Laurent Bétermin

Peng Zhang

Minimization of energy per particle among Bravais lattices in R 2 : Lennard-Jones and Thomas-Fermi cases

Keywords: AMS Classification: Primary 82B20, Secondary 52C15, 35Q40 Lattice energy, Theta functions, Triangular lattice, Crystallization, Lennard-Jones potential, Thomas-Fermi model, Bessel function

We prove in this paper that the minimizer of Lennard-Jones energy per particle among Bravais lattices is a triangular lattice, i.e. composed of equilateral triangles, in R 2 for large density of points, while it is false for sufficiently small density. We show some characterization results for the global minimizer of this energy and finally we also prove that the minimizer of the Thomas-Fermi energy per particle in R 2 among Bravais lattices with fixed density is triangular.

Introduction

Understanding the structure of matter at low temperature has been a challenge for many years. In this case, one of the simplest models is to consider identical points as particles interacting in a Lennard-Jones potential. This model is deterministic, therefore we do not consider either entropy nor other quantum effects. The problem is to find the configuration of the points which minimize the total interaction energy, called the Lennard-Jones energy. Radin, in [START_REF] Gardner | The Infinite-Volume Ground State of the Lennard-Jones Potential[END_REF], studied this problem in one dimension and showed that, in the case of infinite points, the minimizer is periodic. His method is not adaptable in higher dimensions and he studied, in [START_REF] Heitmann | The Ground State for Sticky Disks[END_REF][START_REF] Radin | The Ground State for Soft Disks[END_REF] the case of short range interactions and proved the first result of crystallization in two dimensions for a hard-sphere model. In the meantime, Ventevogel and Nijboer gave in [START_REF] Ventevogel | On the Configuration of Systems of Interacting Particle with Minimum Potential Energy per Particle[END_REF][START_REF] Ventevogel | On the Configuration of Systems of Interacting Particle with Minimum Potential Energy per Particle[END_REF][START_REF] Ventevogel | On the Configuration of Systems of Interacting Particle with Minimum Potential Energy per Particle[END_REF] more general results in one dimension for Lennard-Jones energy per particle. Indeed, they showed that a unique lattice of the form a 0 N minimizes the Lennard-Jones energy and that all lattices aN with a ≤ a 0 minimize this energy when the density of points ρ = a -1 is fixed. Our paper gives some results in the spirit of the latter paper.

After a numerical investigation of Yedder, Blanc, Le Bris, in [START_REF] Blanc | A Numerical Investigation of the 2-Dimensional Crystal Problem[END_REF], about the minimization of the Lennard-Jones and the Thomas-Fermi energy in R 2 , it seemed that the triangular lattice, also called "hexagonal lattice" -which is composed of equilateral triangles -is the minimum configuration for Lennard-Jones energy among any lattices and for Thomas-Fermi energy with nuclei density fixed. Some time after, Theil, in [START_REF] Theil | A Proof of Crystallization in Two Dimensions[END_REF], gave the first proof of crystallization in two dimensions for a "Lennard-Jones like" potential, with a minimum less than one but very close to one and long range interaction. He showed that the global minimizer of the total energy is triangular. His method was adapted by E and Li, in [START_REF] Li | On the Crystallization of 2D Hexagonal Lattices[END_REF], for a three-body potential with long range interactions in order to obtain a honeycomb lattice as global minimizer -see also the works of Mainini, Piovano and Stefanelli in [START_REF] Mainini | Finite Crystallization in the Square Lattice[END_REF][START_REF] Mainini | Crystallization in Carbon Nanostructures[END_REF] about the crystallization in square and honeycomb lattices for three-body potentials with short range interactions -and by Theil and Flatley in three dimensions in [START_REF] Flatley | Face-Centred Cubic Crystallization of Atomistic Configurations[END_REF]. Furthermore Montgomery, in [START_REF] Montgomery | Minimal Theta Functions[END_REF], proved that the triangular lattice is the unique minimizer of theta functions among Bravais lattices with fixed density and hence the unique minimizer of the Epstein zeta function, thanks to the link between these two functions. As the Lennard-Jones potential is a linear sum of Epstein zeta functions, it is natural to study the problem of minimization of the Lennard-Jones energy among Bravais lattices with and without fixed density. However, there are few results about minimization in the general case of periodic systems. For example, Cohn and Kumar described in [START_REF] Cohn | Universally Optimal Distribution of Points on Spheres[END_REF] a method and a conjecture for completely monotonic functions. It is interesting to observe that this kind of problem is connected with the theory of spherical design due to Delsarte, Goethals and Seidel in [START_REF] Delsarte | Spherical Codes and Designs[END_REF] and linked to the layers of a lattice, among others, by Venkov and Bachoc in [START_REF] Venkov | Réseaux et designs sphériques. Réseaux euclidiens, designs sphériques et formes modulaires[END_REF][START_REF] Bachoc | Modular Forms, Lattices and Spherical Designs. Réseaux euclidiens, designs sphériques et formes modulaires[END_REF] and by Coulangeon et al. in [9,[START_REF] Coulangeon | Energy Minimization, Periodic Sets and Spherical Designs[END_REF][START_REF] Coulangeon | Spherical Designs and Heights of Euclidean Lattices[END_REF].

In this paper, our main results are :

Theorem:

• Let V LJ (r) = r -12 -2r -6
be the Lennard-Jones potential, then the minimizer of the energy

E LJ (L) = x∈L\{0} V LJ ( x )
among all Bravais lattices of R 2 with fixed density sufficiently large is triangular and unique, up to rotation.

• A minimizer of E LJ among all Bravais lattices with fixed density sufficiently small cannot be triangular.

• Let W T F : R * + → R be the solution of -∆h + πh = δ 0 which goes to 0 at infinity, then the minimizer of the Thomas-Fermi energy E T F (L) = x∈L\{0} W T F ( x ) among all Bravais lattices of R 2 with density fixed is triangular and unique, up to rotation. This paper is structured as follows : in Section 2, we introduce the notations; in Section 3, we show that the minimizer of the Lennard-Jones energy per particle among Bravais lattices with fixed density, if the density is sufficiently large, it is triangular and unique. Moreover we give numerical results and a conjecture for the minimization with density fixed and we have arguments in order to explain why the global minimizer, among Bravais lattices without fixed density, is triangular; in Section 4, we use proof of Blanc in [START_REF] Blanc | Lower Bound for the Interatomic Distance in Lennard-Jones Clusters[END_REF] to find a lower bound for the interparticle distance of the global minimizer, and finally in Section 5 we study the same kind of problem for the Thomas-Fermi model only when the density is fixed and we prove that the triangular lattice is the unique minimizer of the Thomas-Fermi energy per particle in R 2 .

Preliminaries

A Bravais lattice (also called a "simple lattice") of R 2 is given by L = Zu⊕Zv where (u, v) is a basis of R 2 . By Engel's theorem (see [START_REF] Engel | Geometric Crystallography. An Axiomatic Introduction to Crystallography[END_REF]), we can choose u and v so that u ≤ v and ( u, v) ∈ π 3 , π 2 in order to obtain the unicity of the lattice, up to a rotation. We note |L| = u∧v = u v |sin( u, v)| the area of L which is in fact the area of the lattice primitive cell and L * := L\{0}. The positive definite quadratic form associated with the Bravais lattice L is, for (m,

n) ∈ Z 2 , Q L (m, n) = mu + nv 2 = u 2 m 2 + v 2 n 2 + 2 u v cos( u, v)mn.
For a positive definite quadratic form q(m, n) = am 2 + bmn + cn 2 , we define its discriminant D = 4ac -b 2 ≥ 0. Hence for Q L , we obtain :

D = 4 u 2 v 2 -4 u 2 v 2 cos 2 ( u, v) = 4 u 2 v 2 sin 2 ( u, v) = 4|L| 2 .
In this paper, the term "lattice" will mean a "Bravais lattice", and we define, for s > 2, the Epstein zeta function of the lattice L by

ζ L (s) := x∈L * 1 x s = (m,n) =(0,0) 1 Q L (m, n) s/2 . Let Λ A = 2A √ 3 Z(1, 0) ⊕ Z(1/2, √ 3/
2) be the triangular lattice of area A, also called the hexagonal lattice. Its length is the norm of its vector u, i.e. the minimum distance strictly positive of Λ A , u = 2A/ √ 3. We notice, for any s > 2, that

ζ Λ A (s) = ζ Λ 1 (s) A s/2 (2.1)
and this relation of scaling is true for any lattice L of area A.

We recall the result of Montgomery about theta functions :

Theorem 2.1. (Montgomery, [START_REF] Montgomery | Minimal Theta Functions[END_REF]) For any real number α > 0 and a Bravais lattice L, let

θ L (α) := Θ L (iα) = m,n∈Z e -2παQ L (m,n) ,
where Θ L is the Jacobi theta function of the lattice L defined for Im(z) > 0. Then, for any α > 0, Λ A is the unique minimizer of L → θ L (α) among lattices of area A, up to rotation.

Remark 2.2. The same kind of results were obtained by Nonnenmacher and Voros in [START_REF] Nonnenmacher | Chaotic Eigenfunctions in Phase Space[END_REF]. The previous theorem implies that the triangular lattice is the unique minimizer, up to rotation, of L → ζ L (s) among lattices with density fixed for any s > 2 which is also proved by Rankin (in [START_REF] Rankin | A Minimum Problem for the Epstein Zeta-Function[END_REF]).

We consider the classical Lennard-Jones potential

V LJ (r) = 1 r 12 - 2 r 6
whose minimum is obtained at r = 1, and for L = Zu ⊕ Zv a Bravais lattice of R 2 , we let

E LJ (L) := x∈L * V LJ ( x ) = ζ L (12) -2ζ L (6)
be the Lennard-Jones energy of lattice L. By (2.1) this energy among lattices of area A can be viewed as energy L → E LJ ( √ AL) over lattices of area 1 and we parametrize L with its length u and v by

Q L (m, n) = u 2 m 2 + v 2 n 2 + 2mn u 2 v 2 -1.
It follows that we can write Lennard-Jones energy among lattices of area A as

( u , v ) → (m,n) =(0,0) V LJ √ A u 2 m 2 + v 2 n 2 + 2mn u 2 v 2 -1 . (2.2) Fig. 1: Graph of the Lennard-Jones potential V LJ
The aim of this paper is to study the following two minimization problems, up to rotation :

(P A ) : Find the minimizer of E LJ among lattices L with fixed |L| = A;

(P ) : Find the minimizer of E LJ among lattices.

Proposition 2.3. The minimum of E LJ among lattices is achieved.

Proof. We parametrize a lattice L by x = u , y = v and θ = ( u, v), therefore

f (x, y, θ) := E LJ (L) = (m,n) =(0,0) 1 (x 2 m 2 + y 2 n 2 + 2xymn cos θ) 6 - 2 (x 2 m 2 + y 2 n 2 + 2xymn cos θ) 3 .
First case : minimization without fixed area. If L is the solution of (P ) then x and y cannot be too small, otherwise the energy is too large and a proof of a lower bound for x is given in Section 4. Moreover y ≤ 1 because if y > 1 then a contraction of the line Rv gives smaller energy. Therefore we have

x, y ∈ [m, M ] and θ ∈ [π/3, π/2]. The function (x, y, θ) → f (x, y, θ) is continuous on [m, M ] × [m, M ] × [π/3, π/2]
hence its minimum is achieved. Second case : minimization with fixed area. We can parametrize L with only two variables x and y -as in (2.2) -such that when x → 0 then y → +∞. As L should be a Bravais lattice, it is clear that the minimum of f is achieved.

Minimization among lattices with fixed area 3.1 A sufficient condition for the minimality of E LJ : Montgomery's method

Our idea is to write E LJ in terms of θ L and to use Theorem 2.1 in order to find a sufficient condition for the minimality of the triangular lattice among Bravais lattices with a fixed area.

Theorem 3.1. If A 3 ≤ π 3 120
, then Λ A is the unique solution of (P A ).

Proof. As it is explained in [START_REF] Montgomery | Minimal Theta Functions[END_REF] or [START_REF] Terras | Harmonic Analysis on Symmetric Spaces and Applications[END_REF], we can write the Epstein zeta function in terms of a theta function. Indeed, we have the following identity, where the discriminant of

Q L is D = 1 : for Re(s) > 1, ζ L (2s)Γ(s)(2π) -s = 1 s -1 - 1 s + ∞ 1 (θ L (α) -1)(α s + α 1-s ) dα α . (3.1) Thus, for |L| = A, we write E LJ (L) = ζ L (12)-2ζ L (6) as an integral +∞ 1 g A (α) θ L α 2A -1 dα α ,
up to a constant independent of L and we find A so that g A (α) ≥ 0 for any α ≥ 1. As Λ A is the unique minimizer of θ L (α) for any α > 0, we have for any L such that |L| = A :

E LJ (L) -E LJ (Λ A ) = +∞ 1 θ L α 2A -θ Λ A α 2A g A (α) dα α ≥ 0
and Λ A is the unique solution of (P A ).

In fact (3.1) it is the classic "Riemann's trick" and here we will briefly recall its proof : as

Γ(s)(2π) -s Q L (m, n) -s = ∞ 0 t s-1 e -t (2π) -s Q L (m, n) -s dt
for Re(s) > 1, and by putting t = 2πQ L (m, n)y, we obtain

Γ(s)(2π) -s Q L (m, n) -s = ∞ 0 e -2πyQ L (m,n) y s-1 dy.
Summing over (m, n) = (0, 0) and using the identity θ L (1/α) = αθ L (α) for any α > 0, proved by Montgomery in [START_REF] Montgomery | Minimal Theta Functions[END_REF], we obtain

Γ(s)(2π) -s ζ L (2s) = ∞ 0 (θ L (y) -1)y s-1 dy = 1 0 (θ L (y) -1)y s-1 dy + ∞ 1 (θ L (y) -1)y s-1 dy = ∞ 1 (θ L (1/α) -1)α -1-s dα + ∞ 1 (θ L (α) -1)α s-1 dα = ∞ 1 (αθ L (α) -1)α -1-s dα + ∞ 1 (θ L (α) -1)α s-1 dα = ∞ 1 θ L (α)α -s dα - ∞ 1 α -1-s dα + ∞ 1 (θ L (α) -1)α s-1 dα = ∞ 1 (θ L (α) -1)α -s dα + ∞ 1 (θ L (α) -1)α s-1 dα + ∞ 1 α -s dα - ∞ 1 α -1-s dα = ∞ 1 (θ L (α) -1)α -s dα + ∞ 1 (θ L (α) -1)α s-1 dα + 1 s -1 - 1 s = ∞ 1 (θ L (α) -1)(α s + α 1-s ) dα α + 1 s -1 - 1 s .
Now if |L| = A, by the equality D = (2A) 2 there are two identities :

(2π) -6 (2A) 6 Γ(6)ζ L (12) = 1 5 - 1 6 + +∞ 1 θ L α 2A -1 (α 6 + α 1-6 ) dα α (2π) -3 (2A) 3 Γ(3)ζ L (6) = 1 2 - 1 3 + +∞ 1 θ L α 2A -1 (α 3 + α 1-3 ) dα α and we find ζ L (12) = (2π) 6 30(2A) 6 5! + +∞ 1 θ L α 2A -1 (2π) 6 (2A) 6 5! (α 6 + α -5 ) dα α ζ L (6) = (2π) 3 6(2A) 3 2! + +∞ 1 θ L α 2A -1 (2π) 3 (2A) 3 2! (α 3 + α -2 ) dα α .
Therefore, for any L of area A,

E LJ (L) = C A + π 3 A 3 +∞ 1 θ L α 2A -1 g A (α) dα α
where g A (α) := andC A is a constant depending on A but independent of L. Now we want to prove that if π 3 ≥ 120A 3 then g A (α) ≥ 0 for any α ≥ 1. First, we remark that

π 3 A 3 5! (α 6 +α -5 )-(α 3 +α -2 ),
g A (1) ≥ 0 ⇐⇒ π 3 A 3 5! -1 ≥ 0 ⇐⇒ π 3 ≥ 120A 3 .
Secondly, we compute g ′ A (α) =

π 3 A 3 5! (6α 5 -5α -6 ) -(3α 2 -2α -3
), and if π 3 ≥ 120A 3 then

g ′ A (1) = π 3 A 3 5! -1 ≥ 0.
Finally, we compute g ′′ A (α) =

π 3 A 3 5! (30α 4 + 30α -7 ) -(6α + 6α -4 ). As π 3 A 3 5! ≥ 1 and α ≥ 1, π 3 A 3 5! (30α 4 + 30α -7 ) -(6α + 6α -4 ) ≥ 30α 4 + 30α -7 -6α -6α -4 ≥ 24α + 30α -7 -6α -4 ≥ 0.
Thus, we have shown that, for any A so that

π 3 ≥ 120A 3 , g ′′ A (α) ≥ 0 for any α ≥ 1, g ′ A (1) ≥ 0 and g A (1) ≥ 0. Hence g A (α) ≥ 0 for any α ≥ 1 if π 3 ≥ 120A 3 .
Remark 3.2. We have π 3 120

1/3
≈ 0.63693, hence for A ≤ 0.63692, Λ A is the unique solution of (P A ).

Remark 3.3. We prove below (see Proposition 3.5) that when A is sufficiently large then Λ A is no longer a solution of (P A ). However, our bound π 3 ≥ 120A 3 is likely not to be optimal. If it were, by the Proposition 4.3 and its remark, then the triangular lattice is not the solution to (P ).

This result explains that the behaviour of the potential is important for the interaction between the first neighbours because in this case the reverse power part r -12 is the strongest interaction.

This method can be adapted to any potential of the form V (r) = K 1 r n -K 2 r p with n > p > 2 to obtain similar results in two dimensions.

Remark 3.4. The three-dimensional case is an open problem. Indeed, there is no result related to the minimization of theta and Epstein functions among Bravais lattices of R 3 with fixed volume. Sarnak and Strömbergsson recalled in [START_REF] Sarnak | Minima of Epstein's Zeta Function and Heights of Flat Tori[END_REF] that Ennola had shown in [START_REF] Ennola | On a Problem about the Epstein Zeta-Function[END_REF] the local minimality of the face centred cubic lattice for ζ L (s) and for any s > 0. They also prove that the face centred cubic lattice cannot be the minimizer of ζ L (s) for all s > 0. Hence the problem of minimization of Lennard-Jones energy among lattices of R 3 , and of course in higher dimensions, seems to be very difficult.

A necessary condition for the minimality of the triangular lattice for E

LJ Proposition 3.5. Λ A is a solution of (P A ) if and only if A ≤ inf |L|=1 L =Λ 1 ζ L (12) -ζ Λ 1 (12) 2(ζ L (6) -ζ Λ 1 (6)) 1/3
. Hence if A is sufficiently large, Λ A is not a solution of (P A ).

Proof. We have the following equivalences

E LJ (Λ A ) ≤ E LJ (L) for any L such that |L| = A ⇐⇒ ζ Λ A (12) -2ζ Λ A (6) ≤ ζ L (12) -2ζ L (6) for any L such that |L| = A ⇐⇒ 2(ζ L (6) -ζ Λ A (6)) ≤ ζ L (12) -ζ Λ A (12) for any L such that |L| = A ⇐⇒ 2(ζ L (6) -ζ Λ 1 (6)) A 3 ≤ ζ L (12) -ζ Λ 1 (12) A 6
for any L such that |L| = 1 by the scaling property (2.1). We recall that ζ L (6) > ζ Λ 1 (6) for any L of area A so that L = Λ 1 , as a consequence of Theorem 2.1 and the Riemann's trick (3.1). Then we obtain

E LJ (Λ A ) ≤ E LJ (L) for any L such that |L| = A ⇐⇒ A ≤ inf |L|=1 L =Λ 1 ζ L (12) -ζ Λ 1 (12) 2(ζ L (6) -ζ Λ 1 (6)) 1/3 . It is difficult to study the minimum of function L → ζ L (12) -ζ Λ 1 (12) 2(ζ L (6) -ζ Λ 1 (6)) 1/3
among lattices L = Λ 1 such that |L| = 1. However, we can numerically look for a lower bound. This function can be parametrized with two variables -here the lengths u and v of the lattice L as in (2.2) -and we can plot the level sets of it. We notice that the large differences between the values of the function only give a domain where the function is minimum.

Fig. 2 : Level sets of ( u , v ) → ζ L (12) -ζ Λ 1 (12) 2(ζ L (6) -ζ Λ 1 (6)) 1/3
(black = minimum, white = maximum) Indeed, its minimum seems to be around lattice L of area 1 such that u = v = 1.014 and for this one, we have to the point [START_REF] Abrikosov | The Magnetic Properties of Superconducting Alloys[END_REF][START_REF] Abrikosov | The Magnetic Properties of Superconducting Alloys[END_REF]. In fact it is clear that the point associated with the triangular lattice is a critical point of this energy, because the triangular lattice is the unique minimizer of Epstein zeta function among lattices of area A. Moreover we can prove that the square lattice is also a critical point, by using an other parametrization as ( u , θ). We numerically obtain :

ζ L (12) -ζ Λ 1 (12) 2(ζ L (6) -ζ Λ 1 ( 6
• For A = 1, Λ 1 seems to be its minimizer and Z 2 is a local maximizer.

• For A = 1.13, Λ 1 seems to be its minimizer but Z 2 seems to be not a local maximizer.

• For A = 1.14, Z 2 seems to be its minimizer because we estimate E LJ ( √ 1.14Λ 1 ) ≈ -4.435 is larger than E LJ ( √ 1.14Z 2 ) ≈ -4.437

• For A = 1.16, Z 2 seems to be its minimizer.

• For A = 1.2, Z 2 seems to be its minimizer and Λ 1 is a local maximizer.

• For A = 2 (and more), Z 2 seems to be its minimizer and Λ 1 is a local maximizer.

Hence, we can write the following conjecture based on our numerical study of L → E LJ ( √ AL) among all lattices with area 1 :

Conjecture : If A is sufficiently large, the square lattice is the unique solution of (P A ).

Global minimization of E LJ among lattices

Now we study the problem (P ). We give high properties for the global minimizer among lattices and some indications of its shape.

Characterization of the global minimizer

If L 0 = Zu ⊕ Zv is a solution of (P ) then i) E LJ (L 0 ) = -ζ L 0 (6) = -ζ L 0 (12) < 0, ii) u < 1 and v ≤ 1, iii) ζ L 0 (6) = max{ζ L (6); L such that ζ L (12) ≤ ζ L (6)}. Proof. i) We consider the function f (r) = E LJ (rL 0 ) = r -12 ζ L 0 (12) -2r -6 ζ L 0 (6). As L 0 is a global minimizer of E LJ , r = 1 is the critical point of f and f ′ (r) = -12r -13 ζ L 0 (12) + 12r -7 ζ L 0 (6), hence f ′ (1) = 0 ⇐⇒ ζ L 0 (12) = ζ L 0 (6) and E LJ (L 0 ) = ζ L 0 (12) -2ζ L 0 (6) = -ζ L 0 (6) = -ζ L 0 (12). ii) As ζ L 0 (12) = ζ L 0 (6), it is clear that u < 1 because if r > 1 then r -12 < r -6 . If v > 1, a little contraction of Rv yields a new lattice L 1 such that E LJ (L 1 ) < E LJ (L 0 )
because some of the distances of the lattice decrease while u is constant, therefore the energy decreases.

iii) -ζ L 0 (6) = E LJ (L 0 ) ≤ E LJ (L) ⇐⇒ ζ L (6) -ζ L 0 (6) ≤ ζ L (12) -ζ L (6) and if L is a lattice such that ζ L (12) ≤ ζ L (6), we get ζ L (6) ≤ ζ L 0 (6).
Corollary 4.2. The triangular lattice of length 1 cannot be the solution of (P ) though the minimum of the potential V LJ is achieved for r = 1.

Proposition 4.3. The minimizer of E LJ among triangular lattices is Λ A 0 such that

A 0 = ζ Λ 1 (12) ζ Λ 1 (6) 1/3
.

Proof. As in the above proof, we define the function f (r) = E LJ (rΛ 1 ) and we compute its first derivative f ′ (r) = -12r -13 ζ Λ 1 (12) + 12r -7 ζ Λ 1 [START_REF] Blanc | A Numerical Investigation of the 2-Dimensional Crystal Problem[END_REF]. It follows that :

f ′ (r) ≥ 0 ⇐⇒ r ≥ ζ Λ 1 (12) ζ Λ 1 (6) 1/6 =: r 0 hence Λ A 0 = r 0 Λ 1 , with A 0 = r 2 0 = ζ Λ 1 (12) ζ Λ 1 (6) 1/3
, is the minimizer of E LJ among all triangular lattices.

Remark 4.4. We compute A 0 ≈ 0.84912, therefore the length of this lattice is u ≈ 0.99019. Moreover we notice that E LJ (Λ A 0 ) = -ζ Λ A 0 (6) ≈ -6.76425 (it will be useful for the next part).

Because A 0 > 0.63692, Theorem 3.1 is not sufficient to prove that Λ A 0 is the solution of (P ) but a numerical investigation of L → E LJ ( √ A 0 L) among all lattices of area 1 seems to indicate that the solution of (P A 0 ) is triangular and unique. Moreover it is not difficult to prove numerically that Λ A 0 is a local minimizer among all lattices. Hence we can write the following conjecture for this problem :

Conjecture : The triangular lattice Λ A 0 is the unique solution of (P ).

Minimum length of the global minimizer

Because our method does not show that the triangular lattice of area A 0 is the global minimizer of the Lennard-Jones energy among lattices, we use Blanc's proof, from [START_REF] Blanc | Lower Bound for the Interatomic Distance in Lennard-Jones Clusters[END_REF], in order to find a lower bound for the minimal distance in the globally minimizing lattice. His result was for the Lennard-Jones interaction of N points in R 2 and R 3 . Xue in [START_REF] Xue | Minimum Inter-Particle Distance at Global Minimizers of Lennard-Jones Clusters[END_REF] and Schachinger, Addis, Bomze and Schoen in [START_REF] Addis | New Results for Molecular Formation under Pairwise Potential Minimization[END_REF] improved this. We use Blanc's method because it is well suited to our problem. Proposition 4.5. If L 0 = Zu ⊕ Zv is a solution of (P ), then the minimal distance is greater than an explicit constant c. Furthermore, we have c > 0.74035.

Proof. In [START_REF] Blanc | Lower Bound for the Interatomic Distance in Lennard-Jones Clusters[END_REF], Blanc proved that

E LJ (L 0 ) ≥ V LJ ( u ) -
Q + 2 + (Q + 2) 2 + 4 23 -ζ Λ A 0 (6) (P + 1)     1/6
=: c.

Since P ≈ 0.00988, Q ≈ 1.45918 and ζ Λ A 0 (6) ≈ 6.76425 we get c > 0.74035.

Remark 4.6. As we think that Λ A 0 is the unique solution of (P ), this lower bound is the best that we can find with this method. Moreover, this bound and the second point of Proposition 4.1 imply that 0.47468

< |L 0 | < 1.
5 The Thomas-Fermi model in R 2

In Thomas-Fermi's model for interactions in a solid, we consider N nuclei at positions X N = (x 1 , ..., x N ), with for any 1 ≤ i ≤ N , x i ∈ R 2 , associated with N electrons with total density ρ ≥ 0. Then the Thomas-Fermi energy is given by

E T F (ρ, X N ) = R 2 ρ 2 (x)dx - 1 2 R 2 ×R 2 log x -y ρ(x)ρ(y)dxdy + N j=1 R 2 log x -x j ρ(x)dx - 1 2 j =k log x j -x k .
To introduce this kind of model property in quantum chemistry, refer to [START_REF] Cances | Méthodes Mathématiques en Chimie Quantique. Une introduction[END_REF]. Because the system is neutral, the number of electrons is exactly N and we study the minimization problem

I T F N = inf X N {E T F (X N )} where E T F (X N ) := inf ρ E T F (ρ, X N ), ρ ≥ 0, ρ ∈ L 1 (R 2 ) ∩ L 2 (R 2 ), R 2 ρ = N .
By the Euler-Lagrange equations for this minimization problem, we find -as it is explained in Section 2 of [START_REF] Blanc | A Numerical Investigation of the 2-Dimensional Crystal Problem[END_REF] and Section 4 of [START_REF] Blanc | From Molecular Models to Continuum Mechanics[END_REF] -that the minimizer ρ is the solution of

-∆ρ + π ρ = π N j=1 δ x j .
It is known that the fundamental solution of the modified Helmholtz equation -∆h + h = δ 0also called "screened Poisson equation" -which goes to 0 at infinity, is the radial modified Bessel function of the second kind, also called the Yukawa potential, defined in [START_REF] Gradshteyn | Table of Integrals, Series and Products[END_REF] and [START_REF] Watson | A Treatise on the Theory of Bessel Functions[END_REF], by

K 0 ( x ) = +∞ 0 e -x cosh t dt. Therefore we obtain ρ(x) = π N j=1 W T F ( x -x j ) where W T F ( x ) = 1 2 K 0 ( √ π x ) and finally E T F (X N ) = i =j W T F ( x i -x j ) + N C
where C is a constant independent of N and X N . Now, if we consider that the nuclei are in lattice L, we can study, by taking the mean value of the total energy, the following energy per point

E T F (L) = x∈L * W T F ( x ).
Remark 5.1. This potential W T F decreases. Therefore, it is obvious that the right problem is to minimize this energy among lattices only with a fixed area. We notice that W T F ( √ .) is not completely monotonic on R * + , i.e. (-1) n (W T F ( √ .)) (n) (r) is not positive for any n ≥ 0 and any r > 0. Otherwise, it is explained in [START_REF] Cohn | Universally Optimal Distribution of Points on Spheres[END_REF], by using Bernstein's Theorem (see Theorem 12b of [START_REF] Widder | The Laplace Transform[END_REF]) about the following representation of a completely monotonic function f

f (r) = +∞ 0 e -rt dα(t)
where α is a non decreasing function, and Montgomery's Theorem 2.1 for theta functions, that the triangular lattice is the unique minimizer among lattices of E f (L) := x∈L * f ( x 2 ), provided we have the correct assumptions of convergence, for instance f (r) = O(r -1-η ) at infinity for some η > 0. Nevertheless, a simple idea enables us to use theta functions and we have the following result :

Theorem 5.2. Λ A is the unique minimizer of E T F among all lattices of fixed area A.

Proof. This problem is equivalent to finding the minimizer of x∈L * K 0 ( x ) among lattices with a fixed area. We put y = 1 2 x e t for x = 0 in the integral formula for K 0 ( x ) :

K 0 ( x ) = 1 2 +∞ -∞
e -x cosh t dt = 1 2

+∞ 0 e -x cosh(ln(2y/ x )) dy y Remark 5.3. The Yukawa potential appears in many vortex interaction models, as the α-model in fluid mechanics and in superconductivity (see for example [START_REF] Abrikosov | The Magnetic Properties of Superconducting Alloys[END_REF] and [START_REF] Sow | Measurement of the Vortex Pair Interaction Potential in a Type-II Superconductor[END_REF]). Indeed, the second author recently studied, in [START_REF] Zhang | On the Minimizer of Renormalized Energy related to Ginzburg-Landau Model[END_REF], Ginzburg-Landau's model for the interactions between vortices in superconductors. He proved, by using a more general method -that it can certainly be used for other potentials -the same result was obtained for minimality of the triangular lattice among all lattices with fixed density. The use of results from Number Theory in Ginzburg-Landau's models for vortices can also be seen in [START_REF] Sandier | From the Ginzburg-Landau Model to Vortex Lattice Problems[END_REF].

)) 1 / 3 ≈ 1 . 1 corresponds√ 3 ≈

 13113 1378475, hence numerically the minimum of this function is between 1.13 and 1.14. Actually Fig. 3 gives the Lennard-Jones energy -viewed as a function of two variables u and v over the lattices of area one (see (2.2)) -for ( u , v ) ∈ [1, 1.08] 2 . The triangular lattice Λ (1.075, 1.075) and the square lattice Z 2 corresponds

Fig. 3 :

 3 Fig. 3 : Level sets of ( u , v ) → E LJ ( √ AL) for some interesting values of A (black = minimum , white = maximum)

Proposition 4 . 1 .

 41

Fig. 4 :

 4 Fig. 4 : Level sets of ( u , v ) → E( √ A 0 L) (black = minimum, white = maximum)

2 +∞ 0 E≤ 1 2 +∞ 0 E

 2020 Now, for any y > 0 and any lattice L of area A, we obtainx∈L * e -x 2 4y = θ L 1 8πy-1. Hence, by Montgomery's theorem, the triangular lattice Λ A minimizes θ L (α) for any α > 0, and it is the unique minimizer of L → θ L (α) among all Bravais lattices with a fixed area A.Therefore, for any y > 0, Λ A is the unique minimizer of the energy E y (L) :=x∈L * e -x 2 4y among lattices with a fixed area A. Now it is clear, because E y (Λ A ) ≤ E y (L) for any y > 0 and for any lattice L with area A, that 1 y (Λ A )e -y dy y y (L)e -y dy y .Hence, for any L of a fixed area A :E T F (Λ A ) = x∈Λ * A W T F ( x ) ≤ x∈L * W T F ( x ) = E T F (L).

  As we have ELJ (L 0 ) ≤ E LJ (Λ A 0 ) = -ζ Λ A 0 (6) we obtain 23 -ζ Λ A 0 (6) ≥

									P + 1 u 12 -	Q + 2 u 6 .
	with P :=	k≥2	16k + 8 k 12 and Q :=	k≥2	32k + 16 k 6	.
	Now, setting t = u -6 , we have (P + 1)t 2 -(Q + 2)t -23 + ζ Λ A 0 (6) ≤ 0 which implies
				t ≤	Q + 2 + (Q + 2) 2 + 4(23 -ζ Λ A 0 (6))(P + 1) 2(P + 1)
	and we obtain
				
			u ≥	  				2(P + 1)
							23 +	1 u 12	k≥2	16k + 8 k 12 -	1 u 6	k≥2	32k + 16 k 6	.

Acknowledgements: We are grateful to Etienne Sandier, Xavier Blanc, Yuxin Ge and Henry Cohn for their interest and helpful discussions. We also want to thank our colleague Chieh-Lei Wong for his remarks on the first version of this paper.


Carine Dominguez
email: carine.dominguez@univ-st-etienne.fr

Blandine Ageron

Gilles Neubert

Value Creation in Collaborative Networks: The Case of RFiD Implementation

Keywords: collaborative networks, supply chain, RFiD, value generator, value creation

In this article, we question the way CNs are built around the implementation of RFiD in a supply chain and what are the value creation logics among these CNs. We address three main questions: the way and types of CNs in a RFiD implementation project, the nature of the aggregator and value creation of each actor during the different phases of the project and finally the interconnections between CNs in the value creation logic. We provide answers to theses questions based on a case study that describes the implementation of the RFiD innovative technology in a jewellery supply chain.

Introduction

There are different forms of collaborative networks. In this article, we question the way CNs are built dynamically around the implementation of RFiD in a supply chain and what are the value creation logics among these CNs.

To do so, we observe the dynamic construction of three types of collaborative networks (CNs) that emerge with the implementation of RFiD in jewellery products. At each phase of the RFiD project, we aim at understanding the value generator (investment in the collaboration) and value creation (gains) of each actor. We also question the interconnections of value creation between different CNs based on the fact the RFiD impact studies to date largely focus on a single firm setting. This leads to characterise the specificity of collaboration of each network and the type of value created in this project. We finally open the discussion on the appropriate models that could describe how to create value in RFiD supply chain projects via the different RFiD CNs.

1.1 Inter-organizational information systems supporting collaborative networks [START_REF] Camarinha-Matos | Collaborative networks: a new scientific discipline[END_REF] define a collaborative network (CN) is constituted by a variety of entities (organizations and people) that are largely autonomous, geographically distributed, and heterogeneous in terms of their: operating environment, culture, social capital and goals. Nevertheless these entities collaborate to better achieve common or compatible goals, and whose interactions are supported by inter-organizational information systems. Different forms of collaborative network organizations (CNOs) can be distinguished in the literature [START_REF] Camarinha-Matos | Collaborative networked organizations -concepts and practice in manufacturing enterprises[END_REF] from temporary networks that are goal-oriented (virtual organizations and teams, dynamic supply chains, virtual government) to more long-term strategic networks (virtual organization breeding environment, clusters, business ecosystems, supply chains, districts) that remain relatively stable with a clear definition of members' roles along the value chain.

Supply chains are specific CNOs defined as a network of organizations interlinking suppliers, manufacturers and distributors in the different processes and activities that produce value in the form of products and services delivered to end customer [START_REF] Christopher | Logistics and supply chain management[END_REF].

Given the dynamic and often temporary nature of CNs, it is important to understand their life cycle that can be describe into five main steps [START_REF] Camarinha-Matos | Collaborative networked organizations -concepts and practice in manufacturing enterprises[END_REF]: creation, operation, evolution, dissolution and metamorphosis.

Traditionally, the main inter organizational information systems supporting products in the supply chain are bar codes linked to EDI [START_REF] Hanman | Benchmarking your firm's performance with best practice[END_REF]. One of the key requirements of temporary CN is to develop a generic Information System to control the interoperability of the participating organizations, distributed, heterogeneous and concurrent systems for the purpose of accessing and sharing the necessary data and resources. Because temporary CNs are short terms alliances, formed to address fastchanging opportunities, they should be connected to the IS of the long-term CN via "plug-and-play" options [START_REF] Bolton | Requirements for Plug and Play, Information Infrastructure Frameworks and Architectures to Enable Virtual Enterprises[END_REF]. RFiD technology is emerging in order to improve process efficiency, notably by reducing daily receipts and inventories [START_REF] Yin | Case Study Research: Design and Method[END_REF].

Collaborative networks around RFiD in a supply chain

Supply chain collaboration implies that the chain members become involved in coordinating activities that span boundaries of their organizations in order to fulfil end users needs [START_REF] Bowersox | The strategic benefits of logistics alliances[END_REF]. According to [START_REF] Tzeng | Evaluating the business value of RFiD: Evidence from five case studies[END_REF], the intensity of collaboration depends on the following impact horizons: Short-term, medium-term or long-term.

Collaboration between supply chain partners has been covered extensively in the strategic management literature ( [START_REF] Bowersox | The strategic benefits of logistics alliances[END_REF]; [START_REF] Ivantysynova | RFiD in manufacturing: the investment decision[END_REF]; [11]; [START_REF] Bowersox | How supply chain competency leads to business success[END_REF]). Several research surveys have shown that improvement of inter-enterprise processes is one of the major issues in Supply Chain Management [START_REF] Boyson | Logistics and the Extended Enterprise: Benchmarks and Best Practices for the Manufacturing Professional[END_REF]. [START_REF] Holland | Cooperative supply chain management: the impact of inter-organizational information systems[END_REF] have recognised that the application of information technology in the supply chain management has become one of a major issue of both managers and researchers, as Information technology improve the performance of both individual firm and supply chain as a whole. Information technology plays major role in various supply chain collaboration practices as it enables and integrates information exchanges between supply chain partners. In this context, EDI, Internet, standards and others technologies have been developed in order to support increasing exchanges. By the same time, new collaborative practices emerge between supply chain partners, such as Vendor/Supplier management inventory, Collaborative Planning, Forecasting and Replenishment (CPFR). And as the volume of information exchanged and the intensity of interactions between partners further increased, there has been an amplified requirement towards an adequate underlying technology infrastructure. In order to cope with these new collaboration practices, new ways of information exchanges have emerged [START_REF] Simatupang | The collaborative supply chain: a scheme for information sharing and incentive alignment[END_REF] The recent industry report CGI [START_REF] Gci | EPC: A Shared Vision for Transforming Business Processes, Global Commerce Initiative[END_REF] emphasizes the new collaborative processes that will be empowered through the use of RFID. The emergence of these new technologies revolutionize many of the supply chain operations by reducing costs, improving service levels, etc [START_REF] Yin | Case Study Research: Design and Method[END_REF].

Value creation in RFiD collaborative networks

Different value generators have been highlighted in the literature. [START_REF] Pramatari | Collaborative supply chain practices and evolving technical approaches[END_REF] propose main value generators in CNs that are good indicators of the metric of the collaboration: physical and financial assets, human, organizational and relational capital.

Finally, there are both tangible and intangible types of benefits that can be reached with RFiD in a manufacturing process [START_REF] Kogut | The Network as Knowledge: Generative Rules and the Emergence of Structure[END_REF]. The main benefits are production planning, process optimization, IT management, improving quality and customer service, increasing reputation and improving inter organizational collaboration.

RFID provides multiple benefits for the supply-chain. It can add value along the entire supply-chain and related logistical operations and business relationships for more effective business process design. It provides a means of tracking supplier items from supplier through the distribution network to the point of consumption. The possibilities of RFID let its use in the supply-chain somewhat transformational ( [START_REF] Norman | From value chain to value constellation: designing interactive strategy[END_REF]; [START_REF] Wamba | Integrating RFID Technology and EPC Network into a B2B retail supply chain: a step toward intelligent business processes[END_REF]) have identified the four main processes that can be directly affected by RFID.

Finally, the logic of value creation in these networks included different layers. While the layers may embrace sequential activity chains, other co-producing value creation activities will also take place in parallel, further enhancing value creation at the network level [START_REF] Bolton | Requirements for Plug and Play, Information Infrastructure Frameworks and Architectures to Enable Virtual Enterprises[END_REF]. The notion of value creation is central to strategic management theory and includes value not only in supply and IS flows but also on knowledge sharing, learning and innovation [START_REF] Loebbecke | Piloting RFID along the supply chain: a case analysis[END_REF] in what can be labelled strategic supplier networks. Firms acting in different value constellations [START_REF] Parung | A metric for collaborative networks[END_REF] are characterized by social embeddedness [START_REF] Gunasekaran | Information systems in supply chain integration and management[END_REF], governance mechanism and the development of a sense of trust and reciprocity.

The case study analysis

The majority of research on RFiD projects has focused on quantitative methodologies and there are very few case studies focusing on the description of the complexity of RFID implementation project in a supply chain such as those developed by [START_REF] Loebbecke | Use of innovative content integration information technology at the point of sale[END_REF] and [START_REF] Niedermann | Examining RFiD in Supply Chain Management[END_REF]. We conducted a single case study with 15 semi-structured interviews, to reach an in-depth and insight understanding into this contemporary phenomenon [27].

Our case study was performed with one of the main French retailer. The choice of this case study is largely explained by the fact that:

• The company is a pioneer in the implementation of the RFID technology and no research has been done in France.

• The jewellery supply chain with RFID has never been studied before.

Actors involved in the RFID project and chronological steps

We can present three main CNs in this project (Fig. 2).

• The Product Collaborative Network (PCN): the more stable CN that deals with supplying Casino's stores (POS) with jewellery products. Different actors are part of this CN: suppliers of jewels, the logistics service provider (LSP), Casino's 80 Stores called "Boutique Or".

• The Information System Collaborative Network (ISCN): the network that is in charge of the evolution and maintenance of Casino' IS linked with the jewellery supply chain. Different actors are part of this CN: Casino's CIO (Information Systems Department) and the Information System Service Provider (ISSP)

• The RFiD Collaborative Network (RFiD CN): it is the temporary network that is build with the RFiD project. Different actors are part of this CN: the label supplier, the printer pupplier: this company proposes printing machines to encode and print the RFiD tags inside the LSP warehouse, the reader supplier and the tag supplier. How had these CNs been built over time? Different logics explain the configuration of these networks. The PCN was the initial and historical one: Casino was working with its LSP to supply jewellery products for many years with bar codes. It is interesting to come back to the main criteria that explain the shaping of theses CNs. To do so, let us enter into more details in the description of the phase 1 "seeking opportunities" step 4 (See 3.2.1) and the phase 2 "pilot project and validation" step 7 (See 3.2.2). First, the choice of the ISSP in the IS CN can be surprising as the retailer Casino decided to choose a very small-sized firm with high expertise in IS. Among the different criteria to select an ISSP, the fact that the company that was selected had been working in the past, and for many years, with the LSP brought trust in the relationship. Trust build in the PCN was spread out in the IS CN. Second, the shaping of the RFiD CN did not follow the same logic. Casino sent an invitation to tender to different companies positioned in the RFiD market. Two main RFiD tag suppliers were selected in the final short list and the one that was finally chosen proposed the lower global cost. After that, the RFiD tag supplier managed to convince Casino to work with its own CNS, hence connecting the Label supplier and the RFiD printer and reader supplier.

The case study analysis

In order to analyse the value creation logic in the RFiD project, we use the framework of [START_REF] Wamba | Integrating RFID Technology and EPC Network into a B2B retail supply chain: a step toward intelligent business processes[END_REF] that we adapt and complete to illustrate our case study. Fosso and Wamba's framework, specifically developed for RFID projects, is composed of a sequence of three main phases that describe a RFID project implementation: opportunities seeking, pilot project and validation and RFID project deployment. In the following section, we will present only phase 1 : Opportunity seeking; phases 2 : Pilot project and validation and phase 3: RFID project development will be discussed later during the session.

Phase 1: Opportunities seeking

This initial phase deals with the opportunities to implement the RFID project in a specific value chain. It is composed of six steps. In the table below, we describe for each phase what actors were involved, what did they do, what type of value they invest in the CN called value generator [START_REF] Pramatari | Collaborative supply chain practices and evolving technical approaches[END_REF] and the value created (gains) for each actor in the different CNs.

Phase 2: Pilot project and Validation

The objective of this second phase is to develop a pilot project in order to validate the feasibility of the RFID implementation. The goal is to identify the business opportunities that could come from this new technology and to propose process and IT reconfiguration to fit with RFID. It is made of four steps.

Phase 3: RFID Project Deployment

The last phase concerns the validation of the pilot project and the deployments of the technology on the whole supply chain.

Discussion

The main results of the case study analysis are the following:

• During the project, different CNs are built or broken: the RFiD CN is built upon the project itself, the choice of Casino to externalize the IS part of the project to an ISSP initiated a long-term ISCN working in cooperation with the PCN.

• Some CNs are stable (ISCN and PCN) whereas others are temporary [START_REF] Camarinha-Matos | Collaborative networked organizations -concepts and practice in manufacturing enterprises[END_REF], formed only during the project opportunities seeking and pilot project. The project deployment phase transforms a temporary collaboration between the RFiD CN and the IS CN/PCNs mainly based on an innovative project to a business relationship between the Tag Supplier and Casino.

• The RFiD Temporary CN only invested during the Phases 1 and 2 of the project (Human, Organizational, Physical) and captured value in the last Phase 3 of the project (Brand and Relational above all). Hence, the temporary RFiD CN largely participated to increase the value of the stable CNs

• Value creation (gains) is largely oriented in a CN logic rather than in an individual logic based on criteria such as size and power. For instance, the ISSP, which is a small company, that is working with CIO in the ISCN reinforced its business relationship with Casino (Relational value) and managed to build a long-term ISCN with CIO, owing to its past business relationships with LSP. The configuration of temporary and long-term strategic CNs are based on collaborative characteristics such as personal and historical relationships (the information system service provider get used to work with the LSP, the RFiD tag supplier knows personally some CIO department managers) [START_REF] Gunasekaran | Information systems in supply chain integration and management[END_REF], trust and partners competencies. Moreover, the innovative capacity of all the stakeholders of the collaborative network and their willingness to explore new opportunities are important enablers of these configurations. The RFiD technology engenders organizational change and process optimization ( [START_REF] Norman | From value chain to value constellation: designing interactive strategy[END_REF]; [START_REF] Wamba | Integrating RFID Technology and EPC Network into a B2B retail supply chain: a step toward intelligent business processes[END_REF]). The RFiD technology also generates intangible benefits [START_REF] Kogut | The Network as Knowledge: Generative Rules and the Emergence of Structure[END_REF] but these seem not the same as in manufacturing processes. Knowledge and innovation capacities [START_REF] Loebbecke | Piloting RFID along the supply chain: a case analysis[END_REF] seem to be the main gains for stakeholders participating in a supply chain temporary CNs. This hypothesis had to be developed in further research through a comparison case-study research design.

Fig. 2 .

 2 Fig. 2. Temporary and long-term collaborative networks value creation linked with the RFiD project.

Jewelry Suppliers Logistic Service Provider Point of sale (POS)

					Information
					Systems		Casino CIO
					Service		Department
					Provider
	Jewelry Suppliers	Receiving	Warehousing	Order picking	Delivery	Receiving	Selling	Inventory
	Label Supplier	RFID tag Supplier
		RFiD Supply Chain
		(Temporary CN)

Product Supply Chain (Long-term strategic CN) IT Supply Chain (Long- term strategic CN)

	RFiD Printer supplier
	RFID Reader supplier


Ardavan Dargahi
email: ardavan.dargahi@g2elab.grenoble-inp.fr

Franck Pourroy
email: franck.pourroy@g-scop.inpg.fr

Frédéric Wurtz
email: frederic.wurtz@g2elab.grenoble-inp.fr

Towards Controlling the Acceptance Factors for a Collaborative Platform in Engineering Design

Keywords: acceptance factors, collaborative, platforms, engineering design

This paper might serve as a guide to take step towards a better acceptance of computer-based Knowledge management (KM) tools in institutional setting. At first time, it investigates a set of factors with different origins which are proved to have an effect on usage decision. Secondly, we set a list of candidate factor which are supposed to influence future users of a collaborative KM platform (Dimocode). At the end, we develop a methodology to take into account the selected factors and master their positive or negative impacts. The contents of this paper would be an appropriate framework in the way of Knowledge management systems (KMS) deployment.

Introduction: the controversial success of KMS

Engineers involved in a design process use and create a large amount of technical knowledge during their daily work. Managing this knowledge is claimed to be a way of improving the efficiency of these engineering design activities by fostering knowledge formalization and sharing [START_REF] Gardoni | Valuing Knowledge Management Impact on Engineering Design Activities[END_REF].

Collaborative platforms are often presented as a relevant support for knowledge management activities. They rely on information repositories and provide users with different means of interaction, either with the documents or with other users. Based on the assumption that intensive collaboration between engineers will support knowledge sharing, some research works report successful implementations of such collaborative platforms in the engineering design domain [START_REF] Beylier | A collaboration-centered approach to manage engineering knowledge: a case study of an engineering SME[END_REF]. However, even if the idea of these platforms seems to be attractive and useful to the users, some research works rise into question the relevance of IT systems for applications such as knowledge management or organisational memory systems [START_REF] El Louadi | Do Organizational Memory and Information Technology Interact to Affect Organizational Information Needs and Provision?[END_REF]. The main challenges IT is facing with regarding the potential success of such systems are discussed both in terms of technical challenges and organisational ones [START_REF] Atwood | Organizational Memory Systems: Challenges for Information Technology[END_REF] and despite their importance, the latter seem to be often underestimated, leading to disappointing results.

The issue of identifying and overcoming these organisational challenges is at the heart of our concern, as we are working on the development and the implementation of a new collaborative platform. This platform, namely DIMOCODE, will be dedicated to the sharing of calculation models of electrical components between design engineers and calculation experts. In addition to the sharing of the models' documents, DIMOCODE is also meant to support a knowledge network, thus giving more value to these calculation models. In this paper, we try to identify which factors will play a particular role in the acceptance of a collaborative platform, and which animation policy should be engaged in order to ensure a successful deployment.

The following section gives an overview of the state of the art regarding the acceptance factors of knowledge management systems (KMS). The forthcoming DIMOCODE collaborative platform is then briefly presented in section 3. Finally, section 4 draws some propositions regarding the main acceptance factors of the platform and the technological and organizational leverage for action related with those factors.

Acceptance factors in the literature

The degree in which individuals interact with a system, is seen as the key point in successful implementation of a KMS [START_REF] Turner | Assessing knowledge management system user acceptance with the technology acceptance model[END_REF] and [START_REF] Kankanhalli | Understanding Seeking From Electronic Knowledge Repositories: An Empirical Study[END_REF]. Nevertheless, many studies in the literature report failures in KMS deployment due to their underutilisation by individuals [START_REF] Lin | Understanding knowledge management system usage antecedents: An integration of social cognitive theory and task technology fit[END_REF] and [START_REF] Bourdon | The key success factors of Knowledge management systems: a proposed explanatory model[END_REF]. That is why, knowing how to foster the employees to use such devices and to embrace them in their daily work practices remains a major concern in the institutional setting [START_REF] Goodman | Computer-aided systems and communities: Mechanisms for organizational learning in distributed environments[END_REF]. Since the 70s studies on factors likely to affect human intention to use a system, this issue appears to be a crucial research field needed to promote KMS adoption in organisations [START_REF] Bourdon | The key success factors of Knowledge management systems: a proposed explanatory model[END_REF]. Given that one can deal with a KMS either to feed it with his narratives or to seek suitable information, both knowledge sharing and knowledge seeking situations have been tried to be taken into account in our acceptance factors analysis.

A synthesis of several theoretical models of KMS usage suggested by different studies [START_REF] Kankanhalli | Contributing Knowledge to Electronic Knowledge: An Empirical Investicatio[END_REF] and [START_REF] Lin | Understanding knowledge management system usage antecedents: An integration of social cognitive theory and task technology fit[END_REF], has highlighted a fourfold dimension to what can be assigned each of the acceptance factors. These four dimensions are: 1) Organisational context: some authors argue that the organisational settings, in which teams are embedded, can play an outstanding role on how a technology is framed and enacted [START_REF] Bourdon | The key success factors of Knowledge management systems: a proposed explanatory model[END_REF] and [START_REF] Bernard | A typology of knowledge management system use by teams[END_REF]. 2) Tool: many papers account for the perceived characteristics of the system itself as determinant for an effective KMS usage. 3) Users: the intrinsic characteristics of potential users are also highlighted [START_REF] Lin | Understanding knowledge management system usage antecedents: An integration of social cognitive theory and task technology fit[END_REF] as another dimensions of KMS acceptance 4) Task: the nature of task which is supposed to be assisted by KM tool could alter the form of information exchange, thus that makes it harder or easier contrariwise [START_REF] Lin | Understanding knowledge management system usage antecedents: An integration of social cognitive theory and task technology fit[END_REF]. This classification approach yields to consider four main classes of factors respectively called: organizational factors, system related factors, user related factors and task related factors. A second way has been also derived from our literature review to classify the causal factors might be separating them depending on their inhibitory or stimulating effects on KMS usage behavior into two categories. Table 1 combines both the fourfold dimension and effect-based classification visions by connecting to each category a set of deterrent (italic) and incentive (normal) factors. Each of the factors only comes with a very short description but the related bibliographical references can help those who are seeking to know more about them.

Organisational factors

Cooperative culture An organisational culture conductive to knowledge sharing causes the employees to recognize the benefits of knowledge sharing behavior.

[12], [START_REF] Bernard | A typology of knowledge management system use by teams[END_REF], [START_REF] Goodman | Computer-aided systems and communities: Mechanisms for organizational learning in distributed environments[END_REF] Reciprocity Knowledge providers share their experiences when there will be a great chance of returning from others.

[9], [START_REF] King | Motivating knowledge sharing through knowledge management systems[END_REF], [START_REF] Markus | Toward a Theory of knowledge Reuse: Types of Knowledge Reuse Situation and Factors in Reuse Success[END_REF] Episodic change Under a high rate of episodic context change, people are reluctant to adopt a sharing behavior.

[

11],[17]

Resource availability A sufficient amount of time and a fast access can multiply the interactions with implemented system.

[8], [START_REF] Goodman | Computer-aided systems and communities: Mechanisms for organizational learning in distributed environments[END_REF] Psychological safety Legitimates the errors in well-intentioned actions [START_REF] Bernard | A typology of knowledge management system use by teams[END_REF], [START_REF] Beylier | Implementation of a light knowledge sharing tool in engineering design[END_REF] Pro-sharing norms The rules of teamwork are meant to enhance and facilitate the knowledge sharing climate.

[6], [START_REF] Kankanhalli | Contributing Knowledge to Electronic Knowledge: An Empirical Investicatio[END_REF] Generalised trust The members are more inclined to share their skills those who are trustworthy and whom they believe in their skills and goodwill.

[10], [START_REF] King | Motivating knowledge sharing through knowledge management systems[END_REF], [START_REF] Beylier | Implementation of a light knowledge sharing tool in engineering design[END_REF]

Supervisory control

Feeling monitored by a superior can improve the willingness of people to make use of KMS irrespective of their contribution interest.

[13], [START_REF] Beylier | Implementation of a light knowledge sharing tool in engineering design[END_REF] Organisational reward Moral or material incentives are sometimes proved to promote KMS usage.

[6], [START_REF] Bourdon | The key success factors of Knowledge management systems: a proposed explanatory model[END_REF], [START_REF] Kankanhalli | Contributing Knowledge to Electronic Knowledge: An Empirical Investicatio[END_REF], [START_REF] Leidner | The role of culture in knowledge management: A case study of two global _rms[END_REF], [START_REF] Markus | Toward a Theory of knowledge Reuse: Types of Knowledge Reuse Situation and Factors in Reuse Success[END_REF] User-related factors

Image of self

The information contributed to KMS could bring a good reputation to his author.

[10], [START_REF] Bernard | A typology of knowledge management system use by teams[END_REF], [START_REF] Markus | Toward a Theory of knowledge Reuse: Types of Knowledge Reuse Situation and Factors in Reuse Success[END_REF] Self-confidence Perceived conviction in his abilities to face and solve the problems may foster the contribution.

[5], [START_REF] Goodman | Computer-aided systems and communities: Mechanisms for organizational learning in distributed environments[END_REF], [START_REF] Kankanhalli | Contributing Knowledge to Electronic Knowledge: An Empirical Investicatio[END_REF] Altruism Satisfaction due to benefit another from one's experience and skills.

[10]

Loss of power

Reluctance to disseminate his knowledge using KMS fear of losing the resulting superiority.

[10]

Identification

Commitment to membership in the virtual communities of the KMS.

[7], [START_REF] Goodman | Computer-aided systems and communities: Mechanisms for organizational learning in distributed environments[END_REF], [START_REF] Kankanhalli | Contributing Knowledge to Electronic Knowledge: An Empirical Investicatio[END_REF], [START_REF] Leidner | The role of culture in knowledge management: A case study of two global _rms[END_REF] System-related factors

Contribution cost

The amount of time and energy required to contribute.

[9], [START_REF] Kankanhalli | Contributing Knowledge to Electronic Knowledge: An Empirical Investicatio[END_REF], [START_REF] King | Motivating knowledge sharing through knowledge management systems[END_REF], [START_REF] Markus | Toward a Theory of knowledge Reuse: Types of Knowledge Reuse Situation and Factors in Reuse Success[END_REF], [START_REF] Beylier | Implementation of a light knowledge sharing tool in engineering design[END_REF]

Research cost

The expense of time and energy to find sought information in knowledge base.

[9], [START_REF] Markus | Toward a Theory of knowledge Reuse: Types of Knowledge Reuse Situation and Factors in Reuse Success[END_REF] Perceived usefulness Quality of contents, task-technology-fit and awareness of potential benefits [START_REF] Lin | Understanding knowledge management system usage antecedents: An integration of social cognitive theory and task technology fit[END_REF], [START_REF] Goodman | Computer-aided systems and communities: Mechanisms for organizational learning in distributed environments[END_REF] Ease of use Easier is a to manipulate, more it will be used.

[5], [START_REF] Kankanhalli | Understanding Seeking From Electronic Knowledge Repositories: An Empirical Study[END_REF], [START_REF] Bourdon | The key success factors of Knowledge management systems: a proposed explanatory model[END_REF], [START_REF] Goodman | Computer-aided systems and communities: Mechanisms for organizational learning in distributed environments[END_REF] Task-related factors

Task interdependence When there is an obligation to share resources and expertise with others in order to achieve his aim.

[6], [START_REF] Kankanhalli | Contributing Knowledge to Electronic Knowledge: An Empirical Investicatio[END_REF] Task Tacitness The proportion of tacit knowledge upon explicit one, needed to perform a task using KM tool.

[6], [START_REF] Goodman | Computer-aided systems and communities: Mechanisms for organizational learning in distributed environments[END_REF] However, we believe the factors distribution into a fourfold dimension as suggested by the table.1 gives rise to an issue in the sense that certain factors, whatever deterrent or incentive they are, own a multidimensional nature and could correlate at the same time with more than a single dimension. For instance, we assume that spreading a cooperative culture between staff, having organizational conditions conductive to cooperation will not only be sufficient; but individual intrinsic characteristic might also largely determine the adoption of such a behavior. This type of assumptions led us to draw the model form Fig. 1 taking into account the multidimensionality property of factors. This model will serve to define some requirements of the Dimocode platform which is now briefly presented.

What is the DIMOCODE Platform?

With the DIMOCODE platform, we are interested by the problematic of capitalization and reuse of numerical models for the simulation and optimization of energetic system designed by engineers. We call energetic systems, systems like cars, buildings, or planes, it means systems that uses electric, mechanical, or hydraulic energy. For the simulation, and the optimization of those systems, engineers use physical and mathematical models in order to compute the characteristics of those systems: consumed energy, efficiency, weight, cost... We made the hypothesis that the knowledge around those physical and mathematical models is composed of two aspects: 1) Explicit knowledge: the file containing the models with their physical and mathematical equations, with the associated documentation explaining the assumptions, the limits of use, …2) Implicit or tacit knowledge: this is the skill developed by the expert who has developed the model. So the goal of the DIMOCODE platform is to capitalize all the explicit knowledge and to provide experts and users of model with the opportunity to exchange their knowledge by having possibilities to work in a collaborative way around those models.

Tas k interdependence Codification cost

Ease of use The explicit knowledge can be stored and exchanged thanks to the possibility of creating a workspace for each model. In this workspace, the expert who has developed the model can expose all the files containing the equations and all the information and documentation for his models.

In order to offer a possibility to exchange the implicit or tacit knowledge, some functionalities have been implemented like: -In the workspace of each model, there is a forum that allows users of models to ask question to the expert who has developed the model: this is a semi-formal way in order to offer the possibility to made explicit some knowledge aspects which are not clear, or simply not explained -Experts can also work together around their model, with their simulation and optimization simulation tool, by using virtual meeting rooms in the platform: the idea is here to have the possibility to work together at the same moment on the same problem using the model.

In addition to the model workspace and its associated interaction facilities, the platform is also meant to support communities of practices around the simulation of energetic systems. Thus, it comes with a community workspace in which users have the opportunity to structure a community in order to share their knowledge around a specific practice. One can for example organise online (or offline) events, share documents or stories, define projects. The issue is now to define technical and organizational requirement in order to insure a successful deployment of the platform.

Technological and organizational requirements for the Dimocode platform

It comes from section 2 that many factors may influence the behavior of individuals who contribute to KMS. The approach which was followed in order to take account of these factors for the specification requirement of Dimocode is now presented in three different steps: selecting primary factors, defining strategies in order to manage these factors, and defining leverage for action related with these strategies. The latter will lead us to define both technological and organizational requirements.

Selecting primary factors

From the initial set of factors listed in section 2, a selection of primary ones is proposed, based on the specific aims and context of Dimocode.

Dimocode is an open platform, transverse to existing organizations (laboratories, companies…) whose future users are members of. The organizational context in which they work is so quite varied. Moreover, being external and independent from these organizations, Dimocode's capacity of action on these organizational contexts is very limited. This led us to remove from our selection those of the factors that are mainly related with the organizational context. Nevertheless, it is to be considered that the Dimocode platform will also come with some kind of organization which overlaps with those sometimes highly structured and hierarchical of companies. As an example, it could be proposed to introduce within Dimocode an economic reward system for the contributors, regardless of the local policies of companies. For the particular case of this factor of organizational reward, the choice has yet been made to remove it from the selection, due to some issues in predicting its effect [START_REF] Bourdon | The key success factors of Knowledge management systems: a proposed explanatory model[END_REF]. Again, this characteristic of being independent from already existing organizations led us to remove from our selection those of the factors that are mainly related with the task of the users inside their company. Coming from the outside makes it difficult to have leverage on the purely task related factors.

Finally, a finer evaluation of the relevance of each of the remaining factors, which is not developed in this paper, led us to focus on the list of ten primary factors which are: Reciprocity, Resource availability, Psychological safety, Pro-sharing norms, Generalised trust, Image of self, Altruism, Identification, Contribution cost, Research cost, Perceived usefulness, and Ease of use.

At this stage of the method, the grid of factors form Fig. 1 also proved to be a very useful tool in order to increase both our understanding of the global context of the platform to be developed, and our awareness of threats and opportunities.

Defining the main strategies

Because of the plausible role of the retained factors in using a KMS, controlling for their positive or negative effect is perceived as being a priority in the goal of insuring a dynamic life for the DIMOCODE platform and reaching a mass usage of it. In this way, once the primary factors have been selected, the second step of the approach was to associate each of them with one or more strategies in order either to develop its positive effect or to limit the negative one (see Table 2 for an example).

Identifying leverage for action

To support the defined strategies, we have looked for the levers of action making it possible to put into action and materialise them. The levers of action connected to each factor are considered to be able to act on the impact of primary factors and moderate it and are essentially composed of some platform animation policies and/or some features already built into the last version of the Dimocode platform. Table.2 gives an example of a strategy associated with Image of self and the corresponding levers of action.

Conclusion

Despite collaborative platforms are meant to be a good means for sharing the technical knowledge between engineers, the deployment of such systems often leads to disappointing results, mainly due to a poor contribution level from the users. The paper deals with the case of the future Dimocode platform, which will be dedicated towards the sharing of simulation models for energetic systems.

Based on a literature survey, 20 acceptance factors have been identified and it was suggested a four-dimensional classification to structure these factors, The proposed classification accounts for the multidimensional nature of a large part of these factors, and made it possible to select the most relevant of the factors for the Dimocode context. Finally, a strategy and related leverage for action were associated to each of these primary factors, leading to a more complete set of requirements for the Dimocode platform.

This approach on the Dimocode platform might be considered as the preliminary draft of a more general method to account for acceptance factors in KMS requirement specification.

Among the limitations of this work, one has to consider that the multidimensional nature of some of the factors still has to be consolidated. Another point of importance is the Generalised influence between some of the acceptance factors which was not discussed in this paper.

Fig1.

  Fig1. The multidimensional nature of acceptance factors

Table 1 .

 1 KMS acceptance factors in the literature.

	Factor	Short description	References

Table 2 .

 2 Strategies and levers of action for moderating the effect of retained acceptance factors

	Factor	Strategy	Leverage for action
			Organisational	Technical
	Image of self Making the contributors	Identification of most	Publication on
		visible	relevant contributions	news pages of the
				communities
			Organizing invited
			online conferences from	Visio-conference
			the expert	facilities