Habib Ammari
email: habib.ammari@math.ethz.ch

Alexander Dabrowski
email: alexander.dabrowski@sam.math.ethz.ch

Brian Fitzpatrick

Pierre Millien
email: pierre.millien@espci.fr

Mourad Sini
email: mourad.sini@oeaw.ac.at.

Subwavelength resonant dielectric nanoparticles with high refractive indices

Keywords: Mathematics Subject Classification (MSC2000). 35R30, 35C20 Dielectric nanoparticles, high refractive index, subwavelength resonances, asymptotic expansions

This paper aims at understanding the nature of the subwavelength resonant frequencies of dielectric particles with high refractive indices. It is proved that for an arbitrary shaped particle, these subwavelength resonant frequencies can be expressed in terms of the eigenvalues of the Newtonian potential associated with its shape. The enhancement of the scattered field at the resonant frequencies is shown. The hybridization of the subwavelength resonant frequencies of a dimer consisting of high refractive index dielectric nanoparticles is also characterized.

Introduction

Nanoscale optics is usually associated with plasmonic resonant structures made of metals such as gold or silver. Plasmonic resonances of nanoparticles can be treated as an eigenvalue problem for the Neumann-Poincaré operator, see [START_REF] Ammari | Surface plasmon resonance of nanoparticles and applications in imaging[END_REF][START_REF] Ammari | Shape and size dependence of dipolar plasmonic resonance of nanoparticles[END_REF][START_REF] Ammari | Mathematical analysis of plasmonic nanoparticles: the scalar case[END_REF][START_REF] Ammari | Mathematical analysis of plasmonic resonances for nanoparticles: the full Maxwell equations[END_REF]. However, plasmonic structures suffer from high losses inherent in metals and dissipation due to heating. Recent developments in nanoscale optical physics have led to a new branch of nanophotonics focused on the manipulation of optically induced subwavelength resonances in dielectric nanoparticles with high refractive indices [START_REF] Kuznetsov | Optically resonant dielectric nanostructures[END_REF][START_REF] Yang | Dielectric nanoresonators for light manipulation[END_REF][START_REF] Zhao | Dielectric metamaterials[END_REF]. Resonant high-index dielectric nanostructures form new building blocks which can be used to realize unique functionalities and novel photonic devices [START_REF] Kuznetsov | Optically resonant dielectric nanostructures[END_REF]. Their study has been established as a new research direction in nanophotonics. Nevertheless, despite strong experimental efforts, mathematical modeling of resonant high-index nanoparticles remains limited. Apart from the case where the particles are disks or spheres, their subwavelength resonant frequencies have not been characterized yet.

In this paper, we consider a dielectric high-index nanoparticle of arbitrary shape and characterize its subwavelength resonances in terms of the eigenvalues of the Newtonian potential associated with its shape. Our formula is closely related to the one established in [START_REF] Meklachi | Asymptotic analysis of resonances of small volume high contrast linear and nonlinear scatterers[END_REF]. Then, we provide an asymptotic formula for the field scattered by a dielectric nanoparticle and estimate the scattering enhancement near its resonant frequencies. We also consider the hybridization phenomenon of a dimer consisting of high refractive index dielectric nanoparticles. We derive asymptotic formulas for the hybridized resonant frequencies, which correspond to monopole and dipole modes.

For simplicity of presentation, we consider the Helmholtz equation as a model for the wave propagation. But one should emphasize that the approach developed here can be extended to the full Maxwell's equations. On the other hand, based on the asymptotic formula for the scattered field derived in this paper, one can characterize the temporal response of resonant dielectric nanoparticles and obtain a time-domain resonant-mode-expansion for the scattered field, which generalizes the time-domain asymptotic formula proved in [START_REF] Ammari | Transient anomaly imaging by the acoustic radiation force[END_REF] to the case of a resonant subwavelength particle. This can be easily done by reproducing the calculations presented in [START_REF] Ammari | A mathematical theory of super-resolution by using a system of sub-wavelength Helmholtz resonators[END_REF]Appendix B].

Our results in this paper provide a solid mathematical framework for the analysis of resonant dielectric nanoparticles. They make possible the direct calculation of resonant frequencies and the optimal design of dielectric nanoparticles that resonate at specified frequencies. They can also be applied in the design of dielectric metamaterials and are expected to adavance the applications described in [START_REF] Jahani | All-dielectric metamaterials[END_REF][START_REF] Yang | Dielectric nanoresonators for light manipulation[END_REF][START_REF] Zhao | Dielectric metamaterials[END_REF], in particular those concerned with metasurfaces, double-negative all dielectric materials, super-focusing, and wavefront control at the deep subwavelength scale.

Resonant frequencies of dielectric nanoparticles with high refractive indices

Let D ⋐ R d , for d = 2, 3, be a small particle of the form D = z +δB, where δ is its characteristic size, z its location, and B is a smooth bounded domain containing the origin. Let ω denote the frequency, let ε ≡ τ ε c + ε m inside D and ε ≡ ε m outside D. Here, ε c , ε m , and τ are positive constants. Let E in be an incident plane wave with frequency ω.

Consider the Helmholtz equation

(∆ + ω 2 ε)E = 0 in R d , E -E in satisfies the Sommerfeld radiation condition. From (∆ + ω 2 ε m )(E -E in ) = -ω 2 τ ε c E✶ D in R d ,
where ✶ D is the characteristic function of D, it follows that the following Lippmann-Schwinger representation formula holds:

(E -E in )(x) = -ω 2 τ ε c ˆD E(y)Γ m (x -y; ω)dy for x ∈ R d , (1)
where Γ m is the outgoing (i.e., subject to the Sommerfeld radiation condition) fundamental solution of ∆ + ε m ω 2 in free space.

Let k m = ω √ ε m . Let the volume integral operator K k m D be defined by

K k m D : E ∈ L 2 (D) → -ˆD E(y)Γ m (x -y; ω)dy ∈ L 2 (D).
It is well known that, due to the weak singularity of the fundamental solution,

K k m D is compact. When the norm of τ ω 2 ε c K k m D is smaller than 1, I -τ ω 2 ε c K k m D is invertible, so (3) can be rewritten as E(x) = (I -τ ω 2 ε c K k m D ) -1 [E in ](x) for all x ∈ D, (2)
where I denotes the identity operator. Assume that the characteristic size δ of the particle D is much smaller than the wavelength 2π/k m ), and let ω → 0. The subwavelength resonance problem is then to find an ω ∈ C close to 0 such that (I -τ ω 2 ε c K k m D ) -1 is singular, or equivalently, such that there exists L 2 (D) ∋ E ≡ 0 with E(x) + ω 2 τ ε c ˆD E(y)Γ m (x -y; ω)dy = 0, for x ∈ D;

(3) see [START_REF] Ammari | Super-resolution in high-contrast media[END_REF]. Such an ω would be a subwavelength resonance for the high refractive index dielectric particle D. Through a Taylor series expansion of the fundamental solution we obtain the following result.

Lemma 2.1. Let d = 3. Let K (0)
D be the Newtonian potential on D, i.e., the operator defined by

K (0) D [E](x) = -ˆD E(y)Γ(x -y) dy for x ∈ D,
with Γ(x) being the fundamental solution of the Laplacian in R 3 . The operator K k m D can be rewritten as

K k m D = ∞ i=0 ω i K (i) D , (4)
where the series converges in operator norm if ω is small enough.

Let A i = τ ω 2 ε c K (i) D .
By expanding with a Neumann series, we have

I -A 0 - ∞ i=1 ω i A i -1 = I -(I -A 0 ) -1 ∞ i=1 ω i A i -1 (I -A 0 ) -1 = ∞ k=0 (I -A 0 ) -1 ∞ i=1 ω i A i k (I -A 0 ) -1 = (I -A 0 ) -1 + (I -A 0 ) -1 ωA 1 (I -A 0 ) -1 + O(ω 3 ). ( 5
)
Recall that K (0)

D : L 2 (D) → L 2 (D)
is a compact, self-adjoint operator. Let, for the sake of clarity of the presentation, λ 0 be a simple eigenvalue of K (0) D associated with the normalized eigenfunction φ 0 in L 2 (D). We remark that the eigenvalues of K (0) D are positive. For the analysis of the spectrum of the Newtonian potential, we refer the reader, for instance, to [START_REF] Kalmenov | A boundary condition and spectral problems for the Newton potential[END_REF].

Let ω 0 be a frequency at which I -A 0 becomes singular. In particular, let

ω 0 = 1/ τ ε c λ 0 . (6)
Note that ω 0 is small only for τ large enough. This shows that subwavelength resonances occur only for particles with high refractive indices.

For ω near ω 0 , we have, by a pole-pencil operator decomposition, that

(I -A 0 ) -1 [ψ] = ψ, φ 0 φ 0 1 -τ ω 2 ε c λ 0 + R(ω)[ψ],
where ω → R(ω) is analytic in a neighborhood of ω 0 and •, • denotes the scalar product on L 2 (D). Hence, considering only the first two terms in the expansion (5), we obtain from (2) that an approximation of the resonance must satisfy (see, for instance, [START_REF] Ammari | Mathematical and computational methods in photonics and phononics[END_REF][START_REF] Ammari | Layer potential techniques in spectral analysis[END_REF])

φ 0 1 -τ ω 2 ε c λ 0 + τ ω 3 ε c K (1) D [φ 0 ], φ 0 (1 -τ ω 2 ε c λ 0 ) 2 φ 0 = 0.
Therefore, we have the following approximation for the subwavelength resonances.

Proposition 2.2. Let d = 3 and let τ be large enough. Let ω 0 be defined by [START_REF] Ammari | Shape and size dependence of dipolar plasmonic resonance of nanoparticles[END_REF], where λ 0 is an eigenvalue of the Newtonian potential K (0) D . Then, the O(ω 4 )-approximation of the subwavelength resonant frequencies ω s of the dielectric particle D satisfies

1 -τ ω 2 s ε c λ 0 = -τ ω s s ε c K (1) D [φ 0 ], φ 0 .
Note that, in three dimensions, K

D [φ] = -i √ ε m 4π ˆD φ dy for all φ ∈ L 2 (D). Therefore, ω s satisfies 1 -τ ω 2 s ε c λ 0 = iτ 4π ω s s √ ε m ε c ˆD φ 0 dy 2 . (1)
Since ω s is close to ω 0 , by approximating ω 3 s ≃ ω 3 0 , and since by definition τ ε c λ 0 = 1/ω 2 0 , we obtain

1 - ω 2 s ω 2 0 = iτ 4π ω 3 0 √ ε m ε c ˆD φ 0 dy 2 . Corollary 2.3. Let d = 3.
Then, the O(ω 4 )-approximation of the subwavelength resonant frequencies of the dielectric particle D can be computed as

ω s = ω 0 - i 8π ω 2 0 λ 0 √ ε m ˆD φ 0 dy 2 .
By using the Lippmann-Schwinger representation formula (3), we can also rewrite

E(x) -E in (x) ≃ -ω 2 τ ε c Γ m (x -z; ω) E in , φ 0 ( ´D φ 0 ) 1 -τ ω 2 ε c λ 0 + τ ω 3 ε c K (1) D [φ 0 ], φ 0 E in , φ 0 ( ´D φ 0 ) (1 -τ ω 2 ε c λ 0 ) 2 .
By plugging the expression of ω s obtained in Proposition 2.2 into the above approximation of the scattered field, we arrive at the following result.

Proposition 2.4. For ω (real) near the resonant frequency ω s and E in such that E in , φ 0 L 2 (D) = 0 , the following monopole approximation of the dielectric nanoparticle D holds:

E(x) -E in (x) ≃ - λ 0 ω 2 s ω 2 -1 -i √ ε m 4π ( ´D φ 0 ) 2 ω - ω 3 s ω 2 λ 0 ω 2 s ω 2 -1 -i √ ε m 4π ( ´D φ 0 ) 2 ω 3 s ω 2 2 E in , φ 0 L 2 (D) Γ m (x -z; ω), ( 7
)
for |x -z| ≫ 2π/(ω √ ε m ).
Now, we turn to the two-dimensional case. In this case the problem is complicated by the logarithmic singularity of the operator K k m D as ω → 0 which gives rise to an averaging operator at leading order when asymptotically expanded. This means we cannot expect to frame the resonance frequency in terms of a single eigenvalue of the Newtonian potential. Instead, in two dimensions the resonance frequency takes account of an infinite number of eigenvalues of the Newtonian potential.

From the asymptotic expansion of the Hankel function H

(1) 0 of the first kind of order zero:

H (1) 0 (s) = 2i π ∞ m=0 (-1) m s 2m 2 2m (m!) 2 log(γs) - m j=1 1 j ,
where 2γ = exp(γ -iπ/2) with γ being the Euler's constant (see, for instance, [START_REF] Ammari | Full low-frequency asymptotics for the reduced wave equation[END_REF]), it follows that

K k m D [E] = - 1 2π (log(ω √ ε m γ)) ˆD E(y) dy + K (0) D + (ω 2 log ω) as ω → 0, (8)
where

K (0)
D is the Newtonian potential in dimension two, that is, the operator defined on L 2 (D) by

K (0) D [E](x) = ˆD E(y)Γ(x -y) dy for x ∈ D,
with Γ being the fundamental solution of the Laplacian in R 2 .

Expanding K k m D as in [START_REF] Ammari | Layer potential techniques in spectral analysis[END_REF] and following the same calculations, we obtain the following characterization of subwavelength resonant frequencies in the two-dimensional case; see Appendix A.

Proposition 2.5. Let d = 2 and τ large enough. Then, the o(ω 2 )-approximation of the subwavelength resonant frequencies ω s of the dielectric particle D satisfies

1 -ω 2 s τ ε c - |D| 2π log(ω s γ√ ε m ) + K (0) D [ ✶D ], ✶D = 0,
where |D| is the volume of D and ✶D = ✶ D / |D|.

Hybridization of subwavelength resonant frequencies for a dimer of dielectric nanoparticles

Consider a dimer of two identical particles D 1 and D 2 with the same dielectric parameter as in the above section. Then the field E -E in scattered by the two particles has the following representation formula:

(E -E in )(x) = -ω 2 τ ε c ˆD1 E(y)Γ m (x -y; ω)dy + ˆD2 E(y)Γ m (x -y; ω)dy for x ∈ R d . ( 9
)
Define the operators K k m D i and R k m D i ,D j for i, j = 1, 2, by

K k m D i : E| D i ∈ L 2 (D i ) → - ˆDi E(y)Γ m (x -y; ω)dy D i ∈ L 2 (D i ), and R k m D i ,D j : E| D i ∈ L 2 (D i ) → - ˆDi E(y)Γ m (x -y; ω)dy D j ∈ L 2 (D j ).
Then, from ( 9) we obtain the following system of operator equations:

1 -τ ω 2 ε c K k m D 1 -τ ω 2 ε c R k m D 2 ,D 1 -τ ω 2 ε c R k m D 1 ,D 2 1 -τ ω 2 ε c K k m D 2 E| D 1 E| D 2 = E in | D 1 E in | D 2 . ( 10
)
The scattering resonance problem is to find ω such that the operator in [START_REF] Ammari | Super-resolution in high-contrast media[END_REF] is singular, or equivalently such that there exists

L 2 (D 1 ) × L 2 (D 2 ) ∋ (E 1 , E 2 ) ≡ (0, 0) such that 1 -τ ω 2 ε c K k m D 1 -τ ω 2 ε c R k m D 2 ,D 1 -τ ω 2 ε c R k m D 1 ,D 2 1 -τ ω 2 ε c K k m D 2 E| D 1 E| D 2 = 0 0 . ( 11
)
Note that here we have a coupled system of subwavelength resonators. As in [START_REF] Ammari | Mathematical and computational methods in photonics and phononics[END_REF][START_REF] Ammari | Double-negative acoustic metamaterials[END_REF], the following results hold.

ω ± s = ω 0 ± 1 2 τ ω 3 0 ε c (R ω s D 1 ,D 2 [φ (1) 0 ], φ (2) 0 )(R ω s D 2 ,D 1 [φ (2) 0 ], φ (1)
0 ), [START_REF] Jahani | All-dielectric metamaterials[END_REF] where φ (i) 0 , for i = 1, 2, is the eigenfunction associated to the eigenvalue λ 0 of the Newtonian potential of D i . Moreover, in the far-field, the dimer of dielectric particles behaves as the sum of a monopole and a dipole. Now, let d = 2 and consider for simplicity a dimer of two identical disks D 1 and D 2 with the same dielectric parameters as in the above section.

Define the operators K k m D i and R k m D i ,D j for i, j = 1, 2, by

K k m D i :E| D i ∈ L 2 (D i ) → - ˆDi E(y)Γ k m (x -y)dy D i ∈ L 2 (D i ), R k m D i ,D j :E| D i ∈ L 2 (D i ) → - ˆDi E(y)Γ k m (x -y)dy D j ∈ L 2 (D j ).
Define the operators M k m D i and N k m D i ,D j for i, j = 1, 2, by

M k m D i := Kk m D i + K (0) D i , N k m D i ,D j := Kk m D i ,D j + R (0) D i ,D j ,
where

K (0) D i :E| D i ∈ L 2 (D i ) → ˆDi E(y)Γ(x -y)dy| D i ∈ L 2 (D i ), Kk m D i :E| D i ∈ L 2 (D i ) → log(γk m ) KD i [E]| D i ∈ L 2 (D i ), KD i :E| D i ∈ L 2 (D i ) → - 1 2π ˆDi E(y)dy| D i ∈ L 2 (D i ), R (0)
D i ,D j :E| D i ∈ L 2 (D i ) → ˆDi E(y)Γ(x -y) D j ∈ L 2 (D j ), Kk m D i ,D j :E| D i ∈ L 2 (D i ) → log(γk m ) KD i [E]| D j ∈ L 2 (D j ), KD i ,D j :E| D i ∈ L 2 (D i ) → - 1 2π ˆDi E(y)dy| D j ∈ L 2 (D j ).
We refer to Appendix C for the proof of the following proposition.

1 -ω 2 τ ε c - δ 2 2 log(ωγ √ ε m )(1 ± 1) + K (0) D 1 [ ✶D 1 ] ✶D 1 ± R (0) D 2 ,D 1 [ ✶D 2 ], ✶D 1 = 0.
The following corollary gives more explicit formulae for the hybridized resonances in the case when D 1 and D 2 are unit disks.

ω m (τ ) = - √ 2i τ ε c W (Φ(τ )) , ω d (τ ) = √ 2 τ ε c 1 4 -2 π R (0) D 2 ,D 1 [✶ D 2 ], ✶ D 1 ,
where

Φ(τ ) = - 2 τ ε c exp 2 log( √ ε m γ) - 1 4 - 2 π R (0) D 2 ,D 1 [✶ D 2 ], ✶ D 1 ,
and W is the lower branch of the Lambert W function defined in the interval [-1/e, 0).

Remark 3.4. Note that as τ → ∞ the monopole resonance ω m = O(1/ τ log(τ )) and hence it decays at the same rate as the single particle resonance ω s , however the dipole resonance ω d decays slightly slower, i.e., as O(1/ √ τ ).

Numerical illustrations

Let In Figure 1 we plot the asymptotic resonances along with the corresponding reference resonances and predicted rates of convergence.

ε m = ε c = 1. Let D, D 1 ,
In Table 1

Concluding remarks

In this paper, we have provided the first mathematical model of resonant high-index dielectric nanoparticles. We have characterized their subwavelength resonances in terms of the eigenvalues of the associated Newtonian potential. We have also discussed the hybridization phenomenon of a dimer of dielectric nanoparticles with high refractive indices. Our results in this paper pave the way for the analysis, design, and manipulation of resonant dielectric nanostructures and their use as metamaterials. In particular, they can be used for mathematically and numerically modelling super-focusing in dielectric nanostructures, double-negative dielectric materials, and dielectric metasurfaces. Moreover, following [START_REF] Ammari | Transient anomaly imaging by the acoustic radiation force[END_REF][START_REF] Ammari | A mathematical theory of super-resolution by using a system of sub-wavelength Helmholtz resonators[END_REF], formula [START_REF] Ammari | Mathematical analysis of plasmonic nanoparticles: the scalar case[END_REF] can be easily generalized to the time-domain in order to characterize the temporal response of resonant dielectric nanoparticles and accelerate computations involving the temporal responses of subwavelength dielectric resonators.

A Proof of Proposition 2.5

Expansion (8) can be rewritten as

K k m D = Kk m D + K (0) D + O(ω 2 log(ω)), where Kk m D [E] = log(γk m ) KD [E], KD [E] = - 1 2π ˆD E(y)dy, K (0)
D [E] = ˆD E(y)Γ(x -y)dy = - 1 2π ˆD E(y) log | • -y|dy.
Using this expansion the resonance problem of finding ω ∈ C such that there exists a solution L 2 (D) ∋ E ≡ 0 to (3) becomes

(I -ω 2 τ ε c ( Kk m D + K (0) D ))[E](x) = O(ω 4 log(ω)).
Denote by

M k m D := Kk m D + K (0)
D , which is self-adjoint as it is the sum of bounded self-adjoint operators.

Note that when

ω = 1 √ τ ε c λ 0 ,
where λ 0 belongs to σ(K

(0) D ), the spectrum of K (0)
D , the following equation has a non-trivial solution:

(

I -ω 2 τ ε c K (0) D )[E] = 0. Analogously, when ω = 1 τ ε c ν(ω) , (13)
where ν(ω) ∈ σ(M k m D ) the following equation, which is an approximation of our resonance problem up to order O(ω 4 log(ω)), has a non-trivial solution:

(I -ω 2 τ ε c M k m D )[E] = 0. (14)
Consider the eigenvalue problem for M k m D :

M k m D [Ψ] = ν(ω)Ψ,
where Ψ = Ψ(ω) is normalized on L 2 (D). Using the expansions

Ψ(ω) = Ψ 0 + O( 1 log ω ), ν(ω) = log(ω)ν 0 + ν 1 + O( 1 ω ),
we have

(log(ω) KD + log(γ √ ε m ) KD + K (0) D )[Ψ 0 ] = (log(ω)ν 0 + ν 1 )[Ψ 0 ] + O( 1 log ω ). Equating terms of O(log(ω)) gives KD [Ψ 0 ] = ν 0 Ψ 0 .
As KD is independent of x ∈ D, Ψ 0 must be a constant function, which we normalize on L 2 (D), i.e., Ψ 0 = ✶D . This gives

ν 0 ✶D = KD [ ✶D ] = - |D| 2π
✶D , so ν 0 = -|D|/(2π). Next, equating terms of O(1) we have

ν 1 ✶D = (log(γ √ ε m ) KD + K (0) D )[ ✶D ] = - |D| 2π log(γ √ ε m ) ✶D + K (0) D [ ✶D ],
and so, after taking the inner product with ✶D we get

ν 1 = - |D| 2π log(γ √ ε m ) + K (0) D [ ✶D ], ✶D .
This means that

ν(ω) = - |D| 2π log(γk m ) + K (0) D [ ✶D ], ✶D + O( 1 log ω ). ( 15
)
Using the expansion of ν(ω) in ( 15) we obtain from ( 13) that

1 -ω 2 s τ ε c - |D| 2π log(ω s γ√ ε m ) + K (0) D [ ✶D ], ✶D = O( ω 2 log ω ) = o(ω 2 ). (16)
B The resonance for a unit disk

Let D the unit disk. We can obtain a fully explicit expression for [START_REF] Yang | Dielectric nanoresonators for light manipulation[END_REF] in this case as the eigenvalues of the Newtonian potential have a direct relationship with the zeros of the Bessel function of order zero [START_REF] Kalmenov | A boundary condition and spectral problems for the Newton potential[END_REF]. First we note that K According to [START_REF] Kalmenov | A boundary condition and spectral problems for the Newton potential[END_REF], the eigenvalues of the Newtonian potential for the unit disk are given by

λ 0j = 1 (µ (0) j ) 2 , j = 1, 2, . . . ,
with the associated orthornormal set of eigenfunctions {e j } ∞ j=1 given by e j (r) = β j J 0 µ (0) j r , where

β j = 1 √ πJ 1 (µ (0) j ) . Note that ✶ D , e j 2 = 4π (µ (0) j ) 2 . Then we have K (0) D [✶ D ], ✶ D = ∞ j=1 ✶ D , e j 2 λ 0j = 4π ∞ j=1 λ 0j (µ (0) j ) 2 = 4π ∞ j=1 1 (µ (0) j ) 4 = π 8 ,
where we used the identity ∞ j=1 1/(µ (0) j ) 4 = 1/32 [START_REF] Watson | A treatise on the theory of Bessel functions[END_REF]. Therefore

K (0) D [ ✶D ], ✶D = 1 8
, and ( 16) can be written as

1 - ω 2 τ ε c 2 -log(γk m ) + 1 4 = o(ω 2 ). (17)
The precise dependence of ω on the contrast parameter τ can be found by writing the solution to [START_REF] Zhao | Dielectric metamaterials[END_REF] in terms of the Lambert W function [START_REF] Corless | On the Lambert W function[END_REF].

Proposition B.1. The resonance for a unit disk as the contrast τ → ∞ is given by

ω s (τ ) = - 2i τ ε c W (Φ(τ )) + o 1 τ log(τ ) , (18)
where

Φ(τ ) = - 4 τ ε c exp 2 log( √ ε m γ) - 1 2 ,
and W is the lower branch of the Lambert W function defined in the interval [-1/e, 0).

Proof. Denote by

α 0 (τ ) := τ ε c 2 , α 1 (τ ) := α 0 log( √ ε m γ) - 1 4 , α 2 (τ ) := - 2 α 0 exp 2α 1 α 0 .
Then we can write (17) as 1 + α 0 ω 2 log(ω) + α 1 ω 2 = o(ω) 2 , which leads to

-2 log(ω) = 2 + 2α 1 ω 2 α 0 ω 2 + o(1). Then 1 ω 2 = exp 2 + 2α 1 ω 2 α 0 ω 2 + o(1). Next we have -2 α 0 ω 2 exp - 2 α 0 ω 2 = - 2 α 0 exp 2α 1 α 0 + o(exp(-1/ω 2 )).
The Lambert W function is a map ze z → W (ze z ) = z [START_REF] Corless | On the Lambert W function[END_REF] and as the expression above is in this form, an application of this map leads to

-2 α 0 ω 2 = W 2 α 0 exp - 2α 1 α 0 + o(1) = W (α 2 ) + o(1).
Note that the Lambert W function is double-valued in the interval [-1/e, 0), which is the interval we need to consider when τ → ∞, and we should choose the lower branch, denoted by W -1 in the literature, to obtain a physically meaningful resonance. Now

ω 2 = - 2 α 0 W (α 2 )
+ o(ω 2 ), and so

ω = ± i √ 2 α 0 W (α 2 ) + o(ω 2 ). (19)
Noting that α 2 = O(1/τ ), as τ → ∞ we have the expansion [START_REF] Corless | On the Lambert W function[END_REF] W (α 2 ) = log(-α 2 ) -log(-log(-α 2 )) + log(-log(-α 2 )) log(-α 2 ) + . . . , which implies that W (α 2 ) = O(log(τ )). As α 0 = O(τ ) this means we have

ω + o(ω 2 ) = O 1 τ log(τ ) ,
and squaring both sides gives

ω 2 + o(ω 3 ) = O 1 τ log(τ )
.

Thus, upon taking the term with positive real part in [START_REF] Watson | A treatise on the theory of Bessel functions[END_REF] we obtain [START_REF] Corless | On the Lambert W function[END_REF].

A coarser approximation which gives a clearer qualitative indication of the dependence of the resonance frequency on the system variables is given in the following corollary.

Proposition 3 . 1 .

 31 Let d = 3. The subwavelength resonant frequency ω s is hybridized into two subwavelength resonant frequencies ω ± s approximately given by

Proposition 3 . 2 .

 32 Let d = 2 and τ large enough. Then the monopole and dipole hybridized resonances of the dimer of two identical disks D 1 and D 2 of radius δ are approximately given by

Corollary 3 . 3 .

 33 Let d = 2 and τ large enough. Then the monopole and dipole hybridized resonances of a dimer of two identical unit disks D 1 and D 2 are given by

and D 2

 2 be unit disks with D centered at the origin, D 1 centered at (-2, 0), and D 2 centered at (2, 0) with D being the geometry for the single particle problem (3) and D 1 ∪ D 2 being the geometry for the dimer problem[START_REF] Ammari | A mathematical theory of super-resolution by using a system of sub-wavelength Helmholtz resonators[END_REF].The asymptotic resonances ω s , ω m , and ω d are given by the formulas in Proposition B.1 and Corollary 3.3, with the R (0) D 2 ,D 1 [✶ D 2 ], ✶ D 1 term in the hybridized resonances computed numerically using Python's nquad routine after first putting it in polar coordinates with respect to the center of D 1 . We numerically compute reference solutions to the single particle problem and the dimer problem using boundary integral equation formulations expanded on multipole bases to obtain reference resonances ω s,ref (single particle), ω m,ref (monopole), and ω d,ref (dipole).

Figure 1 :

 1 Figure 1: The asymptotic resonances ω s (single particle), ω m (monpole) and ω d (dipole) given by the formulas in Propositions B.1 and Corollary and the corresponding resonances obtained using reference solutions.

D

  [ ✶D ], ✶D = 1/π K (0) D [✶ D ], ✶ D . Now, let J lbe the Bessel function of order l and define µ j ) = 0, j = 1, 2, . . . .

Corollary B. 2 . 2 τD 2 ,D 1 -τ ω 2 ε c N k m D 1 ,D 2 ID 2 [D 2 ,D 1 [

 222112221 As τ → ∞ it holds that ω s (τ ) = We have W (α 2 ) = log(-α 2 ) + . . .= -log(τ ) + log ε mSubstituting this expression for W (α 2 ) into (18) asserts the claim.CProof of Proposition 3.2We have|D 1 | = |D 2 | = πδ 2 . Since K k m D i = M k m D i + O(ω 2 log(ω)), R k m D i ,D j = N k m D i ,D j + O(ω 2 log(ω)),it holds thatI -τ ω 2 ε c M k m D i -τ ω 2 ε c N k m -τ ω 2 ε c M k m ✶D 2 ], ✶D 2 , η(ω) = N k m D 1 ,D 2 [ ✶D 1 ], ✶D 2 = N k m D 2 ,D 1 [ ✶D 2 ], ✶D 1 ,with these equalities holding due to the symmetry of the dimer. Furthermore, the symmetry of the dimer also means that Kkm D i ,D j [ ✶D i ] = Kk m D j [ ✶D j ].(21)Denote by ν(ω) the eigenvalues of the operators M k m D i such thatM k m D 1 [Ψ D 1 ], Ψ D 1 = ν(ω) = M k m D 2 [Ψ D 2 ], Ψ D 2 , for eigenfunctions L 2 (D i ) ∋ Ψ D i (ω) = Ψ D i ,0 + O( 1 log ω ) = ✶D i + O( 1 log ω ). From Section A we know that M k m D i [Ψ D i ], Ψ D i = ν(ω) = ν(ω) + o(1).(22)Denote by η(ω) = N k m D i ,D j [Ψ D i ], Ψ D j . Note also that η(ω) = η(ω) + o(1).Therefore, we have the following implicit equation for the hybridized resonances, 1 -ω 2 τ ε c -✶D 2 ], ✶D 1 = o(ω 2 ).

Table 1 :

 1 we give values of ω s and ω s,ref and their corresponding relative errors for τ ∈ {2 j } 7 j=3 . The real and imaginary parts of ω s and ω s,ref along with the relative error for τ ∈ {2 j } 7 j=3 .

	τ	Re(ω s,ref )	Re(ω s )	Im(ω s,ref )	Im(ω s )	Rel. err.
	2 3 2.8012e-01 2.8043e-01 1.4476e-01 1.4022e-01 1.44e-02
	2 4 1.9685e-01 1.9649e-01 8.4484e-01 8.3198e-01 6.23e-03
	2 5 1.3602e-01 1.3581e-01 4.9465e-01 4.9189e-01 2.40e-03
	2 6 9.3188e-02 9.3111e-02 2.9242e-02 2.9221e-02 8.22e-04
	2 7 6.3606e-02 6.3587e-02 1.7497e-02 1.7521e-02 4.68e-04


Nassim El Achi

Youssef Bakkour

Wissal Adhami

Julien Molina

Maël Penhoat

Nathalie Azaroual

Laetitia Chausset-Boissarie

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

Laëtitia Chausset-Boissarie

Metal-Free ATRP Catalyzed by Visible Light in Continuous Flow

Keywords: ATRP, continuous flow, visible light, Eosin Y, DOSY NMR

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

Introduction

During the last decade, much attention has been given to reversible deactivation radical polymerization (RDRP) in polymer and material chemistry as it is considered a reliable technique for the production of controlled/living polymers offering wide range of methods like nitroxide-mediated radical polymerization (NMP), reversible addition-fragmentation chain transfer (RAFT), variations of organometallic-mediated polymerizations (OMBP), and atom transfer radical polymerization (ATRP) [START_REF] Gao | Synthesis of star polymers by a combination of ATRP and the "click" co pling ethod[END_REF][START_REF] Chmielarz | Electrochemically mediated atom transfer radical polymerization (eATRP)[END_REF][START_REF] Fantin | Harnessing the interaction between surfactant and hydrophilic catalyst to control e ATRP in miniemulsion[END_REF].

ATRP has become widely employed due to its remarkable performance in the synthesis of polymers of well-defined chemical composition and complex architecture [START_REF] Matyjaszewski | Atom transfer radical polymerization[END_REF][START_REF] Ouchi | Precision control of radical polymerization via transition metal catalysis: From dormant species to designed catalysts for precision functional polymers[END_REF][START_REF] Matyjaszewski | Nanostructured functional materials prepared by atom transfer radical polymerization[END_REF][START_REF] Ouchi | Transition Metal-Catalyzed Living Radical Polymerization: Toward Perfection in Catalysis and Precision Polymer Synthesis[END_REF][START_REF] Matyjaszewski | Atom Transfer Radical Polymerization (ATRP): Current Status and Future Perspectives[END_REF][START_REF] Matyjaszewski | Macromolecular Engineering by Atom Transfer Radical Polymerization[END_REF][START_REF] Boyer | Copper-Mediated Living Radical Polymerization (Atom Transfer Radical Polymerization and Copper(0) Mediated Polymerization): From Fundamentals to Bioapplications[END_REF][START_REF] Ribelli | Atom transfer radical polymerization: billion times more active catalysts and new initiation systems[END_REF].

In an aim to develop more sustainable and visible light mediated ATRP [START_REF] Corrigan | Seeing the light: Advancing materials chemistry through photopolymerization[END_REF], researchers have developed photoredox catalytic systems using catalysts like copper (Cu) (Tasdelen This is a provisional file, not the final typeset article et al., 2011;[START_REF] Konkolewicz | Visible Light and Sunlight Photoinduced ATRP with ppm of Cu Catalyst[END_REF][START_REF] Mosnacek | Photochemically Mediated Atom Transfer Radical Polymerization of Methyl Methacrylate Using ppm Amounts of Catalyst[END_REF][START_REF] Anastasaki | Copper(II)/Tertiary Amine Synergy in Photoinduced Living Radical Polymerization: Accelerated Synthesis of omega-Functional and alpha,omega-Heterofunctional Poly(acrylates)[END_REF][START_REF] Ribelli | How are Radicals (Re)Generated in Photochemical ATRP?[END_REF][START_REF] Yang | Photocatalyzed Cu-Based ATRP Involving an Oxidative Quenching Mechanism under Visible Light[END_REF], cyclometallated Iridium (Ir) [START_REF] Fors | Control of a Living Radical Polymerization of Methacrylates by Light[END_REF][START_REF] Ma | Visible-Light-Induced Controlled Polymerization of Hydrophilic Monomers with Ir(ppy)(3) as a Photoredox Catalyst in Anisole[END_REF]Treat et al., 2014a;[START_REF] Xu | Photoredox catalyst-mediated atom transfer radical addition for polymer functionalization under visible light[END_REF][START_REF] Ma | Visible Light-Controlled Radical Polymerization of Propargyl Methacrylate Activated by a Photoredox Catalyst fac-Ir(ppy)(3)[END_REF] or Ruthenium (Ru) [START_REF] Zhang | Free Radical Polymerization Initiated and Controlled by Visible Light Photocatalysis at Ambient Temperature[END_REF][START_REF] Alfredo | Light-Driven Living/Controlled Radical Polymerization of Hydrophobic Monomers Catalyzed by Ruthenium(II) Metalacycles[END_REF][START_REF] Priyadarshani | Recoverable Reusable Polyisobutylene (PIB)-Bound Ruthenium Bipyridine (Ru(PIB-bpy)(3)Cl-2) Photoredox Polymerization Catalysts[END_REF].

Despite their effectiveness, metal contamination limits their interest for biomedical and electronic applications. significant efforts has therefore been dedicated to develop photoredox metal free ATRP [START_REF] Chen | Light-Controlled Radical Polymerization: Mechanisms, Methods, and Applications[END_REF]Pan et al., 2016b;[START_REF] Hu | Poly (vinylidene fluoride-co-chlorotrifluoroethylene) Modification via Organocatalyzed Atom Transfer Radical Polymerization[END_REF][START_REF] Ryan | Impact of Light Intensity on Control in Photoinduced Organocatalyzed Atom Transfer Radical Polymerization[END_REF][START_REF] Discekici | Evolution and future directions of metal-free atom transfer radical polymerization[END_REF].

Pioneering work by Hawker group demonstrated that 10-phenylphenothiazine (PTH) can be an effective organic photoredox catalyst for the polymerization of methacrylates under ultraviolet irradiation (Treat et al., 2014b). Subsequently, the group of Matyjaszewski has extended the methodology to polyacrylonitrile synthesis [START_REF] Pan | Photoinduced Metal-Free Atom Transfer Radical Polymerization of Acrylonitrile[END_REF].

More importantly, a photoredox ATRP mediated by visible light using organic photoredox catalyst like perylene and fluorescein has been reported with the latter requiring prolonged irradiation times [START_REF] Miyake | Perylene as an organic photocatalyst for the radical polymerization of functionalized vinyl monomers through oxidative quenching with alkyl bromides and visible light[END_REF]) [START_REF] Liu | Metal-free photoinduced electron transfer-atom transfer radical polymerization (PET-ATRP) via a visible light organic photocatalyst[END_REF]. However, some drawbacks are still present like low initiator efficiency that leads to polymers of elevated masses and dispersity (Đ).

To address this challenge {Discekici, 2018 #71}, the group of Miyake introduced diphenyl dihydrophenazine derivatives as metal free photocatalysts [START_REF] Theriot | Organocatalyzed atom transfer radical polymerization driven by visible light[END_REF][START_REF] Ryan | Impact of Light Intensity on Control in Photoinduced Organocatalyzed Atom Transfer Radical Polymerization[END_REF] and recently dimethyl dihydroacridines {Buss, 2020 #79}. Yagci group also recently reported a metalfree photoinduced ATRP using Eosin Y (Figure 1) [START_REF] Kutahya | LED and visible light-induced metal free ATRP using reducible dyes in the presence of amines[END_REF][START_REF] Yilmaz | Photoinduced metal-free atom transfer radical polymerizations: state-of-the-art, mechanistic aspects and applications[END_REF]. Their work gave promising results but with some challenges regarding dispersity and reaction time.

To overcome these issues, continuous photoflow reactor can be used since the short optical lengths of flow microreactors can significantly improve sample irradiation and heat and mass transfer that allow the use of more concentrated photoinitiators lliott et al , Garlets et al , S et al , a bié et al., 2016). Consequently, flow chemistry has recently been used in polymer synthesis since it provides improved initiator efficiency and better-controlled polymers of narrow dispersity compared to classical batch processes [START_REF] Tonhauser | Microflow Technology in Polymer Synthesis[END_REF][START_REF] Chatani | The power of light in polymer science: photochemical processes to manipulate polymer formation, structure, and properties[END_REF][START_REF] Myers | Flow Chemistry Meets Advanced Functional Materials[END_REF][START_REF] Reis | Polymerizations in Continuous Flow: Recent Advances in the Synthesis of Diverse Polymeric Materials[END_REF]. However, this field is not widely studied with only few articles addressing ATRP in flow [START_REF] Wenn | Photo-induced copper-mediated polymerization of methyl acrylate in continuous flow reactors[END_REF][START_REF] Melker | Continuous flow synthesis of poly(methyl methacrylate) via a light-mediated controlled radical polymerization[END_REF][START_REF] Hu | Continuous flow copper-mediated reversible deactivation radical polymerizations[END_REF][START_REF] Corrigan | Controlling Molecular Weight Distributions through Photoinduced Flow Polymerization[END_REF][START_REF] Ramakers | Organocatalyzed Photo-Atom Transfer Radical Polymerization of Methacrylic Acid in Continuous Flow and Surface Grafting[END_REF][START_REF] Rubens | Visible light-induced iniferter polymerization of methacrylates enhanced by continuous flow[END_REF][START_REF] Zhang | Microfluidically mediated atom-transfer radical polymerization[END_REF]; including those using noncommercially available photoredox catalysts [START_REF] Ramsey | Photoinduced Organocatalyzed Atom Transfer Radical Polymerization Using Continuous Flow[END_REF][START_REF] Ryan | Impact of Light Intensity on Control in Photoinduced Organocatalyzed Atom Transfer Radical Polymerization[END_REF][START_REF] Buss | Photoinduced controlled radical polymerizations performed in flow: methods, products, and opportunities[END_REF].

Since our research group focuses on performing wide range of photocatalytic reactions in flow, we herein present the use of Eosin Y as a photocatalyst for ATRP in flow using visible light. The study falls in the category of "proof of concept" where it ai s to determine whether using this cheap and commercially available photocatalyst for ATRP in flow would provide better results than those obtained in batch by Yagci group [START_REF] Kutahya | LED and visible light-induced metal free ATRP using reducible dyes in the presence of amines[END_REF][START_REF] Yilmaz | Photoinduced metal-free atom transfer radical polymerizations: state-of-the-art, mechanistic aspects and applications[END_REF]; and therefore can be used for other reversible-deactivation radical polymerizations. Anhydrous N,N-diisopropylethylamine (i-Pr 2 N t, ≥ 99%) was further distilled over KOH and stored in dark under argon before usage. Methyl methacrylate (99%) was passed over alumina to remove the hydroquinone stabilizer immediately prior to use.

Materials

Chemicals

Flow System

The lab designed microreactor is composed of FEP (fluorinated ethylene propylene) tubing (i.d. 800 μ , length , vol e ≈ L, Cluzeau Info Labo (C.I.L.), Sainte-Foy-La-Grande, France) (Figure S1). Two of these reactors were prepared to fit the UV and the Visible LED irradiation systems. For UV irradiation the tubing was fitted on a metallic grid to allow heat evacuation.

Syringe pump

The different flow rates of the reactions performed were regulated using a Harvard Apparatus (Holliston, MA, USA) PHD ULTRA XF syringe pump fitted with 8 mL stainless steel syringes (THREAD 1/4 -28 inch, Ø = 1/16 inch, PC5 702267, Harvard Apparatus).

LED Systems

Bl e λ = 5 n ) & green λ = 53 n ) high power spots 5 W electrical power, 5 l en, W.cm -2 ) LEDs from Bridgelux (Liver-more, CA, USA) were used for the photoinduced ATRP catalyzed by Ru(bpy) 3 Cl 2 and osin Y respectively O nic re® A 73 L Ds of λ = 365 n and irradiance up to 3 W.cm -2 purchased from Lumen Dynamics (Mississauga, Canada) were used for the photoinduced ATRP catalyzed by CuBr 2 .

Methods

General procedure of ATRP in batch

A Schlenk tube charged with DMF (2 mL, 50% v/v vs ono er) and photocatalyst μ ol) (for CuBr 2 , a ligand (TPMA or PMDETA) was added) was sealed with rubber septum and purged with nitrogen during 20 min. Methyl methacrylate (2 mL, 18.8 mmol), that was freshly passed over alumina, EBiB 7 μL, ol) and freshly distilled i-Pr 2 N t 7 μL, 1 mmol) were then added under nitrogen. The reaction mixture was irradiated with LEDs. UV LEDs (200 mW.cm -2 , 365 nm) were used for CuBr 2 /Ligand catalytic system, blue LEDs (50 W, 4500 lumens, 450 nm) for Ru(bpy) 3 Cl 2 and green LEDs (50 W, 4500 lumens, 530 nm) for Eosin Y. An aliquot was analyzed by 1 H NMR to determine the % of polymerization and by DOSY NMR to determine its M w . The formed polymer was then precipitated in methanol, filtered and dried under vacuum overnight. A solution of 6 mg/mL in THF was prepared from the dry polymer to be analyzed by GPC.

General procedure for ATRP in flow

ATRP using the three different catalytic systems with the same composition were performed under flow conditions (Figure 2, Figure S3). The degassed reaction mixture was then transferred into a lab designed F P t bing reactor i d 8 μ ) that was ill inated by L Ds of specific wavelengths depending on the catalytic system (Figure S1).

ATRP using Eosin Y catalyst illuminated by green LEDs in flow

This is a provisional file, not the final typeset article Using the same composition as that in the batch conditions, [MMA]: [initiator]: [Eosin Y]: [i-Pr 2 NEt] = 200:1:0.02:10 and MMA: DMF= 1:1 (v/v), 4 mL of the reaction mixture was injected within the lab designed icrofl idic reactor F P t bing, i d 8 μ ) that was placed in direct contact with the green LEDs. The irradiation time was varied depending on the flow rate that was adjusted by the syringe pump. For each of the two initiators, EBPA and EBiB, 6 points that correspond to six different irradiation times were performed. For each point the % of conversion was determined by 1 H NMR and the polymers were precipitated, filtered, dried and stored in THF for further analysis by GPC. Note that working under oxygen free conditions was insured, even during the transfer of the reaction mixture from the Schlenk tube to the syringe. The variation of the % of conversion of each polymer vs the irradiation time was used to plot the kinetic curves of ATRP in flow.

Polymers characterization

GPC analysis

Size exclusion chromatography (SEC) was performed at room temperature using a Viscotek GPC max system equipped with a Viscotek guard column (10 × 4.6 mm) and two Viscotek columns LT 5000L mixed medium (300 × 7.8 mm fitted with a Viscotek VE 3580 refractometric detector and a Viscotek VE 3210 UV/Vis detector. THF was used as solvent with a flow rate of 1 mL.min -1 . All molecular weights (M n ) and molecular weight distributions (dispersity, M w /M n , Đ) were deter ined by calibration to known, standard poly (methyl methacrylate) samples purchased from Polymer Laboratories (Church Stretton, United Kingdom).

NMR analysis

Proton and carbon magnetic resonance spectra ( 1 H NMR and 13 C NMR) were recorded on a Bruker AVANCE 300 spectrometer ( 1 H 300 MHz and 13 C 75 MHz) using tetramethylsilane (TMS) as the internal standard he ical shifts, δ, are given in pp and co pling constants, J, in Hz. 1 H NMR data are reported as follows: chemical shift, multiplicity (s = singlet, d =doublet, t = triplet, dd = doublet of doublets, dt = doublet of triplets, td = triplet of doublets, m = multiplet, brs = broad singlet), coupling constants and integration (Figure S4).

DOSY analysis

DOSY experiments were performed on a Bruker AVANCE 500 spectrometer equipped with an ATMA TXI probe with a z-axis gradient coil. All experiments were run without spinning to avoid convection at temperature of 295 K. Maximum gradient strength was 5.35 G.mm -1 and the gradient strengths were varied from 1 to 35 G.cm -1 . The standard Bruker pulse program, dstebpgp3s, employing double stimulated echo sequence and 3 spoil gradients was utilized. Bipolar gradients were used with a total duration of 4 ms. Gradient recovery delay was 100 µs, diffusion time was 200 ms and the number of gradient steps was set to be 32. Diffusion coefficients were calculated from T1/T2 analysis module of Topspin 2.1.

Similar to GPC, 2D-NMR is reported to be used for the determination of the weight average molecular mass M w of the polymer. We run DOSY NMR for PMMA standards of known M w at 20 °C in CDCl 3 instead of benzene-d 6 according to the protocol described by Li and coworkers [START_REF] Alfredo | Light-Driven Living/Controlled Radical Polymerization of Hydrophobic Monomers Catalyzed by Ruthenium(II) Metalacycles[END_REF] Each standard gave a value of diffusion coefficient (D) in m 2 .s -1 whose logarithmic value was plotted as function of the logarithmic value of the corresponding M w . The calibration curve represented in Figure 3 was obtained. The plot has an excellent linearity between log D vs log M w with R 2 = 0.999 showing the efficiency of the used protocol. Seven samples of PMMA polymers chosen randomly whose M w values were determined by GPC were analyzed by DOSY to determine their diffusion coefficients using the same protocol of that of the standards (CDCl 3 , 20°C). Knowing the value of (D) and by using the equation of the calibration curve log D = -0.4656×log M w -7.9116, the M w values of synthesized PMMA were calculated.

Preparation of PMMA-Br macroinitiator

In a Schlenk tube, BPA 7 μL, ol), i-Pr 2 N t 7 μL, ol) and osin Y g, μ ol) were added to a sol tion of MMA in DMF L, : v/v) The reaction mixture was degassed by three freeze-pump-thaw cycles and pumped through the continuous flow photo-microphotoreactor using a syringe pump connected to the flow reactor via a syringe. The solution was pumped through the flow syste , at a flow rate of 3 μL.min -1 (corresponding to 1 hour of residence time) and irradiated with green LEDs (50 W, 4500 Lumens). The polymer was purified by precipitation in methanol. The resulting macroinitiator was dried overnight, and characterized by GPC.

Chain extension of PMMA-Br macroinitiator with styrene (St) and Butyl acrylate (BA)

PMMA macroinitiator (0.24 g, 0.02 mmol), i-Pr 2 N t 35 μL, ol) and Eosin Y (0.3 mg, 0.43 μ ol) were added to a sol tion of styrene 5 μL, ol) in 5 L of DMF (Figure 4). The reaction mixture was degassed by three freeze-pump-thaw cycles and pumped through the continuous flow photo-microphotoreactor using a syringe pump connected to the flow reactor via a syringe. The sol tion was p ped thro gh the flow syste , at a flow rate of μL.min -1 (corresponding to 3 hour of residence time) and irradiated with green LEDs (50 W, 4500 Lumens).The formed copolymer was precipitated, filtered and dried overnight before its analysis by 1 H NMR and GPC.

Si ilarly, copoly erization with b tyl acrylate 57 μL, ol) was perfor ed sing the sa e procedure mentioned above. The formed polymer was then filtered, dried, and analyzed by GPC

Polymerization of MMA with dark periods

In a Schlenk tube, BPA 3 μL, ol), i-Pr 2 N t 3 μL, ol) and osin Y 8 g, μ ol) were added to a sol tion of MMA in DMF 8 L, : v/v) The reaction mixture was degassed by three freeze-pump-thaw cycles and pumped through the continuous flow photo-microphotoreactor using a syringe pump connected to the flow reactor via a syringe.

The reaction mixture was irradiated for 3 disrupted hours; after each period of 1 h of irradiation, the sol tion was kept in dark nder argon for h For every cycle h "L Ds on", h "L Ds off") two sa ples of L each were taken for analysis at the beginning and the end of the "L Ds off" period D ring the "L Ds on" period, the reaction mixture was pumped through the flow reactor at a flow rate of 3 μL.min -1 . Six samples that were taken periodically were precipitated in methanol, filtered and dried overnight for analysis by GPC and NMR.

4

Results and Discussion

Copper catalyzed ATRP using UV light

First, we optimized the flow protocol. The homemade continuous flow photo-microreactor was made of Fluorinated Ethylene Propylene (FEP) tubing (Ø = 800 µm , L = 1.2 m) that can be easily replaced in case of clogging. FEP is transparent to UV so first we performed a classical ATRP in flow catalyzed by CuBr 2 under UV irradiation. Compared to the previously described flow experiment, This is a provisional file, not the final typeset article UV-LEDs (365 nm) with a power of 200 mW.cm -2 were used instead of a medium pressure UV-lamp, and DMSO was replaced by DMF. We also assessed the ligands PMEDTA and TPMA which are commonly used in the literature for Cu catalysed ATRP. Best results were obtained with TPMA (Table S2, entries 1-8) leading to 60% conversion and low dispersity (Đ = 1.27) compared to the previous results in flow (Đ = 1.15-1.25).

Visible light ATRP

We then moved to metallic visible light photoredox catalysis using the Ru(bpy) 3 Cl 2 , i-Pr 2 NEt system in DMF irradiated by blue LEDs in flow. The modest dispersity (Table 1, entry 1; Đ  2) obtained is in agreement with the results obtained in batch [START_REF] Zhang | Free Radical Polymerization Initiated and Controlled by Visible Light Photocatalysis at Ambient Temperature[END_REF]. We then decided to check the activity of Eosin Y, a derivative of fluorescein which is widely used as visible light photoredox catalyst for various organic reactions, [START_REF] Van Bergen | Chemistry of dihydropyridines. 9. Hydride transfer from 1, 4-dihydropyridines to sp3-hybridized carbon in sulfonium salts and activated halides. Studies with NAD (P) H models[END_REF]Hari et al., 2012;Cantillo et al., 2014;[START_REF] Talla | Metal-Free Photocatalytic Aerobic Oxidation of Thiols to Disulfides in Batch and Continuous-Flow[END_REF] to perform the polymerization of MMA under flow conditions. Eosin Y has its maximum absorption at 539 nm in the visible region of the spectrum and with a high absorption coefficient (ε = 60800 M -1 cm -1 ) (Figure S6). The irradiation was afforded by green LEDs with a power of 20 mW.cm -2 .

We originally performed polymerization of MMA in DMF using EBiB as initiator and Eosin Y as photocatalyst in both batch (Table 1, entry 2) and flow (Table 1, entry 3). Interestingly, the change from batch to flow system was enough to improve remarkably both the rate of the reaction and its control. Although 360 min were required in batch to reach a 50% conversion, only 216 min were needed in flow for 70% conversion. Moreover, the dispersity decreased from 1.64 in batch to 1.58 in flow. The results obtained herein in flow are also better than those reported in batch by Yagci group where 120 min of green LEDs irradiation of the reaction mixture gave 28 % conversion but with a dispersity of 1.85 [START_REF] Kutahya | LED and visible light-induced metal free ATRP using reducible dyes in the presence of amines[END_REF].

Following these results, blank experiments were performed where one of the components (catalyst, initiator, i-Pr 2 NEt and light) was removed (Table S3, entry 1). In all of these cases no PMMA was detected showing that all of these components are essential for the polymerization process. Moreover, since Eosin Y has a high extinction coefficient, a large amount of the catalyst in flow (Table 1, entry 4) will lead to a similar situation to that in batch where the light will only illuminate a small portion of the reactor (Figure S2, Table S1), leading to a decrease in the conversion to 43%. Similarly, decreasing the quantity of Eosin Y will lead to a decrease in the conversion to 50% despite having a better light penetration in the system as the quantity of the catalyst is not efficient to activate the initiators. (Table 1, entry 5). Using EBPA as initiator, which has been previously reported to have a higher activation rate k act compared to EBiB due to the radical stability enforced by a phenyl group [START_REF] Braunecker | Controlled/living radical polymerization: features, developments, and perspectives[END_REF], the rate of the polymerization increased remarkably leading to more than 90% of conversion after only 180 min of irradiation (Table 1, entry 6). The polydispersity was also improved to 1.42. To further stabilize the intermediate radical in order to increase the polymerization rate, we used (p-OMe)EBPA as initiator; however, the results were quite disappointing (Table 2, entry 8). The tacticity of the obtained polymer was roughly identical whatever the used ATRP conditions (Table S4, Figure S16).

The best conditions obtained for the ATRP initiated by EBPA (Table 1, entry 6) were further investigated in details (Table 2, entries 1-7). The polymerization follows a first order kinetics during all the course of the reaction with an equation of y= 0.123x for ln ([MMA] 0 /[MMA] t ) versus irradiation time (min). The rate constant is 0.123 min -1 (Figure 5, upper panel) suggesting that the concentration of the propagating radicals is almost constant throughout the polymerization. Molecular weights measured by GPC follow the theoretical values starting from 37% and up to 90% of conversion suggesting a complete initiation (Figure 5, lower panel; Table 2, entries 1-7; Figures S9-S14). Moderate values of Đ (1.36-1.49) indicate relatively slow deactivation though still in the range of controlled polymerization (< 1.5).

However, note that the first point (Table 2, entry 1; Figure S9) does not follow the theoretical M n so that it deviates upwards in (Figure 5, Lower panel). The same results were also observed when using EBiB as an initiator (Figure S8) which suggests that Eosin Y undergoes an induction period before it can enter the catalytic cycle [START_REF] Cornils | Catalysis from A to Z: a concise encyclopedia[END_REF]. As a result, this point was excluded from the kinetic analysis (Figure 5, upper panel).

Livingness of the formed polymer

Controlled "on-off" light switching reg lation of poly erization was st died by collecting eno gh polymer after a given period of irradiation and re-injecting it for the next cycle following a 1 hour on / 1 hour off illumination duration. During the LEDs off periods, the polymerization is paused by having dormant alkyl bromides that are protected from any side radical reaction leading to a stagnant conversion and molecular weight; however, these formed polymers are available for reactivation upon re-exposure to light (Figure 6). The polymerization ceased when the light was turned off and started again in response to the irradiation. The control over the formation and termination of active species can be performed by using a simple on-off operation of light leading to M n (experimental) = M n (theoretical).

The "livingness" of the osin Y photoind ced ATRP and the ter ination of the for ed poly ers by an active bromide ion were further demonstrated by a copolymerization reaction of styrene with a PMMA-Br macroinitiator (Figure 4). The PMMA-Br macroinitiator was firstly synthesized by photoinduced ATRP in flow to get PMMA-Br (Table 2, entry 7, M n = 13100, Đ = 1.42) and then used as a macroinitiator for Eosin Y catalyzed ATRP of styrene (St) and butyl acrylate (BA). The GPC traces of the macroinitiator and the corresponding copolymers are displayed in Figure 7 and the NMR spectrum of PMMA-co-PSt is represented in Figure S15. The results of PMMA-co-PBA (Figure 7, red trace) were not very convenient as the PDI value was 2.1, which signifies a loss in control, that can be attributed with the high reactivity of butyl acrylate that requires strictly anhydrous and oxygen free conditions [START_REF] Roos | Copolymerization of n-butyl acrylate with methyl methacrylate and PMMA macromonomers: comparison of reactivity ratios in conventional and atom transfer radical copolymerization[END_REF]. However, the clear shift of M n of the PMMA-co-PSt to a higher molecular weight while still having a good PDI (Figure 7, blue trace) and the presence of peaks that correspond to PSt and PMMA in the NMR spectrum indicate an effective copolymerization by re-initiation. These results of PSt copolymerization are in accordance with the literature where clear shifts to lower retention volumes were also reported by performing PMMA chain extension and copolymerization with PSt using the same condition but in batch [START_REF] Kutahya | LED and visible light-induced metal free ATRP using reducible dyes in the presence of amines[END_REF][START_REF] Yilmaz | Photoinduced metal-free atom transfer radical polymerizations: state-of-the-art, mechanistic aspects and applications[END_REF].

Mechanism of Eosin Y catalyzed photoinduced ATRP

The suggested mechanism of the Eosin Y photoinduced electron transfer (PET)-ATRP is represented in Figure 8. All the standard potential are in ACN versus SCE Upon irradiation with green LEDs, Eosin Y affords the excited state EY* which has a high oxidation potential (E 0 (EY/EY*) = 1.89 V) [START_REF] Hari | Synthetic applications of eosin Y in photoredox catalysis[END_REF]. In the activation step, the electron donor, i-Pr 2 NEt, reductively quenches EY* by a single electron transfer to form a radical anion EY •-and an amine radical cation i-Pr 2 N •+ Et intermediate (E°(i-Pr 2 NEt/i-Pr 2 N •+ Et)  1.0 V). The latter rearranges almost at diffusion rate to the Ccentered radical i-PrEtNC • (CH 3 ) 2 or i-Pr 2 NCH • CH 3 (Wayner et al., 1986;[START_REF] Romero | Organic Photoredox Catalysis[END_REF]. The radical anion EY •-(E 0 (EY •-/EY) = -1.06 V), or the reductive C-centered radical (E 0 (Ccentered radical/iminium) = -1.12 V), then transfers an electron to the alkyl bromide EBiB or EBPA This is a provisional file, not the final typeset article (E 0 (RX/R • +Br -)= -0.42 V and -0.20 V respectively) [START_REF] Lin | Ab initio evaluation of the thermodynamic and electrochemical properties of alkyl halides and radicals and their mechanistic implications for atom transfer radical polymerization[END_REF]. The EBiB or EBPA radical anion cleaves generating an alkyl radical that adds to the monomer inducing propagation. In the reductive propagation step, the excited Eosin Y recaptures an electron by oxidizing either the bromide ion Br -into bromine radical Br • (E 0 (Br -/Br • ) = 1.75 V) [START_REF] Isse | Estimation of standard reduction potentials of halogen atoms and alkyl halides[END_REF], or the complex propagating radical, bromide ion by a concerted electron transfer (Pan et al., 2016a). The bromine radical Br • deactivates the propagation and forms the dormant polymer. The formed radical anion EY •-is reduced to back to EY by providing an electron to the dormant polymer, which has the same structure as a tertiary -bromoester, that reacts with the monomer. The consumption of the sacrificial amine is to compensate the irreversible terminations of the propagating cycle.

In this mechanism, the very unlikely oxidation of Br -by the radical cation i-Pr 2 N •+ Et, [START_REF] Liu | Metal-free photoinduced electron transfer-atom transfer radical polymerization (PET-ATRP) via a visible light organic photocatalyst[END_REF] or the chain radical oxidation to a cation [START_REF] Xu | Photoredox catalyst-mediated atom transfer radical addition for polymer functionalization under visible light[END_REF] are avoided; only a catalytic amount of the sacrificial amine is required to initiate the reaction. Furthermore, all of the steps have favorable redox potentials. This mechanism shows that Eosin Y has a well-established activation deactivation processes that result in the control of the molecular weights and dispersities of the formed polymers.

Dosy vs GPC

Based on the calibration curve obtained using DOSY NMR (Figure 3), M w values of selected polymers were obtained and presented in Table 3 along with the corresponding M w values obtained by GPC. Interestingly, the two methods almost gave the same M w values in all of the entries. The % of difference between both values varies between 0.08 % in entry 1 to a maximum value of only 5.5 % in entry 7. This shows that DOSY NMR can be a good analytical method for the characterization of polymers. NMR spectroscopy can provide the full characterization of the polymer: % conversion, tacticity, M n by 1 H NMR and M w by DOSY NMR rendering it complementary to GPC.

Conclusion

In conclusion, we have demonstrated that ATRP of MMA using Eosin Y as photocatalyst in a flow reactor illuminated by green LEDs is very efficient affording 90% of conversion in 3 h. Perfect first order kinetics, full initiation or dormant polymer activation, moderate dispersity and masses in agreement with the theoretical values were obtained showing the great mechanistic and synthetic potentials of our conditions. The main reason of this improvement is the homogeneous illumination in flow reactors and hence the shorter reaction time compared to batch a bié et al., 2016). We also have shown that NMR spectroscopy (1D and 2D) could be considered as a reliable tool for characterization of polymers. This work co bines 3 essential co ponents towards a "greener" chemistry: miniaturization, renewable energy and metal free catalysis that can be used later for scaling up. This work is the first in a series of reactions that our lab is assessing that includes: extending the current continuous-flow system to further improve the irradiation efficiency, synthesizing more complex polymers and using this system for other types of RDRPs.

8

This is a provisional file, not the final typeset article M w by DOSY [a] M w by GPC. % diff [b] 1

Figure 1 .Figure 2 .Figure 3 .ÐFigure 5 .Figure 6 .

 12356 Figure 1. Structure of Eosin Y

  Equation of the PMMA standard calibration curve is log D = -0.4656 log M w -7.9116 was used to determine M w using the values of D provided by DOSY NMR. [b] Calculated from M w (GPC) and M

		1.10	24920	24900	0.08
	2	1.20	20740	20270	2.32
	3	1.25	19100	19150	-0.26
	4	1.26	18710	18040	3.70
	5	1.28	18020	17980	0.22
	6	1.30	17550	17290	1.50
	7	1.96	7190	6810	5.58
	[a]

w (DOSY).

Acknowledgments

We would like to thank Dr. Benjamin Nottelet and the team at the Institut des Biomolécules Max Mousseron (IBMM) at Université de Monpellier for providing access to the GPC at their facilities.

Funding

This work was partially funded by LASeR Association-Lebanon in a form of PhD Scholarship to Nassim El Achi. The NMR and Mass Spectrometry facilities used in this study were funded by the European Community (ERDF), Région Haut de France (France), the CNRS, and the Université de Lille, Faculty of Sciences and Technologies.

Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. Author Contributions All authors listed have made a substantial, direct and intellectual contribution to the work, and approved it for publication.

Supplementary Material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles [START_REF] Sharma | Diastereochemical diversity of imidazoline scaffolds via substrate controlled TMSCl mediated cycloaddition of azlactones[END_REF].