Juan Calvo
email: juancalvo@ugr.es

Erwan Hingant
email: erwan.hingant@u-picardie.fr

Romain Yvinec
email: romain.yvinec@inrae.fr

Long-time asymptotic of the Lifshitz-Slyozov equation with nucleation

Keywords: 2020 Mathematics Subject Classification. 35Q92, 35B40, 35L04 Long-time behavior, Lifshitz-Slyozov equation, Entropy functional, Nucleation theory

We consider the Lifshitz-Slyozov model with inflow boundary conditions of nucleation type. We show that for a collection of representative rate functions the size distributions approach degenerate states concentrated at zero size for sufficiently large times. The proof relies on monotonicity properties of some quantities associated to an entropy functional. Moreover, we give numerical evidence on the fact that the convergence rate to the goal state is algebraic in time. Besides their mathematical interest, these results can be relevant for the interpretation of experimental data.

Introduction

In this work we study the long time behavior of the Lifshitz-Slyozov (LS) model with nucleation boundary conditions. The model reads: 𝜕𝑓(𝑡, 𝑥) 𝜕𝑡 + 𝜕{(𝑎(𝑥)𝑢(𝑡) -𝑏(𝑥))𝑓(𝑡, 𝑥)} 𝜕𝑥 = 0 , 𝑡 > 0 , 𝑥 ∈ (0, ∞) ,

with

𝑢(𝑡) + ∫ ∞ 0 𝑥𝑓(𝑡, 𝑥) 𝑑𝑥 = 𝜌 , 𝑡 > 0 , (2)
for some given 𝜌 > 0, subject to the boundary condition ((𝑎(𝑥)𝑢(𝑡) -𝑏(𝑥))𝑓(𝑡, 𝑥)) |𝑥=0 = 𝔫(𝑢(𝑡)) , 𝑡 ∈ {𝑠 > 0 ∶ 𝑢(𝑠) > Φ 0 } ,

where Φ 0 = lim 𝑥→0 + 𝑏(𝑥) 𝑎(𝑥)

, and the initial condition 𝑓(0, 𝑥) = 𝑓 𝑖𝑛 (𝑥) , 𝑥 ∈ (0, ∞) .

The LS model describes the temporal evolution of a mixture of monomers and aggregates that undergo the following interactions: a monomer can join an existing aggregate of size 𝑥, with an attachment rate 𝑎(𝑥), and a monomer can detach from an existing aggregate of size 𝑥, with a detachment rate 𝑏(𝑥). The variable 𝑥 describes the size of the aggregates, so that 𝑓(𝑡, 𝑥) is the number density of aggregates at time 𝑡, whereas 𝑢(𝑡) stands for monomer concentration at time 𝑡. Equation (2) simply encodes the fact that the total mass 𝜌 is preserved.

Depending on the specific rates 𝑎(𝑥) and 𝑏(𝑥), the model may or may not need a boundary condition. No boundary condition is needed for the original Lifshitz-Slyozov version of the model [START_REF] Lifshitz | The kinetics of precipitation from supersaturated solid solutions[END_REF], and this is also the case for the various instances of "Ostwald ripening" that have been analyzed in the literature, e.g. [START_REF] Collet | On solutions of the Lifshitz-Slyozov model[END_REF][START_REF] Collet | Some Recent Results on the Kinetic Theory of Phase Transitions[END_REF][START_REF] Collet | Some remarks on large-time asymptotic of the Lifshitz-Slyozov equations[END_REF][START_REF] Laurençot | Weak solutions to the Lifshitz-Slyozov-Wagner equation[END_REF][START_REF] Niethammer | On the evolution of large clusters in the Becker-Döring model[END_REF][START_REF] Niethammer | On the initial value problem in the Lifshitz-Slyozov-Wagner theory of Ostwald ripening[END_REF][START_REF] Niethammer | Well-posedness for measure transport in a family of nonlocal domain coarsening models[END_REF]. In this article we are interested in situations where the kinetic rates are such that a boundary condition is needed to make sense of the model. Classical examples are power-law rates like 𝑎(𝑥) ∝ 𝑥 𝛼 and 𝑏(𝑥) ∝ 𝑥 𝛽 with 0 ≤ 𝛼 ≤ 𝛽 ≤ 1. To our knowledge, only the special power-law case with equal power (𝛼 = 𝛽) has been considered in the literature, see [START_REF] Collet | On solutions of the Lifshitz-Slyozov model[END_REF] with zero Dirichlet boundary conditions. For the main applications we have in mind the nucleation rate 𝔫 in equation (3) follows a mass action kinetics, that is 𝔫(𝑢) ∝ 𝑢 𝑖 0 with 𝑖 0 ∈ ℕ * .

The boundary condition [START_REF] Calvo | Long-time asymptotics for polymerization models[END_REF] encodes the creation of new aggregates from the available pool of monomers. We understand this as an effective description of a nucleation process, that may be described by a more detailed discrete model. Connections between the discretesize Becker-Döring model and the continuous-size Lifshitz-Slyozov model are indeed well known [START_REF] Collet | The Becker-Döring system and its Lifshitz-Slyozov limit[END_REF][START_REF] Laurençot | From the Becker-Döring to the Lifshitz-Slyozov-Wagner equations[END_REF][START_REF] Schlichting | Macroscopic limit of the Becker-Döring equation via gradient flows[END_REF]. A boundary condition of the form (3) has been deduced from appropriate scaling limits of the Becker-Döring model in the case of a second order nucleation kinetics, see [START_REF] Collet | The Becker-Döring system and its Lifshitz-Slyozov limit[END_REF][START_REF] Deschamps | Quasi steady state approximation of the small clusters in Becker-Döring equations leads to boundary conditions in the Lifshitz-Slyozov limit[END_REF] for details. Higher order mass action kinetics can also be used to describe the nucleation process in a phenomenological way [START_REF] Bishop | Kinetics of nucleation-controlled polymerization. a perturbation treatment for use with a secondary pathway[END_REF][START_REF] Prigent | An Efficient Kinetic Model for Assemblies of Amyloid Fibrils and Its Application to Polyglutamine Aggregation[END_REF][START_REF] Xue | Systematic analysis of nucleationdependent polymerization reveals new insights into the mechanism of amyloid selfassembly[END_REF]; such boundary conditions arise as scaling limits of modified versions of the Becker-Döring model. It is also interesting to note that some nucleation boundary conditions were derived in [START_REF] Deschamps | Quasi steady state approximation of the small clusters in Becker-Döring equations leads to boundary conditions in the Lifshitz-Slyozov limit[END_REF] that do not follow mass action kinetics. This can be relevant for the description of protein polymerization phenomena. All in all, this is a first step towards a number of important applications in the science of materials and in the field of neurodegenerative diseases [START_REF] Prigent | An Efficient Kinetic Model for Assemblies of Amyloid Fibrils and Its Application to Polyglutamine Aggregation[END_REF][START_REF] Ross | Protien aggregation and neurodegenerative disease[END_REF][START_REF] Ross | Prion domains: sequences, structures and interactions[END_REF][START_REF] Szavits-Nossan | Inherent variability in the kinetics of autocatalytic protein self-assembly[END_REF].

We prove below that under generic conditions, when the function 𝑏 𝑎 is strictly increasing, solutions will concentrate at vanishing aggregate sizes, while the concentration of available monomers drops down to the activation threshold Φ 0 in equation [START_REF] Calvo | Long-time asymptotics for polymerization models[END_REF]. Under more specific hypotheses we can get more precise information about the temporal rates at which this dynamics takes place. We complement this with a numerical investigation. The general picture that emerges points to the fact that the concentration dynamics is quite slow, actually taking place with algebraic rates (that depend on 𝛼, 𝛽 and 𝔫). These results are in line with previously known results for the case of outgoing characteristics, which indicate that degenerate steady states are approached at an algebraic rate, see [START_REF] Carrillo | A numerical study on large-time asymptotics of the Lifshitz-Slyozov system[END_REF][START_REF] Collet | Some remarks on large-time asymptotic of the Lifshitz-Slyozov equations[END_REF].

Since the transient behavior for equations ( 1)-( 3) spans a wide temporal scale, this model is thus suitable for comparison with experimental data, e.g. those originating in protein polymerization experiments in vitro. It would be interesting to investigate whether potential asymptotic profiles, after suitable normalisation, could be universal or not, in comparison to the outgoing case [START_REF] Carrillo | A numerical study on large-time asymptotics of the Lifshitz-Slyozov system[END_REF][START_REF] Collet | Some remarks on large-time asymptotic of the Lifshitz-Slyozov equations[END_REF]. Note also that transient oscillatory behavior, for specific initial conditions, cannot be ruled out by our results, see e.g. [START_REF] Lauretano | On the Relation between the Lifshitz-Slyozov and the Lifshitz-Slyozov-Wagner Models for Supersaturated Solutions[END_REF]. However, if we are interested in very long time scales the concentration behavior will take over and then it seems advisable to introduce corrections to the model (e.g. size-diffusion corrections, coagulation-fragmentation operators...) in order to get a more realistic goal state, see for instance [START_REF] Calvo | Long-time asymptotics for polymerization models[END_REF][START_REF] Collet | The Becker-Döring system and its Lifshitz-Slyozov limit[END_REF][START_REF] Conlon | On a diffusive version of the Lifschitz-Slyozov-Wagner equation[END_REF][START_REF] Conlon | A non-local problem for the Fokker-Planck equation related to the Becker-Döring model[END_REF][START_REF] Hariz | A modified version of the Lifshitz-Slyozov model[END_REF][START_REF] Marder | Correlations and Ostwald ripening[END_REF][START_REF] Meerson | Fluctuations provide strong selection in Ostwald ripening[END_REF][START_REF] Schlichting | Macroscopic limit of the Becker-Döring equation via gradient flows[END_REF][START_REF] Velázquez | The Becker-Döring equations and the Lifshitz-Slyozov theory of coarsening[END_REF]. Similarly, if persistent oscillations are required, one should consider extra mechanisms, see e.g. [START_REF] Doumic | A bi-monomeric, nonlinear Becker-Döring-type system to capture oscillatory aggregation kinetics in prion dynamics[END_REF][START_REF] Niethammer | Oscillations in a Becker-Döring Model with Injection and Depletion[END_REF][START_REF] Pego | Temporal oscillations in Becker-Döring equations with atomization[END_REF].

Statement of the problem and results

Here below ℝ + = (0, +∞) and (1 + 𝑥)𝑑𝑥 denotes the measure with density 𝑥 ↦→ (1 + 𝑥) with respect to the Lebesgue measure on ℝ + . The space 𝐿 1 (ℝ + , (1 + 𝑥)𝑑𝑥) denotes the space of integrable function w.r.t. the former measure. This space might be endowed with the weak topology denoted by 𝑤 whose convergence is characterized against bounded functions. Finally, 𝒞([0, ∞), 𝑤 -𝐿 1 (ℝ + , (1 + 𝑥)𝑑𝑥)) is the set of continuous functions from [0, +∞) into 𝐿 1 (ℝ + , (1 + 𝑥)𝑑𝑥) equipped with its weak topology that is, for such 𝑓,

𝑡 ↦→ ∫ ∞ 0 𝑓(𝑡, 𝑥)𝜑(𝑥)(1 + 𝑥)𝑑𝑥 , is continuous on [0, ∞) for all 𝜑 ∈ 𝐿 ∞ (0, ∞).
Here and in the sequel 𝐿 𝑝 refers to the standard Lebesgue spaces and 𝑊 1,𝑝 refers to the standard Sobolev spaces. We may add a subscript to it, so that the integration variable is made clear. 𝒞 𝑘 is the space of continuous function whose 𝑘 𝑡ℎ derivatives are continuous, and 𝒞 𝑘 𝑐 stands for its subspace consisting of compactly supported functions.

In the remainder the rates 𝑎, 𝑏 and 𝔫 are assumed nonnegative and continuous functions on [0, ∞) and such that ∀𝑥 > 0 , 𝑎(𝑥) > 0 and Φ(𝑥) ∶= 𝑏(𝑥) 𝑎(𝑥) → 𝑥→0 + Φ 0 .

(H0)

We deal with global solutions for which the boundary condition is defined for all time, namely solutions defined in the following sense: Definition 2.1. Let 0 ≤ 𝑓 in ∈ 𝐿 1 (ℝ + , (1 + 𝑥)𝑑𝑥) and 𝜌 > 0. We say that 0 ≤ 𝑓 ∈ 𝒞([0, ∞), 𝑤 -𝐿 1 (ℝ + , (1 + 𝑥)𝑑𝑥)) is a (global) solution to the LS equation with nucleation (1)-( 4) provided that:

1. For all 𝑡 ≥ 0 and for every 𝜑 ∈ 𝒞 0 ([0, ∞)) such that 𝜑 ′ ∈ 𝐿 ∞ (0, ∞), we have

Our main hypothesis relies on the monotonicity of Φ. Namely, we suppose that Φ is strictly increasing .

We shall assume some technical hypotheses as well: First, for any 𝜀 > 0, 𝑎 ′ , 𝑏 ′ ∈ 𝐿 ∞ (𝜀, ∞) , (H2) which entails the existence of a constant 𝐶 > 0 such that, for all 𝑥 ≥ 0, 𝑎(𝑥) + 𝑏(𝑥) ≤ 𝐶(1 + 𝑥) .

Next, we suppose

inf 𝑥∈(1,∞)
𝑎(𝑥) > 0 , and

1 𝑎 ∈ 𝐿 1 (0, 1) , (H3)
and that there exists a constant 𝐶 > 0 such that, for all 𝑥 ≥ 0,

𝑏(𝑥)Φ(𝑥) ≥ 1 𝐶 min(1, 𝑥 2 ) . ( H4
)
Concerning the nucleation rate, we assume that there exist two constants 𝑐 > 0 and 𝑘 0 ≥ 1 such that for all 𝑧 ≥ 0 𝔫(𝑧) ≥ 𝑐𝑧 𝑘 0 . (H5)

Finally, we will assume that the initial condition 𝑓 in belongs to 𝐿 1 (ℝ + , (1 + 𝑥)𝑑𝑥), is nonnegative and moreover

∫ ∞ 0 (∫ 𝑥 0 Φ(𝑧)𝑑𝑧 + 𝑥 2 ) 𝑓 in (𝑥)𝑑𝑥 < ∞ . ( H6
)
The well-posedness of equations ( 1)-( 4) was studied in [START_REF] Calvo | The initial-boundary value problem for the Lifshitz-Slyozov equation with non-smooth rates at the boundary[END_REF] under some additional conditions of a technical nature -which are probably non-optimal. At any rate, taking (H1) for granted we know for sure that solutions (should they exist) will be global in time with 𝑢(𝑡) > Φ 0 for all times. Therefore, during the rest of the document we will only concentrate on the set of assumptions needed for our analysis of the long time behavior. Some comments are in order concerning our set of hypotheses. Besides ensuring wellposedness, hypothesis (H1) is the natural counterpart to the typical Ostwald Ripening phenomena where, Φ(𝑥) = 𝑥 -1∕3 for which the long-time asymptotics have been studied e.g. [START_REF] Collet | Some remarks on large-time asymptotic of the Lifshitz-Slyozov equations[END_REF]. Note that hypothesis (H2) is not that demanding; actually, blow-up phenomena may take place for strictly superlinear rates [START_REF] Collet | On solutions of the Lifshitz-Slyozov model[END_REF]. Hypothesis (H3), has been required in connection with the well-posedness of the problem, so that characteristics go back to 𝑥 = 0 in finite time and render the boundary condition relevant. Condition (H4) is purely technical and does not imply a strong restriction, recall the power law rates introduced in Section 1, namely 𝑎(𝑥) ∝ 𝑥 𝛼 and 𝑏(𝑥) ∝ 𝑥 𝛽 with 0 ≤ 𝛼 ≤ 𝛽 ≤ 1. Indeed, such rates satisfy all our hypotheses. Concerning the nucleation rate, any mass action kinetics can be considered under hypothesis (H5). Finally, hypothesis (H6) imposes just a mild technical requirement on the initial condition.

In order to ascertain the temporal evolution of the system, the number of aggregates constitutes an important quantity that is defined as

𝑀 0 (𝑡) ∶= ∫ ∞ 0 𝑓(𝑡, 𝑥) 𝑑𝑥 .
According to the boundary condition [START_REF] Calvo | Long-time asymptotics for polymerization models[END_REF], this evolves in time via 𝑑𝑀 0 𝑑𝑡 = 𝔫(𝑢) .

To discuss concentration phenomena, we define ℳ + 𝜌 ([0, +∞)) the set of nonnegative Radon measures on [0, +∞) with total variation less or equal to 𝜌. This space can be equipped with the weak topology whose convergence corresponds to the convergence against any continuous and bounded function.

Our main results in this document describe concentration phenomena for the long-time evolution of the LS equation with nucleation boundary conditions: Theorem 2.2. Under hypotheses (H0-H6), any global solution in the sense of Definition 2.1 satisfies

• lim 𝑡→+∞ 𝑀 0 (𝑡) = +∞, • lim 𝑡→+∞ 𝑢(𝑡) = Φ 0 , • lim 𝑡→+∞ 𝑥𝑓(𝑡, 𝑥)𝑑𝑥 = (𝜌 -Φ 0 )𝛿 0 , weakly in ℳ + 𝜌 ([0, ∞)).
The proof relies on a Lyapunov functional, which we introduce in Section 3. Right after that we proceed with the proof of Theorem 2.2 in Section 4.

Note that Theorem 2.2 is a generalisation of an earlier result proved in [START_REF] Calvo | Long-time asymptotics for polymerization models[END_REF] for 𝑎(𝑥) = 1 and 𝑏(𝑥) such that 0 < 𝑐 1 ≤ 𝑏 ′ (𝑥) ≤ 𝑐 2 for some constants 𝑐 1 and 𝑐 2 . In that particular case, the rates can be computed explicitly. Those are algebraic and depend on the specific form of the nucleation rate, but are nevertheless quite slow. For more general coefficients, our method of proof cannot provide specific estimates and therefore we cannot ascertain the timescales over which the average aggregate size tends to zero. However, we performed numerical simulations, whose results suggest that this algebraic trend is actually what we should expect generically. This is discussed in Section 5. Our results are complemented with an analysis of the case of constant Φ, which requires a separate treatment. Its long time behavior is analyzed in Section 6 and expands on the results given in [START_REF] Collet | On solutions of the Lifshitz-Slyozov model[END_REF][START_REF] Collet | Some Recent Results on the Kinetic Theory of Phase Transitions[END_REF].

One of the main points underlying the previous results is to discriminate whether the system will be able to fuel nucleation reactions to the extent that the number of fragments grows without control. We actually show that this is the case for a representative number of situations. Since the total mass is preserved, this suggests that the average aggregate size becomes smaller and smaller, which is an instance of dust formation.

Lyapunov functional

We shall introduce a Lyapunov functional in the vein of [START_REF] Calvo | Long-time asymptotics for polymerization models[END_REF][START_REF] Collet | Some remarks on large-time asymptotic of the Lifshitz-Slyozov equations[END_REF]. For 𝑘 a continuous and positive function on [0, +∞) with continuous derivative and 𝑓 a solution of equations ( 1)-( 4), we define for all 𝑡 ≥ 0,

𝐻 𝑘 (𝑡) = ∫ ∞ 0 𝑘(𝑥)𝑓(𝑡, 𝑥)𝑑𝑥 + 𝐾(𝑢(𝑡)) , (8)
with 𝐾(𝑣) = ∫ 𝑣 0 𝑘 ′ •Φ -1 (𝑧)𝑑𝑧. This makes sense since Φ is monotonous. The functional 𝐻 𝑘 is a Lyapunov functional and its time derivative 𝐷 𝑘 is called its dissipation, as the following result makes clear.

Proposition 3.1. Assume 0 ≤ 𝑘 ∈ 𝒞 1 ([0, +∞)) is convex with 𝑘(0) = 0 and 𝑓 is a solution in the sense of Definition 2.1. If 𝐻 𝑘 (0) < ∞, then 𝑡 ↦→ 𝐻(𝑡) is non-increasing, non-negative and for all 𝑡 ≥ 0 𝐻 𝑘 (𝑡) + ∫ 𝑡 𝑠 𝐷 𝑘 (𝑠)𝑑𝑠 ≤ 𝐻 𝑘 (𝑠) , ∀ 0 ≤ 𝑠 < 𝑡 , (9)
where

0 ≤ 𝐷 𝑘 (𝑡) = ∫ ∞ 0 (𝐾 ′ (𝑢(𝑡)) -𝐾 ′ (Φ(𝑥)))) (𝑢(𝑡) -Φ(𝑥)) 𝑎(𝑥)𝑓(𝑡, 𝑥)𝑑𝑥 , belongs to 𝐿 1 𝑡 (0, ∞).
Proof. Let 𝑅 > 0, 𝑘 𝑅 (𝑥) = 𝑘(𝑥) for 𝑥 < 𝑅 and 𝑘 𝑅 (𝑥) = 𝑘 ′ (𝑅)(𝑥 -𝑅) + 𝑘(𝑅) for 𝑥 ≥ 𝑅. Notice that 𝑘 𝑅 can be used as a test function in [START_REF] Carrillo | A numerical study on large-time asymptotics of the Lifshitz-Slyozov system[END_REF]. Moreover, 𝑘 𝑅 is convex. We construct 𝐻 𝑘 𝑅 via formula [START_REF] Collet | The Becker-Döring system and its Lifshitz-Slyozov limit[END_REF] and we compute the dissipative part, which is nonnegative because 𝐾 ′ = 𝑘 ′ 𝑅 •Φ -1 is increasing. Then we conclude by Fatou's lemma.

As a straightforward consequence with 𝑘(𝑥) = 1 2 𝑥 2 we get the following result: Corollary 3.2. Assume hypothesis (H6) and let 𝑓 be a solution in the sense of Definition 2.1. We define

𝐻(𝑡) = 1 2 ∫ ∞ 0 𝑥 2 𝑓(𝑡, 𝑥)𝑑𝑥 + Ψ(𝑢(𝑡)) , (10)
with Ψ(𝑣) = ∫ 𝑣 0 Φ -1 (𝑧)𝑑𝑧. We have that 𝐻(0) < ∞ and 𝑡 ↦→ 𝐻(𝑡) is non-increasing, nonnegative and the dissipation part is given, for all 𝑡 ≥ 0, by

𝐷(𝑡) = ∫ ∞ 0 ( Φ -1 (𝑢(𝑡)) -Φ -1 (Φ(𝑥))
) (𝑢(𝑡) -Φ(𝑥)) 𝑎(𝑥)𝑓(𝑡, 𝑥)𝑑𝑥 .

Remark 3.3.

There are other useful choices of the function 𝑘. We will make use of 𝑘(𝑥) = ∫ 𝑥 0 Φ(𝑦)𝑑𝑦 which leads to 𝑘 ′ •Φ -1 = 𝐼𝑑 and then

𝑑 𝑑𝑡 𝐻 𝑘 ≤ -∫ ∞ 0 (𝑢(𝑡) -Φ(𝑥)) 2 𝑎(𝑥)𝑓(𝑡, 𝑥)𝑑𝑥 ∈ 𝐿 1 𝑡 (0, ∞). (11)
We may also take 𝑘(𝑥) = 𝑥 𝜂 with 𝜂 ≥ 1 to control moments of the form

𝑀 𝜂 (𝑡) ∶= ∫ ∞ 0 𝑥 𝜂 𝑓(𝑡, 𝑥)𝑑𝑥 .
Indeed, for any moment greater than 1 which is initially finite, we have a uniform bound for every positive time.

Proof of the main result 4.1 The number of fragments diverges

We prove first a generic result showing that a shattering phenomenon takes place on long time intervals: the number of fragments diverges.

Proposition 4.1. Assume hypotheses (H0, H2,H4-H6). Then, for every solution in the sense of Definition 2.1, there holds that lim 𝑡→+∞ 𝑀 0 (𝑡) = +∞.

Proof. Taking 𝟏 as a test function in [START_REF] Carrillo | A numerical study on large-time asymptotics of the Lifshitz-Slyozov system[END_REF] we have

𝑀 0 (𝑡) = 𝑀 0 (0) + ∫ 𝑡 0 𝔫(𝑢(𝑠))𝑑𝑠 ,
thus 𝑀 0 is monotonically increasing. The result follows directly in the case Φ 0 > 0 by hypothesis (H5). Therefore, we provide a proof for Φ 0 = 0. We argue by contradiction.

Suppose 𝑀 0 is bounded above independently of time. By equation [START_REF] Carrillo | A numerical study on large-time asymptotics of the Lifshitz-Slyozov system[END_REF] with 𝜑(𝑥) = 𝑥 and noticing inequality [START_REF] Collet | Some Recent Results on the Kinetic Theory of Phase Transitions[END_REF] and 0 ≤ 𝑢(𝑡) ≤ 𝜌, we deduce

𝑢 ′ (𝑡) = -𝑢(𝑡) ∫ ∞ 0 𝑎(𝑥)𝑓(𝑡, 𝑥) + ∫ ∞ 0 𝑏(𝑥)𝑓(𝑡, 𝑥)𝑑𝑥 ∈ 𝐿 ∞ 𝑡 (0, ∞) .
This entails 𝑢 ∈ 𝑊 1,∞ (0, ∞). Moreover, using Cauchy-Schwarz's inequality,

|𝑢 ′ (𝑡)| ≤ (∫ ∞ 0 𝑎(𝑥)𝑓(𝑡, 𝑥)𝑑𝑥) 1 2 (∫ ∞ 0 𝑎(𝑥)(𝑢(𝑡) -Φ(𝑥)) 2 𝑓(𝑡, 𝑥)𝑑𝑥) 1 2
.

By (H6) and Proposition 3.1 with 𝑘(𝑥) = ∫ 𝑥 0 Φ(𝑦)𝑑𝑦 -see also Remark 3.3, we have

𝐷 𝑘 (𝑡) = ∫ ∞ 0 𝑎(𝑥)(𝑢(𝑡) -Φ(𝑥)) 2 𝑓(𝑡, 𝑥)𝑑𝑥 ∈ 𝐿 1 𝑡 (0, ∞) . (12)
Thus, with inequality (7), we deduce that 𝑢 ′ ∈ 𝐿 2 (0, ∞).

Next, we notice that 𝑢 𝑘 0 ∈ 𝐿 1 (0, ∞); this is due to hypothesis (H5) and

𝑐 ∫ ∞ 0 𝑢 𝑘 0 (𝑡)𝑑𝑡 ≤ ∫ ∞ 0 𝔫(𝑢(𝑡))𝑑𝑡 ≤ lim sup 𝑡→∞ 𝑀 0 (𝑡) -𝑀 0 (0),
together with the supposed bound on the 0 𝑡ℎ -order moment. Let 𝑝 > max(2, 𝑘 0 ), we have

∫ ∞ 0 𝑢 𝑝 (𝑡)𝑑𝑡 = ∫ ∞ 0 𝑢 𝑝-𝑘 0 (𝑡)𝑢 𝑘 0 (𝑡) ≤ 𝜌 𝑝-𝑘 0 ‖𝑢 𝑘 0 ‖ 𝐿 1 (0,∞) ,
thus 𝑢 𝑝 ∈ 𝐿 1 (0, ∞). Moreover (𝑢 𝑝 ) ′ = 𝑝𝑢 ′ 𝑢 𝑝-1 belongs to 𝐿 1 (0, ∞) because

∫ ∞ 0 |𝑢 ′ 𝑢 𝑝-1 |𝑑𝑡 ≤ (∫ ∞ 0 |𝑢 ′ | 2 ) 1 2 ∫ ∞ 0 𝑢 2(𝑝-1) 𝑑𝑡 ≤ ‖𝑢 ′ ‖ 𝐿 2 ∫ ∞ 0 𝑢 2(𝑝-1)-𝑘 0 +𝑘 0 𝑑𝑡 .
But 2𝑝 -2 -𝑘 0 > 0, thus 𝑢 2(𝑝-1)-𝑘 0 ≤ 𝜌 2(𝑝-1)-𝑘 0 and therefore (𝑢 𝑝 ) ′ belongs to 𝐿 1 (0, ∞) because 𝑢 𝑘 0 does. This implies that 𝑢 𝑝 ∈ 𝑊 1,1 (0, ∞) and hence 𝑢(𝑡) 𝑝 → 0 as 𝑡 → +∞.

We now turn to the dissipation part, equation [START_REF] Deschamps | Quasi steady state approximation of the small clusters in Becker-Döring equations leads to boundary conditions in the Lifshitz-Slyozov limit[END_REF]. There exists a sequence of times 𝑡 𝑛 → +∞ such that 𝐷 𝑘 (𝑡 𝑛 ) → 0 by integrability. Actually the dissipation reads

𝐷 𝑘 (𝑡) = 𝑢 2 (𝑡) ∫ ∞ 0 𝑎(𝑥)𝑓(𝑡, 𝑥) 𝑑𝑥 + ∫ ∞ 0 𝑏(𝑥)Φ(𝑥)𝑓(𝑡, 𝑥) 𝑑𝑥 -2𝑢(𝑡) ∫ ∞ 0 𝑏(𝑥)𝑓(𝑡, 𝑥) 𝑑𝑥 .
Using inequality [START_REF] Collet | Some Recent Results on the Kinetic Theory of Phase Transitions[END_REF] and the definition of solution, the first and last integrals are continuous and bounded in time. Together with the fact that 𝑢(𝑡) → 0 as 𝑡 → ∞ we have that This contradicts the fact that 𝑢(𝑡 𝑛 ) = 𝜌 -∫ ∞ 0 𝑥𝑓(𝑡 𝑛 , 𝑥)𝑑𝑥 → 0.

Since mass is preserved, the divergence of 𝑀 0 implies that the average aggregate size tends to zero.

Concentration behavior for the mass density

We can use the dissipation to extract some information on the long-time asymptotic. We proceed by standard Lasalle's invariance principle arguments, proving that the orbits are relatively compact and we identify trajectories in the 𝜔-limit set to be time independent. In fact we shall work out the argument from scratch because we lack the continuity of our Lyapunov functional.

In this section we assume that our hypotheses (H0-H5) hold true and we take 𝑓 a solution in the sense of Definition 2.1 with initial data 𝑓 in satisfying hypothesis (H6). We let {𝑡 𝑛 } an arbitrary increasing sequence of times with lim 𝑛→+∞ 𝑡 𝑛 = +∞. Let 𝑇 > 0 arbitrary, we define for 𝑡 ∈ [0, 𝑇] 𝑓 𝑛 (𝑡, 𝑥) = 𝑓(𝑡 + 𝑡 𝑛 , 𝑥) and 𝜇 𝑛 𝑡 (𝑑𝑥) = 𝑥𝑓 𝑛 (𝑡, 𝑥)𝑑𝑥 .

The measures 𝜇 𝑛 𝑡 are bounded nonnegative Radon measures on (0, +∞) with 𝜇 𝑛 𝑡 ((0, +∞)) = 𝜌. Let 𝐻 be given by [START_REF] Conlon | On a diffusive version of the Lifschitz-Slyozov-Wagner equation[END_REF] and 𝐻 𝑛 (𝑡) = 𝐻(𝑡 + 𝑡 𝑛 ).

Let ℳ + 𝜌 (0, +∞) the set of nonnegative Radon measures on (0, +∞) with mass less or equal to 𝜌. The topology induced by the dual of 𝒞 0 𝑐 (0, ∞) on ℳ + 𝜌 (0, ∞) is called vague topology. We denote this space with such topology by 𝑣 -ℳ + 𝜌 (0, ∞). This space is metrizable and compact [START_REF] Brezis | Functional analysis, sobolev spaces and partial differential equations[END_REF], with metric

𝑑(𝜇, 𝜈) = ∑ 𝑘≥0 2 -𝑘 min(1, |⟨𝜇, 𝜑 𝑘 ⟩ -⟨𝜈, 𝜑 𝑘 ⟩|) ,
for all 𝜇, 𝜈 in ℳ + 𝜌 (0, ∞) where (𝜑 𝑘 ) 𝑘≥0 ∈ 𝒞 ∞ 𝑐 (0, ∞) is dense in 𝒞 0 𝑐 (0, ∞) and ⟨𝜇, 𝜑⟩ = ∫ 𝜑 𝑑𝜇 denotes the duality pairing. Proof. Notice that by standard truncation arguments with equations ( 5), [START_REF] Collet | Some Recent Results on the Kinetic Theory of Phase Transitions[END_REF] and hypothesis (H6),

| | | | | | | | 𝑑 𝑑𝑡 ∫ ∞ 0 𝑥 2 𝑓(𝑡, 𝑥)𝑑𝑥 | | | | | | | | ≤ 𝐾(𝜌 + 1) ∫ ∞ 0 𝑥(1 + 𝑥)𝑓(𝑡, 𝑥)𝑑𝑥 .
The latter is uniformly bounded in time. By Arzelà-Ascoli we may extract a subsequence of 𝑡 ↦→ ∫ ∞ 0 𝑥 2 𝑓 𝑛 (𝑡, 𝑥)𝑑𝑥 which converges uniformly on [0, 𝑇]. Moreover, by the monotonicity and nonegativity of 𝐻(𝑡) it has a limit 𝐻 ∞ as 𝑡 → ∞ and 𝐻 𝑛 converges uniformly on [0, 𝑇] to 𝐻 ∞ . This proves that Ψ(𝑢 𝑛 (𝑡)) = 𝐻 𝑛 (𝑡) -∫ ∞ 0 𝑥 2 𝑓 𝑛 (𝑡, 𝑥)𝑑𝑥 converges uniformly to a function Ψ(𝑡) on [0, 𝑇]. Note that Ψ = ∫ 𝑥 0 Φ -1 (𝑦)𝑑𝑦 is continuous and strictly increasing (because Φ -1 > 0), thus 𝑢 𝑛 (𝑡) converges to 𝑢(𝑡) ∶= Ψ -1 (Ψ(𝑡)) pointwise. Remark 4.4. We might replace 𝑥 2 above by 𝑥 1+𝜂 , provided that (𝑎(𝑥) + 𝑏(𝑥))𝑥 𝜃 ≤ 𝐶𝑥 for some 𝜃 ≥ 0 and 𝐶 > 0.

Proposition 4.5. The measure 𝑥𝑓(𝑡, 𝑥)𝑑𝑥 converges to 0 in 𝑣 -ℳ + 𝜌 (0, +∞) as 𝑡 → +∞.

Proof. By the previous two Lemmas we can extract a subsequence such that 𝜇 𝑛 converges to some 𝜇 in 𝒞 0 ([0, 𝑇]), 𝑣 -ℳ + 𝜌 (0, ∞)) and 𝑢 𝑛 converges poinwise to some 𝑢 ∈ 𝒞 0 ([0, 𝑇]). Given that 𝐻 𝑛 (𝑡) converges to 𝐻 ∞ for all 𝑡 ∈ [0, 𝑇], we use Corollary 3.2 and equation ( 9 , 𝑚] and define 𝜒 𝑚 a nonnegative continuous function with compact support in (0, ∞), equal to a positive constant on 𝐾 𝑚 and such that 𝜒 𝑚 (𝑥) ≤ 𝑎(𝑥)∕𝑥 for all 𝑥 > 0. This is possible by the continuity and positivity of 𝑎. Hence,

lim 𝑛→+∞ ∫ 𝑇 0 ∫ ∞ 0 ( Φ -1 (𝑢 𝑛 (𝑡)) -Φ -1 (Φ(𝑥))) ) (𝑢 𝑛 (𝑡) -Φ(𝑥)) 𝜒 𝑚 (𝑥)𝜇 𝑛 𝑡 (𝑑𝑥)𝑑𝑡 = 0 . (13)
Expanding the product, we easily see that space integrals converge uniformly in time on [0, 𝑇]. Moreover, 𝑢 𝑛 converges pointwise and is bounded, hence the above limit can be interchanged with the time integrals and we conclude that

∫ 𝑇 0 ∫ 𝐾 𝑚 ( Φ -1 (𝑢(𝑡)) -Φ -1 (Φ(𝑥))) )
(𝑢(𝑡) -Φ(𝑥)) 𝜒 𝑚 (𝑥)𝜇 𝑡 (𝑑𝑥)𝑑𝑡 = 0 , for all 𝑚 ≥ 1. Thus, a.e. 𝑡 ∈ [0, 𝑇] we have the following measure equality

( Φ -1 (𝑢(𝑡)) -Φ -1 (Φ(𝑥))) ) (𝑢(𝑡) -Φ(𝑥)) 𝜇 𝑡 (𝑑𝑥) = 0 .
As 𝑢 is continuous, the above equality is achieved for all 𝑡 ∈ [0, 𝑇]. Noticing that Φ -1 is increasing, for all 𝑡 ∈ [0, 𝑇], either 𝜇 𝑡 = 0 or both 𝑢(𝑡) > Φ 0 and there exists 𝑚 𝑡 ∶ [0, 𝑇] → (0, 𝜌] such that 𝜇 𝑡 = 𝑚 𝑡 𝛿 Φ -1 (𝑢(𝑡)) .

We should prove the second alternative could not occur. Indeed, let 𝑡 0 such that 𝜇 𝑡 0 = 𝑚 𝑡 0 𝛿 𝑥 0 with 𝑥 0 = Φ -1 (𝑢(𝑡 0 )) > 0 -because 𝑢(𝑡 0 ) > Φ 0 . For all 𝜑 ∈ 𝒞 0 𝑐 (0, ∞), lim 𝑥→+∞ ∫ ∞ 0 𝜑(𝑥)𝑓 𝑛 (𝑡 0 , 𝑥)𝑑𝑥 = 𝑚 𝑡 0 𝜑(𝑥 0 ) 𝑥 0 , which proves that 𝑓 𝑛 (𝑡 0 , 𝑥)𝑑𝑥 converges in ℳ + 𝜌 (0, ∞) and contradicts that the 0 𝑡ℎ -moment 𝑀 0 (𝑡 0 + 𝑡 𝑛 ) goes to ∞ as 𝑛 → ∞, by Prop. 4.1.

Hence, we have shown that lim 𝑛→∞ 𝜇 𝑛 𝑡 = 0. Note that we have 𝜇 𝑛 𝑡 = 𝑥𝑓(𝑡 + 𝑡 𝑛 , 𝑥)𝑑𝑥. Taking 𝑡 = 0 we deduce that lim 𝑛→∞ 𝑥𝑓(𝑡 𝑛 , 𝑥)𝑑𝑥 = lim 𝑛→∞ 𝜇 𝑛 0 = 0 , vaguely in ℳ + 𝜌 ((0, +∞)) , and this holds true for any sequence 𝑡 𝑛 → ∞. Therefore, we have proved that 𝑥𝑓(𝑡, 𝑥)𝑑𝑥 has a limit as 𝑡 → ∞ and this limit is 0.

We conclude that the limit measure concentrates at the origin.

Lemma 4.6.

There exists 𝑚 > 0 and a subsequence (not relabelled) such that lim 𝑛→∞ 𝑥𝑓(𝑡 𝑛 , 𝑥)𝑑𝑥 = 𝑚𝛿 0 , weakly in ℳ + 𝜌 ([0, +∞)) .

Proof. Note that 𝜈 𝑛 = 𝑥𝑓(𝑡 𝑛 , 𝑥)𝑑𝑥 defines a sequence of measures on [0, +∞), such that 𝜈 𝑛 |(0,+∞) = 𝜇 𝑛 . The sequence {𝜈 𝑛 } is bounded and thus admits a subsequence which converges 𝑣 -ℳ + 𝜌 ([0, +∞)); recall that this is the dual of 𝒞 0 𝑐 ([0, ∞)). By the previous results, the limit 𝜈 verifies that 𝜈 |(0,+∞) = 0 as measures. Thus 𝜈 = 𝑚𝛿 0 for some 𝑚 ≥ 0. We recall that ⟨𝜈 𝑛 , 𝑥⟩ is uniformly bounded in 𝑛; this control allows to improve the convergence to ⟨𝜈 𝑛 , 𝜑⟩ → ⟨𝜈, 𝜑⟩ for all 𝜑 ∈ 𝒞 𝑏 ([0, +∞)), the set of continuous and bounded functions. It remains to identify 𝑚 which will be the result of the next section.

Identification of the concentrated mass

This paragraph is devoted to the proof that 𝑢(𝑡) approaches the critical value Φ 0 ; by mass conservation, the concentrated mass 𝑚 in Lemma 4.6 is therefore 𝜌 -Φ 0 . This will conclude the proof of Theorem 2.2. In this section we still assume that our Hypotheses (H0-H5) hold true and we take 𝑓 a solution in the sense of Definition 2.1 with initial datum 𝑓 in satisfying hypothesis (H6). Proof. By hypothesis (H6) and a (refined) de La Vallé Poussin's lemma [START_REF] Laurençot | Weak Compactness Techniques and Coagulation Equations[END_REF] there exists 𝛽 ∈ 𝒞 1 ([0, ∞)), nonegative, increasing, convex, such that lim 𝑥→∞ 𝛽(𝑥)∕𝑥 = +∞ and

∫ ∞ 0 𝛽(𝑥)𝑥𝑓 in (𝑥)𝑑𝑥 < ∞ .
According to Remark 3.3 we may use the Lyapunov functional [START_REF] Collet | The Becker-Döring system and its Lifshitz-Slyozov limit[END_REF] with 𝑘(𝑥) = 𝛽(𝑥)𝑥, which is convex too, to deduce that

sup 𝑡>0 ∫ ∞ 0 𝛽(𝑥)𝑥𝑓(𝑡, 𝑥)𝑑𝑥 < ∞.
Let a sequence of times 𝑡 𝑛 ↗ ∞; Lemma 4.6 ensures that there exists a subsequence (not relabelled) such that 𝑥𝑓(𝑡 𝑛 , 𝑥)𝑑𝑥 → 𝑚𝛿 0 weakly. Thus, we let 𝜒 𝑅 = min(𝑥, 𝑅), so that We take the limit 𝑛 → +∞; the first term on the right hand side goes to 0 since 𝑥𝑓(𝑡 𝑛 , 𝑥)𝑑𝑥 → 𝑚𝛿 0 and 𝜒 𝑅 (0) = 0. Finally we take the limit 𝑅 → +∞ and the remaining term goes to 0. Since this is true for all sequences, we get the full limit.

Remark 4.8. Here again we might deal with 𝑥 1+𝜂 instead of 𝑥 2 as in Remark 4.4.

Lemma 4.9.

There holds that lim 𝑡→+∞ 𝑢(𝑡) = Φ 0 .

Proof. The convergence of 𝐻(𝑡) as 𝑡 → ∞ together with the previous lemma entails that Ψ(𝑢(𝑡)) converges to some constant 𝑐. By the (strict) monotonicity of Ψ and its continuity, Ψ -1 is also continuous and thus 𝑢(𝑡) converges to 𝜃 ∶= Ψ -1 (𝑐). Assume that 𝜃 > Φ 0 . We will prove a contradiction. We introduce the function 𝐴(𝑥) = ∫ This is justified since the function Φ is continuous on [0, 𝜀], thus bounded, and there exists 𝐾 𝜀 > 0 such that Φ(𝑥) = 𝑏(𝑥)∕𝑎(𝑥) ≤ 𝐾 𝜀 𝑥 for all 𝑥 > 𝜀 thanks to hypothesis (H0) and inequality [START_REF] Collet | Some Recent Results on the Kinetic Theory of Phase Transitions[END_REF].

As 𝜃 > Φ 0 , we may find 𝛿, 𝑡 0 > 0 such that 𝑢(𝑡) > Φ 0 + 2𝛿 for all 𝑡 > 𝑡 0 . Next, we may find 𝜖 > 0 such that sup 𝑥∈[0,𝜀] Φ(𝑥) < 𝛿 + Φ 0 . Then Since 𝑀 0 (𝑡) → +∞ we derive that

lim 𝑡→+∞ 1 𝑡 ∫ ∞ 0 𝐴(𝑥)𝑓(𝑡, 𝑥)𝑑𝑥 = +∞.
But, with hypothesis (H3), there exists 𝐾 𝐴 such that 𝐴(𝑥) ≤ 𝐾 (1 + 𝑥) for all 𝑥 > 0, thus

∫ ∞ 0 𝐴(𝑥)𝑓(𝑡, 𝑥)𝑑𝑥 ≤ 𝐾 𝐴 ∫ ∞ 0 (1 + 𝑥)𝑓(𝑡, 𝑥)𝑑𝑥 ≤ 𝐾 𝐴 𝑀 0 (𝑡) + 𝐾 𝐴 𝜌 .
Given that 𝑀 0 (𝑡)∕𝑡 is bounded we have a contradiction. Hence 𝑢(𝑡) → Φ 0 as 𝑡 → ∞.

As a consequence of the previous result and mass conservation, we have 𝑚 = 𝜌 -Φ 0 in the representation of the limit measure 𝜈 = 𝑚𝛿 0 . This concludes the proof of Theorem 2.2. Corollary 4.10. Under the same hypotheses of the theorem, we have that all moments 𝑀 𝜃 (𝑡) with 𝜃 ∈ [0, 1) diverge as 𝑡 → ∞. This follows from Lemma 4.7, by interpolating the first moment between moments of order 𝜃 and two.

Long time behavior for power-law rates and numerics

Rate of convergence for special cases

The purpose of this section is to provide more detailed information of the long time behavior in specific power-law cases. Here we assume 𝑎(𝑥) = 𝑎𝑥 𝛼 and 𝑏(𝑥) = 𝑏𝑥 𝛽 with 𝑎, 𝑏 > 0 and 0 ≤ 𝛼 < 𝛽 ≤ 1. Then Φ(𝑥) = 𝑏 𝑎 𝑥 𝛽-𝛼 verifies Φ 0 = 0 and is monotonically increasing.

We may define the classical moments of 𝑓 as

𝑀 𝑘 (𝑡) ∶= ∫ ∞ 0 𝑥 𝑘 𝑓(𝑡, 𝑥) 𝑑𝑥, 𝑘 ∈ ℝ + .
In the case of power-law kinetic rates, the time derivative of the classical moments reads

𝑑𝑀 𝑘 𝑑𝑡 = 𝑘𝑎𝑢(𝑡)𝑀 𝑘+𝛼-1 (𝑡) -𝑘𝑏𝑀 𝑘+𝛽-1 (𝑡) .
In what follows we consider some particular choices of the exponents 𝛼 and 𝛽 for which more specific information can be given (this works also as a guide to numerical conjectures for the general case, see Section 5.2 below). Lemma 5.1 (Case 𝛼 = 0). Assume that 𝔫(𝑢) = 𝑢 𝑖 0 -a bound from above by a power law works the same way. Then there exists some 𝐶 > 0 such that, for advanced 𝑡 𝑀 0 (𝑡) ≤ 𝐶𝑡

1 1+𝑖 0 𝛽 , 𝑢(𝑡) ≤ 𝐶𝑡 - 𝛽 1+𝑖 0 𝛽 .
Proof. First we prove that 𝑢(𝑡)𝑀 𝛽 0 (𝑡) ≤ 𝐶 for every 𝑡 ≥ 0. For that aim, let 𝑦(𝑡) = 𝑢(𝑡)𝑀 𝛽 0 (𝑡). We readily compute

𝑦 ′ = (𝑏𝑀 𝛽 -𝑢𝑎𝑀 0 )𝑀 𝛽 0 + 𝛽𝑢𝔫(𝑢)𝑀 𝛽-1 0 .
Here we can use that moment interpolation yields the estimate

𝑀 𝛽 (𝑡) ≤ 𝑀 𝛽 1 (𝑡)𝑀 1-𝛽 0 (𝑡) ≤ 𝜌 𝛽 𝑀 1-𝛽 0 (𝑡).
Therefore, 𝑦 ′ ≤ 𝜑(𝑡) + 𝑏𝜌 𝛽 𝑀 0 -𝑎𝑢𝑀 0 𝑀 𝛽 0 = 𝑀 0 (𝑏𝜌 𝛽 -𝑎𝑦) + 𝜑(𝑡) , with 𝜑(𝑡) → 0 as 𝑡 → ∞. This implies that 𝑦(𝑡) belongs to [0, 𝜌 𝛽 𝑏∕𝑎] for advanced times.

Thanks to our assumption on 𝔫 we have

𝑀 ′ 0 = 𝑢 𝑖 0 𝑀 𝛽𝑖 0 0 𝑀 𝛽𝑖 0 0 ≤ 𝜌 𝛽𝑖 0 (𝑏∕𝑎) 𝑖 0 𝑀 𝛽𝑖 0 0
, and therefore 𝑑 𝑑𝑡 𝑀 1+𝑖 0 𝛽 0 ≤ (1 + 𝑖 0 𝛽)𝜌 𝑖 0 𝛽 (𝑏∕𝑎) 𝑖 0 .

Our statements follow easily from here.

We expect to have equality in the previous estimates. Actually, in the general case 𝛼 < 𝛽 we conjecture that:

𝑀 0 (𝑡) ∼ 𝑡 1 1+𝑖 0 (𝛽-𝛼) , 𝑢(𝑡) ∼ 𝑡 - 𝛽-𝛼 1+𝑖 0 (𝛽-𝛼) . (15)
for advanced 𝑡. Likewise, we expect that 𝑢(𝑡)𝑀 𝛽-𝛼 0

(𝑡) will have a finite limit. Compare with the numerical simulations below.

We prove another partial result along the same lines. Proof. In this specific case we have

𝑑𝑀 1 𝑑𝑡 = 𝑎𝑢𝑀 𝛼 -𝑏𝑀 1 .
This is integrated as 𝑀 1 (𝑡) = 𝑀 1 (0)𝑒 -𝑏𝑡 + 𝑎𝑒 -𝑏𝑡 ∫ 𝑡 0 𝑢(𝜏)𝑀 𝛼 (𝜏)𝑒 𝑏𝜏 𝑑𝜏 .

Since 𝑢(𝑡) → 0 as 𝑡 → ∞, we have that 𝑀 1 → 𝜌 as 𝑡 → ∞. Now we argue by contradiction. Assume that lim inf 𝑡→∞ 𝑢𝑀 𝛼 > 𝑏𝜌∕𝑎. Then, given 𝜖 > 0 there exists some 𝑇 > 0 such that Taking the limit 𝑡 → ∞ we obtain lim 𝑡→∞ 𝑀 1 (𝑡) ≥ 𝜌 + 𝑎𝜖∕𝑏, which is a contradiction. We can prove in a similar way that lim sup 𝑡→∞ 𝑢𝑀 𝛼 < 𝑏𝜌∕𝑎 leads to a a contradiction. Thus our statement follows.

𝑢𝑀

Numerical experiments and discussions

To approach numerically the Lifshitz-Slyozov equation we use a standard finite volume scheme with an upwind approximation of the fluxes. The behaviour of the solutions is depicted in Figures 1 to 3.

Figure 1 shows time evolution of the distribution for two distinct initial conditions with rates given by 𝛼 = 1∕3 and 𝛽 = 2∕3, see details in the figure's legend. Note that we have no explicit solution at hand and the rate of convergence is unknown in this case. It seems that, roughly speaking, the particular details of the initial condition are lost as time advances and the concentration behaviour that ensues seems to follow a universal profile. Figure 2 shows the rates of convergence of 𝑢, and divergence of 𝑀 0 , to be polynomial. We compare with the conjecture [START_REF] Laurençot | Weak solutions to the Lifshitz-Slyozov-Wagner equation[END_REF] and the results agree. In fact, this is robust according to various set of coefficients (results not shown). To further capture the limiting profile, we plot in Figure 3 the tail distribution 𝐹(𝑡, 𝑥) = ∫ ∞ 𝑥 𝑓(𝑡, 𝑥)𝑑𝑥, while normalising the mass and the front speed. Specifically, choosing 𝑥 ↦→ 1∕(1 + 𝑀 0 (𝑡))𝐹(𝑡, 𝑥∕(1 + 𝑀 0 (𝑡)), we observe that several initial conditions lead to a similar asymptotic profile. However, we emphasize that our numerical scheme is not specifically designed to robustly capture a potential asymptotic profile, as it has been shown in [START_REF] Carrillo | A numerical study on large-time asymptotics of the Lifshitz-Slyozov system[END_REF] in the outgoing case. Using a sufficiently refined mesh, we hope that at least we capture correctly the rates of convergence.

Long-time behavior for proportional attachment and detachment rates

In this section we discuss the special case of 𝑏(𝑥) = Φ 0 𝑎(𝑥) for some given Φ 0 > 0. Particular instances of this situation for power-law rates (with outflow behavior or zero boundary conditions) have been studied in [START_REF] Collet | On solutions of the Lifshitz-Slyozov model[END_REF][START_REF] Collet | Some Recent Results on the Kinetic Theory of Phase Transitions[END_REF]. Here we investigate the case of nucleation boundary conditions [START_REF] Calvo | Long-time asymptotics for polymerization models[END_REF]. For this section we still assume hypotheses (H0), (H2) and (H3). Note that hypothesis (H1) is replaced by Φ being constant. Here hypothesis (H4) is not required and hypothesis (H5) is replaced by 𝔫 is Lipschitz continuous on [Φ 0 , 𝜌] and 𝔫(𝑧) ≥ 𝔫(Φ 0 ) = 0 , for all 𝑧 ∈ [0, 𝜌] ,

(H5') that is, nucleation cannot fuel anymore at the critical value. Finally, (H6) reduces to just

𝑓 in ∈ 𝐿 1 (ℝ + , (1 + 𝑥)𝑑𝑥) . ( H6

')

We shall assume morevover that 𝑎 ∈ 𝒞 1 (0, ∞), which entails that the characteristic curves are well-defined. Therefore, in this case of constant Φ, existence and uniqueness of global solutions is ensured, see [START_REF] Calvo | The initial-boundary value problem for the Lifshitz-Slyozov equation with non-smooth rates at the boundary[END_REF].

Further, we let

𝐴(𝑥) = ∫ 𝑥 0 1 𝑎(𝑦) 𝑑𝑦 ,
for all 𝑥 ≥ 0, which is an increasing 𝒞 1 -diffeomorphism from ℝ + into ℝ + with 𝐴(0) = 0. We denote

𝑀 𝑎 (𝑡) = ∫ ∞ 0 𝑎(𝑥)𝑓(𝑡, 𝑥)𝑑𝑥 ,
which is finite for all 𝑡 ≥ 0 by [START_REF] Collet | Some Recent Results on the Kinetic Theory of Phase Transitions[END_REF].

The main result of this section rules out concentration phenomena for the density and provides an explicit rate of convergence for 𝑢: Theorem 6.1. Assume 𝑎 ∈ 𝒞 1 (0, ∞) and 𝑏(𝑥) = Φ 0 𝑎(𝑥). Under hypotheses (H0), (H2), (H3), (H5'), (H6'), any global solution in the sense of Definition 2.1 satisfies:

• lim 𝑡→∞ 𝑢(𝑡) = Φ 0 ,
• There exists 𝑓 ∈ 𝐿 1 (ℝ + , (1 + 𝑥)𝑑𝑥) such that lim 𝑡→+∞ 𝑓(𝑡, ⋅) = 𝑓 , 𝑤 -𝐿 1 (ℝ + ) .

Indeed, the limit 𝑓 has a representation formula, given at end of the proof of Theorem 6.1, see [START_REF] Niethammer | On the evolution of large clusters in the Becker-Döring model[END_REF]. This representation depends noticeably on the chosen initial condition. Furthermore, if 𝑎 is non-decreasing in 𝑥 then 𝑀 𝑎 is increasing in 𝑡 and our proof shows that the trend to equilibrium is exponential: 0 < 𝑢(𝑡) -Φ 0 ≤ (𝑢(0) -Φ 0 )𝑒 -𝑀 𝑎 (0)𝑡 .

Proof. We proceed in a number of separate steps. During the proof we assume 𝑓 in ≠ 0. There is no loss of generality in so doing, as the given boundary conditions ensure that starting with 𝑓 in = 0 produces some nonvanising 𝑓(𝑡 0 ) for some 𝑡 0 > 0 small, which we may take as a new initial condition.

Step 1: Mild formulation. We will represent the solution in terms of characteristics. For that aim we will use several results from [START_REF] Calvo | The initial-boundary value problem for the Lifshitz-Slyozov equation with non-smooth rates at the boundary[END_REF]. The equation determining the characteristics reads 𝑑 𝑑𝑠 𝑋(𝑠; 𝑡, 𝑥) = (𝑢(𝑠) -Φ 0 )𝑎(𝑋(𝑠; 𝑡, 𝑥)) ; 𝑋(𝑡; 𝑥) = 𝑥 .

For any given 𝑥 > 0 we can ensure existence and uniqueness of a maximal solution 𝑋(⋅; 𝑡, 𝑥) on (𝜎 𝑡 (𝑥), ∞). Note the following: either 𝜎 𝑡 (𝑥) = 0 and lim 𝑠→0 𝑋(𝑠, 𝑡, 𝑥) > 0, or 𝜎 𝑡 (𝑥) > 0 and lim 𝑠→𝜎 𝑡 (𝑥) 𝑋(𝑠, 𝑡, 𝑥) = 0.

Therefore, for any 𝑠 ∈ (𝜎 𝑡 (𝑥), +∞) we can integrate ( 16) as follows:

𝑋(𝑠; 𝑡, 𝑥) = 𝐴 -1 (𝐴(𝑥) + ∫ 𝑠 𝑡 (𝑢(𝜏) -Φ 0 )𝑑𝜏) .
We define 𝛾(𝑡) for all 𝑡 ≥ 0 through 𝛾(𝑡) ∶= ∫ 𝑡 0 (𝑢(𝜏) -Φ 0 )𝑑𝜏 , and 𝑥 𝑐 (𝑡) = 𝐴 -1 (𝛾(𝑡)) .

The curve 𝑥 𝑐 (𝑡) corresponds to 𝑋(𝑡; 0, 0). In [START_REF] Calvo | The initial-boundary value problem for the Lifshitz-Slyozov equation with non-smooth rates at the boundary[END_REF], it is proved for all 𝑡 > 0, 𝜎 𝑡 is a 𝒞 1diffeomorphism form (0, 𝑥 𝑐 (𝑡)) into (0, 𝑡) and 𝑋(0; 𝑡, ⋅) is also a diffeomorphism from (𝑥 𝑐 (𝑡), ∞) into (0, ∞). These facts provide the following mild formulation, for any bounded 𝜑 ∈

Step 2: 𝛾(𝑡) is bounded. Note that 𝑋(𝑡; 0, 𝑥) = 𝐴 -1 (𝐴(𝑥) + 𝛾(𝑡)) ≥ 𝐴 -1 (𝛾(𝑡)).

Hence, by equation [START_REF] Lauretano | On the Relation between the Lifshitz-Slyozov and the Lifshitz-Slyozov-Wagner Models for Supersaturated Solutions[END_REF] with

𝜑(𝑥) = 𝑥, 𝜌 ≥ ∫ ∞ 0 𝑥𝑓(𝑡, 𝑥)𝑑𝑥 ≥ ∫ ∞ 0 𝑓 𝑖𝑛 (𝑥)𝑋(𝑡; 0, 𝑥)𝑑𝑥 ≥ 𝐴 -1 (𝛾(𝑡)) ∫ ∞ 0 𝑓 in (𝑥)𝑑𝑥 ,
which proves our claim.

Step 3: 𝑢(𝑡) → Φ 0 as 𝑡 → ∞. Note that 𝛾 ′ (𝑡) = 𝑢(𝑡) -Φ 0 and 𝑢 ′ (𝑡) = -(𝑢(𝑡) -Φ 0 )𝑀 𝑎 (𝑡) .

It follows that Φ 0 is a steady state for 𝑢(𝑡). Since 𝑢(0) > Φ 0 and 𝑀 𝑎 (𝑡) is continuous and non-negative (with 𝑀 𝑎 (𝑡) = 0 if and only if 𝑓(𝑡) = 0, that is, if and only if 𝑢(𝑡) = 𝜌), we obtain that 𝑢(𝑡) decreases and 𝑢(𝑡) > Φ 0 for all 𝑡 ≥ 0. Thus 𝑢 has a limit, which is Φ 0 .

Step 4: 𝛾(𝑡) is strictly increasing and has a limit as 𝑡 → ∞. The previous steps ensure that 𝛾 is (strictly) increasing, because 𝛾 ′ (𝑡) = 𝑢(𝑡) -Φ 0 > 0, and 𝛾 is bounded, so 𝛾 has a limit 𝛾 = lim 𝑡→∞ 𝛾(𝑡) .

Step 5: 𝑥 𝑐 (𝑡) has a limit as 𝑡 → ∞. Combining the definition of 𝑥 𝑐 (𝑡) = 𝐴 -1 (𝛾(𝑡)), the continuity of 𝐴 -1 and the limit of 𝛾, we obtain that 𝑥 𝑐 (𝑡) has a limit, 𝑥 𝑐 = lim 𝑡→∞ 𝑥 𝑐 (𝑡) = 𝐴 -1 (𝛾) .

(20)

Step 6: 𝑋(𝑡; 0, ⋅) has a limit as 𝑡 → ∞. We remark that 𝑋(𝑡; 0, 𝑥) = 𝐴 -1 (𝐴(𝑥) + 𝛾(𝑡)) for 𝑥 > 0 which entails that 𝑋(𝑡; 0, 𝑥) has a limit too, since 𝐴 is 𝒞 1 -diffeomorphism, and 𝑋(𝑥) = lim 𝑡→∞ 𝑋(𝑡; 0, 𝑥) = 𝐴 -1 (𝐴(𝑥) + 𝛾) .

This makes clear that 𝑋 is a 𝒞 1 -diffeomorphism from (0, ∞) into (𝑥 𝑐 , ∞).

Step 7: 𝜎 -1 𝑡 has a limit as 𝑡 → ∞. Recall from Step 1 that lim 𝑠→𝜎 𝑡 (𝑥) 𝑋(𝑠, 𝑡, 𝑥) = 0. Using the fact that lim 𝑥→𝜎 -1 𝑡 (𝑠) 𝑋(𝑠; 𝑡, 𝑥) = 0 -see [START_REF] Calvo | The initial-boundary value problem for the Lifshitz-Slyozov equation with non-smooth rates at the boundary[END_REF]-we deduce that We conclude that 𝜎(𝑥) = 𝛾 -1 (𝛾 -𝐴(𝑥)) is a 𝒞 1 -diffeomorphism from (0, 𝑥 𝑐 ) into (0, 𝑡), with reciprocal 𝜎 -1 .

Step 8: The limit density. Let 𝜑 ∈ 𝒞 0 ([0, ∞)) be bounded; we insert it in equation [START_REF] Lauretano | On the Relation between the Lifshitz-Slyozov and the Lifshitz-Slyozov-Wagner Models for Supersaturated Solutions[END_REF]. By the limit in Step 7 and the dominated convergence theorem, we conclude that The result of Theorem 6.1, and in particular formula [START_REF] Niethammer | On the evolution of large clusters in the Becker-Döring model[END_REF], is in line with [START_REF] Collet | On solutions of the Lifshitz-Slyozov model[END_REF] in the case 𝔫 = 0 and 𝑎 and 𝑏 power-law functions, where semi-explicit expressions are available for 𝑥 𝑐 , 𝛾 and 𝑋.

0 (

 0 𝑥)𝑢(𝑠)-𝑏(𝑥))𝜑 ′ (𝑥)𝑓(𝑠, 𝑥) , 𝑥)𝑑𝑥 > Φ 0 .Note that for any smooth test function 𝜑, a solution as defined above satisfies the following moment equation: )𝑓(𝑡, 𝑥) = 𝜑(0)𝔫(𝑢(𝑡)) + ∫ ∞ 𝑎(𝑥)𝑢(𝑡) -𝑏(𝑥))𝜑 ′ (𝑥)𝑓(𝑡, 𝑥) .

𝐶( 1 + 2 .

 12 )Φ(𝑥)𝑓(𝑡 𝑛 , 𝑥)𝑑𝑥 = 0. Thanks to hypothesis (H4) and Cauchy-Swartz's inequality we get ∫ ∞ 0 𝑥𝑓(𝑡, 𝑥)𝑑𝑥 ≤ (∫ ∞ 0 𝑏(𝑥)Φ(𝑥)𝑓(𝑡, 𝑥)𝑑𝑥) 𝑥 2 )𝑓(𝑡, 𝑥)𝑑𝑥) 1Since the 0 𝑡ℎ -order moment is bounded and the second order moment as well by Corollary 3𝑛 , 𝑥)𝑑𝑥 = 0.

Lemma 4 . 2 .Lemma 4 . 3 .

 4243 {𝜇 𝑛 } is relatively sequentially compact in 𝒞 0 ([0, 𝑇]), 𝑣 -ℳ + 𝜌 (0, ∞)).Proof. Let 𝜑 ∈ 𝒞 ∞ 𝑐 (0, ∞), we have by[START_REF] Carrillo | A numerical study on large-time asymptotics of the Lifshitz-Slyozov system[END_REF] that𝑑 𝑑𝑡 ∫ ∞ 0 𝜑(𝑥)𝑥𝑓(𝑡, 𝑥)𝑑𝑥 = ∫ ∞ 0 (𝑥𝜑) ′ 𝑎(𝑥)(𝑢(𝑡) -Φ(𝑥))𝑓(𝑡, 𝑥)𝑑𝑥 .As 𝜑 is compactly supported and 𝑢 is bounded by 𝜌, there exists a constant 𝐶 𝜑 > 0, such that |(𝑥𝜑) ′ 𝑎(𝑥)(𝑢(𝑡) -Φ(𝑥))| ≤ 𝐶 𝜑 𝑥, thus for all 𝑡 ≥ 0, min(1, 𝐶 𝜑 𝑘 𝜌|𝑡 -𝑡 0 |) = 0 , and thus the sequence is equicontinuous on [0, 𝑇]. We conclude by the Arzelà-Ascoli theorem. There exist 𝑢 ∈ 𝒞 0 ([0, 𝑇]) with 0 ≤ 𝑢 ≤ 𝜌 and a subsequence of {𝑢 𝑛 } which converges to 𝑢 pointwise on [0, 𝑇].

  𝑢 𝑛 (𝑡)) -Φ -1 (Φ(𝑥)) ) (𝑢 𝑛 (𝑡) -Φ(𝑥)) 𝑎(𝑥)𝑓 𝑛 (𝑡, 𝑥)𝑑𝑥𝑑𝑡 = 0 . Let 𝑚 ≥ 1 and 𝐾 𝑚 = [ 1 𝑚

Lemma 4 . 7 . 0 𝑥 2

 4702 We have that lim 𝑡→+∞ ∫ ∞ 𝑓(𝑡, 𝑥)𝑑𝑥 = 0.

  )𝑥𝑓(𝑡, 𝑥)𝑑𝑥 .[START_REF] Hariz | A modified version of the Lifshitz-Slyozov model[END_REF]

  and 𝐴 𝑛 (𝑥) = 0 otherwise, we get that 𝐴 𝑛 is continuous and 𝐴 𝑛′ ∈ 𝐿 ∞ . Thus we can use 𝐴 𝑛 as a test function in the moment equation (5) and pass to the limit (𝐴 𝑛 ≤ 𝐴) to get 𝑑 𝑑𝑡 ∫ ∞ 0 𝐴(𝑥)𝑓(𝑡, 𝑥)𝑑𝑥 = 𝑢(𝑡)𝑀 0 (𝑡) -∫ ∞ 0 Φ(𝑥)𝑓(𝑡, 𝑥)𝑑𝑥 .

  )𝑓(𝑡, 𝑥)𝑑𝑥 ≤ (𝛿 + Φ 0 ) 𝑀 0 (𝑡) + 𝐾 𝜀 𝜌 , and collecting both estimates we arrive to )𝑓(𝑡, 𝑥)𝑑𝑥 ≥ 𝛿𝑀 0 (𝑡) -𝐾 𝜀 𝜌 for all 𝑡 > 𝑡 0 .

Lemma 5 . 2

 52 (case 𝛽 = 1). There holds that lim 𝑡→∞ 𝑢(𝑡)𝑀 𝛼 (𝑡) = 𝑏𝜌∕𝑎.

  𝛼 > 𝑏𝜌∕𝑎 + 𝜖 for 𝑡 ≥ 𝑇. Therefore, for each 𝑡 > 𝑇, 𝑀 1 (𝑡) ≥ 𝑀 1 (0)𝑒 -𝑏𝑡 + 𝑎𝑒 -𝑏𝑡 ∫ 𝑇 0 𝑢(𝜏)𝑀 𝛼 (𝜏)𝑒 𝑏𝜏 𝑑𝜏 + 𝑎(𝑏𝜌∕𝑎 + 𝜖)𝑒 -𝑏𝑡 ∫ 𝑡 𝑇 𝑒 𝑏𝜏 𝑑𝜏 = 𝑒 -𝑏𝑡 (𝑀 1 (0) + 𝑎 ∫ 𝑇 0 𝑢(𝜏)𝑀 𝛼 (𝜏)𝑒 𝑏𝜏 𝑑𝜏) + 𝑒 -𝑏𝑡 𝑎(𝑏𝜌∕𝑎 + 𝜖) 𝑒 𝑏𝑡 -𝑒 𝑏𝑇 𝑏 .

  𝑥)𝜑(𝑋(𝑡; 0, 𝑥))𝑑𝑥 .

  𝜏) -Φ 0 )𝑑𝜏) = 𝐴 -1 (𝛾(𝑡) -𝛾(𝑠)) . and thus we have the limit ) = 𝐴 -1 (𝛾 -𝛾(𝑠)) .

1 ′

 1 |𝜑(𝑥)𝑑𝑥[START_REF] Niethammer | On the evolution of large clusters in the Becker-Döring model[END_REF] which proves the desired result.

Figure 1 .

 1 Figure 1. Distribution 𝑓(𝑡, 𝑥) with respect to 𝑥 at different times 𝑡. The rates are given by 𝑎(𝑥) = 𝑥 1∕3 , 𝑏(𝑥) = 𝑥 2∕3 and 𝔫(𝑧) = 𝑧 2 . The total mass is 𝜌 = 1. Simulations were performed with a finite volume scheme (upwind), with ∆𝑡 = 5 ⋅ 10 -5 and ∆𝑥 = 10 -4 . Left column: initial condition is 𝑓 in = 0; Right column: 𝑓 in (𝑥) = (-2000(𝑥 -0.2)(𝑥 -0.3)) + .

Figure 2 .

 2 Figure 2. Up: 𝑢(𝑡) versus time 𝑡 in abscissa (straight line) and 𝑡 -1∕5 (dashed line). Down: 𝑀 0 (𝑡) versus time 𝑡 in abscissa (straight line) and 𝑡 3∕5 (dashed line). Graphics are shown in log-log scale. Parameters are the same as Fig. 1 with 𝑓 in = 0.

Figure 3 .

 3 Figure 3. Normalised distribution 𝑥 ↦→ 1∕(1 + 𝑀 0 (𝑡))𝐹(𝑡, 𝑥∕(1 + 𝑀 0 (𝑡)) with 𝐹(𝑡, 𝑥) = ∫ ∞ 𝑥 𝑓(𝑡, 𝑥)𝑑𝑥, with respect to 𝑥 at different times 𝑡. Parameters and simulations are performed as Fig. 1. Left column: initial condition is in = 0; Right column: 𝑓 in (𝑥) = (-2000(𝑥 -0.2)(𝑥 -0.3)) + .

Acknowledgements: J. C. is partially supported by the spanish MINECO-Feder (grant RTI2018-098850-B-I00) and Junta de Andalucía (grants PY18-RT-2422 and A-FQM-311-UGR18). E. H. and R. Y. have been supported by ECOS-Sud project n. C20E03 (France -Chile) and acknowledge financial support from the Inria Associated team ANACONDA.


Renhe Tang

Julien Petit

Yi Yang

Achim Link

Florian Bächle

Marie-Eve L Perrin

Abderrahmane Amgoune

Anis Tlili

Nickel-Catalyzed Trifluoromethylselenolation of Aryl and Alkenyl-Triflates: Scope and Mechanism

Keywords: Nickel, Cross-Coupling, Trifluoromethylselenolation, Mechanistic Investigation, Xantphos ligand

The nickel-catalyzed trifluoromethylselenolation of aryl and vinyl triflates is described herein, encompassing synthesis method development and reaction understanding. The challenge lies in the propensity of the Csp 2 -SeCF3 product to undergo reversible oxidative addition to nickel(0) intermediates, potentially initiating catalyst deactivation pathways. In this study, we disclose that Xantphos/Ni(COD)2 catalytic system is efficient for the synthesis of aryl and vinyl trifluoromethylselenides. The key elementary steps that delineate the catalytic cycles have been investigated by a joint experimental and computational mechanistic investigation and support a Ni 0 /Ni II process. The Xantphos/Ni 0 system was found to react with both ArOTf and ArSeCF3 to give the corresponding (Xantphos)Ni II Ar(X) intermediates (X = OTf or SeCF3), with the complex (Xantphos)Ni II Ar(OTf) emerging as the most stable. In line with experimental observations, computational studies indicate that oxidative addition of ArSeCF3 to Xantphos/Ni 0 is fastest than the oxidative addition of ArOTf but is thermodynamically much less favorable. Thereof, the oxidative addition of ArSeCF3 is reversible and the product of the ArOTf oxidative addition is yielded as the most stable product. The relative stability of oxidative addition products is attributed to the coordination mode of the Xantphos ligand (k2 vs. k3) at nickel (II) intermediates. Finally, we demonstrate that ArSeCF3 compounds can be prepared using an air-stable (Xantphos)Ni(II) pre-catalyst.

INTRODUCTION

Fluorinated motifs have gained widespread interest and several fluorinated compounds are currently finding applications in life science technology. 1 In this context, the association of chalcogens (O, S and Se) with trifluoromethyl group is covetable due to their interesting physicochemical properties and high lipophilicity. 2 Though several pharmaceuticals and agrochemicals compounds supporting the OCF3 motif are commercialized, [START_REF] Liu | Successful trifluoromethoxy-containing pharmaceuticals and agrochemicals[END_REF] their analogues displaying SCF3 or SeCF3 are more scarcely applied. 1,[START_REF] Ghiazza | Chalcogen OCF3 isosteres modulate drug properties without introducing inherent liabilities[END_REF] Regarding SeCF3, attempts to develop trifluoromethylselenolation reactions through electrophilic, nucleophilic or radical reactions design have been implemented, particularly for the forging of Csp 2 -SeCF3 bonds. 5 From a retrosynthetic perspective, the trifluoromethylselenolation of aryl halides constitutes an attractive route due to the availability of starting material as well as the potential use of the technology in late-stage functionalization. However, the few methodologies reported mostly involve aryl iodide as starting electrophiles. Additionally, these methodologies rely on the arylation of Me4NSeCF3 reagent using either stoichiometric amount of copper under harsh reaction conditions, [START_REF] Chen | Synthesis of Cu I trifluoromethylselenates for trifluoromethylselenolation of aryl and alkyl Halides[END_REF] or palladium-based catalysis. [START_REF] Aufiero | Highly efficient C-SeCF₃ coupling of aryl iodides enabled by an air-stable dinuclear Pd(I) catalyst[END_REF] More recently, interesting nickel catalyzed trifluoromethylselenolation processes have been reported (Scheme 1). On the one hand, the employment of NHC-Ni(I) dimer proved to be an efficient catalyst for the trifluoromethylselenolation of aryl iodides. [START_REF] Dürr | Divergent Reactivity of a Dinuclear (NHC)Nickel(I) Catalyst versus Nickel(0) Enables Chemoselective Trifluoromethylselenolation[END_REF] On the other hand, the use of the Ni(0)/Bpy catalytic system revealed its efficiency toward the synthesis of Ar-SeCF3 starting from aryl-iodides orbromides. [START_REF] Han | Nickel-catalyzed trifluoromethylselenolation of aryl halides using the readily available [Me4N][SeCF3] salt[END_REF][START_REF] Krykun | Nickel-Catalyzed Electrochemical Synthesis of (Hetero)Aryl Trifluoromethyl Selenides[END_REF] Aryl chlorides can also be used as starting electrophiles, but their require the use of Ni(0)/Dppf catalytic loadings up to 40 mol%. [START_REF] Han | Nickel-catalyzed trifluoromethylselenolation of aryl halides using the readily available [Me4N][SeCF3] salt[END_REF] To date, the trifluoromethylselenolation of activated phenol derivatives has not been reported. The main challenge for the development of the trifluoromethylselenolation of Csp 2 -O bonds stands in the inherent reactivity aryl(Csp 2 )-SeCF3 towards oxidative addition to nickel(0) species. [START_REF] Dürr | Divergent Reactivity of a Dinuclear (NHC)Nickel(I) Catalyst versus Nickel(0) Enables Chemoselective Trifluoromethylselenolation[END_REF] Thereof, the trifluoromethylselenolation of aryl electrophiles that are as or less reactive towards Ni(0) than the selenolated awaited product is expected to be intractable.

In this study we report the Ni-catalyzed trifluoromethylselenolation of aryl and vinyl triflates derivatives, under mild conditions. While most of the ancillary ligands provided discouraging results, the use of Xantphos ligand enabled the efficient formation of the desired ArSeCF3 compounds. Mechanistic investigations combining stoichiometric experiments and computational studies shed light on the key parameters governing the reactivity of XantphosNi(0) species towards the formation of trifluoromethylselenated compounds.

Scheme 1. Current development and challenges for nickel catalyzed trifluoromethylselenolation.

Results and Discussion

Catalyst optimization. The first stage of the investigation began with the identification a ligated

Ni complexes that enable the activation of aryl triflates and deliver the desired ArSeCF3 compounds (Scheme 2). Initial tests were devoted to investigating several Josiphos ligands with different electronic properties in conjunction with Ni(COD)2. We have previously observed that this ligand platform efficiently sustains Ni(0)/Ni(II) redox pathways, allowing for the nickelcatalyzed selective monoarylation of acetone. [START_REF] Derhamine | Nickel-Catalyzed Mono-Selective α-Arylation of Acetone with Aryl Chlorides and Phenol Derivatives[END_REF] For preliminary assays, we selected 4-methylphenyltriflate (1a) as a model substrate, along with Me4NSeCF3 (2) as a nucleophile for reactions performed in hexane at 50 °C for 6 h. Importantly, we have used pMeOC6H4CN as an additive in the aim of generating a more robust and reactive Ni(0) (pre)catalyst. [START_REF] Yin | Fundamental Studies and Development of Nickel-Catalyzed Trifluoromethylthiolation of Aryl Chlorides: Active Catalytic Species and Key Roles of Ligand and Traceless MeCN Additive Revealed[END_REF] Under these conditions, Josiphos-type ligands (L1-L5) exhibited poor reactivity towards the desired product (Scheme 1). Other diphosphine-ligands such as Binap, Dppf, and Xantphos-type ligands and which have proven efficient in nickel catalyzed cross-coupling reactions of aryl electrophiles including aryl triflates were then screened. 13 The use of Binap (L6) afforded very low amounts of the desired product. Dppf ligand (L7) showed 90% conversion of 3a but with a maximum yield of 50% for the desired product 3a. This observation contrasts with the reported efficiency of the Dppf/Ni(COD)2 catalyst for the trifluoromethylthiolation of ArOTf, 14 which points out the enhanced reactivity of ArSeCF3 vs. ArSCF3.

Interestingly, the use of Xantphos (L8) offered an optimal balance with 80% conversion and 60% yield of the desired product. A low yield of 12% was obtained with Dppb (L9), and monophosphines (L10 & L11) revealed inefficient as only low conversions were observed without a trace of the desired product. Finally, the use of bipyridine ligand (L12), which was recently reported to be an effective ligand for the activation of aryl halides, [START_REF] Han | Nickel-catalyzed trifluoromethylselenolation of aryl halides using the readily available [Me4N][SeCF3] salt[END_REF] was also ineffective for the trifluoromethylselenolation of aryl triflate 1a.

Scheme 2. Ligand screening for the trifluoromethylselenolation of 1a.

[a] Reactions were performed with 1a (0.1 mmol, 1 equiv.), Ni(COD)2 (0.01 mmol, 0.1 equiv.), LX (0.01 mmol, 0.1 equiv.), 2 (0.15 mmol, 1.5 equiv.), pMeOC6H4CN (0.05 mmol, 0.5 equiv.) and solvent (1 mL) for 16 hours. Yields determined by [START_REF] Standley | A Broadly Applicable Strategy for Entry into Homogeneous Nickel(0) Catalysts from Air-Stable Nickel(II) Complexes[END_REF] F NMR spectroscopy with PhCF3 as an internal standard. [b] LX (0.02 mmol, 0.2 equiv.).

In the second stage, we have investigated other reaction parameters using Xantphos (L8) as an optimal ligand. Increasing the reaction temperature to 60 °C raised the yield of the desired product 3a to 78% (Table 1, entry 2). Further increase in the reaction temperature to 70 °C proved to be detrimental as 3a were obtained in 68% yield. Performing the reaction without the nitrile additive (4-pMeOC6H4CN) was also detrimental to the reaction outcome, in this case only traces of the desired product were observed (Table 1, entry 4). It is noteworthy that using MeCN as an additive (0.5 equiv.) was not as efficient as pMeOC6H4CN for this transformation, as 3a was obtained only in 6% yield. This suggests that the formation of the reactive [Ni(0)-nitrile] adduct is not favorable with acetonitrile (vide infra). [START_REF] Yin | Fundamental Studies and Development of Nickel-Catalyzed Trifluoromethylthiolation of Aryl Chlorides: Active Catalytic Species and Key Roles of Ligand and Traceless MeCN Additive Revealed[END_REF] In contrast, aromatic nitriles such as PhCN and NaphCN proved to be beneficial for the catalytic activity, although slightly lower yields were obtained than with pMeOC6H4CN (Table 1, entries 6 and 7). Performing the reaction in toluene provided the same results as in hexane (Table 1, entry 8), but the combination of hexane and toluene in a 1:1 ratio resulted in the formation of the desired product in 83% yield (entry 9).

Table 1. Optimization of the reaction. [a] Entry [a] Deviation from standard conditions Yield [b] 1 None 60% Toluene/Hexane (1/1) 83%

[a] Reactions were performed with 1a (0.1 mmol, 1 equiv.), Ni(COD)2 (0.01 mmol, 0.1 equiv.), L8 (0.01 mmol, 0.1 equiv.), 2 (0.15 mmol, 1.5 equiv.), pMeOC6H4CN (0.05 mmol, 0.5 equiv.) and solvent (1 mL) for 16 hours. [b] Determined by [START_REF] Standley | A Broadly Applicable Strategy for Entry into Homogeneous Nickel(0) Catalysts from Air-Stable Nickel(II) Complexes[END_REF] F NMR spectroscopy with PhCF3 as an internal standard.

Scope of the reaction. With the best reaction conditions in hand, we explored the scope of the reaction. A set of aryl triflates substituted with electron donating groups have been subjected for trifluoromethylselenolation. When the phenyl ring is substituted with a methyl group in ortho, meta or para position, good yields have been obtained ranging from 65% to 83% (Scheme 3, products 3a-3c). Similarly, when the phenyl ring is substituted with a t-butyl or a cyclohexyl group in para position, the desired products have been obtained in 62% or 97% yield, respectively (Scheme 3, products 3d and 3e). The reaction also proceeded well with ether substituents (Scheme 3, products 3f-3j). The presence of other oxygenated leaving groups on the starting aryl triflates have been also investigated. Interestingly, the aryl triflate is selectively and smoothly converted in the presence of mesylate, tosylate as well as pivalate groups with reaction outcomes up to 97% yield (Scheme 3, products 3k-3m). The naphthene starting material delivered the desired product with 80% yield (Scheme 3, product 3n). Aryl triflates bearing electron withdrawing groups (Scheme 3, 3o-3s) were also found to display good reactivity. However, the transformation was shown to be not compatible with carboxylic acids, amido, amino and hydroxyl functional groups, (unsuccessful substrates are provided in SI). The trifluoromethylselenolation of more complex substrates such as coumarin and chromone core structures (Scheme 3, 3t-3v), an estrone derivative (Scheme 3, 3w), and a derivative of vitamin E (Scheme 3, 3x) proceeded with good to very good yields. The exploration of electrophiles was enlarged by considering alkenyl triflate derivatives 4.

Several alkenyl triflates were converted to their corresponding trifluoromethylselenolated analogues in good to very good yields (Scheme 3, products 5a-5c).

Scheme 3. Substrate scope.

[a] Reactions were performed with 1 or 4 (0.2 mmol, 1 equiv.), Ni(COD)2 (0.02 mmol, 0.1 equiv.), L8 (0.02 mmol, 0.1 equiv.), 2 (0.3 mmol, 1.5 equiv.), pMeOC6H4CN (0.1 mmol, 0.5 equiv.) and solvent (1 mL) for 16 hours. Yields of isolated products.

[a] Determined by [START_REF] Standley | A Broadly Applicable Strategy for Entry into Homogeneous Nickel(0) Catalysts from Air-Stable Nickel(II) Complexes[END_REF] F NMR spectroscopy with PhCF3 as an internal standard. [b] Ni(COD)2 (0.03 mmol, 0.15 equiv.), L7 (0.03 mmol, 0.15 equiv.).

Mechanistic investigations.

Having identified optimal conditions for the trifluoromethylselenolation of aryl-and vinyl-triflates, we were eager to collect mechanistic rationale for the observed reactivity of the Xanthpos/Ni catalytic system, especially the role played by the nitrile additive and the relative reactivity of the catalysts between the triflate reagent and the associated trifluoromethylselenolated product. Based on previous mechanistic observations,

ArSeCF3 was anticipated to be more reactive towards the oxidative addition to [Ni(0)] than ArOTf. [START_REF] Dürr | Divergent Reactivity of a Dinuclear (NHC)Nickel(I) Catalyst versus Nickel(0) Enables Chemoselective Trifluoromethylselenolation[END_REF] However, the trifluoromethylselenolated product can be isolated as the main product and in good yield using the Xantphos/Ni(0) catalytic system. This prompted us to investigate the mechanism of the reaction. More precisely, we were eager to examine the oxidative additions of both ArOTf 1o and of ArSeCF3 3o to Xantphos-ligated Ni(0) complexes.

As previously reported, attempts to prepare XantphosNi(COD) by reacting Ni(COD)2 and 1 equivalent of the Xantphos ligand L8 in toluene resulted in the precipitation of (Xantphos)2Ni as the major species. [START_REF] Staudaher | Synthesis, mechanism of formation, and catalytic activity of Xantphos nickel π-complexes[END_REF] This complex has poor solubility in most solvents, but its formation was confirmed by X-Ray diffraction analysis (See ESI for details). 16 This species is not reactive under our catalytic conditions (see Table 1, entry 4). Additionally, when Ni(COD)2 and 1 equivalent of Xantphos were reacted in the presence of aryl triflate 1o, only precipitation of (Xantphos)2Ni was observed, no oxidative addition product was detected. The beneficial effect of the nitrile additive, as observed during catalysis, was found to arise from the ability of the nitrile to displace one Xantphos ligand and form a catalytically active (Xantphos)Ni(nitrile) complex. [START_REF] Staudaher | Synthesis, mechanism of formation, and catalytic activity of Xantphos nickel π-complexes[END_REF] We have experimentally confirmed that the addition of 5 to 10 equivalents of p-methoxybenzonitrile to the reaction mixture led to an equilibrium between (Xantphos)2Ni and the nitrile adduct (Xantphos)Ni(pMeOC6H4CN) that is formed up to 34%. Notably, it was found that acetonitrile is not efficient in forming the (Xantphos)Ni(MeCN) complex, which explains the poor catalytic activity observed in the presence of acetonitrile (Table 1, entry 5). 17 Unfortunately, all attempts to characterize the corresponding complex by X-ray diffraction analysis were unsuccessful. The characterization data of complex I indicates the formation of a square planar nickel complex with a weakly coordinated OTf counter anion. Then, we investigated the reactivity of p-phenyl-phenylSeCF3 3o towards oxidative addition to XantphosNi(0) in presence of nitrile was investigated. In this case, monitoring the reaction by [START_REF] Standley | A Broadly Applicable Strategy for Entry into Homogeneous Nickel(0) Catalysts from Air-Stable Nickel(II) Complexes[END_REF] F NMR spectroscopy revealed significant consumption of 3o (70 % conversion) after 4h at room temperature in C6D6, along with the formation of a single species appearing as a singlet at δ = -17 ppm with an overall spectroscopic yield of 20%. The 19 F NMR chemical shift of the new species falls within the same region as trifluoromethylselenate metal complexes previously described. 20 Additionally, the new species displays a 31 P NMR resonance signal at δ = 7.5 ppm, very close to that observed for complex I. These NMR data suggest that XantphosNi(0) also readily undergoes oxidative addition of ArSeCF3 to give the corresponding (Xantphos)Ni(II)(Ar)(SeCF3) species (Scheme 4B). The NiSeCF3 intermediate was found to be unstable, as it gradually disappears over time or upon heating to 60°C, while ArSeCF3 is further consumed. Unfortunately, all attempts to isolate and further characterize the NiSeCF3 intermediate and its degradation product(s) were unsuccessful. After 20 hours of reaction the 19 F NMR spectrum of the reaction mixture showed only two minor resonance signals corresponding to residual ArSeCF3 reactant 3o (8%) and to PhSeCF3 (30%). [START_REF] Lee | The formation of PhSeCF3 is proposed to originate from aryl group exchange between Xantphos ligand and PhPhSeCF3. For a related study[END_REF] The reaction of Xantphos/Ni(0) species with aryl halides has been previously

shown to lead to the formation of Ni(I) intermediates concomitantly with high amount of bi-aryl compounds. [START_REF] Diccianni | Mechanistic Characterization of (Xantphos)Ni(I)-Mediated Alkyl Bromide Activation: Oxidative Addition, Electron Transfer, or Halogen-Atom Abstraction[END_REF] Such a pathway may also be operative for ArSeCF3 to explain the decomposition process; however, no traces of bi-aryl compounds could be detected. These observations along with previous reports indicating that DppfNi(I) species are ineffective as catalyst for the formation of C-SCF3 bonds [START_REF] Dürr | Nickel-catalyzed trifluoromethylthiolation of Csp 2 -O bonds[END_REF][START_REF] Kalvet | When Weaker Can Be Tougher: The Role of Oxidation State (I) in P-vs N-Ligand-Derived Ni-Catalyzed Trifluoromethylthiolation of Aryl Halides[END_REF] suggest that the formation of Ni(I) species in catalytic conditions is not predominant.

Overall, these observations indicate that both ArOTf and ArSeCF3 can react readily with XantphosNi(0) species at room temperature. Remarkably, when the reaction is performed in the presence of stochiometric amount of both substrates, only the triflate derivative is converted towards the formation of the corresponding nickel(II) aryl species I (50% 19 F NMR spectroscopic yield). Under the same conditions, the ArSeCF3 substrate remained almost intact. To further confirm the formation of a NiSeCF3 intermediate, a stoichiometric reaction between complex I and NMe4SeCF3 was carried out. Within 10 minutes at room temperature, [START_REF] Standley | A Broadly Applicable Strategy for Entry into Homogeneous Nickel(0) Catalysts from Air-Stable Nickel(II) Complexes[END_REF] F and 31 P NMR spectroscopic analyses of the reaction mixture indicated the formation of complex II (37 % 31 P NMR spectroscopic yield) along with traces of the ArSeCF3 coupling product. Upon further heating the reaction to 60°C for several hours, the signals associated with complexes I and II disappeared, with only a slight increase in the coupling product. However, when the reaction is carried out in the presence of one equivalent of ArOTf 1o, the formation of the desired product 3o is observed in high yield (87% yield, Scheme 4). These observations suggest that salt metathesis is operative to give a selenated nickel intermediate II. The C-SeCF3 reductive elimination from complex II would proceed with an equilibrium constant favoring the reverse oxidative addition.

Under stoichiometric conditions, the addition of ArOTf is necessary to shift the equilibrium toward the formation of thermodynamically stable complex I and the release of the desired product. The catalytic efficiency of complex I was further confirmed, providing the trifluoromethylselenated product in high yields. Notably, similar results were obtained using previously reported XantphosNi(II) aryl chloride analogue complex III. [START_REF] Oechsner | Complex III was recently shown to be very efficient for the Ni(0)/Ni(II) catalyzed C-S bond formation see[END_REF] All together, these results support a Ni(0)/Ni(II) pathway, which is initiated by the oxidative addition of ArOTf to a mono-Xantphos-ligated Ni(0) complex, followed by salt metathesis and subsequent reductive elimination.

To complement the experimental mechanistic investigation, a computational study was conducted at the M06/CPCM(PhMe)/def2-TZVP:SDD-ECP//B3PW91/6-311G(2d,p):LANL2-ECP DFT level (see ESI for computational details) to confirm the key intermediates and to assess the energetics of the most relevant reaction steps. Xantphos ligand was modeled without structural simplification, and PhOTf was used as a typical electrophile. Unless specified ionic species were considered separated from their counter ion. The (Xantphos)Ni(nitrile) adduct A was optimized as a minimum on the potential energy surface. Its formation is computed endergonic by only 7.8 kcal mol -1 relative to separated (Xantphos)2Ni, Ni(COD)2 and nitrile (Figure 1). This result is consistent with detailed experimental mechanistic observations and previous reports, [START_REF] Yin | Fundamental Studies and Development of Nickel-Catalyzed Trifluoromethylthiolation of Aryl Chlorides: Active Catalytic Species and Key Roles of Ligand and Traceless MeCN Additive Revealed[END_REF][START_REF] Staudaher | Synthesis, mechanism of formation, and catalytic activity of Xantphos nickel π-complexes[END_REF] but also with the equilibrium in solution of (Xantphos)2Ni, Ni(COD)2 and complex A as previously discussed.

The oxidative addition of PhOTf to complex A requires overcoming a Gibbs energy barrier of 16.8 kcal.mol -1 to yield the nickel(II) aryl complex B whose formation is computed exergonic by 19.4 kcal.mol -1 relative to A and the associated reactants. In comparison, the oxidative addition of PhSeCF3 to complex A is much faster, as indicated by a computed Gibbs energy barrier of 11.4

kcal.mol -1 . This is consistent with the observed experimental reactivity trend and in line with the computed (Ar)C-SeCF3 bond dissociation energies that is weaker than the (Ar)C-OTf one by 21 kcal mol -1 (See ESI). Thermodynamically, formation of the resulting (Xantphos)Ni II (Ph) (SeCF3) complex in its singlet (C) or triplet state (D) is exergonic by ca. 4 kcal mol -1 , which is less stable by ca. 15 kcal.mol -1 than the PhOTf oxidative addition product. Therefore, it is likely that the For comparative analysis, we have conducted the same computational investigation using Dppf as a bidentate ligand (see ESI for details). Starting from the (Dppf)Ni(nitrile) adduct, the computed Gibbs energy barriers for the oxidative addition of PhOTf and PhSeCF3 stand at 14.4 and 10.6

kcal.mol -1 , respectively. Both processes were also found to be thermodynamically favorable, by 13.8 kcal.mol -1 for (Dppf)Ni II (Ph)(SeCF3) and by 21.2 kcal.mol -1 for (Dppf)Ni II (Ph)(Otf).

Analogously to that observed with Xantphos ligand, these results indicate that both reactions are feasible and the kinetic preference for PhSeCF3 oxidative addition. The resulting oxidative addition intermediates are thermodynamically stable, with (Dppf)Ni II (Ph)(SeCF3) being more stable than the corresponding (Xantphos)Ni II (Ph)(SeCF3) intermediate.

It is worth noting that the oxidative addition of ArSCF3 to (Dppf)Ni(0) species was reported to lead to a catalytically inactive species. [START_REF] Dürr | Nickel-catalyzed trifluoromethylthiolation of Csp 2 -O bonds[END_REF] This insight raises the possibility that the computed Finally, to further confirm that oxidative addition of ArSeCF3 occurs during catalysis, we hypothesized that trifluoromethylselenolation of Ar 2 OTf may be observed using one equivalent of Ar 1 SeCF3 as the -SeCF3 precursor. This process would entail the concomitant oxidative addition of both Ar 1 SeCF3 and Ar 2 OTf substrates, followed by ligand exchange and reductive elimination to give some Ar 2 SeCF3 product. To test this hypothesis, we conducted a catalytic reaction using a 1/1 ratio of 4-MePhOTf 1a and 4-PhPhSeCF3 3o under identical reaction conditions. We were delighted to observe the formation of the desired product 3a 4-MePhSeCF3 albeit in low yield of 13% and recovering of 75% of the starting 3o (Scheme 5). This experiment evidences the occurrence of some ArSeCF3 oxidative addition during catalysis.

Scheme 5. trifluoromethylselenolation of ArOTf 1a using 3o as SeCF3 precursor.

Finally, we sought to evaluate the use of Ni(II) precatalysts instead of air sensitive Nickel(0) precursors. While the combination of Xantphos with nickel(II) salts in the presence of reductants (Zn, Mn or TDAE) is not operative for this transformation (see ESI for details), we found that the isolated (Xantphos)Ni(II)(OTf)PhPh complex II was very efficient for the trifluoromethylselenolation of 1a affording the desired product in 90% yield. However, as complex II was found to be air sensitive, our attention turned to complex III that is air stable. Using III, the trifluoromethylselenolation of aryl and vinyl triflates were explored. To our delight, complex III performs well the trifluoromethylselenolation independently of the stereo-electronic features of the electrophile substituents, encompassing more complex structures (Scheme 6). Interestingly, starting with III enabled us to decrease the catalyst loading down to 5 mol% with similar reaction outcome (compound 3a, or 3q Scheme 6). Notably, we also observed interesting reactivity with alkenyl triflates, and were able to obtain compound 5a with a yield of 63% (Scheme 6).

Scheme 6. Substrate scope with complex III [a] Reactions were performed with 1 or 4 (0.2 mmol, 1 equiv.), III (0.02 mmol, 0.1 equiv.), 2 (0.3 mmol, 1.5 equiv.), pMeOC6H4CN (0.1 mmol, 0.5 equiv.) and solvent (1 mL) for 16 hours. Yields of isolated products.

  Then, the oxidative addition of p-phenyl-phenyltriflate 1o to Ni(0) complexes yielded from Xantphos and Ni(COD)2 in the presence of 5 eq. of p-methoxybenzonitrile has been investigated. The reaction proceeded at room temperature and resulted in the rapid formation of the oxidative addition complex I, which was isolated in 78% yield (Scheme 4A). The formation of complex I was confirmed in solution by NMR spectroscopy. The 31 P{ 1 H} NMR spectrum of complex I displayed a single resonance signal at δ = 11.4 ppm indicating a trans-symmetric coordination of both phosphorus atoms.[START_REF] Yin | Pd-Catalyzed Intermolecular Amidation of Aryl Halides: The Discovery that Xantphos Can Be Trans-Chelating in a Palladium Complex[END_REF] In addition, the[START_REF] Standley | A Broadly Applicable Strategy for Entry into Homogeneous Nickel(0) Catalysts from Air-Stable Nickel(II) Complexes[END_REF] F NMR resonance signal of the triflate moiety is shifted to δ = -78.8 ppm, suggesting a weak coordination to the Ni center. ESI mass spectrometry analyses of I clearly indicated the formation of a cationic (Xantphos)nickel(II) aryl species (mass peak at 789.1981 m/z for [C51H41OP2Ni + ]).

19

 19

Scheme 4 .

 4 Scheme 4. Mechanistic investigations: A, B. Comparative reactivity of Xantphos/Ni(COD)2/p-methoxybenzonitrile with ArOTf 1o and

(

  Xantphos)Ni II (Ph)(SeCF3) (C) intermediate is formed but undergoes rapid reductive elimination (DG # = 15.4 kcal mol -1 ) to regenerate the Ni(0) adduct A, which ultimately performs the oxidative addition of the triflate with an overall Gibbs energy barrier of 20.8 kcal.mol -1 and a global exergonicity of 15.4 kcal.mol -1 . This mechanistic picture is consistent with the experimental observation.

Figure 1 .

 1 Figure 1. Simplified Gibbs energy profile for the oxidative addition of PhOTf vs. PhSeCF3 (see level of theory in ESI).

(

  Dppf)Ni(II)SeCF3 intermediate may undergo decomposition rather than reversible reductive elimination. Additionally, the oxidative addition of ArOTf to (Dppf)Ni(0) species may lead to (DPPF)Ni(I) species, via a subsequent comproportionation pathway.[START_REF] Kalvet | When Weaker Can Be Tougher: The Role of Oxidation State (I) in P-vs N-Ligand-Derived Ni-Catalyzed Trifluoromethylthiolation of Aryl Halides[END_REF] These species have been shown to lead to inactive species in the context of trifluoromethylthiolation of aryl halides.[START_REF] Kalvet | When Weaker Can Be Tougher: The Role of Oxidation State (I) in P-vs N-Ligand-Derived Ni-Catalyzed Trifluoromethylthiolation of Aryl Halides[END_REF] Thus, on one hand the thermodynamic stability of (Xantphos)Ni II (Ph)(Otf) may prevent from the competitive formation of Ni(I) species. On the other hand, the comparatively diminished stability of (Xantphos)Ni II (Ph)(SeCF3) enables facile reverse reductive elimination and thus do not eventually lead to catalytically inactive species.To get more insights into the factors that account for the important difference of thermodynamic stability between (Xantphos)Ni II (Ph)(OTf) and (Xantphos)Ni II (Ph)(SeCF3) intermediates, optimized structures of XantphosNi(II) complexes B and C were analyzed. In complex B, the nickel center exhibits a square pyramidal geometry in which the tridentate k 3 -P,O,P Xantphos ligand and the phenyl ring form the base of the pyramid, while the OTf ligand occupies the apical position. The distance between the OTf anion and the metal center is relatively short (2.321 Å), much shorter than the sum of the van der Waals radii of Ni and O atoms (3.15 Å). The deviation of Ni from the base is only around 0.28 Å relative to an ideal square-based pyramid geometry, indicating a slightly distorted geometry. In contrast, complexes (Xantphos)Ni II (Ph)(SeCF3) (C or D) display tetrahedral geometries in which the Xantphos ligand is no longer tridentate but k 2 -P,P bidentate as illustrated in Table 2. The optimized Ni-Se distances of 2.44 Å in C and 2.40 Å in D are short relative to the sum of the Ni and Se van der Waals radii (3.53 Å), potentially indicating a strong interaction. Table 2. Top: From left to right: computed structures of complexes B, C and D. Bottom: angles and distances of computed structures. The weaker coordination of the OTf moiety to Ni compared to SeCF3, induces a different coordination mode of the Xantphos ligand, which significantly impacts the coordination geometry around the nickel(II) center.Overall, this computational mechanistic study suggests that the formation of compound C is kinetically favored, yet reversible, while formation of compound B is thermodynamically favored.Consequently, the activation of PhSeCF3 is unlikely to significantly disturb the catalytic trifluoromethylselenolation of PhOTf under the experimental conditions used. The important Gibbs free energy difference observed between compound C and compound B (DG = 15.4 kcal.mol -1 ) may originate from the different coordination geometry around the nickel(II) center induced by the Xantphos ligand (k 3 -P,O,P vs. k 2 -P,P). This may explain the superior catalytic efficiency of Xantphos compared to other bidentate ligands.[START_REF]The bite angle might also contribute to ligand effectiveness as observed with DPEphos and DPPF. However, the catalytic performance doesn't align strictly with bite angle trends[END_REF]

  Scheme 6. Substrate scope with complex III [a] Reactions were performed with 1 or 4 (0.2 mmol, 1 equiv.), III (0.02 mmol, 0.1 equiv.), 2 (0.3 mmol, 1.5 equiv.), pMeOC6H4CN (0.1 mmol, 0.5 equiv.) and solvent (1 mL) for 16 hours. Yields of isolated products. [a] Determined by[START_REF] Standley | A Broadly Applicable Strategy for Entry into Homogeneous Nickel(0) Catalysts from Air-Stable Nickel(II) Complexes[END_REF] F NMR spectroscopy with PhCF3 as an internal standard.[b] III (0.01, 0.05 equiv.)

AUTHOR INFORMATION

Corresponding Author marie-eve.perrin@univ-lyon1.fr abderrahmane.amgoune@univ-lyon1.fr anis.tlili@univ-lyon1.fr

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. The CCIR of ICBMS is acknowledged for a generous allocation of computational resources and providing technical support. We thank abcr for providing Me4NF.