J-F Mercier

C Mietka
email: colin.mietka@gmail.com

F Millot

V Pagneux
email: cent.pagneux@univ-lemans.fr

ACOUSTIC PROPAGATION IN A VORTICAL HOMENTROPIC FLOW *

Keywords: Acoustics, Hydrodynamics, Goldstein's Equations, Finite Element Method, Discontinuous Galerkin Element Method, Perfectly Matched Layers AMS subject classifications. 65J10, 65N30, 65Z05, 35J50, 35Q35, 35Q31

This paper is devoted to the theoretical and the numerical studies of the radiation of an acoustic source in a general homentropic flow. As a linearized model, we consider Goldstein's Equations, which extend the usual potential model to vortical flows. The equivalence between Linearized Euler's Equations with general source terms and Goldstein's Equations is established, and the relations between unknowns, in each model, are analysed. A closed-form relation between the hydrodynamic phenomena and the acoustics is derived. Finally, numerical results are presented and the relevance of using Goldstein's Equations compared to the potential model is illustrated.

1. Introduction. Aeroacoustics consists in determining the acoustic perturbations propagating in an imposed flow. It is mostly the need of noise reduction in aeronautics which creates an increasing interest in the field of aeroacoustics, the main application concerning the sound propagation inside the radiation and outside a plane engine. But let us cite also the need to reduce the sound emitted by exhaust pipes in the car industry or by ventilation ducts in the domestic industry. In the present paper, we focus on the propagation of the sound created by a known source in a given flow. In particular we do not address the mechanisms responsible of noise generation [START_REF] Ting | On vortical flow and sound generation[END_REF][START_REF] Ting | Vortical flow outside a sphere and sound generation[END_REF], involving non-linear models. We consider linearized equations and we focus on the time harmonic regime.

The most studied case of acoustic propagation in a flow is the case of a curl free carying flow. This case is simpler because for a uniform flow [START_REF] Bonnet-Ben Dhia | Acoustic diffraction by a plate in a uniform flow[END_REF] or even for a flow v 0 potential and homentropic, the acoustic perturbations are also potential and the velocity potential ϕ satisfies the convected Helmholtz equation [START_REF] Blokhintzev | The propagation of sound in an inhomogeneous and moving medium I[END_REF][START_REF] Pierce | Wave equation for sound in fluids with unsteady inhomogeneous flow[END_REF]:

(1) D Dt 1 c 2 0 Dϕ Dt = 1 ρ 0 div(ρ 0 ∇ϕ),
where ρ 0 and c 0 are respectively the density and the sound velocity of the flow and where

(2) D Dt = ∂ ∂t + v 0 • ∇,
is the convective derivative. This model is well adapted to a classical Finite Element discretization and is widely used in some industrial applications [START_REF] Coyette | Manuel théorique ACTRAN, Free Field Technologies[END_REF][START_REF] Duprey | Etude mathématique et numérique de la propagation acoustique d'un turboréacteur[END_REF] or in the analysis of the influence of liners on the acoustic propagation [START_REF] Eversman | The Boundary condition at an Impedance Wall in a Non-Uniform Duct with Potential Mean Flow[END_REF][START_REF] Gabard | A full discrete dispersion analysis of time-domain simulations of acoustic liners with flow[END_REF][START_REF] Rienstra | A numerical comparison between multiple-scales and finiteelement solution for sound propagation in lined flow ducts[END_REF].

For a more complex flow, a scalar description of acoustic perturbations is no longer possible because of the coupling between acoustic and hydrodynamic phenomena. Such coupling requires more sophisticated vector models. For the time-domain problem, several methods have been developed to solve the Linearized Euler Equations, but the treatment of the artificial boundaries still raises open questions. On the other hand, the time-harmonic problem in an undounded domain has not been considered for a general flow, excepting thanks to the Galbrun equation [START_REF] Bonnet-Ben Dhia | Time-Harmonic Acoustic Scattering in a Complex Flow: A Full Coupling Between Acoustics and Hydrodynamics[END_REF]. However, the Galbrun equation [START_REF] Galbrun | Propagation d'une onde sonore dans l'atmosphère terrestre et théorie des zones de silence[END_REF][START_REF] Gabard | Stability and accuracy of finite element methods for flow acoustics: II. Two-dimensional effects[END_REF][START_REF] Treyssede | A mixed finite element method for acoustic wave propagation in moving fluids based on an Eulerian-Lagrangian description[END_REF] revealed not to be adapted to 3D configurations. To go beyond this limitation, in this paper we consider the Goldstein equations [START_REF] Goldstein | Unsteady vortical and entropic distortion of potential flows round arbitrary obstacles[END_REF][START_REF] Bergliaffa | Wave Equation for Sound in Fluids with Vorticity[END_REF].

They have been widely used to model the development of perturbations in a swirling flow [START_REF] Golubev | Sound propagation in an annular duct with mean potential swirling flow[END_REF][START_REF] Golubev | Acoustic-vorticity waves in swirling flows[END_REF][START_REF] Cooper | Propagation of unsteady disturbances in a slowly varying duct with mean swirling flow[END_REF][START_REF] Atassi | Computing the sound power in non-uniform flow[END_REF][START_REF] Cooper | Effect of mean entropy on unsteady disturbance propagation in a slowly varying duct with mean swirling flow[END_REF][START_REF] Tam | The wave modes in ducted swirling flows[END_REF][START_REF] Heaton | Algebraic and exponential instability of inviscid swirling flow[END_REF] and we will show that they are well adapted for aeroacoustics applications.

The main advantage of using Goldstein's equations is that they are simpler than alternative models, among which the Linearized Euler equations [START_REF] Prax | Control of the vorticity mode in the linearized Euler equations for hybrid aeroacoustic prediction[END_REF][START_REF] Seo | Linearized perturbed compressible equations for low Mach number aeroacoustics[END_REF][START_REF] Nataf | A new approach to perfectly matched layers for the linearized Euler system[END_REF][START_REF] Munz | Linearized acoustic perturbation equations for low Mach number flow with variable density and temperature[END_REF], the Galbrun equation or the Möhring equations [START_REF] Möhring | A well proposed acoustic analogy based on a moving acoustic medium, Proceedings 1st Aeroacoustic Workshop (in connection with the german research project SWING)[END_REF][START_REF] Legendre | Sound Propagation in a sheared flow based on fluctuating total enthalpy as generalized acoustic variable[END_REF], because they are mainly scalar and close to a usual wave equation. Indeed, they can be seen as a perturbation of the simple convected Helmholtz equation [START_REF] Ting | On vortical flow and sound generation[END_REF] and they even exactly reduce to this equation in the potential areas of the carrier flow, this last point being particularly important for 3D applications.

In this paper, we study theoretically and numerically the radiation of general sources in a general flow, using the Goldstein equations. A general flow means a flow at least not potential, but also not only a parallel shear flow, for which Pridmore-Brown equation [START_REF] Brown | Sound propagation in a fluid flowing through an attenuating duct[END_REF] is more commonly used. In addition to the velocity potential, Goldstein's equations involve a supplementary unknown, namely the hydrodynamic vector field ξ, which satisfies a transport equation. Our main contribution is to determine the closed-form expression ( 14) of the Goldstein hydrodynamic unknown ξ. In particular we will show that for a potential flow and potential sources, ξ = 0 and Eq. ( 1) is recovered. Note that Goldstein derived his equations only in the case of a potential flow in presence of incident convected disturbances but without explicit source term (see Eq. ( 9)). Since we are interested in a general configuration, we derive the equations in the general case. Starting from Euler's equations, we first derive a generalization of the Goldstein equations that are valid in presence of compactly supported source terms and of a vortical flow as well. Finally, since the Goldstein equations are exact, we will quantify the errors when the convected Helmholtz equation ( 1) is used in a non potential flow, such approximation being often used for its simplicity.

The outline of the paper is the following. The generalized Goldstein equations are derived in section 2 and we show they are equivalent to Euler's equations. This section ends up with the determination of the general expression of ξ. This manuscript is for review purposes only.

Euler's equations and Goldstein's equations in presence of source terms and of a vortical flow. Although we are primary interested in the time harmonic regime, such proof will be done in the time regime because the initial conditions enable us to have uniqueness results when integrating transport equations. We start by recalling the Linearized Euler Equations.

2.1. Euler's equations. Since the theory is not limited to a 2D configuration, we work in 3D and we consider a propagation domain Ω = R 3 or Ω = R 3 \B where B represents obstacles or walls guide, on which a rigid boundary condition is imposed ṽ • n = 0 with ṽ the total fluid velocity. We consider a subsonic inviscid flow of a perfect compressible fluid with constant specific heats capacity c p and c v at constant pressure and volume. The flow is the superposition of a steady flow and of small timedependent acoustic perturbations: the velocity v, the density ρ, the pressure p and the entropy s. The small perturbations satisfy in Ω the Linearized Euler Equations.

Following Goldstein [START_REF] Goldstein | Unsteady vortical and entropic distortion of potential flows round arbitrary obstacles[END_REF], it is assumed that the carrier flow is homentropic: s 0 is constant. This is an hypothesis commonly used [START_REF] Golubev | Sound propagation in an annular duct with mean potential swirling flow[END_REF][START_REF] Golubev | Acoustic-vorticity waves in swirling flows[END_REF][START_REF] Cooper | Propagation of unsteady disturbances in a slowly varying duct with mean swirling flow[END_REF][START_REF] Atassi | Computing the sound power in non-uniform flow[END_REF], despite effects of a mean entropy have been sometimes taken into account [START_REF] Cooper | Effect of mean entropy on unsteady disturbance propagation in a slowly varying duct with mean swirling flow[END_REF][START_REF] Tam | The wave modes in ducted swirling flows[END_REF]. For such a carrier flow, Goldstein showed that the Linearized Euler Equations simplify in a system that we extend by including general source terms f , g and h:

(3)

                         Dv ⋆ Dt + (v ⋆ • ∇)v 0 + ∇ p ρ 0 = g, D Dt p ρ 0 c 2 0 + 1 ρ 0 div (ρ 0 v) = -f, Ds Dt = h, v ⋆ = v - sv 0 2c p .
D/Dt is defined in Eq. ( 2) and the boundary conditions are v • n = 0 on ∂Ω. Source terms are chosen compactly supported and such that f, h ∈ L 2 (Ω), g ∈ (L 2 (Ω)) 3 . We consider causal sources and null initial conditions:

f (x, t ≤ 0) = 0 = g(x, t ≤ 0) = h(x, t ≤ 0), v ⋆ (x, 0) = 0 = p(x, 0) = s(x, 0).
In (3), the entropy perturbation s is decoupled from the other perturbations. Note that this is no longer true if the carrier flow is not homentropic. Then the entropy satisfies Ds Dt + v • ∇s 0 = h and as soon as s 0 = 0, s depends on v. For a homentropic carrier flow, s can be eliminated by solving the following decoupled problem:

(4) Ds Dt = h, with s(x, 0) = 0, whose unique solution is noted H(x, t) (see remark 2). Then, noting

ω ⋆ = curl v ⋆
This manuscript is for review purposes only.

the vorticity perturbation, we rewrite (3) in the more convenient form

∂v ⋆ ∂t + ∇(v 0 • v ⋆ ) + ω 0 × v ⋆ + ω ⋆ × v 0 + ∇ p ρ 0 = g, (5a) D Dt p ρ 0 c 2 0 + 1 ρ 0 div(ρ 0 v) = -f, (5b) v ⋆ = v - Hv 0 2c p , (5c) v ⋆ (x, 0) = 0 = p(x, 0) = s(x, 0). (5d) with (6) ω 0 = curl v 0 ,
the vorticity of the carrier flow and where we have used the vector identity Eq. ( 31).

2.2. Goldstein's equation with general sources in a general flow. In this paragraph, we aim at obtaining the Goldstein equations from Eq. ((5a),(5b),(5c),(5d)), with a general flow (ω 0 = 0 or = 0) and general sources f and g. The Helmholtz decomposition indicates that g can be written

g = ∇g 1 + curl g 2 , with g 2 = 0 if curl g = 0.
In 3D, the condition divg 2 = 0 is added whereas in 2D, g 2 becomes a scalar. We will prove the theorem:

Theorem 1. Euler's Equations ( (5a),(5b),(5c),(5d)) are equivalent to the generalized Goldstein equations

(7)        D Dt 1 c 2 0 Dϕ Dt - 1 ρ 0 div [ρ 0 (∇ϕ + ξ)] = F + D Dt g 1 c 2 0 , Dξ Dt + (ξ • ∇)v 0 -∇ϕ × ω 0 = curl g 2 ,
where

F = f + v 0 • ∇H 2c p , associated to the initial conditions ϕ(x, 0) = 0 = (∂ϕ/∂t)(x, 0) = ξ(x, 0), with (8)
       v = ∇ϕ + ξ + Hv 0 2c p , p = ρ 0 g 1 - Dϕ Dt .
Before proving this theorem, let us make some remarks.

Remark 1.

• If all the sources are vanished g = 0 = f = h and if ω 0 = 0, Eq. (7) and Eq.

(8) degenerate to the result of Goldstein [START_REF] Goldstein | Unsteady vortical and entropic distortion of potential flows round arbitrary obstacles[END_REF]:

(9)        D Dt 1 c 2 0 Dϕ Dt = 1 ρ 0 div [ρ 0 (∇ϕ + ξ)] , Dξ Dt = -(ξ • ∇)v 0 .
• The definition usually used [START_REF] Goldstein | Unsteady vortical and entropic distortion of potential flows round arbitrary obstacles[END_REF][START_REF] Golubev | Acoustic-vorticity waves in swirling flows[END_REF][START_REF] Atassi | Computing the sound power in non-uniform flow[END_REF][START_REF] Cooper | Effect of mean entropy on unsteady disturbance propagation in a slowly varying duct with mean swirling flow[END_REF], p = -ρ 0 Dϕ/Dt, is no longer valid if g 1 = 0 (see Eq. ( 8)).

• the "acoustical" part g 1 of the source g appears as a source term only in the equation for the acoustic field ϕ. In a same way the "hydrodynamic" part g 2 of the source g appears as a source term only in the equation for the hydrodynamic field ξ.

• the decomposition of the velocity in Eq. ( 8) is not a Helmholtz decomposition since we have not imposed div ξ + Hv 0 2c p = 0. In fact in practice this quantity is not found to be equal to zero and it happens that the Helmholtz decomposition is not well-adapted to aeroacoustics. We have chosen to use the Goldstein's decomposition v ⋆ = ∇ϕ + ξ, but an alternative is to use Clebsch potentials [START_REF] Bergliaffa | Wave Equation for Sound in Fluids with Vorticity[END_REF].

Proof. It is straightforward to check that if ϕ and ξ are solutions of the coupled system of Goldstein's equations [START_REF] Duprey | Etude mathématique et numérique de la propagation acoustique d'un turboréacteur[END_REF], then p = ρ 0 (g 1 -Dϕ/Dt) and v ⋆ = ∇ϕ + ξ are solutions of Eq. ((5a),(5b),(5c),(5d)). It has been already done without any source term, for isentropic perturbations (s = 0) [START_REF] Golubev | Sound propagation in an annular duct with mean potential swirling flow[END_REF][START_REF] Golubev | Acoustic-vorticity waves in swirling flows[END_REF][START_REF] Cooper | Propagation of unsteady disturbances in a slowly varying duct with mean swirling flow[END_REF][START_REF] Atassi | Computing the sound power in non-uniform flow[END_REF] or for general perturbations in a cylindrical geometry [START_REF] Cooper | Effect of mean entropy on unsteady disturbance propagation in a slowly varying duct with mean swirling flow[END_REF].

To prove the converse implication, that a solution of Euler's equations is a solution of Goldstein's equations, is more complicated and has never been done in the general case. A first difficulty is to define uniquely the Goldstein unknowns (ϕ, ξ) from the Euler unknowns (p, v ⋆ ). It has been done only for a potential flow and with incident fields as source terms [START_REF] Goldstein | Unsteady vortical and entropic distortion of potential flows round arbitrary obstacles[END_REF]. To extend the results to a vortical flow and to the presence of sources, first we define ϕ, starting from ( 8) with the initial condition ϕ(x, 0) = 0 (see remark 2). Then we deduce ξ. More explicitly we use the relations (10)

   Dϕ Dt = g 1 - p ρ 0 , with ϕ(x, 0) = 0, ξ = v ⋆ -∇ϕ,
to define the Goldstein's unknowns (ϕ, ξ).

Applying the change of unknowns Eq. ( 10), the mass conservation equation (5b) leads to

(11) D Dt 1 c 2 0 Dϕ Dt = 1 ρ 0 div[ρ 0 (∇ϕ + ξ)] + f + D Dt g 1 c 2 0 + v 0 • ∇H 2c p .
Moreover, using ω ⋆ = curl ξ, deduced form Eq. ( 10), (5a) leads to

∇ ∂ϕ ∂t + v 0 • v ⋆ + g 1 - Dϕ Dt + ∂ξ ∂t + ω 0 × v ⋆ + curl ξ × v 0 = g,
which simplifies in:

∇ (v 0 • ξ) + ∂ξ ∂t + ω 0 × (∇ϕ + ξ) + curl ξ × v 0 = curl g 2 .
Finally using once again Eq. ( 31) is obtained the hydrodynamic equation

(12) Dξ Dt = ∇ϕ × ω 0 -(ξ • ∇)v 0 + curl g 2 .
This manuscript is for review purposes only.

Remark 2. An efficient method to solve the first equation of [START_REF] Rienstra | A numerical comparison between multiple-scales and finiteelement solution for sound propagation in lined flow ducts[END_REF] or to solve Eq. ( 4) is to use a change of variable in order to get a family of ordinary differential equations along the streamlines of the flow. For instance, for a parallel shear flow v 0 (x) = v 0 (y)e x with x = (x, y, z), the streamlines are the lines y = cste and z = cste and we get the explicit unique solutions:

ϕ(x, t) = t 0 ζ[x -v 0 (y)(t -u), y, z, u] du where ζ(x, t) = g 1 (x, t) - p(x, t) ρ 0 (x) , H(x, t) = t 0 h[x -v 0 (y)(t -u), y, z, u] du.
2.3. General expression of the hydrodynamic unknown. The perturbation ϕ is governed by a classical convected wave-like equation [START_REF] Duprey | Etude mathématique et numérique de la propagation acoustique d'un turboréacteur[END_REF], with source terms.

The main difficulty is to determine ξ, which satisfies a less classical transport-like equation. However there is a simplification: it is possible to find a simple expression of ξ which in particular predicts where ξ vanishes. We present now the derivation of such expression.

In the case of a potential flow ω 0 = 0 and of potential sources g 2 = 0,it is easy to get from Eq. ( 12) that ξ = 0 and Goldstein's equations [START_REF] Duprey | Etude mathématique et numérique de la propagation acoustique d'un turboréacteur[END_REF] reduce to the potential model (Eq. ( 1) with the addition of source terms). In the general case of a nonpotential flow in presence of non-potential sources, ξ = 0 but a simple expression of ξ can also be obtained. Indeed Eq. ( 12) may be written

(13) Dξ Dt + (ξ • ∇)v 0 = ∇ϕ × ω 0 + curl g 2 .
This means that ξ is induced by two different sources: the coupling between acoustic perturbations ∇ϕ and the flow vorticity ω 0 and the vorticity source term curl g 2 .

More precisely we will prove the following theorem:

Theorem 2. The general solution of Eq. ( 13) is

(14) ξ = u × ω 0 + σ,
where u is the displacement perturbation and where σ(g 2 ) is a term only due to the hydrodynamic source g 2 (or equivalently to curl g).

u is defined from the velocity through (see [START_REF] Bonnet-Ben Dhia | Time-Harmonic Acoustic Scattering in a Complex Flow: A Full Coupling Between Acoustics and Hydrodynamics[END_REF])

(15) v ⋆ = Du Dt -(u • ∇)v 0 ,
with the initial condition u(x, 0) = 0. To prove the Theorem 2 and the decomposition [START_REF] Treyssede | A mixed finite element method for acoustic wave propagation in moving fluids based on an Eulerian-Lagrangian description[END_REF], we introduce the temporary unknown

(16) ξ = u × ω 0 ,
and we prove first the following lemma:

Lemma 3. The quantity ξ defined in Eq. ( 16) satisfies the following equation

(17) D ξ Dt = (v ⋆ -ξ) × ω 0 -( ξ • ∇)v 0 .
This manuscript is for review purposes only.

Note that this is simply a vectorial relation, which has nothing to do with the problems satisfied by u and v ⋆ . We give the proof in the Appendix A.

Note that since ∇ϕ = v ⋆ξ, ξ in Eq. ( 12) with curl g 2 = 0 satisfies the same equation than ξ in Eq. [START_REF] Golubev | Sound propagation in an annular duct with mean potential swirling flow[END_REF]. Thanks to this remark, from Eq. ( 12) and Eq. ( 17) we

deduce that ζ = ξ -ξ satisfies (18) Dζ Dt = -ζ × ω 0 -(ζ • ∇)v 0 + curl g 2 , with ζ(x, 0) = 0.
We can now prove the Theorem 2 by solving Eq. ( 18), the solution depending of the value of the source term:

• if curl g 2 = 0, then the solution of Eq. ( 18) is zero (and thus ξ = u × ω 0 ), since, it is easy to get that

dE dt ≤ 2 ( ω 0 + ∇v 0 ) E, where E = Ω ρ 0 2 |ζ| 2 dx
. Since E(t = 0) = 0, we deduce thanks to Gronwall lemma that ξ = ξ at any time.

• if curl g 2 = 0, if we note σ(g 2 ) the unique solution (see the previous item for the homogeneous problem) of Eq. ( 18), then ξ = ξ + σ.

Remark 3. In the case g = 0 and for homentropic perturbations s = 0, the first part ξ = u × ω 0 has already been found [START_REF] Bergliaffa | Wave Equation for Sound in Fluids with Vorticity[END_REF] whereas Goldstein [START_REF] Goldstein | Unsteady vortical and entropic distortion of potential flows round arbitrary obstacles[END_REF], considering the case ω 0 = 0 but with an incident field, acting like a source term curl g = 0, found the second part ξ = σ.

Numerical experiments.

To illustrate the superiority of the Goldstein equations with respect to the potential model (1), we will now solve in the time harmonic regime the radiation problem defined by equations [START_REF] Duprey | Etude mathématique et numérique de la propagation acoustique d'un turboréacteur[END_REF]. We use a Finite Element method, which, contrary to a Finite Differences method, has the main advantage to be well suitable for unstructured meshes, adapted to complex geometries. We will restrict to the 2D case.

v 0 (x, y) = v 0 (y)e x ,
with a velocity v 0 continuously differentiable on R. We impose an homentropic flow (s 0 = cst) and we get from the Linearized Euler Equations that the density ρ 0 , the pressure p 0 and the sound velocity c 0 are constant. We suppose the flow subsonic |v 0 | < c 0 and we introduce the Mach number M 0 (y) = v 0 (y)/c 0 . We consider harmonic noise sources F (x)e -iωt and g(x)e -iωt with the frequency ω and we solve [START_REF] Duprey | Etude mathématique et numérique de la propagation acoustique d'un turboréacteur[END_REF] with (2) replaced by

D Dt = -iω + v 0 • ∇.
This manuscript is for review purposes only.

Time harmonic regime seems easier to solve than transient regime since the time variable has disappeared, but it introduces an extra major difficulty when solving numerically a radiation problem: to consider a bounded domain, we have to define some well-suited radiation conditions. In this aim, following the treatment of Galbrun's equation [START_REF] Bonnet-Ben Dhia | Time-Harmonic Acoustic Scattering in a Complex Flow: A Full Coupling Between Acoustics and Hydrodynamics[END_REF], we surround the computational domain with perfectly matched layers (PMLs) [START_REF] Bécache | Perfectly matched layers for timeharmonic acoustics in the presence of a uniform flow[END_REF][START_REF] Hu | A perfectly matched layer absorbing boundary condition for linearized Euler equations with a non-uniform mean flow[END_REF]. The computational domain, schematized on Fig. 1, is Ω = Ω L ∪ Ω b

where Ω b =]x m , x p [×]0, h[ is the physical bounded domain containing at least the

sources, Ω =]x m -L 1 , x p + L 1 [×] -L 2 , h + L 2 [
∀(x, y) ∈ Ω,          ∂ ∂x → α1 (x) ∂ ∂x ∂ ∂y → α2 (y) ∂ ∂y
where the piecewise functions αi are defined by

α1 (x) = 1 if x ∈ ]x m , x p [, α ∈ C otherwise. α2 (y) = 1 if y ∈ ]0, h[, α ∈ C otherwise.
The complex number α in the PMLs is chosen such that [START_REF] Bonnet-Ben Dhia | Time-Harmonic Acoustic Scattering in a Complex Flow: A Full Coupling Between Acoustics and Hydrodynamics[END_REF] (20) Re(α) > 0 and Im(α) < 0. conditions ϕ = 0 on Σ ± and ξ = 0 on Σ -(since ξ satisfies a transport equation, it just needs to be set upstream [START_REF] Bonnet-Ben Dhia | Time-Harmonic Acoustic Scattering in a Complex Flow: A Full Coupling Between Acoustics and Hydrodynamics[END_REF]).

The Goldstein equations, in time harmonic regime, with PMLs and in a parallel shear flow become ( 21)

             D 2 α ϕ = div α (∇ α ϕ + ξ) + F + D α g 1 c 0 , D α ξ x = -M ′ 0 (y) α2 ∂ϕ ∂y + ξ y + α2 ∂g 2 ∂y , D α ξ y = M ′ 0 (y)α 1 ∂ϕ ∂x -α1 ∂g 2 ∂x ,
where D α = -ik + M 0 (y)α 1 ∂/∂x with k = ω/c 0 . Thanks to the PMLs, we can prove that problem ( 21) is of Fredholm type, which in particular ensures the convergence of a Finite Element approximation of the solution (outside an eventual set of discrete resonance frequencies). More precisely we can prove that (the demonstration is rather technical and not given here):

Theorem 4. Problem (21) is of Fredholm type if (22) min ℜe (α) (1 -s 2 0 ), ℜe 1 α -s 1 C α,Ω > 0,
where

s 0 = max y∈[0,h] |M 0 (y)|, s 1 = max y∈[0,h] M ′ 0 (y) M 0 (y)
and C α,Ω is a constant depending of the PML parameter and of the geometry of Ω.

Remark 4. Note that thanks to Eq. ( 20), both ℜe (α) and ℜe (1/α) are positive.

Therefore the theorem requires that s 0 < 1 (subsonic flow) and that s 1 is small enough (low shear flow). It is known that an incompressible shear flow with an inflection point in the velocity profile may be unstable if the slope is large enough. The condition s 1 small is certainly linked to this instability, although the condition M ′′ 0 = 0 does not appear explicitly in the condition [START_REF] Tam | The wave modes in ducted swirling flows[END_REF]. The stability of compressible shear flows has been less studied, results can be found in the low frequency limit [START_REF] Bonnet-Bendhia | Stability of Acoustic Propagation in a 2D Flow Duct: A Low Frequency Approach[END_REF], but no explicit results on the influence of max y∈[0,h] |M ′ 0 | are given is this reference.

Remark 5. The condition [START_REF] Tam | The wave modes in ducted swirling flows[END_REF] is impossible to fulfill if M 0 vanishes, but we have found an alternative condition that replaces the condition [START_REF] Tam | The wave modes in ducted swirling flows[END_REF] for vanishing or even low Mach number values. If M 0 (y 0 ) = 0, then let us introduce the small layer y ∈ [y 0 -ε, y 0 + ε] where ε is chosen such that M 0 is small in this layer. Then the radiation problem ( 21) is found to be globally well posed if it is well-posed both

• in the "low Mach" layer y ∈ [y 0 -ε, y 0 + ε], which is the case under the condition:

(23) min Re (α) (1 -s 2 0 ), Re 1 α - max y∈[y0-ε,y0+ε] |M ′ 0 (y)| k > 0,
• outside the "low Mach" layers, which is the case if the condition ( 22) is satisfied by

s 1 = max y / ∈[y0-ε,y0+ε] M ′ 0 (y) M 0 (y) .
Remark 6. Conditions ( 22) and ( 23) are sufficient conditions to get a well-posed radiation problem. In practice, the numerical method is stable for less restrictive conditions.

This manuscript is for review purposes only.

A numerical method for solving the Galbrun Equation has been developed in the case of shear flows [START_REF] Bonnet-Ben Dhia | Time-harmonic acoustic propagation in the presence of a shear flow[END_REF][START_REF] Bonnet-Ben Dhia | Acoustic propagation in a flow: numerical simulation of the time-harmonic regime[END_REF], and then extended to slow [START_REF] Bonnet-Ben Dhia | A low Mach model for time harmonic acoustics in arbitrary flows[END_REF] and general flows [START_REF] Bonnet-Ben Dhia | Time-Harmonic Acoustic Scattering in a Complex Flow: A Full Coupling Between Acoustics and Hydrodynamics[END_REF]. This numerical approach relies on a Finite Element method coupling continuous and Discontinuous elements [START_REF] Delorme | Computational Aeroacoustics applications based on a discontinuous Galerkin method[END_REF][START_REF] Gabard | Discontinuous Galerkin methods with plane waves for time-harmonic problems[END_REF][START_REF] Ern | Theory and practice of finite Element[END_REF], these latter leading to a stable method to deal with harmonic transport problems. The numerical scheme to solve Goldstein's equations is not new, we have adapted the numerical scheme developped to solve Galbrun's equation. Here we just sum up the numerical approach, for more details, see [START_REF] Bonnet-Ben Dhia | Time-Harmonic Acoustic Scattering in a Complex Flow: A Full Coupling Between Acoustics and Hydrodynamics[END_REF]. To solve the Goldstein equations, we couple two Finite Element schemes: the potential equation of ( 21) is discretized with Lagrange Finite Elements whereas the hydrodynamic equations of ( 21) are discretized with Discontinuous Galerkin Elements. Both quantities are discretized on the same mesh with around 70 000 nodes. They are determined conjointly and time calculations are of the order of a few minutes.

3.1.2. Parameters for the numerical simulations. We work with variables and unknowns without dimension which leads to ρ 0 = c 0 = 1. In most of the numerical results, we consider a jet flow of Mach number given by ( 24)

M 0 (y) = M ∞ + µ exp - y 2 R 2 ,
where µ, M ∞ and R are parameters. We consider first F = 0 in Eq. ( 21), g 2 = 0

(thus the hydrodynamic field ξ is equal to u × ω 0 , see Eq. ( 14)) and two types of acoustic sources g 1 , such that ∇g 1 is a dipolar or a quadripolar source:

g * 1 (x, y) = exp - (x -x c ) 2 + (y -y c ) 2 r 2 S , g 1 (x, y) = (x -x c )(y -y c ) r 2 exp - (x -x c ) 2 + (y -y c ) 2 r 2 S . (25)
(x c , y c ) is the center of the source and r S is its characteristic radius. For simplicity we consider isentropic disturbances s = 0, which implies v ⋆ = v in Eq. ( 3), and we will compare three models, where D/Dt = M 0 (y)∂/∂x -ik with k = ω/c 0 :

• the potential model

D 2 ϕ Dt 2 = div(∇ϕ) + Dg 1 Dt ,
• the Goldstein model ( 26)

         D 2 ϕ Dt 2 = div(∇ϕ + ξ) + Dg 1 Dt , Dξ Dt = ∇ϕ × ω 0 -(ξ • ∇)M 0 , with ω 0 = curl M 0 ,
• the Galbrun model (see [START_REF] Galbrun | Propagation d'une onde sonore dans l'atmosphère terrestre et théorie des zones de silence[END_REF][START_REF] Bonnet-Ben Dhia | Time-Harmonic Acoustic Scattering in a Complex Flow: A Full Coupling Between Acoustics and Hydrodynamics[END_REF])

(27) D 2 u Dt 2 -∇(∇ • u) = ∇g 1 .
Thanks to the introduction of the source term ∇g 1 in Eq. ( 27), it is possible to prove that Eq. ( 26) and Eq. ( 27) are equivalent. Our aim is to illustrate the efficiency of the Goldstein model. The Galbrun model is introduced as a reference method and the potential model is used for its simplicity. More precisely:

This manuscript is for review purposes only.

• Since the Galbrun model has been implemented and tested on many cases [START_REF] Bonnet-Ben Dhia | Time-Harmonic Acoustic Scattering in a Complex Flow: A Full Coupling Between Acoustics and Hydrodynamics[END_REF], it will be used to produce reference solutions to test the validity of the numerical resolution of the Goldstein equations. To compare the numerical results from the three models, we need to consider common quantities: the pressure and the velocity. These quantities are given by

p = g 1 - Dϕ Dt = -div u and v = ∇ϕ + ξ = Du Dt -(u • ∇)M 0 .
• The potential model is very attractive because its resolution is simple: only continuous Finite Element are required. Although it is not valid for nonpotential flows, it can be solved for any flows and the comparison with Goldstein's model will indicate the importance of the induced error. The results will be presented first for low shear flows, for which the error when using the potential model is expected to be small, then for stronger shear flows and finally for a potential flow with an hydrodynamic source. Eq. [START_REF] Rienstra | A numerical comparison between multiple-scales and finiteelement solution for sound propagation in lined flow ducts[END_REF]. Indeed the vorticity ω 0 = -dM 0 /dye z is bounded by

ω 0 ≡ |ω 0 | = 2µ R 2 y exp - y 2 R 2 ≤ 2 e µ R .
From the second equation of ( 26), we get the rough estimate ω|ξ| ∼ ω 0 |∇ϕ| and therefore |ξ|/|∇ϕ| ∼ ω 0 /ω ≤ 0.04.

On Fig. 2 and Fig. 3 are represented respectively the horizontal acoustic velocity and the pressure radiated by the source for the three models. The horizontal white lines delimit the area with a strong shear of the flow. We obtain very similar numerical solutions, which confirms that the potential model may be used as a good approximation in this low shear case. Actually, there are small differences in the vorticity areas (between the white lines) on the acoustic velocity but no difference on the pressure.

The impact of these differences on the far-fields is found very limited.

3.1.4. Comparisons for a larger vorticity flow. We consider now for the same geometry and the same source a jet flow with a stronger vorticity, R = 0.15 and This manuscript is for review purposes only. capture the hydrodynamic phenomena, neglected by the potential model: oscillations of v x (inclined lines of small wavelength downstream the source, due to oscillations of ξ) are seen on Fig. 4 left part, as a result of the transport phenomenon (see [START_REF] Bonnet-Ben Dhia | Time-Harmonic Acoustic Scattering in a Complex Flow: A Full Coupling Between Acoustics and Hydrodynamics[END_REF]).

Indeed the solution of Dξ/Dt = 0 is ξ = A(y) exp(ikx/M 0 (y)): for each y value, it corresponds to patterns moving at M 0 (y) with a wavelength 2πM 0 (y)/k.

In the previous example, the impact of neglecting ξ is found very limited: indeed, the radiated patterns for the Goldstein and potential models in Fig. 4 This manuscript is for review purposes only. from the Galbrun Equation are found exactly equal. Note that ξ is not only convected by the flow, it takes non zero values upstream the source. This is due to the fact that the acoustic field u is radiated in all directions by the source.

3.2. Case of a 2D flow. In this section, we extend our illustrations to nonshear flow and non-potential sources. We focus on Goldstein's configuration [START_REF] Goldstein | Unsteady vortical and entropic distortion of potential flows round arbitrary obstacles[END_REF] and we consider a potential flow with a vortical source (g 2 = 0). We consider the flow around a circular obstacle of radius R, M 0 (x, y) = ∇ϕ 0 with

ϕ 0 = M ∞ r R + R r cos θ,
This manuscript is for review purposes only. We have introduced the general source term

S = f + D Dt g 1 c 2 0 + v 0 • ∇H 2c p ,
where H denotes the entropic perturbation. Moreover the solution ξ of the hydrodynamic equation of Eq. ( 7) may be written in the form (Eq. ( 14)):

ξ = u × ω 0 + σ(g 2 ),
where σ is the solution to (Eq. ( 18)):

Dσ Dt + (σ • ∇)v 0 + σ × ω 0 = curl g 2 .
Our conclusions depend on the value of the vorticity of the flow ω 0 and of the value of curl g in the momentum conservation law of Euler's equations. An overview of these conclusions is summarized in the following table:

curl g = 0 curl g = 0 Aϕ = S Aϕ = S -Bσ ω 0 = 0 ξ = 0 ξ = σ Blokhintzev's model [START_REF] Blokhintzev | The propagation of sound in an inhomogeneous and moving medium I[END_REF] Goldstein's model [START_REF] Goldstein | Unsteady vortical and entropic distortion of potential flows round arbitrary obstacles[END_REF] Aϕ + Bξ = S Aϕ + Bξ = S ω 0 = 0 Cξ + Dϕ = 0 Cξ + Dϕ = curl g 2 ξ = u × ω 0 ξ = u × ω 0 + σ Visser's model [START_REF] Bergliaffa | Wave Equation for Sound in Fluids with Vorticity[END_REF] New model

After the first theoretical section, we have presented a numerical method to solve the Goldstein equations in the time harmonic regime. PMLs have been introduced in order to bound the computational domain in the case of a two dimensional flow. The numerical method has been validated by a comparison with the Galbrun Equation and the importance of the hydrodynamic unknown has been emphasized by a comparison with the potential model.

A natural extension of this work is to consider more complicated boundary conditions than rigid boundaries, like impedance boundary conditions. Another perspective is to extend the numerical method to the 3D case. A possible strategy to decrease the computational cost, due to the introduction of many degrees of freedom by the Discontinuous Galerkin Element method, would be to compute ϕ and ξ on different meshes: ξ would be determined on a coarser mesh, since it is associated to small wavelengths, but on a mesh of small extension since only restricted to the vortical areas of the carrier flow.

This manuscript is for review purposes only.

  Section 3 is mainly devoted to some 2D numerical experiments. First, we consider a parallel shear flow in an unbounded domain. Such simple flow facilitates the presentation of the numerical method, based on the introduction of Perfectly Matched Layers (PMLs) to bound the calculation domain and on the coupling between continuous and discontinuous Finite Elements. We also give some conditions under which the radiation problem with PMLs is well-posed, ensuring the convergence of the Finite Element scheme. Then we use Goldstein's equations to study numerically the acoustic radiation of a source in presence of such a flow. The solutions are validated by comparison with the solutions obtained from the Galbrun equation and the influence of a vortical flow is illustrated. Finally we consider a non-parallel flow and we illustrate the closed-form relation giving ξ. 2. Derivation of the Goldstein equations from the Euler equations. Our first purpose is to prove in a simple and comprehensive way the equivalence between

3. 1 . 3 . 1 . 1 .

 1311 Acoustic radiation in a parallel shear flow. We start by considering the simple case of a shear flow for two reasons. First it is a validation case because for such flow, the link between different alternative models is simple. Second it is a simple way to control the vorticity of the carrier flow (it is the derivative of the velocity profile) and thus to quantify the domain of validity of the potential model Eq. (1). Numerical scheme. Let us consider a two dimensional shear flow in R 2

  is the computational domain and Ω L = Ω\Ω b is the PML domain. The velocity v 0 is supposed uniform outside [0, h] in order to get no reflected waves from the domain outside Ω b (otherwise the PMLs could not be introduced). The introduction of PMLs corresponds to define the scale changes

Fig. 1 .

 1 Fig. 1. Description of the two dimensional problem with PMLs

3. 1 . 3 .

 13 Comparisons for a low vorticity flow. First we consider a jet flow associated to a low vorticity |ω 0 |, defined by R = 0.35, M ∞ = 0.1 and µ = 0.2 in the domain Ω b =]0, 3[×] -1, 1[. The source is g * 1 (see Eq. (25)) with (x c , y c ) = (1, 0), r S = 0.05 and k = 12. For small |ω 0 | values, ξ = u × ω 0 is small and the potential model is expected to be a rather good approximation. More quantitatively, the term ∇ϕ is dominant over the hydrodynamic term ξ in the acoustic velocity decomposition

Fig. 2 .

 2 Fig. 2. Real part of the velocity perturbation Re(vx) in a low shear jet flow Eq. (24) for Goldstein (left), Galbrun (center) and the potential model (right). The shear area is between the white lines. The source corresponds to the central disc. For such a low shear flow, the solutions of the three different equations are found very similar

Fig. 3 .

 3 Fig. 3. Imaginary part of the pressure perturbation Im(p) for the same configuration than in Fig. 2 for: Goldstein (left), Galbrun (center) and the potential model (right). Since the pressure in not an hydrodynamic quantity, the three solutions are nearly the same.

Fig. 4 .

 4 Fig. 4. Real part of the velocity perturbation Re(vx) in a strong shear jet flow for Goldstein (left) and the potential model (right). Hydrodynamics patterns, appearing as inclined lines with a small wavelength, are forgotten by the potential model.

  are very similar. This is because |ξ|/|∇ϕ| ≤ 0.1. But in general it is important not to neglect hydrodynamic phenomena since strong differences in the far field can occur. Such differences are easier to see, considering a new geometry: we consider a larger domain Ω b =] -3, 3[ 2 , k = 12 and we place the quadripolar source g 1 of Eq. (25) at (x c , y c ) = (-1, 1.5) with r S = 0.1. The jet flow corresponds to R = 0.1, M ∞ = 0 and µ = 0.3. The results obtained from the three models are presented on Fig. 5. The far fields obtained for the Goldstein and the Galbrun models are equivalent and both different from the result given by the potential flow model. The far field of the potential solution is clearly different downstream the source, in particular for the rays reflected by the jet core, highlighted by dashed lines.

Fig. 5 .Fig. 6 .

 56 Fig. 5. Real part of the velocity perturbation Re(vx) for the radiation of a quadripolar source for Goldstein (left), Galbrun (center) and the potential model (right). A large domain enables us to see the far field. The dashed black lines highlight the main differences between the potential model and the other models. In particular an extra reflected "ray", non physical, is obtained by the potential model.

  The operators A, B, C and D are defined as: • ∇)v 0 , Dϕ = -∇ϕ × ω 0 .

J-F. MERCIER, C. MIETKA, F. MILLOT, AND V. PAGNEUX

This manuscript is for review purposes only.

Acknowledgments. The authors gratefully acknowledge the Agence Nationale de la Recherche (AEROSON project, ANR-09-BLAN-0068-02 program) for financial support.

* This work was funded by the Fog Research Institute under contract no. FRI-454.

in polar coordinates (x = r cos θ, y = r sin θ). Such flow and source lead to the following equations:

with D/Dt = M 0 • ∇ -ik and where we take g 1 = g * 1 = g 2 (see Eq. ( 25)). From Eq. ( 14), since for a potential flow u × ω 0 = 0, we have ξ = σ. On Fig. 7, for a flow coming horizontally from the left, is represented

for R = 0.3, M ∞ = 0.3, k = 6 and for two different positions of the source. We see ξ, created by the source whose location is indicated by a black circle, and following the flow streamlines. On Fig. 7 (left) the source is off-centered from the flow whereas on Fig. 7 (right) the source is centered on the stream line associated to a stop point on the disc. In the second case the pattern of Re(ξ x ) is rather complex and behind the disc, the wavelength of Re(ξ x ) is very small because the velocity is weak. Note that the potential model would ignore these patterns since this model consists in taking ξ = 0 in the Goldstein equation and therefore the potential model is a bad approximation when g 2 is not small. Firstly, by taking some acoustic and hydrodynamic source terms into account, we have proved that the Goldstein equations are equivalent to the Linearized Euler model. For source terms f and g = ∇g 1 + curl g 2 introduced in Eq. (5a), we have established that the generalized Goldstein equations can be written (see Eq. ( 7)) in the following way:

This manuscript is for review purposes only.

Appendix A. Hydrodynamic field.

Here we prove the lemma 3:

Proof. The carrier flow satisfies the Euler Equations:

associated to the state law of the fluid p 0 = ν(ρ 0 ), given by ( 29)

where κ is a constant.

First the usual vorticity equation for ω 0 is recovered, by writing the second equation of (28), using Eq. ( 29), in the form

where we have used the vector identity we get (it is also Eq. ( 52) of [START_REF] Bergliaffa | Wave Equation for Sound in Fluids with Vorticity[END_REF])

in which we have used ∇, the nabla operator applying only on v 0 (while u and ω 0 are considered constant)(see [START_REF] Bergliaffa | Wave Equation for Sound in Fluids with Vorticity[END_REF]). Therefore we get

The vector identity [START_REF] Bécache | Perfectly matched layers for timeharmonic acoustics in the presence of a uniform flow[END_REF], valid also for the operator ∇, applied to ξ and v 0 leads us to the result of the lemma.

This manuscript is for review purposes only.


Muhammad Khokhar

Emmanuel Rozière

Frédéric Grondin

Ahmed Loukili
email: ahmed.loukili@ec-nantes.fr

Muhammad Irfan

Ahmad Khokhar

Effect of Mineral Additives on Some of Durability Parameters of Concrete

Keywords: CO 2 emissions, Durability, Mineral additives

published or not. The documents may come   L'archive ouverte pluridisciplinaire

INTRODUCTION

The construction industry must address certain consequential issues in the process of achieving sustainable development as it consumes considerable resources and has a significant impact on the environment. Carbon dioxide (CO 2 ) is a by-product of a chemical conversion process used in the production of clinker, a component of cement, in which limestone (CaCO 3 ) is converted to lime (CaO). CO 2 is also emitted during cement production by combustion of fossil fuel and is accounted for elsewhere and making one tonne of cement produces approximately one tonne of CO 2. Further, the manufacturing of Portland cement is a highly energy-intensive process requiring about four Giga Joules (4GJ) of energy per tonne of finished product. One of the most significant activities stressed by the engineers and scientists related to concrete industry aims at a high percentage of replacement of clinker in cement with secondary raw materials, with the possibility of improvement of cement characteristics and durability of concrete. With this kind of approach, the consumption of natural raw materials, thermal and electric energy, together with the reduction of CO 2 emissions, are decreasing [START_REF] Roskovic | Role of mineral additions in reducing CO2 emissions[END_REF]. The less energy intensive materials being sought are industrial by-products that are easily available, require little or no pyro-processing and have inherent or latent cementitious properties and are called as supplementary cementitious materials or mineral additives. Mineral additives are available in large quantities that can be used to replace portland cement in concrete. These include fly ash, granulated blast-furnace slag, silica fume, natural pozzolans, rice-husk ash, and metakaolin.

These mineral additives if used in combination with high range water reducing agents (HRWRA) can lead to economical high performance concrete with enhanced durability and by reducing cement content in mixture proportioning contribute to CO 2 emissions reduction [START_REF] Malhotra | Reducing CO2 Emissions-The role of fly ash and other supplementary cementitious materials[END_REF]. These mineral additives are added in the cement as extra binder materials and the benefits of using these materials in terms of workability are well established [START_REF] Lachemi | Long-term performance of silica fume concretes[END_REF][START_REF] Bijen | Benefits of slag and fly ash[END_REF][START_REF] Malhotra | Fly ash, slag, silica fume and rice-husk ash in concrete -A review[END_REF]. Some studies have been performed to determine mechanical properties, thermal properties, transport mechanisms and the influence of mineral additions on the durability of blended concrete [START_REF] Nielsen | Mechanical Properties for Green Concrete[END_REF][START_REF] Sobolev | Mechano-chemical modification of cement with high volumes of blast furnace slag[END_REF][START_REF] Park | Rheological properties of cementitious materials containing mineral admixtures[END_REF][START_REF] Barnett | Strength development of mortars containing ground granulated blast-furnace slag: Effect of curing temperature and determination of apparent activation energies[END_REF][START_REF] Bhatty | Sulfate resistance of concrete using blended cements or supplementary cementitious materials[END_REF][START_REF] Shehata | The effect of fly ash composition on the expansion of concrete due to alkali-silica reaction[END_REF].

The project suggested here aims at studying the impact of the massive mineral additions on the strength, workability and permeability of concrete to assess their durability. The additions like the slag of blast furnace and the fly-ash are known to have rather slow kinetics of hydration and consequently delay the time setting and the hardening of concrete. However the short-term strength is a major parameter of the management of the building sites. Results of this study should make it possible to promote the use in high quantities of mineral additions, directly in the manufacturing of concrete and either only as addition in cement. From an economic point of view, that means to develop industrial by-products. That passes by the standardization of the concrete to high proportioning in additions [START_REF] Bisaillon | High-volume fly ash system: concrete solution for sustainable development[END_REF].

Tested concretes in this work contain a high volume of mineral additives. First of all, the experimental program is detailed with the description of the tested materials and of the experimental procedure. Then, results on the carbonation test and on the acid attack are presented and compared to ordinary concrete.

EXPERIMENTAL PROGRAM

Mixtures compositions and materials

Two sets of mix were prepared along with a reference mix called Ref (V A /V C = 0), with V A the volume of additions and V C the volume of cement. The concrete constituents are presented in Table 1. In the first set of mix, the addition/cement volume ratio (V A /V C ) of Ref, equal to 0.8 was used and in the second set, V A /V C was reduced to 0.4, keeping the paste volume constant. Limestone filler (LF) and fly ash (FA) were used for each value of V A /V C to investigate the influence of different addition type. For each mixture, super plasticizer content was adjusted in order to achieve a slump flow between 60 and 80 cm and to avoid segregation, which was controlled using the sieve test. The concrete mixtures are described in Table 2.

No addition A = Limestone filler A = Fly ash

V A /V C = 0 V A /V C = 0.80 V A /V C = 0.40 V A /V C = 0.80 V A /V C =

Experimental procedure

Compressive strength was determined at 2, 7 and 28 days on 11 x 22 cm cylinders, stored in a room at 20°C and 95% of relative humidity (RH). The elastic modulus was determined at 28 days from compressive tests on the same cylinders, loaded at 30 % of the peak value.

Free shrinkage was determined on 7 x 7 x 28.4 cm concrete prisms, kept at 20°C and 95% RH during 24 hours after mixing. From the age of 1 day, total shrinkage (i.e. drying + autogenous) was measured using LVDT sensors, in a room at a temperature 20±2°C and relative humidity 50±5 %. Autogenous shrinkage was measured in the same way, on prisms sealed with a double layer of adhesive aluminium.

Restrained shrinkage tests were performed with the help of rings. The total shrinkage of a concrete ring (width = 7 cm; thickness = 7 cm) was hindered by a steel ring (inner radius = 8.5 cm; outer radius = 11 cm). A 2 cm notch was placed on the steel ring to promote cracking.

The test to analyse the effects of leaching and acid attack on concrete consist of immersion of concrete sample during at least 60 days in a solution maintained at pH = 5 using nitric acid solution (HNO 3 ) of concentration 0.5 mol/L. The solution was renewed after 30 ml of acid used, and concentration of calcium ions was analysed for the samples of each solution. The quantity of acid added gives the value of leached hydroxide anions (OH) -. Acid used and cumulated calcium ions dissolved with reference to time give access to the kinetics of leaching.

Measurements of carbonated portion of concrete were carried out after moisturising the surface of concrete sample with dimensions 7 x 7 x 28.4 cm and spraying over the phenolphthalein solution. The phenolphthalein solution reveals the interface between the healthy zone and the carbonated zone. Measurements correspond to the distances (in mm) between the external surface of the concrete and the face of colouring. They were carried out on the face of colouring except in the corners and where the face of carbonation passes through an interface aggregate-matrix. The aggregates with D>3mm were given the importance. Five measurements were taken for each face. The value corresponding in the results is the average of the average values of the four faces.

RESULTS AND ANALYSIS

Compressive strength of concrete

The compressive strength decreases when addition is substituted to cement, whatever the type of addition is limestone filler or fly ash (Figure 1). This is mainly due to the increase in W/C ratio with the increase in the substitution rate i.e. V A /V C , since the paste volume and the water content remain constant [START_REF] Barnett | Strength development of mortars containing ground granulated blast-furnace slag: Effect of curing temperature and determination of apparent activation energies[END_REF][START_REF] Poon | A study on high strength concrete prepared with large volumes of low calcium fly ash[END_REF]. The reduction in strength is not the same for LF and FA, what would reveal the difference of chemical activity between these two additions [START_REF] Barnett | Strength development of mortars containing ground granulated blast-furnace slag: Effect of curing temperature and determination of apparent activation energies[END_REF].

Influence of the mineral additives on the shrinkage of concrete

In the case of limestone filler, the total shrinkage and the autogenous shrinkage decrease with an increase in V A /V C (Figure 2). The reduction in autogenous shrinkage is due to the increase in W/C with V A /V C . The reduction in total shrinkage can be partly explained by this reduction in autogenous shrinkage. Besides, it appears that limestone filler allows also reducing the drying shrinkage (since the reduction in total shrinkage is higher than the reduction in autogenous shrinkage). This could mean that the use of limestone filler leads to an increase in the packing density and then to a decrease in the transfer of humidity. The reduction in autogenous shrinkage due to the substitution of cement by fly ash is not as high as in the case of limestone filler (Figure 3). Moreover, fly ash has low influence on the total shrinkage (and so on the drying shrinkage) until the age of 50 days. After this, the shrinkage curves of the mixtures with and without fly ash diverge. The tested fly ash seems to affect long-term drying shrinkage. The ages of cracking in the ring tests are given in Table 2 and the tensile stress curves in Figure 4 and in Figure 5. The increase in V A /V C (for both fly ash and limestone filler) leads to a reduction of the age of cracking. Thus, the cracking susceptibility can be said to increase with the cement content of the paste. For a given substitution rate, the mixtures containing fly ash cracked later than those containing limestone. This result appears first paradoxical, because the shrinkage (and so the tensile stress) is higher for the mixtures with fly ash than for those with limestone. This means that the cracking phenomenon involves the cracking resistance as much than the tensile stress due to restrained shrinkage.

Effect of carbonation

During the service-life of concrete, CO 2 from the atmosphere penetrates in the concrete pores. This penetration is more fast when concrete is more porous. But this gas can be dissolved in the residual water in some pores. It can then react with cement and form carbonates. This reaction called as carbonation decreases the concrete pH down to a value close to 9.5. So, carbonation starts on concrete surface, and concerns some thickness (called carbonation depth) of this material.

From 24 hours after the batching, specimens were stored at 20°C and 50% RH to assess the carbonation depths (after 18 months). The results of carbonation depths are plotted in Figure 6. Compared to reference concrete (V A /V C =0), the concrete incorporating mineral admixtures showed lower resistance to carbonation, possibly due to the dominating effect of the reduction in calcium hydroxide over pore refinement [START_REF] Papadakis | Effect of supplementary cementing materials on concrete resistance against carbonation and chloride ingress[END_REF]. This drop of resistance to carbonation is observed whatever type of addition is used, so pozzolanic activity of fly ash would not have any effect [START_REF] Ganesh Babu | Efficiency of GGBS in concrete[END_REF].

Acid attack (Leaching)

The quantity of leached hydroxide ions was assessed through the added volumes of nitric acid HNO 3 solution, and leached calcium by the titration of the solutions. They are plotted against square root of time in Figure 7 and in Figure 8. Parameters of leaching kinetics may be deduced from quantity of leached ions versus square root of time curves. Steady state was assumed to be reached after one day. Corresponding parameters are given in Table 3 for selfcompacting concrete and in table 4Erreur ! Source du renvoi introuvable. for vibrated concrete for the following equation:

t a t N . ) ( = +b.
Table 3: Parameters of leaching kinetics for self compacting concrete From the parameters of the leaching kinetics for self compacting concrete (see table 3) and the uncertainty on the results (approximately 10%), only better behaviour of V A /V C = 0 can be deduced for calcium leaching and V A /V C = 0.80FA for hydroxide leaching. Higher value for V A /V C = 0.80LF concrete mixture could come from leaching of limestone filler. But the amounts of leached ions at the end of the tests are very close to each other. As limestone aggregates were used, they are likely to be attacked and the corroded depth cannot be deduced from the amounts of leached ions. After immersion, the corroded depth can be measured using Phenolphthalein solution. Results in Figure 9 show that the layer which is corroded by leaching is very thin, for the three concrete mixtures. In Figure 9.a and Figure 9.c it can be seen that the interface of aggregates has drawn back faster than the interface of the matrix. Degradation of the concrete matrix with V A /V C = 0 is so thin that it cannot be measured, and corroded layer of V A /V C = 0.80FA is thinner than corroded layer of V A /V C = 0.80LF. Better behaviour of concrete incorporating fly ash could come from chemical compositions, because fly ash is mainly composed of silica, whereas limestone filler is mainly composed of calcite CaCO 3 , which is likely to be leached.

V A /V C =0 V A /V C =0.80 V A /V C =0
The same test was carried on vibrated concrete mixtures, incorporating siliceous aggregates (S1, S2, S3 and S4) and limestone aggregates (B4 and B5). The concrete mixtures are presented in table 4. The corroded depths of two out of three self-compacting concrete (SCC) mixtures (V A /V C = 0 and V A /V C = 0.80FA) are in the range of the vibrated concrete mixtures of the closest W eff /Binder ratios. V A /V C = 0.80LF concrete mixture shows faster corrosion. So good compactness of SCC mixtures ensures equivalent behaviour of concrete exposed to leaching, provided that chemical composition of binder is adequate for this exposure.

CONCLUSIONS

The results presented in this paper are part of an ongoing study on durability of concrete made using mineral additives and exposed to chemically aggressive environments. The volume of paste is kept constant and the aim is to assess performances of different selfcompacting concretes using mineral additives like Fly ash and Lime stone Filler.

Results of compressive strength show the difference in strength for both type of additions compared to concrete without any addition which may be due to late activation of the mineral content. Hence, adequate curing is recommended for enhancing the resistance of concrete containing FA to carbonation. Concrete containing high proportions of fly ash and limestone filler are likely to have lower portlandite content and they can show higher carbonation rates.

As far as leaching is concerned, the test could provide relevant data. It may be considered as representative, as the mechanism of degradation is not changed. Taking into account the uncertainty, some significant differences may be deduced. Cement and binders containing fly ash shows better behaviour than binder containing limestone filler, because of the chemical composition of the filler. SCC shows the same resistance to leaching as vibrated concrete, for close water/binder ratios. So, as it occurs for other exposures, durability of concrete exposed to leaching depends on two main parameters: compactness and reactivity of binder. The two parameters may be varied to achieve a level of performance for one exposure, but curing has also to be taken into account to ensure durability of concrete for some exposures, such as carbonation.

Figure 1 :

 1 Figure 1: Compressive strength vs. V A /V C

Figure 2 :Figure 3 :Figure 4 :Figure 5 :

 2345 Figure 2 : Influence of limestone filler on the shrinkage of concrete.

Figure 6 :

 6 Figure 6: Natural carbonation depths

Figure 7 :

 7 Figure 7 : Leached hydroxide ions.

Figure 8 :

 8 Figure 8 : Leached calcium ions

Figure 9 :

 9 Figure 9: Profiles of degradation by leaching.

Table 1 :

 1 Materials of the concrete mixtures

	Gravel	3/8 mm rolled gravel
	Sand	0/4 mm sea sand
	Cement	CEMI 52.5 type ; 0/100µm ; Blaine = 405 m 2 /kg
	Limestone filler (LF) 0/100µm ; Blaine = 397 m 2 /kg
	Fly ash (FA)	Median diameter: 14 µm ; SiO 2 : 53 %

Table 2 :

 2 Compositions and properties

Table 4 :

 4 Parameters of leaching kinetics for vibrated concrete

	.40


M T Dau

C Vergnaud

A Marty

F Rortais

C Beigné

H Boukari

E Bellet- Amalric

V Guigoz

O Renault

C Alvarez

H Okuno

P Pochet

M Jamet

Millimeter-scale layered MoSe2 grown on sapphire and evidence for negative magnetoresistance

Keywords: molecular beam epitaxy, 2D materials, transition metal dichalcogenides MoSe2, variable range hopping, negative magnetoresistance

Molecular beam epitaxy technique has been used to deposit a single layer and a bilayer of MoSe2 on sapphire. Extensive characterizations including in-situ and ex-situ measurements show that the layered MoSe2 grows in a scalable manner on the substrate and reveals characteristics of a stoichiometric 2H-phase. The layered MoSe2 exhibits polycrystalline features with domains separated by defects and boundaries. Temperature and magnetic field dependent resistivity measurements unveil a carrier hopping character described within two-dimensional variable range hopping mechanism. Moreover, a negative magnetoresistance was observed, stressing a fascinating feature of the charge transport under the application of a magnetic field in the layered MoSe2 system. This negative magnetoresistance observed at millimeter-scale is similar to that observed recently at room temperature inWS2 flakes at a micrometer scale [Zhang et al., Appl. Phys. Lett. 108, 153114 (2016)]. This scalability highlights the fact that the underlying physical mechanism is intrinsic to these two-dimensional materials and occurs at very short scale.

I. INTRODUCTION

Layer-structured transition metal dichalcogenides (TMDs) have drawn much attention recently since they are being considered as a new class of semiconducting twodimensional (2D) materials with thickness-tunable band-gap modulation [1][2][3]. In the singlelayer configuration, the TMDs offer a unique platform to explore not only the carrier transport in an ultrathin channel but also the control of 2D excitonic systems and the spinvalley physics [4][5][6]. These may be achieved owing to their electronic structure and spatialinversion symmetry that give rise to a strong spin-obit coupling, i.e. a large spin-splitting of the valence band, and to a versatile control of the spin and valley properties of the carriers [7,8].

Chemical vapor deposition has been employed for fabrication of TMDs layers. This technique enables to a large coverage of species on substrates [9][10][11]. However, the TMDs layers obtained by this method show an inhomogeneity and discontinuous nature due to the absence of precise thickness control. Molecular beam epitaxy (MBE) is of particular interest because one can achieve high purity of the layers, precise control of the thickness, flexible choice of metals and scalability [12,13]. Since a single layer of TMDs is separated from each other by an out-of-plane van der Waals gap, interactions between TMDs layers and a chemically passivated substrate are expected to be of van der Waals type. In this regard, kinetic energies of deposited atoms are in agreement with the dynamism of van der Waals assembly by employing MBE technique.

Here, we report the achievement of MBE MoSe2 layers on sapphire-c(0001) combined with electrical demonstration. MoSe2, which belongs to TMDs family, exhibits abundant functionalities related to its atomically thin crystal and electronic structure, leading to a wide range of applications in electronics, optoelectronics, valley-spintronics and photocatalysis [1,14,15]. Very recently, MoSe2 gained more attention and has been claimed as a potential candidate for room temperature exciton-polaritonic devices [16]. In this letter, we focus on growth and characterizations of single layer MoSe2 and on electrical properties of bilayer grown at 550 °C. In-plane X-ray diffraction (XRD) with respect to the azimuths of the substrate shows that the layer does not have any preferential in-plane orientation.

Polycrystalline character of the layer results from disoriented domains those size is estimated about 8.0 nm. A clear spatial separation of 6.5 Å between the layer and the substrate, which corresponds to a van der Waals gap, was evidenced by cross-sectional scanning transition electron microscopy (STEM). The layer was found to be stoichiometric based on X-ray photoemission spectroscopy (XPS) and its homogeneity was demonstrated by using Raman spectroscopy. The electrical measurements as a function of temperature and magnetic fields were performed on the bilayer MoSe2 with the contacts separated by milimeter-scaled distance. The carrier transport exhibits a clear temperature dependence which is well described by the two-dimensional Mott's variable range hopping (VRH) model.

Interestingly, a decrease in resistance, i.e. negative magnetoresistance (MR), was found up to room temperature when an external magnetic field was applied. With the transport study in WS2 flakes [17], the observation of negative MR in the MBE MoSe2 clearly points to intrinsic feature of transport properties of 2D TMDs which is extended down to very small scale.

II. RESULTS AND DISCUSSION

Reflection high-energy electron diffraction (RHEED) patterns and intensity profiles of the post-annealed monolayer are shown in Fig. 1(a). The RHEED diagrams are streaky and remain the same when rotating the samples around the growth axis. That indicates a good crystalline quality of the surface and a presence of multi-domains having no preferential inplane orientation. Full width at half maximum (FWHM) of MoSe2 (01 ̅ ) streak normalized over streak-to-streak distance is plotted and displayed in Fig. 1(b). It can be seen that the layer grown at 550 °C has a smallest FWHM, indicating a largest coherent size of domains in this layer compared to those of other layers. The structural properties of MoSe2 were investigated by in-plane XRD (Fig. 1(c)). The reflections of the MoSe2 monolayer grown at 550 °C are identified as (100), ( 110) and (200), and are consistent with a 2H-MoSe2 phase [18]. The extracted in-plane lattice parameter is 3.28 ± 0.01 Å, which is identical to the bulk value. This is a strong indication of a van der Waals epitaxy which permits a structural adaptation despite a large misfit (31.3%) between the substrate and the layer. The presence of both (100) and (110) in-plane reflections of the MoSe2 along both the (h00) and (hh0) directions of the substrate is in agreement with the polycrystalline nature of the layer as observed in RHEED patterns. From the broadening of the MoSe2 peaks taken in Fig. 1(c) and using the Scherrer equation, one can estimate the size of the monodomains to about 8 nm. Such a small domain size implies the presence of many structural defects, boundaries in a continuous layer. substrate and deposited adatoms could be a possible reason for the formation of this buffer layer.

High-resolution Mo3d and Se3d XPS spectra are shown in Fig. 3(a) and 3(b). The Mo3d peak fitting highlights a main doublet (component I) with the 3d5/2 peak located at 229.2 ± 0.1 eV binding energy, and is assigned to stoichiometric MoSe2: indeed, the energy is in agreement with the value of 229.3 eV reported for bulk MoSe2 [19]. The energy separation of 174.4 eV for the main Se3d peak is also consistent with stoichiometric MoSe2 systems [19,20]. In the Mo3d spectrum, a second, much weaker doublet, denoted as component II, is shifted by 1 eV to higher binding energies and is most likely due Mo suboxides [21]. Oxidized Se species are also found with the weak Se3d component II. Besides the Se3s line, the Mo3d spectrum fitting also evidences a third doublet measured at 233.7 eV (3d5/2 binding energy) and assigned to MoO3 species. We attribute these states to the interfacial layer as observed from STEM. We determined the stoichiometry of the MoSe2 layer by considering the area of I and II Mo3d components, the area of the Se3d peak and the suitable relative sensitivity factors: by this we infer an overall composition of the layer of MoSe1.97, which confirms the excellent quality of the sample, close to that of bulk MoSe2.

The Raman results are presented in Fig. 3(c) and 3(d). In Fig. 3(c), the main contributions are assigned to the first-order Raman peaks Ag 1 and E2g 1 , which correspond respectively to the out-of-plane and in-plane vibration modes of 2H-MoSe2. The Raman shifts are 241.1 cm -1 for Ag 1 and 289.7 cm -1 for E2g 1 . These values are in good agreement to the ones reported for bulk and exfoliated MoSe2 [22]. The E2g 1 peak is significantly broadened. This might be related to the structural defects in the layer including Se vacancies and domains [23] or to multi-peaks arising from the activation of the second-order Raman processes [24]. A much weaker peak is also detected around 338 cm -1 and attributed to multi-phonon scattering [24]. It is known that the breathing mode B2g 1 can be either Raman or an infrared active due to the spatial symmetry of the TMDs [25,26]. This mode is Raman active for few monolayers and the corresponding peak appears at around 352 cm -1 , whereas it becomes Raman inactive for one monolayer. For our monolayer MoSe2, the absence of B1 2g mode in the spectrum confirms that the layer consists of one-layer sheet within the spatial resolution of the laser spot. Additionally, the two characteristic peaks Ag 1 and E2g 1 of MoSe2, are homogenously observed and their position does not shift wherever the acquisition takes place. The intensity ratio of these two peaks, which is used as an indicator of a homogeneous thickness, is also mapped over a size of 25x25 μm 2 and displayed in Fig. 3(d). The mapping shows an homogeneity of the layer and thoroughly corroborates the macroscopic observations. It is worth noting that the measurements were carried out at different positions of the sample and showed similar results.

The structural and spectroscopic properties of one monolayer MoSe2 on sapphire were found to be comparable to that of bulk. The relevant attributes of the layer, that are polycrystalline feature with disoriented domains and nanoscopic disruption, were evidenced by analyzing XRD and STEM. Owing to the disruption, the current injection could not be achieved in the monolayer. To study the transport with a macroscopic contacts length, we grew a bilayer MoSe2 to make sure continuity in the layer. The carrier transport was investigated with four points geometry illustrated in Fig. 4(b). The cross-sectional STEM image (Fig. 4(a)) shows that the interface between Mo/Pd and MoSe2 is continuous and smooth. Fig. 4(c) displays a plot of temperature-dependent resistivity and the corresponding conductivity. An insulating characteristic of the layer with exponential dependence of resistivity was found for the studied range of temperature. As discussed previously for the single layer MoSe2, the bilayer is also polycrystalline (isotropic RHEED patterns), hence, it is expected to be composed of disoriented domains with high density of vacancies and boundary defects. The structural defects could give rise to localized states that are in-gap electronic states and located near the Fermi level as revealed by density functional theory calculations and scanning tunneling spectroscopy in layered-TMDs systems [9,27]. In such a strongly disordered layer, VRH is expected to be dominant among transport mechanisms [28]. The logarithm of conductivity is also plotted (Fig. 4(c)) and found to be linearly dependent on T -1/3 , which is indeed in good agreement with the VRH transport theory [28].

Note that the fit was made by assuming that the MoSe2 bilayer is of two-dimensional nature.

The conductivity can therefore be expressed as: 𝜎 ~𝑒𝑥𝑝[-(𝑇 0 /𝑇 1/3 )] where T0 is the Mott characteristic temperature. The value of T0 obtained by the fit is 1.1x10 6 K. This high value, which is comparable to the one recently reported in a thick layer of MoSe2 [29], implies a strong localization of carriers mediated by the localized states. Furthermore, the Mott characteristic temperature in 2D system can be written: 𝑇 0 = 13.8/(𝑁 𝜇 𝑘 𝐵  2 ) where N is density of states near the Fermi level, kB is Boltzmann constant and  is localization length [30]. By assuming temperature-independent 𝑁 𝜇 = 𝑔 𝑣 𝑚 * /(𝜋 ħ 2 ) = 4.77 x 10 14 eV -1 cm -2 with inplane effective mass m * =0. 57m0 and valley degeneracy gv = 2 for MoSe2 [31],  can be written:  = [13.8/(𝑁 𝜇 𝑘 𝐵 𝑇 0 )] The magnetoresistance, MR(%)=[R(T)-R(0)]/R(0), was found to be negative over the whole range of magnetic fields. The representing data obtained at 250 K are shown in Fig.

4(d).

One can see that the MR is negative throughout the magnetic field range and does not show any indication of saturation at 0H = 5 T. We note that the negative MR was also observed while the magnetic field was applied parallel to the sample surface, indicating that the negative variation of the resistance is related as well to the anisotropic orbital effect.

Earlier reports on the magneto-transport properties of highly defected graphene [32,33] and of exfoliated multilayers TMDs WS2 [17] pointed out the negative MR in layered system.

The finding of negative MR in bilayer MoSe2 clearly stresses the pivotal implication of defects in the magnetic field dependent carrier transport. In our bilayer, the negative MR, which has never been observed so far for MBE or CVD TMDs, likely results from the hopping character of the VRH transport at high temperatures where the quantum corrections may not set in due to the predominance of inelastic scattering events. In a system exhibiting a low charge carrier density and a high degree of disorder, the hopping rate between localized states writes:

𝑃~𝑒𝑥𝑝 ( 2𝑅  - 1 𝜋𝑅 2 𝑁 𝜇 𝑘 𝐵 𝑇
), where R is the hopping length [34]. The optimal hopping length corresponding to a maximum hopping rate is deduced:

𝑅 0 = (  𝜋𝑁 𝜇 𝑘 𝐵 𝑇 ) 1/3
. The maximum hopping rate P is therefore proportional to localization length . Consequently, the decrease of resistance seems to result from a change of  due to magnetic field.

Negative MR in VRH regime was theoretically studied and found linear or quadratic in magnetic field on the basis of the hopping lengths  and R0 [35]. The linear negative MR of bilayer MoSe2 in low magnetic fields can be simply described by these models where the hopping is associated with interference of various paths. In the moderate-field region, MR is linearly dependent on magnetic field and is given: 𝑀𝑅~𝑅 0 3/2  1/2 𝐵 = 𝑓()𝑇 -1/2 𝐵. The slope values obtained with the MR curves in the range of 0.05-0.75 T at different temperatures were obtained and then plotted versus temperature (see Supplementary Material). With the data in hand, the linear fit of ln(slope-MR) vs ln(T) yields -0.46 ± 0.12 which is somewhat different from -0.5. That is probably due to a significant dependence of the localization length on the magnetic field as we aforementioned. It is noted that a possible contribution on negative MR stems from polaron-type interactions of local magnetic moments coupled with localized carriers. Spin polarization occurred in metal orbitals due to a large density of Se vacancies would enable long-range coupling of magnetic polarons with an application of magnetic fields [36]. This could increase the carrier hopping rate, thus giving rise to the decrease in resistance of the layer. For the studied range of temperatures, this kind of interaction is probably weak in MoSe2 layer due to thermal fluctuations and phonon scattering. Finally, the origin of the negative MR related to an enhancement of mean free path for carriers scattering at boundaries can be ruled out in our layer for the following reason. The domain size of MoSe2 layer is estimated about 8 nm which is in the length range where boundaries might scatter the charge carriers over a long distance as a result of the enhanced mean free path when applying magnetic fields. However, a quadratic dependence of resistance on magnetic field was predicted in this case and that clearly does not fit with the observed linear tendency of the MR [37]. Therefore, the magneto-transport in MoSe2 layer is mainly attributed to the VRH process via localized states associated with the enhancement of the hopping rate, giving rise to a negative MR.

III. CONCLUSION

In conclusion, we have investigated the fabrication of a single layer and a bilayer of MoSe2 on sapphire using MBE. We found that the MoSe2 layers are homogeneous over the substrate dimensions. By combining in-situ RHEED, in-plane XRD and STEM, we have shown that the layers are of polycrystalline nature and composed of nanometer-sized monodomains with free rotational angles. XPS and Raman spectroscopy confirm their stoichiometry and the homogeneous character over a large scale. The insulating characteristic of layered MoSe2 was pointed out and interpreted in the theoretical framework of VRH. An unexpected negative magnetoresistance was evidenced at high temperatures and high magnetic fields. In the low charge carrier density system, this transport nature probably stems from the effect of magnetic field on the hopping characteristic through localized states induced by vacancy defects. This finding has revealed the intrinsic transport properties of layered TMDs which are valid for both natural and artificial sources at sub-micro scale.

Raman measurement were done with a Horiba Raman set-up using a laser excitation source f 632 nm with spot size of 0.5 μm. The signal was collected by choosing a 1800 grooves/mm grating. XPS measurements were carried out with a PHI 5000 VersaProbe II spectrometer delivering a micro-focused (100 m), monochromatized Al-K radiation (1486.6 eV). The overall energy resolution (X-ray bandwidth and electron analyser broadening) was 0.4 eV.

Electrical measurements

For electrical measurements, it is well known that metallic contacts deposited on exfoliated TMDs exhibit a large Schottky barrier due to the large bandgap Eg of the TMDs layer and its ultra-thin thickness. When TMDs layers are undoped, it is extremely difficult to inject charge carriers in the layers. To circumvent this, we have fabricated in-situ Mo(5nm)/Pd(10 nm) contacts (2 mm x 3 mm) that were deposited directly on the as-grown

MoSe2 layer using a movable mechanical mask underneath the sample in the UHV chamber.

The sample was then annealed at 200 °C for few minutes to obtain a good quality of contacts. The Mo/Pd pads were connected to a chip carrier with silver paste and Au wires.

The temperature and magnetic field dependent transport measurements were performed with input currents of 10 nA and 50 nA respectively.

I-V curve is shown in Fig. S1 indicating an Ohmic nature of the contacts. Figure S2 shows

Figure 2

 2 Figure2displays the cross-sectional STEM micrograph of the MoSe2/sapphire interface. The bright line, which is parallel to the surface of the substrate, corresponds to the monolayer of MoSe2. It can be seen that the interface between the layer and the sapphire is abrupt with a separation of about 6.5 Å wide as for the interlayer van der Waals gap in MoSe2. This underlines the expected weak chemical coupling between the MoSe2 and the substrate, resulting in a quasi-free-standing monolayer of MoSe2 on sapphire. The STEM image also reveals a disrupted region (dotted arrow in Fig.2(b)) where defects or domain edges might be located and they supposedly result from domain merging, Mo depletion or Se vacancies. Finally, one clearly distinguishes a regular bright point network at the topmost layer of the substrate. This seems to be of different chemical nature from the substrate due to different contrasts. Interfacial interactions involving oxygen of the first layer of the

FIG. 1 .

 1 FIG. 1. (a) RHEED patterns of post-annealed samples grown at temperatures from 400 °C to 650 °C. The corresponding intensity profiles are shown on the top panel of the patterns. (b) FWHM of the (01 ̅ ) streaks taken from (a) normalized by the distance between (01 ̅ ) and (00) streaks as a function of growth temperature. (c) In-plane XRD of the monolayer MoSe2 grown at 550 °C along two principal reflections (h00) and (hh0) of the substrate.

FIG. 2 .

 2 FIG. 2. (a) Ball-and-stick model represents the interface between the single layer and the substrate. (b) Cross-sectional STEM image of the MoSe2/sapphire interface. The white arrows indicate the van der Waals gap and the dotted arrow indicates a disruption of the single layer.

FIG. 3 .

 3 FIG. 3. (a) and (b) are respectively, Mo3d and Se3d XPS core-level spectra. (c) Positiondependent Raman shift in the 210-360 cm -1 region. The intensity was normalized by Ag 1 peak intensity. (d) Intensity map of the E2g 1 /Ag 1 ratio with a standard deviation  0.012.

1/ 2 FIG. 4 .

 24 FIG. 4. (a) Cross-sectional STEM images of the bilayer MoSe2/sapphire interface and the entire stack Mo/Pd/MoSe2/sapphire (the inset). (b) Sketch of the measurement set-up with I the electrical current, V the bias voltage and H the applied magnetic field. (c) Carrier conductivity against temperature and the logarithm of the conductivity against T -1/3 fitted with the 2D-VRH model (solid red line). (d) Magnetic field dependence of the magnetoresistance measured at 250 K with an external magnetic field applied parallel (H//) and perpendicular (H) to the sample surface.

  MR at different temperatures with the linear fitting window of MR curves. The inset is the logarithmic dependence of fitted slopes-MR against temperature that allows us to infer the relationship slope-MR  T -0.46

Fig

  Fig. S1. I-V curve of the contact MoSe2/Mo/Pd measured at 250 K.

Fig. S2 .

 S2 Fig. S2. Magnetoresistance of the bilayer at different temperatures. The yellow window represents the mediated-field region where the slope-MR was obtained. The inset is the plot of slope-MR versus temperature.

ACKNOWLEDGMENTS

The authors thank Vincent Mareau, Laurent Gonon and João Paulo Cosas Fernandes for their assistance and discussions with Raman spectroscopy. The authors acknowledge Nicolas Mollard for preparation of TEM samples and Lucien Notin for technical assistance with MBE. This work is supported by the CEA-project 2D FACTORY and by the Agence Nationale de la Recherche within the ANR MoS2ValleyControl and 2D Transformers contracts. The LANEF framework (No. ANR-10-LABX-51-01) is also acknowledged for its support with mutualized infrastructure.

SUPPLEMENTAL INFORMATION

MBE growth and substrate preparation

All samples were grown in a MBE chamber with a base pressure of about 5x10 -10 mbar which increased up to about 2x10 -8 mbar during the co-deposition of Mo and Se. The Mo and Se are respectively evaporated by an e-beam gun and a home built evaporation cell.

The direct Se flux measured at the sample position with a retractable Bayard-Alpert gauge is about 2x10 -6 mbar. The deposition rate of the Mo measured with a quartz micro-balance was 1.5 Å.min -1 . The post-annealing was performed at 720 °C during 15 min under Se flux.

The as-received sapphire (0001) substrates were first cleaned in ultrasonic bath using acetone and iso-propanol, then annealed in a quartz tube oven in air at 300 °C for 30 minutes to desorb the organic species from the surface. The substrates were then heated up to 1000•C for 1h to smoothen the surface. They were finally loaded in the MBE system and outgassed in the UHV chamber at 700 °C for 15 minutes just before the deposition of the MoSe2. After the preparation process, the sapphire surface is expected to be chemically inert with an Oxygen-Aluminium plane polarity.

Structural analysis

X-ray diffraction analysis was performed with a SmartLab Rigaku diffractometer equipped with a Copper Rotating anode beam tube (K = 1.54 Å) and operated at 45 kV and 200 mA. A parabolic mirror and a parallel in-plane collimator of 0.5° of resolution was used in the primary optics and a second parallel collimator was used in the secondary side. A K filter was used for all the measurements. A Nanoscope Atomic force microscopy was used in tapping mode to image the samples. Scanning transmission electron microscopy was operated at an accelerated voltage of 80-200 kV using a probe aberration corrected microscope (Titan Themis FEI). The ball-and-stick model is built with VESTA software.

Spectroscopic characterization