Gabriele Scrivanti

Émilie Chouzenoux

Jean-Christophe Pesquet

A CNC approach for Directional Total Variation

Keywords: Directional Total Variation, Denoising, Nonconvex and non-smooth regularisation, Primal-Dual Algorithm

The core of many approaches for the resolution of variational inverse problems arising in signal and image processing consists of promoting the sought solution to have a sparse representation in a well-suited space. A crucial task in this context is the choice of a good sparsity prior that can ensure a good trade-off between the quality of the solution and the resulting computational cost. The recently introduced Convex-Non-Convex (CNC) strategy appears as a great compromise, as it combines the high qualitative performance of non-convex sparsity-promoting functions with the convenience of dealing with convex optimization problems. This work proposes a new variational formulation to implement CNC approach in the context of image denoising. By suitably exploiting duality properties, our formulation allows to encompass sophisticated directional total variation (DTV) priors. We additionally propose an efficient optimisation strategy for the resulting convex minimisation problem. We illustrate on numerical examples the good performance of the resulting CNC-DTV method, when compared to the standard convex total variation denoiser.

I. INTRODUCTION

A wide class of problems arising in image and signal processing can be described by sparsity-regularised variational models. The most natural sparsity inducing penalty is the ℓ 0pseudo-norm, but it induces the related problem to be NP hard and non-convex. A popular convex variational surrogate is the ℓ 1 -norm, though it has the drawback of under-estimating the high amplitude components of the considered signal. Nonconvex variational regularisers manage to overcome this issue, at the cost of possibly introducing suboptimal local minima in the objective function. An efficient solution to keep only the best traits of these regularisers is represented by Convex-Non-Convex (CNC) strategies [START_REF] Selesnick | Convex 1-d total variation denoising with non-convex regularization[END_REF]- [START_REF] Selesnick | Non-convex total variation regularization for convex denoising of signals[END_REF]. They consist of building convex objective functionals that include non-convex regularisation terms. Suitable CNC strategies have been designed for more and more general classes of problems in different fields of data processing (see [START_REF] Lanza | Sparsity-inducing nonconvex nonseparable regularization for convex image processing[END_REF] and the references therein). For the well-known Total Variation (TV) regularisation model [START_REF] Rudin | Nonlinear total variation based noise removal algorithms[END_REF], a CNC modification has shown to get around the notorious problems of boundary reduction and staircasing effect for image restoration [START_REF] Selesnick | Convex 1-d total variation denoising with non-convex regularization[END_REF], [START_REF] Lanza | Sparsity-inducing nonconvex nonseparable regularization for convex image processing[END_REF], [START_REF] Selesnick | Non-convex total variation regularization for convex denoising of signals[END_REF] and image segmentation [START_REF] Chan | Convex non-convex image segmentation[END_REF].

In this work we propose a new variational formulation extending the CNC strategy to a popular TV-based model, namely the Directional TV (DTV). DTV was firstly introduced in [START_REF] Bayram | Directional total variation[END_REF] for the restoration of images whose structures or textures This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 861137. follow a single dominant direction. The idea is to provide the model with a suitable directional prior information in order to allow the regularisation to operate more efficiently. The model was then extended in [START_REF] Zhang | Edge adaptive directional total variation[END_REF] to handle multiple dominant directions by means of a spatially varying directional information that locally coincides with the edge directions of the image. In [START_REF] Kongskov | Directional total generalized variation regularization[END_REF] and [START_REF] Parisotto | Higher-order total directional variation: Analysis[END_REF] the authors analysed the possibility to extend the directional approach to the Total Generalised Variation (TGV) model from [START_REF] Bredies | Total generalized variation[END_REF], which takes into account higher-order derivatives of the image and allows to promote piecewise-affine reconstructions, in particular. A possibly nonconvex version of DTV was proposed in [START_REF] Calatroni | A flexible spacevariant anisotropic regularization for image restoration with automated parameter selection[END_REF] by considering a space-variant exponent for the DTV regulariser that adapts to the local smoothness of the image.

Our contributions in this work are the following. We infer a formulation for DTV that allows us to incorporate this regulariser into a CNC denoising framework. We then define a numerical procedure to efficiently address the resulting optimisation problem. We finally provide experimental results that support the use of the proposed regularisation method.

The paper is organised as follows. In Section II, we introduce a general TV-based image denoising problem. We describe the construction of a CNC sparsity-promoting function, and provide a sufficient condition for the convexity of the overall problem. Section III is dedicated to the derivation of our proposed CNC-DTV regularisation approach and Section IV provides the description of a dedicated optimisation procedure to tackle it. Section V presents our numerical results showing the interest of the proposed approach.

II. IMAGE DENOISING VIA CNC TOTAL VARIATION

A. Notation

Throughout this paper we denote by ⟨• | •⟩ the scalar product over R n and ∥ • ∥ 2 the associated Euclidean norm. I n states for the n × n identity matrix.

The spectral norm is defined as ∞, +∞] that are proper (i.e., with a nonempty domain), lower semicontinuous and convex. For a function f ∈ Γ 0 (R n ), function f * : R n → [-∞, +∞] represents its convex conjugate that is defined as

|||L||| = sup{∥Lz∥ 2 | z ∈ R n , ∥z∥ 2 ≤ 1}. Γ 0 (R n ) indicates the class of functions f : R n → (-
(∀u ∈ R n ) f * (u) = sup x∈R n {⟨x, u⟩ -f (x)}.

B. Image denoising

We focus on solving an image denoising problem, i.e. on restoring a source image x ∈ R n from an acquired

measurement o ∈ R n that is related to the sought image through o = x + e, (1)
where e ∈ R n is an additive noise, here assumed to be i.i.d. zero-mean Gaussian. Then, a simple denoising strategy consists of defining an estimate x ∈ R n of x by solving the penalized least squares problem minimize

x∈R n J (x) = ψ(x) + λ 2 ∥x -o∥ 2 2 . (2)
Hereabove, function ψ : R n → (-∞, +∞] is the regularization term, associated to the regularization factor λ -1 , with λ > 0.

C. Total-variation based regularizers

When dealing with images, a common approach consists of choosing ψ so as to sparsify the sought image in a transformed space obtained via a linear transformation. This amounts to defining ψ = Ψ • D where D ∈ R m×n is a linear operator and Ψ : R m → (-∞, +∞] is a sparsity-promoting term. Among possible choices for Ψ and D, the total variation (TV) model [START_REF] Rudin | Nonlinear total variation based noise removal algorithms[END_REF] is probably the most celebrated one. TV regularizer promotes sparsity in the space of the vertical/horizontal gradients of the image, thus allowing piece-wise constant functions in the solution space. Matrix D ∈ R 2n×n is set as the linear operator defined as 2 are the discrete horizontal and vertical 2D gradient operators obtained with a finite difference scheme. Then,

D = [D ⊤ h D ⊤ v ] ⊤ , where (D h , D v ) ∈ (R n×n )
(∀x ∈ R n ) TV(x) = ∥Dx∥ 1,2 , = n i=1 ∥(Dx) i ∥ 2 . (3)
Here, for every u ∈ R 2n and i ∈ {1, . . . , n}, we use the compact notation:

u i = (u i , u n+i ) ∈ R 2 .
We can also provide another definition of TV, based on duality [START_REF] Chambolle | An algorithm for total variation minimization and applications[END_REF]:

(∀x ∈R n ) TV(x) = max u∈R 2n {⟨Dx | u⟩ | u i ∈ B 2 , i ∈ {1, . . . , n}}, (4) with B 2 = {υ ∈ R 2 | ∥υ∥ 2 ≤ 1} the unit closed ball of R 2 .
One drawback of TV is that it gives an isotropic role to vertical and horizontal directions, that might not be welladapted for natural images. In [START_REF] Bayram | Directional total variation[END_REF], a modified version of TV is proposed, that is more suitable for images containing objects with a (possibly non vertical/horizontal) dominant direction. The idea is to introduce an affine transformation in the dual space (i.e., the space of the image gradients). This transformation is parametrized by an expansion factor α ≥ 1 and a rotation angle θ ∈ [-π/2, π/2]. We then define the transition matrices:

R θ = cos θ -sin θ sin θ cos θ , Λ α = 1 0 0 α .
These are used to build the elliptic set

E α,θ ⊂ R 2 given by E α,θ = R θ Λ α B 2 , that is then substituted for B 2 in (4).
Actually, shapes present in images might have more than one directional orientation. In order to adapt the previous regularizer to each edge directions in the image, following [START_REF] Zhang | Edge adaptive directional total variation[END_REF], we can rely on a set of local ellipses E αi,θi ⊂ R 2 for each pixel i ∈ {1, . . . , n}. This leads to the so-called directional TV (DTV) regularizer, defined as

(∀x ∈ R n ) DTV(x; α, θ) = n i=1 max u∈E α i ,θ i ⟨(Dx) i |u⟩. (5)
Here, α = (α i ) 1≤i≤n , θ = (θ i ) 1≤i≤n are a predefined set of scaling factors/angles describing the edge direction information at each pixel of the image.

It is worth noticing that DTV is actually a generalized version of TV. Indeed, by choosing α i = 1 and θ i = 0 for every i ∈ {1, . . . , n}, one retrieves the classic TV functional as expressed in (4), since

E 1,0 = R 0 Λ 1 B 2 = B 2 .
DTV can be described in a more compact manner, by introducing the operator A α,θ : R 2n → R 2n that encodes the underlying local affine transformations. This linear operator is such that, for every u ∈ R 2n , A α,θ u = (R θi Λ αi u i ) 1≤i≤n . We can also express its inverse, through

A -1 α,θ u = Λ 1/αi R -θi u i 1≤i≤n for every u ∈ R 2n . Then, an equivalent definition of DTV is (∀x ∈R n ) DTV(x, α, θ) = max u∈R 2n {⟨Dx | u⟩ | (A -1 α,θ u) i ∈ B 2 , i ∈ {1, . . . , n}}. (6)
The aforementioned (D)TV prior relies intrinsically on the convex sparsity measure ℓ 1,2 . However, as emphasized in [START_REF] Zhang | Nearly unbiased variable selection under minimax concave penalty[END_REF], [START_REF] Soubies | A continuous exact \ell 0 penalty (cel0) for least squares regularized problem[END_REF], non-convexity sparsity penalties might be key to obtain high quality results. The goal of this paper is to incorporate a CNC approach with the aim to obtain an enhanced image denoiser without complexifying the optimization procedure.

III. PROPOSED CNC-DTV METHOD

A. CNC approach

The idea of CNC, initially proposed in [START_REF] Lanza | Sparsity-inducing nonconvex nonseparable regularization for convex image processing[END_REF], [START_REF] Selesnick | Non-convex total variation regularization for convex denoising of signals[END_REF], is to use the following construction for the regularization term ψ = Ψ M •D in (2), with D the discrete gradient linear operator and

(∀u ∈ R 2n ) ΨM (u) = φ(u) -inf t∈R 2n φ(t) + 1 2 ∥M (u -t)∥ 2 2 φM (u) . ( 7
)
Function Ψ M is the non-convex modification of a convex sparsity promoting term φ ∈ Γ 0 (R 2n ), parametrized by a matrix M ∈ R k×2n . It is obtained by subtracting from function φ its so-called generalised Moreau envelope φ M : R 2n → R depending on matrix M [5, Definition 6]. This matrix plays a fundamental role, as it can be designed so as to guarantee both the efficiency of a non-convex regularisation approach and, at the same time, the overall convexity of the objective function, as we will discuss hereafter.

In [START_REF] Lanza | Sparsity-inducing nonconvex nonseparable regularization for convex image processing[END_REF], two main examples for the choice of function φ are reported, which lead to different instances of CNC-TV regularisations, namely φ = ℓ 1 yields an anisotropic version of the TV while φ equal to the Hessian Schatten norm [START_REF] Lefkimmiatis | Hessian schatten-norm regularization for linear inverse problems[END_REF] leads to a second-order extension of the TV regularizer. However, up to our knowledge, DTV (as defined in (6)) has not been explored in the context of CNC, and this is the aim of this work.

B. Convexity condition

A key feature of CNC approach is that, despite the nonconvexity of the introduced penalty Ψ M , it is still possible to recast the minimization problem as a convex one. More precisely, in accordance with [18, Example 75], Problem (2) with ψ = Ψ M • D and Ψ M given in [START_REF] Chan | Convex non-convex image segmentation[END_REF], can be reformulated as the following minimisation problem by introducing the dual variable y ∈ R 2n :

minimize (x,y)∈R n ×R 2n JM (x, y) = λ 2 ∥x -o∥ 2 2 - 1 2 ∥M Dx∥ 2 2 (8) + 1 2 ∥M y∥ 2 2 + φ(Dx) + φ * (M ⊤ M (Dx -y)).
Remark 1. The equivalent problem in ( 8) is slightly different from the one proposed in [4, Eqs (88)-( 89)] and [3, Eqs (52)-( 53)], which relied on a saddle point formulation.

Since φ is convex, so is its conjugate φ * , as well as their compositions with linear mappings. It results that the only non-convex term in the objective function in [START_REF] Bayram | Directional total variation[END_REF] is the concave quadratic one x → -1 2 ∥M Dx∥ 2 2 . In order to guarantee that JM is convex, it is sufficient to ensure the convexity of the twice continuously differentiable function

x → λ 2 ∥x -o∥ 2 2 -1 2 ∥M Dx∥ 2 2 [1]
, [START_REF] Lanza | Sparsity-inducing nonconvex nonseparable regularization for convex image processing[END_REF], [START_REF] Selesnick | Non-convex total variation regularization for convex denoising of signals[END_REF], that is equivalent to impose that the Hessian

H = λI 2n -D ⊤ M ⊤ M D (9)
is a positive semidefinite matrix. In order to guarantee the existence of a unique minimizer, we further require H to be positive definite and M ⊤ M to be a full-rank matrix, thus making the objective function in (8) coercive.

C. Proposed CNC Directional Total Variation

Let us now present our main contribution, that is a CNC formulation able to encompass the DTV regulariser (6) (itself being a generalisation of ( 4)).

First, by Fenchel-Rockafellar duality [START_REF] Bauschke | Convex analysis and monotone operator theory in hilbert spaces[END_REF], from (6), we obtain an equivalent primal formulation for DTV, by noticing that the adjoint operator of A -1 α,θ is given, for every u ∈ R 2n , by A - * α,θ u = R θi Λ 1/αi u i 1≤i≤n . This yields

(∀x ∈ R n ) DTV(x, α, θ) = min v∈R 2n {∥v∥ 1,2 | A - * α,θ v = Dx} = (A - * α,θ ▷ ∥ • ∥ 1,2 )(Dx), (10)
where (A - * α,θ ▷ g) denotes the Exact Infimal Postcomposition of a function g by A - * α,θ [START_REF] Bauschke | Convex analysis and monotone operator theory in hilbert spaces[END_REF]Chapter 12]. Then the proposed CNC modification of DTV parametrised by some matrix M ∈ R k×2n reads

(∀x ∈R n ) CNC-DTV(x) = φ A (Dx) -inf t∈R 2n φ A (t) + 1 2 ∥M (Dx -t)∥ 2 2 ,
where

φ A = A - * α,θ ▷∥•∥ 1,2 .
According to properties of the infimal postcomposition by a bounded operator [START_REF] Bauschke | Convex analysis and monotone operator theory in hilbert spaces[END_REF]Proposition 13.24(iv)], the conjugate of φ A is expressed as

(∀y ∈ R 2n ) φ * A (y) = ι B∞,2 (A -1 α,θ y), (11)
where B ∞,2 is the ℓ ∞,2 ball with center 0 and radius 1.

Hence, choosing φ = φ A in ( 8) and introducing a constraint on the range of x lead to the following optimisation problem where we set

H = R n × R 2n × R 2n : minimize (x,y,v)∈H F (x, y, v) + ∥v∥ 1,2 + ι S (x, y, v) + ι E (Dx, A - * α,θ v) + ι B∞,2 (A -1 α,θ M ⊤ M (Dx -y)). (12)
Here,

S = [0, 1] n × R 2n × R 2n is the additional constraint set, E is the vector space {(w, z) ∈ R 2n × R 2n | w = z} and, for every (x, y, v) ∈ H, F (x, y, v) = λ 2 ∥x -o∥ 2 2 - 1 2 ∥M Dx∥ 2 2 + 1 2 ∥M y∥ 2 2 . ( 13
)
Since φ A is convex, the positive semidefiniteness condition for H in (9) can also be applied as a convexity guaranty for our new objective function in [START_REF] Bredies | Total generalized variation[END_REF]. In the next section, we investigate how to address Problem [START_REF] Bredies | Total generalized variation[END_REF] by means of an optimisation scheme that exploits the structure of this objective function.

IV. OPTIMISATION ALGORITHM

A. Primal-dual splitting

The Primal-Dual (PD) method in [START_REF] Condat | A Primal-Dual splitting method for convex optimization involving lipschitzian, proximable and linear composite terms[END_REF]- [START_REF] Komodakis | Playing with duality: An overview of recent primal-dual approaches for solving large-scale optimization problems[END_REF] allows to efficiently deal with problems involving several Lipschitzian, proximable, and linear composite terms. In [START_REF] Bredies | Total generalized variation[END_REF] we identify the following functions and linear operators: for every

(y, v) ∈ R 2n × R 2n , h 1 (v) = ∥v∥ 1,2 L 1 = [0 0 I 2n ] h 2 (v) = ι B∞,2 (v) L 2 = [A -1 α,θ M ⊤ M D -A -1 α,θ M ⊤ M, 0] h 3 (y, v) = ι E (y, v) L 3 = D 0 0 0 0 A - * α,θ , so that (12) amounts to minimizing F + 3 i=1 h i • L i + ι S .
The iterations defining a sequence (z ℓ ) ℓ∈N = (x ℓ , y ℓ , v ℓ ) ℓ∈N which, starting from point z 0 ∈ H, converges to a solution z ∞ ∈ H to [START_REF] Bredies | Total generalized variation[END_REF], are presented in Algorithm 1, where δ > 0 is a Lipschitz constant of function F . For f ∈ Γ 0 (R n ), prox f corresponds to the proximity operator of f . When f = ι S , prox f reduces to proj S , the projection onto S.

Algorithm 1 Primal-Dual Algorithm to solve (12) Initialize z 0 ∈ H, w 1,0 ∈ R n , w 2,0 ∈ R 2n ,w 3,0 ∈ R 2n Set τ > 0 and σ > 0 s.t. 1 τ -σ||| 3 i=1 L ⊤ i L i ||| ≥ δ 2 for ℓ = 0, 1, . . . do z ℓ+1 = proj S (z ℓ -τ ∇F (z ℓ ) -τ 3 i=1 L ⊤ i w i,ℓ ) for i = 1, 2, 3 do w i,ℓ+1 = prox σh * i (w i,ℓ + σL i (2z ℓ+1 -z ℓ )) end end Since |||D||| 2 = 8, and |||A -1 α,θ ||| ≤ 1 (resp. |||A - * α,θ ||| ≤ 1) as A -1 α,θ (resp. A - * α,θ
) is the combination of rotations and contractions along one axis, the norm of the involved linear operator can be upper bounded as follows:

|||L ⊤ 1 L 1 + L ⊤ 2 L 2 + L ⊤ 3 L 3 ||| ≤ 1 + (1 + 8)(|||M ||| 4 ) + 8. (14)
It is worthy to note that Problem [START_REF] Bredies | Total generalized variation[END_REF] and Algorithm 1 represent a unified framework for convex / non-convex, classical / directional TV, since proper choices for matrix M and operator A α,θ allow us to model the four different instances of the denoising problem at hand. For M = 0 2n ∈ R 2n×2n , i.e. the null operator, we retrieve the convex TV formulation, whereas the classical TV formulation is obtained by setting, for every i ∈ {1, . . . , n}, α i = 1 and θ i = 0.

In the proposed framework, all the proximal computations are exact since the PD method allows us to decouple the functions that are defined by composing a convex function with a linear operator. This represents an advantage with respect to the Forward-Backward strategies proposed in [START_REF] Selesnick | Non-convex total variation regularization for convex denoising of signals[END_REF]Algorithm 3] and in [4, Proposition 10], which involve nested optimisation procedures, i.e. a subroutine has to be used to compute the proximity operator of the classical TV functional. We dedicate the next subsection to the description of the proximity operators of the involved terms.

B. Practical implementation

The proximity operator of function h * 2 = h 1 = ∥ • ∥ 1,2 can easily be inferred from the one of the ℓ 2 norm by applying the rule for a separable sum of terms:

(∀u ∈ R 2n ) prox σh * 2 (u) = u i - u i max( ∥ui∥2 σ , 1) 1≤i≤n
, while the proximity operator of function h * 1 = h 2 is obtained by applying Moreau's identity: [START_REF] Bauschke | Convex analysis and monotone operator theory in hilbert spaces[END_REF] By applying again Moreau's formula, the proximity operator of σh * 3 is given by

(∀u ∈ R 2n ) prox σh * 1 (u) = u -σprox h 2 σ ( u σ ) = u i max(∥u i ∥ 2 , 1) 1≤i≤n .
(∀(y, v) ∈ (R 2n ) 2 ) prox σh * 3 (y, v) = (y, v) -σ proj E y σ , v σ = 1 2
y -v v -y .

V. NUMERICAL RESULTS

We now evaluate the proposed approach for the restoration of noisy grayscale images. In accordance with [START_REF] Lanza | Sparsity-inducing nonconvex nonseparable regularization for convex image processing[END_REF], we choose matrix M = √ γI 2n , so that M ⊤ M = γI 2n and therefore the convexity condition reduces to ensuring the positive semidefiniteness of H = λI 2n -γD ⊤ D. Since |||D||| 2 = 8, the convexity condition is satisfied as soon as γ < λ/8. We therefore set γ = (ρλ)/8, with ρ ∈ {0, 0.99}. For ρ = 0, we retrieve the convex formulation of TV, whereas for ρ = 0.99 we get a high degree of non-convexity for ψ M while ensuring the convexity of the global problem. We illustrate the performance of CNC-DTV on three synthetic images, texture, barcode, and geometric, shown in Figure 1. We added white zero-mean Gaussian noise with standard deviation σ e = 0.1. In order to extract the directional information θ in DTV, we exploited the strategy proposed in [START_REF] Parisotto | Higher-order total directional variation: Analysis[END_REF]Section 5]. For the definition of parameter α, we set α i = α for every i as in [START_REF] Zhang | Edge adaptive directional total variation[END_REF]. We then chose the best combination of the two parameters α and λ by means of a grid search to optimise the Peak Signal-to-Noise Ratio (PSNR) of the restored image.

In Table I, we report the best PSNR obtained when running our PD algorithm for the classic convex (i.e, ρ = 0) TV (C-TV), the non-convex (i.e, ρ = 0.99) TV (NC-TV), convex directional (C-DTV), and non-convex directional (NC-DTV) TV. This quantitative assessment shows that the combination of non-convex prior with a DTV-based space-variant regularisation yields an improvement w.r. We also provide in Figure 2 a visual illustration of the performance of the four approaches on texture image, showing the absolute error between the best estimated solution x and the original image x. The reconstructions involving a nonconvex regulariser or directional information show peculiar structures in the distribution of its absolute residual error, whereas the classic convex approach yields a rather dense error distribution. NC-DTV inherits the high coherence w.r.t the directions in the image from C-DTV and the high accuracy in noise removal and sharp transition reconstruction from NC-TV. Eventually, in Figure 3, we illustrate for image geometric the PSNR evolution along 3000 iterations (left) and the distance from the iterates z ℓ to the solution z ∞ in logarithmic scale (right) of the four approaches, which shows the fast and stable convergence behaviour of the proposed PD algorithm. VI. CONCLUSIONS In this work we investigated the extension of the CNC approach to a directional version of the TV regularisation model for image denoising. We proposed to address the resulting minimisation problem with a primal-dual procedure that efficiently exploits the structure of the objective function and we provided numerical results supporting the interest of the proposed approach.

Fig. 1 :

 1 Fig. 1: Original images: texture (a), barcode (b), and geometric (c).

Fig. 2 :

 2 Fig. 2: Residual absolute error for texture.

Fig. 3 :

 3 Fig. 3: PSNR versus iterations (left) and distance from the iterates to the solution versus iterations in logarithmic scale (right) for geometric.

TABLE I :

 I Best PSNR for σ e = 0.1 optimised over a grid search for parameters λ and α.


Dimitri Bréard
email: dimitri.breard@univ-angers.fr

Thibault Barrit

Daniel Sochard

Sophie Aligon

Elisabeth Planchet

Béatrice Teulat

Josiane Le Corff

Claire Campion

David Guilet

Development of a quantification method for routine analysis of glucosinolates and camalexin in brassicaceous small-sized samples by simultaneous extraction prior to liquid chromatography determination

Keywords: Glucosinolates, Camalexin, Small-sized samples, Simultaneous extraction, Absolute quantification

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

Introduction

Plants produce several secondary metabolites that play a crucial role as defence compounds against pathogens and herbivores. In the Brassicaceae family, most species produce phytoanticipins (namely glucosinolates) as constitutive defences (Fig. 1). Some species further produce phytoalexins as inducable defences among which camalexin is the most studied compound [START_REF] Kliebenstein | Secondary metabolites and plant/environment interactions: a view through Arabidopsis thaliana tinged glasses[END_REF] (Fig. 1). Fungal pathogens including A. brassicicola, a Brassicaceae specialist that causes black spot disease, are more or less sensitive to camalexin, glucosinolates and glucosinolates-breakdown products, present in leaves [START_REF] Thomma | Deficiency in phytoalexin production causes enhanced susceptibility of Arabidopsis thaliana to the fungus Alternaria brassicicola[END_REF][START_REF] Sellam | In vitro antifungal activity of brassinin, camalexin and two isothiocyanates against the crucifer pathogens Alternaria brassicicola and Alternaria brassicae[END_REF]. However, A. brassicicola also spreads by seeds [START_REF] Oliver | Vertical disease transmission in the Cakile-Alternaria host-pathogen interaction[END_REF][START_REF] Pochon | The Arabidopsis thaliana-Alternaria brassicicola pathosystem: A model interaction for investigating seed transmission of necrotrophic fungi[END_REF] and little is known about the transmission of the pathogen from seed to seedling during germination and post-germinative growth as well as the role of defence compounds in this process. One reason for that is that the detection of secondary metabolites from the complex matrix of seed reserves [START_REF] Mira | Lipid Thermal Fingerprints of Long-term Stored Seeds of Brassicaceae[END_REF] is quite difficult and that only small amounts of sample material [START_REF] Al-Shehbaz | The tribes of Cruciferae (Brassicaceae) in the southeastern United States[END_REF] is available at the early stages of plant development. To accurately determine secondary metabolites in brassicaceous seeds and seedlings, an analytical procedure suitable for small-sized samples and adapted to maximize the recovery of glucosinolates and camalexin is therefore required. Furthermore, both glucosinolates and camalexin play important but different roles in plant protection against various pathogens. As their biosynthetic pathways are tightly regulated and interconnected [START_REF] Yang | Co-regulation of indole glucosinolates and camalexin biosynthesis by CPK5/CPK6 and MPK3/MPK6 signaling pathways[END_REF], a method allowing identification and quantification of both types of metabolites from a single sample may assist the elucidation of their relative roles during and after germination.

Because of the great ongoing interest in glucosinolates as biologically active compounds [START_REF] Maina | Human, Animal and Plant Health Benefits of Glucosinolates and Strategies for Enhanced Bioactivity: A Systematic Review[END_REF][START_REF] Traka | Health Benefits of Glucosinolates[END_REF][START_REF] Razis | Glucosinolates and Isothiocyanates: Cancer Preventive Effects[END_REF], many protocols for extraction, identification and quantification have been published for five decades. They can be basically classified into two groups [START_REF] Clarke | Glucosinolates, structures and analysis in food[END_REF][START_REF] Mawlong | A simple spectrophotometric method for estimating total glucosinolates in mustard de-oiled cake[END_REF].

The first group focuses on the total content of glucosinolates and including destructive methods as well as non-destructive methods will not be discussed.

Conversely, the second group allows quantification of individual glucosinolates, like desulfoglucosinolates, TMS-ether of desulfo-glucosinolates, isothiocyanates or intact form. The conversion of glucosinolates to isothiocyanates by myrosinase or the derivatization of desulfo-glucosinolates to TMSethers for GC-MS analysis is no longer conducted [START_REF] Clarke | Glucosinolates, structures and analysis in food[END_REF]. Current analytical determinations are now conducted on the desulfo-glucosinolates or intact forms by LC-UV, GC-FID or LC-MS, however it is uncommon to perform UV quantification on the intact form [START_REF] Francisco | Simultaneous identification of glucosinolates and phenolic compounds in a representative collection of vegetable Brassica rapa[END_REF] because the desulfation process typically acts as an essential clean-up step. HPLC-UV and GC-FID yield comparable results for desulfoglucosinolates and are therefore considered validated approaches [START_REF] Hrncirik | Comparison of HPLC and GLC methodologies for determination of glucosinolates using reference material[END_REF]. However, with the advanced development of hyphenated MS based chromatographic methods, GC-FID is no longer used [START_REF] Kiddle | Highperformance liquid chromatographic separation of natural and synthetic desulphoglucosinolates and their chemical validation by UV, NMR and chemical ionisation-MS methods[END_REF]. The detection of glucosinolates has been reported for many ionization techniques (FAB, ESI, APCI or MALDI) [START_REF] Bianco | Investigation of Glucosinolates by Mass Spectrometry[END_REF], and detector configurations such as single/triple quadrupole, IT or TOF [START_REF] Bianco | Investigation of Glucosinolates by Mass Spectrometry[END_REF]. The quantification can be accurate and precise if executed correctly. However, the lack of universal response factors in MS which is linked to several parameters (individual behaviour of analyte with settings for ionization and/or fragmentation, matrix effects, differences in linear range of detection among instruments) limits accurate quantification to few commercially available standards [START_REF] Clarke | Glucosinolates, structures and analysis in food[END_REF][START_REF] Crocoll | Analysis and Quantification of Glucosinolates[END_REF].

For the moment, the conversion of glucosinolates to desulfo-glucosinolates by sulfatase followed by HPLC-UV analysis [START_REF] Minchinton | Separation of desulfoglucosinolates by reversed-phase high-performance liquid-chromatography[END_REF] represents the official analytical approach outlined in European Union regulations (ISO 9167) [START_REF] Anon | Rapeseed and rapeseed meals -Determination of glucosinolates content -Method using high-performance liquid chromatography, International Organisation for Standardization[END_REF] and therefore remains widely used [START_REF] Crocoll | Analysis and Quantification of Glucosinolates[END_REF]. It still provides a robust, direct and relatively low-cost approach that can be easily performed on standard laboratory equipment.

Meanwhile, suggestions to improve and simplify the method have been outlined in several publications [START_REF] Wathelet | Determination of glucosinolates in rapeseed improvement of the official HPLC ISO method (precision and speed)[END_REF][START_REF] Grosser | A Straightforward Method for Glucosinolate Extraction and Analysis with High-pressure Liquid Chromatography (HPLC)[END_REF][START_REF] Doheny-Adams | Development of an efficient glucosinolate extraction method[END_REF]. Moreover, desulfo-glucosinolates are chemically quite stable when kept airtight in cool water [START_REF] Grosser | A Straightforward Method for Glucosinolate Extraction and Analysis with High-pressure Liquid Chromatography (HPLC)[END_REF] and most response factors of quantification (relative to sinigrin) are well defined and accessible by open access in publications [START_REF] Clarke | Glucosinolates, structures and analysis in food[END_REF]. In many cases, retention times linked to reported data are used for compound identification [START_REF] Blazevic | Glucosinolate structural diversity, identification, chemical synthesis and metabolism in plants[END_REF], however MS detection provides significant added analytical value by allowing a more accurate analysis [START_REF] Bianco | Investigation of Glucosinolates by Mass Spectrometry[END_REF].

Camalexin is an important component of inducible defences and contributes to the partial resistance against several pathogens [25]. Therefore, the quantification of this phytoalexin is also very important for understanding processes of plant physiology and phytopathology. The extraction protocols for camalexin are quite similar in most publications. In general 70 to 80% methanol is used as solvent at a temperature range of 20-80°C. Extraction time may last from ten minutes to one hour [START_REF] Beets | Quantification of camalexin, a phytoalexin from Arabidopsis thaliana: A comparison of five analytical methods[END_REF][START_REF] Tewes | Metal hyperaccumulation in the Brassicaceae species Arabidopsis halleri reduces camalexin induction after fungal pathogen attack[END_REF] and sometimes residue is re-extracted with chloroform [START_REF] Glazebrook | Isolation of phytoalexin-deficient mutants of Arabidopsis thaliana and characterization of their interactions with bacterial pathogens[END_REF][START_REF] Mert-Turk | Quantification of camalexin in several accessions of Arabidopsis thaliana following inductions with Peronospora parasitica and UV-B irradiation[END_REF][START_REF] Staal | Transgressive segregation reveals two Arabidopsis TIR-NB-LRR resistance genes effective against Leptosphaeria maculans, causal agent of blackleg disease[END_REF]. Fluorometric analyses are performed using 318 nm excitation and 385 nm emission wavelength [START_REF] Mert-Turk | Quantification of camalexin in several accessions of Arabidopsis thaliana following inductions with Peronospora parasitica and UV-B irradiation[END_REF] but for an accurate quantification of camalexin chromatographic purification is required in order to eliminate quenching interference [START_REF] Beets | Quantification of camalexin, a phytoalexin from Arabidopsis thaliana: A comparison of five analytical methods[END_REF].

For ten years, LC-MS has been the predominant technique because of short analysis times and is especially effective in identifying of camalexin based on accurate mass as well as determining low concentrations [START_REF] Beets | Quantification of camalexin, a phytoalexin from Arabidopsis thaliana: A comparison of five analytical methods[END_REF][START_REF] Beets | Activation of camalexin biosynthesis in Arabidopsis thaliana in response to perception of bacterial lipopolysaccharides: a gene-to-metabolite study[END_REF].

On the other hand, the simultaneous detections of glucosinolates and camalexin has hardly been conducted from one sample. When done, it was either by a single method based only on relative quantification by UPLC-MS [START_REF] Shih | Differential roles of glucosinolates and camalexin at different stages of Agrobacterium-mediated transformation[END_REF][START_REF] Izquierdo Zandalinas | A Fast and Precise Method To Identify Indolic Glucosinolates and Camalexin in Plants by Combining Mass Spectrometric and Biological Information[END_REF] or by an absolute quantification with two separate methods requiring twice as much raw material [START_REF] Shih | Differential roles of glucosinolates and camalexin at different stages of Agrobacterium-mediated transformation[END_REF][START_REF] Cargnel | Linking phytochrome to plant immunity: low red : far-red ratios increase Arabidopsis susceptibility to Botrytis cinerea by reducing the biosynthesis of indolic glucosinolates and camalexin[END_REF][START_REF] Geu-Flores | Cytosolic gamma-Glutamyl Peptidases Process Glutathione Conjugates in the Biosynthesis of Glucosinolates and Camalexin in Arabidopsis[END_REF]. A procedure allowing the identification and quantification of glucosinolates and camalexin from a single extraction would be beneficial for the analysis of samples that are generally limited in quantity, like small seeds and seedlings. The aim of this work was therefore to provide a validated method analysis for the determination of glucosinolates and camalexin in small-sized samples (less than 250 mg of dry matter (DM) for a triplicate analysis).

The proposed method comprises the simultaneous extraction of both constituents from seed and leaf samples prior to HPLC analysis and targets a broad Brassicaceae phylogenetic group with potentially contrasting profiles. The approach is based on the specificity of the desulfation process of glucosinolates and the specificity of fluorescence detection for camalexin analysis.

Experimental

Chemicals and reagents

HPLC grade acetonitrile (MeCN) and methanol (MeOH) were purchased from HoneyWell (Charlotte, USA). Absolute ethanol (EtOH) was purchased from Carlo Erba Reagents (Val de Reuil, France).

Anhydrous sodium acetate (NaOAc) and 37% hydrochloric acid (HCl) were purchased from Fisher Chemical (Illkirch, France). Sinigrin potassium salt (≥ 99%) was purchased from Extrasynthese (Genay, France). Camalexin (≥ 98%), DEAE-Sephadex A-25 and sulfatase type H-1 from Helix pomatia were purchased from Sigma-Aldrich (Darmstadt, Germany). Ultrapure water (H2O) was obtained by a MilliQ Advantage system from Millipore (Darmstadt, Germany).

Standards and reagents solutions

Stock solution of sinigrin was prepared in H2O at a concentration of 5050 µM. A stock solution of camalexin was prepared in 70% MeOH at a concentration of 38.8 µg.mL -1 . The enzyme solution was prepared by dissolving and mixing 10000 units of sulfatase in 30 mL of H2O and 30 mL of EtOH before centrifugation at 2650 g for 20 min at room temperature. The supernatant was taken and mixed with 90 mL of EtOH before centrifugation at 1030 g for 15 min at room temperature. Finally, the pellet was dissolved in a total of 25 mL of H2O and aliquoted in 1.5 mL microtubes. All solutions were stored at -20°C before use.

The ion exchange medium was prepared by mixing 10 g of DEAE-Sephadex A-25 resin with 125 mL of H2O and the resulting mixture was stored at 4°C. The solution of 70% MeOH was prepared by mixing 350 mL of MeOH and 150 mL of H2O. The 20 mM NaOAc buffer was prepared by dissolving 410 mg of NaOAc in 250 mL of H2O and adjusting the pH at 5.5 with HCl. These solutions were prepared extemporaneously and conserved one week at 4°C.

Samples collection and preparation

Three certified rapeseed reference materials by European Commission called ERM-BC366 (seeds with low level of glucosinolates), ERM-BC190 (medium level) and ERM-BC367 (high level) were purchased from Sigma-Aldrich and stored at 4°C. Seeds of Arabidopsis thaliana (L.) Heynh (= natural population), Barbarea vulgaris Aiton, Brassica nigra (L.) Koch, Capsella bursa-pastoris (L.) Medik., Cardamine hirsuta L., Coincya monensis (L.) Greuter and Burdet, Sinapis arvensis L., and Sisymbrium officinale (L.) Scop. were collected in 2018 in natural populations around Angers (France). Species identification was verified by Josiane Le Corff.

Seeds from these populations were sown in the spring in a substrate of 50% sand, 37.5% blond peat, and 12.5% black peat, and grown in a glasshouse under natural daylight for 4-5 weeks. Mean daily temperature was 14°C and ranged from 10°C up to 22°C for a few hours on sunny days.

In addition, seeds of A. thaliana of the ecotypes Columbia (Col) and Wassilewskija (Ws) were harvested in 2018 from a culture in a growth chamber in a substrate of 75% blond peat and 25% black peat, under controlled conditions (16h/8h light/dark, 22°C/19°C light/dark) for 3 months. Seeds were sown in a substrate of 75% blond peat and 25% black peat and grown in a growth chamber under controlled conditions (8h/16h light/dark, 22°C/19°C light/dark) for 4 weeks.

Before harvest, a subsample of rosettes (at least 5 per genotype) from A. thaliana natural population, Col and Ws were inoculated on five leaves per plant with one 5 µL drop of an A. brassicicola (previously described Abra43 isolate [START_REF] Sellam | In vitro antifungal activity of brassinin, camalexin and two isothiocyanates against the crucifer pathogens Alternaria brassicicola and Alternaria brassicae[END_REF]) conidial suspension (10 5 conidia.mL -1 in water). Other plants as controls (at least 5 per genotype), were treated with one 5 µL drop of water on 5 leaves per plant. All plants were then maintained for two days in saturating humidity (100% relative humidity) and collected seven days after inoculation. For all samples (seeds from natural populations and ecotypes and leaves from plants that had been inoculated or not), material was immersed and grinded in liquid nitrogen upon harvest, freeze-dried then stored at -20°C prior to analysis.

Extraction procedure

Extraction was carried out following a protocol previously published by Grosser et al. [START_REF] Grosser | A Straightforward Method for Glucosinolate Extraction and Analysis with High-pressure Liquid Chromatography (HPLC)[END_REF] with minor modifications. One Pasteur pipette (150 mm) per sample and sinigrin reference was filled with approximatively 1 cm of glass wool, then set up on a homemade rack. Each column was filled with 0.5 mL of resin (40 mg dry weight equivalent) then washed with 1 mL of H2O. Finally, 15 mL vials were placed on a rack to collect the column washes including camalexin.

Dry samples were weighed in round-bottom microtubes and two metal balls (3 mm in diameter) as boiling retardants were added. In parallel, seven calibration standards of sinigrin were prepared in the concentration range of 50 to 3500 µM by diluting the stock solution in H2O.

A volume of 1 mL of 70% MeOH (not pre-heated) was added on each sample, vortexed and heated 5 min at 91°C. Then, the microtubes were placed in an ultrasonic bath during 15 min before centrifugation at 2700 g for 10 min at room temperature. The supernatants and the calibration standards were deposited on separate columns. Another 1 mL of 70% MeOH was added to the remaining pellets and to the calibration standards microtubes. The samples were vortexed and placed in an ultrasonic bath during 15 min then centrifugated at 2700 g for 10 min at room temperature. The supernatants and the 7 calibration standards were added to their respective columns.

Each column was sequentially washed with 2 x 1 mL of 70% MeOH, 1 mL of H2O and 2 x 1 mL of 20 mM NaOAc buffer waiting for the column to run dry between each washing. At this moment, 15 mL vials (camalexin fractions) were replaced by 2 mL microtubes and placed overnight under an air stream to remove MeOH. A volume of 20 µL of sulfatase solution was added onto each column followed by 50 µL of 20 mM NaOAc buffer. The columns were covered with aluminium foil and let stand overnight.

The next day, the desulfo-glucosinolates were eluted in 2 mL microtubes by pipetting 2 x 0.75 mL of H2O onto each column (desulfo-glucosinolates fractions).

Camalexin and desulfo-glucosinolates fractions were frozen in a -80°C freezer for 4 hours then freezedried for 16 hours (8h/8h, primary/secondary desiccation) to remove water.

Liquid chromatography 2.5.1. HPLC-UV-MS conditions for desulfo-glucosinolates

Chromatographic analyses were performed on a 2695 separation module coupled to a 2489 UV/Visible detector (Waters, Saint-Quentin-en-Yvelines, France) and to an Esquire 3000 Plus ESI-IT-MS (Bruker, Wissembourg, France) assisted by Empower (Waters) and Data Analysis software (Bruker).

After the freeze-drying process, the desulfo-glucosinolates fractions were dissolved in 1 mL of H2O (high content samples, e.g. seeds) or 0.2 mL of H2O (low content samples, e.g. leaves). The injection volume was 10 µL.

Samples were injected onto a Gemini C18 column (150 mm x 4.6 mm, 3 µm, Phenomenex, Le Pecq, France) heated at 40°C. The analytes were separated by a gradient elution with H2O (A) and MeCN (B) at a flow rate of 0.75 mL.min -1 (2% to 35% B in 35 min). The UV detection was monitored at 229 nm and MS n detection was monitored in positive and negative mode.

HPLC-FLD conditions for camalexin

Chromatographic analyses were performed on a 2695 separation module coupled to a 2475 multi λ fluorescence detector assisted by Empower software (Waters).

After freeze-drying process, the camalexin fractions were dissolved in 2 mL of 70% MeOH. Eight calibration standards of camalexin were prepared in the concentration range of 0.025 to 5 µg.mL -1 by diluting the stock solution in 70% MeOH. The injection volume was 10 µL.

Samples were injected onto a Gemini C18 column (150 mm x 4.6 mm, 3 µm, Phenomenex) heated at 40°C. The analytes were separated by a gradient elution with H2O (A) and MeCN (B) at a flow rate of 0.75 mL.min -1 (2% to 100% B in 35 min). The fluorescence detection was monitored at 318 and 385 nm as excitation and emission wavelengths, respectively.

Method validation

The calibration curves were set to the concentration range expected for each compound. The linearity of the calibration curves was assessed by injecting calibration standards of sinigrin (7 levels) or camalexin (8 levels) in three replicates. Residuals (difference between nominal concentration and calculated concentration by the linear model) and their distribution (normally distributed around the mean) were monitored. LOD and LOQ were defined as 3 x and 10 x SD of five replicate measurements of analyte close to the detection limit, respectively.

Precision was evaluated by repeated analysis of both, standards and samples, within different analytical batches. For glucosinolates, accuracy was assessed using three certified rapeseed reference materials with three different levels of glucosinolates. Some samples were spiked with different levels of camalexin at the start of extraction procedure in order to evaluate the recovery. All statistical work was conducted on Microsoft Excel 365® and the Minitab 19® software.

Method calculation

Glucosinolates analysis

The HPLC-UV-MS chromatograms at 229 nm were integrated respecting the defined LOQ and glucosinolates were identified by MS data, retention time and comparison with literature data.

Peak areas of desulfo-glucosinolates were corrected with an UV response factor at 229 nm compared to desulfo-sinigrin provided reference data were available [START_REF] Clarke | Glucosinolates, structures and analysis in food[END_REF][START_REF] Grosser | A Straightforward Method for Glucosinolate Extraction and Analysis with High-pressure Liquid Chromatography (HPLC)[END_REF] (Table A1). For all other cases including unidentified glucosinolates the response factor was set to 1. Results were expressed in mmol.kg -1 of DM (following the standard) and calculated according to the formula presented in appendix (Eq. A1).

Camalexin analysis

The HPLC-FLD chromatograms acquired at an emission wavelength of 385 nm and an excitation wavelength of 318 nm were integrated according to the defined LOQ. The results were expressed in µg.g -1 of DM following the formula presented in appendix (Eq. A2).

Results and discussion

Glucosinolates analysis validation

The evaluation of the modified method [START_REF] Grosser | A Straightforward Method for Glucosinolate Extraction and Analysis with High-pressure Liquid Chromatography (HPLC)[END_REF] was the first step of this study. For that, three samples of certified rapeseed reference material containing the three classes of glucosinolates, aliphatic (progoitrin, epiprogoitrin, glucoalyssin, gluconapoleiferin, gluconapin and glucobrassicanapin), benzenic (gluconasturtiin) and indolic (4-Hydroxyglucobrassicin, glucobrassicin and neoglucobrassicin) [START_REF] Fahey | The chemical diversity and distribution of glucosinolates and isothiocyanates among plants[END_REF] and with three levels of glucosinolates were taken as references: BCR-366R at 11.9 ± 1.3, BCR-190R at 23 ± 4 and BCR-367R at 99 ± 9 mmol.kg -1 [START_REF] Linsinger | The certification of the total glucosinolate and sulphur contents of three rapeseed (colza) materials BCR-190R, -366R and -367R, Office for Official Publications of the European Communities[END_REF].

Previously, the linear relationship between area under the curve at 229 nm and the concentration of sinigrin was validated in the concentration range evaluated for glucosinolates with a correlation coefficient higher than 0.999 (Fig. A1) and was repeated for different batch analyses at different times of the study. The range of linearity from 50 to 3500 µM was assessed after control of the standardized residuals. LOD and LOQ were statistically calculated at 3.8 and 12.6 µM, respectively (Table A2).

Next, each certified rapeseed sample was tested in triplicate at about 80 mg following the protocol described in the experimental section and the results were calculated with the calibration curve of sinigrin and UV response factor at 229 nm of identified glucosinolates (Fig. 2). The results (total and individual glucosinolates contents) were in accordance with the certification report [START_REF] Linsinger | The certification of the total glucosinolate and sulphur contents of three rapeseed (colza) materials BCR-190R, -366R and -367R, Office for Official Publications of the European Communities[END_REF] (Fig. 3).

Finally, the procedure was down scaled to 10 mg of DM for the three certified rapeseed samples. Fig. 4 shows that results were similar in a mass range of 10-80 mg for the three certified rapeseeds for total and individual glucosinolates contents.

The method was precise and accurate and can therefore be used from 10 to 80 mg samples with high or low glucosinolates contents.

Camalexin analysis validation

To verify whether the quantitative extraction of camalexin and glucosinolates together with desulfation can be performed simultaneously, rapeseed samples were spiked with camalexin. Indeed, the absence of camalexin in the rapeseed samples was checked beforehand.

Previously, the linear relationship between the area under the curve at 385 nm (after excitation at 318 nm) and the concentration of camalexin was validated in the concentration range evaluated for the samples with a correlation coefficient higher than 0.999 (Fig. A2) and was repeated for different batch analysis at different times of the study. The range of linearity was assessed after control of the standardized residuals. LOD and LOQ were statistically calculated at 0.014 and 0.046 µg.mL -1 , respectively (Table A3).

Next, different rapeseed samples were supplemented with camalexin from 0.05 to 2 µg before the extraction procedure. The extracts containing camalexin and glucosinolates were deposited on anionexchange columns. Columns were sequentially washed with some eluents to prepare them to the desulfation process of glucosinolates. All these columns washes were collected then freeze-dried giving fractions containing camalexin. These fractions were analysed in HPLC-FLD (Fig. 2) and the percentage of recovery was calculated with the calibration curve between 98 and 117% with an RSD between 3 and 21% (Table 1).

Finally, a down scaled was carried out from 50 to 10 mg of DM of one of the A. thaliana natural population leaf samples containing a medium quantity of camalexin. Quantification of camalexin were homogeneous for starting materials from 50 to 20 mg while below 20 mg the sampling appeared too small to be reproducible (Fig. 5). This method was linear from 0.025 to 5 µg.mL -1 with an LOD and LOQ of 0.014 and 0.046 µg.mL -1 , respectively. It was precise and trueness and can therefore be used from 20 to 50 mg of samples.

Application of the method

The applicability of the method was evaluated by screening for glucosinolates and camalexin in different Brassicaceae genera (Arabidopsis, Barbarea, Brassica, Capsella, Cardamine, Coincya, Sinapis and Sisymbrium), genotypes of A. thaliana (natural population, ecotypes Col and Ws) and organs (leaves and seeds) and with plants inoculated or not with A. brassicicola.

According to previous results (see 3.1 and 3.2), these investigations were performed on 20 to 50 mg of DM as starting material and each sample was tested 2 to 11 times (depending on the sample availability). Glucosinolates were identified by MS data and retention times compared to the references, considering both rapeseed reference materials and A. thaliana Col seeds as authentic references (Table A4). Apparent glucosinolates not available as references were tentatively identified from MS data, reasonable retention times compared with the literature [START_REF] Olsen | Glucosinolate diversity within a phylogenetic framework of the tribe Cardamineae (Brassicaceae) unraveled with HPLC-MS/MS and NMR-based analytical distinction of 70 desulfoglucosinolates[END_REF], and comparison with previous investigations of these species, as discussed below. Concentrations (total and individual) were calculated with the calibration curve of sinigrin and the UV response factor acquired at 229 nm.

Camalexin was identified according to its retention time and its concentration was calculated with the calibration curve.

The range in total glucosinolates content was quite large from 0.2 to 92.4 mmol.kg -1 with an average RSD of 7% (Fig. 6). As expected [START_REF] Bennett | Screening crucifer seeds as sources of specific intact glucosinolates using ion-pair high-performance liquid chromatography negative ion electrospray mass spectrometry[END_REF], the total concentration of glucosinolates was higher in seeds (18.4 to 92.4 mmol.kg -1 ) compared to leaves (0.2 to 18.2 mmol.kg -1 ). For a given Brassicaceae species, seeds contained 2 to 20 times more glucosinolates than leaves. Brassica nigra was the species with the highest concentration of glucosinolates in the leaves (18.2 mmol.kg -1 ) and C. monensis the species with the highest concentration in the seeds (92.4 mmol.kg -1 ). Similarly, the seeds of the two A. thaliana ecotypes Col and Ws, showed a 100-fold higher concentration of glucosinolates than leaves, while the concentration in the seeds was similar between the two ecotypes, Col (51.1 mmol.kg -1 ) and Ws (58.6 mmol.kg -1 ).

From 89 to 100% of the individual glucosinolates were identified or tentatively identified in different samples (Table 2). More precisely, 40 glucosinolates were (tentatively) identified and quantified in 21 samples including 31 response factors. In accordance with previously published reports, experiment data confirmed that leaves and seeds of S. arvensis mostly contained sinalbin and long chained methionine derived glucosinolates [START_REF] Bennett | Screening crucifer seeds as sources of specific intact glucosinolates using ion-pair high-performance liquid chromatography negative ion electrospray mass spectrometry[END_REF][START_REF] Agerbirk | Sinapis phylogeny and evolution of glucosinolates and specific nitrile degrading enzymes[END_REF], while sinigrin was predominant in leaves and seeds of B. nigra [START_REF] Bennett | Screening crucifer seeds as sources of specific intact glucosinolates using ion-pair high-performance liquid chromatography negative ion electrospray mass spectrometry[END_REF][START_REF] Agerbirk | Sinapis phylogeny and evolution of glucosinolates and specific nitrile degrading enzymes[END_REF]. Similarly, seeds of C. bursa-pastoris mostly contained glucoarabin and glucocamelinin [START_REF] Daxenbichler | Glucosinolate composition of seeds from 297 species of wild plants[END_REF], seeds of S. officinale exhibited high yields of glucoputranjivin [START_REF] Bennett | Screening crucifer seeds as sources of specific intact glucosinolates using ion-pair high-performance liquid chromatography negative ion electrospray mass spectrometry[END_REF], and leaves of C. hirsuta mostly gluconapin, glucobrassicanapin and glucotropaeolin [START_REF] Agerbirk | Glucosinolate profiles and phylogeny in Barbarea compared to other tribe Cardamineae (Brassicaceae) and Reseda (Resedaceae), based on a library of ion trap HPLC-MS/MS data of reference desulfoglucosinolates[END_REF]. Leaves and seeds of B. vulgaris contained mostly one glucosinolate. Four isomers with the side chain formula C8H9O have been reported from B. vulgaris [START_REF] Agerbirk | Glucosinolate profiles and phylogeny in Barbarea compared to other tribe Cardamineae (Brassicaceae) and Reseda (Resedaceae), based on a library of ion trap HPLC-MS/MS data of reference desulfoglucosinolates[END_REF], but usually the dominant isomer is glucobarbarin [START_REF] Griffiths | Identification of glucosinolates on the leaf surface of plants from the Cruciferae and other closely related species[END_REF][START_REF] Badenes-Perez | Insect Attraction versus Plant Defense: Young Leaves High in Glucosinolates Stimulate Oviposition by a Specialist Herbivore despite Poor Larval Survival due to High Saponin Content[END_REF]. However, for some samples, the composition in glucosinolates differed from results reported in previous studies. Mostly sinigrin was found in the leaves of C. bursa-pastoris as it is sometimes referenced in seed samples [START_REF] Lockwood | Glucosinolate spectrum of some Algerian Cruciferae[END_REF]. Leaves of S. officinale mostly contained glucobrassicin and not a 65/35 ratio of glucoputranjivin and glucobrassicin as previously described [START_REF] Griffiths | Identification of glucosinolates on the leaf surface of plants from the Cruciferae and other closely related species[END_REF]. In a previous study, on leaves of C. monensis, sinalbin, gluconapin, progoitrin, epiprogoitrin and sinigrin were detected [START_REF] Agerbirk | Sinapis phylogeny and evolution of glucosinolates and specific nitrile degrading enzymes[END_REF]. Our seed samples mostly contained progoitrin and epiprogoitrin while in the leaves, besides small quantities of sinalbin and gluconapin, the main glucosinolate was tentatively identified as glucoaubrietin generally found in the genera Tropaeolum and Aubretia [START_REF] Bennett | Screening crucifer seeds as sources of specific intact glucosinolates using ion-pair high-performance liquid chromatography negative ion electrospray mass spectrometry[END_REF][START_REF] Agerbirk | Sinapis phylogeny and evolution of glucosinolates and specific nitrile degrading enzymes[END_REF].

Unlike the previously discussed species, seeds of A. thaliana exhibit a diverse array of individual glucosinolates, with principally sulfur-containing side-chains [START_REF] Brown | Variation of glucosinolate accumulation among different organs and developmental stages of Arabidopsis thaliana[END_REF][START_REF] Petersen | Composition and content of glucosinolates in developing Arabidopsis thaliana[END_REF][START_REF] Hogge | HPLC separation of glucosinolates from leaves and seeds of Arabidopsis thaliana and their identification using thermospray liquidchromatography mass-spectrometry[END_REF]. The main compound, glucoerucin or glucoibervirin, representing 40% of the total glucosinolates content was different in the ecotypes Col and Ws, respectively. Although in trace quantities, presence of late-eluting acylated glucosinolates is well-established in seeds of A. thaliana [START_REF] Reichelt | Benzoic acid glucosinolate esters and other glucosinolates from Arabidopsis thaliana[END_REF] and B. vulgaris [START_REF] Bianco | Acylated glucosinolates with diverse acyl groups investigated by high resolution mass spectrometry and infrared multiphoton dissociation[END_REF]. We searched for the expected peaks using extracted ion chromatograms with negative result. We concluded that either our material did not contain these glucosinolates, or our conditions were not suitable for detection of acylated glucosinolates, which are known to require higher elution volumes from anion-exchange columns after desulfation process than other glucosinolates.

Leaves of A. thaliana natural population showed a high diversity of glucosinolates with indol-3ylmethyl backbone, methylthio and methylsulfinylalkyl side-chains. Glucoraphanin was the main compound in all samples (50%) [START_REF] Brown | Variation of glucosinolate accumulation among different organs and developmental stages of Arabidopsis thaliana[END_REF][START_REF] Hogge | HPLC separation of glucosinolates from leaves and seeds of Arabidopsis thaliana and their identification using thermospray liquidchromatography mass-spectrometry[END_REF][START_REF] Bianco | Acylated glucosinolates with diverse acyl groups investigated by high resolution mass spectrometry and infrared multiphoton dissociation[END_REF]. There was no significant change in the total glucosinolates content nor in the proportion of the 12 identified glucosinolates, following inoculation with A.

brassicicola. The concentration of glucosinolates in the ecotypes Col and Ws remained extremely low after inoculation and did not differ from non-inoculated samples. This was a quite interesting observation as the pathogen was reported to trigger glucosinolates production [START_REF] Hanschen | Differences in the enzymatic hydrolysis of glucosinolates increase the defense metabolite diversity in 19 Arabidopsis thaliana accessions[END_REF]. However, this is in line with some reports mentioning no accumulation of glucosinolates after fungal inoculation. The effect might be linked to several factors, such as the kinetics of induction, environmental conditions, and the pathogen [START_REF] Doughty | Variation in the glucosinolate content of oilseed rape (Brassica napus L.) leaves[END_REF][START_REF] Andini | Modulation of Glucosinolate Composition in Brassicaceae Seeds by Germination and Fungal Elicitation[END_REF].

Camalexin was not detected in any seed sample, which is in line with results from by Rauhut et al. [START_REF] Buxdorf | The Effects of Glucosinolates and Their Breakdown Products on Necrotrophic Fungi[END_REF].

Likewise and confirmed by previous reports for some, camalexin was not found in the leaves of B. vulgaris, B. nigra, C. bursa-pastoris [START_REF] Rauhut | Inducible expression of a Nep1-like protein serves as a model trigger system of camalexin biosynthesis[END_REF], C. hirsuta [START_REF] Jimenez | Phytoalexins produced in the leaves of Capsella bursapastoris (shepherd's purse)[END_REF], C. monensis, S. arvensis and S. officinale.

Conversely, all leaf samples of A. thaliana (except for the non-inoculated leaves of ecotype Ws) contained some camalexin. As expected, the inoculation with A. brassicicola triggered camalexin production, increasing its overall content from less than 2.8 µg.g -1 to 30.9 µg.g -1 (Fig. 7). With regards to the natural population and the ecotype Col of A. thaliana, non-infected samples showed very similar basal concentrations of camalexin of 2.7 and 2.8 µg.g -1 , respectively. After inoculation, the concentration increased 11-fold to 30.9 µg.g -1 for the natural population and 7-fold to 20.2 µg.g -1 for ecotype Col, respectively. In the Ws ecotype, no camalexin was detected before inoculation. Inoculated samples then showed 3.3 µg.g -1 of the compound, which is in the range of the basal level of the two other ecotypes. The constitutive presence of camalexin in some ecotypes of A. thaliana has been documented [START_REF] Bednarek | Conservation and clade-specific diversification of pathogen-inducible tryptophan and indole glucosinolate metabolism in Arabidopsis thaliana relatives[END_REF] as well as the lower camalexin content for Ws compared to the Col ecotype [START_REF] Zhang | Constitutive camalexin production and environmental stress response variation in Arabidopsis populations from the Iberian Peninsula[END_REF]. In addition to the variability between genotypes, the levels of camalexin measured in this study and the increase after infection were consistent with data from the literature [START_REF] Izquierdo Zandalinas | A Fast and Precise Method To Identify Indolic Glucosinolates and Camalexin in Plants by Combining Mass Spectrometric and Biological Information[END_REF][START_REF] Kagan | Arabidopsis ecotype variability in camalexin production and reaction to infection by Alternaria brassicicola[END_REF][START_REF] Bottcher | The Biosynthetic Pathway of Indole-3-Carbaldehyde and Indole-3-Carboxylic Acid Derivatives in Arabidopsis[END_REF]. It should nevertheless be kept in mind that the production of camalexin seems greatly dependent on a number of factors, including plant development stage, time of harvest after inoculation, and environmental conditions [START_REF] Buxdorf | The Effects of Glucosinolates and Their Breakdown Products on Necrotrophic Fungi[END_REF][START_REF] Zhang | Constitutive camalexin production and environmental stress response variation in Arabidopsis populations from the Iberian Peninsula[END_REF][START_REF] Yaguchi | Induction of plant disease resistance upon treatment with yeast cell wall extract[END_REF].

Conclusion

The present study proposes a quantification method for the simultaneous routine analysis of glucosinolates (total and individual) and camalexin adapted to small-sized samples from Brassicaceous plants. The protocol is based by simultaneous liquid extraction and combined purification prior to HPLC-UV-MS and HPLC-FLD approach. The linearity of the method for glucosinolates and camalexin were in the range of 50 to 3500 µM and 0.025 to 5 µg.mL -1 , respectively. LOD/LOQ based on UV absorption and fluorescence detection were 3.8/12.6 µM and 0.014/0.046 µg.mL -1 . This method was proven precise, accurate and true and can therefore be used from 20 to 50 mg of DM starting material, requiring as little as 60 mg for a triplicate analysis. The applicability of the proposed method was evaluated by the identification and quantification of glucosinolates and camalexin in different Brassicaceae genera, A. thaliana genotypes, organs (leaves and seeds), and after pathogen inoculation.

The results were comparable to those found in the literature, which confirms the precision of our method.

The proposed concept not only allows characterizing glucosinolates and camalexin from single smallsized samples it also significantly reduces experimental time and the overall costs of analysis. Obtaining qualitative and quantitative information on phytoalexins and phytoanticipins from a single small-sized sample is a crucial element for a better understanding of the role of these compounds in the A. brassicicola-Brassicaceae interactions. This particularly concerns early developmental stages of the black spot cycle, in which important changes in the content of glucosinolates and camalexin are expected. Such analyses are essential for all Brassicaceae species including those with small seeds, or with high concentrations of seed reserves.

CRediT authorship contribution statement

Appendix A. Supplementary data

Supplementary data to this article can be found online.

Figures and Tables

Fig. 1. General structure of glucosinolates with biological variation predominantly due to a substitution of side group R but derivatives at the thioglucose moiety are also known (137 identified whose 88 conclusively characterized) [START_REF] Blazevic | Glucosinolate structural diversity, identification, chemical synthesis and metabolism in plants[END_REF] and camalexin.

E P R O A L Y G N L G N A 4 O H G B N G B C N A S N E O (C) Certification report (n=2-7) Results (n=3) 0 1 2 3 4 5 6 7 P R O E P R O A L Y G N L G N A 4 O H G B N G B C N A S N E O (D) Certification report (n=2-7) Results (n=3)

Dimitri Bréard:

  Conceptualization, Methodology, Validation, Investigation, Writing -Original Draft Preparation, Visualization. Thibault Barrit: Resources. Daniel Sochard: Resources. Sophie Aligon: Resources. Elisabeth Planchet: Writing -Review & Editing, Supervision. Béatrice Teulat: Writing -Review & Editing, Supervision. Josiane Le Corff: Resources, Writing -Review & Editing, Supervision. Claire Campion: Writing -Review & Editing, Supervision, Project administration, Funding acquisition. David Guilet: Writing -Review & Editing, Supervision, Project administration, Funding acquisition.

Fig. 2 .

 2 Fig. 2. Examples of HPLC analysis of desulfo-glucosinolates and camalexin fractions. (A) HPLC-UV

Fig. 3 .

 3 Fig. 3. Comparison of glucosinolates content in mmol.kg -1 of DM between certification report of

Fig. 4 .

 4 Fig. 4. Down scaled results in mmol.kg -1 of DM and comparison with certification report of European

Fig. 6 . 7 AFig. 7 .

 677 Fig. 6. Total glucosinolates content in mmol.kg -1 of DM in our sample panel. (A) Comparison of seed

Table 1

 1 Recovery of camalexin from rapeseed samples spiked at different levels (n=3). Down scaled results in µg.g -1 of DM for A. thaliana natural population leaf samples containing a medium quantity of camalexin. Camalexin content with starting material from 8 to 45 mg. The value between brackets was not included in the average.

	Spiked quantity (µg) Recovery (%) RSD (%)
	2.00	110	18
	1.50	101	3
	1.03	99	8
	0.77	104	9
	0.39	117	9
	0.26	108	4
	0.10	98	21
	0.05	100	4

Acknowledgments

This research was funded by SFR Quasav (BRAF project 2018-2020). The authors also acknowledge Michaël Anjuere for manufacturing the columns rack, Pascale Satour for use of the freeze-dryer, PT Phyto (SFR Quasav) and PT Astral (SFR Matrix) for instruments, Andreas Schinkovitz and Fanny Schurrer for English corrections as well as Elza Neau and Marie-Anne Portela for inspiring the graphical abstract.

Glucosinolate (GSL). Below limit of quantification (blq) is between LOD and LOQ. Tentatively identified (*).


PhD Thierry Auger
email: thierry.auger@ensam.eu

PhD Bassem Barkia
email: bassem.barkia@ensam.eu

PhD Eva Héripré
email: eva.heripre@centralesupelec.fr

M. Sc Vincent Michel
email: vincent.michel@ensam.eu

M. Sc Denis Mutel
email: denis.mutel.1@ulaval.ca

PhD Ivan Guillot
email: ivan.guillot@cnrs.fr

PhD Zehoua Hamouche
email: zehoua.hamouche@ensam.eu

PhD Liliana Medina-Almazàn
email: liliana.medina@cfe.gob.mx

Crack path and liquid metal embrittlement specificity of austenitic steels in mercury at room temperature

A liquid metal embrittlement specificity of three austenitic steels with increasing nickel content (304 L, 316 L and 316L(N)) is studied in liquid mercury in the axisymmetric notched geometry. Only the low nickel alloys are susceptible to LME. The crack path of an austenitic steel fracture induced by liquid mercury has been elucidated at microstructural scale. Deformation induced martensite (γ(fcc) → α'(bcc)) of the low nickel steels induces numerous α'/α' interfaces at small scale that are susceptible to be embrittled. Because the only steel that resists LME is the one that shows stability over α' phase change due mostly to its higher nickel content, a point confirmed by X Ray fractography, it is inferred that the major factor contributing to the LME sensitivity at room temperature is the α' phase formation in unstable austenitic steels during plastic strain. This provides a sound rationale on how to prevent mercury induced embrittlement with austenitic steels.

Introduction

Austenitic stainless steels are the structural materials of choice for containment of liquid metals in many applications. Understanding their mutual chemical or physical interactions such as corrosion or liquid metal embrittlement (LME) is therefore highly desirable. While austenitic steels have been rather thoroughly investigated for liquid metal corrosion (in PbBi for example [1]), the LME phenomenology of austenitic steels is still not satisfactorily understood. A screening study in mercury by Krupowitz with several austenitic stainless steels indicates that the 304, 304 L and 321 steels are susceptible to LME in mercury at room temperature but not the 316 and 316 L steels [START_REF] Stp | Slow Strain Rate Testing for the Evaluation of Environmentally Induced Cracking: Research and Engineering Applications[END_REF]. The study was focused on uniaxial mechanical testing and the fractographic study that was carried out gave no particular rationale for the observed variation in LME sensitivity. In a work on oligocyclic fatigue of 316LN at room temperature, a mild reduction in the lifetime of the samples was observed when exposed to liquid mercury [START_REF] Tian | [END_REF]. Another study showed significant LME in plane stress fracture mechanics experiments with 316 L in contact with mercury [4]. This spread in the sensitivity with 316 L steels seems to obfuscate the issue of LME in this environment with contradicting statements. A matter, maybe not given sufficient consideration, is the unstable behavior of some of the austenitic steels when plastically strained leading to a γ(fcc) to α'(bcc) phase transformation [5]. One can indeed notice when considering the steel's composition, that those showing LME sensitivity are low nickel austenitic steels (see compositions given in [START_REF] Stp | Slow Strain Rate Testing for the Evaluation of Environmentally Induced Cracking: Research and Engineering Applications[END_REF][START_REF] Tian | [END_REF][4]). Therefore, one may infer that the LME sensitivity could be connected to or triggered by the Deformation Induced Martensite (DIM) transformation, a behavior mainly controlled by the nickel content when varying composition [6]. This possibility was already hinted by the behavior of the 304 L steel in sodium where detailed small-scale analysis showed that γ/α' and α'/α' interfaces are the fracture crack path [7,8]. A strong correlation of the LME trend with the amount of DIM was also recently unveiled on a low nickel version of the 316 L steel in contact with eutectic gallium-indium melt [9]. The LME sensitivity of austenitic steels with Hg would therefore have to be attributed essentially to a LME sensitivity of the bcc phase formed during plastic deformation.

The work described in this paper allows to settle this question in mercury by specifically varying the amount of nickel around the critical concentration for a drastic change in the DIM at room temperature. An experimental investigation at small-scale is first carried out to characterize further mercury induced microstructural crack path. We then chose to test and analyze the LME behavior of 304 L (9wt% Ni), a low nickel version of 316 L (10wt% Ni) and a nuclear grade of nitrogen stabilized 316 L, the 316L(N) (with 12 wt% Ni). Increasing the nickel content with these three materials procures a higher stability relative to DIM. Nitrogen has also the ability to stabilize austenite relative to the phase diagram but does not contribute to suppress DIM [6]. The fraction of γ to α' phase transform upon fracture was assessed by XRay fractography, a technique used up to now mainly for characterizing samples in fatigue [10] that we repurpose here in the context of LME. The results presented here point to the conclusion that the α' phase formation in unstable austenitic steels is associated with LME sensitivity. We thereby demonstrate here that the LME sensitivity can also be suppressed by tailoring the composition for stability relative to the DIM behavior giving for the first time a sound way out LME concerns in mercury.

Experimental procedures

The composition of the three stainless austenitic steels (304, 316 L and 316L(N)) used in this study is given in Table 1.

The three steels were provided in the annealed metallurgical state. The mean grain size was 40 ± 4 µm for 304 L, 15 ± 2 µm for 316 L and 43 ± 5 µm for 316L(N). The Md 30 temperature (temperature level at which 50% of the material's volume has undergone martensite transform after 30% plastic deformation) for each of these steels is, respectively of 285 K, 219 K and 129 K according to the Nohara correlation [12].

Table 1. 304 L, 316 L and 316L(N) elemental composition from suppliers (* not measured). Note that the equivalent nickel content is given in parenthesis (Pryce and Andrew formulae [11]). A Center Cracked in Tension (CCT) type sample was tested in contact with mercury. The geometry (thickness of 1.5 mm, width of 50 mm with a centered elliptic void of 10 mm length and 2 mm height) allows to study crack propagation in plane stress condition. The CCT samples were wetted at the notch level and were mechanically loaded at a low crosshead speed in the brittle regime (see schematic in Fig. S10 in Supplementary Materials) [START_REF] Medina-Almazàn | Etudes Expérimentale Et Numérique De L'effet Du Mercure Sur Le Comportement Mécanique Des Aciers 316 L et T91[END_REF]. A liquid mercury brittle sub-critical crack propagation occurs inducing a reduction of the order of 50% of the energy to fracture [START_REF] Medina-Almazàn | Etudes Expérimentale Et Numérique De L'effet Du Mercure Sur Le Comportement Mécanique Des Aciers 316 L et T91[END_REF][START_REF] Auger | [END_REF]. The fracture surface induced by the contact with mercury is atypical in this geometry as it leads to a river pattern that has remained unexplained so far [4]. Such a pattern was a strong motive for the small-scale investigation of the fracture crack path reported here. In addition, Notched Axisymmetric (NA) samples were used for the composition variation study (initial diameter of 4 mm, gauge length of 15 mm and a 60 • angle notch of 600 µm depth with a tip radius of the order of 100 µm). Two NA samples of each composition were wetted by mercury using the chemical etching technique known from previous studies [4]. An exposure of the notch to diluted hydrochloric acid (5%) was first applied followed by mercury insertion using a variable microliter volume pipette. Mercury readily wets the notch tip ensuring also adherence of the liquid metal drop during mounting manipulations. All mechanical tests were carried out at room temperature in air. The NA wetted samples were tested at two strain rates (1.67 × 10 -7 m.s -1 and 5.0 × 10 -8 m.s -1 ). Reference NA samples were also tested at 1.67 × 10 -7 m.s -1 . This notched geometry allows to measure quantitatively the degree of embrittlement when in contact with the liquid metal by a comparative measurement of the energy to fracture. The ratio of that energy by the one for the reference test gives a direct measure of the degree of embrittlement (the mean value of the two wetted measurements is taken here).

After mechanical testing, all the samples were cleaned from mercury by ultrasonic cleaning in distilled water repeated 2 to 3 times to remove mercury. The CCT sample, to be analyzed by EBSD, was electrochemically covered with a nickel deposit of roughly 100 µm. Transverse cuts on the CCT and NA samples were prepared using masked argon ion beam milling (Cross-Polisher from Jeol). EBSD orientation mapping was carried out using a FEG-SEM (FEI Helios 650) in the vicinity of the fracture surface. The gallium focused Ion Beam column of the Helios 650 was used to machine a transmission electron microscope (TEM) lamella at a selected location on a CCT sample (Fig. 1a). The TEM lamella was investigated using EBSD in transmission mode (t-EBSD) at a sample tilt of +38 • relative to the EBSD setup. The scanning step could then be reduced down to 10 nm. Two phases (bcc and fcc iron) were allowed in EBSD pattern indexing analysis.

The NA samples were also characterized by X-Ray fractography. The phases in the volume immediately beneath the fracture surface are analyzed using the small X-Ray penetration depth of 40 keV X-Rays in a θ-2θ diffraction geometry (it does not exceed a few µm) [10]. The X-Ray diffraction spectra were acquired by focusing the beam onto the fracture surface using a blocking collimator. An amorphous polysiloxane film was inserted covering the notch areas to avoid counting machining induced phase change. A semi-quantitative analysis of the spectra can be conveyed to differentiate the γ (fcc) and α' (bcc) phases. In spite of the unavoidable spread in half-width due to height unevenness, the peaks from the two phases are sufficiently separated to be quantified without ambiguities (here between 44 • and 100 • ). The error estimate of this semi-quantitative analysis is then estimated according to standard procedure at 3% level [START_REF]Standard Practice for X-Ray Determination of Retained austenite in steel with near random crystallographic orientation[END_REF].

Results and discussion

The unusual fracture surface is shown by SEM in Fig. 1a. One can notice the river pattern along which crack propagation proceeds (perpendicularly to the platinum deposit used for the FIB lift-out, indicated by the white arrow). The result of t-EBSD mapping for the 316L-Hg CCT material on the FIB lift-out is shown in Fig. 1b with the phase map (α' iron in red) superposed with the image quality in grey. One can notice that in the probed area, the zone immediately below the fracture surface has almost entirely turned up into α' phase relative to the initial microstructure. The IPF maps (Fig. S1 in Supplementary Materials) demonstrate that fracture proceeds in an interfacial manner in a heavily deformed microstructure. The larger view obtained by conventional EBSD (CCT specimen) reveals that numerous twinning interfaces as well as grain refinement compared with the initial grain size of 15 µm both develops during straining (Fig. 1c, the black spots in the scan are due to ductile cavities growth). The phase map in Fig. 1d shows the banded structure of α' martensite phase transforms in this sample. This is reminiscent of the fluctuations in local chemical composition in the through-thickness direction, an effect of the chemical inhomogeneities resulting from solidification that are not annealed out in fast casting The NA samples fractographic analysis is shown in Fig. 2. The mechanical tests results are shown in Fig. 3.

Clearly only the 304 L (9%Ni) and the 316 L (10.2%Ni) show a reduction in energy to fracture (Fig. 3) associated with a change in fracture surface (Fig. 2). At high magnification, one would classify the embrittlement as a "quasi-clivage" fracture mode in the NA geometry. The embrittlement is rated at a factor of two decrease for both steels. This is similar to the ratio found by fracture mechanics experiments for the same 316 L steel (although these tests were carried out in plane stress condition using the CCT geometry) [4]. The onset of embrittlement is seen also after a significant amount of plastic deformation notwithstanding the geometry (CCT or NA). After a sudden crack initiation at macroscopic scale, further crack propagation is sub-critical in the NA geometry as well.

The stunning result of this work is the absence of LME for 316L(N). Neither fractography nor the energy to fracture hints at any LME. The fracture surface looks entirely ductile even at the notch tip location. When considering the X-Ray fractography analysis, one sees a clear difference between the two embrittled steels (304 L and 316 L) and the immune steel (316L(N)). The raw diffractogram for the three steels can be found in supplementary (Fig. S6 in supplementary). The embrittlement is clearly correlated with the amount of DIM probed immediately beneath the fracture surface by X-Rays (Fig. 3, right scale). Overall, these results provide strong evidences that the deformation induced martensitic transformation is the key element in the LME sensitivity.

The EBSD analysis on transverse cuts on the three steels in the NA geometry are shown in Fig. 4 (the corresponding orientation maps can be found in Supplementary Materials, Figs. S7-S9).

The local EBSD analysis for the three steels confirms completely the large difference between the two low nickel austenitic steels and the higher nickel content one (316L(N)). Transformed areas are nearly absent with 316L(N) as indicated by the absence of α' bcc phase detected by EBSD or its low amount revealed by X-Ray fractography. This is in complete accordance with the known behavior of austenitic steels when varying the nickel content [6]. In the phase maps shown here, the γ/α' interfaces were not found to constitute interfaces able to be weakened by mercury while it formally cannot be excluded.

State-of-the-art local samples preparation techniques (Cross-Polisher and FIB lift-out) allowed local EBSD analysis up to the fracture surface enlightening the LME case that was reported in [4]. It is confirmed by the global phase change quantification obtained by X-Ray fractography. The scenario for the LME mechanism with austenitic steels in mercury is therefore the following: the material is plastically deformed until DIM has produced sufficient density of α'/α' interfaces. Beyond some threshold, crack propagation becomes possible in a 3D complex martensite network. In the course of the crack propagation process, mercury does mostly embrittle these interfaces as it can be seen that cracking takes place mostly in transformed area. It is then clear that because no such phase transformation towards a bcc structure is activated with 316L(N), LME is no longer possible to a significant extent.

The findings of this paper are in full agreement with the work of Krupowicz when considering the steel's composition given by the author [START_REF] Stp | Slow Strain Rate Testing for the Evaluation of Environmentally Induced Cracking: Research and Engineering Applications[END_REF]. No LME is found with mercury on austenitic steels with a nickel content higher than 12% and as such deemed stable austenitic steels (stability is here to be understood relative to the crystallographic ground state that remains in the fcc structure at any plastic strain). This implies that a sufficient amount of nickel in austenitic steels is able to prevent LME sensitivity. This provides the rationale behind the spread in LME sensitivity response in the literature and emphasizes the critical role of DIM with austenitic steel's LME. A similar conclusion was reached with the eutectic gallium-indium-tin liquid melt environment when varying the temperature [9]. When one reaches a temperature high enough to offset the DIM plastic response, LME sensitivity is suppressed. Temperature is not the only variable therefore one can adjust to avoid LME concern; the steel's composition can also be adjusted to suppress this sensitivity. Whether this procedure is sufficient by itself to maintain an LME insensitivity is to be pondered by the fact that liquid metal corrosion of austenitic steels can lead to nickel depletion by selective leaching in austenitic steel potentially inducing ferritization [START_REF] Hosemann | [END_REF]. This therefore does not preclude to take other measures such as corrosion control in safeguarding against LME.

Conclusion

It has been demonstrated here that a high correlation exists with the occurrence of DIM upon plastic strain in unstable austenitic steels and LME sensitivity at room temperature in mercury. The mechanism of LME requires the formation of α' martensite volume in sufficient amount to manifest itself. The implication for fracture is that principally the bcc crystallographic phase can be embrittled, an empirical finding yet to explain thoroughly. This also raises interesting prospects as to avoid LME concerns in mercury and potentially other liquid metal such as Galinstan or sodium-potassium near room temperature. A careful selection of the material such as a minimum nickel content as in the 316L (N) steel is protective. Structural integrity concerns then shift to ensuring that materials are stable enough in the selected liquid metal environment against ferritization.

Fig. 1 .

 1 Fig. 1.. 316L-Hg (a) Tilted SEM view of the FIB sampling area with the protective platinum deposit before FIB sectioning (b) t-EBSD phase map of the FIB lamella with Image Quality (IQ) gray coding (α' iron in red) (c) Inverse pole figure (IPF) + IQ map of the coarse-grained cross section d) Corresponding coarse-grained phase map with IQ gray coding relative to c) (α' iron in red). All images are taken from slow strain rate CCT samples at room temperature.

Fig. 2 .

 2 Fig. 2.. Fractographic analysis by SEM of NA fracture. Upper image taken at low magnification. Lower image is a small-scale view of the upper fracture surface -(a) 304L-Hg (b) 316L-Hg (c) 316L(N)-Hg.

Fig. 3 .

 3 Fig. 3.. Left) Energy to fracture normalized to the reference test (in black). Right) Quantification of the γ phase by X-Ray fractography (in blue). Upper left insert) Force versus Cross-head displacement curves for 304 L austenitic steel (Blue and green in contact with mercury). Lower right insert) Force versus Cross-head displacement curves for 316L(N) austenitic steel.

Fig. 4 .

 4 Fig. 4.. EBSD phase maps + IQ on cross-sections for 304 L, 316 L and 316L(N) steels in the NA geometry (α' iron in red).

Acknowledgments

The Helios 660 FIB-SEM work was carried out using the equipment of the MATMECA consortium at CentraleSupélec partially funded by the French ANR under contract number ANR-10-EQPX-37. Sokona Konaté is thanked for her help with the Jeol Cross Polisher. Jean Louis Courouau is thanked for providing the 316L(N) steel. It is a pleasure for one of the authors (T.A.) to acknowledge many pleasant and inspiring discussions on this topic with André Pineau from the Ecole des Mines de Paris (Mines-ParisTech).

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Supplementary materials

Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.scriptamat.2022.114733.