M Raous

L Cangémi

M Cocu

A consistent model coupling adhesion, friction, and unilateral contact

A model considering both unilateral contact, Coulomb friction, and adhesion is presented. In the framework of continuum thermodynamics, the contact zone is considered as a material boundary and the local constitutive laws are derived by choosing two specific surface potentials : the free energy and the dissipation potential. Because of the non regular properties of these potentials, convex analysis is used to derive the local behavior laws from the state and the complementary laws. The adhesion is characterized by an internal variable β, introduced by Frémond, which represents the intensity of adhesion. The continuous transition from a total adhesive condition to a possible pure frictional one is enforced by using elasticity coupled with damage for the interface. Non penetration conditions and Coulomb law are strictly imposed without using any penalty. The variational formulation for quasistatic problems is written as the coupling between an implicit variational inequality, a variational inequality, and a differential equation. An incremental formulation is proposed. An existence result under a condition on the friction coefficient is given. A numerical method is derived from the incremental formulation and various algorithms are implemented : they solve a sequence of minimization problems under constraints. The model is used to simulate a micro-indentation experiment conducted to characterize the behavior of fiber/matrix interface in a ceramic composite. Identification of the constitutive parameters is discussed.

Introduction

This paper is concerned with the coupling of unilateral contact, friction, and adhesion. The present model considers the interface as a material surface. The unilateral condition and the Coulomb friction law are strictly imposed without using any penalty. Adhesion and friction are strongly coupled through a specific compliance with damage that acts only in traction or shear and that disappears when the contact displacements increase. This insures a continuous transition between total adhesive and pure frictional states [START_REF] Cangémi | Frottement et adhérence : modèle, traitement numérique et application à l'interface fibre/matrice[END_REF][START_REF] Raous | Un modèle couplant adhérence et frottement pour le contact entre deux solides déformables[END_REF]. Adhesion is characterized by an intensity of adhesion β, previously introduced by Frémond [START_REF] Frémond | Adhérence des solides[END_REF]. In the present model, evolution of this intensity of adhesion β includes a viscosity effect. The thermodynamics basis of the model are given in Section 2 and a variational formulation is considered in Section 3. It is written under the form of the coupling between two variational inequalities (of which one is implicit) and a differential equation. Introducing an incremental formulation allows us to give an existence theorem when the friction coefficient is small. This incremental formulation suggests two numerical schemes to solve the problem. Several algorithms are presented in Section 4. In Section 5, the model is used to simulate the micro-indentation of a fiber of a composite material. Identification of the model parameters is presented, and ability of the model to describe the behavior of the fiber/matrix interface is discussed.

The present work is based on Frémond's work on adhesion [START_REF] Frémond | Adhérence des solides[END_REF] and on previous works on quasistatic problems with friction [START_REF] Chabrand | Various numerical methods for solving unilateral contact problems with friction[END_REF][START_REF] Cocu | Formulation and approximation of quasistatic frictional contact[END_REF][START_REF] Raous | Un modèle couplant adhérence et frottement pour le contact entre deux solides déformables[END_REF][START_REF] Raous | Numerical methods for frictional contact problems and applications[END_REF]. Most of the works on interface modelling have been conducted on fiber/matrix interface or on delamination. Some of them are analytical [START_REF] Hutchinson | Models of fiber debonding and pull-out in brittle composites with friction[END_REF][START_REF] Marshall | An indentation method for mesuring matrix fiber frictional stresses in ceramic composites[END_REF][START_REF] Péres | Analyse théorique et expérimentale du rôle des paramètres de microstructure sur le comportement des composites à matrice fragile[END_REF][START_REF] Shetty | Shear-lag analysis of fiber push-out (indentation) tests for establishing interfacial friction stress in ceramic matrix composites[END_REF]. They give a global analysis of the decohesion, and friction modelling (when it is considered) is elementary (Tresca's law). Lemaitre [START_REF] Lemaitre | Formulation de l'endommagement des interfaces[END_REF] has given a formulation of interface damage based on a viscoplastic analysis restricted to a surface. Needleman [START_REF] Needlman | A continuum model for void nucleation by inclusion debonding[END_REF][START_REF] Needlman | An analysis of tensile decohesion along an interface[END_REF] has proposed a decohesion model without friction, based on atomistic considerations. An exponential representation is used for the interface potential. A few years later, Tvergaard [START_REF] Tvergaard | Effect of fiber debonding in a whisker-reinforced metal[END_REF] introduced a non dimensional damage parameter to include a non reversible evolution of decohesion. The same author added the coupling with a frictional behavior, which is active only when adhesion is totally broken. This induces a non smooth evolution of the contact forces, which turn back to zero before friction starts. Recently Chaboche et al [START_REF] Chaboche | On the Interface Debonding Models[END_REF] have introduced a way, slightly different of ours [START_REF] Cangémi | Adhesion and friction model for the fiber/matrix interface of a composite[END_REF], to couple the two phenomena ; it insures the continuity of the stress and displacement evolutions. Stupkiewicz [START_REF] Stupkiewicz | Fiber sliding model accounting for interfacial micro-dilatancy[END_REF] adds the influence of micro-asperities in a strip model [START_REF] Mroz | Hysteretic effects and progressive delamination at composite interfaces[END_REF] governed by energy or stress criteria. In these models, the contact conditions (both the non penetration condition and the friction law) are regularized and viscosity is not considered for the evolution of adhesion. For homogenization problems, decohesion between inclusions and matrix has been considered in [START_REF] Needlman | A continuum model for void nucleation by inclusion debonding[END_REF] and in [START_REF] Michel | Une modélisation du rôle des interfaces dans le comportement des composites à matrice métallique[END_REF], where Michel et al use a Griffith criterion derived from interface damage model with elastic domain. For delamination problems, adhesion has been widely studied by Allix and Ladeveze [START_REF] Allix | Interlaminar interface Modelling for prediction of delamination[END_REF][START_REF] Allix | Damage analysis of interlaminar fracture specimens[END_REF]. Contrary to the previous progressive decohesion models, several approaches have considered brittle break down for the interface adhesion (either analytical or numerical treatment) [START_REF] Hutchinson | Models of fiber debonding and pull-out in brittle composites with friction[END_REF][START_REF] Leguillon | Un exemple d'intéraction singularité-couche limite pour la modélisation de la fracture dans les composites[END_REF][START_REF] Martin | Influence des conditions de couplage fibre-matrice sur le comportement à la rupture d'un microcomposite fragile-fragile[END_REF][START_REF] Martinez | Energy criterion for crack deflection at an interface between two orthotropic media[END_REF]. On the basis of material boundary hypothesis Klarbring [START_REF] Klarbring | Derivation and analysis of rate boundary problems of frictional contact[END_REF] proposed a thermodynamic analysis of frictional contact and Strömberg et al [START_REF] Strömberg | Derivation and analysis of a generalized standard model for contact, friction and wear[END_REF] gave an extension to take wear into account.

Some works have been done considering an interphase model with the presence of a third body [START_REF] Chaboche | Numerical analysis of composite systems by using interphase/interface models[END_REF]. Although a third body approach would be attractive for the sake of simplicity of the concept, numerical simulations are difficult for two reasons : the parameters and the real geometry of the third body are very hard to determine, and the numerical treatment (mesh,...) is difficult because the interphase domain is one hundred or one thousand times smaller than the other domains under consideration. On the contrary, asymptotic methods and theoretical studies on the equivalent behavior of the third body when the thickness decreases to zero are very constructive [START_REF] Eldund | Surface adhesive joint description with coupled elastic-plastic damage behavior and numerical applications[END_REF][START_REF] Ganghoffer | Modèles asymptotiques de joints collés par un adhésif viscoplastique[END_REF][START_REF] Klarbring | Derivation of a model of adhesively bonded joints by the asymptotic expansion method[END_REF][START_REF] Licht | Comportement asymptotique d'une bande dissipative mince de faible rigidité[END_REF][START_REF] Licht | Une modélisation du comportement d'un joint collé élastique[END_REF][START_REF] Sanchez-Palancia | Passage à la limite de l'élasticité tridimensionnelle à la théorie asymptotique des coques minces[END_REF].

A thermodynamic formulation for the coupling of adhesion and friction

Hypotheses and notations

Let Ω 1 and Ω 2 be two domains of R d (d = 2, 3) occupied by two continuous bodies, with a boundary separated in three disjoint parts for each domain :

∂Ω α = Γ α U ∪ Γ α φ ∪ Γ α C α = 1, 2 .
Let Γ 1 φ , Γ 2 φ denote the parts of the boundary where external forces φ 1 , φ 2 are

C U Γ 1 U Γ φ 2 C Γ 2 φ Γ 1 n Γ 2 φ 1 φ 1 2 n f C Γ Ω 2 1 Γ 2 f Ω 1 1 2
Fig. 1. Contact between two elastic bodies respectively applied (see Fig. 1). In the same manner, Γ 1 U and Γ 2 U are the parts of ∂Ω 1 , ∂Ω 2 where the displacements are prescribed and f 1 and f 2 denote the imposed volume force densities. Let Γ 1 C , Γ 2 C denote the parts of the boundary where the two solids are initially in contact. Introducting a gap in the unilateral conditions would allow the treatment of a case where contact may occur on a part larger than the present Γ C . Assuming small displacements hypothesis, we have n 1 = -n 2 , where n α denotes the outward normal unit vector to ∂Ω α , α = 1, 2. Thus each particle of Γ 1

C is coupled with a particle of Γ 2

C

in a single valued correspondence. Consequently, the two material boundaries Γ α C ⊂ ∂Ω α (α = 1, 2) define at the beginning a common zone of contact denoted Γ

C : Γ 1 C ≃ Γ 2 C ≃ Γ C .
The relative displacement between the two bodies is defined on Γ C by [u] = u 2 -u 1 where u 1 and u 2 measure the displacements of two corresponding points. For the normal/tangential decomposition, the vector

n 1 is chosen (n = n 1 = -n 2 ) : [u] = U N n + U T with U N = [u]
.n. According to this choice, the non penetration condition will be written U N ≥ 0 (convenient for a gap condition). When the second solid is rigid (u 2 = 0), the usual Signorini condition

U N = u 1 .n = u 1 .n 1 ≤ 0 is obtained.
The internal force on Γ C is denoted R. By using the principle of virtual power (see [START_REF] Daher | The method of virtual power in continuum mechanics. Application to media presenting singular surfaces and interfaces[END_REF]), we obtain : R = σ 1 n 1 = -σ 2 n 2 , according to the previous choice for [u].

The general framework

Considering the contact area Γ C as a material boundary [START_REF] Daher | The method of virtual power in continuum mechanics. Application to media presenting singular surfaces and interfaces[END_REF][START_REF] Frémond | Adhérence des solides[END_REF][START_REF] Klarbring | Derivation and analysis of rate boundary problems of frictional contact[END_REF][START_REF] Strömberg | Derivation and analysis of a generalized standard model for contact, friction and wear[END_REF], we introduce a surface density of internal energy E and a density of entropy S associated to the pseudo domain Γ C . Then the Helmholtz free energies can be written Ψ = E -ST on Γ C and ψ = e -sT in Ω 1 ∪ Ω 2 (where e is the specific internal energy, s the specific entropy and T the absolute temperature). According to Frémond [START_REF] Frémond | Equilibre des structures qui adhèrent à leur support[END_REF][START_REF] Frémond | Adhérence des solides[END_REF], we introduce the internal state variable β, which represents the intensity of adhesion (β = 1 means that the adhesion is total, β = 0 means that there is no adhesion and 0 < β < 1 is the case of partial adhesion). If adhesion is described with a damage parameter D, we have β = 1 -D. As Michel et al [START_REF] Michel | Une modélisation du rôle des interfaces dans le comportement des composites à matrice métallique[END_REF] and Needlman [START_REF] Needlman | A continuum model for void nucleation by inclusion debonding[END_REF], let us introduce the normalized relative displacements u N = U N /l and u T = U T /l where l is a characteristic length associated to the geometry such that u N < 1 and u T < 1. Numerous choices are possible : in the example of Section 6, we take l equal to the radius of the fiber. The following state variables are chosen : ǫ = (ǫ ij ) , the strain tensor in Ω 1 ∪ Ω 2 , the normalized relative displacement u T and u N and the intensity of adhesion β on Γ C . Classically, let σ r denote the thermodynamic force associated to ǫ, where the superscript (.) r means reversible part. In the same manner, we take R r N and R r T as the thermodynamic forces associated to u N and u T . The internal variable β is then related to the thermodynamic force of decohesion denoted by G β (see Table 1).

Table 1 The thermodynamic variables.

State variables

ǫ u N u T β Thermodynamic forces σ r R r N R r T G β

Free energies and state laws

Into the domain Ω 1 ∪ Ω 2 , the free energy ψ is the potential of elasticity (1), and the classical state law (2) is written as follows.

ψ α (ǫ) = 1 2ρ A α ijkl ǫ ij ǫ kl (1)
(σ r ) α = ρ ∂ψ α (ǫ) ∂ǫ (2)
In the following, we will consider only what concerns the contact boundary Γ C , the part concerning Ω 1 ∪ Ω 2 being a classical elasticity problem. Onto Γ C , the surface density of free energy Ψ is chosen as follows :

Ψ(u N , u T , β) = 1 l CN 2 U 2 N β 2 + CT 2 U T 2 β 2 -w h(β) + I K (U N ) + I P (β) (3)
where K = {v / v ≥ 0} and P = {γ / 0 ≤ γ ≤ 1}.

Introducing the new variables u N = U N /l and u T = U T /l, the potential (3) can be written as :

Ψ(u N , u T , β) = C N 2 u 2 N β 2 + C T 2 u T 2 β 2 -w h(β) + I K (u N ) + I P (β) (4)
where C N = l CN and C T = l CT .

The introduction of the indicator functions I K and I P imposes the unilateral condition u N ≥ 0 and the condition β ∈ [0, 1]. The interfacial forces induced by the adhesion are introduced under the form of a compliance law depending on the current state of adhesion β and characterized by the initial stiffness C N and C T . In the work of Truong Dinh Tien [START_REF] Truong Dinh Tien | Contact avec adhérence[END_REF], concerning a frictionless problem with adhesion, the penalty coefficients introduced for algorithmic reasons can be considered as large limit values of C N and C T . The term w h(β) is a general form of the energy of decohesion which is supposed to be differentiable. With h(β) = β, w could be considered as Dupré's energy. Other forms for h(β) could be considered.

The pseudo-potential Ψ can be separated into :

• a part (the first three terms) that is differentiable but not convex according to the couple (u, β) (it is only convex according to u or to β separately), • a part (the last two terms) that is convex but not differentiable.

To write the state laws, the two difficulties (lack of convexity and lack of differentiability) are overcome by using local or partial subdifferentiation [START_REF] Cangémi | Frottement et adhérence : modèle, traitement numérique et application à l'interface fibre/matrice[END_REF][START_REF] Daher | The method of virtual power in continuum mechanics. Application to media presenting singular surfaces and interfaces[END_REF][START_REF] Frémond | Adhérence des solides[END_REF][START_REF] Raous | Un modèle couplant adhérence et frottement pour le contact entre deux solides déformables[END_REF]. Then the state laws can be written as :

R r N ∈ ∂ u N Ψ(u N , u T , β) (5) R r T ∈ ∂ u T Ψ(u N , u T , β) (6) -G β ∈ ∂ β Ψ(u N , u T , β) (7)
where ∂ u and ∂ β denote respectively the subdifferential with respect to the variables u and β.

The following relation ( 8) is easily deduced from [START_REF] Chaboche | Numerical analysis of composite systems by using interphase/interface models[END_REF]. By making explicit the subdifferentials ∂ u N and ∂ β in ( 5) and ( 7), we obtain the following relations on the normal components ( 9) and on the thermodynamic force G β [START_REF] Cocu | Approximation results for a class of quasistatic contact problems including adhesion and friction[END_REF].

R r T = C T u T β 2 (8)
u N ≥ 0 -R r N + C N u N β 2 ≥ 0 -R r N + C N u N β 2 u N = 0 (9)              G β ≥ w h ′ (β) if β = 0 G β = w h ′ (β) -(C N u 2 N + C T u T 2 ) β if β ∈ ]0, 1[ G β ≤ w h ′ (β) -(C N u 2 N + C T u T 2 ) if β = 1. (10)
The state laws show that :

-the reversible (elastic) part of the tangential force depends on the square of the adhesion β, -the reversible parts of the normal components of R and u satisfy a generalized Signorini condition (unilateral contact), -the thermodynamic force G β , if β ∈]0, 1[, is composed of the adhesive energy minus the elastic energy of the interface.

Dissipation and complementary laws

The two laws of thermodynamics give the following form of the Clausius-Duhem inequalities (in an isothermic evolution) :

     ρ ψ ≤ σ • ǫ in Ω 1 ∪ Ω 2 Ψ ≤ R N uN + R T . uT on Γ C . (11)
In Ω α the behavior is totally reversible (elastic) and so σ = σ r . On Γ C , using the state laws ( 5), ( 6), [START_REF] Chabrand | Programmation mathématique pour le contact avec frottement et comparaison avec d'autres méthodes, Actes 2ième Colloque de Calcul des Structures[END_REF] to evaluate the time derivative of Ψ in the second Clausius Duhem inequality [START_REF] Cangémi | Frottement et adhérence : modèle, traitement numérique et application à l'interface fibre/matrice[END_REF], the following expression for the dissipation D is obtained.

D = (R N -R r N ) uN + (R T -R r T ). uT + G β β ≥ 0 ( 12
)
The irreversible part of the contact forces are

R ir N = R N -R r N and R ir T = R T -R r T .
The only dissipative processes under consideration are friction and adhesion. To formulate the following complementary laws where the irreversible forces verify [START_REF] Daher | The method of virtual power in continuum mechanics. Application to media presenting singular surfaces and interfaces[END_REF], we have to choose a pseudo-potential of dissipation Φ = Φ( uT , β; χ N ) with χ N = (R r N , u N , β) , positive, convex in ( uT , β) , and zero in (0, 0) (extension of standard generalized materials). We choose the following form for the pseudo-potential of dissipation :

Φ uT , β; χ N = µ R N -C N u N β 2 uT + b p + 1 β p+1 + I C -( β) ( 13
)
with C -= {γ ∈ W/γ ≤ 0} and p ≤ 1. A power law is considered for the evolution of the adhesion. The indicator function imposes that β ≤ 0 : the adhesion is allowed only to decrease and cannot be regenerated (not reversible). This is in agreement with our application but other choices could be made for other situations. The parameter µ is the friction coefficient of the Coulomb law. The parameter b characterizes a time dependent evolution of the adhesion.

The complementary laws are then written :

R ir N = 0 (14) (R ir T , G β ) ∈ ∂Φ( uT , β; χ N ) (15)
and ( 15) can be written as :

R ir T ∈ ∂ uT Φ( uT , β; χ N ) (16)
G β ∈ ∂ β Φ( uT , β; χ N ) (17)
The normal behavior has been supposed to be totally elastic. Making explicit the subdifferentials in ( 16) and ( 17), we obtain on Γ C :

R T -C T u T β 2 ≤ µ R N -C N u N β 2 (18)
with :

R T -C T u T β 2 < µ R N -C N u N β 2 ⇒ uT = 0 R T -C T u T β 2 = µ R N -C N u N β 2 ⇒ ∃ λ ≥ 0 , uT = λ(R T -C T u T β 2 ) β = -G - β /b 1/p , (19)
where G - β denotes the negative part of G β .

The local model

Putting together the various relations obtained in the previous sections, the local model coupling adhesion, friction, and unilateral contact is written on Γ C as follows.

Unilateral conditions with adhesion

-R N + C N u N β 2 ≥ 0 , u N ≥ 0 , (-R N + C N u N β 2 ) u N = 0 (20)
Coulomb friction with adhesion

R r T = C T u T β 2 R T -R r T ≤ µ |R N -C N u N β 2 | R T -R r T < µ |R N -C N u N β 2 | ⇒ uT = 0 R T -R r T = µ |R N -C N u N β 2 | ⇒ ∃λ ≥ 0 , uT = λ(R T -R r T ) (21)
Evolution of adhesion intensity

β = -(1/b) (w h ′ (β) -(C N u 2 N + C T u T 2 )β ) -1/p if β ∈ [0, 1[ β ≤ -(1/b) (w h ′ (β) -(C N u 2 N + C T u T 2 )β ) -1/p if β = 1 . (22)
The contact variables are u N , u T , β, R N , R T . The six parameters of the model are C N , C T (initial normal and tangential stiffness of the interface if adhesion is complete), µ (the friction coefficient), b (viscosity of the adhesion evolution), w (limit of decohesion energy) and p (the power coefficient). In what follows, we take h(β) = β.

Let us now analyze the interface behavior for a 2D case. The initial conditions are supposed to be complete adhesion (β = 1) and zero displacement (u T = u N = 0). Considering first the normal behavior (see Fig. 2), under compressive action, the non penetration condition is strictly verified (u N = 0).

Under traction (u N ≥ 0), an adhesive resistance (R (elasticity with damage). The intensity of adhesion starts to decrease when the displacement is sufficiently large such that the elastic energy becomes larger than the limit of adhesion energy w. Evolution of the adhesion is then governed by equation [START_REF] Leguillon | Un exemple d'intéraction singularité-couche limite pour la modélisation de la fracture dans les composites[END_REF]. When adhesion is totally broken, the classical Signorini problem is obtained.

N = C N u N β 2 )
Considering now the shear behavior (see Fig. 3), note first that friction acts only if a normal compression is applied ; if a normal traction is applied (u N > 0), the sliding limit (µ [START_REF] Klarbring | A mathematical programming approach to contact problems with friction and varying contact surface[END_REF] and the tangential behavior is elastic with damage (R T = C T u T β 2 ). Under compression, the sliding limit is (µ |R N |) (because u N = 0). As long as the norm of the tangential force R T is smaller than the sliding limit, sliding does not occur (u T = 0 as initial condition and uT = 0 in relation ( 21)). When the sliding limit is reached, an elastic tangential displacement occurs. The adhesion begins to decrease when the adhesive limit is reached and evolution of β is then governed by [START_REF] Leguillon | Un exemple d'intéraction singularité-couche limite pour la modélisation de la fracture dans les composites[END_REF]. When adhesion is lost (β goes to zero), the usual Coulomb friction conditions are obtained.

|R N -C N u N β 2 |) is zero because of
If the loading remains constant (when the adhesion limit is overcome), the adhesion keeps decreasing (by relaxation). If the tangential loading is now backward, an opposite tangential displacement occurs only when the other side of the Coulomb cone is reached.

3 Variational formulation of the quasistatic problem

The quasistatic problem

Problem P1.

Find the displacements u α , the stresses σ α (α = 1, 2), the strains ǫ, the contact force R such that :

ǫ = grad s u α in Ω α (23) div σ α + f α = 0 in Ω α (24) σ α n α = φ α on Γ α φ (25) u α = 0 on Γ α U (26) [u] = u 2 -u 1 = u N n + u T on Γ C (27) σ 1 n 1 = -σ 2 n 2 = R N n + R T on Γ C (28) σ α = K α : ǫ in Ω α (29)
and on Γ C :

-R N + C N u N β 2 ≥ 0 , u N ≥ 0 , -R N + C N u N β 2 u N = 0 (30) R r T = C T u T β 2 R T -R r T ≤ µ R N -C N u N β 2 (31) R T -R r T < µ R N -C N u N β 2 ⇒ uT = 0 R T -R r T = µ R N -C N u N β 2 ⇒ ∃λ ≥ 0 , uT = λ(R T -R r T ) , β = -(1/b) w h ′ (β) -(C N u 2 N + C T u T 2 )β -1/p if β ∈ [0, 1[ , (32)
β ≤ -(1/b) w h ′ (β) -(C N u 2 N + C T u T 2 )β -1/p if β = 1 .
The time dependence of the loading is slow enough to make the inertial terms negligible.

The variational formulation

In what follows, we make the simplest hypothesis of a linear dissipation for the adhesion evolution (case p = 1).

We denote by V α the following spaces :

V α = v α ∈ [H 1 (Ω α )]
d ; v α = 0 a.e. on Γ α U , α = 1, 2, and we set

V = V 1 × V 2 , K = {v = (v 1 , v 2 ) ∈ V ; v N ≥ 0 a.e. on Γ C } , H = L ∞ (Γ C ).
For all v ∈ V the norm on V is given by v

V = v 1 V 1 + v 2 V 2 and ., . shall denote the duality pairing on H 1 2 (Γ C ) × H -1 2 (Γ C
) . We suppose that :

f ∈ W 1,2 (0, T ; [L 2 (Ω 1 )] d × [L 2 (Ω 2 )] d ), φ ∈ W 1,2 (0, T ; L 2 (Γ 1 φ ) d × L 2 (Γ 2 φ ) d ),
which imply that F ∈ W 1,2 (0, T ; V ), where

(F, v) = α=1,2 Ω α f α .v α dx + Γ α φ φ α .v α ds ∀v ∈ V.
We shall adopt the following notations :

• a : V × V -→ IR, a(u, v) = a 1 (u 1 , v 1 ) + a 2 (u 2 , v 2 ) ∀u = (u 1 , u 2 ), v = (v 1 , v 2 ) ∈ V, where a α (u α , v α ) = Ω α A α ijkl ǫ ij (u α )ǫ kl (v α )dx, α = 1, 2 . • j : H × V × V -→ IR , j(β, u, v) = Γ C µ|σ * N (P u 1 ) + C N β 2 u N | v T ds,
where P is the projection of W 1,2 (0, T ;

V 1 ) on V 1 0 with V 1 0 = w 1 ∈ W 1,2 (0, T ; V 1 ); T 0 a 1 (w 1 , ψ) dt = T 0 (f 1 , ψ) [L 2 (Ω 1 )] d dt + T 0 (φ 1 , ψ) [L 2 (Γ 1 φ )] d dt, ∀ ψ ∈ L 2 (0, T ; V 1 ), ψ = 0 a.e. on Γ C ×]0, T [ , µ ∈ L ∞ (Γ C ) and µ ≥ 0, ( .
) * : H -1 2 (Γ C ) → L 2 (Γ C
) is a linear and compact mapping.

• c N , c T : H × V × V -→ IR , c N (β, u, v) = Γ C C N β 2 u N v N ds and c T (β, u, v) = Γ C C T β 2 u T .v T ds. • y(β, u) = - 1 b w -(C N u 2 N + C T u T 2 )β -.
Then the local problem P1 admits the following variational formulation (see [START_REF] Cangémi | Frottement et adhérence : modèle, traitement numérique et application à l'interface fibre/matrice[END_REF][START_REF] Cocu | Formulation and approximation of quasistatic frictional contact[END_REF][START_REF] Raous | Un modèle couplant adhérence et frottement pour le contact entre deux solides déformables[END_REF]).

Problem P2. Find (u, β) ∈ W 1,2 (0, T ; V ) × W 1,2 (0, T ; H) such that u(0) = u 0 , β(0) = β 0 and for almost all t ∈ [0, T ], u(t) ∈ K and a(u, v -u) + j(β, u, v) -j(β, u, u) + c T (β, u, v -u) ≥ (F, v -u) + σ N (u 1 ), v N -uN ∀ v ∈ V (33) σ N (u 1 ), z N -u N + c N (β, u, z -u) ≥ 0 ∀ z ∈ K, (34)
β = y(β, u) a.e. on Γ C , (35)
where the initial conditions u 0 ∈ K, β 0 ∈ H, β 0 ∈ [0, 1[ a.e. on Γ C and satisfy the following compatibility condition :

a(u 0 , w -u 0 ) + j(β 0 , u 0 , w -u 0 ) + c T (β 0 , u 0 , w -u 0 ) ≥ (F (0), w -u 0 ) ∀w ∈ K. ( 36
)
This problem is a generalization of the quasistatic unilateral contact problem with friction considered by Cocu et al in [START_REF] Cocu | Formulation and approximation of quasistatic frictional contact[END_REF].

Incremental formulation and mathematical results

First, let us focus on the properties of the bilinear form a and the mapping j involved in the problem P2 : a(., .) is continuous on V × V and coercive i.e. it satisfies

∃M > 0 ∀u ∈ V ∀v ∈ V |a(u, v)| ≤ M u V v V , ( 37
) ∃m > 0 ∀u ∈ V a(u, u) ≥ m u 2 V , (38)
and the mapping j(., ., .) satisfies the following property :

∃C > 0 ∀u, ū, v, v ∈ V 1 0 |j(β, u, v) -j(β, u, v) -j(β, ū, v) + j(β, ū, v)| ≤ μC u -ū V v -v V , (39)
with μ = |µ| L ∞ (Γ C ) .
An incremental formulation is obtained by operating a time discretization of problem P2, taking n ∈ IN * and setting ∆t = T /n, t i = i ∆t and F i = F (t i ) for i = 0, ..., n. Using an implicit scheme we obtain the following sequence of problems (P n i ), i = 0, ..., n -1, defined for a given (u 0 , β 0 ) ∈ K × H .

Problem P n i . Find (u i+1 , β i+1 ) ∈ K × H such that : a(u i+1 , v -u i+1 ) + j(β i+1 , u i+1 , v -u i ) -j(β i+1 , u i+1 , u i+1 -u i ) +c(β i+1 , u i+1 , v -u i+1 ) ≥ (F i+1 , v -u i+1 ) ∀v ∈ K (40) β i+1 -β i = ∆t y(β i+1 , u i+1 ) a.e. on Γ C , (41)
where

c(•) = c N (•) + c T (•).
The incremental problem P n i solved using a fixed point method as follows.

For every ū ∈ K let us denote by s(ū) = β the solution of

β = ∆t y(β, ū) + β i . (42)
For every β ∈ H we denote by u(β) the solution of

u ∈ K a(u, v -u) + j(β, u, v -u i ) -j(β, u, u -u i ) +c(β, u, v -u) ≥ (F i+1 , v -u) ∀v ∈ K. ( 43
)
The existence of the solution for problem ( 42) is clear and inequalities such as ( 43) have a unique solution if the friction coefficient is sufficiently small i.e. μ < m C (see [START_REF] Cocu | Existence of solutions of Signorini problems with friction[END_REF]). We suppose from now on that µ satisfies this condition.

We define the mapping T :

K -→ K by ∀ū ∈ K T (ū) = u(s(ū)).
By a straightforward computation, we obtain that ∃ k 1 > 0 such that for all ū1 , ū2 ∈ K

|s(ū 1 ) -s(ū 2 )| ≤ k 1 ∆t ( ū1 + ū2 ) [ū 1 -ū2 ] a.e. on Γ C . ( 44
)
Let us now set u 1 = u(s(ū 1 )), u 2 = u(s(ū 2 )). Adding the inequalities [START_REF] Stupkiewicz | Fiber sliding model accounting for interfacial micro-dilatancy[END_REF] with u = u 1 , v = u 2 and u = u 2 , v = u 1 and using the properties of a, j and b one can show that ∃ k 2 > 0 such that for all ū1 , ū2 ∈ K

||u 1 -u 2 || 2 ≤ k 2 Γ C |u 2 | |u 1 -u 2 | |s(ū 1 ) -s(ū 2 )|ds. ( 45
)
From the inequalities ( 44), [START_REF] Tvergaard | Effect of fiber debonding in a whisker-reinforced metal[END_REF] and the relation ||u|| ≤ k 3 (with k 3 independent of ū), satisfied by u(s(ū)), it follows that T is a contraction mapping for sufficiently small ∆t. Then T has a unique fixed point u and (u, s(u)) is the solution of the incremental problem P n i (see also [START_REF] Cocu | Approximation results for a class of quasistatic contact problems including adhesion and friction[END_REF]).

Numerical methods and algorithms

The previous sections showed that the two variational inequalities (one is implicit) coupled with the differential equation on β can be reduced for the incremental formulation to the only one variational inequality [START_REF] Sanchez-Palancia | Passage à la limite de l'élasticité tridimensionnelle à la théorie asymptotique des coques minces[END_REF]. By extending the results of [START_REF] Cocu | Formulation and approximation of quasistatic frictional contact[END_REF], it is evidenced that this variational inequality is very similar to the one obtained for the static problem of Coulomb friction with unilateral contact without adhesion. As shown in [START_REF] Cocu | Formulation and approximation of quasistatic frictional contact[END_REF], two alternative discrete formulations (one set on the variables u i+1 and the other one set on the increments ∆u i+1 = u i+1 -u i ) can be associated : they include extra terms related to the previous time steps (velocity formulation of the friction) and to the variable β (coupling with adhesion).

For the time discretization, an implicit Euler method was introduced in the previous section. The implicit or semi-implicit character of the integration is important because the intensity of adhesion β may decrease very fast during the loading. A θ-method is also used ; it improves the accuracy of the solution. For the sake of simplicity, only the Euler method is presented here. A finite element approximation is used to solve the problem P n i at each time step t i+1 . Only 2D problems are considered here. In Note 2, a convenient treatment for 3D problems is given.

The approximate problem P h i associated to problem P n i is written:

Problem P h i .
For each time step t i+1 , find βh fixed point of the application s(•) :

β h -→ s u h (β h ) = β i h + ∆y β h , u h (β h ) , where u h (β h ) is solution of Problem Q h i . Problem Q h i . Find u h (β h ) ∈ K h such that ∀w h ∈ K h , a(u h , w h -u h ) + π h j h (β h , u h , w h -u i h ) -π h j h (β h , u h , u h -u i h ) + c(β h , u h , w h -u h ) ≥ (F i+1 h , w h -u h ) , where : -K h = v h ∈ U h / v h ≥ 0 on Γ C
is the set of the admissible displacements with (U h ) h a family of finite dimensional spaces which constitutes a finite element approximation of V , -π h denotes the projection on the finite element discretization; the choice of this extra projection for the approximate problem makes the approximation of the absolute value in j(•, •, •) much simpler (as shown in [START_REF] Licht | Remarks on a numerical method for unilateral contact including friction[END_REF]), -β i h , u i h are the approximate solutions computed for the previous step

t i , -y(β h , u h (β h )) = - 1 b w -(C N u h N 2 + C T u h T 2 )β h - -c(β h , u h , w h -u h ) = Γ C C N β h 2 u h N (w h N -u h N )ds+ Γ C C T β h 2 u h T .(w h T -u h T )ds , -j h (β h , u h , w h -u i h ) = Γ C µ R N (u h ) -C N β h 2 u h N w h T -u i hT ds .
For solving the implicit variational inequality in problem Q h i , let us introduce a fixed point method on the sliding limit. As show in [START_REF] Lebon | Multibody contact problems including friction in structure assembly[END_REF][START_REF] Licht | Remarks on a numerical method for unilateral contact including friction[END_REF][START_REF] Raous | Numerical methods for frictional contact problems and applications[END_REF], problem

Q h i is then equivalent to the problem R h i .
Problem R h i . Find ḡ fixed point of the application t(•) :

g -→ t u h (g) = µ R N (u h (g)) -C N β h 2 u h N (g) , where u h (g) is solution of problem S h i . Problem S h i . Find u h (g) ∈ K h such that ∀w h ∈ K h , a(u h (g), w h -u h ) + π h j ⋆ (w h -u i h ) -π h j ⋆ (u h (g) -u i h ) + c(β h , u h (g), w h -u h (g)) ≥ (F i+1 h , w h -u h (g)) ,
where

j ⋆ (v) = Γ C g v T ds.

Now, problem S h

i is a classical variational inequality problem associated to a Tresca problem with a given sliding limit g. A minimum principle can be associated, thus problem S h i is equivalent to the following problem

T h i . Problem T h i . Find u h g such that J u h g ≤ J (v) ∀v ∈ K h with J (v) = 1 2 a(v, v) + j ⋆ (v -u i h ) + 1 2 c(β h , v, v) -(F i+1 h , v).
This is a minimization problem under constraint (v ∈ K h ) of a quadratic functional including a non differentiable part (j ⋆ (v -u i h )). The discrete problem is then written as follows :

Problem U h i . Find u ∈ IK such that J (u) ≤ J (v) ∀v ∈ IK with J (v) = 1 2 v T Av + G T v -u i h + 1 2 v T C(β)v -F i+1 h T v,
where :

-

IK = {ΠK i with K i = R + if i ∈ I c N and K i = R if not } (if
the shape functions are strictly positive the approximation is internal : that is the case when linear elements (T3 or Q4) are used).

-I c N is the set of the number of degrees of freedom concerning the normal components of the contact nodes, -A is the matrix of dimension N = dim(V ) : A ij = a(w i , w j ), -C is the diagonal matrix of dimension M (M is the number of contact nodes) :

C kl = c(β h , w k , w l ), -G is the vector of dimension M : G j = Γ C g w j ds .
Problem U h i is very similar to the one solved in [START_REF] Cocu | Formulation and approximation of quasistatic frictional contact[END_REF][START_REF] Lebon | Multibody contact problems including friction in structure assembly[END_REF][START_REF] Raous | Numerical methods for frictional contact problems and applications[END_REF]. The main difference is that, at each step, the terms of A concerning the contact variables have to be modified by adding the term C kl related to the current contact condition. The memory of the loading history (velocity formulation of the friction) is given by the term w h -u i h in the functional j ⋆ of the variational inequality of problem S h i , where u i h is the solution at the previous time step. Various solvers of problem U h i have been implemented :

-Successive Over-Relaxation with Projection (see [START_REF] Lebon | Multibody contact problems including friction in structure assembly[END_REF][START_REF] Raous | Numerical methods for frictional contact problems and applications[END_REF]), -Gauss Seidel with Aitken acceleration and Projection, -Projected Conjugate Gradient with Preconditioning (see [START_REF] Raous | Preconditioned conjugate gradient method for a unilateral problem with friction[END_REF]).

These algorithms are robust, and the average number of iterations is smaller than the number of degrees of freedom. The convergence of the fixed point of the sliding limit g (Problem R h i ) is fast (generally 7 or 8 iterations), and a diagonal process (coarse resolution for the first values of g) improves the efficiency of the algorithm. Computational times are given in the next section.

Note 1 : An alternative choice of the unknowns. As given in [START_REF] Cocu | Formulation and approximation of quasistatic frictional contact[END_REF], problem S h i can also be written under the form of an equivalent problem where the unknown is the increment of displacement ∆u i+1 = u i+1 -u i ∆t ( velocity formulation). In that case, the convex IK changes at each step, and the memory of the loading appears in the contribution to the loading term of the contact forces at the previous time step.

Note 2 : An alternative solver (see [START_REF] Chabrand | Programmation mathématique pour le contact avec frottement et comparaison avec d'autres méthodes, Actes 2ième Colloque de Calcul des Structures[END_REF][START_REF] Chabrand | Various numerical methods for solving unilateral contact problems with friction[END_REF][START_REF] Klarbring | A mathematical programming approach to contact problems with friction and varying contact surface[END_REF][START_REF] Panagiatopoulos | Contact problems : theory, methods, applications[END_REF]). The incremental problem P h i can also be written as a complementarity problem. Two extra variables must be introduced (see [START_REF] Klarbring | A mathematical programming approach to contact problems with friction and varying contact surface[END_REF]) : sliding is separated into right and left slidings. After condensation of the problem, which is a reduction of the problem to only the contact variables, various algorithms of mathematical programming can be used. We have implemented Lemke's method (see [START_REF] Chabrand | Programmation mathématique pour le contact avec frottement et comparaison avec d'autres méthodes, Actes 2ième Colloque de Calcul des Structures[END_REF][START_REF] Chabrand | Various numerical methods for solving unilateral contact problems with friction[END_REF][START_REF] Panagiatopoulos | Contact problems : theory, methods, applications[END_REF][START_REF] Raous | Numerical methods for frictional contact problems and applications[END_REF]). It is a powerful pivoting direct method. Comparison with other methods can be found in [START_REF] Chabrand | Various numerical methods for solving unilateral contact problems with friction[END_REF]. By making the Coulomb cone polygonal, 3D problems can be treated (see [START_REF] Klarbring | A mathematical programming approach to contact problems with friction and varying contact surface[END_REF]).

6 An application : modelling of the fiber/matrix interface of a composite material

We have used the present model to describe the behavior of the fiber/matrix interface of composite materials. The global study deals with the interaction of a crack in the matrix with a fiber/matrix interface. The final topic is to optimize the characteristics of the interface in order to enforce the resistance of the composite to macro-and micro-crack progression. The first step of this study consists of the validation of the model by considering simulations on micro-indentation experiments carried out at the ONERA (Office National d'Etudes et de Recherches Aerospatiales). This is widely discussed from a mechanical point of view in [START_REF] Leroy | Etude et modélisation de l'essai de micro-indentation sur composite SIC/SIC[END_REF]. Some corrections of the experiment data have to be made to take into account the plastic zone under the indentor, and the global elasticity of the testing bench. Various geometries are considered, boundary conditions are discussed, and residual thermal stresses are taken into account. To complete the validation, different loadings are simulated : single loading, cycles, relaxation, different loading velocities, etc.

In the present paper, we present only the ability of the model to describe the complex behavior of a fiber/matrix interface during a micro-indentation experiment (push-in). A qualitative identification of the model parameters is conducted on a single experiment, and the numerical results are discussed in relation to the fundamental choices of the model : smooth evolution from adhesion to friction, strict unilateral conditions, etc.

Experiment and model

A push-in experiment carried out at the ONERA on a SIC/SIC composite is used to test the model. The parameters are identified on the plot of the evolution of the force on the indentor relatively to its prescribed displacement. Figure 4 shows the model geometry. The radius of the fiber is R f = 8.6µm. By analyzing the volume density of fibers (V f = 40%), the radius R m = 12.6µm is chosen to define a zone of pure matrix. The extra zone is defined as an equivalent homogeneous material equivalent to the composite one by using a mixture rule. Various global dimensions were tested. Results are given for L = 200µm (radius of the domain 23 times larger than the fiber one) and h = 1600µm. The boundary conditions are given on Fig. 4. The elasticity coefficients of the different materials are given on Table 2. The characteristic length l is taken equal to the fiber radius : l = 8.6µm. A vertical prescribed displacement is applied on the indentor. A frictionless unilateral contact is considered between the indentor and the upper part of the fiber.

Simulation and identification of the parameters

The finite element mesh is given on Fig. 5 and6. Triangle T 3 are used. The mesh has 1419 nodes of which 300 are contact nodes of the interface.

Results of the identification of the model parameters are given on Table 3 : the change of the slope is strongly related to the lost of adhesion and the final slope to the residual friction between the fiber and the matrix when adhesion is broken. On Fig. 8, the contact condition is presented for a given step of the load- ing. The values of the ratio R T /R N (scale on the left), the adhesion intensity β, and the tangential sliding u T (scales on the right) are plotted along the interface. Three zones are clearly characterized :

• zone 1 : close to the indentor, adhesion is totally broken (β ≃ 0) and only friction remains active (R T /R N = µ), • zone 2 : in this transition zone, adhesion is partial (0 < β < 1) and the elasticity acts (R T /R N > µ), • zone 3 : on this part, the interface is still weakly affected, the adhesion is total (β = 1).

The smooth evolution of the solution underlines the good property of the model, which gives a continuous transition from total adhesion to pure frictional contact.

Computational aspects

In that case, the Aitken -Gauss Seidel solver has been used. The number of iterations is quite the same as it is for the successive overrelaxation solver but the determination of the optimal relaxation parameter is not needed in that case. For the time discretization, the number of steps is 20. The time steps are chosen so as to correctly follow the changes in the condition of the contact: there is no convergence problem with large increments. For each step of the loading, the average number of iterations of the fixed point on the sliding limit g is 7 ; the one on the fixed point on the adhesion intensity β (treatment of the implicit integration) is 3. The average number of Aitken -Gauss Seidel solver is 600. This includes the fixed point iterations on g because a diagonal process is used : for the first values of g, a coarse resolution is conducted (the convergence test is coarse), and when more precise values of g are computed the accuracy of the resolution is enforced. The global CPU time, for the complete computation of the results given on Fig. 7 is 40 mn on a VAX 8400/bi-processor 350MHz.

A model coupling adhesion and friction that is based on the choice of convenient thermodynamics potentials has been elaborated. Damage, through traction and shear but not compression, that is introduced in the initial elasticity of the interface provides a smooth transition between the adhesive and frictional states. The unilateral contact conditions are strictly imposed without penalty regularization. It remains possible to add some elasticity in compression in case of mechanical necessity because the problem is then more regular. Viscosity of the evolution of the intensity of adhesion is considered. The limit case when viscosity may be neglected can be treated without numerical difficulties.

The variational formulation given as the coupling between an implicit variational inequality, a regular variational inequality, and a differential equation is constructive to establish an existence result and a condition of uniqueness, and to propose a discretization scheme.

The numerical methods based on optimization methods are robust. A direct mathematical programming method, presented in [START_REF] Chabrand | Programmation mathématique pour le contact avec frottement et comparaison avec d'autres méthodes, Actes 2ième Colloque de Calcul des Structures[END_REF][START_REF] Chabrand | Various numerical methods for solving unilateral contact problems with friction[END_REF][START_REF] Panagiatopoulos | Contact problems : theory, methods, applications[END_REF], may also be used.

This application to the simulation of fiber/matrix interface is only a test on the ability of the model to describe this kind of phenomena. A more complete analysis has been conducted in collaboration with ONERA [START_REF] Leroy | Etude et modélisation de l'essai de micro-indentation sur composite SIC/SIC[END_REF]. An extended paper will be presented. The role of viscosity and of anisotropy in the behavior of such an interface should be investigated. Our undergoing research concerns the interaction of a crack with an interface so as to better understand crack progression in composite materials [START_REF] Martinez | Energy criterion for crack deflection at an interface between two orthotropic media[END_REF].

Fig. 2 .

 2 Fig. 2. Normal behavior (for u T = 0 here)

Fig. 3 .

 3 Fig. 3. Tangential behavior (R N is constant)

Fig. 4 .

 4 Fig. 4. The micro-indentation test

Fig. 7 .

 7 Fig. 7. Simulation and experimental results.

Fig. 8 .

 8 Fig. 8. Contact forces, adhesion intensity β and tangential displacement u T along the interface for a given loading step.

Table 2

 2 Elasticity coefficients.

		E (Gpa)	ν
	fiber	200	0.25
	matrix	350	0.2
	composite	290	0.22

Acknowledgments : The experimental result in Fig. 7 was provided by the Office National d'Etudes et de Recherches Aerospatiales (ONERA).

Zoom

and on Fig. 7. The same contact stiffness was chosen for the normal and the tangential interface behavior : 7 shows the ability of the model to fit the force/displacement evolution of the push-in experiment.

A more realistic identification is conducted in [START_REF] Leroy | Etude et modélisation de l'essai de micro-indentation sur composite SIC/SIC[END_REF] on a collection of experiments. Mechanical considerations give a specified range of variations for each parameter. For the contact stiffness, the range is evaluated by considering the composition and the thickness of the layer of carbon and oxyde in the interface. On Fig. 7, we have also plotted the results of the simulation by making µ = 0 (no friction) and w = 0 (no adhesion) in the values of the parameters given on Table 3. This is helpful to better understand the experimental results


Dr Aleix Comas-Vives

Agustí Lledós

Rinaldo Poli
email: rinaldo.poli@lcc-toulouse.fr

Prof A Lledόs

Prof R Poli

A Computational Study of the Olefin Epoxidation Mechanism Catalyzed by Cyclopentadienyloxidomolybdenum(VI) Complexes

Keywords: epoxidation catalysis, DFT calculations, molybdenum, organometallic oxido complexes

whether they are published or not. The documents may come

Introduction

The olefin epoxidation reaction is extremely important as a relatively easy access way to functionalized organic materials from crude oil fractions. The reaction occurs with organic peroxides such as peracids and dioxiranes without the need of a catalyst. [START_REF] Schwesinger | Stereoselective synthesis[END_REF] The metalcatalyzed version has attracted much attention due to the possibility to perform more selective, notably enantioselective, transformations. Among the possible oxygen delivering agents, H2O2 is the subject of the greatest amount of investigations [START_REF] Lane | [END_REF] given its low cost and easy availability. However, tert-butylhydroperoxide (TBHP) is still heavily used in the research laboratory and industrially because it generally outperforms hydrogen peroxide. Many transition metal catalysts have been used to carry out this transformation, including high oxidation state oxido complexes (methyltrioxidorhenium, dioxido derivatives of Mo and W), [3][4][5][6][7] bis(peroxido) complexes [(L1)(L2)MO(O2)2] (M = Mo, W), [8] polyoxometallates, [9][10][11] a variety of oxido complexes generated in situ from Fe and Mn porphyrin, salen, and other coordination compounds. [12][13][14][15] The mechanism of this reaction has been and continues to be the subject of controversy. In many cases, a mechanism related to that accepted for organic peracids is proposed, [START_REF] Sheldon | Metal-catalyzed organic reactions[END_REF] see Scheme 1(a). However, this mechanism is easily understood only when H2O2 is the oxidant, since the active M-OOH species, Scheme 1(b), can be regenerated from the hydroxido product, M-OH, by simple ligand exchange. [START_REF] Faller | [END_REF] It cannot be the operative mechanism for the reactions using TBHP, unless a second oxygen atom transfer takes place to regenerate the active M-OOH species from TBHP and M-OH.

The fact that a large number of peroxido complexes of molybdenum and tungsten have been isolated and fully characterized [START_REF] Wilkinson | Comprehensive Coordination Chemistry[END_REF] has led to the consideration that a peroxido ligand is capable of transferring an oxygen atom to the olefin. Two reference mechanisms are based on this idea, usually referred to as the "Mimoun" [START_REF] Mimoun | [END_REF] and the "Sharpless" [20] mechanisms, see Scheme 2. Once again, for the reasons outlined above, these two mechanisms are easily understood only when H2O2 is used, less so when TBHP is involved. A number of theoretical investigations have addressed the two competing mechanisms shown in Scheme 2, mostly for Mo, [21][22][23][24][25][26][27][28][29] but also for other metals, [30,31] leading to a clear preference for the Sharpless scheme. The need to modify these mechanisms in order to account for the activity of TBHP has been presented by Thiel, [32][33][34] but to the best of our knowledge no thorough theoretical investigation has followed. Thiel's proposal is outlined in Scheme 3. Essentially, the peroxido ligand serves as a depository of the reactant's proton. The oxidizing agent is activated by coordination and becomes susceptible to nucleophilic attack by the olefin at the electrophilic oxygen atom. Note that this mechanism, proposed by Thiel for the specific oxidation with TBHP (R = tBu), may also operate for H2O2 (R = H). Note also that the basic principle is identical to that proposed by Sharpless (exogenous attack of the olefin at an electrophilic oxygen atom, without coordination). However, the oxygen is not transferred from a coordinated peroxido ligand, rather from the tbutylperoxido (or hydroperoxido) ligand. Finally, note that other metal-bonded functionalities may exert, in principle, the same proton depository function (for instance, oxido ligands).

M O O Thiel's pathway O O R H M O O O O H R M O O O O H R -ROH - O Scheme 3
Extensive experimental studies carried out with [MoO2X2L2]type catalysts and TBHP as oxidant have shown that the source of the oxygen atom for the epoxidation is TBHP and not the catalyst's oxido ligands, ruling out the possible involvement of direct oxygen atom transfer from Mo(O2) moieties and invalidating both Mimoun and Sharpless mechanisms as originally proposed. [35] Thus, the existence and stability of peroxido complexes must be related to side processes such as the deprotonation of hydroperoxido ligands, Mo-OOH. Indeed, a study by Bergman (the first catalytic study using an organometallic oxido derivative of molybdenum) has shown that the [Cp*MoO2Cl]/TBHP system is effective for olefin epoxidation, whereas the peroxido analogue, [Cp*MoO(O2)Cl], is catalytically inactive. [36] Thus, the peroxido compound cannot be implicated as a catalytic intermediate in the [Cp*MoO2Cl]-catalyzed epoxidation with TBHP. These findings have later been confirmed by Roesky, who also reported the x-ray structure of the [Cp*MoO(O2)Cl] compound. [37] Subsequent studies by Kühn and Romão have shown that related complexes with different cyclopentadienyl ligands, as well as alkyl derivatives of type [(Ring)MoO2R] (Ring = substituted cyclopentadienyl ligand, R = alkyl) are also catalytically active. [38][39][40] In a recent collaborative study, some of us have shown that the dinuclear oxido-bridged [(Ring)2Mo2O5] systems are also catalytically active when using TBHP in an organic solvent and are also active with the same reagent under aqueous biphasic conditions. [41] However, they do not yield significant amounts of epoxidation product when TBHP is replaced by H2O2. [START_REF]these conditions is currently unknown[END_REF] Only recently, the use of MeOOH as a model of the tBuOOH reagent has been considered in a theoretical study, based on compound MoO2Br2(MeN=CHCH=NMe) as catalyst. [35,43] The activation step of the oxidant reported by this study resembles that proposed by Thiel, except that an oxido ligand is used as the proton depository, see Scheme 4. Hydrogen bonding between the hydroxido proton and the -O atom of the organoperoxido ligand was found to stabilize this intermediate. However, the subsequent step of the mechanism is quite different than that proposed by Thiel and involves an insertion into the metal-peroxido bond, similar to the pathway proposed by Mimoun (though preliminary olefin coordination does not occur). The most puzzling feature of this mechanism is that the olefin insertion transition state is calculated as 52 kcal/mol higher in energy (63 kcal/mol higher in free energy) than the intermediate, which is itself 16 kcal/mol higher in energy (29 kcal/mol higher in free energy) than the starting materials (catalyst + MeOOH + olefin). This is a prohibitive activation barrier for an efficient catalytic process. For this reason, we thought that a new theoretical investigation of the epoxidation mechanism was warranted. [START_REF][END_REF] M Calhorda et al.

O O R H O M O O O H R M O O O H R -ROH - O Scheme 4
We have been interested in the aqueous chemistry of the Cp*Mo VI system [45] and have shown that compound [Cp*2Mo2O5] self-ionizes in water to yield a 1:1 mixture of [Cp*MoO2(H2O)] + and [Cp*MoO3] -, then evolving to a different ratio by spontaneous hydrolysis or by adjusting the pH with a buffer. [46] While the dinuclear compound may exert the same mechanistic function as [Cp*MoO2Cl] (the Cl ligand being replaced by the oxido-bridged Cp*MoO3 group), only the cationic complex is likely to exert a catalytic function among the charged species, since the water ligand can dissociate rather easily [47] and the resulting coordination site may be used for the oxidant activation. Useful background information comes from our recent computational study of hydration and proton transfer processes for the Cp*MoO2 + (aq) system. [47] Therefore, we have decided to examine the mechanism of the olefin epoxidation process by both Bergman's [Cp*MoO2Cl] system (which may also serve as a model for -indentify a low-energy pathway for oxygen transfer from TBHP (or H2O2) to the olefin; -understand the difference in catalytic activity between the oxido and peroxido derivatives for Bergman's system, [Cp*MoO2Cl] and [Cp*MoO(O2)Cl]; -find possible reasons for a different performance of TBHP vs.

H2O2 as an oxidant.

It is to be noted that the [Cp*MoO2X] system (X = Cl, CH3, Cp*MoO3) is isoelectronic with the [MoO2X2L2] system, see Scheme 5. Thus, the considerations resulting from our calculations on this system may be extrapolated to the more traditional class of [MoO2X2L2] catalysts. It is also to be noted that in a recent contribution, Colbran et al. have shown that a (perarylcyclopentadienyl)molybdenum(VI) dioxido complex, although catalytically active in cyclooctene epoxidation by TBHP, decomposes to a more active noncyclopentadienyl containing catalyst as the reaction proceeds. [48] However, an equivalent protonolysis is not necessarily associated also to the more robust [45] Cp*Mo bond. Furthermore, the isolobal relationship indicated in Scheme 5 makes a mechanistic investigation carried out for the Cp*Mo species also relevant to a putative product of hydrolytic decomposition. A more recent mechanistic study by Kühn et al. on the CpMoO2(CH3)-catalyzed epoxidation does not report any catalyst hydrolytic decomposition. [49]

Results and Discussion

Ethylene was used as a model substrate for our computational study (larger olefins, mostly cyclooctene, have been used for the experimental studies), while H2O2 ans MeOOH were the models for the oxidant. On the other hand, the full Cp* ligand was maintained in all calculations. The same numbering scheme will be used for the two cycles with different ROOH oxidants. Explicit reference to an individual system will be made by adding R as superscript, e.g. 3 H and 3 Me refer to the specific examples of intermediate 3. Unless otherwise stated, only the solvated energies (E CPCM ) will be reported in the figures and discussed. Related diagrams reporting the relative gas phase energies are available as Supporting Information. As stated in the Introduction, compound [Cp*MoO2Cl] was the first reported organomolybdenum olefin epoxidation catalyst but a theoretical investigation of the catalytic cycle using this compound has not been reported, to the best of our knowledge. In addition, the corresponding peroxido complex, [Cp*Mo(O2)OCl], 7, was described as catalytically inactive, [36,37] but a clear mechanistic interpretation of this phenomenon does not appear to be available. Calculations on this system may also be considered to model the action of other [Cp*MoO2X] catalysts (X = CH3 or Cp*MoO3). [38][39][40][41] These molecules are excellent epoxidation catalysts provided TBHP is used as the oxidant and the solvent is apolar. [START_REF]these conditions is currently unknown[END_REF] Chloroform is frequently used, therefore the calculations have been carried out by introducing the solvent effect by the CPCM in CHCl3.

On the basis of all previously reported experimental and computational investigations on Mo-catalyzed olefin epoxidation, [2, 21-26, 28-31, 35, 43] it is rather clear that the first step of the catalytic cycle consists of the activation of the oxidant molecule. Therefore, the first step of our investigation was an analysis of the coordination and activation of the ROOH molecule by 1. The energy profile is shown in Figure 1 while the optimized geometries of the key species are given in Figure 2 for R = Me (corresponding views of the systems with R = H are in the Supporting Information). The starting species 2 features an H bond between the oxidant as proton donor and an oxido ligand as proton acceptor, slightly stabilizing the system relative to the two separate molecules. CHCl3 solution; 24.9 for R = H). We have found a similar situation in a recent study of the intramolecular proton transfer leading from [Cp*MoO(OH)2] + to [Cp*MoO2(H2O)] + . [47] In that case, we found that the barrier could be dramatically reduced by the proton relay action of a water molecule, which is the reaction solvent, since this allows a reduced angular distortion. In CHCl3, the solvent molecules cannot assure a proton relay mechanism, but the same role may be exerted by additional ROOH molecules and also by the corresponding ROH co-product, once this has started to form, as shown in Scheme 6. Nevertheless, the related proton-relay mechanism did not led to a significant decrease in the energy barrier. This step leads to the formation of the activated complex [Cp*MoOCl(OH)(OOR)], 3. Note that 3 displays a significant interaction between the methyl(hydro)peroxido -O atom and the metal center (Mo-O  = 2.830 Å, and Mo-O  -O  = 109.0º for 3 Me ; 2.446 Å and 88.7° for 3 H ), forming what could be formally described as an alkylated/protonated metal peroxido [Mo( 2 -O2)] ligand. It also features an H-bond between the OH ligand as a proton donor and the -O atom of the OOH ligand as a proton acceptor. The nature of this product differs substantially from that obtained in the calculations by Calhorda et al., [43] where the two terminally bonded hydroxido and tert-butylperoxido ligands in H-O-Mo-O-O-Me establish a H-bond between OH as a proton donor and the OOMe O  atom as a proton acceptor. It seems that the O  atom prefers to donate its electrons to the electrophilic Mo center than to engage an H-bond with the OH ligand. Attempts to optimize a structure related to that reported by Calhorda et al. did not lead to a stable minimum for this system. Only two additional minima at slightly higher energy could be located for the R = H system (3' H and 3" H ) where the Mo-OOH moiety is oriented in the opposite direction (namely with a pseudo-equatorial O  atom and a pseudo-axial O  atom) and still featuring a weak Mo•••O  interaction (ca. 2.8 Å). Views relative energies of these minima are given in the Supporting Information. The next step is the oxygen atom transfer from 3 to ethylene. The energy profile is also included in Figure 1 and the transition state geometry is shown in Figure 2. The process starts with the establishment of a weak H-bond between the olefin and the OH ligand (which is essentially thermoneutral: endothermic by 0.6 kcal mol -1 for Me system, exothermic by 1.0 kcal mol -1 for the H system in CHCl3), forming the adduct 4. The subsequent step is transfer of the alkyl(hydro)peroxido O  atom to the olefin through state TS(4-5), while the Mo atom strengthens its interaction with the O  atom. The final products, ethylene oxide and [Cp*MoCl(O)(OH)2], 6, are formed via a H-bonded intermediate 5. The barrier height (17.2 and 12.9 kcal mol -1 for the Me and H systems, respectively), appears as reasonably small for the rate-determining step of an efficient catalytic cycle and much lower than those estimated earlier for other model systems on the basis of a different mechanism. [35,43] Note that this barrier is significantly affected by the nature of R, most probably because the greater electron donating power of the Me group renders the O  atom less electrophilic. In order to proceed to a new catalytic cycle, product 6 now needs to eliminate ROH. This process presents an energy barrier of 13.6 and 17.0 kcal mol -1 for the Me and H systems in CHCl3, for an exothermic process by 9.7 and 9.2 kcal mol - 1 , respectively. Thus, it is easier than the initial activation of the catalyst.

To conclude, this pathway for the oxygen transfer step is quite reasonable once ROOH has been activated. The rate-limiting step calculated for the isolated system is the ROOH activation for R = H and the O atom transfer for R = Me, according to the energy profile. However, even for the H system the transition-state energy of TS(4-5) is very close to that of TS(2-3) and entropic effects disfavour TS(4-5) respect to TS(2-3) because two species have been added to the system (ROOH and ethylene) in TS(4-5) and only one (ROOH) in TS(2-3) (see Supporting Information). From the experimental point of view, all previous kinetic studies yield results consistent with the rate-determining step being oxygen transfer to the olefin, [50][51][52] including a recent study on a cyclopentadienyl substituted Mo VI catalyst, CpMoO2(CH3). [49] Thus, our calculated pathway is consistent with the experimental observations. Given the low barriers obtained for the mechanism outlined in Figure 1, we did not consider it worthwhile to explore other pathways, notably those involving the insertion of the ethylene molecule into the Mo-O bond according to the pathway explored by Calhorda et al. [43] For clarity, the two key steps are compared in Scheme 7. Note that a more recent contribution by Calhorda et al. estimates a much reduced energy for the rate-determining transition state at the MP2 level, [START_REF][END_REF] which remains however higher than the barrier for our pathway. We believe that the -O atom coordination with formation of the strained three-membered MoOO(R) cycle is important for the activation of the The mechanism of Figure 1 can therefore be described as a variant of the Sharpless mechanism, where the oxygen atom is transferred from an alkylperoxido ligand after activation of the oxidant by protonation of an oxido ligand. It is closely related to the mechanism proposed by Thiel (Scheme 3), [32][33][34] with the peroxido function being replaced by a simpler oxido function as the proton accepting functionality.

(b) Study of the [Cp*Mo(O2)OCl] system, 8.

Next, we proceeded to analyze a possible oxygen transfer process from the peroxido complex [Cp*Mo(O2)(O)Cl] (8), which was found catalytically inactive by the Bergman study. [36] If the peroxido ligand in this complex is already sufficiently activated to transfer an oxygen atom to the exogenous olefin substrate, an elementary process leading to the epoxide product and to the dioxide complex 1 can be envisaged. The latter would then need to be transformed back to 8 by interaction with another oxidant molecule (H2O2 or TBHP). The lower energy pathway found for the oxygen atom transfer process involves the attack of ethylene at the exo oxygen atom (further away from the Cp* ligand).

The transition state TS(8-1) is illustrated in Figure 3. The relative energy barrier height for this process is 23.3 kcal mol -1 in the gas phase and 23.3 kcal mol -1 in CHCl3, namely ca. 10 kcal mol -1 higher than for the hydroperoxido complex and 5 kcal mol -1 higher than for the methylperoxido complex. This result is in good agreement with the experimental observation. A possible reason for the higher oxygen transfer barrier for the peroxido (O2 2-) ligand is the higher energy of the O-O * orbital relative to that of the ROO -ligand in complex 3, as already commented above. In previous theoretical work by the groups of Rösch and Frenking it was also concluded that the hydroperoxido mechanisms is competitive or superior to the peroxido mechanism. [27,29] Figure 3.

Optimized geometry and main structural data for system TS(8-1).

The above result is not sufficient to discard the action of complex 8 as a catalyst, because it can still be envisaged that the peroxido ligand serves as the depository of a proton for the activation of another molecule of ROOH in the same way as shown above for the oxido ligand in 1, as in the pathway suggested by Thiel (Scheme 3). [32- 34] The corresponding energy pathway has also been calculated and is reported in Figure 4, while the relevant optimized geometries are shown in Figure 5. A quick comparison of Figure 4 and Figure 1 shows that the Thiel mechanism (involving compound 8 as the active catalyst) is much less favorable than the mechanism involving compound 1. Both the ROOH activation, TS(9-10), and oxygen transfer, TS(11-12), steps have much greater activation barriers than the corresponding steps for complex 1, TS(2-3) and TS(4-5). The high energy of TS(9-10) is somewhat unexpected, since the system is geometrically quite similar to TS(2-3) (cf. Figure 2 12 The above results also give a hint as to why certain complexes of type Cp # MoO2X (Cp # = substituted cyclopentadienyl ring; X = Cl, alkyl, etc.), as well as isoelectronic MoO2X2L2 analogues, are efficient for the epoxidation reaction when using TBHP as the oxidant and much less so when using H2O2. A direct comparison of the calculated pathways for H2O2 and MeOOH shows in fact smaller barriers for the reaction with H2O2 in each case. However, the H2O2 activation yields a Cp # MoO(OH)(OOH)X intermediate such as 3, which may then eliminate water and afford electron-richer peroxido complexes Cp # Mo(O2)(O)X such as 8, the latter being less active catalysts according to the above calculations. This water elimination process has not been investigated computationally for the chloride system, but has been considered for the related cationic system that is discussed in the next section. Note that in some cases, such as the recently investigated CpMoO2(CH3) system, [49] the resulting peroxo complex is also catalytically active. However, it was shown that the catalytic activity for the TBHP-based epoxidation of complex CpMoO(O2)(CH3) is lower than that of CpMoO2(CH3), by an estimated factor of 3-5. [49] An analysis of the reason for the catalytic activity of CpMoO(O2)(CH3), while Cp*MoO(O2)Cl was found inactive, is beyond the scope of the present investigation but is likely due to a fine tuning of the activation barriers by the nature of X and to the different size of the cyclopentadienyl ring.

Mo Cl O O O H O O R Mo Cl O O O H O O R Mo Cl O O O R O H O Mo Cl O O O R O H O Mo Cl O O O R O   O H Mo Cl O O O H O R O Mo Cl O O O H O R -38.

(c) Study of the [Cp*MoO2] + system

This investigation was prompted by our knowledge of the speciation of compound [Cp*2Mo2O5] in an aqueous medium [46,47] and by our recent finding that the compound catalyzes cyclooctene epoxidation by TBHP in the presence of water. [41,[START_REF]these conditions is currently unknown[END_REF] [Cp*2Mo2O5] maintains a dinuclear structure in organic solvents, including polar ones such as MeCN and MeOH, but behaves as a weak electrolyte in water producing [Cp*MoO2(H2O)] + and [Cp*MoO3] -. In addition, water dissociation from [Cp*MoO2(H2O)] + was found to be rapid and reversible. [47] The electron richness of the anionic complex is likely to preclude H2O2 (or TBHP) activation and olefin epoxidation catalysis, but the cationic complex, which is isoelectronic with compound 1, could lead to epoxidation catalysis. Therefore, we have repeated the study described above using this cationic system as a catalyst.

Taking complex [Cp*MoO2(H2O)] + (14) as the starting point, the energy profile limited to the ROOH activation phase is reported in Figure 6. The dissociation of water from [Cp*MoO2(H2O)] + to yield the coordinatively unsaturated [Cp*MoO2] + species has already been reported in our previous study [47] and is shown again here for the purpose of comparison. It requires only 8.7 kcal mol -1 in water. There is, therefore, a great difference between this cationic complex in water and the isoelectronic [Cp*MoO2Cl] complex in CHCl3: Cl - dissociation from [Cp*MoO2Cl] is very energy demanding (61.7 kcal/mol) in a non-polar solvent such as CHCl3. This dissociation leaves the coordination position in [Cp*MoO2] + available for H2O2 binding and activation. Coordination of H2O2 to this unsaturated species leads to a substitution product 16 that is only slightly destabilized relative to the aqua complex 14. This different stability reflects the greater donor power of H2O relative to H2O2 and MeOOH. The isomeric [Cp*MoO(OH)(OOR)] + species (17a) is even lower in energy, being located at -20.6 (for R = H) and -21.2 (for R = Me) kcal/mol from 16 in water solution. The optimized geometries for the Me systems are shown in Figure 8. The Mo-O  distance of 2.391 Å and the Mo-O  -O  angle of 89.2º suggest a significant interaction between Mo and O  , similar to and even stronger that that experienced by complex 3, as expected from the unsaturation of this system. The intramolecular proton transfer leading from 16 to 17a requires a rather high activation, 28.7 (R = H) or 29.2 (R = Me) kcal mol -1 in water solution. In this activation, one proton of the ROOH molecule migrates to one oxo ligand, whereas the O-O-R moiety becomes covalently bonded to the Mo atom. The migrating proton is located approximately midway between the two oxygen atoms in the transition state TS(16-17) (also shown in Figure 8). Note that this transition state is related to that obtained for the ROOH activation by 1, TS(2-3) (Figure 2). However, the Mo-O(H)OR bond is already fully formed in the precursor to TS(16-17), whereas it is in the process of being established in TS(2-3). The O-Mo-O angle involving the donating and accepting O atoms in TS(16-17) has quite significantly narrowed relative to 16. This is most probably the main reason for such a high activation barrier. Following the same strategy previously used to investigate a similar intramolecular proton transfer process leading from [Cp*MoO(OH)2] + to [Cp*MoO2(H2O)] + , [47] we considered the participation of additional water molecules as proton relay agents. [47,[53][54][55][56] Figure 8. Optimized geometries and main structural data for the systems in Figure 6 (R = Me). Those with R = H are in the Supporting Information.

Indeed, the addition of just one water molecule is sufficient to dramatically decrease the barrier to only 1.6 kcal/mol for both R = H and Me in solution, see Figure 6. Therefore, the water-assisted pathway should be the preferred one. The additional water lowers the energy of both starting (16 to 16') and final (17a to 17') systems through the establishment of a hydrogen bond. Thermodynamically, the process is exothermic by 13.9 (R = H) or 13.7 (R = Me) kcal mol - 1 in water solution. The related optimized geometries are shown also in Figure 8, including that of the transition state TS(16'-17'), which is relaxed with a O-Mo-OOH angle of 94.8º. An analysis of the various O-H distances shows that the donating O-H bond has already largely broken, whereas the incipient O-H bond has not yet formed to a great extent. Therefore, the transition state may be more correctly described as having a [Cp*MoO2(OOR)]•(H3O + ) character. To conclude this part, the calculations suggest that the ROOH activation process is very facile. The slowest step is the H2O dissociation from 14, which requires only 8.7 kcal mol -1 in solution, considering a dissociative mechanism.

We now turn to the oxygen transfer step from the activated oxidant to the olefin. Starting from 17a H , we were able to locate two pathways with practically equivalent activation barriers, where the transferred oxygen atom is the O  atom in one case (transition state TS(18-19) H ) and the O  atom in the other case (transition state TS(21-22) H ). The energy profiles of these two pathways are comparatively shown in Figure 8. The first pathway starts with an H bonding interaction between the ethylene  electron density and the hydroperoxido proton, leading to the 18 H structure accompanied by a significant stabilization, and continues with a 1,2-proton shift from the O  to the O  atom leading to the bis(hydroxido) derivative 20 through an intermediate 19 with an H-bonded epoxide product. This pathway would certainly occur through a prohibitively high energy transition state when R = Me, therefore calculations along this pathway were not done.

Concerning the second pathway (optimized geometries in Figure 9), the first step is an endoergic reorientation of the O-O-R group (7.4 and 6.9 kcal mol -1 for R = H and Me, respectively) from a perpendicular to a parallel arrangement (17b), to which the ethylene may attack at the O  atom after an initial approach through a slightly stabilizing (for 21 H ) or thermoneutral (for 21 Me ) H-bond with the O-H group. A direct olefin attack at 17a did not allow us to locate an oxygen transfer pathway. Like for compound 1 (Figure 1), the olefin attack is exogenous, without formation of metal-carbon bonds, through the transition state TS(21-22) in which the Mo center strengthens its interaction with the O  atom. The transition state TS(21-22) for this cationic system is closely related to TS(4-5) of the neutral chlorido system 1 Figure 1. Like for TS(4-5), the barrier is slightly higher for the Me system than for the H system, for the same reasons discussed above. The barrier height (16.1 and 17.7 kcal mol - 1 for the H and Me systems in water with respect to 17a) is also close to the barrier leading to TS(4-5) (12.9 and 17.2 kcal mol -1 in CHCl3). This pathway leads to the same products, [Cp*Mo(O)(OH)2] + (20) and ethylene oxide, as the 1,2 proton shift discussed above, through the H-bonded intermediate 22, only slightly different than 19. The regeneration of the starting complex 14 by intramolecular proton transfer, assisted by additional water molecules, occurs by a low- energy pathway, as shown in our previous study. [47] Clearly, both pathways are energetically viable, although only that through TS(21-22) may take place when using TBHP as the oxidant.

Conclusion

The present study has unveiled a low-energy pathway for olefin epoxidation catalyzed by cyclopentadienyl Mo VI systems. The initial H2O2 activation follows the same general path described in other recent computational studies, [35,43] with protonation of an oxido ligand and generation for a hydroxide hydroperoxido intermediate, but unlike the previous study the latter is found to adopt an asymmetric  2 coordination mode, with a weak interaction between the -O atom and the metal center. This interaction is critical for the activation of the -O atom toward an exogenous nucleophilic attack by the olefin substrate, leading to significantly lower activation barriers for the oxygen atom transfer to the olefin relative to the previous study. This mechanism closely corresponds to what has been proposed by Thiel et al., except that an oxido ligand is the depository of the oxidant proton rather than a peroxido ligand. An analogous mechanism is found both for [Cp*MoO2Cl] (1) and [Cp*MoO2(H2O)] + (13) complexes. The study also provides a rationale for the lower activity of peroxido derivatives relative to the oxido analogues, and for the lower activity of H2O2 relative to THBP. The former oxidant may undergo isomerization of the reactive hydroperoxido intermediate through proton transfer, leading to less reactive peroxido derivatives. Therefore, the design of an efficient catalytic system for olefin epoxidation by H2O2 should only provide a basic site for transfer of one of the two H2O2 protons.

Computational Details

Calculations were carried out using the Gaussian 03 package [START_REF] Frisch | Gaussian 03, Revision D.01[END_REF] at the DFT level by means of the B3LYP functional. [START_REF] Becke | [END_REF][59][60] For the Mo atom, the LANL2DZ pseudopotential [61] was used with the addition of f polarisation functions. [62] The 6-31G(d) basis set was used for C atoms whereas additional diffuse functions [6-31+G(d)] were added for O and Cl atoms due to their anionic character. For the hydrogen atoms, the 6-31G(d,p) basis set was employed. IRC calculations were made in order to get the two minima linked by every transition-state. [63][64][65] Solvent effects were included by means of CPCM single point calculations using chloroform ( = 4.9 at 25°C) for the neutral systems and water ( = 78.39 at 25°C) for the cationic catalyst. [66,67] For these calculations, an extended basis set was used for C, O and H: 6-311++g(d,p). Additional spheres were included for all the hydrogens except for the Cp* hydrogens by means of the SPHEREONH option. Thus, all the presented energies both in the figure and in the text are electronic calculations taking into account solvent effects unless explicitly stated. Both free energy values in the gas phase and in solution can be found in the Supporting Information. Frequency calculations were made in order to check the presence of one imaginary frequency in transition-state geometries.

Entry for the Table of Contents

  Scheme 1

  [Cp*MoO2R], R = alkyl, and [Cp*2Mo2O5]) and the [Cp*MoO2] + cation. Points of interest are:

  Scheme 5

( a )

 a Study of the [Cp*MoO2Cl] system, 1.

Figure 1 .

 1 Figure 1. Energy profiles in CHCl3 solution (in kcal mol -1 ) for the ROOH activation and C2H4 epoxidation by [Cp*MoO2Cl] (R = H, dashed lines; Me, plain lines). The reference energy corresponds to the separate reagents ([Cp*MoO2Cl] + ROOH + C2H4).

  *MoO 2 Cl] (1) (+ROH+C 2 H 4 O)

Figure 2 .

 2 Figure 2. Optimized geometries and main structural data for the systems in Figure 1 (R = Me). Those with R = H are in the Supporting Information.

Scheme 7 .

 7 Scheme 7. Comparison of the key (rate-determing) steps of our mechanism and that reported by Clahorda et al. The relative energies shown are from the gas phase calculations for comparative purposes.

1 )Figure 4 .

 14 Figure 4. Energy profiles in CHCl3 solution (in kcal mol -1 ) for the ROOH activation by [Cp*Mo(O2)OCl] (R = H, dashed lines; Me, plain lines). The reference energy corresponds to the separate reagents ([Cp*Mo(O2)OCl] + ROOH + C2H4).

2 13- 34 . 8 - 38 . 3 electron

 2348383 richer than in the oxido system (O2 2-is a better electron donor than O 2-), rendering the Mo•••O  interaction weaker. Indeed, in system 10 the Mo-O  distance is much longer (3.008 Å for 10 Me and 2.976 Å for 10 H ) and the Mo-O  -O  angle wider (120.9 for 10 Me and 119.2° for 10 H ) relative to 3. A related effect is an energy increase for the O-O * orbital, thus rendering the O  atom less electrophilic. Indeed, the molecular orbitals with highest σ* O-O contribution are located at 0.144, 1.250 and 1.494 eV for 10 H , i.e. much higher than for 3 H . Once again, the O transfer step is much less favorable for the Me system than for the H system because the O atom electrophilicity is reduced by the Me donor power.

Figure 5 .

 5 Figure 5. Optimized geometries and main structural data for the systems in Figure 4 (R = Me). Those with R = H are in the Supporting Information.

Figure 6 .

 6 Figure 6. Energy profile in water solution (in kcal mol -1 ) for the activation of ROOH (R = Me, plain lines; H, dashed lines) by [Cp*MoO2(H2O)] + (14).

4 [

 4 Cp*MoO 2 (ROOH)] + •H 2 O [Cp*MoO(OH)(OOR)] + •H 2

Figure 9 .

 9 Figure 9. Optimized geometries and main structural data for the systems in Figure 8 (R = Me). Those with R = H are in the Supporting Information.

  Olefin epoxidation mechanism --AleixComas-Vives, Agustí Lledós,* Rinaldo Poli* ……… Page -Page A computational study of the olefin epoxidation mechanism catalyzed by cyclopentadienyloxidomolybdenum(VI) complexes Sharpless of Mimoun? The mechanism of olefin epoxidation catalyzed by cyclopentadienyloxidomolybdenum(VI) complexes is revised on the basis of DFT calculations. A pathway related to a proposition of Thiel, with oxidant activation and proton transfer to an oxido ligand and subsequent exogenous electrophilic attack of the MoOOR -O atom by the olefin, is found the preferred pathway for this system.

16 Me TS(16-17) Me 17a Me 16' Me TS(16'-17') Me 17' Me Figure

  7. Energy profiles in water solution (in kcal mol -1 ) for the oxygen atom transfer from 17a to ethylene (R = Me, plain lines; H, dashed lines).

										3.4
			1.0
								17a
										2.2
			TS(18-19)			-6.5	-6.7	TS(21-22)
									-7.4	-7.3
								-13.9
									17b
										21
						-13.3	-14.3
						18
										Mo
										O	O
			O	Mo	O			O	Mo	O	O	O	R
			O					O		R	H
			H		O	H		H	O
						O	Mo	O		O	Mo	O	R	Mo
						O	O		O	O	O	O
										O
	O	Mo	O			H	H			H	H	R
	O		H							O
	H
										-54.3
	-57.8		O
										-59.9
										-57.8
	20
			-65.2
										20
			19
										-69.9
										22

Acknowledgements

We gratefully acknowledge the European Commission for funding this work through the AQUACHEM Research Training Network (Contract n° MRTN-CT-2003-503864). Support from the Spanish MICINN (Projects CTQ2008-06866-CO2-01 and Consolider Ingenio 2010 CSD2007-00006 and FPU fellowship to A.C.-V.), CNRS (LEA Toulouse-Barcelone project), Generalitat de Catalunya (2009/SGR/68 and LEA project), and CICT (project CALMIP) is gratefully acknowledged.

Supporting information for this article is available on the WWW under http://www.chemeurj.org/ or from the author.