Béchir Amri

Jean-Philippe Anker
email: anker@univ-orleans.fr

Mohamed Sifi
email: mohamed.sifi@fst.rnu.tn

Échir Amri

Three results in Dunkl theory

Keywords: . 2000 Mathematics Subject Classification. Primary 33C52 ; Secondary 42B10, 43A32, 33C80, 22E30 Dunkl operator, Paley-Wiener theorem, generalized translations

come

Introduction

Dunkl theory generalizes classical Fourier analysis on R N . It started twenty years ago with Dunkl's seminal work [START_REF] Dunkl | Differential-Difference operators associated to reflextion groups[END_REF] and was further developed by several mathematicians. See for instance the surveys [START_REF] Rösler | Dunkl operators : theory and applications[END_REF][START_REF]Harmonic and stochastic analysis of[END_REF] and the references cited therein.

In this setting, the Paley-Wiener theorem is known to hold for balls centered at the origin. In [START_REF] De Jeu | Paley-Wiener theorems for the Dunkl transform[END_REF], a Paley-Wiener theorem was conjectured for convex neighborhoods of the origin, which are invariant under the underlying reflection group, and was partially proved. Our first result in Section 3 is a proof of this conjecture in the crystallographic case, following the third approach in [START_REF] De Jeu | Paley-Wiener theorems for the Dunkl transform[END_REF].

Generalized translations were introduced in [START_REF] Rösler | Generalized Hermite polynomials and the heat equation for Dunkl operators[END_REF] and further studied in [START_REF] Trimèche | Paley-Wiener theorems for the Dunkl transform and Dunkl translation operators[END_REF][START_REF] Rösler | A positive radial product formula for the Dunkl kernel[END_REF][START_REF] Thangavelu | Convolution operator and maximal function for the Dunkl transform[END_REF]. Apart from their abstract definition, we lack precise information, in particular about their integral representation (τ x f )(y) = R N f (z) dγ x,y (z) , which was conjectured in [START_REF] Rösler | Generalized Hermite polynomials and the heat equation for Dunkl operators[END_REF] and established in few cases, for instance in dimension N = 1 or when f is radial. Our second result in Section 4 is an optimal bound for the integral R |dγ x,y (z)| in dimension N = 1 , improving upon earlier results in [START_REF] Rösler | Bessel-type signed hypergroup on R, in Probability measures on groups and related structures XI[END_REF][START_REF] Thangavelu | Convolution operator and maximal function for the Dunkl transform[END_REF]. Our bound depends on the multiplicity k ≥ 0 and tends from below to √ 2 , as k → +∞ . Our third result in Section 5 deals with the support of the distribution γ x,y in higher dimension, that we determine rather precisely in the crystallographic case.

Background

In this section, we recall some notations and results in Dunkl theory and we refer for more details to the articles [START_REF] Dunkl | Differential-Difference operators associated to reflextion groups[END_REF][START_REF] De Jeu | The Dunkl transform[END_REF] or to the surveys [START_REF] Rösler | Dunkl operators : theory and applications[END_REF][START_REF]Harmonic and stochastic analysis of[END_REF].

Let G ⊂ O(R N ) be a finite reflection group associated to a reduced root system R and k : R → [0, +∞) a G-invariant function (called multiplicity function). Let R + be a positive root subsystem, Γ + the corresponding open positive chamber, Γ + its closure, Γ + = α∈R + R + α the dual cone, and let us denote by x + the intersection point of any orbit G.x in R N with Γ + .

The Dunkl operators T ξ on R N are the following k-deformations of directional derivatives ∂ ξ by difference operators :

T ξ f (x) = ∂ ξ f (x) + α∈R + k(α) α, ξ f (x)-f (σα. x) α, x
, where σ α . x = x-α, x 2 |α| 2 α denotes the reflection with respect to the hyperplane orthogonal to α. The Dunkl operators are antisymmetric with respect to the measure w(x) dx with density

w(x) = α∈R + | α, x | 2 k(α)
. The operators ∂ ξ and T ξ are intertwined by a Laplace-type operator

V f (x) = R N f (y) dµ x (y) (1)
associated to a family of compactly supported probability measures { µ x | x ∈ R N } . Specifically, µ x is supported in the the convex hull

C x = co(G.x) .
For every λ ∈ C N , the simultaneous eigenfunction problem

T ξ f = λ, ξ f ∀ ξ ∈ R N has a unique solution f (x) = E(λ, x) such that E(λ, 0) = 1, which is given by (2) E(λ, x) = V (e λ, . )(x) = R N e λ,y dµ x (y) ∀ x ∈ R N .
Furthermore λ → E(λ, x) extends to a holomorphic function on C N and the following estimate holds :

|E(λ, x)| ≤ e (Re λ) + , x + ∀ λ ∈ C N , ∀ x ∈ R N .
In dimension N = 1, these functions can be expressed in terms of Bessel functions. Specifically,

E(λ, x) = j k-1 2 (λ x) + λ x 2k+1 j k+ 1 2 (λ x) , where j ν (z) = Γ(ν +1) +∞ n=0 (-1) n n! Γ(ν+n+1) z 2 ) 2n are normalized Bessel functions.
The Dunkl transform is defined on L 1 (R N , w(x)dx) by

Df (ξ) = 1 c R N f (x) E(-i ξ, x) w(x) dx , where c = R N e -|x| 2 2 w(x) dx .
We list some known properties of this transform : (i) The Dunkl transform is a topological automorphism of the Schwartz space S(R N ).

(ii) (Plancherel Theorem) The Dunkl transform extends to an isometric automorphism of L 2 (R N , w(x)dx). (iii) (Inversion formula) For every f ∈ S(R N ), and more generally for every f ∈ L 1 (R N , w(x)dx) such that Df ∈ L 1 (R N , w(ξ)dξ), we have

f (x) = D 2 f (-x) ∀ x ∈ R N .
(iv) (Paley-Wiener theorem) The Dunkl transform is a linear isomorphism between the space of smooth functions f on R N with supp f ⊂ B(0, R) and the space of entire functions h on C N such that

(3) sup ξ∈C N (1+|ξ|) M e -R | Im ξ | |h(ξ)| < +∞ ∀ M ∈ N .

A geometric Paley-Wiener theorem

In this section, we prove a geometric version of the Paley-Wiener theorem, which was looked for in [START_REF] De Jeu | Paley-Wiener theorems for the Dunkl transform[END_REF][START_REF] Trimèche | Paley-Wiener theorems for the Dunkl transform and Dunkl translation operators[END_REF][START_REF] De Jeu | Some remarks on a proof of geometrical Paley-Wiener theorems for the Dunkl transform[END_REF], under the assumption that G is crystallographic. The proof consists merely in resuming the third approach in [START_REF] De Jeu | Paley-Wiener theorems for the Dunkl transform[END_REF] and applying it to the convex sets considered in [START_REF] Anker | L p Fourier multipliers on Riemanian symmetric spaces of the noncompact type[END_REF][START_REF] Anker | The spherical Fourier transform of rapidly decreasing functions (a simple proof of a characterization due to Harish-Chandra, Helgason, Trombi and Varadarajan)[END_REF][START_REF] Anker | Handling the inverse spherical Fourier transform[END_REF][START_REF] Anker | Sharp estimates for some functions of the Laplacian on noncompact symmetric spaces[END_REF] instead of the convex sets considered in [START_REF] Opdam | Harmonic analysis for cetain representations of graded Hecke algebras[END_REF]. Recall that the second family consists of the convex hulls

C Λ = co(G.Λ)
of G-orbits G.Λ in R N , while the first family consists of the polar sets

C Λ = { x ∈ R N | x, g.Λ ≤ 1 ∀ g ∈ G } . Λ Λ Figure 1. The sets C Λ and C Λ for the root system A 1 ×A 1 Λ Λ Figure 2.
The sets C Λ and C Λ for the root system B 2

Before stating the geometric Paley-Wiener theorem, let us make some remarks about the sets C Λ and C Λ . Firstly, they are convex, closed, G-invariant and the following inclusion holds :

C Λ ⊂ |Λ| 2 C Λ .
Secondly, we may always assume that Λ = Λ + belongs to the closed positive chamber Γ + and, in this case, we have

C Λ ∩ Γ + = Γ + ∩ Λ -Γ + , C Λ ∩ Γ + = { x ∈ Γ + | Λ, x ≤ 1 } .
Thirdly, on one hand, every G-invariant convex subset in R N is a union of sets C Λ while, on the other hand, every G-invariant closed convex subset in R N is an intersection of sets C Λ . For instance,

B(0, R) = |Λ|=R C Λ = |Λ|=R -1 C Λ .
Fourthly, we shall say that Λ ∈ Γ + is admissible if the following equivalent conditions are satisfied :

(i) Λ has nonzero projections in each irreducible component of (R N , R), (ii) C Λ is a neighborhood of the origin, (iii) C Λ is bounded.
In this case, we may consider the gauge

χ Λ (ξ) = max x∈C Λ x, ξ = min { r ∈ [ 0, +∞) | ξ ∈ r C Λ } on R N .
Theorem 3.1. Assume that Λ ∈ Γ + is admissible. Then the Dunkl transform is a linear isomorphism between the space of smooth functions f on R N with supp f ⊂ C Λ and the space of entire functions h on C N such that

(4) sup ξ∈C N (1+|ξ|) M e -χ Λ (Im ξ) |h(ξ)| < +∞ ∀ M ∈ N .
Proof. Following [START_REF] De Jeu | Paley-Wiener theorems for the Dunkl transform[END_REF], this theorem is first proved in the trigonometric case, which explains the restriction to crystallographic groups, and next obtained in the rational case by passing to the limit. The proof of Theorem 3.1 in the trigonometric case is similar to the proof of the Paley-Wiener Theorem in [START_REF] Opdam | Harmonic analysis for cetain representations of graded Hecke algebras[END_REF][START_REF] Opdam | Lecture notes on Dunkl operators for real and complex reflection groups[END_REF], and actually to the initial proof of Helgason for the spherical Fourier transform on symmetric spaces of the noncompact type. This was already observed in [START_REF] Br | Contributions to the hypergeometric function theory of Heckman and Opdam (sharp estimates, Schwartz space, heat kernel)[END_REF] and will be developed below for the reader's convenience. The limiting procedure, as far as it is concerned, is described thoroughly in [START_REF] De Jeu | Paley-Wiener theorems for the Dunkl transform[END_REF] and needs no further explanation. Thus assume that h is an entire function on C N satisfying (4) and, by resuming the proof of [START_REF] Opdam | Harmonic analysis for cetain representations of graded Hecke algebras[END_REF]Theorem 8.6 (2)], let us show that its inverse Cherednik transform [START_REF] Dunkl | Differential-Difference operators associated to reflextion groups[END_REF] f (x) = const.

R N h(ξ) E(iξ, x) w(ξ) dξ
vanishes outside C Λ . Firstly, one may restrict by G-equivariance to x = g 0 .x + , where x + ∈ Γ + C Λ and g 0 denotes the longest element in G, which interchanges Γ + and -Γ + . Secondly, by expanding

α∈R + ( α, ξ -k α ) E(ξ, x) = g∈G q∈Q + c(-g.ξ) E q (g, g.ξ) e g.ξ+̺+q, x (5)
becomes

f (x) = const. g∈G det g q∈Q + f g,q (x) e ̺+q, x
, where (6)

f g,q (x) = R N h(g -1 .ξ) E q (g, iξ) e i ξ, x α∈R + Γ(i α, ξ +kα ) Γ(i α, ξ +1)
dξ .

Thirdly, one shows that all expressions (6) vanish, by shifting the contour of integration from R N to R N + i t g 0 .Λ with t > 0, which produces an exponential factor e -ct with c = Λ, x + -1 > 0 , and by letting t → +∞ .

Since every G-invariant convex compact neighborhood of the origin in R N is the intersection of admissible sets C Λ , Theorem 3.1 generalizes as follows.

sup ξ∈C N (1+|ξ|) M e -χ(Im ξ) |h(ξ)| < +∞ ∀ M ∈ N . Remark 3.3. Notice that the Dunkl transform D always maps C ∞ C (R N ) into H χ (C N
) and that the assumption that G is crystallographic is only used to prove that D is onto.

L p bounds for generalized translations in dimension 1

Dunkl translations are defined on S(R N ) by

(τ x f )(y) = 1 c R N Df (ξ) E(iξ, x) E(iξ, y) w(ξ) dξ ∀ x, y ∈ R N .
They have an explicit integral representation [START_REF] Rösler | Bessel-type signed hypergroup on R, in Probability measures on groups and related structures XI[END_REF] in dimension N = 1 : Proposition 4.1. The following inequality holds, for every x, y ∈ R :

(τ x f )(y) = R f (z) dγ x,y (z) , where (7) dγ x,y (z) =      γ(x, y, z) |z| 2k dz if x, y ∈ R * dδ y (z) if x = 0 dδ x (z) if y = 0 is a signed measure such that R dγ x,y (z) = 1 . Specifically, γ(x, y, z) = d σ(x, y, z) ρ(|x|, |y|, |z|) 1l I |x|,|y| (|z|) ∀ x, y, z ∈ R * , where d = Γ(k+ 1 2 ) √ π Γ(k) , σ(x, y, z) = 1 -x 2 + y 2 -z 2 2 x y + z 2 + y 2 -x 2 2 z y + x 2 + z 2 -y 2 2 x z = (z+x+y) (z+x-y) (z-x+y) 2 x y z ∀ x, y, z ∈ R * , ρ(a, b, c) = { c 2 -(a-b) 2 } k-1 { (a+b) 2 -c 2 } k-1 ( 2 a b c ) 2k-1 = ( 2 b 2 c 2 + 2 a 2 c 2 + 2 a 2 b 2 -a 4 -b 4 -c 4 ) k-1 ( 2 a b c ) 2k-1
(9) R dγ x,y (z) ≤ A k = √ 2 { Γ(k+ 1 2 )} 2 Γ(k+ 1 4 ) Γ(k+ 3 4 ) .
Actually there is equality if

x = y ∈ R * . Moreover A k < -→ √ 2 as k → +∞ .
Remark 4.2. This result improves earlier bounds obtained in [START_REF] Rösler | Bessel-type signed hypergroup on R, in Probability measures on groups and related structures XI[END_REF] and [START_REF] Thangavelu | Convolution operator and maximal function for the Dunkl transform[END_REF], which were respectively 4 and 3 . Case 2 : Assume that x y > 0 . By symmetry, we may reduce to 0 < x ≤ y . Then

R |dγ x,y (z)| = 0 -∞ |dγ x,y (z)| + +∞ 0 |dγ x,y (z)| = 2 d y+x y-x x + y 2 x y z z 2 -x 2 -y 2 + 2 x y 2 x y z k x 2 + y 2 + 2 x y -z 2 2 x y z k-1 z 2k dz .
After performing the change of variables z = x 2 + y 2 -2 x y cos θ and setting y = s x , we get

(10) R |dγ x,y (z)| = Γ(k+ 1 2 ) √ π Γ(k) (1+s) π 0 (1 -cos θ) sin 2k-1 θ √ 1 + s 2 -2 s cos θ dθ .
Denote by F (s) the right hand side of [START_REF] De Jeu | Some remarks on a proof of geometrical Paley-Wiener theorems for the Dunkl transform[END_REF]. Since

F ′ (s) = Γ(k+ 1 2 ) √ π Γ(k) (1-s) π 0 sin 2k+1 θ (1 + s 2 -2 s cos θ) 3 2
dθ is nonpositive, F (s) is a decreasing function on [ 1, +∞ ), which reaches its maximum at s = 1 . Let us compute it :

A k = F (1) = √ 2 Γ(k+ 1 2 ) √ π Γ(k) π 0 (1-cos θ) k-1 2 (1+cos θ) k-1 sin θ dθ = 2 2k Γ(k+ 1 2 ) √ π Γ(k) 1 0 t k-1 2 (1-t) k-1 dt = 2 2k Γ(k+ 1 2 ) √ π Γ(k) B(k+ 1 2 , k) = 2 2k { Γ(k+ 1 2 )} 2 √ π Γ(2k+ 1 2 ) = √ 2 Γ(k+ 1 2 ) Γ(k+ 1 4 ) Γ(k+ 1
2 ) Γ(k+ 3 4 ) , after performing the change of variables t = 1-cos θ 2 and using standard properties of the beta and gamma functions.

Finally let us show that

A k < -→ √ 2 as k → +∞ . Write A k = √ 2 G(k+ 1 4 ) G(k+ 1 2 ) , where G(u) = Γ(u+ 1 4 ) Γ(u) ∀ u > 0 .
Since the logarithmic derivative Γ ′ Γ of the gamma function is a strictly increasing analytic function on (0, +∞), the logarithmic derivative

G ′ (u) G(u) = Γ ′ (u+ 1 4 ) Γ(u+ 1
is positive. Hence G is an strictly increasing function and A k < √ 2 . On the other hand, using Stirling's formula

Γ(u) ∼ √ 2π u u-1 2 e -u as u → +∞ , we get G(k+ 1 4 ) ∼ G(k+ 1 2 ) hence A k → √ 2 , as k → +∞ .
As a first consequence, we obtain the L 1 → L 1 operator norm of Dunkl translations in dimension N = 1 .

Corollary 4.3. Let x ∈ R * . Then τ x is a bounded operator on L 1 (R, |x| 2k dx), with τ x L 1 →L 1 = A k .
Proof. The inequality τ x L 1 →L 1 ≤ A k follows from ( 9), together with [START_REF] De Jeu | The Dunkl transform[END_REF], and it remains for us to prove the converse inequality. By symmetry, we may assume that x > 0 . Since

A k = lim y→x R |γ(x, y, z)| |z| 2k dz ,
for every 0 < ε < A k , there exists 0 < η < x such that, for every y

∈ [x-η, x+η], (11)
R |γ(x, y, z)| |z| 2k dz > A k -ε .
Let f be a nonnegative measurable function on R such that

supp f ⊂ [-x-η, -x+η ] and f L 1 = R f (z) |z| 2k dz = 1 . Since γ(x, y, z) ≥ 0 ∀ y < 0 , ∀ z < 0 , γ(x, y, z) ≤ 0 ∀ y > 0 , ∀ z < 0 , we have (τ x f )(y) = -x+η -x-η f (z) |γ(x, y, z)| |z| 2k dz .
Hence, using ( 8) and [START_REF] Opdam | Harmonic analysis for cetain representations of graded Hecke algebras[END_REF], Proof. The proof is straightforward, via the Plancherel formula, and generalizes to higher dimensions. On one hand, the inequality τ x L 2 →L 2 ≤ 1 follows from the estimate |E(iξ, x)| ≤ 1 . On the other hand, let

τ x f L 1 = R (τ x f )(y) |y| 2k dy = -x+η -x-η R |γ(x, -z, -y)| |y| 2k dy f (z) |z| 2k dz is bounded from below by A k -ε . Consequently τ x L 1 →L 1 ≥ A k -ε
f ε (x) = ε k+ 1 2 f (εx)
be a rescaled normalized function in L 2 (R, |x| 2k dx). Then

f ε L 2 = f L 2 = 1 while τ x f ε 2 L 2 = R |E(iξ, x)| 2 ε -2k-1 |Df (ε -1 ξ)| 2 |ξ| 2k dξ = R |E(iεξ, x)| 2 |Df (ξ)| 2 |ξ| 2k dξ tends to R |Df (ξ)| 2 |ξ| 2k dξ = f 2 L 2 = 1
as ε → 0 . This concludes the proof of the lemma.

Eventually, Corollary 4.3 and Lemma 4.4 imply the following result, by interpolation and duality.

τ x L p →L p ≤ A 2 | 1 p -1 2 |N k
for every x ∈ R N and 1 ≤ p ≤ ∞ .

A support theorem for generalized translations

As mentioned in the introduction, we lack information about Dunkl translations in general. In this section, we locate more precisely the support of the distribution γ x,y , f = (τ x f )(y) which is known [START_REF] Trimèche | Paley-Wiener theorems for the Dunkl transform and Dunkl translation operators[END_REF] to be contained in the closed ball of radius |x|+|y| . (ii) If G is crystallographic, then the support of γ x,y is more precisely contained in z ∈ R N z + x + + y + , z + y + + g 0 .x + and x + + g 0 .y + .

Here g 0 denotes the longest element in G, which interchanges the chambers Γ + and -Γ + , and the partial order on R N associated to the cone Γ + :

a b ⇐⇒ b -a ∈ Γ + .
x + y

xy Figure 3. Support of γ x,y for the root system A 1 ×A 1

x + y

xy Proof. Let h ∈ C ∞ c (R N ) be an auxiliary radial function such that

R N h(x) w(x) dx = 1
and supp h ⊂co(G.u), where u ∈ Γ + is a unit vector. For every ε > 0 and x, y, z ∈ R N , set If G is crystallogaphic, we use Corollary 3.2 (actually the third version of the Paley-Wiener theorem in [START_REF] De Jeu | Paley-Wiener theorems for the Dunkl transform[END_REF]), and deduce from ( 12) that the function z -→ γ ε (x, y, z) is supported in co {G.(x + + y + + ε u)} = co(G.x) + co(G.y) + ε co(G.u) .

γ ε (x, y, z) = 1 c 2 R N Dh(ε ξ) E(i ξ, x) E(i ξ, y) E k (-i ξ, z) w(ξ) dξ .

Equivalently,

γ ε (x, y, z) = 0 =⇒ z + ≺ x + + y + + εu .

Using the symmetries (13), we see that γ ε (x, y, z) = 0 implies also -g 0 .x + ≺ -g 0 .z + + y + + εu i.e. z + ≻ x + + g 0 .y + + εg 0 .u , -g 0 .y + ≺ -g 0 .z + + x + + εu i.e. z + ≻ g 0 .x + + y + + εg 0 .u . We conclude again by letting ε → 0 .

The conclusion of

Corollary 3 . 2 (

 32 Geometric Paley-Wiener Theorem). Let C be a G-invariant convex compact neighborhood of the origin in R N and χ(ξ) = max x∈C x, ξ the dual gauge. Then the Dunkl transform is a linear isomorphism between the space C ∞ C (R N ) of smooth functions f on R N with supp f ⊂ C and the space H χ (C N ) of entire functions h on C N such that

∀

  a, b, c > 0 , and I a,b denotes the interval [ | a-b |, a+b ] . Notice the symmetries , x, z) , γ(-x, -y, -z) , γ(-z, y, -x) = γ(x, -z, -y) .

Proof.Case 1 :

 1 Let x, y ∈ R * . Assume that x y < 0 . Then | |x|-|y| | = | x+y | and |x|+|y| = | x-y | , hence σ(x, y, z) 1l I |x|,|y| (|z|) = z + x + y z (x-y) 2 -z 2 -2 x y 1l I |x|,|y| (|z|) and γ x,y are positive. Thus R |dγ x,y (z)| = R dγ x,y (z) = 1 .

Lemma 4 . 4 .

 44 and we conclude by letting ε → 0 . Let us next compute the L 2 → L 2 operator norm of Dunkl translations. Let x ∈ R . Then τ x is a bounded operator on L 2 (R, |x| 2k dx), with τ x L 2 →L 2 = 1 .

Corollary 4 . 5 ..Remark 4 . 6 .

 4546 Let x ∈ R and 1 ≤ p ≤ ∞ . Then τ x is a bounded operator on L p (R, |x| 2k dx), with τ x L p →L p ≤ A 2 |1/p-1/2| k In the product case, where G = Z N2 acts on R N , we have

Theorem 5 . 1 .

 51 (i) The distribution γ x,y is supported in the spherical shell z ∈ R N |x|-|y| ≤ |z| ≤ |x|+|y| .

Figure 4 .

 4 Figure 4. Support of γ x,y for the root system B 2

Firstly, according to ( 3 )

 3 and (2), ξ -→ Dh(ε ξ) E(i ξ, x) E(i ξ, y) is an entire function on C N satisfying[START_REF] Opdam | Lecture notes on Dunkl operators for real and complex reflection groups[END_REF] Dh(ε ξ) E(i ξ, x) E(i ξ, y) ≤ C M (1+|ξ|) -M e -g 0 .(x + +y + +εu), (Im ξ) + , where g 0 is the longest element in G, which interchanges the chambers Γ + and -Γ + . Secondly,γ x,y , f = 1 c R N Df (ξ) E(i ξ, x) E(i ξ, y) w(ξ) dξ = lim ε→0 1 c R N Dh(ε ξ) Df (ξ) E(i ξ, x) E(i ξ, y) w(ξ) dξ = lim ε→0 R N f (z) γ ε (x, y, z) w(z) dzi.e. the distribution γ x,y is the weak limit of the measures γ ε (x, y, z) w(z) dz . Thirdly, notice the symmetries(13) γ ε (x, y, z) =     γ ε (y, x, z) , γ ε (g.x, g.y, g.z) ∀ g ∈ G ∪ {-Id} , γ ε (-z, y, -x) = γ ε (x, -z, -y) .

  Theorem 5.1 in the crystallographic case is obtained by letting ε → 0 . If G it not crystallographic, we can only use the spherical Paley-Wiener theorem and we obtain this way that γ ε (x, y, z) + |y| + ε , |x| ≤ |z| + |y| + ε , |y| ≤ |x| + |z| + ε , hence |x| -|y|ε ≤ |z| ≤ |x| + |y| + ε .

) -Γ ′ (u) Γ(u)

Authors partially supported by DGRST project 04/UR/15-02 and CMCU program 07G 1501.


R E G I S G U E G A N

M A T H I E G Au T I E R, J E A N -M I C H E L B E N Y

ADSORPTION OF A C10E3 NON-IONIC SURFACTANT ON A Ca-SMECTITE

Keywords: Ca-smectite, Infrared Spectroscopy, Organo-clay, Surfactant, X-ray Diffraction

The transformation of clay minerais into organo-clays by surfactant intercalation is of great environmental and industrial importance because it causes the clay to attract hydrophobic contaminants and other non-polar organic compounds, but a better understanding is needed of the mechanisms by which different classes of surfactants are intercalated. The purpose of this study was to synthesize and characterize an organo-clay comprising triethylene glycol monodecyl ether (C10E3) non-ionic surfactant, which has a lamellar phase at room temperature, intercalated into Ca-montmorillonite from Wyoming (SWy-2). The C10E3 non-ionic surfactant differed from previous non-ionic surfactants used in the formation of a lamellar phase in that it consisted of the stacking of molecules by hydrophobic interaction. C10E3-clay composites were characterized by complementary techniques (adsorption isotherms, X-ray diffraction, and infrared spectroscopy) and were compared to benzyldimethyltetradecyl ammonium chloride (BDTAC) cationic surfactant-clay composites for different loadings of the surfactant. For large loadings, the amount of C10E3 adsorbed, which can be described by the Langmuir equation, seemed to reach a steady state close to that of the cationic surfactant. The adsorption processes of the two surfactants were different. For the cationic surfactant, the adsorption, as described in the literature, was due to ion exchange between organic cations and Ca 2+ counterions. The adsorption of C10E3 did not depend on electrostatic interaction but rather was due to several interaction mechanisms (H-bonding, ion-dipole, and hydrophobic interaction). For both surfactants, the expansion was limited to two adsorbed monolayers parallel to the clay surface. The expansion of the basal spacing to 17 Å suggested a complete dissociation of the C10E3 lamellar phase when adsorbed on the Ca-smectite. Organo-clays made using the non-ionic surfactant were stable, changing the chemical nature of clay to hydrophobic, and allowing for other cations to be exchanged, which has importance in the manufacture of new nanocomposites or geochemical barriers.

INTRODUCTION

Smectite clay minerais such as Wyoming montmorillonite exhibit, in the presence of water, cation exchange capacities (CEC), swelling and hydration properties, and a large surface area which lead to strong adsorption properties [START_REF] Dentel | Sorption of tannic acid, phenol, and 2,4,5-trichlorophenol on organo-clays[END_REF][START_REF] Breen | Acid -activated organo -clays: Preparation, characterization and catalytic activity of acidtreate d tetraalkylammonium-exchanged smectites[END_REF][START_REF] Breen | Acid -activated organo -clays: Preparation, characterization and catalytic activity of acidtreate d tetraalkylammonium-exchanged smectites[END_REF][START_REF] He | Location and migration of cations in Cu2+-adsorbed montmorillonite[END_REF] which have been highlighted in several applications such as catalysis, nanocomposite materials, and geochemical barriers (as in the case of waste landfills) [START_REF] Lebaron | Polymerlayered silicate nanocomposites: an overview[END_REF][START_REF] Prost | Use of modified clays for controlling soil environmental quality[END_REF][START_REF] Kim | Sequential sorption and desorption of chlorinated phenols in organo-clays[END_REF]2003). The structure of dioctahedral smectite is characterized by two SiO4 tetrahedral sheets sandwiching an Al or a Mg octahedral sheet and it displays a hydrophilic surface which is inconsistent with the adsorption of organic compounds [START_REF] Prost | Use of modified clays for controlling soil environmental quality[END_REF][START_REF] Kim | Sequential sorption and desorption of chlorinated phenols in organo-clays[END_REF]2003). The inability to adsorb organic compounds can lead to environmental problems as the majority of contaminants are non-ionic organic compounds (NOCs) which emanate from products such as household detergents and cosmetics [START_REF] Portet | Polydispersity of a non-ionic surfactant as related to its adsorption characteristics on porous silica particles in water[END_REF]. One solution is to use ion exchange between the * E-mail address of corresponding author: regis.guegan@univ-orleans.fr DOI: 10.1346/ CCMN.2009.0570411 interlayer inorganic and organic cations to create organo-clays. Numerous studies have been devoted to the synthesis of organo-clays [START_REF] Mortland | Clayorganic complexes as adsorbents for phenol and chlorophenols[END_REF][START_REF] Boyd | Pentachlorophenol sorption by organo-clays[END_REF][START_REF] Jaynes | Hydrophobicity of siloxane surfaces in smectites as revealed by aromatic hydrocarbon adsorption from water[END_REF][START_REF] Tahani | Adsorption of a cationic surfactant on Na-Montmorillonite: Inspection of adsorption layer by X-ray and fluorescence spectroscopies[END_REF][START_REF] Lee | Microstructural changes of reference montmorillonites by cationic surfactants[END_REF][START_REF] Xi | Modification of Wyoming montmorillonite surfaces using a cationic surfactant[END_REF]. Major features of the organo-clays provided by the intercalation of cationic surfactants, such as quaternary ammonium compounds into the interlayer space, are the adsorption of organic cations through ion exchange. The consequences for the hybrid clay are the changes in the interlayer height and the clay surface which then switches from organophobic to organophilic. Depending on the nature of the inorganic cation and the amount of cationic surfactant loaded, the basal spacing increases considerably, allowing the confinement of surfactant molecules to self-organize in onelayer, two-layer, paraffin, or micellar structures [START_REF] Othmani-Assmann | Physico-chemical characterizations of Tunisian organophilic bentonites[END_REF]. The interlayer thus becomes organophilic, which changes the properties of the hybrid material, enhancing the capacity of the resulting organo-clays to remove NOCs. The insertion of the surfactant is, however, irreversible and prevents any further cationic exchange with other cationic contaminants such as heavy metals.

Synthesis of organo-clays by intercalation of nonionic surfactants has been suggested [START_REF] Deng | Intercalation and surface modification of smectite by two non-ionic surfactants[END_REF](Deng et al., , 2006)). The mode of grafting non-ionic surfactants onto the clay surface depends on various interactions such as hydrogen-bond and ion-dipole interactions. The availability of these organo-clays to remove both NOCs and heavy metals has been demonstrated [START_REF] Deng | Intercalation and surface modification of smectite by two non-ionic surfactants[END_REF](Deng et al., , 2006)). All of the aforementioned applications depend on the structure and properties of the organo-clays and knowledge of these properties is essential for industrial application. Numerous studies have been carried out on the adsorption of various cationic surfactants onto clays [START_REF] Lee | Microstructural changes of reference montmorillonites by cationic surfactants[END_REF][START_REF] Othmani-Assmann | Physico-chemical characterizations of Tunisian organophilic bentonites[END_REF]. In contrast, few studies have explored the adsorption of non-ionic surfactants on smectite [START_REF] Hackett | Computer simulation studies of PEO/Layer silicate nanocomposites[END_REF][START_REF] Deng | Intercalation and surface modification of smectite by two non-ionic surfactants[END_REF]Deng et al., , 2006)). In particular, no study has focused on understanding the surfactant phase in terms of adsorption onto smectite. Non-ionic surfactants such as polyethylene oxide (PEO), Brij 56, or Igepal CO 720, used in previous studies [START_REF] Deng | Intercalation and surface modification of smectite by two non-ionic surfactants[END_REF](Deng et al., , 2006)), display common micellar phases above the critical micellar concentration (CMC) and adsorption of nonionic surfactants is similar in behavior [START_REF] Deng | Intercalation and surface modification of smectite by two non-ionic surfactants[END_REF](Deng et al., , 2006)). Triethylene glycol monodecyl ether (C10E3) is a non-ionic surfactant which, at room temperature and in the presence of water, comprises several membrane phases such as lamellar and sponge phases [START_REF] Le | Nonionic amphiphilic bilayer structures under shear[END_REF]. The lamellar phase is characterized by a stack of surfactant molecules aggregated into a two-dimensional structure (membrane) and layers of water which display a structure at longrange order [START_REF] Le | Nonionic amphiphilic bilayer structures under shear[END_REF]. The lamellar phase is largely stabilized by steric interactions due to the loss of entropy when molecules are stacked together by hydrophobic interaction.

The purpose of the present study was to partly overcome the paucity of information available regarding the intercalation behavior of non-ionic organic compounds in expandable clay minerais by investigating the adsorption of the C10E3 lamellar phase onto a Camontmorillonite, by characterizing the structural changes in C10E3-clay composites with different nonionic surfactant loadings, and by comparing the results obtained with the adsorption of a commonly used cationic surfactant. Adsorption isotherms were used to evaluate the non-ionic surfactant loading. The degree of intercalation for different C10E3 loadings and the changes in the surface structure of the Ca-montmorillonite were investigated by Fourier-transform infrared (FTIR) spectroscopy and X-ray diffraction (XRD). New insights into the structures and properties of organo-clays synthesized by the intercalation of nonionic surfactants were gained.

Purdue University, West Lafayette, Indiana) This smectite was Ca-exchanged by washing five times with a 1 M CaC12 solution. The resulting clay was washed with distilled water and centrifuged at 5000 rpm for 10 min (ROUCAIRE, Hermes Z383). The process was repeated five times. The clay obtained was dried at 100°C for 2 days in order to remove adsorbed water and Ca 2+ was assumed to be the only counter ion between the layers. The resulting clay was dialyzed against distilled water in order to remove excess Ca 2± . The cation exchange capacity (CEC) of the resulting clay, determined by the adsorption of a Cu triethylenetetramine complex (so-called Cu-trien) [START_REF] Meier | Determination of the cation exchange capacity (CEC) of clay minerals using the complexes of copper (II) ion with triethylenetetramine and tetraethylenepentamine[END_REF], was ~82 meq/100 g.

One ionic and one non-ionic surfactant were used, namely, the benzyldimethyltetradecyl ammonium chloride cation (BDTAC) and triethylene glycol monodecyl ether (C10E3), which were purchased from Sigma-Aldrich Chemical Company and used without further purification (Table 1). Their length and height were determined from crystallographic data in relation to the C-C, C-0, and C-N bonds as well as the C-C-C, C-O-C, and C-N-C angles [START_REF] Rodier | Bromure de Benzododécinium Monohydraté[END_REF]. Above the CMC, the C10E3 shows, at room temperature and in the presence of water, a lamellar phase, the basal spacing of which depends on its concentration and a constant bilayer thickness of 27.6 Å [START_REF] Le | Nonionic amphiphilic bilayer structures under shear[END_REF]. Explicit C10E3 concentration-temperature phase diagrams have been published elsewhere [START_REF] Le | Nonionic amphiphilic bilayer structures under shear[END_REF].

Synthesis of organo-clays

Aqueous solutions of BDTAC and C10E3 were prepared in demineralized water at room temperature at concentrations >CMC. The concentrations of the solutions used in the determination of the adsorption as well as in the physicochemical characterizations of organo-clays were calculated according to the CEC values of the clays which varied from 6.5 x 10 -4 to 1.31 x 10 -2 mol L -1 (i.e. 0.2 to 4 x CEC for the cationic surfactant). The average pH value of the solutions was ~6.5±0.2 and remained constant during the synthesis. Clay powder was dispersed into the surfactant solutions. The suspensions obtained were stirred for 24 h at 250 rpm. The mixtures were then centrifuged at 5000 rpm for 20 min. The organo-clays obtained were dried at 70°C for 48 h and crushed using an agate mortar.

The CEC of both organo-clays was determined using the Cu-trien method [START_REF] Meier | Determination of the cation exchange capacity (CEC) of clay minerals using the complexes of copper (II) ion with triethylenetetramine and tetraethylenepentamine[END_REF].

Adsorption isotherms

Batch equilibration isotherms were determined using 20 mg of each of the organo-clay samples. The concentrations of N and C were measured using an element analyzer and were carried out at the Service Central d'Analyse du CNRS (Solaire, France). The amount of surfactants adsorbed was determined by the difference between the surfactant added and that remaining in the solution.

Infrared spectroscopy

Infrared spectra were collected using a NICOLET Magna IR 760 Fourier-transform infrared spectrometer with a resolution of 2 cm -1 . Spectra were collected over the range 400-4000 cm -1 . Samples consisted of pellets of anhydrous KBr mixed with 0.33% clay powder.

X-ray diffraction

The d001 basal spacings of the Ca-smectite and the surfactant-smectite composites were measured using an ARL X'TRA Diffractometer (Thermo Electron Corporation, Ecublens, Switzerland) operating at 40 kV and 40 mA with CuKa r adiation ( λ 1.54 Å). Measurements were carried out at room temperature in the step-scanning mode with a step size of 0.05°2θ. Samples were dried at 70°C, crushed using an agate mortar and pestle, and then placed directly into the sample holder.

RESULTS AND DISCUSSION

Adsorption and surface properties of the organo-clays

The BDTAC adsorption isotherm exhibited features usually encountered for cationic surfactants on clay minerais and presented two main regions of interest (Figure 1). The first region corresponded to a linear variation of the amount adsorbed as a function of its equilibrium concentration and can be explained as an exchange between the organic and the Ca-exchangeable cations [START_REF] Trompette | Scanning of the cationic surfactant adsorption on a hydrophilic silica surface at low surface coverages[END_REF][START_REF] Xu | Cationic surfactant adsorption by swelling and nonswelling layer silicates[END_REF]. The analysis performed on the resulting solution showed a Ca molar quantity of 1.6 x 10 -5 mol L -1 which confirmed the removal of the Ca 2± from the interlayer spaces. The second region showed a steady state where the amount adsorbed is ~9.0 x 10 -4 mol g -1 . The adsorption isotherm of the BDTAC on the Casmectite was well described by a Langmuir equation (1):

θ=αθfC/(1 + αC) ( 1
)
where θ is the amount adsorbed, θf is the maximum adsorption, α is a constant, and C is the equilibrium concentration of the surfactant. The maximum adsorption isotherm obtained, θf, was 8.8 x 10 -4 mol g -1 and highlighted the good affinity of the cationic surfactant for the clay. The C10E3 adsorption isotherm behaved differently and showed a graduai increase in the amount adsorbed with the equilibrium solution concentration. No Ca 2+ was found in the resulting solution which suggested that no cation exchange occurred. Indeed, for small concentrations, the amount of C10E3 adsorbed did not present any linear variation with the concentration. However, for large concentrations, the adsorption of C10E3 seemed to reach a steady state and the final amount of C10E3 adsorbed was close to that of the cationic surfactant. 9.67 x 10 -4 mol g -1 . Previous studies of the adsorption of NOC surfactants in the micellar phase onto smectite reported much less affinity for smectite [START_REF] Deng | Intercalation and surface modification of smectite by two non-ionic surfactants[END_REF]. Further work is, therefore, required on other cation-exchanged montmorillonites, especially Nasmectite. Another interesting point was the difference in the maximum adsorption isotherms between the two surfactants. The Ca-smectite showed a greater affinity for C10E3 than for BDTAC. The adsorption of C 10 E 3 could result from several interaction processes with the surface and between the confined molecules, including the hydrophobic interactions. This interaction process involving hydrophobic molecules stacks the molecules together in a lamellar phase in solution. When molecules were adsorbed, other molecules were stacked together by hydrophobic interaction and covered the clay surface, increasing, as a result, the amount of surfactant adsorbed on clay, and explaining the greater maximum adsorption isotherm than achieved in other studies on non-ionic surfactants and BDTAC.

The CEC for the Ca-smectite was ~82 meq/100 g (Table 2). Determination of the CEC for both samples used in this study allowed identification of the anchoring bonding mode between the surfactant and the clay surface. The BDTAC-clay did not exchange any BDTA + cations with the Cu-trien complex, underlining the irreversibility of the ion exchange and the affinity of BDTA + for smectite. The BDTA + was ion-exchanged with Ca 2+ and stabilized the negative charge surface of the clay by electrostatic interaction. The CEC results suggested that BDTA + was adsorbed irreversibly and once the surface charge was neutralized, no other cations could be exchanged but only adsorbed by other interaction processes (e.g. van der Waals, hydrophobic interaction, etc.) and may explain the amount of BDTAC adsorbed on Ca-smectite.

In the case of the C 10 E 3 , Ca 2+ cations in the interlayer were exchanged with the Cu-trien complex and the C 10 E 3 -smectite composite preserved almost all of its initial CEC (Table 2). Ca 2+ was removed from the interlayer space by the exchange reaction. The difference in CEC values of the two organo-clays can, therefore, be explained by their different adsorption mechanisms. Ion exchange did not occur between C 10 E 3 and the exchanged Ca 2± in the smectite so this non-ionic surfactant must have been absorbed by another mechan- ism [START_REF] Deng | Intercalation and surface modification of smectite by two non-ionic surfactants[END_REF](Deng et al., , 2006)). In previous models, the non-ionic surfactant formed H-bonds with water in the hydration shells of exchangeable cations, or bound directly to the inorganic cations through ion-dipole interactions. Another possible scenario is that direct H-bonding of the surfactant to the surface occurred.

Further analysis, such as by nuclear magnetic resonance (NMR), needs to be performed in order to clarify this particular point.

FTIR study

The FTIR spectra of Ca-smectite and of both organoclays at the greatest concentration of 1.31 x 10 -2 mol L -1 , unwashed, washed once, and washed twice (with distilled water, for 12 h) were normalized with respect to the band at ~3620 cm -1 (Figures 2,3). This band, arising from the OH group of the smectite, is expected to be unaltered (according to [START_REF] Deng | Intercalation and surface modification of smectite by two non-ionic surfactants[END_REF] by the surfactant adsorption and can, therefore, be used as a reference band.

The spectra for both the Ca-smectite and organoc l a y s s h o w e d a b r o a d b a n d i n t h e r a n g e 3000-3600 cm -1 which corresponds to the OH-stretching of adsorbed H 2 O and OH groups of the C 10 E 3 . The presence of this broad band relative to the OH-stretching emphasized that for organo-clay samples, even after heat treatment at 70°C for 48 h, water molecules remained adsorbed, indicating that the adsorption mechanism of both surfactants must involve H-bonding, ion-dipole, and hydrophobic interaction. The adsorption bands at ~2850 and 2920 cm -1 are related to the symmetric and antisymmetric CH 2 stretching modes. The intensities of these stretching adsorption bands are related to the density of the alkyl chains within the interlayer space [START_REF] Peker | Adsorption behavior of a cationic surfactant on montmorillonite[END_REF]. In order to estimate the strength of the surfactant-clay interaction, the organo-clays were washed once and then a second time. The integrated intensity of the CH 2 stretching mode, linked to the density of the confined surfactants [START_REF] Peker | Adsorption behavior of a cationic surfactant on montmorillonite[END_REF], remained constant, demonstrating the strong interactions and affinity for Ca-smectite, even for different adsorption mechanisms for the two surfactants. Some small differences did occur, however. For BDTAC, after one or two washings the integrated intensity of the CH 2 stretching mode remained the same, confirming the irreversibility of the ion exchange between Ca 2+ and the organic cations and the strong electrostatic interaction with the surface [START_REF] Tahani | Adsorption of a cationic surfactant on Na-Montmorillonite: Inspection of adsorption layer by X-ray and fluorescence spectroscopies[END_REF][START_REF] Othmani-Assmann | Physico-chemical characterizations of Tunisian organophilic bentonites[END_REF]. In the case of C 10 E 3 , the CH 2 integrated intensity decreased with washing (~5% each time), without calling into question the strong interaction between the non-ionic surfactant and the clay surface. The adsorption isotherms showed the good affinity of the C 10 E 3 for Ca-smectite leading to a maximum adsorption isotherm which was greater than for BDTAC. This difference was explained by the hydrophobic interaction which confined more molecules, though this interaction is rather weak when exposed to a washing procedure. The washing treatments should remove C10E3 molecules adsorbed by hydrophobic interaction.

Limited expansion of the basal spacing at two adsorbed surfactant layers

The XRD patterns revealed that the d001 reflections of the BDTAC-clay were well defined compared to that of the Ca-smectite (Figure 4). The evolution of the basal spacing depends on the initial amount of surfactant and conformed with results from previous studies [START_REF] Peker | Adsorption behavior of a cationic surfactant on montmorillonite[END_REF][START_REF] Tahani | Adsorption of a cationic surfactant on Na-Montmorillonite: Inspection of adsorption layer by X-ray and fluorescence spectroscopies[END_REF][START_REF] Othmani-Assmann | Physico-chemical characterizations of Tunisian organophilic bentonites[END_REF]. For small concentrations, the basal spacing increased due to the adsorption of a BDTAC monolayer (insert in Figure 4). At a concentration of 1.96 x 10 -3 mol L -1 Figure 3. FTIR spectra of pure C10E3, C10E3-clay, C10E3-clay (washed once), C10E3-clay (washed twice) at the concentration of 1.31 x 10 -2 mol L -1 , and Ca-smectite, over the wavenumber range 2700-3700 cm -1 . XRD patterns exhibited two (001) reflections, corresponding to two interlayer distances of 14.2 and 16 Å, suggesting a regularly interstratified BDTAC-smectite. For large concentrations, the basal spacing reached ~18 Å, corresponding to the insertion of two BDTA + layers (Deng et al., 2006).

Evolutions of C10E3 diffraction patterns and basal spacing were similar to those of the cationic surfactant (Figure 5 and insert). For small concentrations, the basal spacing grew from 11.7 to 14 Å, corresponding to the adsorption of a surfactant monolayer (Figure 5 insert). At 1.96 x 10 -3 mol L -1 , two (001) reflections described the XRD patterns, corresponding to basal spacings of 14 and 16.5 Å, respectively, proving a regularly interstratified C10E3-smectite with one lateral and two adsorbed layers in the interlayer space. Thus, for large concentration, the d spacing increased to ~17.2 Å and then remained constant. The limited expansion of the interlayer space was interesting, considering the constant 27.6 Å bilayer thickness of the C10E3 lamellar phase. Preparation of the organo-clay disrupted the arrangement of the lamellar phase, leading to the adsorption of a monolayer parallel to the surface. The adsorption isotherm showed the affinity of the C10E3 for the clay mineral surface. Ca-smectite disturbed the lamellar phase completely, and attracted isolated molecules which are stabilized with the clay surface.

However, no ion exchange was achieved, proving that adsorption did not take place by a surface-binding mechanism. The adsorption should result from an aggregation process. When molecules were adsorbed, other molecules could be stacked on each other by hydrophobic interaction and cover the clay surface, thereby increasing the amount of surfactant adsorbed. However, bilayers cannot be formed for either surfactant. Previous studies of the adsorption of cationic surfactants on Na-montmotillonite showed an increase in the basal spacing, allowing the confined surfactant molecules to self-organize into one-layer, two-layer, paraffin, or micellar structures. The nature of the counterion in the interlayer space is of great importance for the swelling of the clay mineral. Synthesis of organoclays was undertaken in clay suspension, where the Casmectite was very swollen, so the hydration process for Ca 2+ in the smectite allowed the adsorption of two or three layers of water, leading to a maximum interlayer space of ~18 Å. The swelling of the clay then allowed the intercalation of surfactant into the structure either by ion exchange or by H-bonding with the clay surface. Experiments and numerical simulations demonstrated the large increase in the interlayer space for Li + and Na + compared to other interlayer cations, allowing the organization of cationic surfactants in paraffin or micelle structures [START_REF] Othmani-Assmann | Physico-chemical characterizations of Tunisian organophilic bentonites[END_REF]. Studies of the adsorption of non-ionic surfactant on Na-smectite showed the confinement of one or two organic layers parallel to the clay surface [START_REF] Tahani | Adsorption of a cationic surfactant on Na-Montmorillonite: Inspection of adsorption layer by X-ray and fluorescence spectroscopies[END_REF][START_REF] Othmani-Assmann | Physico-chemical characterizations of Tunisian organophilic bentonites[END_REF]. Investigation of the adsorption of the C10E3 lamellar phase on a Nasmectite would help to understand further the mechanisms of adsorption of non-ionic surfactant on clay.

CONCLUSIONS

The adsorption of a lamellar phase made from C10E3 non-ionic surfactant on a Ca-smectite was studied, Vol. 57, No. 4, 2009 Adsorption of surfactant on a Ca-smectite 507 resulting in the formation of organo-clays, which were characterized by several complementary techniques (adsorption isotherms, XRD, and IR spectroscopy) and compared with organo-clays synthesized using BDTAC cationic surfactant. The adsorption mechanisms differed for the two surfactants. The BDTAC was ion exchanged in stoichiometric proportions with the Ca 2+ counter ions present in the interlayer space. For C10E3, the adsorption process did not result in ion exchange. The interaction between the adsorbed C10E3 and Ca-smectite depended on several possible mechanisms (ion-dipole, H-bonding, van der Waals forces, and hydrophobic interactions).

The adsorption isotherm demonstrated good affinity of the C10E3 for the clay-mineral surface, and the narrow opening of the basal spacing led to the complete reorganization of the lamellar phase into isolated molecules which were adsorbed parallel to the surface. The hydrophobic interaction allowed C10E3 molecules to be retained, increased the amount of surfactant adsorbed in the clay, and explained the greater maximum adsorption isotherm than for the cationic version. The hydrophobic interaction was, however, rather weak as indicated by the relative ease with which molecules adsorbed by this mechanism were removed by washing treatments. This explains the 5% decrease in the integrated intensity of the CH2 stretching bands in the infrared. For both surfactants, the expansion was limited to two adsorbed layers parallel to the surface and was mainly due to the nature of the counterion of the montmorillonite. Synthesis of the organo-clay was undertaken in clay suspension, where the Ca-smectite was very swollen. The hydration process for Ca 2+ cations in smectite allowed the adsorption of two or three layers of water, leading to a maximum opening for the interlayer space of ~18 Å and which permitted the retention of two monolayers of either surfactant. The synthesis of organo-clays using C10E3 non-ionic surfac-tant to produce stable composites changed the chemical nature of clay to hydrophobic and allowed other cations to be exchanged, which could lead to the realization of new nanocomposites or geochemical barriers in waste landfills.

Figure 1 .

 1 Figure 1. Adsorption of BDTAC (gray circles) and C10E3 (black circles) on SWy2-Ca montmorillonite vs. their initial concentration in solution. Langmuir fits were performed for both surfactants and are indicated by the black lines.

Figure 2 .

 2 Figure 2. FTIR spectra of pure BDTAC, BDTAC-clay, BDTAC-clay (washed once), and BDTAC-clay (washed twice) at the concentration of 1.31 x 10 -2 mol L -1 (i.e. 4 x CEC) and Ca-smectite, over the wavenumber range 2700-3700 cm -1 .

Figure 4 .

 4 Figure 4. XRD patterns of BDTAC for several initial concentrations. The insert represents the change in basal spacing, d001, for BDTAC as a function of the initial concentration (CuKα radiation).

Figure 5 .

 5 Figure 5. XRD patterns of C10E3 for several initial concentrations. The insert represents the change in basal spacing, d001, for C10E3 as a function of the initial concentration (CuKα radiation).

Table 1 .

 1 Characteristics of the BDTAC cationic and the C10E3 non-ionic surfactants.

		Commercial Molecular	CMC Length of Height of
	MATERIALS AND METHODS	name	weight	(mol L -1 ) surfactant surfactant
	Materials				(Å)	(Å)
	The smectite used in this study was the SWy-2-BDTAC	368	2 x 10 -3	23	5.5
	montmorillonite (Wyoming) obtained from the Source C10E3	290	6 x 10 -4	23

Table 2 .

 2 CEC values of Ca-smectite and the two organoclays.

	Sample	CEC	Percentage relative
		(meq/100 g)	to Ca-smectite
	Ca-smectite	82	100
	BDTAC-clay	5	6
	C 10 E 3 -clay	76	92

Clays Repository of The Clay Minerais Society (hosted byVol. 57, No. 4, 2009 Adsorption of surfactant on a Ca-smectite


Cédric Augonnet

Samuel Thibault

Raymond Namyst

Automatic Calibration of Performance Models on Heterogeneous Multicore Architectures

. A deep knowledge of the application is usually required to model per-task performance models, based on the algorithmic complexity of the underlying numeric kernel. We present an alternate, auto-tuning performance prediction approach based on performance history tables dynamically built during the application run. This approach does not require that the programmer provides some specific information. We show that, thanks to the use of a carefully chosen hash-function, our approach quickly achieves accurate performance estimations automatically. Our approach even outperforms regular algorithmic performance models with several linear algebra numerical kernels.

Introduction

Multicore architectures are now widely adopted throughout the computer ecosystem. There is also clear evidence that solutions based on specialized hardware, such as accelerator devices (e.g. GPGPUs) or integrated coprocessors (e.g. Cell's SPUs) are offering promising answers to the physical limits met by processor designers. Future processors will therefore not only get more cores, but some of them will be tailored for specific workloads.

In spite of their promising performance in terms of computational capabilities and power efficiency, such heterogeneous multicore architectures require appropriate tools. This introduces challenging problems at all levels, ranging from programming models and compilers to the design of libraries with a real support for heterogeneity. As they offer dynamic support for what has become hardly doable in a static fashion, runtime systems have a central role in this software stack. In previous work, we have therefore developed StarPU [START_REF] Augonnet | StarPU: A Unified Platform for Task Scheduling on Heterogeneous Multicore Architectures[END_REF], a unified runtime system that offers support for heterogeneous multicore architectures. Its specificity is that it not only targets accelerators (GPUs, Cell's SPUs, etc.) but also multicore processors at the same time, in a portable fashion. StarPU also provides portable performance thanks to a high-level framework for designing portable scheduling policies.

Performance modeling is a very common technique in the scheduling literature. Whenever doable, practically building such models usually requires consequent efforts along with a certain knowledge of both the application algorithm and the underlying architecture. This is even more difficult in the case of heterogeneous platforms. But without an appropriate interface, such knowledge is not available from the runtime system's perspective: describing a task as a function pointer and pointers to the data (similar to OpenMP 3.0 tasks) does not really give much information to the runtime system in charge of the scheduling. (Un)fortunately, current accelerators reintroduce the problem of data management across a distributed memory model, so that we have to adopt much more expressive task APIs anyway. The majority of the programming models that target accelerators (and that do not just delegate data movements to the programmer!) require to explicitly describe which data is accessed by a task [START_REF] Duran | Extending the openmp tasking model to allow dependant tasks[END_REF][START_REF] Fatahalian | Sequoia: Programming the memory hierarchy[END_REF][START_REF] Mccool | Data-Parallel Programming on the Cell BE and the GPU using the RapidMind Development Platform[END_REF][START_REF] Augonnet | A unified runtime system for heterogeneous multicore architectures[END_REF]. While this adds constraints on the programmer who has to adapt its applications to those expressive programming interfaces, the underlying runtime system gets much more information.

In this paper, we explain how StarPU takes advantage of that expressiveness to seamlessly build performance models on heterogeneous multicore architectures. Then, we illustrate how this systematic approach performs in terms of prediction accuracy and regarding its impact on the actual performance. Finally, we show that StarPU not only grabs information from the programming interface to perform better scheduling, but it also returns performance feedback information thanks to convenient tools which are helpful for instance in the context of auto-tuned libraries or when analyzing performance.

StarPU, a runtime system for heterogeneous machines

In this section, we briefly present StarPU, our unified runtime system designed for heterogeneous multicore platforms, described in more details in a previous paper [START_REF] Augonnet | StarPU: A Unified Platform for Task Scheduling on Heterogeneous Multicore Architectures[END_REF]. It distributes tasks onto both accelerators and processors simultaneously while offering portable performance thanks to generic scheduling facilities.

A unified runtime system

The design of StarPU is organized around three main components: a portable offloable-task abstraction, a library that manages data movements across heterogeneous platforms, and a flexible framework to design portable scheduling policies. A unified execution model. StarPU exposes the structure of codelet, which is the set of implementations of the same computation kernel (e.g. a vector sum) for different computation units (e.g. CPU and GPU). A StarPU task is then an instance of a codelet applied to some data. Figure 1 shows the path followed by tasks in StarPU. The programmer explicitly submits (graphs of) tasks to StarPU which maps them as efficiently as possible on the eligible processing units. Instead of hard-coding all the interactions between the processing units, StarPU makes it possible to concentrate on the design of efficient computational kernels and algorithmic problems instead of being stuck by low-level concerns.

GPU

A data management library. Maintaining data coherency (and availability) is a crucial issue with accelerators. In a previous paper [START_REF] Augonnet | A unified runtime system for heterogeneous multicore architectures[END_REF], we have designed a high-level data management library that is integrated in StarPU. Mapping data statically is not necessarily sufficient when multiple processing units access the same pieces of data. The resulting data transfers are critical for the overall performance so that integrating data management within StarPU made it possible to apply optimizations (e.g. prefetching, reordering, asynchronous memory transfers) and to guide the scheduler.

A scheduling framework. StarPU not only executes tasks, but it also maps them as efficiently as possible thanks to its expressive scheduling interface. Hence, StarPU offers a flexible framework to implement portable scheduling policies [START_REF] Augonnet | StarPU: A Unified Platform for Task Scheduling on Heterogeneous Multicore Architectures[END_REF]. Such policies are portable in the sense that they are directly applicable to platforms as different as a Cell processor and a hybrid GPU/CPUs machine.

Scheduling strategies based on performance models

In a previous paper [START_REF] Augonnet | StarPU: A Unified Platform for Task Scheduling on Heterogeneous Multicore Architectures[END_REF], we have presented various scheduling policies implemented in StarPU with relatively little effort. For instance, one of these policies is similar to the HEFT scheduling strategy [START_REF] Topcuoglu | Performance-effective and lowcomplexity task scheduling for heterogeneous computing[END_REF]. As shown in Figure 2, the scheduler keeps track of the expected duration until the different processing units are available. When a task is submitted to the scheduler, it is attributed to the processing unit that minimizes termination time according to the expected duration of the task on the different architectures (depicted by hatchings).

We have for instance used this rather simple strategy successfully to obtain superlinear speedups on an LU decomposition thanks to per-architecture performance models that take into account the (lack of) affinity of tasks with the different processing units. However this strategy requires that we can approximate the execution time of the tasks on the various architectures.

Dynamically building Performance Models

In this section, we discuss how we can build performance models, and we give a systematic approach to dynamically construct and query a performance model based on historical knowledge, seamlessly for the programmer.

In the context of dynamic task scheduling, we do not need perfectly accurate models, but we need to take appropriate decisions when assigning the tasks onto the different processing units. Our performance models should for instance capture the relative speedups as well as the affinities between tasks and processors.

How to define a performance model?

In order to define a performance model, we need to decide which parameters the model should depend on.

The most obvious parameters to describe a task are the kernel and the architecture: in the case of a matrix product on CUDA, we could for instance identify a task by the pair (SGEMM, CUDA) and associate it with its predicted execution time. A trivial refinement is to consider the total size of the tasks' data, so we can also associate this pair with a parametric cost function depending on that size (e.g. O S 3/2 in the case of SGEMM applied on a matrix of size S).

The total size is often not sufficient: in the case of a kernel handling a (n×m) matrix in O(n 2 m), we must make a distinction between (1024×512) and (512 × 1024) matrices (for example). Such multivariate models are however only applicable if we have sufficient knowledge of the algorithm, which a runtime system could hardly infer automatically in a generic way. Finding an explicit model of the execution time can also be awkward because of architectural concerns such as the size of caches. Using piecewise models is possible, but it requires to delimit the boundaries of the pieces, which can be time demanding, especially for a multivariate model and in a heterogeneous environment.

In many classes of algorithms, we can reasonably make some extra regularity assumption such as most tasks handling blocks of fixed size (e.g. in tiled algorithms), or a limited set of sizes (e.g. in divide and conquer algorithms). In this case, explicitly modeling the performance as a function of the data size can be unnecessarily complicated. A history-based approach would be much simpler: instead of using a complicated multivariate model to differentiate between a (1024 × 512) matrix and a (512 × 1024) one, we simply store the execution time that was measured for those different input configurations. The advantage of this approach is that it is transparent to the programmer as long we have some mechanism to match a task with those previously executed. This method is however not applicable to irregular applications: if we know the performance of a kernel on a (1024 × 512) matrix, no prediction can be made for a (1026 × 510) matrix for instance. In the next section, we present how StarPU implements history-based models with sufficient performance feedback to help programmer easily decide whether this is an appropriate model or not.

How to build performance models?

There are various ways to determine the parameters required to build the performance models that we have described in the previous section: either completely manual, or completely automated, depending on the type of the adopted model.

Building a performance model by hand (e.g. using the ratio between the number of operations and the speed of the processor) is hardly applicable to modern processors and require a detailed knowledge of both the application and the architecture. In the case of heterogeneous multicore processors, with multiple processing units to handle, this becomes rather unrealistic. It is however possible to design a model based on the amount of computations per task, and to calibrate the parameters by the means of a regression.

It is common to use specific precalibration programs to build those models. While this may be suited for kernels that are widely used (e.g. BLAS), this requires a specific test-suite and the corresponding inputs, which often represents an important programming overhead. In the context of multicore architectures, it is even harder to create a realistic workload: independently benchmarking the various processing units without taking into account the various interactions (e.g. cache sharing or bus contention) may not result in reliable measures.

On the other hand, it is possible to measure the performance of the different tasks during an actual execution. This does not require any additional programs, and it provides realistic performance measurements. StarPU can therefore automatically calibrate parametric models, either at runtime using linear regression models (e.g. of the form O(n α )) or offline in the case of non-linear models (e.g. of the form αn β + γ, as shown in Figure 7). StarPU also builds history-based performance models by storing the performance of the tasks on different inputs, transparently for the application.

A generic approach for building history-based performance models dynamically

This section shows how StarPU keeps track of the performance obtained by the tasks on the different input, and how it is possible to match a task with its similar predecessors. As shown in Figure 3, this process involves three main steps: measuring the actual duration of the tasks when they are executed and integrating these measurements in the history log of the task; being able to lookup the performance of some task according to the previous measurements; and offering some performance feedback to the application.

Measuring tasks' duration. Measuring the time spent to compute a task is usually simple thanks to the cycle counter facility provided by most manufacturers. In the case of Cell processors, which lacks such functionality, we had to make the SPUs transmit those measurements to the PPU along with the output data, this is not intrusive since DMA transfers are overlapped.

Identifying task kinds. We use the layout and size of the data to distinguish the different kind of instances of a computational kernel. We now explain how to compute a hash value to characterize the data layout of a task.

StarPU's data management library not only manipulates buffers described by a pointer and its length, but it also handles a mixture of various high-level data interfaces [START_REF] Augonnet | A unified runtime system for heterogeneous multicore architectures[END_REF]. In Figure 4, a matrix-vector product accesses a set of matrices and vectors. There can also be much more complex data interfaces (e.g. compressed sparse matrices), but the size of any piece of data must be characterized by a k-tuple of parameters (p 1 , . . . , p k ) where k and the parameters depend only on the data interface. A matrix is for instance described by a pair (n, m), and a single parameter is sufficient to describe the length of a vector.

We now define a hash function that computes a unique identifier for such a set of parameters. As shown in Figure 4, we characterize the size of each piece of data by applying a hash function1 to the parameters p 1 , . . . , p k-1 describing it. By then applying the hash function to the different per-data hashes, we get a characterization of the data layout and size for the whole task. Applying this method on a tiled algorithm would for instance result in having as many hash values as there are tile sizes.

Feeding and looking up from the model. It is now extremely simple to implement a model based on the history in StarPU: each computational kernel is associated with a hash table per architecture. When a task is submitted to StarPU, it computes its hash, and consults the hash table corresponding to the proper kernel-architecture pair to retrieve the average execution time previously measured for this kind of task. The average execution time and other metrics such as standard deviation are updated when a new measure is available. Hash tables can be saved (or loaded) to (from) a file so that these performance models are persistent between different runs. We therefore rapidly calibrate models by running small problems that have the same granularity as the actual problems.

Experimental validation

We have implemented these automatic model calibration mechanisms in StarPU which runs on multicore CPUs, GPUs and Cell processors. In this section, we give evidence that they have a significant impact on performance; we also illustrate the performance feedback offered by StarPU, and how StarPU provides some tools to help the programmer to understand the obtained performance, and to select the most appropriate models in consequence. We here show how these mechanisms perform in the case of a hybrid platform with a nVidia Quadro FX4600 GPU and a E5410 Xeon quad-core CPU.

Sharpness of the performance prediction

Figure 5 shows the results obtained on an LU decomposition for two different problem sizes. The first line exhibits the average and standard deviation of the reference performance obtained when using a greedy scheduling policy to distribute tasks to CPUs and the GPU. The second line shows the results obtained when calibrating the history-based performance model after either one, two or three runs and the average performance (and standard deviation) obtained after 4 runs. During the first execution, the greedy strategy clearly outperforms the non-calibrated strategy based on performance models. But once the model is calibrated, the performance obtained by the model-based strategy gets better, not only in terms of average speed, but also with respect to the standard deviation. The improvement between the runs is explained by the fact that the application Speed (GFlop/s) runs on a hybrid CPU/GPU platform: the better the accuracy, the better the load balancing. Until the models are properly calibrated, some processing units receive too much work while others are not kept busy enough.

Figure 6 depicts the evolution of the prediction inaccuracies depending on the number of collected samples. More precisely, the error is computed by taking the sum of the absolute differences between prediction and measurements, for all tasks, and by dividing this total prediction error by the total execution time. As suggested by Figure 5, the accuracy of the models becomes better as we keep collecting measurements. We finally obtain an accuracy of an order of 1% for multicore CPUs, and below 0.1% for a GPU. This difference is due to complex interactions occuring within multicore CPUs (e.g. cache sharing and contention) while computations are not perturbed on GPUs. The large majority of tasks in an LU decomposition are matrix products, whose performance is especially regular even on CPUs, so that we obtain a relatively good overall accuracy.

Performance feedback tools

StarPU provides tools to detect tasks that are not predictable enough (e.g. BLAS1 kernels). Figures 7 and8 are automatically generated by StarPU, which can collect performance measurements at runtime.

Figure 7 summarizes the behaviour of a kernel on all input sizes and the performance variations observed for the different sizes, and for the different architectures; it also shows the non-linear regression-based performance models automatically generated by StarPU so that we can figure out whether such a model is applicable or not. It also illustrates in which situation it is worth using accelerators or CPUs, therefore helping to select the most appropriate granularity. Using a small grain size on CPUs results in variable execution times, certainly explained by a poor cache use which makes performance very sensitive to the bus contention for instance. This problem disappears as we take large tiles, or if we use a GPU that is much less sensitive to such variations.

Figure 8 shows the actual distribution of the measurements that were collected for a given hash value. This not only gives a precise idea of the performance dispersion, but it can also be used to understand the actual performance issues: on the very predictable GPUs, we obtain a very thin peak, while on the CPUs, the distribution exhibiting two hills suggests that there may be some contention issue which should be further analyzed.

Related Works

Auto-tuning techniques have been successfully used to automatically generate the kernels of various high-performance libraries such as ATLAS [START_REF] Whaley | Automatically Tuned Linear Algebra Software[END_REF], FFTW, OSKI or SPIRAL; and similar results are obtained in the context of GPU computing by the MAGMA project [START_REF] Li | A Note on Auto-tuning GEMM for GPUs[END_REF]. While performance models permit to generate efficient computational kernels even on heterogeneous systems, computations are usually mapped statically on the different processing resources when dealing with hybrid systems [START_REF] Tomov | Towards Dense Linear Algebra for Hybrid GPU Accelerated Manycore Systems[END_REF].

Iterative compilation frameworks also use performance feedback to take the most appropriate optimization decisions. Jimenez et al. [START_REF] Jiménez | Predictive runtime code scheduling for heterogeneous architectures[END_REF] keep track of the relative speedups of the applications on the different architectures to decide which processing unit should be assigned to an application. Their approach is much less flexible since it does not allow to actually schedule interdependent tasks within an application.

Different runtime systems currently offer support for accelerators [START_REF] Bellens | Cellss: a programming model for the cell be architecture[END_REF], or even hybrid systems. Similarly to StarPU, the Harmony runtime system targets hybrid platforms while proposing some scheduling facilities, possibly based on performance modeling [START_REF] Diamos | Harmony: Runtime Techniques for Dynamic Concurrency Inference, Resource Constrained Hierarchical Scheduling, and Online Optimization in Heterogeneous Multiprocessor Systems[END_REF]. Its performance is modeled by the means of (possibly multivariate) regression models. This approach is hardly applicable without any support from the programmer, and possibly requires a large number of samples to have a reliable model. Thanks to the high-level support for data management integrated within StarPU, the history-based solution that we propose in this paper is simpler as it is completely transparent for the programmer.

Conclusion

We have proposed a generic approach to seamlessly build history-based performance models. It has been implemented within the StarPU runtime system with the support of its integrated data management library, and we have shown how StarPU's performance feedback tools help the programmer to analyze whether the resulting performance prediction are relevant or not.

Such history-based performance models naturally rely on some regularity hypothesis since it cannot predict the behaviour of a task if all its predecessors had different sizes: in that case, a parametric performance model calibrated by the means of regressions is more suitable. Our history-based approach also requires computational kernel with a static flow control. Tasks' execution time should be independent from the actual content of the data, the latter is often unknown when the scheduling decisions are taken anyway. This does not require any effort from the programmer who can easily use our auto-tuning mechanisms to see whether such models results into performance improvements or not.

This technique is directly applicable to the case of complex hybrid setups (e.g. heterogeneous multi-GPU). This work could also be extended to model the performance of memory transfers to schedule them as well. Scheduling policies could take advantage of performance models that depend on the actual state of the underlying machine: using hardware performance counters, the historybased models could for instance keep track of contention or cache usage. Finally, performance feedback can be valuable: this not only helps to understand the behaviour of an application during a post-mortem analysis, but this is also useful for iterative compilation environments and auto-tuned libraries.

Fig. 1 .

 1 Fig. 1. Execution of a Task within StarPU. Applications submit tasks that are dispatched onto the different drivers by the scheduler. The driver offloads the computation, using the proper implementation from the codelet, and the DSM (Distribution Shared Memory) ensures the availability of coherent data. A callback is executed when the task is done.

Fig. 2 .

 2 Fig. 2. The ''Earliest Finish'' Scheduling Strategy.

Fig. 3 .hFig. 4 .

 34 Fig. 3. Performance feedback loop.

Fig. 5 .Fig. 6 .Fig. 7 .Fig. 8 .

 5678 Fig. 5. Impact of performance sampling on the speed of an LU decomposition (in GFlop/s)

For example, we can use the usual CRC hash functions: h(p1, . . . , p k ) = CRC(p1, . . . , CRC(p k-1 , CRC(p k , 0))).