Habib Ammari

Pierre Millien

Shape and size dependence of dipolar plasmonic resonance of nanoparticles

Keywords: Mathematics Subject Classification (MSC2000). 35R30, 35C20 plasmonic resonance, volume integral equation, singular integrals

Introduction

Position of the problem

The optical properties of metallic nanoparticle have been a subject of great interest in the past decades. They have the ability to exhibit plasmonic resonances, which are strong enhancement of the scattering and absorption cross sections at certain frequencies. This capacity to interact strongly with light is a key to many major innovations in nanophotonics [START_REF] Novotny | Principles of Nano-Optics[END_REF][START_REF] Atwater | Plasmonics for improved photovoltaic devices[END_REF][START_REF] Sarid | Modern Introduction to Surface Plasmons: Theory, Mathematical Modeling, and Applications[END_REF], in biomedical imaging [START_REF] Jain | Calculated absorption and scattering properties of gold nanoparticles of different size, shape, and composition: Applications in biomedical imaging and biomedicine[END_REF][START_REF] Link | Shape and size dependence of radiative, non-radiative and photothermal properties of gold nanocrystals[END_REF], cancer treatment therapy [START_REF] Baffou | Mapping heat origin in plasmonic structures[END_REF]. For a nice review of some of these applications we refer the reader to [START_REF] Dreaden | The golden age: gold nanoparticles for biomedicine[END_REF].

These resonances have been theoretically and experimentally studied by the physics community. It has been experimentally shown [START_REF] Link | Size and temperature dependence of the plasmon absorption of colloidal gold nanoparticles[END_REF] (via measurements of the extinction and absorption cross sections) and numerically (simulations of the Maxwell equations, often via a coupled dipoles method, see [START_REF] Kelly | The optical properties of metal nanoparticles: the influence of size, shape, and dielectric environment[END_REF][START_REF] Hao | Synthesis and optical properties of anisotropic metal nanoparticles[END_REF]) that the frequency at which a metallic nanoparticle resonates depends on (i) the shape of the particle; (ii) the type of metal; (iii) the surrounding medium; (iv) the size of the particle. Plasmonic resonances have been the subject of some theoretical work as well in the physics community. In the case of a spherical particle, the classical Mie theory explains points (ii) to (iv). In the case where the particle is not spherical, using the quasi-static approximation and solving Laplace equation, computations of the polarizability for some simple shapes have given a lot of insights on points (i), (ii), and (iii); see, for instance, [START_REF] Su | Interparticle coupling effects on plasmon resonances of nanogold particles[END_REF]. Moreover, the conservation of energy fails in the quasi-static theory, due to the absence of radiative loss. This issue has been dealt by adding a radiative correction [START_REF] Albaladejo | Radiative corrections to the polarizability tensor of an electrically small anisotropic dielectric particle[END_REF].

The size dependence has been more problematic. Some corrections of the quasi-static approximation, sometimes called the modified long-wavelength approximation, or computations of a dynamic polarizability have tackled this issue [START_REF] Meier | Enhanced fields on large metal particles: dynamic depolarization[END_REF][START_REF] Schatz | Theoretical studies of surface enhanced raman scattering[END_REF][START_REF] Kelly | The optical properties of metal nanoparticles: the influence of size, shape, and dielectric environment[END_REF][START_REF] Moroz | Depolarization field of spheroidal particles[END_REF]. Nevertheless, they heavily rely on strong assumptions and are valid only for spheroidal shapes.

In the mathematical community, plasmonic resonances are a more recent subject of interest. In the quasi-static approximation, plasmonic resonances were shown to be an eigenvalue problem linked to the Neumann Poincaré operator [START_REF] Grieser | The plasmonic eigenvalue problem[END_REF][START_REF] Ando | Analysis of plasmon resonance on smooth domains using spectral properties of the neumann-poincaré operator[END_REF][START_REF] Kang | Spectral properties of the neumann-poincaré operator and uniformity of estimates for the conductivity equation with complex coefficients[END_REF]. It was then showed that Maxwell's equation yields a similar type of eigenvalue problems, and a computation of the polarizability for small plasmonic particle was given, solving items (i) to (iii) for a general regular shape [START_REF] Habib Ammari | Surface plasmon resonance of nanoparticles and applications in imaging[END_REF]. Note that these studies were all done in the case where the shape of the particle is assumed to have some regularity, and the theory breaks down when the particle has corners. Some recent progress has been made on this topic [START_REF] Bonnetier | Characterization of the essential spectrum of the neumann-poincaré operator in 2d domains with corner via weyl sequences[END_REF][START_REF] Helsing | Classification of spectra of the neumann-poincaré operator on planar domains with corners by resonance[END_REF][START_REF] Perfekt | The essential spectrum of the neumann-poincaré operator on a domain with corners[END_REF].

The size dependance has been justified in [START_REF] Habib Ammari | Mathematical analysis of plasmonic nanoparticles: the scalar case[END_REF][START_REF] Ando | Plasmon resonance with finite frequencies: a validation of the quasi-static approximation for diametrically small inclusions[END_REF] in the scalar case (transverse electric or transverse magnetic) and in [START_REF] Habib Ammari | Mathematical analysis of plasmonic resonances for nanoparticles: the full maxwell equations[END_REF] in the Maxwell setting. However, practical computations of this size dependency remains complicated. We aim here at presenting a new approach, based on a singular volume integral equation, to compute this size dependency. Our integral volume approach can be extended to the case where the shape of the particle has corners.

Main contribution

In this work, using a volume integral equation, we show that the resonant frequencies at which a nanoparticle of characteristic size δ exhibits plasmonic resonances occurs can be written as a nonlinear eigenvalue problem:

Find ω such that f (ω) ∈ σ T (ωδ) (1.1)
for some nonlinear function f and some operator T (ωδ) (see Definition 2.2). These types of problems are extremely difficult to handle in their generality. In this work, we add some assumptions arising from experimental observations and classical electromagnetic theory to compute solutions of (1.1) in a regime that corresponds to practical situations.

The perturbative analysis presented in this work is based on the following assumptions:

(i) The size δ of the particle is small compared to the wavelenght of the lights at plasmonic frequencies:

δ ω c 1;
(ii) The particle is constituted of metal, whose permittivity can be described by a Drude-Lorentz type model [START_REF] Ordal | Optical properties of the metals al, co, cu, au, fe, pb, ni, pd, pt, ag, ti, and w in the infrared and far infrared[END_REF].

In this regime, we show that (Theorem 5.1):

∂ ∂ω σ(T (ωδ) ) ∼ δ c 1.
And using that we give the following procedure for solving (1.1):

• Find ω 0 such that f (ω 0 ) ∈ σ(T (0) );

• Compute σ(T (δω 0 ) ) by a perturbative method;

• Find ω 1 such that f (ω 1 ) ∈ σ(T (δω 0 ) ).
Since, in practical situations ∂ ∂ω f (ω) δ c (this comes from the fact that the particle is metallic and can be checked numerically, see Appendix Appendix A for more details), one can see that ω 1 is a good approximated solution of problem (1.1).

Additional contributions

In this paper, we also show that in the case where the particle has an elliptic shape, the dipole resonance of the nanoparticle (and its dependence on the size of the particle) can be very easily computed using the L dyadic that can be found in the physics literature [START_REF] Van Bladel | Singular electromagnetic fields and sources[END_REF][START_REF] Yaghjian | Electric dyadic green's functions in the source region[END_REF]. This dyadic L is often incorrectly derived in the literature. In Appendix Appendix B we give a correct derivation of L, as well as some precisions on some common misconceptions about singular integrals found in the classical literature on electromagnetic fields. We also give formulas for the computations of some observable quantities such as the extinction and absorption cross sections for elliptical particles (see Section 6). To the best of our knowledge, this is the first time that a formal proof is given for these type of computations. We consider the scattering problem of a time-harmonic wave incident on a plasmonic nanoparticle. Denote by ε 0 and µ 0 the electric permittivity and the magnetic permeability of the vacuum and by c 0 = (ε 0 µ 0 ) -1/2 the speed of light in the vacuum. The homogeneous medium is characterized by its relative electric permittivity ε m and relative magnetic permeability µ m , while the particle occupying a bounded and simply connected domain of center of mass z 0 :

Model and definition

Maxwell's equations

z 0 E i (x, ω) ε m ε c (ω) D = z 0 + δB E s (x, ω) = ? δ δω 1
D = z 0 + δB R 3
with C 1,α boundary is characterized by its electric permittivity ε c and its magnetic permeability µ c , both of which may depend on the frequency. We assume that ε c < 0, ε c > 0 and define

k m = ω c 0 √ ε m µ m , k c = ω c 0 √ ε c µ c , (2.1)
and

ε D (ω) = ε m χ(R 3 \ D) + ε c (ω)χ( D), µ D = µ m χ(R 3 \ D) + µ c χ(D), (2.2)
where χ denotes the characteristic function. We assume that the particle is nonmagnetic, i.e., µ m = µ c . Throughout this paper, we assume that ε m is real and strictly positive and that k c < 0 and k c > 0.

For a given plane wave solution (E i , H i ) to the Maxwell equations

∇ × E i = iωµ m H i in R 3 , ∇ × H i = -iωε m E i in R 3 ,
let (E, H) be the solution to the following Maxwell equations:

   ∇ × E = iωµ D H in R 3 \ ∂D, ∇ × H = -iωε D E in R 3 \ ∂D, [ν × E] = [ν × H] = 0 on ∂D, (2.3)
subject to the Silver-Müller radiation condition:

lim |x|→∞ |x|( √ µ m (H -H i ) × x - √ ε m (E -E i )) = 0,
where x = x/|x|. Here, [ν × E] and [ν × H] denote the jump of ν × E and ν × H along ∂D, namely,

[ν × E] = (ν × E) + -(ν × E) -, [ν × H] = (ν × H) + -(ν × H) -. Proposition 2.1. If I εc εm = 0 , then problem (2.
3) is well posed. Moreover, if we denote by (E, H) its unique solution, then (E, H) D ∈ H(curl, D) and (E, H)

R 3 \D ∈ H loc (curl, R 3 \ D).
Proof. The well-posedness is addressed in [START_REF] Torres | Maxwell's equations and dielectric obstacles with lipschitz boundaries[END_REF][START_REF] Costabel | The essential spectrum of the volume integral operator in electromagnetic scattering by a homogeneous body[END_REF][START_REF] Habib Ammari | Mathematical analysis of plasmonic nanoparticles: the scalar case[END_REF].

We also denote by G km the scalar outgoing Green function for the homogeneous medium, i.e., the unique solution in the sense of distributions of

∆ + ω 2 c 2 0 ε m µ m G km (•, z) = δ z in R 3 , (2.4)
subject to the Sommerfeld radiation condition. G km is given by (see [START_REF] Nédélec | Acoustic and electromagnetic equations: integral representations for harmonic problems[END_REF]):

G km (x, z) = e ikm|x-z| 4π|x -z| .
(2.5)

Volume integral equation for the electric field

We start by defining a singular integral operator, sometimes known as the magnetization integral operator [START_REF] Mark | Spectral properties for the magnetization integral operator[END_REF].

Definition 2.1. Introduce T k D : L 2 (D, R 2 ) -→ L 2 (D, R 2 ) f -→ k 2 D G k (x, y)f (y)dy -∇ D ∇G k (•, y) • f (y)dy.
We then give the equation satisfied by the electric field: Proposition 2.2. The electric field inside the particle satisfies the volume integral equation (or Lippmann-Schwinger equation):

ε m ε m -ε c I -T k D E = ε m ε m -ε c E i . (2.6)
Proof. See [START_REF] Colton | Inverse acoustic and electromagnetic scattering theory[END_REF]Chapter 9] or [START_REF] Costabel | Volume and surface integral equations for electromagnetic scattering by a dielectric body[END_REF].

Plasmonic resonances as an eigenvalue problem

Definition 2.2. We say there is a plasmonic resonance if

ε c ε m -ε c ∈ σ T k D .

Dipole resonance

Definition 2.3. The dipole moment of a particle is given by

P = D p(x)dx = D ε m χ(x)E(x)dx = D (ε c -ε m )E(x)dx.
We say that there is a dipolar plasmonic resonance if the dipole moment P satisfies

|P| (ε c -ε m ) D E i (x)dx .
Therefore, we want to compute the values of ε c and ε m such that (i)

λ := ε c ε m -ε c ∈ σ T k D ;
(ii) One of the eigenvectors ϕ λ associated with λ has non zero average:

1 |D| D ϕ λ = 0.

The quasi-static approximation

In this section, we study the case when the particle has finite size δ = 0 and δk m 1. This corresponds to the usual quasi-static approximation. It has already been shown in [START_REF] Habib Ammari | Surface plasmon resonance of nanoparticles and applications in imaging[END_REF][START_REF] Habib Ammari | Mathematical analysis of plasmonic nanoparticles: the scalar case[END_REF] that the solution of Maxwell's or Helmholtz equation converge uniformly when δk m → 0 to the solution of the quasi-static problem in the case of negative index materials. Proposition 3.1. In the quasi-static approximation, the excitation field E i becomes constant, the electric field can be written as the gradient of a potential E = ∇u and the scattering problem described by (2.6) becomes:

Find u such that ∇ • (ε D (x)∇u) = 0, u(x) -E i • x = O |x| -1 .
Equivalently, E = ∇u is a solution of the following integral equation:

ε m ε m -ε c I -T 0 D E = ε m ε m -ε c E i . (3.1)
Remark 3.1. These types of transmission / exterior problems have been extensively treated in the literature. For more details on the well posedness, the appropriate functional spaces, and the study of small conductivity inhomogeneities we refer to [START_REF] Nédélec | Acoustic and electromagnetic equations: integral representations for harmonic problems[END_REF][START_REF] Ammari | Reconstruction of small inhomogeneities from boundary measurements[END_REF].

Proposition 3.2. Let y = z 0 + δ ỹ and write ũ(ỹ) = u(y), and ũi (ỹ) = u i (y). Then ũ solves:

ε c ε m -ε c I -T 0 B ∇ũ = ε c ε m -ε c ∇ũ i .
Proof. This is a direct consequence of Theorem Appendix B.3.

Spectral analysis of the static operator, link with Neumann-Poincaré operator

It has been shown in [START_REF] Habib Ammari | Surface plasmon resonance of nanoparticles and applications in imaging[END_REF][START_REF] Habib Ammari | Mathematical analysis of plasmonic nanoparticles: the scalar case[END_REF] that the plasmonic resonances are linked to the eigenvalues of the Neumann-Poincaré operator. In this subsection, we show that the surface integral approach and the volume integral approach are coherent. The link between the volume integral operator and the Neumann Poincaré operator is summed up in Corollary 3.1. We first recall the definition of the Neumann-Poincaré operator

Definition 3.1. The operator K * D : L 2 (∂D) → L 2 (∂D) is defined by K * D [ϕ](x) := 1 ω d ∂D (x -y) • ν(x) |x -y| d ϕ(y)dσ(y) , (3.2)
with ν(x) being the outward normal at x ∈ ∂D, ω d the measure of the unit sphere in dimension d, and σ the Lebesgue measure on ∂D. We note that K * D maps L 2 0 (∂D) onto itself (see, for instance, [START_REF] Ammari | Reconstruction of small inhomogeneities from boundary measurements[END_REF]).

Recall the orthogonal decomposition

L 2 (D, R 3 ) = ∇H 1 0 (D) ⊕ H(div 0, D) ⊕ W,
where H(div 0, D) is the space of divergence free L 2 vector fields and W is the space of gradients of harmonic H 1 functions. We start with the following result from [START_REF] Costabel | The essential spectrum of the volume integral operator in electromagnetic scattering by a homogeneous body[END_REF]:

Proposition 3.3. The operator T 0 D is a bounded self-adjoint map on L 2 (D, R 2 ) with ∇H 1 0 (D), H(div 0, D) and W as invariant subspaces. On ∇H 1 0 (Ω), T 0 D [ϕ] = ϕ, on H(div 0, D), T 0 D [ϕ] = 0 and on W: ν • T 0 D [ϕ] = 1 2 + K * D [ϕ • ν] on ∂D.
Proof. The proof can be found in [START_REF] Mark | Spectral properties for the magnetization integral operator[END_REF][START_REF] Costabel | The essential spectrum of the volume integral operator in electromagnetic scattering by a homogeneous body[END_REF].

From this, it immediately follows that the following corollary holds.

Corollary 3.1. Let λ = 1. Let ϕ D ≡ 0 be such that λϕ D -T 0 D [ϕ D ] = 0 in D.
Then,

ϕ D ∈ W, ∇ • ϕ D = 0 in D, λϕ D = ∇S D [ϕ D • ν] in D, λϕ D • ν = 1 2 + K * D [ϕ D • ν] on ∂D. Letting u D = S D [ϕ D • ν],
we have:

∆u D = 0 in R 3 \ ∂D, [∂ n u D ] = ϕ D • ν on ∂D. (3.3) Proposition 3.4. If the boundary of D is C 1,α , then T 0 D W : W -→ W is a compact operator.
Proof. The operator T 0 D is a bounded map from W to H 1 (D, R 3 ) [START_REF] Mark | Spectral properties for the magnetization integral operator[END_REF][START_REF] Grigorevich | Multidimensional singular integrals and integral equations[END_REF]. The C 1,α regularity of ∂D and the usual Sobolev embedding theorems ensure its compactness (see [START_REF] Brezis | Functional analysis, Sobolev spaces and partial differential equations[END_REF]Chapter 9]).

Proposition 3.5. The set of eigenvalues (λ n ) n∈N of T 0 D W is discrete, and the associated eigenfunctions (ϕ n ) form a basis of W. We have:

T 0 D W = n λ n ϕ n , • ϕ n .
3.2. Link between σ(L B ) and σ(T 0 B ), and the dipole resonances for an ellipse As explained in Section 2.4, to understand the dipole resonances of the particle, we need to compute the eigenvectors of T 0 B that have a non zero average, i.e., that are not orthogonal in the L 2 (B) sense to every e i , i ∈ {1, 2, 3}, where (e 1 , e 2 , e 3 ) is an orthonormal basis of R 3 .

In general, constant vector fields over B are not eigenvectors of T B . Nevertheless in the case where B is an ellipse, then constant vector fields can be eigenvectors for T B . This is essentially a corollary of Newton's shell theorem. Theorem 3.1. If B is an ellipse centered at the origin, then the following holds: Let ϕ ∈ L 2 (B, R 3 ) and let λ ∈ R \ {0, 1} be such that

   λϕ -T 0 B [ϕ] = 0, B ϕ = 0.
Then,

(λI -L B ) B ϕ 0 = 0.
Remark 3.2. The operator we are considering is essentially the double derivative of a classical Newtonian potential. When the domain is an ellipse, the Newtonian potential of a constant is a second order polynomial. Therefore, its second derivative is a constant. Hence, the possibility to have constant eigenvectors for T B occurs. This property characterizes ellipses. In fact, it is the weak Eshelby conjecture; see [START_REF] Kang | Solutions to the pólya-szegö conjecture and the weak eshelby conjecture[END_REF] for more details.

Proof. For the proof we need the following lemma from [START_REF] Di | The newtonian potential and the demagnetizing factors of the general ellipsoid[END_REF]:

Lemma 3.1. If B is an ellipse, then, for any ϕ 0 ∈ R 3 , T B [ϕ 0 ] = L B ϕ 0 .
Combining this with Proposition 3.5, and using the orthogonality between the eigenvectors of T 0 B , one gets the result.

Corollary 3.2. Let E ∈ R 3 \ {0} be such that λE = L B E. Then, T 0 [E] = λE.
Proof. This is a direct consequence of Lemma 3.1.

Static polarizability of an ellipse

In this subsection, we assume that B is an ellipse.

Definition 3.2. The polarizability M is the matrix linking the average incident electrical field to the induced dipolar moment. It is defined by

p = M 1 |D| D E i .
Theorem 3.2. The static polarizability M of the particle B is given by

M = δ 3 ε 0 (ε c -1) ε m ε m -ε c + L B -1
.

(3.4)

Remark 3.3. The polarizability is used to compute different observables such as the scattering and extinction cross sections of the particle (see Section 6).

Proof. We recall equation (3.1) for the electric field inside the particle

ε m ε m -ε c I -T 0 D ∇u = ε m ε m -ε c ∇u i .
We now remark that the operator

P : L 2 (D, R 3 ) -→L 2 (D, R 3 ) f -→ D f
is the projector onto the subspace of L 2 -spanned by constant functions. Compose the previous equation with P we get:

P • ε c ε m -ε c I -T 0 D [∇u] = P[∇u i ].
We now use Proposition 3.5 to diagonalise T 0 :

P i ε c ε m -ε c + λ i ϕ n , ∇u ϕ n = P[∇u i ].
Since the particle is an ellipse, we know by Corollary 3.2 that E i ∈ R 3 , the eigenvectors of L B associated with the eigenvalues λ L,i , are also eigenvectors for T 0 for some eigenvalues λ i 1 , λ i 2 and λ i 3 . Moreover,

λ i 1 = λ L,1 , λ i 2 = λ L,2 and λ i 3 = λ L,3
We can also note that (E 1 , E 2 , E 3 ) span the image of the projector P. By orthogonality of the eigenvectors of T 0 , we obtain that

P i ε c ε m -ε c + λ i ϕ n , ∇u ϕ n = i ε m ε m -ε c + λ L,i E i , ∇u E i
Noticing that E i , ∇u = E i , P(∇u) = E i • P(∇u), we obtain that the right-hand side of the previous equation is exactly the expression of

ε c ε m -ε c + L B P(∇u)
in the basis (E 1 , E 2 ). Therefore, we have shown that

P • ε c ε m -ε c I -T 0 D [∇u] = ε c ε m -ε c + L B P(∇u). Thus D ∇u = ε c ε m -ε c + L B -1 D ∇u i .
Using Definition 2.3 of the induced dipole moment, we get the result.

Static polarizability of an arbitrary particle

In the case where the particle occupies an arbitrary C 1,α domain, the volume integral approach does not yield a simple expression for the polarizability. Nevertheless, the layer potential approach gives the well known polarization tensor. The validity of the polarization tensor formula for negative index material has been shown in [START_REF] Habib Ammari | Mathematical analysis of plasmonic nanoparticles: the scalar case[END_REF][START_REF] Habib Ammari | Surface plasmon resonance of nanoparticles and applications in imaging[END_REF].

We recall here the formula for completeness:

Theorem 3.3. [4, 2]
The static polarizability is given by

α = δ 3 ε 0 (ε -1) ∂B y ε + 1 2(ε -1) I -K * B -1
[ν](y)dσ(y),

where ε = εc εm .

Perturbative approach: spectral analysis of the dynamic operator

In this section, we aim at finding λ such that there exists some

f ≡ 0 ∈ L 2 (B, R 3 ) such that λI -T δk B [f ] = 0.
Let λ n 0 be an eigenvalue of T 0 . Let V ⊂ C be a neighborhood of λ n 0 such that λI -T 0 is invertible for every λ ∈ V . Let ϕ n 0 ∈ L 2 (B, R 3 ) be a unitary eigenvector associated with λ n 0 . Lemma 4.1. For any λ ∈ V , the following decomposition holds:

λI -T 0 -1 = ϕ n 0 , • λ -λ n 0 ϕ n 0 + R(λ),
where

C -→ L 2 (B, R 3 ) → L 2 (B, R 3 ) λ -→R(λ)
is holomorphic in λ.

Proof. Denote by

P 1 : L 2 (B, R 3 ) → L 2 (B, R 3 ) and P 2 : L 2 (B, R 2 ) → L 2 (B, R 3
) the orthogonal projections on ∇H 1 0 (B) and H(div 0, B), respectively. Using Propositions 3.3 and 3.5, we can write:

λI -T 0 = (λ -λ n ) ϕ n , • ϕ n + (λ -1)P 1 + λP 2 .
The result immediately follows.

Lemma 4.2. Let λ n 0 be an eigenvalue for T 0 . Then, if |k| is small enough, there exists a neighborhood V ⊂ C of λ n 0 such that T δk B has exactly one eigenvalue in V .

Proof. We start by writing:

λI -T δk = λI -T 0 + T 0 -T δk .
Recall that there exists V ⊂ C such that λI -T 0 is invertible for every λ ∈ V \{λ n 0 }. Therefore, for λ ∈ V \ {λ n 0 }, λI -T δk = λI -T 0 I + λI -T 0 -1 T 0 -T δk .

Using Lemma 4.1, we get

λI -T δk [f ] = f + ϕ n 0 , T 0 -T δk [f ] λ -λ n 0 ϕ n 0 + R(λ) T 0 -T δk [f ].
We can show that

T 0 -T δk -→ 0 (δk → 0). Since λ → R(λ) is holomorphic, the compact operator λ -→ R(λ) T 0 -T δk
converges uniformly to 0 with respect to λ when k goes to 0. Since the operator

K k : L 2 (B, R 3 ) -→ L 2 (B, R 3 ) f -→ ϕ n 0 , T 0 -T δk [f ] λ -λ n 0 ϕ n 0
is a rank one linear operator, the operator I + K δk B is invertible. Therefore, there exists K > 0 such that λI -T k = I + K δk + R(λ) T 0 -T δk is invertible for every λ ∈ V \ {λ n 0 } and every |δk| < K.

We can now give an asymptotic formula for the perturbed eigenvalues λ of T δk : Proposition 4.1. The following asymptotic formula for the perturbed eigenvalues holds:

λ ∼ λ n 0 -T 0 -T δk ϕ n 0 , ϕ n 0 L 2 (B,R 3 ) . ( 4

.1)

Proof. We use the same notations as in the previous lemmas. We have:

λ ∈ σ T δk ∪ V \ {λ n 0 } ⇔ ∃f ≡ 0 such that λI -T δk [f ] = 0 ⇔ ∃f ≡ 0 such that I + T 0 -T δk λI -T 0 -1 [f ] = 0.
Using the decomposition stated in Lemma 4.1 for λI -T 0 -1 , we get the following equation for f and λ:

f + ϕ n 0 , f λ -λ n 0 T 0 -T δk [ϕ n 0 ] + T 0 -T δk R( λ)[f ] = 0. (4.2)
We start by proving that ϕ n 0 , f = 0. Indeed, if one has ϕ n 0 , f = 0, then (4.2) becomes

I + T 0 -T δk R( λ) [f ] = 0.
If k is close enough to 0, then T 0 -T δk R( λ) < 1 and then I + T 0 -T δk R( λ) is invertible and we have f = 0, which is a contradiction. We then note that f and

ϕ n 0 ,f λ-λn 0 T 0 -T δk [ϕ n 0 ] are terms of order O(|f |) whereas the regular part, the term T 0 -T δk R( λ)[f ] is of order O(δk|f |).
We drop the regular part, and take the scalar product against ϕ n 0 to obtain that

ϕ n 0 , f + ϕ n 0 , f λ -λ n 0 T 0 -T δk [ϕ n 0 ], ϕ n 0 = 0. (4.3) Finally, dividing equation (4.3) by ϕ n 0 , f = 0 yields λ = λ n 0 -T 0 -T δk [ϕ n 0 ], ϕ n 0 .
We also prove the following lemma, giving an approximation of the resolvent of T k D along the direction of the eigenfunction ϕ n 0 :

Proposition 4.2. Let g ∈ L 2 (B, R 3 ). If f ∈ L 2 (B, R 3 ) is a solution of λI -T δk B f = g,
then, for λ ∼ λ n 0 , the following holds:

f , ϕ n 0 L 2 (B,R 3 ) ∼ g, ϕ n 0 L 2 (B,R 3 ) λ -λ n 0 + (T 0 -T δk ) [ϕ n 0 ], ϕ n 0 L 2 (B,R 3 ) .
Proof. The result follows directly from Lemma 4.1 and identity (4.2) with g in the right-hand side.

5. Dipolar resonance of a finite sized particle

Computation of the perturbation via change of variables

By the change of variables: y = z 0 + δ ỹ, Ẽ(ỹ) = E(y), Ẽi (ỹ) = E i (y), and (2.6) becomes:

ε c ε m -ε c I -T δk B Ẽ = ε c ε m -ε c E i .
We are now exactly in the right frame to apply the results of Section 4. We know that there is a neighborhood

V i ⊂ C of λ (0) i , i ∈ {1, 2, 3} such that T δkm B
has exactly one eigenvalue in V i (Lemma 4.2) and that the perturbed eigenvalue is given by Theorem 5.1. We have

λ ∼ λ (0) i -T 0 B -T δkm B ϕ i , ϕ i L 2 (B,R 3 ) , (5.1)
where ϕ i is a unitary eigenvector of T 0 B associated with λ (0) i .

The case of an ellipse 5.2.1. The perturbative matrix

In the case where B is an ellipse, since the eigenmodes associated with a dipole resonance are constant ϕ i ≡ E i ∈ R 3 (see Section 3.2) and therefore formula (5.1) simplifies to: Proposition 5.1. We have

T 0 B -T δkm B [ϕ i ], ϕ i = E i • M δkm B E i with M δkm B := ∂B×∂B G 0 (x, z) -G δkm (x, z) ν(x)ν(z) dσ(x)dσ(z).
Proof. From

T 0 B -T δkm B [ϕ i ], ϕ i = E i • B ∇ B ∇ G 0 (x, y) -G δkm (x, y) dydxE i ,
an integration by parts yields the result.

The algorithmic procedure

We now give a practical way to compute this perturbation in the case of an elliptical particle:

(i) Compute the resonant value associated with the static problem:

• Compute the matrix L B ∈ M 3 (R); • Compute its spectrum λ 1 , λ 2 ,
= λ i -E i • M δkm B E i .

Computation of observables for an elliptical nanoparticle

Dipole moment beyond the quasi-static approximation

Denote by λ j , j = 1, 2, 3, the three eigenvalues of L B . Denote by E j the three eigenvectors (∈ R 3 ) associated with λ j such that (E 1 , E 2 , E 3 ) forms an orthonormal basis of R 3 . Denote by 3) the matrix associated with this basis.

Q = E 1 , E 2 , E 3 ∈ O(
Since E j are eigenmodes for T 0 , we can use Lemma 4.2 to find that

E, E j ∼ E i , E j λ -λ j + (T 0 -T δk ) [E j ], E i .
We can then write:

P ∼ δ 3 ε 0 (ε -1)Q          1 λ -λ 1 + (T 0 -T δk ) [E 1 ], E 1 0 0 0 1 λ -λ 2 + (T 0 -T δk ) [E 2 ], E 2 0 0 0 1 λ -λ 3 + (T 0 -T δk ) [E 3 ], E 3 0          Q t 1 |D| D E i .
(6.1) Remark 6.1. This expression is valid near the resonant frequencies when the corresponding mode is excited, i.e., when λ ∼ λ i and ∇u i (z 0 )

• E i ∼ |∇u i (z 0 )|.
Remark 6.2. The expression

M dyn := δ 3 ε 0 (ε -1)Q     1 λ-λ1+ (T 0 -T δk )[E1],E1 0 0 0 1 λ-λ2+ (T 0 -T δk )[E2],E2 0 0 0 1 λ-λ3+ (T 0 -T δk )[E3],E3 0     Q t
is a dynamic version of the usual quasi-static polarization tensor.

Far-field expansion

Assume that the incident fields are plane waves given by

E i (x) = E i 0 e ikmd•x , H i (x) = d × E i 0 e ikmd•x , with d ∈ S 2 and E i 0 ∈ R 3 , such that E i 0 • d = 0.
Since we have an approximation of the dipole moment of the particle we can find an approximation of the electric field radiated far away from the particle. The far-field expansion written in [START_REF] Habib Ammari | Mathematical analysis of plasmonic resonances for nanoparticles: the full maxwell equations[END_REF] is still valid (Theorem (4.1) in the aforementioned paper), one just has to replace the dipole moment M (λ, D)E i where M is the usual polarization tensor defined with the Neumann Poincaré operator by the new corrected expression obtained in (6.1).

The scattered far field has the form

E(x) -E i (x) ∼ k 2 m 4π e ikm|x| |x| M dyn 1 |D| D E i (y)dy (|x| → ∞), (6.2)
and the scattering amplitude is given by

k 2 m 4π M dyn 1 |D| D E i (y)dy = k 2 m 4π P.

Scattering and absorption cross sections

Having an approximation of the dipole moment, we can compute the extinction and scattering cross sections of the particle. Proposition 6.1. ([12, Chapter 13]) The power radiated by an oscillating dipole P can be written

P r = µ m ω 4 12πc 0 |P| 2 ,
and the power removed from the incident plane wave (absorption and scattering) can be written

P e = 4π k m I E i 0 • k 2 m 4π P |E i 0 | 2 .
In the same spirit as in [START_REF] Habib Ammari | Mathematical analysis of plasmonic nanoparticles: the scalar case[END_REF][START_REF] Habib Ammari | Mathematical analysis of plasmonic resonances for nanoparticles: the full maxwell equations[END_REF] we can then give upper bounds for the cross sections as follows.

Proposition 6.2. Near plasmonic resonant frequencies, the leading-order term of the average over the orientation of the extinction (respectively absorption) cross section of a randomly oriented nanoparticle is bounded by

Q ext m ∼k m I [TrM dyn ] , Q a m ∼ k 4 m 6π |TrM dyn | 2 ,
where Tr denotes the trace.

Proof. We start from equation (6.1) and get that

P = M dyn E i 0 [1 + f (D, k m , d)] with f (D, k m , d) = 1 |D| D 1 -e ikmd•x dx.
Here, f represents the correction of the average illuminating field over the particle due to the finite ratio between the size of the particle and the wavelength. Its magnitude is of the order of δk. If we take the average over all directions for E i 0 and d, then we obtain that

Q ext m = 1 (4π) 2 S 2 4π k m k 2 m 4π I [e 0 • M dyn e 0 (1 + f (D, k m , d)] dσ(e 0 )dσ(d).
The term f (D, k m , d) is a small correction of the order of k m δ that is due to the fact that what determines the dipole moment is not the incident field at the center of the particle, but the Remark Appendix B.1. These conditions are not necessary, and singular integrals can be defined for a much larger class of functions. f can be replaced by f (x, θ) and u does not need to be chosen as Hölder continuous. One can choose u in some Lebesgue space u ∈ L p (D).

Example 1. Consider the Green function for the free space Laplace equation in two and three dimensions:

G(x, y) =      1 2π log |x -y| if d = 2, 1 4π 1 |x -y| if d = 3, then ∂ x i ,x j G(x, y) = f x-y |x-y| 1 |x-y| d with f x -y |x -y| =        1 2π δ ij -2 (x i -y i )(x j -y j ) |x -y| 2 if d = 2, - 1 4π δ ij -3 (x i -y i )(x j -y j ) |x -y| 2 if d = 3.
One can check that S d-1 f (θ)dθ = 0. Therefore, for u ∈ C 0,α (D) one can write:

D ∂ x i ,x j G(x, y)u(y)dy = lim ε→0 D\B(x,ε) ∂ x i ,x j G(x, y)u(y)dy.

Appendix B.2. Non spherical volume of exclusion

One common misconception found in the physics literature is that the limit of the integral over the domain minus a small volume around the singularity does not depend on the shape of the volume when the maximum cord of the volume of exclusion goes to zero. The limit does depend on the shape of the volume. This issue has been dealt with by Mikhlin [34, p. 40]. We include here the formula for the limit, using the notations used in physics literature. Assume that V (x, ε) ⊂ D is a small volume of exclusion such that its boundary is given, in polar coordinates by:

∂V (x, ε) = y ∈ D, |x -y| = εβ x -y |x -y| .
Theorem Appendix B.2. Under the assumptions of Theorem Appendix B.1,

lim ε→0 D\V (x,ε) f x -y |x -y| 1 |x -y| d u(y)dy = D f x -y |x -y| 1 |x -y| d u(y)dy -u(x) S d-1 f (θ) log β(θ)dθ. Example 2. Let d = 2 and let f x -y |x -y| = 1 2π 1 -2 (x 1 -y 1 ) 2 |x -y| 2
be corresponding to the angular term of ∂ 1,1 G(x, y). If V (x, ε) is an ellipse of semi-axis ε and eccentricity e where x is at one of the focal point

∂V (x, ε) = y ∈ D, |y -x| = ε (1 -e 2 ) 1 -e x 1 -y 1 |x-y| , then the correction term is u(x) 2π θ∈S 1 1 -2θ 2 1 log 1 -e 2 1 -eθ 1 dθ = u(x) 2π 2π 0 (1 -2 cos 2 (t)) log 1 -e 2 1 -e cos(t) dt.
Remark Appendix B.2. Note that the correction term does not only depend on the shape of the volume of exclusion, but also on the position of x inside it. In the previous example, if x is at the center of the ellipse instead of being one of the focal point, the polar equation, hence the correction term, is modified.

Appendix B.3. Change of variables

This issue has also been dealt with by Seeley [START_REF] Robert | Singular integrals on compact manifolds[END_REF] and Mikhlin [34, p. 41]. The classical formula for a change of variables in an integral cannot be applied in a straightforward way, and some precautions have to be taken into account. Consider a region D and an homeomorphism

ψ : D -→ D. Consider f = f • ψ -1 , ũ = u • ψ -1 ,
and denote by J(x) the non vanishing Jacobian of ψ -1 . The corrective term to the usual change of variables formula is given by the reciprocal image of the unit sphere by ψ. One can establish formulae of the form:

|x -y| 2 = |ψ(x) -ψ(y)| 2 F ψ(x), ψ(x) -ψ(y) |ψ(x) -ψ(y)| + O |ψ(x) -ψ(y)| 3 ,
and then the change of variables can be written as follows:

Theorem Appendix B.3. We have D f x -y |x -y| 1 |x -y| d u(y)dy = D f x - ỹ |x -ỹ| 1 |x -ỹ| d ũ(ỹ)J(ỹ)dỹ + ũ(x)J(x) S d-1 f ( θ) log F (x, θ)d θ.
Remark Appendix B.3. For a dilation : x = x-z 0 δ , the image of the unit sphere is still a sphere and therefore, F = 1 and the usual change of variables formula is valid.

Appendix B.4. Differentiation of weakly singular integrals, integration by parts

We want to differentiate integrals of the type

D g x -y |x -y| 1 |x -y| d-1 u(y)dy, x ∈ D.
The following results can be found in [START_REF] Robert | Singular integrals on compact manifolds[END_REF][START_REF] Grigorevich | Multidimensional singular integrals and integral equations[END_REF]:

Theorem Appendix B.4.
If u is Hölder continuous and if g and its first derivative are bounded, then:

(i) Differentiation formula under the integral sign:

∂ ∂x i D g x -y |x -y| 1 |x -y| d-1 u(y)dy = D ∂ ∂x i g x -y |x -y| 1 |x -y| d-1 u(y)dy + u(x) S d-1 g(θ)θ i dθ;
(ii) Integration by parts formula:

D g x -y |x -y| 1 |x -y| d-1 ∂ ∂x i [f (y)] dy = - D ∂ ∂x i g x -y |x -y| 1 |x -y| d-1 f (y)dy + ∂D g x -y |x -y| f (y) |x -y| d-1 ν(y) • e i dσ(y) + f (x) S d-1 g(θ)θ i dθ.
Remark Appendix B.4. Once again we only give sufficient conditions for the validity of theses formulas, corresponding to our framework. These formulas are valid for u ∈ L p and for more general kernels. x j -y j |x -y| 3 u(y)dy.

We can apply Theorem Appendix B.4 with g(θ) = θ j and get: Before the computation we also recall that ∂ x i ,x j G(x, y) can be written as Remark Appendix B.5. There are several issues and misconceptions in the literature with this formula:

∂ x i ,x j
∂ x i ,
(i) The shape V * cannot be completely arbitrary as often mentioned. It has to satisfy some regularity condition, since the construction of L V * uses some integration on the boundary of V * involving the normal vector. (ii) The exclusion volume V (x, ε) needs to be taken small in the numerical evaluation of the integral. Only if the test function u is constant then ε does not need to be small. (iii) The derivation of this formula often contains mistakes. One common derivation of this formula is through a splitting of the integral of the form ∂ x i D ∂ x j G(x, y)u(y)dy = D\V (x,ε) ∂ x i ,x j G(x, y)u(y)dy + V (x,ε) ∂ x i ,x j G(x, y) [u(y) -u(x)] dy + u(x)∂ x i V (x,ε) ∂ x j G(x, y)dy , which is a wrong application of the differentiation under the sign theorem. The reason why it is wrong is that, if the limit when ε → 0 is to be taken, then one has to take into account the dependency of the volume of integration on D \ V (x, ε) on the variable x and use Reynold's transport theorem to compute the derivative and add some boundary integral terms. The correct splitting would be: ∂ x i ,x j G(x, y)dy .

∂ x i D ∂ x j G(x,
In the limit ε → 0, the extra terms compensate each other and the first (wrong) splitting gives the same (correct) result as the second one.

Figure 1 :

 1 Figure 1: Schematic representation of the scattering problem.

λ 3

 3 and corresponding unitary eigenvectors E 1 , E 2 and E 3 ; (ii) Compute the perturbative matrix M δkm B and the perturbed eigenvalues λi

Figure A. 2 :

 2 Figure A.2: Numerical values of the real part of f (ω) = λ(ω) = εc(ω)εm-εc(ω) for a gold nanoparticle in water.

Figure A. 3 :

 3 Figure A.3: Numerical values of the real part of df dω (ω) for a gold nanoparticle in water.

Example 3 .

 3 Let d = 3 and consider the second derivative of a Newtonian potential:∂ x i ,x j D 1 4π |x -y| -1 u(y)dy = -∂ x i D 1 4π

D 1 4π 3 .

 13 |x -y| -1 u(y)dy = D 1 4π ∂ x i ,x j |x -y| -1 u(y)dy + u(x) δ ijProof. Let x ∈ D and let V (x, ε) ⊂ D. Assume that V (x, ε) can be described by some polar equation:V (x, ε) = y ∈ D, |x -y| ≤ ερ x -y |x -y| . (B.2)

∂S d- 1 f+ θ∈S d- 1 f

 11 x j G(x, y) = f ij x -y |x -y| |x -y| -d ,as it was seen in Example 1. Then we have∂ x i ,x j D G(x, y)u(y)dy =∂ x i D ∂ x j G(x, y)u(y)dy (x i ,x j G(x, y)u(y)dy. (B.4)Using Theorem Appendix B.2 we obtain:D ∂ x i ,x j G(x, y)u(y)dy = lim ε→0 D\V (x,ε) ∂ x i ,x j G(x, y)u(y)dy -u(x) ij (θ) log ρ(θ)dθ,and therefore,∂ x i D ∂ x j G(x, y)u(y)dy = lim ε→0 D\V (x,ε) ∂ x i ,x j G(x, y)u(y)dy -u(x) -1 d i,j (θ) log ρ(θ)dθ .Finally, using Lemma Appendix B.1 we arrive at∂ x i D ∂ x j G(x,y)u(y)dy = lim ε→0 D\V (x,ε) ∂ x i ,x j G(x, y)u(y)dy -(L V * ) ij u(x).

  y)u(y)dy = D\V (x,ε)∂ x i ,x j G(x, y)u(y)dy-∂V (x,ε) ∂ x j G(x, y)u(y)ν(y)•e i dσ(y) + V (x,ε) ∂ x i ,x j G(x, y) [u(y) -u(x)] dy + ∂V (x,ε) ∂ x j G(x, y) [u(y) -u(x)] ν(y) • e i dσ(y) +∂ x i u(x) V (x,ε) ∂ x j G(x, y)dy+u(x) ∂V (x,ε) ∂ x j G(x, y)ν(y) • e i dσ(y) + 1 d + V (x,ε)

average of the field over the particle. Therefore, it reduces the dipole response of the particle. Ignoring it and considering only the leading order term gives: A similar computation gives the leading term of the absorption cross section.

Appendix A. Justification of the asymptotic regime

To quickly justify the model and the regime we are working in, we give some values for the physical parameters used in the model corresponding to practical situations.

In practice:

Therefore, one can see that we have δk ≤ 10 -2 for very small particles, and δk ∼ 1 for bigger particles (of size 100nm).

For the permittivity of the metal, one can use a Lorentz-Drude type model:

with

• τ = 10 -14 s;

This model is enough to understand the behavior of ε but for numerical computations, it is better to use the tabulated parameters that can be found in [START_REF] Aleksandar | Optical properties of metallic films for vertical-cavity optoelectronic devices[END_REF]. We plot f (ω) on Figure 2 and df dω on Figure 3. One can see that df dω is of order 10 -15 while δ c ∼ 10 -16 for size particle under 100nm. So the procedure described in Section 1.2 is justified.

Appendix B. Singular integrals, Calderón Zygmund type operators

There is an abundant literature on singular integral operators, yet these types of principal value integrals are misunderstood and misused in some of the physics literature. We include here some properties that are well known for people who are familiar with these types of operators, but seem to be often misunderstood.

There have been numerous contributions in the twentieth century. Some notable ones are: [START_REF] Tricomi | Equazioni integrali contenenti il valor principale di un integrale doppio[END_REF] [START_REF] Tricomi | Equazioni integrali contenenti il valor principale di un integrale doppio[END_REF]; Kellogg (1929) [START_REF] Oliver Dimon | Foundations of potential theory[END_REF]; Calderón-Zygmund (1952) [START_REF] Alberto | On the existence of certain singular integrals[END_REF]; Seeley (1959) [START_REF] Robert | Singular integrals on compact manifolds[END_REF]; Gel'fand-Shilov (1964) [21], and Mikhlin (1965) [START_REF] Grigorevich | Multidimensional singular integrals and integral equations[END_REF].

In the following, we do not state the results in their most general settings and assumptions. We use some notations and hypotheses that are adapted to our problem (Green's function method). Assume that D can be written in polar coordinates as

Then,

with f i,j being defined in Example 1.

Proof. We start by using the integration by part formula from Theorem Appendix B.4, with g(θ) = 1 2π θ j if d = 2 and g(θ) = 1 4π θ j if d = 3, and f = 1. We obtain

In order to compute D ∂ x i ,x j G(x, y)dy, we use the change of variables [START_REF] Robert | Singular integrals on compact manifolds[END_REF][START_REF] Grigorevich | Multidimensional singular integrals and integral equations[END_REF] :

f i,j (θ) log ρ(θ)dθ.

Appendix B.6. Second derivative of a Newtonian potential

In this section we give a correct simple derivation of the formula found in [47, p. 73] and [START_REF] Yaghjian | Electric dyadic green's functions in the source region[END_REF][START_REF] Lee | Singularity in green's function and its numerical evaluation[END_REF][START_REF] Asvestas | Comments on" singularity in green's function and its numerical evaluation[END_REF].

Proposition Appendix


Laëtitia Chausset-Boissarie

Nicolas Cheval
email: nicolas.cheval@gmail.fr

Christian Rolando
email: christian.rolando@univ-lille.fr

Vito Capriati

Yurii Yagupolskii

Palladium-Catalyzed Cross-Coupling of Gem-Bromofluoroalkenes with Alkylboronic Acids for the Synthesis of Alkylated Monofluoroalkenes

Keywords: monofluoroalkenes, gem-bromofluoroalkenes, alkylboronic acids, Suzuki-Miyaura-cross-coupling

Palladium-Catalyzed Cross-Coupling of Gem-Bromofluoroalkenes with Alkylboronic Acids for the Synthesis of Alkylated Monofluoroalkenes

Laetitia Chausset-Boissarie, Nicolas Cheval, Christian

Introduction

The incorporation of fluorine atoms into bioactive molecules hugely impacts their physicochemical and pharmacokinetic properties, prevents oxidative metabolism and, more important, modulates their overall biological activities [1,2]. Accordingly, fluorinated compounds are abundant scaffolds found in a large variety of materials, agrochemicals and pharmaceuticals [3][4][5][6][7][8]. In particular, monofluoroalkenes are highly valuable fluorinated synthons in organic synthesis, in high-performance materials and in medicinal chemistry as they are excellent peptide bond mimics with enhanced stability towards proteases and stable conformation, improving the molecule stability and lipophilicity [9,[START_REF] Osada | Fluoroalkene modification of mercaptoacetamide-based histone deacetylase inhibitors[END_REF]. Despite the importance of alkylated monofluoroalkenes, limited methodologies have been developed for their modular synthesis. Pioneering studies to obtain alkyl-substituted monofluoroalkenes were focused on classical olefination (Wittig, Horner-Wadsworth-Emmons or Julia Kocienski reaction) [START_REF] Zajc | Synthesis of Fluoroolefins via Julia-Kocienski Olefination[END_REF]12], electrophilic fluorination or fluorination of alkynes [13][START_REF] Gorske | Regio-and Stereoselective Synthesis of Fluoroalkenes by Directed Au(I) Catalysis[END_REF][START_REF] Liu | Divergent reactivities in fluoronation of allylic alcohols: Synthesis of Z-fluoroalkenes via carbon-carbon bond cleavage[END_REF][START_REF] Gauthier | Gold-Catalyzed Hydrofluorination of Internal Alkynes Using Aqueous HF[END_REF]. More recently, transition metal-catalyzed defluorinative alkylation of gem-difluoroalkenes [START_REF] Dai | Sterically Controlled Cu-Catalyzed or Transition-Metal-Free Cross-Coupling of gem-Difluoroalkenes with Tertiary, Secondary, and Primary Alkyl Grignard Reagents[END_REF][START_REF] Lu | Nickel-Catalyzed Defluorinative Reductive Cross-Coupling of gem-Difluoroalkenes with Unactivated Secondary and Tertiary Alkyl Halides[END_REF][START_REF] Yang | Synthesis of Alkylated Monofluoroalkenes via Fe-Catalyzed Defluorinative Cross-Coupling of Donor Alkenes with gem-Difluoroalkenes[END_REF][START_REF] Yu | Zinc-Mediated Decarboxylative Alkylation of Gem-difluoroalkenes[END_REF][START_REF] Zhou | Ni-catalyzed migratory fluoro-alkenylation of unactivated alkyl bromides with gem-difluoroalkenes[END_REF] or gem-difluorocyclopropanes [START_REF] Xu | ChemInform Abstract: Pd-Catalyzed Regioselective Activation of gem-Difluorinated Cyclopropanes: A Highly Efficient Approach to 2-Fluorinated Allylic Scaffolds[END_REF][START_REF] Wenz | ChemInform Abstract: Catalytic C-F Bond Activation of Geminal Difluorocyclopropanes by Nickel(I) Complexes via a Radical Mechanism[END_REF][START_REF] Ahmed | Palladium-Catalyzed Stereoselective Defluorination Arylation/Alkenylation/Alkylation of gem-Difluorinated Cyclopropanes[END_REF] with various carbon nucleophiles has proven to be efficient strategies to access alkylated monofluoroalkenes. In the meantime, photoredox monofluoroalkenylation of gem-difluoroalkenes has also been successfully applied for their syntheses [START_REF] Xie | Monofluoroalkenylation of Dimethylamino Compounds through Radical-Radical Cross-Coupling[END_REF][START_REF] Li | Visible light photocatalytic decarboxylative monofluoroalkenylation of α-amino acids with gem-difluoroalkenes[END_REF][START_REF] Yang | Organic photoredox catalytic decarboxylative cross-coupling of gem-difluoroalkenes with unactivated carboxylic acids[END_REF][START_REF] Du | Synthesis of Monofluoroalkenes through Visible-Light-Promoted Defluorinative Alkylation of gem-Difluoroalkenes with 4-Alkyl-1,4-dihydropyridines[END_REF]. Despite these remarkable achievements, defluorinative cross-coupling towards the C(sp 3 )-C(sp 2 ) bond formation is still limited by the use of expensive catalytic systems, moderate Z/E selectivity, air-sensitive reagents or specific alkyl sources bearing a heteroatom at the α-position. Gem-bromofluoroalkenes, which are easily accessible, starting materials from aldehyde or ketones via a Wittig-Burton reaction, can also be efficient substrates for the selective formation of alkyl-substituted monofluoroalkenes [START_REF] Chelucci | ChemInform Abstract: Synthesis and Metal-Catalyzed Reactions of gem-Dihalovinyl Systems[END_REF][START_REF] Landelle | Synthetic approaches to monofluoroalkenes[END_REF]. In this regard, Pannecoucke's group reported the selective synthesis of stereo-defined butylated Z-(fluoro)alkene by Pd-catalyzed cross-coupling of (E/Z)-gem-bromofluoroalkenes with an in situ-generated organozinc intermediate [START_REF] Lei | A Facile and Mild Method for the Synthesis of Terminal Bromofluoroolefins via Diethylzinc-Promoted Wittig Reaction[END_REF]. Following up, the group of Wnuk reported an elegant pallado-catalyzed Negishi cross-coupling of gem-bromofluoroalkenes with alkyl organozinc derivatives as coupling partners to selectively produce (Z)-monofluoroalkenes [START_REF] Andrei | Synthesis of the Multisubstituted Halogenated Olefins via Cross-Coupling of Dihaloalkenes with Alkylzinc Bromides[END_REF]. Nevertheless, one of the drawbacks of these pathways is a low functional group tolerance and the use of sensitive reagents. Therefore, despite great successes achieved, the development of mild and practical methodologies to monofluoroalkenes, especially 2-fluoroalkyl scaffolds, remains an appealing task. Continuing our research directed towards the development of new methodology for the synthesis of functionalized monofluoroalkenes [START_REF] Zemmouri | Palladium-Catalyzed Stereoconvergent Formylation of (E/Z)-β-Bromo-β-fluorostyrenes: Straightforward Access to (Z)-α-Fluorocinnamic Aldehydes and (Z)-β-Fluorocinnamic Alcohols[END_REF][START_REF] Eddarir | Stereoselective cyanation of β-bromo-β-fluorostyrenes using potassium cyanide and promoted by (CH3CN)4Cu+ BF4[END_REF][START_REF] Kajjout | An efficient access to (Z)-β-fluoroallyl alcohols based on the two carbon homologation of aromatic aldehydes by Horner-Wadsworth-Emmons reaction with 2-(diethoxyphosphinyl)-2-fluoro-ethanethioic acid, S-ethyl ester followed by reduction with sodium borohydride[END_REF][START_REF] Kajjout | A new approach to the synthesis of (Z)-2-fluoro-2-alkenals via Wittig-type carbonyl condensation reactions of 2-(fluoromethyl)-4,4,6-trimethyl-1,3-oxazine phosphonium bromide[END_REF] Herein, we report the first example of a stereospecific Suzuki-Miyaura-cross-coupling reaction with readily available alkyl boronic acids that is adaptable across a range of gem-bromofluoroalkenes providing a large array of alkylated monofluoroalkenes with retention of configuration and in good yields under mild conditions.

Results and Discussion

At the outset of the study, coupling reactions were investigated with the easily accessible (E/Z)-1-(2-bromo-2-fluorovinyl)-4-nitrobenzene 1a [START_REF] Lei | A Facile and Mild Method for the Synthesis of Terminal Bromofluoroolefins via Diethylzinc-Promoted Wittig Reaction[END_REF] and butylboronic acid 2a.

To establish the best reaction conditions, a broad range of palladium catalyst precursors, bases, solvents, temperatures and phosphine ligands were evaluated (Table 1). An initial survey demonstrated that the use of PdCl2dppf as catalyst gave the desired product as a mixture of E/Z isomers in 95% yield (entries 1-3). Among the bases, Cs2CO3 proved to be the most effective (entries 3-7). Subsequently, the solvents were screened, and the original biphasic mixture of toluene/H2O (9:1) was the best of choice (entries 3, 8-10). Further examination revealed that a decrease in the reaction temperature reduces the reaction efficiency (entries [START_REF] Zajc | Synthesis of Fluoroolefins via Julia-Kocienski Olefination[END_REF][12]. Common ligands of palladium were tested (entries [13][START_REF] Gorske | Regio-and Stereoselective Synthesis of Fluoroalkenes by Directed Au(I) Catalysis[END_REF][START_REF] Liu | Divergent reactivities in fluoronation of allylic alcohols: Synthesis of Z-fluoroalkenes via carbon-carbon bond cleavage[END_REF][START_REF] Gauthier | Gold-Catalyzed Hydrofluorination of Internal Alkynes Using Aqueous HF[END_REF][START_REF] Dai | Sterically Controlled Cu-Catalyzed or Transition-Metal-Free Cross-Coupling of gem-Difluoroalkenes with Tertiary, Secondary, and Primary Alkyl Grignard Reagents[END_REF], and bidentate bisphosphines and, above all, those with large P-Pd-P bite angles appeared to be essentials [START_REF] Tan | Widely Applicable Pd-Catalyzed trans-Selective Monoalkylation of Unactivated 1,1-Dichloro-1-alkenes and Pd-Catalyzed Second Substitution for the Selective Synthesis of (E)-or (Z)-Trisubstituted Alkenes[END_REF]. Under some conditions, (E)-isomer reacts faster than the corresponding (Z)-isomers in Pd-catalyzed coupling reactions (entries 3,5,[START_REF] Gauthier | Gold-Catalyzed Hydrofluorination of Internal Alkynes Using Aqueous HF[END_REF]. The best catalytic system was found to be Pd2(dba)3 (2 mol%) with xantphos (2 mol%) as the catalyst and Cs2CO3 as the base in a mixture of toluene/H2O (9:1) at 80 °C under nitrogen affording the desired product in almost quantitative yield (entry 15).

With the optimized conditions in hand, we investigated the substrate scope of the cross-coupling reaction on the gem-bromofluoroalkene part (Scheme 1). A large range of (E/Z)-gem-bromofluoroalkenes was successfully cross-coupled to afford the related Z and E monofluoroalkenes in good to excellent isolated yield. The electronic effects of the substituents on the aromatics rings showed no obvious influence on this transformation since (E/Z)-gem-bromofluoroalkenes possessing electron neutral (3ba-ca), electron-donating (3da) and electron-withdrawing groups (3ea-ga) provided the (E/Z)-monofluoroalkenes in high yields. Several sensitive or valuable functional groups, notably for further post-functionalizations, such as esters, trifluoromethyl and nitro groups, were well tolerated throughout the coupling reactions. In all cases, no sterodifferentiation was observed since a mixture of the corresponding E/Z isomers was obtained with the same isomeric composition of the starting material. The cross-coupling reaction of isomerically pure (Z)-1-bromo-1-fluoroalkene 1a led stereospecifically to a corresponding (E)-monofluoroalkene 3aa with complete retention of the stereochemistry confirming the stereospecificity of the reaction. Interestingly, gem-bromofluoroalkene that are meta-substituted (3ha) or sterically hindered at the ortho position (3ia) were suitable coupling partners for the reaction albeit, aryl gem-bromofluoroalkenes bearing substituents in the para position showed better reactivity. Gratifyingly, in the case of symmetric and unsymmetric gem-bromofluoroalkenes derived from ketones, the corresponding tetrasubstituted monofluoroalkenes 3ja and 3ka are obtained in excellent isolated yield. Unfortunately, the reaction is not compatible with nitrogen or sulfur hetaryl gem-bromofluoroalkenes (3la-ma) mainly due to the degradation of the starting material. In addition, when alkylated gem-bromofluoroolefin 1n was used as the substrate, the reaction also failed to give any coupling product. a All reactions unless specified were carried out using 1a (1 eq), 2a (1.2 eq), Pd source (2 mol%), ligand (2 mol%) and base (3 eq), in solvent ([0.09 M]) under N2 for 6 h. b Ratio determined by 1 H NMR. c Yield was determined using 1,3,5-trimethoxybenzene as an internal standard. Pd2dba3 = tris(dibenzylideneacetone)dipalladium. Dppf = 1,1'-bis(diphenylphosphino)ferrocene. Dppe = 1,2-bis(diphenylphosphino)ethane. Xantphos = 4,5-bis(diphenylphosphino)-9,9-dimethylxanthene. Xphos = 2-dicyclohexylphosphino-2′,4′,6′-triisopropylbiphenyl. TFP = tri(2-furyl)phosphine.

We then examined the coupling reactions with different primary and secondary alkyl boronic acids and (E/Z)-1-(2-bromo-2-fluorovinyl)-4-nitrobenzene 1a using the same set of reaction conditions developed (Scheme 2). All of the primary aliphatic alkyl boronic acids 2a-e provided the desired product 3aa-ae in good to excellent isolated yields. In the case of secondary alkyl substituents such as isopropyl or cyclohexyl boronic acids 2f-g, no reaction occurred; only starting materials were recovered. Cyclopropyl boronic acid 2 h was shown to undergo a cross-coupling reaction giving the product in 83% yield. This could be due to the geometry of the substrate, which suppresses β-hydride elimination. Scheme 1. Palladium-catalyzed cross-coupling of butylboronic acid 2a with various gem-bromofluoroalkenes 1a-n. Scheme 2. Palladium-catalyzed cross-coupling of (E/Z)-1-(2-bromo-2-fluorovinyl)-4-nitrobenzene 1a with various alkyl boronic acids 2a-i.

Materials and Methods

General Methods

All reagents were purchased from commercial suppliers and were used without further purification unless otherwise indicated. Thin-layer chromatography (TLC) was performed on silica gel 60 F254 plates (Merck, Pfizer, Sanofi) and visualized under UV (254 nm) or by staining with potassium permanganate or phosphomolybdic acid. The purification of the obtained products was performed by flash chromatography using silica gel (230-400 mesh, 0.040-0.063 mm).

NMR spectra were recorded on a Bruker AVANCE 300 spectrometer (Bruker Corporation, Billerica, MA, USA) at 300 MHz (75 MHz). Chemical shifts are given in parts per million relative to the solvent signal. Multiplicities of the signals are reported using the standard abbreviations: singlet (s), doublet (d), triplet (t), quartet (q) and multiplet (m). Coupling constants are reported in hertz (Hz). Coupling constants (J) are reported in hertz (Hz). High-resolution mass spectra (HRMS) were performed on a ThermoFisher Scientific LTQ Orbitrap XL mass spectrometer (Thermo Fischer Scientific, Bremen, Germany) using electrospray ionization (ESI).

Synthesis of Gem-Bromofluoroalkenes 1

Gem-bromofluoroalkenes 1a-n were synthesized according to known procedures reported by Pannecoucke's group 31 from the appropriate aldehyde and tribromofluoromethane.

General Procedure for the Synthesis of Monofluoroalkenes 3

In a Schlenk tube, gem-bromofluoroalkene 1 (1.0 equiv), boronic acid 2 (1.2 equiv), Pd2dba3.CHCl3 (2 mol%), xantphos (2 mol%) and Cs2CO3 (3 equiv) were added. The vial was flushed under nitrogen, then filled with a mixture of toluene/H2O (9:1) (0.09 M). The reaction mixture was heated for 6 h at 80 °C then cooled to r.t., filtered through Celite ® and washed with EtOAc. The filtrate was concentrated under vacuum, and the residue was purified by flash chromatography on silica gel (petroleum ether/EtOAc = 100:0 to 95:5) to afford the pure product 3. 1 H-, 13 C-and 19 F-NMR spectra of products can be found in Supplementary Materials. (0.23 mmol, 56.32 mg), butyl boronic acid 2a (0.28 mmol, 28.45 mg), Pd2dba3.CHCl3 (4.710 -3 mmol, 4.25 mg), xantphos (4.710 -3 mmol, 2.69 mg), Cs2CO3 (0.70 mmol, 227.10 mg), toluene/H2O (2.53 mL) were reacted according to general procedure. The crude product was purified by silica gel column chromatography (petroleum ether/EtOAc = 100:0 to 95:5) affording compound 3aa in 99% yield (50.80 mg) as a yellow solid. 1 2), 140.5 (d, 3 J E C-F = 14.8 Hz), 139.6 (d, 3 J Z C-F = 2.7 Hz), 127.9 (2), 127.7, 127.6, 122.8 (2), 122.7 (2), 106.1 (d, 2 J Z C-F = 31.0 Hz), 103.5 (d, 2 J E C-F = 8.2 Hz), 31.9 (d, 2 J Z C-F = 25.6 Hz), 28.0 (d, 2 J E C-F = 26.6 Hz), 27.3 (2), 21.2, 21.0, 12.7 (2). 19 F NMR (282.5 MHz, CDCl3): δ -99.23 (q, J E = 23.2 Hz), -101.65 (dt, J Z = 39. 7,18.1 Hz). HRMS (ESI): m/z [M + H] + calc. for C12H15FNO2, 224.1081 found 224.1080. (0.23 mmol, 56.32 mg), butyl boronic acid 2a (0.28 mmol, 28.45 mg), Pd2dba3.CHCl3 (4.710 -3 mmol, 4.25 mg), xantphos (4.710 -3 mmol, 2.69 mg), Cs2CO3 (0.70 mmol, 227.10 mg), toluene/H2O (2.53 mL) were reacted according to general procedure. The crude product was purified by silica gel column chromatography (petroleum ether/EtOAc = 100:0 to 95:5) affording compound E-3aa in 99% yield (50.80 mg) as a yellow solid. 1 H NMR (300 MHz, CDCl3): δ 8.09 (t, J = 9.1 Hz, 2.0H), 7.51 (d, J = 8.9 Hz, 2.0 H), 6.15 (d, J E H-F = 21.1 Hz, 1.0H), 2.47-2.23 (m, 2.0H), 1.59-1.50 (m, 2.0H), 1.37-1.27 (m, 2.0H), 0.90-0.81 (m, 3.0H). 13 C NMR (75 MHz, CDCl3): δ 164.1 (d, 1 J E C-F = 257. 5 Hz), 145.3, 140.5 (d, 3 J E C-F = 14.8 Hz), 127.9, 127.7, 122.8, 122.7, 103.5 (d, 2 J E C-F = 8.2 Hz), 28.0 (d, 2 J E C-F = 26.6 Hz), 27.3, 21.2, 12.7. 19 F NMR (282.5 MHz, CDCl3): δ -99.23 (q, J E = 23. 2 Hz). HRMS (ESI): m/z [M + H] + calc. for C12H15FNO2, 224.1081 found 224.1082. (0.23 mmol, 46.39 mg), butyl boronic acid 2a (0.28 mmol, 28.45 mg), Pd2dba3.CHCl3 (4.710 -3 mmol, 4.25 mg), xantphos (4.710 -3 mmol, 2.69 mg), Cs2CO3 (0.70 mmol, 227.10 mg), toluene/H2O (2.53 mL) were reacted according to general procedure. The crude product was purified by silica gel column chromatography (petroleum ether/EtOAc = 100:0 to 95:5) affording compound 3ba in 95% yield (39.25 mg) as a yellow solid. 1 H NMR (300 MHz, CDCl3): δ 7.43-7.34 (m, 1.00H), 7.28-7.20 (m, 2.00H), 7.16-7.05 (m, 2.00H), 6.10 (d, J E H-F = 22.0 Hz, 0.53H), 5.38 (d, J Z H-F = 39.5 Hz, 0.47H), 2.43-2.19 (m, 2.00H), 1.57-1.46 (m, 2.00H), 1.31 (ddd,J = 15.1,9.5,7.4 Hz,2.00H), 0.87 (t, J = 6.3 Hz, 1.40H), 0.82 (t, J = 6.4 Hz, 1.60H). 13 C NMR (75 MHz, CDCl3): δ 162.8 (d, 5 Hz),128.6,128.6 (3),128.5 (2),128.4,128.3,126.7 (2), 108.1 (d, 2 J Z C-F = 28.6 Hz), 105.7 (d, 2 J E C-F = 8.8 Hz), 32.8 (d, 2 J Z C-F = 26.3 Hz), 28.7 (d, 2 J E C-F = 29.2 Hz), 28.5 (2), 22.3, 22.1, 13.8 (2). 19 F NMR (282 MHz, CDCl3): δ -98.35 (dd, J E = 45.7, 23.4 Hz), -100.61 (dt, J Z = 39.4, 18.0 Hz). HRMS (ESI): m/z [M + H] + calc. for C12H16F 179.1230, found 179.1233. (0.23 mmol, 49.64 mg), butyl boronic acid 2a (0.28 mmol, 28.45 mg), Pd2dba3.CHCl3 (4.710 -3 mmol, 4.25 mg), xantphos (4.710 -3 mmol, 2.69 mg), Cs2CO3 (0.70 mmol, 227.10 mg), toluene/H2O (2.53 mL) were reacted according to general procedure. The crude product was purified by silica gel column chromatography (petroleum ether/EtOAc = 100:0 to 95:5) affording compound 3ca in 96% yield (42.79 mg) as a yellow solid. 1 H NMR (300 MHz, CDCl3): δ 7.39 (d, J = 8.2 Hz, 1.00H), 7.21-7.08 (m, 3.00H), 6.18 (d, J E H-F = 22.1 Hz, 0.51H), 5.45 (d, J Z H-F = 39.8 Hz, 0.49H), 2.54-2.29 (m, 2.00H), 2.36 (s, 1.47H), 2.35 (s, 1.53H), 1.64-1.56 (m, 2.00H), 1.47-1.37 (m, 2.00H), 0.98 (t, J = 5.5 Hz, 1.47H), 0.93 (t, J = 5.5 Hz, 1.53H). 13 C NMR (75 MHz, CDCl3): δ 162.5 (d, 1 J E C-F = 250.2 Hz), 160.9 (d, 1 J Z C-F = 263. 9 Hz), 136.4 (2), 131.6 (d, 3 J E C-F = 14.0 Hz), 131.3 (d, 3 J Z C-F = 2. 5 Hz), 129.3 (3), 129.2, 128.5, 128.4, 128.3, 128.2, 108.0 (d, 2 J Z C-F = 28.6 Hz), 105.6 (d, 2 J E C-F = 9.0 Hz), 32.9 (d, 2 J Z C-F = 26.4 Hz), 28.9 (d, 2 J E C-F = 28.6 Hz), 28.7 (2), 22.5, 22.2, 21.3, 21.2, 13.9 (2). 19 F NMR (282 MHz, CDCl3): δ -99.23 (q, J E = 23.2 Hz), -101.65 (dt, J Z = 39. 7,18.1 Hz). HRMS (ESI): m/z [M + H] + calc. for C13H18F 193.1387, found 193.1388. 3.3.5. (0.23 mmol, 53.35 mg), butyl boronic acid 2a (0.28 mmol, 28.45 mg), Pd2dba3.CHCl3 (4.710 -3 mmol, 4.25 mg), xantphos (4.710 -3 mmol, 2.69 mg), Cs2CO3 (0.70 mmol, 227.10 mg), toluene/H2O (2.53 mL) were reacted according to general procedure. The crude product was purified by silica gel column chromatography (petroleum ether/EtOAc = 100:0 to 95:5) affording compound 3da in 96% yield (46.35 mg) as a yellow solid. 1 H NMR (300 MHz, CDCl3): δ 7.42 (d, J = 8.9 Hz, 0.96H), 7.13 (d,J = 8.3 Hz,1.04H),2.00H), 6.14 (d, J E H-F = 22.0 Hz, 0.52H), 5.41 (d, J Z H-F = 39.8 Hz, 0.48H), 3.81 (s, 3.00H), 2.49-2.28 (m, 2.00H), 1.64-1.56 (m, 2.00H), 1.45-1.36 (m, 2.00H), 0.96 (t, J = 5.4 Hz, 1.44H), 0.92 (t, J = 5.5 Hz, 1.56H). 13 C NMR (75 MHz, CDCl3): δ 162.1 (d, 1 J E C-F = 249.3 Hz), 160.1 (d, 1 J Z C-F = 262. 2 Hz),158.6,158.4,129.7,129.6 (2), 129.5, 126.9 (d, 3 J E C-F = 13.9 Hz), 126.8 (d, 3 J Z C-F = 2.3 Hz), 114.0 (4), 107.6 (d, 2 J Z C-F = 28.9 Hz), 105.1 (d, 2 J E C-F = 9.2 Hz), 55.4 (2), 32.9 (d, 2 J Z C-F = 26.5 Hz), 28.8 (d, 2 J E C-F = 27.1 Hz), 28.7 (2), 22.4, 22.2, 13.9 (2). 19 (0.23 mmol, 59.84 mg), butyl boronic acid 2a (0.28 mmol, 28.45 mg), Pd2dba3.CHCl3 (4.710 -3 mmol, 4.25 mg), xantphos (4.710 -3 mmol, 2.69 mg), Cs2CO3 (0.70 mmol, 227.10 mg), toluene/H2O (2.53 mL) were reacted according to general procedure. The crude product was purified by silica gel column chromatography (petroleum ether/EtOAc = 100:0 to 95:5) affording compound 3ea in 89% yield (48.75 mg) as a yellow solid. 1 H NMR (300 MHz, CDCl3): δ 7.90 (dd, J = 8.4, 6.0 Hz, 2.00H), 7.43 (d, J = 8.5 Hz, 0.92H), 7.17 (d, J = 8.1 Hz, 1.08H), 6.12 (d, J E H-F = 21.6 Hz, 0.54H), 5.43 (d, J = 38.9 Hz, 0.46H), 3.83 (s, 1.62H), 3.82 (s, 1.38H), 2.44-2.21 (m, 2.00H), 1.57-1.48 (m, 2.00H), 1.36-1.24 (m, 2.00H), 0.87 (t, J = 7.0 Hz, 1.38H), 0.81 (t, J = 7.3 Hz, 1.62H). 13 28.5 (2), 22.4, 22.2, 13.9, 13.8. 19 F NMR (282 MHz, CDCl3): δ -94.29 (dd, J E = 45.3, 23.4 Hz), -96.35 (dt, J Z = 38.8, 18.3 Hz). HRMS (ESI): m/z [M + H] + calc. for C14H18FO2, 237.1285 found 237.1287. (0.23 mmol, 62.16 mg), butyl boronic acid 2a (0.28 mmol, 28.45 mg), Pd2dba3.CHCl3 (4.710 -3 mmol, 4.25 mg), xantphos (4.710 -3 mmol, 2.69 mg), Cs2CO3 (0.70 mmol, 227.10 mg), toluene/H2O (2.53 mL) were reacted according to general procedure. The crude product was purified by silica gel column chromatography (petroleum ether/EtOAc = 100:0 to 95:5) affording compound 3fa in 95% yield (54.24 mg) as a yellow solid. 1 H NMR (300 MHz, CDCl3): δ 7. 35-7.26 (m, 3.00H), 7.00 (d, J = 8.6 Hz, 1.00H), 5.93 (t, J E H-F = 15.5 Hz), 6.2, 6.1, 5.8 (2). 19 F NMR (282 MHz, CDCl3): δ -108.50 (dd, J = 37.9, 22.7 Hz). HRMS (ESI): m/z [M + H] + calc. for C11H11FNO2, 208.0768 found 208.0765.

3.3.1. (E/Z)-1-(2-fluorohex-1-en-1-yl)-4-nitrobenzene (3aa) (E/Z)-1-(2-bromo-2-fluorovinyl)-4-nitrobenzene 1a
3.3.2. (E)-1-(2-fluorohex-1-en-1-yl)-4-nitrobenzene (E-3aa) (Z)-1-(2-bromo-2-fluorovinyl)-4-nitrobenzene 1a
3.3.3. (E/Z)-(2-fluorohex-1-en-1-yl)benzene (3ba) (E/Z)-1-(2-bromo-2-fluorovinyl)-benzene 2b
1 J E C-F = 251.3 Hz), 161.3 (d, 1 J Z C-F = 265.1 Hz), 134.5 (d, 3 J E C-F = 14.1 Hz), 134.0 (d, 3 J Z C-F = 2.
3.3.4. (E/Z)-1-(2-fluorohex-1-en-1-yl)-4-methylbenzene (3ca) (E/Z)-1-(2-bromo-2-fluorovinyl)-4-methylbenzene 2c
(E/Z)-1-(2-fluorohex-1-en-1-yl)-4-methoxybenzene (3da) (E/Z)-1-(2-bromo-2-fluorovinyl)-4-methoxybenzene 2d

Conclusions

In conclusion, an efficient palladium-catalyzed carbon-carbon coupling reaction of readily available gem-bromofluoroalkenes with primary and strained secondary alkyl boronic acid derivatives was successfully achieved under mild conditions. This methodology demonstrates its applicability for the synthesis of alkyl trisubstituted or tetrasubstituted monofluoroalkenes with a broad range of gem-bromofluoroalkenes and alkyl boronic acids with good group compatibility, stereospecificity and excellent yields. Such reactions may be useful for the synthesis of fluoroolefins of interest for life and material sciences.

  F NMR (282 MHz, CDCl3): δ -100.27 (dd, J E = 45.9, 23.2 Hz), -103.58 (dt, J Z = 39.8, 18.3 Hz). HRMS (ESI): m/z [M + H] + calc. for C13H18FO, 209.1336 found 209.1336. 3.3.6. (E/Z)-methyl 4-(2-fluorohex-1-en-1-yl)benzoate (3ea) (E/Z)-methyl 4-(2-bromo-2-fluorovinyl)benzoate 2e

  C NMR (75 MHz, CDCl3): δ 167.0 (2), 164.2 (d, 1 J E C-F = 254.7 Hz), 163.3 (d, 1 J Z C-F = 269.0 Hz), 139.5 (d, 3 J E C-F = 14.3 Hz), 138.7 (d, 3 J Z C-F = 2.6 Hz), 129.9 (3), 129.8 (3), 128.4 (2), 128.2, 128.1, 107.8 (d, 2 J Z C-F = 29.7 Hz), 105.3 (d, 2 J E C-F = 8.5 Hz), 52.2, 52.1, 33.0 (d, 2 J Z C-F = 26.0 Hz), 29.0 (d, 2 J E C-F = 26.8 Hz),

3. 3

 3 .7. (E/Z)-1-(2-fluorohex-1-en-1-yl)-4-(trifluoromethyl)benzene (3fa) (E/Z)-1-(2-bromo-2-fluorovinyl)-4-trifluoromethylbenzene 2f

Table 1 .

 1 Optimization of the cross-coupling between E/Z-1a and butylboronic acid 2a a .

	Entry E/Z 1a b	[Pd]	Ligand	Base	Solvent	T (°C) Z/E 3aa b Yield (%) c
	1	55:45	Pd(Ph3)4	-	Cs2CO3	toluene/H2O	80	-	-
	2	55:45	Pd2dba3 CHCl3	-	Cs2CO3	toluene/H2O	80	-	-
	3	56:44	PdCl2dppf	-	Cs2CO3	toluene/H2O	80	58:42	95
	4	55:45	PdCl2dppf	-	K2CO3	toluene/H2O	80	46:54	76
	5	55:45	PdCl2dppf	-	Na2CO3 toluene/H2O	80	57:43	44
	6	55:45	PdCl2dppf	-	K3PO4	toluene/H2O	80	51:49	93
	7	55:46	PdCl2dppf	-	Ba(OH)2 toluene/H2O	80	-	-
	8	39:61	PdCl2dppf	-	Cs2CO3	toluene	80	38:62	89
	9	39:61	PdCl2dppf	-	Cs2CO3	Dioxane	80	-	-
	10	39:61	PdCl2dppf	-	Cs2CO3	THF	80	-	-
	11	55:45	PdCl2dppf	-	Cs2CO3	toluene/H2O	RT	-	-
	12	55:45	PdCl2dppf	-	Cs2CO3	toluene/H2O	60	45:55	51
	13	55:45	Pd2dba3.CHCl3	Dppf	Cs2CO3	toluene/H2O	80	51:49	91
	14	48:52	Pd2dba3 CHCl3	Dppe	Cs2CO3	toluene/H2O	80	48:52	41
	15	48:52	Pd2dba3 CHCl3 Xantphos	Cs2CO3	toluene/H2O	80	49:51	99
	16	48:52	Pd2dba3 CHCl3	Xphos	Cs2CO3	toluene/H2O	80	60:40	35
	17	48:52	Pd2dba3 CHCl3	TFP	Cs2CO3	toluene/H2O	80	42:58	78

  H NMR (300 MHz, CDCl3): δ 8.09 (t, J = 9.1 Hz, 2.0H), 7.51 (d, J = 8.9 Hz, 0.8 H), 7.26 (d, J = 8.4 Hz, 1.2H), 6.15 (d, J E H-F = 21.1 Hz, 0.6H), 5.49 (d, J Z H-F = 38.1 Hz, 0.4H), 2.47-2.23 (m, 2.0H), 1.59-1.50 (m, 2.0H), 1.37-1.27 (m, 2.0H), 0.90-0.81 (m, 3.0H). 13 C NMR (75 MHz, CDCl3): δ 164.1 (d, 1 J E C-F = 257.5 Hz), 163.6 (d, 1 J Z C-F = 271.8 Hz), 145.3 (

Funding: This research received no external funding.

Acknowledgments:

The authors gratefully acknowledge the CNRS for financial support. The NMR facilities used in this study were funded by the European Community (FEDER), Région Haut de France (France), the CNRS and Lille University.

Supplementary Materials: 1 H-, 13 C-and 19 F-NMR spectra of products associated with this article are available online.

Author

Conflicts of Interest:

The authors declare no conflicts of interest.


Urelle Biapo

Alessio Ghisolfi

Geoffrey Gerer

Denis Spitzer

Valérie Keller

Thomas Cottineau
email: cottineau@unistra.fr

Nanostructured and functionalized cantilever for sensing organophosphorous compounds

Keywords: TiO2 nanostructure, Organophosphorous agents Micromechanical sensor, Functionalization, Oxime molecule. I

Vertically aligned TiO2 nanorods have been successfully synthesized on the micrometric surface of a cantilever by a two-step procedure involving solvothermal method. Adjusting TiO2 synthesis conditions, first optimized on centimetric silicon substrate, allowed to well-control the morphology of the nanorods, aimed to increase the collection surface of this micromechanical sensor. By using the nanostructured microcantilever for the detection of organophosphorous agent (OPs), a frequency shift of 833 Hz corresponding to a remarkable enhancement of the sensor sensitivity by a factor of 75 compared to a pristine cantilever is achieved. Moreover, functionalization of this nanostructured cantilever with a tailored functional oxime molecule having strong affinity with OPs compounds significantly improved the performance of the sensor in vapor phase. A frequency shift of 1319 Hz is obtained at 500 ppm of DMMP with the functionalized sensor.

INTRODUCTION

The use of microcantilevers as chemical sensors has demonstrated promising results on various applications including medical and defense fields [START_REF] Alvarez | Microcantilever-based platforms as biosensing tools[END_REF]- [START_REF] Hansen | Microcantilever biosensors[END_REF]. While in biology microcantilevers can be used to follow the hybridization of DNA molecules [START_REF] Calleja | Highly sensitive polymer-based cantilever-sensors for DNA detection[END_REF], in the field of chemistry this technology can be employed to control the presence of moisture [START_REF] Thundat | Vapor Detection Using Resonating Microcantilevers[END_REF], volatile organic compounds [START_REF] Pinnaduwage | Sensitive detection of plastic explosives with self-assembled monolayer-coated microcantilevers[END_REF], [START_REF] Lee | Sensitive and selective detection of hydrocarbon/water vapor mixtures with a nanoporous silicon microcantilever[END_REF], or others specific target in environment [START_REF] Yang | Nerve Agents Detection Using a Cu2+/l-Cysteine Bilayer-Coated Microcantilever[END_REF], [START_REF] Ahmad | Nanodiamond-coated silicon cantilever array for chemical sensing[END_REF]. Thanks to an actuation system, the cantilever oscillates at its natural resonant frequency, which is directly related to the spring constant and to the mass of the beam. Any change of the mass of the cantilever, for instance due to adsorption of molecules will result in a variation of the resonance frequency (f0). In this study, we employed this type of micromechanical sensor for the chemical detection of organophosphorous compounds (OPs) taking advantage of its gravimetric sensor properties. The OPs molecules used as chemical warfare agents (Sarin or Soman) but also as pesticides (Glyphosate) are highly toxics and are considered as one of the most dangerous threats for animals and humans, because they are effective at very low concentration [START_REF] Kassa | Review of oximes in the antidotal treatment of poisoning by organophosphorus nerve agents[END_REF]. In order to improve the detection of these compounds, we propose a novel strategy to improve both the sensitivity and the selectivity of a cantilever by chemically modifying its surface.

A TiO2 nanostructure having high specific surface area was grown on cantilever for increasing its active surface. This concept has been highlighted in the work described by Spitzer and al., where TiO2 nanotubes prepared using anodization method were employed to increase the sensitivity of the cantilever and resulted in a detection limit of 0.8 ppb of trinitrotoluene vapors [START_REF] Spitzer | Bio-Inspired Nanostructured Sensor for the Detection of Ultralow Concentrations of Explosives[END_REF]. In this work, such nanostructure was grown on the cantilever using a versatile solvothermal route, easier to achieve on small substrate than the anodization. In addition, the ability of the nanostructure to interact with chemicals was further improved by grafting specific molecules that selectively bind OPs. The detection tests were carried on a common OPs simulant, the DiMethylMethylphosphonate (DMMP).

II. EXPERIMENTAL PROCEDURE

A. Wafer nanostructuration

The nanostructure was initially achieved on a model surface (silicon wafer), and the optimized synthesis parameters were applied to the cantilevers surfaces. Before solvothermal synthesis, a thin layer of metallic titanium (~ 50 nm) was first deposited on silicon surface by magnetron sputtering at a DC power density of 1.37 W.cm -2 in order to compensate the lattice mismatch between silicon and TiO2. This layer was finally annealed at 800 °C during 6 h to oxidize the metallic film for improving the adherence and the growth of TiO2 nanostructure at the surface. The growth solution containing a mixture of 15 ml hydrochloric acid (37 %), absolute ethanol (15 ml) and 0.25 ml of titanium isopropoxide (97 %) is prepared and loaded in the Teflon liner of autoclave. Afterwards, the coated substrate was dipped in the solution and the solvothermal process was done at 150 °C during 8 hours. The obtained sample was thoroughly washed with distilled water and dried under nitrogen flow. All the samples were characterized by x-ray diffraction (XRD) and scanning electron microscopy (SEM).

B. Functionalization of the nanostructure

A bifunctional molecule called (E)-4-((hydroxyimino) methyl)-benzoic acid is used to functionalize TiO2 nanostructure. The grafting mechanism involves a reaction between the hydroxyl function present at titanium dioxide 978-1-7281-1634-1/19/$31.00 ©2019 IEEE surface and the carboxylic group of the molecule. The other part of the molecule contains an oxime function, well-known as nucleophile group that aid for the recognition of OPs molecules [START_REF] Kassa | Review of oximes in the antidotal treatment of poisoning by organophosphorus nerve agents[END_REF]. A mixture (1:1) of 5 ml acetonitrile and tertbutanol is employed to dissolve 0.01 g of the molecule. The nanostructured silicon was immersed in the obtained solution and heat up for 24 hours at 70 °C. After the reaction, the sample was rinsed and analyzed by Raman and UV spectroscopy.

C. Detection measurements

The vapor generation system is composed of a DMMP saturator which delivers a precise concentration of the nerve simulant vapor (500 ppm). This vapor is then carried by a stream of air and sent to the detection chamber where the cantilever is located. Tipless silicon microcantilevers (TL-NCL-50 type) having 7 µm of thickness, 225 µm length, 38 µm width and a nominal resonant frequency range of 146-236 kHz were purchased from NanoSensors. The cantilever covered with the nanostructure is mounted in the head of a PicoSPM AFMmultimode to track the resonance frequency during the detection tests.

III. RESULTS

A. TiO2 synthesis on wafer

The diffractogram of oxidized titanium seed layer depicted on figure 1a showed the presence of pure tetragonal rutile phase of TiO2. This layer provides initial nucleation sites for growing a rutile nanostructure. On the same figure, we can observe the XRD pattern of the nanostructure film obtained after the solvothermal reaction. A single intense peak localized at 62.7° is evidenced on the pattern, as expected; this peak is attributed to rutile phase. The absence of others main rutile peaks indicates that the nanostructure is highly oriented and has a preferred growth direction along the c axis. Silicon diffraction peaks arise from the substrate can be also seen on this figure. The morphology of the nanostructure is exhibited on the SEM image reported on figure 1b andc. The rutile TiO2 nanostructure which composed the film is highly dense and revealed vertically aligned nanorods with top square facet, commonly observed in tetragonal crystal structure. The nanorods are uniformly distributed on the Si surface, with an average length of 2.5 μm and a width around 12 nm. The preferential orientation of the nanorods is due to the adsorption of chloride ions on the lateral facets (110) of rutile crystal that hinders their growth and promotes the anisotropic growth of the nanostructure [START_REF] Qin | Hydrothermal Growth and Photoelectrochemistry of Highly Oriented, Crystalline Anatase TiO2 Nanorods on Transparent Conducting Electrodes[END_REF]. The monocrystalline structure of the film was also confirmed by the presence of few diffractions spots shown on the selected area diffraction pattern. Moreover, the measured interplanar distance (d110 = 0.324 nm) using TEM image (Figure 1d) is in good agreement with the standard space between two (110) plans of rutile crystal [START_REF] Kumar | Growth of Aligned Single-Crystalline Rutile TiO2 Nanowires on Arbitrary Substrates and Their Application in Dye-Sensitized Solar Cells[END_REF].

B. TiO2 functionalization

We functionalized the as-synthesized nanorods surface with oxime molecules and its loading ability was estimated.

Figure 2a shows the Raman spectra of the reference nanostructure (Rut-EtOH) and the functionalized one (Rut-EtOH-Oxime). It can be found on both spectra, peaks at 443 and 609 cm -1 corresponding to Eg and A1g active modes of rutile TiO2 polymorph respectively [START_REF] Burungale | Surfactant free single step synthesis of TiO2 3-D microflowers by hydrothermal route and its photoelectrochemical characterizations[END_REF]. This result is in accordance with that observed in XRD analysis where the presence of rutile TiO2 phase has been already revealed. The intense narrow peak at 520 cm -1 and the large one peak at 943 cm -1 arise from silicon substrate [START_REF] Li | Raman spectral study of silicon nanowires[END_REF]. In addition, for the grafted sample, two supplementary peaks located at 1412 and 1610 cm -1 can be observed. These peaks are typical vibration bands of COO group and C=C from the aromatic ring [START_REF] Pérez León | Characterization of the Adsorption of Ru-bpy Dyes on Mesoporous TiO2 Films with UV-Vis, Raman, and FTIR Spectroscopies[END_REF], thereby suggesting the presence of oxime molecule onto rutile TiO2 nanorods surface of this sample. The grafted nanostructure was then immersed in NaOH solution at pH = 12, to desorb the molecules from the surface for UV visible spectroscopy quantification. The absorbance of the pi-pi* transition of the phenyl group (from oxime molecule) desorbed in solution was followed by UV spectroscopy to evaluate the amount of grafted oxime on the nanostructure surface. Using a calibration curve obtained via oxime titration, the estimated molecule density at the nanorods surface was ca. 70 nmol.cm -2 .

C. Effect of TiO2 synthesis solvent

In order to find the best type of nanostructuration for chemical functionalization, various synthesis parameters were adjusted to control the morphology, the width and the length of the nanorods. We describe the effect of one of the most influential parameters (the solvothermal solvent).

The ethanol was replaced by ethylene glycol (EG) or water during the preparation of the growth solution. The SEM images of these samples (Figure 2b,c), showed that the morphology of rutile TiO2 phase obtained with water and EG differs significantly from the one prepared in ethanol medium. In the case of EG, well separated vertically oriented needles-like structure was obtained instead of the TiO2 NRs with square facets given by ethanol. In hydrothermal conditions, randomly oriented nanorods involving low density of nanorods at silicon surface was observed. Nevertheless, the rods-like structure which composed this new nanoarchitecture are agglomerated and should result in lower active surface for grafting. The XRD peaks of both films still indicate the presence of rutile phase. However, the intensity of resulting XRD peaks denotes that the orientation of the nanorods is strongly dependent on the nature of solvent. The needles structure morphology achieved with EG are less oriented compared to the nanorods with square facets. After performing the functionalization, the Raman spectra of the nanostructure synthesized in EG exhibited the two peaks related to the oxime molecule while they are not observed for the one prepared in water (Figure 2a). The number of adsorbed molecules estimated for the needles structure obtained in EG solvent is 34 nmol.cm -2 , which corresponds to the half of the oxime density of the nanostructure obtained in ethanol (70 nmol.cm -2 ). As expected for the TiO2 NRs synthesized in water, the molecules density is only 8 nmol.cm -2 due to the lower surface area.

D. Sensor surface modification and detection tests

Since TiO2 NRs synthesis in ethanol gives the nanostructure with the higher adsorption capacity, we applied this optimized synthesis for cantilevers nanostructuration. From the SEM image of this sample depicted on Figure 3a, we can see that a homogeneous layer of TiO2 nanostructure completely covers the surface of the microcantilever. The morphology of TiO2 obtained at this micrometric surface is very similar to TiO2 nanostructure synthesized on centimetric silicon wafers. The growth of similar nanostructure on both types of support highlighted the reproducibility and the versatility of this optimized TiO2 NRs synthesis. The modified cantilever was tested for sensing DMMP vapor. For comparison, the detection tests were also performed with cantilevers nanostructured in the others solvents (EG and water). Figure 3b shows the evolution of the resonance frequency over the time of pristine cantilever and cantilevers covered with different nanostructure. All the tested sensors showed a resonance frequency drop when 500 ppm of DMMP vapors was generated near their surfaces within a few seconds. The frequency shifts correspond to 11, 256, 556 and 833 Hz for uncoated microcantilever and the ones nanostructured in ethylene glycol, water and ethanol respectively. These results indicate the adsorption of DMMP on the cantilevers surfaces and therefore the detection of the target molecule. When the air containing the DMMP vapor is replaced after 10 minutes by the reference air flux, the DMMP molecules are desorbed from the nanostructured cantilevers, and the resonance frequency returns to the baseline value after a short period. A cycle measurement of each modified sensor was performed and resulted in a good repeatability.

The higher frequency shift obtained with the cantilever nanostructured in ethanol, confirms that a large part of its surface area is accessible for the analyte even though the film appears highly dense. The same sensor exhibits the highest frequency shift of 1319 Hz when the oxime molecules are adsorbed on its surface, as expected for the high surface area and the high loading of grafting offered by the nanostructure (Figure 3c). A theoretical detection limit of 2.25 ppm of DMMP with this sensor is obtained. Although this concentration remains high compared to the OPs detection limits (5 and 7.2 ppb) reported for some gravimetric sensors [START_REF] Cai | In situ construction of metal-organic framework (MOF) UiO-66 film on Parylene-patterned resonant microcantilever for trace organophosphorus molecules detection[END_REF], [START_REF] Liu | Hyper-branched sensing polymer directly constructed on a resonant micro-cantilever for the detection of trace chemical vapor[END_REF], this result provides a kind of guideline for developing new nanostructures with higher sensitivity.

IV. CONCLUSION

The synthesis of TiO2 NRs on microcantilever surface was demonstrated using a solvothermal method. The control of the morphology of TiO2 was also evidenced and the results showed that the surface area developed by the nanostructure has a high impact on the oxime loading. The proposed approach to increase the active surface of cantilever was very beneficial to significantly improve by a factor of 75 the sensitivity of the cantilever for sensing OPs compounds. We also found that the presence of grafted molecules at the surface almost doubles the response of the nanostructured cantilever even though oxime was only aimed to enhance sensor selectivity.

Figure 1 :

 1 Figure 1: (a) XRD patterns of different films (b and c) SEM images of TiO2 nanorods grown on silicon wafer by solvothermal method (d) TEM image of a single rutile TiO2 nanorod.

Figure 2 :

 2 Figure 2: (a) Raman spectra of rutile nanorods and oxime-functionalized nanorods. SEM images of rutile prepared by changing ethanol with (b) Ethylene glycol and (c) Water (d) XRD of the nanostructure synthesized in water, glycol ethylene and ethanol.

Figure 3 :

 3 Figure 3 : (a) TiO2 nanostructured cantilever prepared in ethanol. Measured sensing frequency responses to 500 ppm of DMMP vapor of (b) pristine microcantilever and the one prepared in different solvents (c) oxime functionalized cantilevers.

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