Phuc Do
email: phuc.do@univ-lorraine.fr

Christophe Bérenguer

Residual life-based importance measures for predictive maintenance decision-making

Keywords: Reliability, maintenance decision-making, importance measure, mean residual life, multicomponent system, economic dependence

Importance measures have been widely used as meaningful decision-aiding indicators in reliability engineering, risk management and maintenance optimization. However, few importance measures integrates the actual condition (working states or degradation levels) of components that dynamically evolves with time. This work develops a novel time-dependent importance measure defined as the capacity of a component (or group of components) to improve, when it is replaced, the system residual life. The proposed IM M RL measure can help to better prioritize a component or group of components regarding to its improvement ability in the system life time while considering the actual conditions of all components of the system. The originality and complementarity of the proposed measure when compared to existing importance measures is also investigated. The proposed importance measure is then extended to integrate the economic dimension of the maintenance decision, through the maintenance costs, the benefit gained by the maintenance operations and as well as the economic dependence between components. It is finally shown how the proposed IM M RL measure and its extension can "optimally" suggest a component or a group of several components for preventive maintenance decision-making, based on both the technical criterion (residual life of the system) and the economic aspects (benefit and costs). The use and the advantages of the proposed importance measure and its extension are illustrated on a 4-component system.

Introduction

Within the framework of system engineering, importance measures are used to quantify the importance of a component or a group of components on the key performances of interest of a system, such as its reliability, its availability, its residual life, its productivity or its safety. These importance metrics can support decision-making to choose between several system upgrade options or design modifications for correcting identified weaknesses, to select the most relevant maintenance actions or to identify and avoid operation bottlenecks. Different importance measures have been proposed in the field of reliability, risk and maintenance and they have been successfully used to solve many real-world problems, see [START_REF] Kuo | Importance Measures in Reliability, Risk, and Optimization: Principles and Applications[END_REF] for a comprehensive overview of the classical importance measures and their applications. In risk analyses and reliability engineering, importance measures have been used for risk-informed decision-making [START_REF] Borgonovo | A new importance measure for risk-informed decision making[END_REF][START_REF] Cheok | Use of importance measures in risk informed regulatory applications[END_REF] or for ranking components within a system for reliability improvement purposes [START_REF] Beeson | Importance measures for non-coherent-system analysis[END_REF][START_REF] Birnbaum | On the importance of different elements in a multielement system. multivariate analysis[END_REF][START_REF] Borgonovo | A new time-independent reliability importance measure[END_REF][START_REF] Do | Conditional reliability-based importance measures[END_REF]. Recently, several applications of importance measures have been developed aiming at maintenance optimization and spare parts management [START_REF] Nguyen | Joint predictive maintenance and inventory strategy for multicomponent systems using birnbaum's structural importance[END_REF][START_REF] Vu | A stationary grouping maintenance strategy using mean residual life and the birnbaum importance measure for complex structures[END_REF][START_REF] Wu | Linking component importance to optimisation of preventive maintenance policy[END_REF]. For example, for complex multi-component systems, a maintenance optimization approach based on the Birnbaum structural importance measure has been developed in [START_REF] Nguyen | Condition-based maintenance for multi-component systems using importance measure and predictive information[END_REF][START_REF] Vu | A stationary grouping maintenance strategy using mean residual life and the birnbaum importance measure for complex structures[END_REF]. [START_REF] Borgonovo | Differential importance and comparative statics: An application to inventory management[END_REF] has proposed to resort to a differential importance measure for inventory management. More recent works have addressed the issue of basing the preventive maintenance decisions on importance metrics of the maintained system's components [START_REF] Do | Conditional reliability-based importance measures[END_REF][START_REF] Wu | Linking component importance to optimisation of preventive maintenance policy[END_REF][START_REF] Zhang | A heuristic policy for maintaining multiple multi-state systems[END_REF].

With the aim of optimizing condition-based or predictive maintenance decisions on a system, the available information on the actual condition of components (states or degradation levels) is obviously a critical input to the decision process and should be considered in the synthesis of decision-aiding indicators. Following this idea, a data-driven framework integrating the deterioration monitoring information has been recently introduced to identify the critical components of a complex technical infrastructure in [START_REF] Lu | A data-driven framework for identifying important components in complex systems[END_REF] and a new importance measure, called condition-based importance measure, has been recently proposed in [START_REF] Do | Conditional reliability-based importance measures[END_REF]. It relies on the quantification of the reliability improvement for a given finite horizon (t, t+u) brought by a component replacement and it integrates the available information on the actual condition of all the system components at decision time point t. This importance measure proved to be relevant for ranking the system components with respect to a reliability metric on a finite horizon. However, it cannot be directly applied for an infinite horizon reliability metric such as the system residual life (or the remaining useful life) that is nevertheless considered as an important reliability metric for various decision-making, see for instance [START_REF] Do | Remaining useful life (rul) based maintenance decision making for deteriorating systems[END_REF][START_REF] Rausand | System Reliability Theory -Models, Statistical methods and Application[END_REF][START_REF] Vu | MRL, T) grouping policy for complex structure systems[END_REF]]. To answer this gap, this work proposes a novel importance measure based on the system residual life, namely IM M RL measure, that can be used to rank components/groups of components with respect to their ability to improve the system residual life considering all the available information of the actual conditions of all components at the decision time point. To highlight the originality and complementarity of the proposed IM M RL measure when compared to existing ones, some direct links between these already existing measures and the proposed one are also highlighted and investigated.

Moreover, when implementing maintenance operations in practice, an economic dependence often arises : in this case, the joint maintenance of several components turns out to be cheaper than maintaining separately the components. Such an economic dependence in maintenance should be integrated appropriately in the maintenance decision-making. However, to the best of our knowledge, very few existing importance measures designed to support maintenance decisions actually integrate this kind of economic dependence between components. To fill this second gap, an extension of the proposed IM M RL measure is thus proposed in this work to take into account the economic aspects related to maintenance, i.e. costs incurred by maintenance actions, with the economic dependence, and benefit gained from maintained system operation. The proposed IM M RL measure and its extension allow ranking and selecting the best candidate component or group of components for maintenance according to both a technical criterion (residual life of the system) and/or economic considerations (benefit and cost).

In summary, the contribution of the work presented in this paper is twofold : (i) it first proposes a component importance measure elaborated from the estimated improvement in the mean system residual life brought by the as-good-as new replacement of the component, given the available information on the actual condition of all the system components ; (ii) the proposed importance measure is then extended to integrate economic aspects related to maintenance costs and system profit. This paper develops an extended version of the work presented in [START_REF] Do | MRL-based importance measures[END_REF].

The remainder of this paper is organized as follows. Section 2 is devoted to the presentation of the main modelling assumptions of this work. The different considered levels for the monitoring information and the mean residual life are presented and discussed. In addition, the impact of the information levels on the estimated system residual life is also highlighted. The proposed IM M RL measure is introduced in Section 3. The link with two existing importance measures in different contexts related to given components' information levels is discussed. In Section 4, an extension of IM M RL measure integrating economic aspects is developed. The economic dependence between components is also studied. To illustrate the use and the advantages of IM M RL measure and its extension, Section 5 presents a numerical example of a 4-component system. In addition, various numerical studies are herein analyzed and discussed. Finally, the conclusions drawn from this work and some perspectives are presented in the last section. We consider a coherent structured system consisting of n non-identical components that are organised according to a structure assumed to be known (e.g., series, parallels or mix, etc.). Each component may fail because of an underlying random ageing phenomenon or a stochastic degradation process.

The failures of the components are assumed to be independent.

Only perfect maintenance actions or replacements are assumed to be implementable on the system, i.e., performing a maintenance action on a component brings the maintained component into an "asgood-as new" state. In addition, each maintenance action incurs a maintenance cost.

Component failure modelling and system reliability assessment

Component failure modelling In reliability engineering, the reliability of an entity (component or system) is defined as the probability that the entity performs its mission without any failure until time t. To assess the reliability of a component, two main approaches are classically used: lifetime models or degradation-based failure models [START_REF] Nguyen | Joint predictive maintenance and inventory strategy for multicomponent systems using birnbaum's structural importance[END_REF][START_REF] Rausand | System Reliability Theory -Models, Statistical methods and Application[END_REF]. The use of either model depends on the level of information available on the state of the considered component. Lifetime models are suitable to describe the situation where the information on the component degradation levels is not available for technical and/or economical reasons (e.g., it is not allowed or impossible to inspect in-depth the component to reveal its degradation level; its inspection operation is too expensive or the component is highly reliable and its inspection is not considered to be worthwhile). In this situation, the component is only known to be working or failed. On the contrary, when the degradation level of the component can be monitored more closely and finely, a degradation-based failure model based on a stochastic process is more informative and is often preferable. Accordingly, the two following cases can be considered:

• Components whose failure behavior is described by a lifetime model are referred to as "ageing components". The reliability of the ageing component i can be written

r i (t) = P(τ i > t) = 1 - t 0 f τ i (x) dx (1)
where τ i is the time to failure of component i and follows a probability distribution with probability density function f τ i (x). As an example, the Weibull distribution which has been widely used in the literature for reliability modeling at component level [START_REF] Rausand | System Reliability Theory -Models, Statistical methods and Application[END_REF], is used for reliability assessment of certain components in this work. Of course, other probability laws could be used in a similar manner. A detailed description of the Weibull distribution is given in Appendix A.

• Components that gradually degrades with wear and tear, and whose degradation can be monitored are referred to as 'degrading components". Assuming that the degradation evolution of the degrading component i is described by a stochastic process X i t , its reliability can be then written as:

r i (t) = P(X i t < L i ) = L i 0 f X i t (x)dx, (2)
where L i is the degradation failure threshold above which the component i is considered as failed (i.e., when X i t ≥ L i , the component i is failed) and f X i t (u) is the probability density function of the degradation level of component i at time t. In the literature, different degradation stochastic processes are available such as gamma, Wiener, etc [START_REF] Rausand | System Reliability Theory -Models, Statistical methods and Application[END_REF][START_REF] Van Noortwijk | A survey of the application of Gamma processes in maintenance[END_REF]. In this work, a gamma stochastic process is used as an example for reliability assessment of degrading components. A detailed description is given in Appendix B.

System reliability Because the components failures are assumed to be statistically independent, the system reliability can be computed from the components' reliabilities using the system reliability structure function. More precisely, let R(t) be the system reliability. R(t) is a function of the reliabilities of all the system components r i (t) with i = 1, 2, ...n : R(t) = ϕ(r 1 (t), r 2 (t), ..., r n (t)).

(

) 3
The structure function ϕ(.) can be obtained from the minimal path sets or the minimal cut sets of the system [START_REF] Rausand | System Reliability Theory -Models, Statistical methods and Application[END_REF]. As an example, the reliability of a series structure of two component is given by Eq.( 11)

and the reliability of four-component system with a mixed structure is provided in Section 5.

Level of information and mean residual life

Level of information and conditional reliability assessment

Let S t be the state of the system at time t. As only binary states are considered at the system level, S t can be expressed as follows

S t =    1 if the system is functioning at time t; 0 otherwise.
It is assumed that at initial time, t = 0, the system is new S 0 = 1. At time t, two cases can be distinguished:

• if the system is in a failed state, S t = 0: R(t + u|S t = 0) = 0.

• if the system is functioning at time t, S t = 1, the predicted conditional reliability of the system at time t + u (with u > 0) can be computed using the available information at time t from Eq.( 3) by replacing the components reliability r i (t), with i = 1, 2, ..., n, by the components conditional reliability r i (t+u|Z i t ) where Z i t represents the available information about the state of component i at time t.

It should be noticed that the available information about the state of components is crucial for predicting the components' reliability. For degrading components, four levels of information can be considered:

Z i t =               
0 if component i is known to be in a failed state at time t;

1 if component i is known to be functioning at time t but its degradation level is not available;

2 if component i is known to be functioning and its degradation level is measured at time t;

3 if component i is known to be replaced by a new component at time t.

Accordingly, given the information level available at time t on a component state, the conditional reliability of the component can be calculated by distinguishing the following 4 cases :

• If component i is failed at time t, Z i t = 0, its conditional reliability at t + u is equal to zero (no replacement is assumed between t and t + u): r i (t + u|Z i t = 0) = 0;

• If component i is still working at time t but no information on its degradation level is available (because of a technical or economical reason), Z i t = 1, its conditional reliability at t + u writes

r i (t + u|Z i t = 1) = r i (t + u) r i (t) . (4)
It is worth noting that in the case of an exponential lifetime law for component i, i.e. its failure rate is time-independent, its conditional reliability can be calculated by r i (t + u|Z i t = 1) = r i (u), which means that component i can be considered as new if it is known that it has survived time t, and thus no preventive maintenance is required.

• If component i is still working and its monitored degradation level is available at time t, Z i t = 2, its conditional reliability at t + u is then given by

r i (t + u|Z i t = 2) = P(X i t+u < Z i |X i t = x i t ) = Z i x i t f X i t+u (x)dx, (5)
where X i t = x i t denotes the degradation level of component i at time t.

• If component i is replaced by a new one at time t, Z i t = 3, its conditional reliability writes simply

r i (t + u|Z i t = 3) = r i (u). ( 6
)
The conditional reliability assessment of an ageing component remains similar to that of a degrading component, noting however that for an ageing component i, according to the definition used in this work, only three levels of information on the component state are considered, i.e., Z i t can be 0, 1 or 3 (no degradation level information is ever available for degrading components).

It must be noticed that for industrial systems, the replacement of component i may require disassembling other components, this phenomenon is called structural dependence [START_REF] Nicolai | Optimal maintenance of multi-component systems: A review[END_REF]. The disassembly operations may affect the failure/degradation processes of certain disassembled components. Such a structural dependence is not considered in this work, but for further discussion on the disassembly impacts, the interested reader can consult [START_REF] Dinh | Degradation modeling and reliability assessment for a multicomponent system with structural dependence[END_REF].

Mean residual life

The residual life (or remaining useful life) of a system is an important reliability metric of the system since it provides the information about the duration left before the system fails given the condition of its components at time t [START_REF] Byington | Prognostic enhancements to diagnostic systems for improved condition-based maintenance[END_REF][START_REF] Rausand | System Reliability Theory -Models, Statistical methods and Application[END_REF][START_REF] Xia | Recent advances in prognostics and health management for advanced manufacturing paradigms[END_REF]. Even though the full probabilistic characterization of the random resiudal life contains more information, the mean residual life (MRL) can still be used as an appropriate reliability metric for various decision-making [START_REF] Do | Remaining useful life (rul) based maintenance decision making for deteriorating systems[END_REF][START_REF] Rausand | System Reliability Theory -Models, Statistical methods and Application[END_REF][START_REF] Vu | MRL, T) grouping policy for complex structure systems[END_REF]. Indeed, the mean residual life of a system at time t is defined as the mean time left before the system failure.

Let Z t = (Z 1 t , Z 2 t ), .
..Z n t ) be the vector representing the information about all components of the system at time t. The mean residual life of a system at time t is mathematically expressed as follows:

M RL(t|Z t ) = ∞ 0 R(t + u|Z t )du. ( 7
)
Obviously M RL(t|Z t ) depends not only on the time t but also on level of the available information on the components' state at time t. In addition, M RL(t|Z t ) has the following properties:

• M RL(t|Z t ) ≥ 0 and M RL(t|Z t ) = 0 if the system fails at time t, i.e., S t = 0;

• At the initial time point t = 0, all the components are new ( Z i t = 3 with i = 1, ..., n), we get:

M RL(0|Z 1 0 = 3, Z 2 0 = 3, ..., Z n 0 = 3) = ∞ 0 R(u)du = M T T F, (8)
where M T T F represents the mean time to failure of the system [START_REF] Rausand | System Reliability Theory -Models, Statistical methods and Application[END_REF];

• For a coherent system, without any maintenance intervention M RL(t|Z t ) is a decreasing function over time, i.e., M T T F ≥ M RL(t

1 |Z t 1 ) ≥ M RL(t 2 |Z t 2 ) with t 1 ≤ t 2 ;
• If all the components of the system are replaced simultaneously at time t, i.e. the system is renewed at time t, we get:

M RL(t|Z 1 t = 3, Z 2 t = 3, ..., Z n t = 3) = M T T F. (9)
Assume that at time t, only component i is replaced, the mean residual life of a system is then:

M RL(t|Z t , Z i t = 3) = ∞ 0 R(t + u|Z t , Z i t = 3)du. ( 10
)
Example 1 For illustration purposes, consider the simple example of a 2-component series system.

Both components of the system, C1 and C2, are assumed to undergo a degradation process that can be described by a gamma process with a shape and a scale parameter (α i , β i ) with i = 1, 2 (see Appendix B for a detailed description of the gamma process). Both components fail as soon as their degradation level reach their failure threshold, resp L 1 = L 2 = 100. The system reliability block diagram and the values of the parameters are given in Figure 1.

C1 C2 a1 b1 a2 b2

Figure 1: A 2-component system.

The reliability of the system is obtained readily as:

R(t) = r 1 (t).r 2 (t). ( 11
)
To illustrate the impact on the estimated mean residual life of the level of the available information on the components states, we consider the three following cases:

• Both components are monitored and their degradation state is measured and available at each time unit, i.e. Z 1 t = Z 2 t = 2;

• Component C2 is monitored and its degradation state is measured and available at each time unit, however no inspection operation is ever realised on component C1 due to a technical or economical reason, i.e. Z 1 t = 1, Z 2 t = 2;

• Neither of the two components C1 and C2 is monitored and their state of degradation remains unknown, i.e.

Z 1 t = Z 2 t = 1.
Figure 2 shows an example of the degradation paths of each component and the estimated mean residual life M RL(t|Z t ) of the system according to the three mentioned cases of information level.

This example illustrates the fact that the estimated mean residual life M RL(t|Z t ) depends on the information levels of both components C1 and C2. Ignoring the degradation levels of one or two components may lead to a significant error in the evaluation of M RL(t|Z t ). As the mean residual life is an important reliability metric of the system and as it allows to embed in its estimation process different levels of information, in this study it is used to define a novel importance measures. The detailed discussion is given in the following section.

3 Importance measures based on the mean residual life

Definition of IM M RL measure

An importance measure based on the mean residual life can be defined as follows for component i:

IM i M RL (t) = M RL(t|Z t , Z i t = 3) -M RL(t|Z t ). ( 12
)
IM i M RL (t) quantifies the improvement in the system mean residual life time brought by the replacement of component i at time t, given the available information of the actual condition of all the other system components. The IM M RL measure is characterized by the following properties:

• For a coherent system M RL(t|Z t , Z i t = 3) ≥ M RL(t|Z t ), and IM i M RL (t) is thus non negative and IM i M RL (t) ≤ M T T F :

-IM i M RL (0) = 0;
-IM i M RL (t) = 0 if component i is known to be still working at time t and its failure rate is time-independent. This means that the replacement of component i does not have any impact on the residual life time of the system, which leads to the well-known result that it is not necessary to replace component i with a constant failure rate if it is kown to be still working at time t;

-IM i M RL (t) = M T T F if the system is composed of a single component (n = 1
) and the component is in a failed state at time t.

• IM i M RL (t) depends on the available information on the actual state or degradation level of all the components at time t;

• Finally, IM i M RL (t) is time-dependent if the failure/degradation process of one or several components is time-dependent.

From Eq.( 7) and Eq.( 10), IM i M RL (t) can be written as:

IM i M RL (t) = ∞ 0 R(t + u|Z t , Z i t = 3)du - ∞ 0 R(t + u|Z t )du = ∞ 0 [R(t + u|Z t , Z i t = 3) -R(t + u|Z t )]du (13)
In the purpose of condition-based maintenance decision-making, as IM M RL measure allows integrating the available information on the current actual condition (state or degradation level) of all components, it can be a relevant indicator to identify the most important component that should be replaced, at given time t, to improve the system residual life.

Example 1 (continued) Reconsider now the example of 2-component system shown in Figure 1 and assume that the system is still functioning at time t, i.e. both components have survived t, Z i t = 0 with i = 1, 2. The system mean residual life in the absence of any maintenance action at time t is given by:

M LR(t|Z 1 t =, Z 2 t = 0) = ∞ 0 r 1 (t + u|Z 1 t = 0).r 2 (t + u|Z 2 t = 0)du.
The IM M RL measure of component i (i = 1, 2) writes as:

IM i M RL (t) = ∞ 0 r 3-i (t + u|Z 3-i t = 0).[r i (u) -r i (t + u|Z i t = 0)]du. ( 14
)
Two different cases can be distinguished depending on the level of the available information on both components at time t:

Case 1 Both components are functioning at time t but their degradation levels are not available, i.e.

Z 1 t = Z 2 t = 1.
In this case, Table 1 shows the value of the IM M RL measure for each component obtained using Eq.( 14) and Eq.( 4) : component C2 turns out to be more important than C1, which leads to conclude that the replacement of C2 is more effective than the replacement of C1 to improve the system mean residual life.

Case 2 Both components C1 and C2 are monitored and the information on their degradation level is available, i.e. Z 1 t = Z 2 t = 2. Both components being subject to a stochastic degradation process, the degradation level of each component at time can be random and two different degradation path realizations are considered in this example, refered to as respectively Case 2a and 2b. The different degradation levels of both components at time t = 10 in each case (2a and 2b) are reported in Table 1 and are used to compute in each case the value of IM M RL measure for both components at time t = 10, using Eq.( 14) and Eq.( 5). For the first realization of the degradation path (Case 2a), IM 1 M RL (t) is greater than IM 2 M RL (t) and component C1 is thus more important than component C2 in improving the system mean residual life. It should be noticed that this importance ranking is not the same as the one observed when the components' degradation levels are unknown. On the contrary, for the second realization of the degradation path (case 2b), IM 1 M RL (t) < IM 2 M RL (t), i.e. C2 turns out to be more important than C1.

Information at t = 10 IM M RL measure at t = 10 Ranking These experimental results show that IM M RL is time-dependent and both the information level of the components and their degradation levels are crucial for the IM M RL 's values and the associated components importance ranking.

Z 1 t X 1 t Z 2 t X 2 t IM 1 M RL (t) IM 2 M RL (t) C1 C2 Case 1 1 - 1 - 2.

Link with existing importance measures

In this section, the link between the proposed importance measure IM M RL with two existing importance measures, Birnbaum importance measure [START_REF] Birnbaum | On the importance of different elements in a multielement system. multivariate analysis[END_REF] and conditional reliability-based importance measure [START_REF] Do | Conditional reliability-based importance measures[END_REF], is discussed to highlight the originality and the complementarity of the proposed IM M RL measure with respect to existing ones. In addition, constructing these links allows also evaluating one measure when other ones are known.

Birnbaum importance measure

It is shown [START_REF] Rausand | System Reliability Theory -Models, Statistical methods and Application[END_REF] that the reliability of the system can be expressed as a linear function of the reliability of component i as follows:

R(t) = r i (t).[R(t|Z i t = 1) -R(t|Z i t = 0)] + R(t|Z i t = 0), (15)
where,

• R(t|Z i t = 1) is the probability that the system is still surviving at time t when component i is functioning at time t. For a parallel structure system, R(t|Z i t = 1) = 1;

• R(t|Z i t = 0) is the probability that the system is functioning at time t given that component i is in a failed state. R(t|Z i t = 0) = 0 if component i is a critical one i.e., a failure of component i leads to a failure of the whole system [START_REF] Rausand | System Reliability Theory -Models, Statistical methods and Application[END_REF].

The Birnbaum' importance measure, that is defined as the partial differentiation of the system reliability with respect to the reliability of a component, is the most popular importance measure in ranking components for various decision-making purposes [START_REF] Birnbaum | On the importance of different elements in a multielement system. multivariate analysis[END_REF][START_REF] Kuo | Importance Measures in Reliability, Risk, and Optimization: Principles and Applications[END_REF][START_REF] Rausand | System Reliability Theory -Models, Statistical methods and Application[END_REF].

From Eq.( 15), Birnbaum's importance measure for component i is mathematically defined as:

IB i (t) = ∂R(t) ∂r i (t) = R(t|Z i t = 1) -R(t|Z i t = 0). ( 16
)
By definition, IB measure is bounded, i.e., 0 ≤ IB i (t) ≤ 1 with i = 1, .., n. The larger IB i (t), the bigger is the impact in the reliability of the system when the reliability of component i changes.

Therefore, IB measure can be seen as a sensitivity importance measure [START_REF] Rausand | System Reliability Theory -Models, Statistical methods and Application[END_REF].

Note also that IB measure is time-dependent. If the system information is available at time t, IB measure can be extended as follows:

IB i (t + u|Z t ) = ∂R(t + u|Z t ) ∂r i (t + u|Z i t ) = R(t + u|Z i t+u = 1, Z t ) -R(t + u|Z i t+u = 0, Z t ). ( 17
)
An illustration of the evolution of IB measure for a two-component system is shown in Figures 4 and5.

By using Eq.( 15) and Eq.( 17), the conditional reliability of the system at time t + u given the condition/information of system components at time t, Z t , can be written as:

R(t + u|Z t ) = r i (t + u|Z t ).IB i (t + u|Z t ) + R(t|Z i t = 0). ( 18
)
From Eq.( 13) and Eq.( 18), we obtain a relationship between the proposed IM M RL measure and Birnbaum's measure as follows:

IM i M RL (t) = ∞ 0 IB i (t + u|Z t ).(r i (u) -r i (t + u|Z i t )du (19)
This allows evaluating IM M RL measure from Birnbaum's measure.

Conditional reliability-based importance measure

Recently, an importance measure, namely conditional reliability-based importance measure, has been proposed for ranking components and preventive maintenance decision-making for a given mission [START_REF] Do | Conditional reliability-based importance measures[END_REF]. Indeed, the condition reliability-based importance measure for a component i is mathematically defined as follows:

RIM i (u, t) = R(t + u|Z t , Z i t = 3) -R(t + u|Z t ). ( 20
)
By definition, RIM measure quantifies the potential improvement in the system reliability within the horizon (t, t + u) if component i is replaced at time t [START_REF] Do | Conditional reliability-based importance measures[END_REF] considering the available information related to the current states of all components at time t. Furthermore, RIM measure is time-dependent and bounded, i.e., 0 ≤ RIM i (u, t) ≤ 1 with i = 1, .., n. RIM i (u, t) = 0 means that the replacement of component i at time t does not lead to any change in the system reliability at time t+u. RIM i (0, t) = 1 if component i is critical and it is in a failed state at time t.

It is also pointed out in [START_REF] Do | Conditional reliability-based importance measures[END_REF], RIM measure is an adequate measure for ranking components and preventive maintenance decision-making at time t regarding to a given reliability level of the system at t + u. For further discussions, the interested reader can refer to [START_REF] Do | Conditional reliability-based importance measures[END_REF].

From Eq.( 13) and Eq.( 20), we get a direct link between the proposed importance measure IM M RL and RIM measure as follows:

IM i M RL (t) = ∞ 0 RIM i (u, t)du (21)
This relationship shows that RIM i (u, t) quantifies the impact on the system reliability at time t + u when component i is replaced at time t while IM i M RL (t) measures the impact of this replacement on the whole residual life time of the system, infinite horizon. This means that RIM and IM M RL measures are complementary in ranking components and/or maintenance decision-making since both reliability and residual life time are two important reliability metrics of industrial systems [START_REF] Rausand | System Reliability Theory -Models, Statistical methods and Application[END_REF].

Example 1 (continued) As an example, we reconsider the system of two components shown in Figure 1. The mathematical formulation of IB, RIM and IM M RL measures for each component and the relationships between the three measures are reported in Table 2.

Component IB i (t|Z t ) RIM i (u, t) IM i M RL (t) C1 r 2 (t + u|Z 2 t ) IB 1 (t|Z t ).(r 1 (u) -r 1 (t + u|Z 1 t )) ∞ 0 RIM 1 (u, t)du C2 r 1 (t + u|Z 1 t ) IB 2 (t|Z t ).(r 2 (u) -r 2 (t + u|Z 2 t )) ∞ 0 RIM 2 (u, t)du
Table 2: IB, RIM and IM M RL measures for a series structure system of two components.

Figure 4 sketches IB and RIM measures for each component at time t + u given that two components are functioning at time t = 10 but their degradation levels are unknown. The results show that, according to IB measure or RIM measure, When 30 ≤ u ≤ 50, both IB and RIM measures lead to the same importance ranking, C2 > C1. However, However, it should be pointed out that RIM measure leads to a more contrasted decision for the ranking of both components. This can be explained by the fact that RIM of a component allows considering the current condition of the component while IB does not. However, when u < 30 or u > 50, both IB and RIM do not allow ranking the two components while the proposed IM M RL measure provides only one importance ranking, C2 > C1, see again Table 1 (Case 1). In the same manner, we assume now the degradation levels of two components at time t = 10 are measured. Figure 5 shows IB and RIM measures for each component at time t + u given degradation levels of two components at time t = 10: X 1 t = 21.76, X 2 t = 12.82 (see again Case 2a of Table 1). Similar results are herein obtained. Note however that, according to IB or RIM measure, when 25 ≤ u ≤ 55, component C1 is more important than C2 (C1 > C2). In addition, when u < 25 or u > 55 both IB and RIM do not allow clearly ranking the two components. When compared to the result given in Table 1 for this case, the proposed IM M RL measure provides only one importance ranking, C1 > C2. According the obtained results, it is pointed out that, in this example, IM M RL measure seems to be better than IB and RIM measures in ranking components since IM M RL leads to an unique components ranking for any value of u while IB and RIM measures may lead to different components ranking with different values of u.

IM M RL of a group of components

The proposed IM M RL measure can also applied to a group of several components. Indeed, without loss of generality, we consider a group G k including k components, denoted {j 1 , ..j k } with k = 2, 3, ..., IM M RL measure of group G k at time t can be mathematically defined as follows:

IM {j 1 ,..j k } M RL (t) = M RL(t + u|Z t , {Z j 1 t = 3, ..., Z j k t = 3}) -M RL(t|Z t ), (22)
where M RL(t + u|Z t , {Z j 1 t = 3, ..., Z j k t = 3}) is the system mean residual life when k components {j 1 , ..j k } are jointly replaced at time t considering the actual condition of the other components of the system.

According to the definition, IM {j 1 ,..jk} M RL (t) quantifies the residual life improvement of the system thanks to the joint replacement of group G k at time t. Note that if all components of the system are jointly replaced at time t, i.e., the system is renewed, the maximum improvement in the system life time is reached.

It should be noticed that that if components are structurally dependent, the joint replacement of several component can reduce the disassembly impacts on the failure/degradation processes of other components [START_REF] Dinh | Degradation modeling and reliability assessment for a multicomponent system with structural dependence[END_REF].

Due to joint effects in the system residual life time when group of components are replaced together,

IM M RL is not additive, i.e., IM {j 1 ,..j k } M RL (t) = k l=1 IM {j 1 } M RL (t).
It is pointed out in [START_REF] Do | Conditional reliability-based importance measures[END_REF] that, RIM measure can also be applied to a group of several components. Therefore, the direct link between the proposed IM M RL measure and RIM can be extended as follows:

IM {j 1 ,..j k } M RL (t) = ∞ 0 RIM {j 1 ,..j k } (u, t)du (23)
The use and the advantages of IM M RL measure in ranking a component/group of components will be discussed in more detail in Section 5.

4 An extension of IM M RL measure for maintenance decision-making

Maintenance cost and benefit from maintenance interventions are strongly interrelated and often plays a significant role in maintenance decision-making. Therefore, they should be considered in the components ranking process. To this aim, IM M RL measure is herein extended to incorporate both the benefit gained by the replacement of a component and its replacement cost. Indeed, the extension of IM M RL measure for a component i at time t is mathematically defined as follows:

IM i M RL-c (t) = h(IM i M RL (t)) C i , (24)
where:

• h(IM i M RL (t)) is the benefit associated to the replacement of component i, which is considered as a function of the residual life improvement of the system thanks to the replacement of component i at time t, IM i M RL (t);

• C i is the replacement cost of component i and can be divided into two parts:

C i = c i + c d • d i
where c d •d i is the downtime cost due to production loss during replacement of component i, that takes d i time units. Note that c i includes all other costs related to spare part, labour, set-up, etc.

The benefit function h(IM i M RL (t)) may have different forms, i.e., a linear or non-linear function with respect to IM i M RL (t). As an example, in this work a linear form is considered as follows:

h(IM i M RL (t)) = Q.IM i M RL (t), ( 25
)
with Q is positive number and can be seen as the benefit resulting in each operating time unit of the system. The two following cases can be specified for Q:

• Q = 1, i.e., h(IM i M RL (t)) = IM i M RL (t)
; in this case, IM i M RL-C (t) represents the ratio between the improvement in the system life time, gained from the replacement of component i at time t, and the replacement cost of component i;

• Q > 1, h(IM i M RL (t)
) can be expressed as a benefit gained from the improvement in the system residual life that is provided by the replacement of component i at time t. In such a way, IM M RL-C measure can assist in selecting the most cost-effective component for preventive maintenance regarding to a specific threshold O (O ≥ 1) for the benefit-cost. More precisely:

if IM i M RL-C (t) ≥ O, component i is considered as cost-effective one at time t, i.e., component i could be admissible for the preventive maintenance selection; -0 ≤ IM i M RL-C (t) < O means that the cost benefit given by the replacement of component i is not enough. Consequently, component i is not cost-effective at time t, i.e., it should not be preventively replaced at time t at least from a financial point of view.

It is worth noting that the value of O may be determined from a technical or economical issue, e.g., maintenance optimisation phase.

Just like IM M RL measure, IM M RL-C can be applied for a group of several components as follows:

IM {j 1 ,..j k } M RL-C (t, u) = h(IM {j 1 ,..j k } M RL (t, u)) C {j 1 ,..j k } , (26)
where C {j 1 ,..j k } is the total maintenance cost for the replacement of the group components (j 1 , ..j k ). It must be noticed that in a multi-component system, economic dependence, whereby the joint maintenance of several components is cheaper than when these components are separately maintained, often exists between components [START_REF] Hong | Joint reliability importance of k-out-of-n systems[END_REF][START_REF] Nicolai | Optimal maintenance of multi-component systems: A review[END_REF]. Indeed, it is pointed out in [START_REF] Do | Modelling and application of condition-based maintenance for a two-component system with stochastic and economic dependencies[END_REF] that when several components are simultaneously replaced, the maintenance cost saving arises from the sharing of the replacement set-up cost and the reduction of replacement duration.

Therefore, when k components {j 1 , ..j k } are jointly replaced, the total cost saving can be defined as follows:

CS {j 1 ,..j k } = a • k i=1 c i + b • c d • k i=1 d i , (27)
where:

• a (0 ≤ a) is the cost-saving factor for the joint replacement of k components {j 1 , ..j k }. As it is pointed out in [START_REF] Wildeman | A dynamic policy for grouping maintenance activities[END_REF], the cost saving is commonly equal to 5% of the total maintenance cost of the components (a = 0.05);

• b (0 ≤ b) is the duration-saving factor for the joint replacement of k components.

By definition, a, b represent the economic dependence degree between components. The bigger are a and b, the stronger is the economic dependence between the components. When a = 0 and b = 0, the components are economically independent. A detailed investigation on the impact of the economic dependence between components in condition maintenance decision-making process is discussed in [START_REF] Do | Modelling and application of condition-based maintenance for a two-component system with stochastic and economic dependencies[END_REF][START_REF] Wildeman | A dynamic policy for grouping maintenance activities[END_REF].

IM M RL of groups of components

To illustrate the use of the proposed IM M RL measure for groups of components, IM M RL measure is herein applied to two-component groups given the available information on components at time t = 10, which are presented in the previous section (see case 2). Of course, IM M RL can be implemented to three-component groups or four-component groups in a similar way. The results shows that group (C1,C2) having the highest IM M RL value is the most important group and the less important one is group (C3,C4).

As RIM measure can be extended for groups of several components, RIM measure is herein applied to two-component groups. The results in Figure 8 show again that the importance ranking It is shown that, when compared to the results reported in Table 7, IM M RL-C values change and a new groups ranking appears. This means that the economic dependence degree between components has an important impact on ranking groups of components.

Joint consideration of IM M RL and IM M RL-C for importance rankings

In this section, both IM M RL and IM M RL-C measures are jointly considered in finding the most relevant component/group of components for preventive maintenance decision-making.

Conclusions

In this work, a residual life-based importance measure, namely IM M RL , defined as the improvement ability in the system life time when replacing a component or a group of several components, is introduced. The proposed IM M RL measure can help to rank a component/group of components according to its importance in the system residual life improvement by considering not only the available information related to the current condition (state or/and degradation level) of all components at a given time point but also the system structure into a singe technical metric. The links with existing importance measures (Birnbaum's importance measure and conditional reliability-based importance measure) are also established and discussed to highlight the originality and complementarity of the proposed IM M RL measure. Furthermore, to consider economic aspects (e.g., economic dependence between components, preventive/corrective maintenance costs and the benefit gained from maintenance operations), an extension of IM M RL measure is then developed. Thereby, IM M RL-C can be used in finding the most cost-effective component/group of several components. To illustrate the use and advantages of the proposed measures, a numerical example of a 4-non repairable component system is investigated. The numerical results reveals that, at a given time point, IM M RL and IM M RL-C depend strongly on the components degradation/failure process and the available information levels on the current condition of the components. In addition, the economic dependence degree has also an important impact on ranking groups of component. Importantly, IM M RL and IM M RL-C may lead to two different importance rankings. Nevertheless, from a practical point of view, IM M RL and IM M RL-C measures are complementary and should be jointly considered in finding the most appropriate component/group of components to be preventively maintained.

Figure 2 :

 2 Figure 2: Components' degradation paths of a two-component system and the evolution of the estimated system mean residual life for different levels of information.

Figure 3 :

 3 Figure 3: IM M RL measure of individual components with respect to time t in two cases (Case 1 and Case 2a).

Figure 4 :

 4 Figure 4: IB and RIM measures at time t+u when the degradation levels at time t = 10 are unknown.

Figure 5 :

 5 Figure 5: IB and RIM measures at time t + u given the degradation levels at time t = 10.

Figure 7 :

 7 Figure 7: IB and RIM measures at time t + u given information available at time t = 10.

Figure 9 1 .Figure 9 :

 919 Figure 9 sketches the IM M RL with respect to IM M RL-C of individual component, evaluated at time t = 10. Note that the degradation levels of C1 and C2 at time t = 10 are measured, i.e.,Z 1 t = Z 2 t = 2, X 1 t = 15.91, X 2 t = 22.16 and Z 3 t = Z 4 t = 1.

Figure 10 :

 10 Figure 10: Joint consideration of IM M RL and IM M RL-C for ranking two-component groups.

Table 1 :

 1 Example of IM M RL measure and components ranking.The evolution of the IM M RL measure for both components as a function of time t in both considered cases (1 and 2a) are sketched in Figure3. Because IM M RL measures depend on the information levels available for both components and because the actual degradation paths of each component evolves stochastically with time, IM M RL measures change stochastically with time in Case 2a (degradation levels of both components are measured). It should be noticed that IM M RL measure of each individual component is evaluated based on the degradation paths of both components that are shown in Figure2. In Case 1 (degradation levels of both components are unknown), IM M RL measures evolve deterministically with respect to the average evolution of each component degradation process.

	69	3.06	2	1

Table 5

 5 reports the obtained results of IM M RL for two-component groups.

		At time t = 10: Z 1 t = Z 2 t = 2, X 1 t = 15.91, X 2 t = 22.16; Z 3 t = Z 4 t = 1
	Group	IM G M RL (t)	Ranking
	(C1,C2)	8.24	1
	(C1,C3)	3.37	2
	(C1,C4)	3.17	3
	(C2,C3)	2.91	4
	(C2,C4)	2.89	5
	(C3,C4)	0.60	6

Table 5 :

 5 IM M RL measure and 2-component groups ranking.

The total maintenance cost when k components are jointly replaced can be finally evaluated by:

From an economical point of view, the IM M RL-C measure seems to be an adequate indicator for the preventive maintenance selection because it considers not only the benefit resulting from the system life time improvement given by the replacement of a group of components, the corresponding replacement cost but also the economic dependence between components. An illustration on the use of IM M RL-C measure for ranking components and preventive maintenance decision-making is discussed in the following section.

Numerical example

In this section, we show how the proposed IM M RL measure and its extension IM M RL-C can be used for ranking components/groups of components and preventive maintenance decision-making. For this purpose, a 4-non repairable component system whose reliability block diagram is given in Figure 6, is considered. It is important to note that proposed importance measure IM M RL and its extension IM M RL-C can be applied for any kind of systems for which a reliability structure function is available.

Figure 6: Example of a 4-component system.

From the system's reliability block diagram and the underlying structure function, the reliability of the system can be calculated from the system components' reliability as follows: R(t) = r 1 (t).r 2 (t) + r 1 (t).r 3 (t).r 4 (t) -r 1 (t).r 2 (t).r 3 (t).r 4 (t).

For components reliability modelling, both failure time laws and degradation process are considered. It is assumed that the failure behaviors of components C3 and C4 are described by Weibull ditribution with shape and scale parameters γ i , η i (i = 3, 4), see Appendix A. For components C1 and C2, their reliability behaviors are assumed to be described by a degradation-threshold failure model with homogeneous gamma stochastic processes with shape and scale parameters α i , β i (i = 1, 2), see Appendix B. Component C1 or C2 fails when its degradation level searches a failure threshold L 1 or L 2 respectively. Table 3 reports the numerical values of α i , β i , L i (with i = 1, 2) and γ i , η i (with i = 3, 4). Based on the failure/degradation process' parameters, the MTTF (mean time to failure) of each component is evaluated (see Table 3). Note that these parameters are in arbitrary time unit (atu).

Unit

IM M RL measure and importance ranking

It is considered in this section that the maintenance costs are not available due to whatever reason.

IM M RL measure is applied to rank a component/group of components.

IM M RL of individuals components

At time t = 10, the condition/state of each component is delivered by an inspection. Suppose that components C3 and C4 are in a working state, i.e., Z 3 t = Z 4 t = 1. For components C1 and C2, two cases are considered:

• Case 1: Components C1 and C2 are working at time t but their degradation levels are not measured due to a technical or economical reason, Z 1 t = Z 2 t = 1. Based on the available information, the conditional reliability of each component is predicted. Then, IM M RL of each component is evaluated, the associated components ranking is then obtained according to IM M RL 's values. The results reported in Table 4 show that C1 is the most important component and the importance ranking is C1>C2>C3>C4. Therefore, if only one component can be preventively replaced at time t, component C1 should be selected.

• Case 2: Components C1 and C2 are working at time t and their degradation levels are measured, i.e., Z 1 t = Z 2 t = 2. For the degradation levels at time t = 10, we reuse herein a numerical result presented in Section 3, i.e., X The obtained results point out that the information about the components' states at time t is crucial on IM M RL measure and, as a consequence, on the components ranking. The absence of the degradation levels of components may lead to an inaccurate importance ranking.

In all studies presented the remainder of the paper, only the case 2 is considered, i.e., at time

As a comparison study, Birnbaum measure and RIM measure are applied for each individual component. The obtained results are sketched in Figure 7. It can be seen again that both IB and RIM of each component change with time. Consequently, the components ranking may change. This is especially true with RIM measure. Furthermore, the importance ranking provided by IB measure is not the same as those given by RIM measure or the proposed IM M RL one. More precisely, when u < 44, RIM provides the same the importance ranking given by IM M RL . This is no longer true when u ≥ 44. These differences can be explained by the fact that IB measure of a given component does not allow considering the current condition/state of the component while both RIM and IM M RL do. In addition, RIM is evaluated with an finite horizon while IM M RL is defined with an infinite horizon. associated with RIM measure may change with time. In addition, it is difficult to point out the groups ranking when u < 26 or u > 60. RIM(t+u|z1=2,z2=2)

Figure 8: RIM measures at time t + u given information available at time t = 10.

IM M RL-C measure for importance ranking and selective maintenance

We suppose now the maintenance costs are known: It is apparent that the IM M RL-C measure leads to an importance ranking that is not the same as the one provided by IM M RL (see again Table 4, case 2). Furthermore, according to IM M RL criteria, component C2 is the most important one while, regarding to IM M RL-C measure, C1 is the most important component. This is because IM M RL focuses only on the improvement of the system residual life, i.e., the maintenance cost is not considered, while IM M RL-C considers both the benefit associated with the system life time improvement and the related maintenance cost.

IM M RL-C of individual components

IM M RL-C of groups of components

IM M RL-C measure is now applied for all groups of two components and the obtained results are shown in Table 7.

At time t = 10: It is not surprising again that the two-component groups ranking provided by IM M RL-C measure is not the same as the one given by IM M RL , see again Table 5. More precisely, according to IM M RL measure, the groups ranking is (C1,C2)>(C1,C3)>(C1,C4)>(C2,C3)>(C2,C4)>(C3,C4) while IM M RL-C leads to another importance ranking: (C1,C2)>(C1,C3)>(C1,C4)>(C2,C4) >(C2,C3)>(C3,C4).

Impact of economic dependence on ranking groups of components

To study the impact of economic dependence in IM M RL-C and the associated importance ranking, we change the economic dependence degree between component. More precisely, it is assumed that the duration-saving factor is equal to 30% (b = 30%). The obtained results are reported in Table 8. This paper presents the development of our research in the framework of importance measures for predictive maintenance decision-making presented in part in [START_REF] Do | MRL-based importance measures[END_REF]. Our future research will focus on the development of the proposed importance measures for joint inspection and maintenance decisionmaking with imperfect inspection quality. Another perspective could be a comparison study with others importance measures for components ranking and maintenance decision-making of complex systems.

Appendix

Appendix A. Reliability assessment with Weibull lifetime distribution

If the failure behavior of a component is described by a Weibull law, its lifetime follows a Weibull distribution, f τ i (t), given by

, with parameters γ i > 0 (shape or form parameter) and η i > 0 (scale parameter). The reliability of the component i is expressed as

If component i is working at time t, i.e., Z i t = 1, the conditional reliability of component i is written as

Appendix B. Reliability assessment with gamma degradation process

It is assumed that the degradation behavior of component i is described by a gamma processes ( Xt ) t≥0

with the following characteristics:

• ( Xt ) t≥0 has independent increments;

• for all 0 ≤ l < t, the random increment Xt -Xl follows a gamma probability density function (pdf) with shape parameter α i (t -l) and scale parameter β i :

x α i (t-l)-1 e -β i x I {x≥0} , where, I {.} is an indicator function. I {x} = 1 if x ≥ 0, I {x} = 0 and otherwise;

The mean degradation speed is α i /η i and its variance is α i /(β i ) 2 . Noting that various degradation behaviors can be modeled by changing the couple of parameters α i , β i .

According to the degradation process, the reliability of a component i can be assessed by:

If component i is working at time t and the degradation level of component i is measured at time t, Z i t = 2, X i t = x i t (x i t < L i ), the conditional reliability of component i can be evaluated by:


Mariam Abd

Dr El Sater

Dr Mohamed Mellah

Diana Dragoe

Dr Emilie Kolodziej

Nada Jaber
email: nada.jaber@ul.edu.lb

Emmanuelle Schulz
email: emmanuelle.schulz@universite-paris-saclay.fr

] M Abd

Chiral Chromium Salen@rGO as Multipurpose and Recyclable Heterogeneous Catalyst

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

Introduction

Carbon materials have been used extensively as highly stable supports, ideally porous and mechanically resistant in catalysis, in particular for the dispersion and stabilization of metal particles or for the immobilization of organometallic complexes. [1] Their particular surface chemistry allows in fact either covalent grafting by complementary functionalization of the support and of the ligand, or non-covalent bonding via π-π-interactions between a polyaromatic functionalized compound and the graphitic sp 2 carbon network. In the context of homogeneous supported catalysis, and precisely when valuable enantiopure chiral catalysts are engaged, their non-covalent interactions with the support are based on simple synthetic procedures avoiding its prior modification. The strength of these supramolecular interactions can be adjusted as a function of the solvent and the temperature, allowing both easy recovery of the catalyst and recycling of the support, in the event of deactivation. On the other hand, precautions must be taken to suppress metal leaching. [2] Some successes have already been reported using this immobilization strategy on different carbon supports, including the immobilization of Ru-carbene complexes to perform metathesis reactions. [3] The Au-catalyzed enyne cyclization reaction has been achieved on multi-walled CNTs [4] and Pd-based catalysts were non covalently anchored onto magnetic Co/carbon nanoparticles for Suzuki-Miyaura couplings. [5] rGO was further used for the Pd-catalyzed heterogeneous hydrogenation of alkenes and the Ru-catalyzed alcohol oxidation. [6] The diversity of application is moreover demonstrated by the Fe-catalyzed ethylene polymerization on multi-walled CNTs, [7] the Rh-promoted 1,4-addition reaction and hydrosilylation of alkynes, [8] and the Cu-catalyzed CH activation on rGO. [9] In the latter case, the carbon support not only contributes to make the catalyst insoluble but also exacerbates its activity by reducing the electron density of the copper site. Mention should be made, in each case, of the functionalization of the ligand by a pyrene group as a demonstration of its high affinity with the carbon surface by π-π-interactions. Such noncovalent anchoring on carbon surfaces proved also its utility by enhancing the cooperativity between two pyrene-tagged Pd-and Ru-complexes for hydrodefluorination. [10] In contrast, this immobilization strategy has been much less developed in the context of asymmetric catalysis, with only four examples reported, to our knowledge. In the field of Rh-promoted asymmetric hydrogenation, chiral phosphine-containing catalysts have been deposited on CNT, [11] or assembled onto rGO. [12] We have immobilized copper bis(oxazoline) complexes for the enantioselective formation of C-C bonds, [13] and more recently, the non-covalent immobilization of Rh complexes was reported for the asymmetric hydroformylation and implemented in a flow mode. [14] The use of salen-complexes is ubiquitous in asymmetric catalysis. [15] These valuable enantiopure catalysts have been immobilized on various supports or their structures modified to become insoluble species, in order to be easily recovered by filtration. [16] We describe here the first immobilization of chromium salen complexes on carbon supports [START_REF] Li | A salophen-Co complex was already immobilized on the surface of MWCNT through π-π-interactions, towards electrocatalytic water oxidation[END_REF] through π-π interactions with rGO. The high efficiency of these chromium catalysts [18] is demonstrated both in the heterogeneous asymmetric ring-opening (ARO) [19] of a meso epoxide and in the hetero-Diels-Alder reaction (HDA) [20] of various aldehydes. Their stability in recycling procedures is also evaluated. [START_REF] Sellner | For immobilization procedures implying Cr-salen derivatives used in ARO or HDA reactions see[END_REF] These reactions were chosen as benchmarks for the diversity of the products to which they lead. While the ARO reaction provides precursors of cyclic 1,2-amino alcohols, the HDA reaction between aldehydes and the electron-rich Danishefsky's diene delivers enantioenriched dihydro-pyranones, important scaffolds for the preparation of various natural compounds. [22]

Results and Discussion

We therefore immobilized the salen complexes by π-interactions with the rGO support by placing the pyrene group far from the active chromium site (Scheme 1). This strategy aims to preserve the selectivity of the homogeneous transformation achieved in the presence of the conventional Jacobsen catalyst. We tackled the synthesis of salen catalysts tagged with one or two pyrene groups via links of different lengths, to compare both the accessibility to the active sites and their stability to recycling. Compounds A and B, as key intermediates for the synthesis of Sym and Unsym ligands were easily obtained through click reactions between 1-[(2-propyn-1-yloxy)methyl]-pyrene [23] or 1-(4-(prop-2-yn-1-yloxy)butyl)pyrene [24] and 5-(azidomethyl)-3-(tert-butyl)-2-hydroxybenzaldehyde [25] in two steps and high isolated yield (see Scheme 1 and SI). Then, classical condensation with 0.5 equivalent of (1S,2S)-cyclohexane-1,2diamine directly afforded Sym-A and Sym-B, the latter with a metal coordinating site remote from pyrene by four methylene groups. After the preliminary preparation of the diamine monoammonium salt C, [26] the high yield formation of both Unsym ligands also occurred by successively reacting 3,5-ditert-butyl-2-hydroxybenzaldehyde then A (or B), without isolating the intermediate ammonium mono-imine. Insertion of chromium in the coordinating sites was finally performed from CrCl2 in THF under argon atmosphere and then oxidation in air. [20] Scheme 1. Synthesis of pyrene-tagged salen ligands.

The four corresponding chromium complexes Sym-Cr and Unsym-Cr (A and B) were prepared in almost quantitative yield and fully characterized by IR, UV-Vis and HRMS spectroscopy. They were first tested under homogeneous conditions, to assess the influence of their structure on their reactivity and selectivity. Cyclohexene oxide ring opening with TMSN3 was chosen as the first test reaction carried out in diethyl ether. Different results in terms of activity and selectivity were obtained depending on the structure of the catalysts used (see Table 1). In our hands, the Jacobsen catalyst delivered the ring-opening product in 88 % ee, but full conversion was not achieved after 24 h of reaction. The pyrene-tagged catalysts were all more active in promoting this reaction, but differed in their selectivity. Indeed, both Sym-Cr-A and Sym-Cr-B were found to be poor enantioselective catalysts while the unsymmetric analogues were much better, with Unsym-Cr-A giving the target product in 81 % ee. In addition, further experiment showed that almost complete conversion (96 %) was already achieved in just 6 hours using this catalyst.

Catalyst

Conv (%) [b] ee (%) [c] Jacobsen Cat. 78 88

Sym-Cr-A 86 26

Unsym-Cr-A >99 81

Sym-Cr-B 98 8

Unsym-Cr-B >99 70

[a] substrate, 0.5 mmol, [2.9 M] in Et2O. [b] determined by GC with dodecane as internal standard. [c] determined by GC (see SI).

We propose that the absence of two tBu groups in the para position of the phenol, in the Sym-Cr catalysts, due to the bifunctionalization by the pyrene groups is the cause for this significant decrease in enantioselectivity. On the other hand, the accessibility of the catalytic sites in these symmetrical species can be constrained by a possible π-π intramolecular stacking between the two adjacent pyrene groups. In contrast, Unsym-Cr-A allowed complete conversion and led to the best enantioselectivity value for the series. This catalyst was therefore chosen for the rest of the study. Its immobilization onto the carbon surface was achieved by impregnation of the rGO suspension with a solution of Unsym-Cr-A in DCM with a mass ratio of 4/1. After 24 hours of stirring at room temperature, the solid was filtered off, washed with DCM and vacuum dried. The efficiency of the immobilization was monitored by UV-Vis analyses (see Figure 1 and SI for details). A calibration line curve for Unsym-Cr-A was indeed obtained by measuring the absorbance (Abs) of solutions with different concentrations at the λmax (345 nm, see blue trace in Figure 1). After 24 h, analysis of the supernatant (orange trace) indicated that 90% of the complex has been deposited, what was confirmed by weighing the resulting solid. Furthermore, X-ray photoelectron spectroscopy was conducted on powders of both unmodified rGO support and Unsym-Cr-A@rGO. The wide-scan spectra are shown in Figure 2a. The presence of signals arising from Cr2p, N1s and Cl2p core-levels was clearly noticed on the spectrum of Unsym-Cr-A@rGO, proving that the catalyst had been successfully grafted. The narrow-scan spectra for Cr2p, N1s and Cl2p were then measured (Figure 2b), which allowed to determine the position in energy of the three elements, as well as their atomic ratios.

The Cr2p core-level spectrum consists of two main peaks corresponding to the spin-orbit components Cr2p3/2 and Cr2p1/2, situated at 577.2 eV and 586.9 eV, respectively, and separated by 9.7 eV. These values allowed to assign the Cr 3+ according to the literature. [27] The Cl2p and N1s spectra are centred at 198.7 eV and 400.5 eV respectively. Expecting information on the catalyst structure, experimental determination of the Cr/Cl content in Unsym-Cr-A@rGO was performed and is comparable to the theoretical value (1), as well as the N/Cl ratio ( 5), undoubtedly revealing the presence of intact Unsym-Cr-A on the surface. The heterogeneous catalysis was then performed under conditions exactly matching the one used for the tests with the non-supported catalysts. A black suspension (3M in substrate) was obtained, corresponding to the introduction of a 2 mol % catalytic ratio. Pleasingly, the first Unsym-Cr-A@rGO catalysis in the ARO of cyclohexene oxide afforded the ring-opened product with complete conversion after 6 h, indicating no detrimental effect of the support on the catalytic activity, which is often the case in heterogeneous reactions. However, the target product was isolated with a decreased enantioselectivity value than that observed under homogeneous conditions (see Table 2, entry 1). After completion, the supernatant from the first run of heterogeneous catalysis was filtered off and the supported complex was washed with diethyl ether, the reaction solvent. It was then dried under vacuum, before its reuse by further addition of substrate. The second use of the supported catalyst was also successful, as were subsequent ones, allowing 10 uses of the same batch of catalyst. The enantiomeric excess reached 79 %, fortunately almost equal to the high values obtained under homogeneous conditions (Table 2).

Table 2. Heterogeneous ARO of cyclohexane oxide with Unsym-Cr-A@rGO, recycling experiments [a] Catalyst Conv (%) [b] ee (%) [c] Run 1 >99 63 [a] substrate, 0.5 mmol, [3M] in Et2O. [b] determined by GC with dodecane as internal standard. [c] determined by GC (see SI)

The immobilization of Unsym-Cr-A on rGO thus did not alter its activity and the enantioselectivity values remained also very high in this procedure. Altogether, an important increase in the calculated TON of the catalyst is thus obtained by its simple reuse after filtration.

A second reaction, namely the HDA reaction between aldehydes and trans-1-methoxy-3-trimethylsilyloxy-buta-1,3-diene, was tested to evaluate the extent of application of Unsym-Cr-A@rGO in asymmetric catalysis. To ensure the efficiency of the modified complexes, the performances of Sym-Cr-A and Unsym-Cr-A were again compared with the Jacobsen complex in the HDA reaction involving cyclohexane carboxaldehyde in DCM (Table 3). Catalyst Conv 1a (%) [b] Yield 2a (%) [c] ee 2a (%) [d] Jacobsen Cat. 84 69 78

Sym-Cr-A

-80

Unsym-Cr-A 73 61 78

[a] 0.5 mmol, [5M] in DCM. [b] determined by GC with dodecane as internal standard. [c] isolated yield. [d] determined by GC (see SI).

In this case again, the unsymmetrical pyrene-derived complex Unsym-Cr-A provided activity and enantioselectivity values very close to those obtained by the Jacobsen catalyst, with Sym-Cr-A far less active. The study therefore continued using Unsym-Cr-A with rapid screening for solvents, but neither Me-THF nor diethyl ether outperformed DCM in terms of activity (61 and 67 % yield, respectively), nor of selectivity (78% ee). A new batch of Unsym-Cr-A@rGO was prepared and the catalysis was first carried out in DCM as was the homogeneous reaction. The first Unsym-Cr-A@rGO catalysis afforded the dihydropyranone with conversion and enantioselectivity values close to those obtained in homogeneous conditions. In this case also and ideally, the presence of the support did not impact the chiral catalyst efficiency. This was verified by using unmodified rGO in the reaction conditions, which resulted in no conversion after 24 h of stirring. After complete conversion, the heterogeneous catalyst was recovered by filtration and several solvents were tested to wash the supported complex between each reuse (see Table 4). The reaction solvent, cold DCM, was initially used during the first three cycles which led to the conservation of the enantioselectivity but to a constant decrease in the conversion with each recycling (entries 1-4). The even more important decrease (14 % conversion) during the 4 th use of the supported catalyst led to carrying out the following washes with cold acetone. Interestingly, this resulted in the complete recovery of the catalyst activity and enantioselectivity for its 5 th use (entry 5), values which were maintained for the next four runs. We propose that acetone is both a polar solvent which promotes π-stacking type interactions, and a favourable medium for the dispersion of the rGO carbon sheets, which allows a better accessibility to the catalytic sites. [3a] The recovery procedure was however not facilitated by the use of acetone as washing solvent because of settling difficulties. Thus, after the 8 th run, this same catalytic batch was washed with cold diethyl ether, and gratifyingly, the next run afforded the same results in terms of activity and enantioselectivity (entries 8-9).The validity of this optimized washing procedure in diethyl ether was verified by the preparation of a new batch of Unsym-Cr-A@rGO, which was involved in a new recycling experiment (entries 10-16); efficient catalysis occurred for seven reuses of the supported catalyst, with excellent stability of the enantioselectivity values of the transformation. The scope of this recovery procedure via non-covalent interactions was tested for the HDA of various aldehydes. Consequently, three aromatic aldehydes reacted with Danishefsky's diene under homogeneous conditions with Unsym-Cr-A and the corresponding dihydropyranones were isolated with good enantioselectivity values (see SI). Then, another batch of Unsym-Cr-A@rGO was prepared and used to catalyse the heterogeneous HDA reaction of these aldehydes successively, by changing the structure of the substrate at each reuse of the supported complex (Table 5). A first run was thus devoted to the transformation of benzaldehyde 1b, resulting in the isolation of the corresponding dihydropyranone 2b with a good yield and an enantioselectivity value of 64 % ee, even slightly better than that obtained under homogeneous conditions (Table 5, run 1 and SI for comparison). After washing with diethyl ether, the recovered catalyst was then engaged in the transformation of cyclohexane-carboxaldehyde 1a (Table 5, run 2). Run Conv (%) [b] ee (%) [c] Yield (%) [d]

Run

Conv (%) [b] ee (%) [ The subsequent recycling procedure involved consecutively two other aromatic aldehydes (1c-d, runs 3-4). If the heterogeneous catalysis turned out to be slightly less effective in terms of activity compared to the results obtained under homogeneous conditions (Table 5 and SI), similar enantioselectivity values could nevertheless be obtained. The products were easily recovered without being contaminated by traces of dihydropyranones prepared in the previous runs. Finally, recovered Unsym-Cr-A@rGO after the 4 th run was reassigned to the HDA of benzaldehyde (run 5), interestingly offering improved activity and the same enantioselectivity for the target product 2b. Recycling was continued again with the other three substrates (1a, 1c and 1d) while maintaining its efficiency (runs 6-8), as clear evidence of the stability of the procedure. Spent Unsym-Cr-A@rGO was also subjected to XPS-analyses and the Cr to C ratio was compared to that determined for the fresh supported catalyst (see SI). Within the limits of precision of the analysis, no leaching of active species could be determined; with regard to the stability of the catalytic values, no degradation of the chromium complex has occurred, obviously. In addition, Unsym-Cr-A@rGO was stirred for 18 h in DCM and the resulting solution was collected after filtration. No catalytic activity was detected after adding substrates to this solution, proving that there is no release of the active species during the heterogeneous catalysis reaction.

Conclusion

To conclude, non-covalent attachment of a Cr-salen complex on rGO was proved to afford an efficient supported catalyst to produce highly enantioenriched compounds when engaged in recycling procedures. Using two different asymmetric catalytic transformations as a test, the ARO and the HDA reactions, various valuable scalemic compounds were easily isolated in their pure form. This also applies when the supported catalyst is engaged in a multi-substrate procedure. For both types of reactivity, we demonstrated that a single pyrene group covalently linked to the enantiopure salen unit was sufficient to retain the activity values of the parent Cr-salen complex and allow its grafting to the support surface by non-covalent π-π interactions. As expected however, the nature of the solvent used for the reaction and for the products recovery after completion was of major importance. In this context, the use of ether as a polar solvent allowed the dispersion of the rGO sheets, therefore maintaining accessibility to the catalytic sites. Numerous stable recycling operations have been carried out, highlighting the possibility of using chiral organometallic catalysts heterogenized by weak interactions. Such noncovalent bindings result from easy synthesis of the catalyst and from the absence of support modification. As far as we know, this is the first immobilization of salen derivatives on carbon surfaces by π-π interactions, and its success opens the way for many extensions, given the wide applicability of these complexes in asymmetric catalysis. Work is in progress to further extend this procedure to different asymmetric catalytic transformations with other pyrene-tagged salen complexes, both in batch and in flow conditions.

Figure 1 .

 1 Figure 1. Immobilization of Unsym-Cr-A on rGO.

Figure 2 .

 2 Figure 2. a) XPS spectra for rGO and Unsym-Cr-A@rGO. b) narrow-scan spectra for Cr2p, N1s and Cl2p.

Table 1 .

 1 Homogeneous ARO of cyclohexane oxide, catalysts comparison

	[a]

Table 3 .

 3 Homogeneous HDA of cyclohexane carboxaldehyde, catalysts comparison

[a]

Table 4 .

 4 Heterogeneous HDA of cyclohexane-carboxaldehyde with Unsym-

	Cr-A@rGO in DCM	[a]
	Entry	Run	Conv 1a (%) [b]	ee 2a (%) [c]	Washing Solvent
	1	1	55	81	DCM
	2	2	61	81	DCM
	3	3	49	81	DCM
	4	4	14	71	Acetone
	5	5	85	79	Acetone
	6	6	82	81	Acetone
	7	7	79	79	Acetone
	8	8	79	79	Et2O
	9	9	79	80	-
	10	1	82	80	Et2O
	11	2	85	81	Et2O
	12	3	86	80	Et2O
	13	4	78	80	Et2O
	14	5	74	81	Et2O
	15	6	76	81	Et2O
	16	7	68	81	-
	[a] substrate, 0.5 mmol, [5M] in DCM, rt, 24 h. [b] determined by GC with

dodecane as internal standard (see SI).

[c]

determined by GC (see SI).

Table 5 .

 5 Heterogeneous HDA of various aldehydes with Unsym

-Cr-A@rGO

	[a]

  ] substrate, 0.8 mmol, [5M] in DCM, rt, 24 h, washing with Et2O.[b] determined by GC with dodecane as internal standard (see SI).[c] determined by GC (see SI).[d] isolated yield

	[a
							c]
	1	68	64	60	5	83	64
	2	63	81	30	6	70	81
	3	76	63	65	7	85	60
	4	75	64	71	8	84	63

Acknowledgements

We acknowledge Université Paris Saclay, CNRS, Ministère de l'Europe et des Affaires Etrangères (bourse d'excellence "Eiffel") and Charm3At LabEx (ANR-11-LABEX-0039) for financial support. This project has been jointly funded with the support of the National Council for Scientific Research in Lebanon CNRS-L and Lebanese University. The authors are also grateful to Drs. Philippe Dauban and Olivier David for fruitful discussions.

Conflict of interest

The authors declare no conflict of interest.

Entry for the Table of Contents

A pyrene-tagged Cr-salen complex immobilized by π-π interactions at rGO surface delivers a robust, reusable, heterogeneous catalyst efficient in several asymmetric catalytic reactions


Yu-Chao Yuan

Mariam Abd

El Sater

Mohamed Mellah

Nada Jaber

Olivier R P David
email: olivier.david@uvsq.fr

Emmanuelle Schulz
email: emmanuelle.schulz@universite-paris-saclay.fr

Enantiopure Isothiourea@Carbon-Based Support: Stacking Interactions for Recycling a Lewis Base in Asymmetric Catalysis

Keywords:

Dedicated to Dr. Christian Bruneau for his outstanding contribution to

Introduction

The use of enantiopure catalysts in synthesis is nowadays obvious for the preparation of valuable scalemic substrates. For cost and environmental motivations, these precious promoters must be readily available and exhibit high activity at low catalytic loading. In this context, their easy recovery for an efficient reuse is also of great value. Numerous procedures have already reported the heterogenization of chiral catalysts involving either covalent or non-covalent binding to many supports.

1 As highly stable, ideally porous and mechanically resistant supports, carbon materials have been used extensively in catalysis.

2 They can indeed be modified by covalent bindings 3 or be involved in π-π interactions. 4 In the first case, a covalent link of the chiral catalyst on the support ensures robustness and decreases any risk of leaching, but it can lead to a tedious synthesis for the necessary modification of both the ligand and the support. On the contrary, immobilization by non-covalent interactions often means simpler processes and easy recovery of the support, when the catalyst is deactivated. A possible catalyst release must however be taken under consideration. 5 The chemical modification of carbon supports has already been described for the subsequent covalent grafting of enantiopure organometallic catalysts, 6 but very few examples have proposed immobilization by non-covalent bonds. 7 They include the deposit of chiral phosphine-containing catalysts on CNT 7a or their assembly onto rGO 7d to realize Rh-promoted asymmetric hydrogenation. Our group described the immobilization of copper bis(oxazoline) complexes on various carbon supports for the enantioselective formation of C-C bonds.

7b More recently, the non-covalent immobilization of Rh complexes was reported for the asymmetric hydroformylation and implemented in a flow mode. 7c This year, imidazolium salts have also been immobilized by π-stacking interactions onto carbon materials and these organocatalysts were efficiently reused in the cycloaddition of epoxides with CO 2 .

8

Worth mentioning is, in each case, the functionalization of the ligand by a pyrene group as a demonstration of its high affinity with the carbon surface by very stable π-π-interactions. However, and as far as we know, there are no examples dealing with the immobilization of an enantiopure organocatalyst on carbon supports by such reversible interactions. This journal is © The Royal Society of Chemistry 20xx

Please do not adjust margins Please do not adjust margins Chiral isothioureas (ITUs) [START_REF] Birman | Benzotetramisole: A Remarkably Enantioselective Acyl Transfer Catalyst[END_REF] are enantiopure Lewis bases active to promote numerous asymmetric catalytic processes. 10,11 These catalysts were already immobilized mostly on organic polymeric supports by covalent grafting; they were stable in many efficient batches and even flow processes. 12 In this context, we describe here the first immobilization of an enantiopure ITU organocatalyst (hyperBTM) on reduced graphene oxide (rGO) as carbon support through π-π noncovalent interactions. We selected the enantioselective formal [3+2] cycloaddition of ammonium enolates with oxaziridines described by the group of Smith, [START_REF] Smith | Asymmetric Isothiourea-Catalysed Formal [3+2] Cycloadditions of Ammonium Enolates with Oxaziridines[END_REF] to test our procedure. This reaction indeed leads to the synthesis of oxazolidin-4-ones, valuable scaffolds present in natural products such as lipoxazolidinones A-C 14 or simply precursors of chiral αhydroxyacids. While this reaction has never been described under recycling conditions, we report here our efforts to achieve it under heterogeneous conditions on a carbonaceous support, and to evaluate the robustness of the chiral organic catalyst to recovery.

Results and discussion

We targeted the synthesis of pyr-hyperBTM for its immobilization on rGO (see Scheme 1). The synthesis of 3 was realized following the procedure described by the group of Smith, starting from benzaldehyde 1. [START_REF] Morrill | Isothiourea-mediated asymmetric Michaellactonisation of trifluoromethylenones: a synthetic and mechanistic study[END_REF] The introduction of the stereogenic centers occurred via the use of (S)-proline leading to the formyl-carbamate 2 (see Supporting Information for details). The anchoring point for the pyrene ring is chosen far from the active center of the organocatalyst, on the phenyl ring of the benzotetramisole. This strategy has already been illustrated by the immobilization of the enantiomer of compound 3 on a Merrifield-type resin, dedicated to the acylative kinetic resolution of alcohols. 12c To this aim, compound 3 is functionalized by a propargyl group after demethylation, to give 4 in 42% yield. On the other side, commercially available pyren-1-yl-methanol is transformed via two consecutive nucleophilic substitutions into the azido derivative 7, possessing a four methylene spacer. A copper(I)promoted azide-alkyne cycloaddition between ITU-alkyne 4 and pyrene-azide 7 afforded the desired functionalized pyr-hyperBTM. This pyrene-modified organocatalyst was prepared at the gram scale with an overall yield of 4.1 % from benzaldehyde (see Supporting Information for the detailed complete procedure and characterization). Immobilization of the pyrene-tagged hyperBTM onto the rGO surface was achieved by impregnation of the carbon suspension with a solution of pyr-hyperBTM in DCM with a mass ratio of 4/1. [START_REF] Sater | Chiral Chromium Salen@rGO as Multipurpose and Recyclable Heterogeneous Catalyst[END_REF] After sonication for a few minutes and 24 h

of stirring at room temperature, the solid was filtered off, washed with DCM and vacuum-dried. The efficiency of the immobilization was monitored by UV-Vis analyses (see Figure 1 and Supporting Information for details). The comparison with a calibration curve indicated that the entire complex was totally deposited, which was further confirmed by differential weighing. The organocatalyst dispersed as a fine black powder was therefore tested as heterogeneous catalyst.

The catalytic efficiency and the recyclability of pyr-hyperBTM@rGO was tested on the asymmetric formal [3+2] cycloaddition of 2-phenylacetic anhydride and rac-3-phenyl-2tosyl-1,2-oxaziridine. [START_REF] Smith | Asymmetric Isothiourea-Catalysed Formal [3+2] Cycloadditions of Ammonium Enolates with Oxaziridines[END_REF] Pyr-hyperBTM was firstly engaged under homogeneous conditions to assess its efficiency against the published values of its unmodified analogue. The reaction is indeed described to deliver the oxazolidin-4-ones anti-10a and syn-10a from racemic 9 and Cs 2 CO 3 as base, in high yield, with excellent enantiocontrol but a poor diastereoselectivity when hyperBTM is used (Table 1, entry 1, ref 13). We explored the activity of the pyr-hyperBTM precursor 3 under the same conditions, to evaluate the changes induced by the presence of a methoxy substituent on the phenyl ring of the organocatalyst (entry 2).

Please do not adjust margins Please do not adjust margins This species delivered the target products with similar values in terms of yield and diastereoselectivity but a reduction of the enantioselectivity was observed. Fortunately, high selectivity values were again obtained when pyr-hyperBTM was tested as catalyst in the presence of Cs 2 CO 3 as base, although both oxazolidin-4-ones were synthesized in only 74% yield (entry 3). The use of an organic base, the Hünig base, was then investigated as soluble species in the medium for limiting mass transfer in the context of the subsequent heterogeneous catalysis. These operating conditions led to an improved activity (92 % yield, entry 4) to the detriment of selectivity values. In this last case, both diastereoisomers anti-10a and syn-10a were recovered but contaminated by a slight amount of N-tosylbenzylimine, as by-product presumably arising from oxidation of the organic base.

The reaction was finally performed under heterogeneous conditions, with pyr-hyperBTM@rGO as supported catalyst in the presence of both bases and successful enantioselective catalysis still occurred. The yields were recorded at the same reaction time as under homogeneous conditions, revealing a small decrease in activity as is often observed with supported catalysts. The catalyst deposit on rGO slightly improved the diastereoselectivity, but decreased the enantioselectivity value, specifically when Cs 2 CO 3 was concerned (entries 5-6).

The quantity of organocatalyst introduced to carry out this catalytic transformation is not negligible (10 mol %) and we therefore explored the possibility of its recycling to artificially increase the TON. Both the use of the organic or the inorganic base was evaluated for the recovery process and the detailed results are summarized in the Supporting Information (see tables, ESI S14). In all cases, and after the first run, the suspension was left to settle, the supernatant solution was removed by filtration and the remaining solid was washed with DCM, the reaction solvent. A second run was performed after drying the immobilized catalyst under vacuum and addition of new substrates. For instance, Cs 2 CO 3 possesses the advantage to avoid the formation of the by-product, but as an insoluble species, it is recovered together with the immobilized catalyst.

The second run conducted under these conditions and without addition of supplementary base, delivered the desired mixture of diastereoisomers 10a, rid of N-tosylbenzylimine but in only 20 % yield. Addition of Cs 2 CO 3 before a third use of the catalyst unfortunately did not result in any formation of the targets. A new batch of pyr-hyperBTM@rGO was prepared and a second attempt was completed with the inorganic base, but the workup implied a washing with a mixture of H 2 O/acetone to eliminate all salts. A significant decrease in yield was also observed in this case (2 nd run, 40 % yield) but the products were still isolated in 23% yield after the third cycle. Nevertheless, the decrease in the enantioselectivity values along with recycling is too important under these conditions This journal is © The Royal Society of Chemistry 20xx

Please do not adjust margins Please do not adjust margins and the recycling procedure was thus renewed by using the soluble Hünig base. Both acetone and DCM were tested as solvents to wash the recycled supported catalyst leading to better results. Although a slight decrease in activity could not be avoided, the preservation of the enantioselectivity values was noticeable. Unfortunately, however, the mixture of both diastereoisomers remained contaminated with slight amount of inseparable N-tosylbenzylimine, as by-product.

The procedure implying several washings of the solid catalyst with DCM as reaction solvent and using iPr 2 NEt as base was thus chosen to carry the reaction with pyr-hyperBTM@rGO in heterogeneous, recycling conditions (Table 2). The recycling tests were performed the 0.1 mmol scale, and the supported catalyst pyr-hyperBTM@rGO was engaged seven successive times in reaction. The continuous formation of the target products is observed with each recycling, even if their quantity decreases; after the 2 nd run, the imine side product appears as traces together with the expected oxazolidinones. It is important to note that the enantioselectivity values are maintained at a high level, at least until the 5th use of the catalyst. Finally, and before performing the 7 th cycle, a small amount (1/5 of the quantity initially introduced) of fresh pyr-hyperBTM was added on the supported catalyst in reuse. This notably led to a renewed increase in both the yield and also the selectivity of the reaction. Although some performance instability during recycling was observed, possibly due to catalyst degradation over the process by some hydrolysis, this procedure made it possible to carry out seven runs with notable selectivity, which is unprecedented for this formal [3 + 2] cycloaddition. Finally, pyr-hyperBTM@rGO was used in a similar transformation under these optimized conditions, but different arylacetic anhydrides possessing various structures were successively engaged in the reaction, at each recycling, in a challenging multi-substrate procedure, to widen the scope of the methodology and to further test the stability of the supported catalyst. The three new arylacetic anhydrides were first subjected separately to the homogeneous asymmetric reaction in the presence of pyr-hyperBTM as organocatalyst and yield as selectivity values were determined for comparison (Table 3, entries 2-4). The transformation of these substrates was productive as the target aryl-substituted oxazolidiones were in each case isolated with good to high yields, using Hünig's base. The diastereoselectivity of these reactions are in agreement with published results but the enantioselectivity values turned out to be much weaker than the results reported in the presence of Cs 2 CO 3 . [START_REF] Smith | Asymmetric Isothiourea-Catalysed Formal [3+2] Cycloadditions of Ammonium Enolates with Oxaziridines[END_REF] The ability of pyr-hyperBTM@rGO to catalyze these enantioselective formal cycloadditions by changing the structure of the ammonium enolates at each reuse was finally investigated. After preparation of anti-10a and syn-10a in good isolated yield with high enantioselectivity values, the supported catalyst was washed with DCM and reused for the reaction with 2-(naphthalen-1-yl)acetic anhydride, 8b. Anti-10b and syn-10b were easily recovered in their enantioenriched forms, with 65 % yield, importantly free of any trace of oxazolidinones prepared in the previous cycle.

A third recycling of this batch of pyr-hyperBTM@rGO led to the oxazolidinones from the reaction with 8c, however affected by a reduction of the expected values. A final reuse, carried out in the presence of Cs 2 CO 3 as base with 8d anhydride bearing an electron donor substituent, still made it possible to isolate the targeted products anti-10d and syn-10d with high enantioselectivity values.

Conclusions

The functionalization of hyperBTM by a pyrene group therefore allowed the proposal of a new method for its recycling, which involves the formation of non-covalent bonds

Please do not adjust margins

Please do not adjust margins with rGO. These π-stacking interactions advantageously make it possible to avoid any modification of the support. We have here discussed the fine-tuning of the procedure allowing for the formation of various oxazolidinones from a racemic oxaziridine with arylacetic anhydrides. Two bases were evaluated: the Hünig's base led to high yield and enantioselectivity values in the target products; the use of Cs 2 CO 3 , which is insoluble and accumulates during the recycling of the heterogeneous catalyst made it possible, on the other hand, to avoid the formation of the secondary product, the N-tosylbenzylimine. The catalytic reaction is nevertheless carried out under operating conditions matching those used in the homogeneous phase while maintaining high activity and selectivity. The recovery of the catalyst before its re-engagement in a new reaction is simply carried out by its filtration and further washing with the reaction solvent. Its effectiveness has been demonstrated by its reuse in 7 successive cycles of catalysis and also in a multi-substrate procedure (4 cycles) during which the structure of the anhydride reagent was changed at each recycling. Although a decrease in the activity and selectivity values of the spent catalyst could not be avoided, this is the first demonstration of its reuse in this formal cycloaddition reaction. To our knowledge, this study is also the first use of a chiral organic catalyst under these conditions, among the very few reports which deal with non-covalent immobilization by π-stacking of asymmetric organometallic catalysts. This study paves the way for the application of this procedure to promote other catalytic reactivities by these chiral isothiourea species, immobilized alone, or with other catalysts, towards asymmetric multicatalysis. This journal is © The Royal Society of Chemistry 20xx

Please do not adjust margins Please do not adjust margins

Scheme 1 .Figure 1 .

 11 Scheme 1. pyr-hyperBTM synthesis procedure

Table 1 .

 1 Reaction tests in homogeneous and heterogeneous conditions

	Entry	Cat	Base	Yield a	d.r. anti/syn b	ee anti/syn b
	1 c		Cs2CO3	83	57 / 43 97 / 97
	2		Cs2CO3	81	57 / 43 85 / 81
	3		Cs2CO3	74	61 / 39 96 / 98
	4	pyr-hyperBTM	iPr2NEt	92	55 / 45 89 / 89
	5		Cs2CO3	64	69 / 31 87 /81
	6	pyr-hyperBTM@rGO	iPr2NEt	75	59 / 41 88 / 92

a isolated yield of anti-10 + syn-10. b determined by HPLC or SFC analyses (see Supporting Information).

Table 2 .

 2 Recycling tests with pyr-hyperBTM@rGO and iPr 2 NEt as a base.

		Yield a d.r.anti/syn b eeanti/syn b
	1 st run	74	57/43	92/96
	2 nd run	66	64/36	90/93
	3 rd run	56	62/38	87/92
	4 th run	40	66/34	85/87
	5 th run	29	63/37	83/95
	6 th run	20	68/32	74/89
	7 th run c	51	67/33	91/93

a isolated yield of anti-10a + syn-10a. b determined by analysis. c 2 mg of pyr-hyperBTM added to the spent supported catalyst before performing the 7 th run.

Table 3 .

 3 Homoanhydride substrate scope and recycling tests with pyr-hyperBTM@rGO in a multi-substrate procedure. determined by HPLC and SFC analyses. c 10c contaminated with an impurity preventing an accurate ee determination of syn-10c. d Cs2CO3 is used as base.

	Entry Run	Cat	Substrate Yield a	d.r. anti/syn b	ee anti/syn b
	1	-		8a	92	55/45 89/89
	2	-	pyr-hyper	8b	85	58/42 79/53
	3	-	BTM	8c	82	63/37 85/60
	4	-		8d	90	44/56 76/88
	5 6 7 8 d	1 st 2 nd 3 rd 4 th	pyr-hyperBTM @rGO	8a 8b 8c 8d	75 65 60 30	62/38 96/91 80/20 86/69 44/56 72/6 c 81/19 87/64

a isolated yield of anti-10 + syn-10. b

This journal is © The Royal Society of Chemistry 20xx J. Name., 2013, 00, 1-3 | 3

Acknowledgements

We acknowledge Université Paris Saclay, CNRS, Ministère de l'Europe et des Affaires Etrangères (bourse d'excellence "Eiffel") and Charm3At LabEx (ANR-11-LABEX-0039) for financial support.

Conflicts of interest

There are no conflicts to declare.