization using NaNO2 was investigated from 5 different chitosans. 3 of them, obtained from crustaceous, have the same DA (i.e. 16%) and only differ by their Mw: 30, 150 and 250 kg/mol.

They will be labeled as CH30, CH150 and CH250. The fourth chitosan, also obtained from crustaceous and labeled CHDA1, has a Mw of 150 kg/mol with DA of 1%. The last studied chitosan, labeled CHY, has a Mw of 240 kg/mol, DA of 20% and was provided from insects.

During the course of depolymerization reaction, molecular weight evolution was followed by means of SEC using PEO/PEG standards. We have first decided to validate this method by also assessing Mw values by triple detection SEC. It was shown that dn/dc values are smaller than those of respective crude chitosans (Table 1). Moreover, even if the dn/dc values are very close for any oligomers of similar molar masses, this study shows that dn/dc values are different for all samples.

Therefore, it is very important to determine dn/dc before doing further analyses. Indeed, a small variation of the dn/dc value has a significant impact on the molar masses obtained, specifically with small objects.

Mw and Mn values for each chitosan oligomer, as well as the corresponding crude ones, are given in Table 1. It was especially shown that both methods afford rather close values when targeted DP below 20. Above this value and also for crude chitosans, SEC using PEO/PEG standards seems to underestimate the Mn values (25% lower than triple detection values). Nevertheless, this works aims at synthesizing COS oligomers with DP lower than 20. Thus we have decided to use SEC with PEO/PEG standards to monitor the Mn evolution for all chitosans.

At first, each chitosan was reacted with 0.2eq of NaNO2 at room temperature (30 °C) and molar masses decrease was followed over time ( g/mol after 30 min. According to the mechanism (scheme 1), this faster decrease can be easily explained by the fact that CHDA1 has more free amines available for reaction with NaNO2, thus leading to more glycosidic bond cleavages. Consequently, CHDA1 reaches a plateau after only 6 h, whereas 24h are needed for the four other chitosans. The kinetics of the reaction also depends on crude chitosan Mn: CH30 molar masses decreases faster than that of CH150 and CH250. This is probably due to the higher viscosity for higher Mn chitosans in solution. It was also shown that dispersity of the different chitosans decreases during the depolymerization process to reach interesting values ranging between 1.3 and 1.6 (Figure 2). This reaction was carried out at room temperature, as already described in the literature [START_REF] Tømmeraas | Preparation and Characterisation of Oligosaccharides Produced by Nitrous Acid Depolymerisation of Chitosans[END_REF] .

Nevertheless, an increase of the temperature should impact the reaction kineticss but also may lead to reaction completion, until reaching lower molar masses COS, i.e. DP<20.

The reaction was carried out with CH150, with 0.2 eq of NaNO2 at 4 different temperatures, i.e. 10, 30, 50 and 70°C (Figure 3). At 10°C, the reaction is not completed even after 24h. Indeed, at low temperature, viscosity of crude chitosan solution is high, which therefore decreases the reaction kinetics. Furthermore, NaNO2 could be consequently decomposed in nitrogen oxide by acetic acid instead [START_REF] Williams | Nitrosation Reactions and the Chemistry of Nitric Oxide[END_REF] . With a similar NaNO2/NH2 ratio, a temperature increase leads to faster depolymerization reaction. As expected, the reaction kinetics proceeded very fast at 70°C, and lower Mn COS were obtained. This could be explained by additional acidic depolymerization, due to the presence of acetic acid (pH 4,5) [START_REF] Tsao | Kinetic Study of Acid Depolymerization of Chitosan and Effects of Low Molecular Weight Chitosan on Erythrocyte Rouleaux Formation[END_REF] . When heated at 50°C, the plateau was reached after only 3h. The reaction was still carried out for 24h to ensure that no further degradation occurred. Consequently, the following experiments were carried out at 50°C during 3h. Allan et al. [START_REF] Allan | Molecular Weight Manipulation of Chitosan II: Prediction and Control of Extent of Depolymerization by Nitrous Acid[END_REF] already showed that the resulting Mw of COS was directly related to the content of NaNO2. Then CH30, CH150, CH250, CHDA1 and CHY were reacted at 50°C for 3h with different NaNO2/NH2 equivalents, and their Mn were assessed (Figure 4). First, we can note that the five chitosans behave similarly, i.e. the evolution of Mn changed at 0.2 eq. below this value, a strong Mn decrease was observed from crude chitosans to reach values above 2,500 g/mol, i.e. DP > 20. These To check their stability over time, the resulting COS with DP ranging from 5 to 20 were analyzed by SEC after 6 months (Figure 5). No decrease was observed even after 6 months, which proved that no further degradation occurred. The 10% difference between the two molar masses was explained by the change of calibration curve and evolution of the column. Finally, COS obtained with 0.2 eq. of NaNO2/NH2 were analyzed by FTIR and compared to the corresponding crude chitosans (SI 6). The carbonyl band at 1650 cm -1 is still visible in the oligomer spectra meaning that the acetylated units are still present. The similarity of both crude chitosan and oligomers by 1 H NMR analysis proves that no sugar units degradation occurred, as expected (SI 4 & 5). As a matter of fact, the DA is conserved during depolymerisation. mainly furan derivatives (peak 18, 19, 20, 21, 23, 24, 25, 27) (SI 8). However, no correlation between glucosamine and COS emissions were noticed, meaning no D-glucosamine was left at the end of the depolymerization process (SI 9).

Then, to assess the presence of those small molecules, 2D DOSY was performed (Figure 6).

This experiment allows differentiating the NMR signals of different species according to their diffusion coefficient. In ideal case of non-overlapping component lines and no chemical exchange, the 2D peaks align themselves along horizontal lines, each corresponding to one sample component.

This means that on the spectrum every signal belonging to the same molecule will appear on the same line. NaOH. 1 H NMR spectra were then recorded straight away at pH 2, 5.5 and 12 (Figure 7). At pH 5.5 both aldehyde and imine functions are clearly visible at 8.3 and 7.9 ppm, respectively. At pH 2, the peak corresponding to imine group totally disappeared, whereas the presence of HMF was characterized by the furanic protons peaks (7.4 and 6.5 ppm) as well as by the aldehyde proton peak (9.3 ppm), in agreement with Tømmeraas et al. [START_REF] Tømmeraas | Preparation and Characterisation of Oligosaccharides Produced by Nitrous Acid Depolymerisation of Chitosans[END_REF] work. The aldehyde peaks characteristic of both AMF and COS AMF chain-end were still visible (8.2 ppm). Intensity of both aldehyde peaks allows determining that 50% of aldehydes were converted into free HMF. Finally, when increasing pH to 12, the aldehyde signal (8,1 ppm) only remained and no imines were formed. and CH3CO2Na but also free AMF and HMF. Furthermore, during COS dialysis, acetic acid was extracted, and then the pH increased. 2D DOSY NMR spectrum of COS was performed after dialysis (Figure 8) and almost no free aldehyde was observed, i.e. from AMF (<5%) (SI 10).

Imine signals

After dialysis, ecotoxicity of 2 COS, i.e. DP 5 and 10, was evaluated. The experiment was reproduced 4 times on a population of 5 daphnia. It was shown that after 48h, the daphnia used for the test are still mobiles for DP5 and DP10 at concentration of 100 mg/L and below (Figure 9 & SI 11).

Those results assess that DP5 and DP10 are not acute toxic/ecotoxic and their CE50-24h and CE50-48h are above 100 mg/L.

1 H

 1 NMR of 1600 g/mol oligomer, D2O, 400 MHz, 298 K, pH 5,5: δ 8.2-8.5 (s, Haldehydes ), 8-7.1 (Himines ), 5.0 (d, 5 Hz, 1H H1 gemdiol), 3.2-4.4 (m, 1H H2 acetyl, 1H H3, 1H H4, 1H H5, 2H H6), 2.5 (m, 1H, H2), 2.0 (s, 3H, HCH3), 4.8 (s, HOD). 23 % of the amines are in imines form. 51% of the aldehydes (free or chain end) have reacted with an amine to form imines, 21% are in gem diol form and 28% are in aldehyde form (SI 3).

Figure 1 a

 1 Figure 1 a)). Mn of CH30, CH150, CH250 and CHY showed a similar trend, i.e. a very fast decrease during the first 30 min until approximately reaching 6,000 g/mol, then Mn slightly decreased until stabilization. It was shown that the reaction is immediate, the most important drop of molar masses occurs during the first minutes (

Figure 1 Figure 2

 12 Figure 1 a) Evolution of molar masses log (Mn) as a function of time for five different chitosans (1% w/w) with 0.2eq of NaNO2/NH2 b) Evolution of molar masses as a function of time for CH30 with 0.2 eq of NaNO2/NH2 during the first 30 min

Figure 3

 3 Figure 3 Evolution of Mn vs time at different temperatures for the reaction of CH150 with 0.2eq of NaNO2/NH2

Figure 4 .

 4 Figure 4. Evolution of Mn vs NaNO2/NH2 equivalents for the depolymerization of CH30, CH150, CH250, CHDA1 and CHY at 50°C for 3h.

Figure 5

 5 Figure 5 Evolution of Mn of oligochitosan CH30 with 0.4eq of NaNO2/NH2 from t0 to t0+6months

Figure 6 a

 6 ) shows 2D DOSY NMR spectrum of a 1600 g/mol COS obtained by nitrous deamination of CH30 at 50°C (0.4eq of NaNO2). Aldehyde signals are clearly highlighted, evidencing the presence of both COS AMF chain-end and free AMF in equilibrium with the gem-diol form of both products (4,5-5 ppm). These signals were then analyzed by 1 H-13 C 2D HSQC NMR (Figure 6 b)). The carbon signals near 180 ppm were specifically attributed to aldehydes function. Carbon signals between 140 and 180 ppm correlating to proton signals between 7 and 8 ppm are characteristic of imine groups. These imines resulted from the reaction between AMF (free or chain end) and amine functions of CH30. Then, if the pH drops, free HMF can be obtained, as already proved by Tømmeraas et al 21 .

Figure 6 a

 6 Figure 6 a) 2D DOSY NMR spectrum of a 1600 g/mol oligomer obtained from COS from CH30 + 0.4 eq NaNO2/NH2 (3h at 50°C) b) 1 H-13 C 2D HSQC NMR spectrum of the aldehyde zone

Figure 7 a) 1 HFigure 8

 718 Figure 7 a) 1 H NMR spectra of COS from CH30 + 0.4 eq NaNO2/NH2 (1600 g/mol) recorded in D2O at pH 12 (a), at pH 5.5 (b) and at pH 2 (c)

Figure 9 .

 9 Figure 9. Mobility of daphnia after 48h against sample concentration

Table 1 .

 1 DA, dn/dc, molar masses and dispersity of crude chitosan and chitosan oligomers obtained with action of NaNO2 during 3h at 50°C.

		DA (%)	dn/dc	Mw (kg/mol)	Mn (kg/mol)	Dispersity	DPn
		Labeled Measured		PEO/PEG equivalent	Triple detecti on	PEO equivalent	Triple detection	PEO/PEG equivalent	Triple detection	PEO/PEG equivalent	Triple detection
	CH30	≥10	16 ± 2	0.1782 ± 0.0022	40 ± 4	44 ± 4,4	13 ± 1	21 ± 2	3 ± 0.3	2 ± 0.2	77 ± 6	125 ±
	CH150	≥10	16± 2	0.1828 ± 0.0031	163 ± 16	150 ± 15	96 ± 10	80 ± 8	3 ± 0.3	2 ± 0.2	571 ± 60 476 ±
	CH250	≥10	16± 2	0.1893 ± 0.0012	227 ± 23	370 ± 37	200 ± 20	192 ± 19	1.9 ± 0.2 2 ± 0.2	1190 ± 120	1143 ± 113
	CHY	≥10	20± 2	0.1911 ± 0.0027	120 ± 12	240 ± 24	120 ± 12	170 ± 17	1.7 ± 0.2 1.6 ± 0.2 706 ± 70	1000 ± 100
	CHDA1	≥1	1± 1	0.1980 ± 0.0009	100 ± 10	150 ± 15	62 ± 6	80 ± 8 2.1 ± 0.2 2 ± 0.2	383 ± 36 493 ±
	Oligomer
		Equivalent of
		NaNO2/GlcN		Mw(g/mol)	Mn (g/mol)	Dispersity	DPn
			unit
	CH30		0.4	0.1449 ± 0.0017	2,500 ± 250	2,750 ± 275	1,850 ± 185	2,500 ± 250	1.4 ± 0.1	1.1 ± 0.1	11 ± 1	15 ± 2
	CH30		0.2	0.1510 ± 0.0045	3,900 ± 390	4,150 ± 415	2,350 ± 235	3,500 ± 350	1.6 ± 0.2	1.2 ± 0.1	14 ± 1	21 ± 3
	CH30		0.1	0,1539 ± 0.0009	7,950 ± 795	7,900 ± 790	5,050 ± 505	8,100 ± 810	1.6 ± 0.2	1.2 ± 0.1	30 ± 3	48 ± 5
	CH150		0.4	0.1439 ± 0.0038	2,600 ± 260	2,800 ± 280	1,850 ± 185	2,500 ± 250	1.4 ± 0.1	1.1 ± 0.1	11 ± 1	15 ± 1
	CH150		0.2	0.1576 ± 0.0007	4,400 ± 440	3,450 ± 345	2,850 ± 285	3,000 ± 300	1.6 ± 0.2	1.2 ± 0.1	17 ± 2	18 ± 2
	CH150		0.1	0.1512 ± 0.0015	10,200 ± 1020	11,800 ± 1180	6,200 ± 620	10,100 ± 1010	1.6 ± 0.2	1.2 ± 0.1	37 ± 3	60 ±6
	CH250		0.4	0.1366 ± 0.0034	2,600 ± 260	3,000 ± 300	1,850 ± 185	2,500 ± 250	1.5 ± 0.2	1.1 ± 0.1	11 ± 1	15 ± 1
	CH250		0.2	0.1423 ± 0.0027	4,200 ± 420	3,800 ± 380	2,700 ± 270	3,500 ± 350	1.6 ± 0.2	1.2 ± 0.1	16 ± 2	21 ± 2
	CH250		0.1	0.1399 ± 13,600 17,400	7,600 ±	13,900	1.8 ± 0.2	1.2 ±	45 ± 5	83 ± 8

Toxicity and ecotoxicity of oligomers post dialysis (DP5 and DP10) were evaluated according to acute tests on Daphnia magna (NF EN ISO 6341, 2012) performed by Eurofins. This test aims at finding the concentration immobilizing 50% of the daphnia population after 24h and 48h (CE50). A solution of K2Cr2O7 CE50-24h = 0,94 mg/L (this value corresponds to standard between 0.60 mg/L and 2,1 mg/L according to NF EN ISO 6341 : 2012) is used as a reference substance.

determined above. DPn were calculated according to the formula (1):

Funding Sources

This work was funded by a private funding and a CIFRE grand from the ANRT.

Notes

The authors declare no conflict of interest.

ACKNOWLEDGMENT

The authors want to thank Agnes CREPET for triple detection SEC and data analysis and Aurelien LEBRUN for performing 2D NMR analysis. ABBREVIATIONS SEC: size exclusion chromatography; FTIR: Fourier transform infrared spectroscopy; DA: degree of N-acetylation; COS: chitooligosaccharides; AMF: 2,5 anhydromannofuranose HMF: 5hydroxymethyl-2-furfural; DP degree of polymerization

the end, the purified oligomers show no acute toxicity or ecotoxicity, which make them suitable for various interesting applications.

SUPPORTING INFORMATION

 1 H NMR spectrum of oligomer 1600 g/mol from CH30 + 0.4 eq NaNO2 /NH2

 1 H NMR spectrum of crude chitosan CH30  1 H NMR spectrum of chitosan oligomer from CH30 + 0.2 eq NaNO2 /NH2

 IR spectra of both chitosan CH30 and chitosan oligomers of 1,600 g/mol  Chromatograms from headspace chromatography of AMF, glucosamine and chitosan oligomers and their respective attribution data.