Greqam

Jean-Philippe Garnier

Kazuo Nishimura

Alain Venditti

CAPITAL-LABOR SUBSTITUTION AND INDETERMINACY IN CONTINUOUS-TIME TWO-SECTOR MODELS

Keywords: Sector-specific externalities, constant returns, capital-labor substitution, indeterminacy. Journal of Economic Literature Classification Numbers: C62, E32, O41

The aim of this paper is to discuss the role of the elasticity of capital-labor substitution on the local determinacy properties of the steady state in a two-sector economy with CES technologies and sector-specific externalities.

Introduction

Recently, a large number of papers have established the fact that locally indeterminate equilibria and sunspots fluctuations easily arise within twosector infinite-horizon growth models with sector specific external effects in production and linear preferences. [START_REF] Benhabib | Indeterminacy and Sunspots in Macroeconomics[END_REF] For instance, considering Cobb-Douglas technologies, Benhabib and Nishimura [2] prove within a continuous-time model that the existence of local indeterminacy is obtained if and only if there is a reversal of factor intensities between the private and social levels.

The consumption goods has indeed to be capital intensive from the private perspective but labor intensive from the social perspective.

When CES technologies are introduced, a much larger set of configurations is compatible with local indeterminacy. In particular, Leontief or linear technologies for the consumption good sector may be considered. Therefore, we will consider CES technologies in a continous-time model and provide a complete analysis of the local determinacy properties of equilibria. We will show that, even with asymmetric factor substitutability properties, local indeterminacy is still based on a reversal of factor intensities between the private and social levels, what we call quasi factor intensity. We will also show that local indeterminacy is compatible with a Leontief technology in the consumption good sector, and that this result is preserved if the elasticity of capital-labor substitution in that sector is not too large.

The paper is organized as follows: Section 2 presents the basic model with the production structure, the intertemporal equilibrium and the steady state. In Section 3 we provide all the results on the existence of local indeterminacy depending on different configurations for the CES coefficients at the private and social levels. Section 4 finally contains some concluding comments. All the proofs are gathered in a final Appendix.

The model 2.1 The production structure

We consider an economy producing a pure consumption good y 0 and a pure capital good y 1 . Each good is assumed to be produced by capital x 0j and labor x 1j , j = 0, 1, through a CES technology which contains sector specific externalities. The representative firm in each industry indeed faces the following function:

y j = β 0j x -ρ j 0j + β 1j x -ρ j 1j + e j (X 0j , X 1j ) -1/ρ j , j = 0, 1 (1)
with β ij > 0, ρ j > -1 and σ j = 1/(1 + ρ j ) ≥ 0 the elasticity of capital-labor substitution. The positive externalities are equal to

e j (X 0j , X 1j ) = b 0j X -ρ j 0j + b 1j X -ρ j 1j
with b ij ≥ 0 and X ij denoting the average use of input i in sector j. We assume that these economy-wide averages are taken as given by each individual firm. At the equilibrium, since all firms of sector j are identical, we have X ij = x ij and we may define the social production functions as follows

y j = β0j x -ρ j 0j + β1j x -ρ j 1j -1/ρ j (2)
with βij = β ij + b ij . The returns to scale are therefore constant at the social level, and decreasing at the private level. We assume that in each sector j = 0, 1, β0j + β1j = 1 so that the production functions collapse to Cobb-Douglas in the particular case ρ j = 0. Labor is normalized to one, i.e.

x 00 + x 01 = 1, and the total stock of capital is given by x 1 = x 10 + x 11 . Choosing the consumption good as the numeraire, i.e. p 0 = 1, a firm in each industry maximizes its profit given the output price p 1 , the rental rate of capital w 1 and the wage rate w 0 . Its profit is:

π j = p j y j -w 0 x 0j -w 1 x 1j
The first order conditions subject to the private technologies (1) are

p j β ij (y j /x ij ) 1+ρ j = w i , i, j = 0, 1 (3)
From (3) we have

x ij /y j = (p j β ij /w i ) 1 1+ρ j ≡ a ij (w i , p j ), i, j = 0, 1 (4)
We call a ij the input coefficients from the private viewpoint. If the agents take account of externalities as endogenous variables in profit maximization, the first order conditions subject to the social technologies (2) are p j βij (y j /x ij ) 1+ρ j = w i , i, j = 0, 1 and the input coefficients become āij (w i , p j ) = p j βij /w i 1/(1+ρ j )

, i, j = 0, 1

We call āij the input coefficients from the social viewpoint. We also define

âij (w i , p j ) ≡ ( βij /β ij )a ij (w i , p j ) (6)
as the quasi input coefficients from the social viewpoint, and it is easy to derive that âij (w i , p j ) = āij (w i , p j ) βij /β ij ρ j /(1+ρ j )

Notice that âij = āij if there is no externality coming from input i in sector j, i.e. b ij = 0, or if the production function is Cobb-Douglas, i.e. ρ j = 0. As we will show below, the factor-price frontier, which gives a relationship between input prices and output prices, is not expressed with the input coefficients from the social viewpoint but with quasi-input coefficients from the social viewpoint.

Based on these input coefficients straightforward computations allow to establish various Lemmas. We first show that the factor-price frontier is determined by the quasi input coefficients from the social viewpoint. Note that the rental rate is a function of the output price only, w 1 = w 1 (p 1 ), while outputs are functions of the capital stock and the output price, y j = y j (x 1 , p 1 ), j = 0, 1.

We now examine some comparative statics. The factor-price frontier satisfies the Stolper-Samuelson theorem:

Lemma 3. dw 1 /dp 1 = â00 â11 â00 -â10 â01
.

The factor market clearing equation finally satisfies the Rybczynski theorem:

Lemma 4. dy 1 /dx 1 = a 00 a 11 a 00 -a 10 a 01 .

Without external effects, i.e. b ij = 0, we have Â(w, p) = A(w, p). The Rybczynski and Stolper-Samuelson theorems are equivalent since [∂y/∂x] = [∂w/∂p]. However, in presence of externalities, the Rybczynski effects depend on the input coefficients from the private perspective while the Stolper-Samuelson effects depend on the quasi input coefficients from the social perspective. The duality between these two effects is thus destroyed. Local indeterminacy of equilibria will be a consequence of this property.

Intertemporal equilibrium and steady state

A representative agent optimizes a linear additively separable utility function with discount rate δ ≥ 0. This problem can be described as: max {x ij (t)} +∞ 0 β 00 x 00 (t) -ρ 0 + β 10 x 10 (t) -ρ 0 + e 0 (X 00 (t), X 10 (t))

-1
ρ 0 e -δt dt s.t. y 1 (t) = β 01 x 01 (t) -ρ 1 + β 11 x 11 (t) -ρ 1 + e 1 (X 01 (t), X 11 (t))

-1 ρ 1 ẋ1 (t) = y 1 (t) -gx 1 (t) 1 = x 00 (t) + x 01 (t)
x 1 (t) = x 10 (t) + x 11 (t)

x 1 (0) and {e j (X 0j (t), X 1j (t))} t≥0 , j = 0, 1, given where g > 0 is the depreciation rate of the capital stock. We can write the modified Hamiltonian in current value as: H = β 00 x 00 (t) -ρ 0 + β 10 x 10 (t) -ρ 0 + e 0 (X 00 (t), X 10 (t))

-1 ρ 0 + w 0 (t) (1 -x 00 (t) -x 01 (t)) + w 1 (t) (x 1 (t) -x 10 (t) -x 11 (t)) + p 1 (t) β 01 x 01 (t) -ρ 1 + β 11 x 11 (t) -ρ 1 + e 1 (X 01 (t), X 11 (t)) -1 ρ 1 -gx 1 (t)
The static first order conditions are given by equations (3). The necessary conditions which describe the solution to the optimization problem are given by the following equations of motion:

ẋ1 (t) = y 1 (x 1 (t), p 1 (t)) -gx 1 (t) ṗ1 (t) = (δ + g)p 1 (t) -w 1 (p 1 (t)) (7)
Any solution {x 1 (t), p 1 (t)} t≥0 that also satisfies the transversality condition lim t→+∞ e -δt p 1 (t)x 1 (t) = 0 is called an equilibrium path.

A steady state is defined by a pair (x * 1 , p * 1 ) solution of

y 1 (x 1 , p 1 ) = gx 1 w 1 (p 1 ) = (δ + g)p 1 (8)
We introduce the following restriction on parameters' values which will ensure the existence of an unique non trivial steady state:

Assumption 1. β 11 > δ + g and ρ 1 ∈ (ρ 1 , +∞) with ρ1 ≡ ln β11 ln " β 11 δ+g " -ln β11 ∈ (-1, 0) (9)
Considering the fact that, within continuous-time models, the discount rate δ and the capital depreciation rate g are quite small, the restriction β 11 > δ +g does not appear to be too demanding. Assumption 1 precisely guarantees positiveness and interiority of all the steady state values for input demand functions x ij . Moreover it allows to prove the following result:

Proposition 1. Under Assumption 1, there exists a unique steady state

(x * 1 , p * 1 ) > 0 such that x * 1 = " β 10 β 01 β 00 β 11 " 1 1+ρ 0 " β 11 δ+g « ρ 1 1+ρ 1 -β11 β01 1+ρ 1 ρ 1 (1+ρ 0 ) 1- " β 11 δ+g " 1 1+ρ 1 g 2 6 6 6 4 1- " β 10 β 01 β 00 β 11 " 1 1+ρ 0 " β 11 δ+g « ρ 1 1+ρ 1 -β11 β01 ρ 1 -ρ 0 ρ 1 (1+ρ 0 ) 3 7 7 7 5 p * 1 = β 10 δ+g    β00 β 10 β 01 β 00 β 11 ρ 0 1+ρ 0 " β 11 δ+g " ρ 1 1+ρ 1 -β11 β01 ρ 0 (1+ρ 1 ) ρ 1 (1+ρ 0 ) + β10    - 1+ρ 0 ρ 0

Main results

We start by linearizing the dynamical system (7) around (x * 1 , p * 1 ):

J =   ∂y 1 ∂x 1 (x * 1 , p * 1 ) -g ∂y 1 ∂p 1 (x * 1 , p * 1 ) 0 -∂w 1 ∂p 1 (p * 1 ) + δ + g  
Any solution from (7) that converges to the steady state (x * 1 , p * 1 ) satisfies the transversality condition and is an equilibrium. Therefore, given x 1 (0), if there is more than one initial price p 1 (0) in the stable manifold of (x * 1 , p * 1 ), the equilibrium path from x 1 (0) will not be unique. In particular, if J has two roots with negative real parts, there will be a continuum of converging paths and thus a continuum of equilibria. Definition 1. If the locally stable manifold of the steady state

(x * 1 , p * 1 ) is two-dimensional, then (x * 1 , p * 1
) is said to be locally indeterminate.

The roots of J are given by the diagonal terms. We know from Lemmas 3 and 4 that ∂y 1 /∂x 1 corresponds to the factor intensity difference from the private viewpoint and ∂w 1 /∂p 1 corresponds to the quasi factor intensity difference from the social viewpoint. Using the definitions of input coefficients given in Section 2, we may indeed interpret the elements of ∂y 1 /∂x 1 and ∂w 1 /∂p 1 as follows:

Definition 2. The consumption good is said to be: i) capital intensive at the private level if and only if a 11 a 00 -a 10 a 01 < 0, ii) quasi capital intensive at the social level if and only if â11 â00 -â10 â01 < 0, iii) capital intensive at the social level if and only if ā11 ā00 -ā10 ā01 < 0.

We may thus relate the input coefficients to the CES parameters:

Proposition 2. Let Assumption 1 hold. At the steady state: i) the consumption good is capital (labor) intensive from the private perspective if and only if

b ≡ 1 -β 10 β 01 β 00 β 11 1 1+ρ 0 " β 11 δ+g " ρ 1 1+ρ 1 -β11 β01 ρ 1 -ρ 0 ρ 1 (1+ρ 0 ) < (>)0
ii) the consumption good is quasi capital (labor) intensive from the social perspective if and only if b ≡ 1 -β10 β01 β00 β11

β 00 β 11 β 10 β 01 ρ 0 1+ρ 0 " β 11 δ+g " ρ 1 1+ρ 1 -β11 β01 ρ 1 -ρ 0 ρ 1 (1+ρ 0 ) < (>)0
iii) the consumption good is capital (labor) intensive from the social perspective if and only if b ≡ 1 -β10 β01 β00 β11

1 1+ρ 0 " β11 δ+g « ρ 1 1+ρ 1 -β11 β01 ρ 1 -ρ 0 ρ 1 (1+ρ 0 ) < (>)0
The following Proposition establishes that local indeterminacy requires a capital intensity reversal from the private input coefficients to the quasi input coefficients. Proposition 3. Let Assumption 1 hold. The steady state is locally indeterminate if and only if the consumption good is capital intensive from the private perspective, but quasi labor intensive from the social perspective.

To get indeterminacy in a framework with constant returns to scale at the social level, we need a mechanism that nullify the duality between the Rybczynski and Stolper-Samuelson effects. As shown in Section 2.1, the Rybczynski effect is given by the input coefficients from the private perspective while the Stolper-Samuelson effect is given by the quasi input coefficients from the social perspective. In the presence of external effects, the duality between these coefficients is broken and local indeterminacy may appear.

This mechanism is very similar to the one exhibited in the contribution of Benhabib and Nishimura [2]. However in the current paper, it follows that depending on the value of the elasticities of capital-labor substitution, the capital intensity reversal from the private input coefficients to the quasi input coefficients does not necessarily requires a capital intensity reversal from the private to the social level.

Propositions 2 and 3 show that the stability properties of the steady state depend, among all the parameters, on whether the ratios β 10 β 01 /β 11 β 00 and β10 β01 / β11 β00 are lower or greater than 1. Around the steady state, it can be easily shown that if the elasticities of capital-labor substitution are identical across sectors, the consumption good is capital intensive at the private level if and only if β 10 β 01 /β 11 β 00 > 1 while it is capital intensive at the social level if and only if β10 β01 / β11 β00 > 1. With asymmetric elasticities, as it is shown in Proposition 2, the capital intensity differences between sectors also depend on the parameters ρ 0 and ρ 1 .

Notice also from Proposition 2 that the capital intensity differences at the private and quasi social level are linked as follows: b = 1 -β10 β01 β11 β00

β 10 β 01 β 11 β 00 (1 -b)
We know from Proposition 3 that local indeterminacy requires the consumption good to be capital intensive at the private level, i.e. b < 0, but quasi labor intensive at the social level, i.e. b > 0. It follows from the above expression that when b < 0, a necessary condition for b to be positive is given by the following Assumption:

Assumption 2. β10 β01 β11 β00 < β 10 β 01 β 11 β 00
Let us first consider the configuration β 10 β 01 /β 11 β 00 < 1 which is known in the case with symmetric elasticities of capital-labor substitution to imply local determinacy of the steady state. [START_REF] Benhabib | Indeterminacy and Sunspots with Constant Returns[END_REF] The following Proposition shows on the contrary that as soon as the elasticities are sufficiently asymmetric, there is room for local indeterminacy.

Then there are ρ 0 ∈ (-1, 0), ρ0 > 0 such that the following results hold: i) for any given ρ 0 ∈ (ρ 0 , ρ0 ), there exist ρ 1 > max{0, ρ 0 } and ρ1 > ρ 1 such that the steady state is locally indeterminate when ρ 1 ∈ (ρ 1 , ρ1 );

ii) for any given ρ 0 > ρ0 , there exists ρ 1 > ρ 0 such that the steady state is locally indeterminate when ρ 1 > ρ 1 .

Before giving interpretations of our results, we need to justify the part of condition (10) which concerns the CES coefficients at the social level: this inequality ensures the existence of the bounds ρ0 and ρ 0 , and the possible occurrence of local indeterminacy when both elasticities of capital labor substitution are close to zero.

All the various conditions for the existence of indeterminacy in Proposition 4 are based on the restriction ρ 1 > ρ 0 which implies a lower elasticity of capital-labor substitution in the investment good sector than in the consumption good sector. Since under Assumption 1 the elasticity of capital-labor substitution in the investment good sector is necessarily finite, it clearly appears that the same condition has to be considered for the investment good sector. In cases i), if the elasticity of capital-labor substitution in the consumption good sector is large enough, the extreme configuration of a Leontief technology in the investment good sector is also ruled out. On the contrary, if the factor substitutability of the consumption good sector is low enough, local indeterminacy becomes compatible with a Leontief technology in the investment good sector.

Let us now consider the configuration with β 10 β 01 /β 11 β 00 > 1 and β10 β01 / β11 β00 < 1 which is known to imply the existence of local indeterminacy in the case with Cobb-Douglas technologies in both sectors. [START_REF] Nishimura | Indeterminacy and the Role of Factor Substitutability[END_REF] Notice that Assumption 2 is then necessarily satisfied. Then there are ρ 0 ∈ (-1, 0) and ρ0 > 0 such that the following results hold: i) for any given ρ 0 ∈ (ρ 0 , ρ0 ), there exist ρ 1 ∈ (ρ 1 , 0) and ρ1 > 0 such that the steady state is locally indeterminate when ρ 1 ∈ (ρ 1 , ρ1 );

ii) for any given ρ 0 > ρ0 , there exists ρ 1 ∈ (0, ρ 0 ) such that the steady state is locally indeterminate when ρ 1 > ρ 1 .

Proposition 5 covers the formulation with symmetric elasticities of capitallabor substitution across sectors previously analyzed in Benhabib and Nishimura [2] with Cobb-Douglas technologies and Nishimura and Venditti [3] with CES technologies having symmetric elasticities of capital-labor substitution. The part of condition (11) which concerns the CES coefficients at the social level is introduced to ensure the possibility of local indeterminacy when the consumption good technology is Cobb-Douglas while the investment good technology is Leontief (see case ii)). This explains why such a condition does not occur under the assumption of symmetric elasticities. However, since the plausible values of δ and g are close to zero, this condition does not imply a strong restriction on the CES coefficients.

Contrary to the previous Proposition, the existence of local indeterminacy does not require the elasticity of capital-labor substitution in the consumption good sector to be larger than the one in the investment good sector. The extreme configuration of an infinite factor substitutability in the consumption good sector is again ruled out. Moreover, as shown in case ii), the occurrence of local indeterminacy with factors complementarity in the investment good sector is not compatible with a Cobb-Douglas technology in the consumption good sector and requires an elasticity of capital-labor substitution significantly lower than 1.

Let us finally consider the configuration with β10 β01 / β11 β00 > 1 which is known in the case with symmetric elasticities of capital-labor substitution to be compatible with the existence of local indeterminacy only when the common elasticity is lower than one, or ρ 0 = ρ 1 = ρ > 0.4 In the next Proposition, we show that in the asymmetric case, ρ 0 has to be positive but under Assumption 1, ρ 1 can be negative, equal to 1 or positive. Proposition 6. Under Assumptions 1-2, let

β 10 β 01 β 11 β 00 > 1 and β10 β01 β11 β00 > 1 (12)
Then there are ρ 0 > 0 and ρ0 > ρ 0 such that the following results hold: i) for any given ρ 0 ∈ (ρ 0 , ρ0 ), there exist ρ 1 ∈ (ρ 1 , ρ 0 ) and ρ1 > 0 such that the steady state is locally indeterminate when ρ 1 ∈ (ρ 1 , ρ1 ); ii) for any given ρ 0 > ρ0 , there exists ρ 1 > 0 such that the steady state is locally indeterminate when ρ 1 > ρ 1 .

Proposition 6 confirms only part of the conclusions obtained by Nishimura and Venditti [3] under symmetric elasticities of capital-labor substitution across sectors: local indeterminacy requires some elasticity lower than one but only in the consumption good sector. Indeed, the existence of multiple equilibria does not rely on particular restrictions for the factor substitutability in the investment good sector. In particular, some elasticities greater than 1 are compatible with local indeterminacy in case i). However, as in the previous Propositions, a Leontief technology for the investment good can be reached only if the elasticity in the consumption good sector is low enough.

Concluding comments

Within the framework of CES technologies, we precisely show how much the elasticities of capital-labor substitution can differ across the industries while leading to local indeterminacy of equilibria. For instance, given a value for the elasticity of substitution in the consumption good sector, the elasticity of substitution in the investment good sector can be arbitrarily large.

Appendix

Proof of Lemma 1

Substituting (4) into the social production functions (2) gives

y j = β0j y -ρ j j p j β 0j w 0 -ρ j 1+ρ j + β1j y -ρ j j p j β 1j w 1 -ρ j 1+ρ j -1/ρ j It follows that p ρ j 1+ρ j j = β0j β 0j β 0j w 0 1 1+ρ j w 0 + β1j β 1j β 1j w 1 1 1+ρ j w 1 -1/ρ j
Multiplying both sides by p 1/(1+ρ j ) j then gives

p j = â0j w 0 + â1j w 1
The result follows considering that p 0 = 1.

Proof of Lemma 2

By definition x ij = a ij y j and thus

x i = a i0 y 0 + a i1 y 1
The result follows considering that x 0 = 1.

Proof of Lemma 3

The result immediately follows from the fact that the function Â(w, p) is homogeneous of degree zero in w and p.

Proof of Lemma 4

The result follows from a direct differenciation of x i = a i0 y 0 + a i1 y 1 under the assumption that prices are constant.

Proof of Proposition 1

From equation (3) considered at the steady state with y 1 = gx 1 and w 1 = (δ + g)p 1 , we get

x 11 = β 11 δ+g 1 1+ρ 1 gx 1
Using now the social production function (2) for the investment good we derive

x 01 = " β 11 δ+g " ρ 1 1+ρ 1 -β11 β01 -1 ρ 1 β 11 δ+g 1 1+ρ 1 gx 1
and thus

x 01

x 11 = " β 11 δ+g " ρ 1 1+ρ 1 -β11 β01 -1 ρ 1 (13)
Finally we easily obtain from (3):

β 10 β 01 β 00 β 11 = x 01
x 11 1+ρ 1 x 10

x 00 1+ρ 0 ⇔ x 10 x 00 = β 10 β 01

β 00 β 11 1 1+ρ 0 " β 11 δ+g " ρ 1 1+ρ 1 -β11 β01 1+ρ 1 ρ 1 (1+ρ 0 ) (14)
Considering ( 13), ( 14) and the fact that x 00 + x 01 = 1, x 1 = x 10 + x 11 , we get the final expression of x * 1 . Equation ( 3) for i = 1 and j = 0 gives

w 1 = β 10 β11 x 10 x 00 ρ 0 + β10 - 1+ρ 0 ρ 0
The final expression of p * 1 is then derived from ( 14) and the fact that w 1 = (δ + g)p 1 .

Proof of Proposition 2

i) From (4) we derive a 11 a 00 -a 10 a 01 = a 00 a 01 a 11 a 01 -a 10 a 00 = a 00 a 01

x 11

x 01 1 -x 10 x 01

x 00 x 11 and the result follows after substitution of (13) and ( 14) into the previous expression.

ii) From (6) we derive â11 â00 -â10 â01 = â00 â01 x 11

x 01 1 -β10 β01 β00 β11

β 00 β 11 β 10 β 01

x 10 x 01 x 00 x 11 and the result follows after substitution of (13) and ( 14) into the previous expression.

iii) The result is derived as in i) but considering instead the input coefficients at the social level (5).

Proof of Proposition 3

From Lemma 2, x 1 = a 10 y 0 +a 11 y 1 . Moreover, at the steady state, y 1 = gx 1 , and it follows

a 10 y 0 + ga 11 x 1 = x 1 ⇔ a 10 y 0 = [1 -ga 11 ]x 1 > 0 Therefore ∂y 1 ∂x 1 -g = a 00
a 11 a 00 -a 10 a 01 -g = a 00 [1-ga 11 ]+a 10 a 01 g a 11 a 00 -a 10 a 01 < 0 if and only if a 11 a 00 -a 10 a 01 < 0. From Lemma 1, p 1 = â01 w 0 + â11 w 1 . Moreover, at the steady state, (δ + g)p 1 = w 1 , and it follows

(δ + g)â 01 w 0 + (δ + g)â 11 w 1 = w 1 ⇔ (δ + g)â 01 w 0 = [1 -(δ + g)â 11 ]w 1 > 0 Therefore δ + g -∂w 1 ∂p 1 = δ + g - â00 â11 â00 -â 10 â01 = -[1-(δ+g)â 11 ]â 00 +(δ+g)â 10 â01 â11 â00 -â 10 â01
< 0 if and only if â11 â00 -â10 â01 > 0.

Proof of Proposition 4

Proposition 3 shows that local indeterminacy occurs if and only if b < 0 and b > 0. Under Assumption 1, β 11 > δ + g and ρ 1 ∈ (ρ 1 , +∞), while ρ 0 > -1 without particular restriction. We will then study the sign of b and b for different values of ρ 0 and ρ 1 over these intervals. Our strategy consists in considering a fixed value for ρ 0 and varying ρ 1 in order to find intervals of values in which local indeterminacy occurs. To simplify the analysis, we have to impose some restrictions on the parameters ρ 0 and ρ 1 such that the following function

g(ρ 1 ) = " β 11 δ+g " ρ 1 1+ρ 1 -β11 β01 ρ 1 -ρ 0 ρ 1 (1+ρ 0 ) (15)
is monotone increasing.

Lemma 5. Under Assumption 1, there is a ρ0 ∈ (-1, 0) such that for any given ρ 0 > ρ0 there exists ρ1 ∈ [ρ 1 , 0) such that g(ρ 1 ) is a monotone increasing function for all ρ 1 > ρ1 .

Proof : Notice first that

∂ ∂ρ 1 ρ 1 1+ρ 1 = 1 (1+ρ 1 ) 2 > 0, ∂ ∂ρ 1 ρ 1 -ρ 0 ρ 1 (1+ρ 0 ) = ρ 0 ρ 2 1 (1+ρ 0 )
It follows that for any given ρ 0 ≥ 0, g(ρ 1 ) is a monotone increasing function when ρ 1 ≥ 0. Consider now the case ρ 0 ∈ (-1, 0). We can write g(ρ 1 ) as follows:

g(ρ 1 ) = exp ρ 1 -ρ 0 ρ 1 (1+ρ 0 ) ln exp h ρ 1 1+ρ 1 ln " β 11 δ+g "i -β11 β01 ≡ exp{f (ρ 1 )} If f (ρ 1
) is monotone increasing, so in g(ρ 1 ). We easily compute

f (ρ 1 ) = ρ 0 (1+ρ 1 ) 2 " β 11 δ+g " ρ 1 1+ρ 1 -β11 ln " β 11 δ+g « ρ 1 1+ρ 1 -β11 β01 +ρ 1 (ρ 1 -ρ 0 ) " β 11 δ+g " ρ 1 1+ρ 1 ln " β 11 δ+g " ρ 2 1 (1+ρ 1 ) 2 (1+ρ 0 ) " β 11 δ+g " ρ 1 1+ρ 1 -β11
We have now to compute the sign of lim ρ 1 →0 f (ρ 1 ). A Taylor expansion of order two allows to show that lim ρ 1 →0 f (ρ 1 ) > 0 for any ρ 0 > ρ0 with

ρ0 = -1 + ln β 11 δ+g β11 2 β01 -1 ∈ (-1, ρ1 ) Notice finally that lim ρ 1 →ρ 1 f (ρ 1 ) = -∞ and lim ρ 1 →ρ 1 f (ρ 1 ) = +∞ if ρ 0 < ρ1 , while lim ρ 1 →ρ 1 f (ρ 1 ) = +∞ and lim ρ 1 →ρ 1 f (ρ 1 ) = -∞ if ρ 0 > ρ1 .
We then conclude from all this that for any given ρ 0 > ρ0 , there exists ρ1 ∈ [ρ 1 , 0) such that g(ρ 1 ) is a monotone increasing function for all ρ 1 > ρ1 . then lim ρ 1 →+∞ b| ρ 0 →+∞ = 0 -and lim ρ 1 →+∞ b| ρ 0 ≤0 < 0. It follows that under (19), lim ρ 1 →+∞ b < 0 for any ρ 0 > -1. Notice finally that if ρ 0 = ρ 1 we have b > 0. Considering Lemma 5 under (19), we then conclude that for any given ρ 0 > ρ 1 0 , there exists ρ 1 1 > max{0, ρ 0 } such that b < 0 for all ρ 1 > ρ [START_REF] Benhabib | Indeterminacy and Sunspots in Macroeconomics[END_REF] 1 . Consider now the quasi capital intensity difference at the social level b. Equation ( 16

Notice that the right-hand-side of (20) is a monotone function of ρ 0 . Under Assumption 2 and β10 β01 / β11 β00 < 1, we derive lim ρ 1 →0 b| ρ 0 ≥0 > 0 so that there exists ρ 2 0 ∈ [-1, 0) such that lim ρ 1 →0 b > 0 for any ρ 0 > ρ 2 0 . Since b > b, we get when ρ 0 = ρ 1 0 , b| ρ 1 =0 = 0 while b| ρ 1 =0 > 0 so that ρ 2 0 < ρ 1 0 . Similarly, we have lim we get with lim ρ 1 →+∞ b| ρ 0 =0 < 0 and lim ρ 1 →+∞ b| ρ 0 →+∞ > 0. Therefore, under (22), there exists ρ 3 0 > 0 such that lim ρ 1 →+∞ b > 0 for any ρ 0 > ρ 3 0 . Considering Lemma 5 under (22), we then conclude the following results:

-for any given ρ 0 ∈ (ρ 2 0 , ρ 3 0 ), there exists ρ 2 1 > 0 such that b > 0 for all ρ 1 ∈ [0, ρ 2 1 ). Notice that since b > b, ρ 2 1 > ρ 1 1 ; -for any given ρ 0 > ρ 3 0 , b > 0 for all ρ 1 > ρ 0 .

Notice that under Assumption 2, (22) implies (19). Therefore, under (22), we conclude that local indeterminacy occurs in the following cases: i) for any given ρ 0 ∈ (max{ρ 1 0 , ρ 2 0 }, ρ 3 0 ) and ρ 1 ∈ (ρ 1 , ρ1 ) with ρ 1 = ρ 1 1 > max{0, ρ 0 } and ρ1 = ρ 2 1 > ρ 1 ; ii) for any given ρ 0 > ρ 3 0 and ρ 1 > ρ 1 with ρ 1 = ρ 2 1 > ρ 0 .

Proof of Proposition 5

We use the same kind of arguments as in Proposition 4. Consider first the capital intensity difference at the private level b. Under Assumption 2 and β 10 β 01 /β 11 β 00 > 1, we derive from (17) that lim ρ 1 →0 b| ρ 0 ≤0 < 0 while lim ρ 1 →0 b| ρ→+∞ > 0. Therefore, there exists ρ 1 0 > 0 such that lim ρ 1 →0 b < 0 for any ρ 0 ∈ (-1, ρ 1 0 ). Similarly, we derive from (18) that lim ρ 1 →+∞ b < 0 for any ρ 0 > -1. Notice also that b|

ρ 1 =ρ 0 = 1 -β 10 β 01 β 00 β 11 1 1+ρ 0 < 0 (23)
Considering Lemma 5, we then conclude the following results: a) for any given ρ 0 ∈ (-1, ρ 1 0 ), b < 0 for all ρ 1 ≥ 0; b) for any given ρ 0 > ρ 1 0 , there exists ρ 1 1 ∈ (0, ρ 0 ) such that b < 0 for all ρ 1 > ρ [START_REF] Benhabib | Indeterminacy and Sunspots in Macroeconomics[END_REF] 1 . Consider now the quasi capital intensity difference at the social level b. Under Assumption 2 and β10 β01 / β11 β00 < 1, we derive from (20) that lim ρ 1 →0 b| ρ 0 ≥0 > 0 while lim ρ 1 →0 b| ρ 0 →-1 < 0. Therefore, there exists ρ 2 0 ∈ (-1, 0) such that lim ρ 1 →0 b > 0 for any ρ 0 > ρ 2 0 . Similarly, we derive from (21) that lim ρ 1 →+∞ b| ρ 0 =0 < 0 if (22) holds, while lim ρ 1 →+∞ b| ρ 0 →+∞ > 0. Therefore, under (22), there exists ρ 3 0 > 0 such that lim ρ 1 →+∞ b > 0 for any ρ 0 > ρ 3 0 . Notice also that b| ρ 1 =ρ 0 = 1 -β10 β01 β11 β00 Therefore b| ρ 1 =ρ 0 > 0 for any ρ 0 ≥ ρ 2 0 . Considering Lemma 5 under (22), we then conclude the following results: c) for any given ρ 0 ∈ (ρ 2 0 , ρ 3 0 ), there exists ρ 2 1 > max{0, ρ 0 } such that b > 0 for all ρ 1 ∈ [0, ρ 2 1 ); d) for any given ρ 0 > ρ 3 0 , b > 0 for all ρ 1 ≥ 0. Therefore, under (22), we conclude from points a)-e) that local indeterminacy occurs in the following cases: i) for any given ρ 0 ∈ (ρ 2 0 , max{ρ 1 0 , ρ 3 0 }) and ρ 1 ∈ (ρ 1 , ρ1 ) with ρ 1 ∈ (ρ 1 , 0) and ρ1 = ρ 2 1 > max{0, ρ 0 }; ii) for any given ρ 0 > max{ρ 1 0 , ρ 3 0 } and ρ 1 > ρ 1 with ρ 1 = ρ 1 1 > 0.

β

Proof of Proposition 6

We use the same kind of arguments as in Proposition 5. Consider first the capital intensity difference at the private level b. Under Assumption 2 and β 10 β 01 /β 11 β 00 > 1, we have already shown in the proof of Proposition 5 that there exists ρ 1 0 > 0 such that: a) for any given ρ 0 ∈ (-1, ρ 1 0 ), b < 0 for all ρ 1 ≥ 0; b) for any given ρ 0 > ρ 1 0 , there exists ρ 1 1 ∈ (0, ρ 0 ) such that b < 0 for all ρ 1 > ρ [START_REF] Benhabib | Indeterminacy and Sunspots in Macroeconomics[END_REF] 1 . Consider now the quasi capital intensity difference at the social level b. Under Assumption 2 and β10 β01 / β11 β00 > 1, we derive from (20) that lim ρ 1 →0 b| ρ 0 =0 < 0 while lim ρ 1 →0 b| ρ 0 →+∞ > 0. Therefore, there exists ρ 2 0 > 0 such that lim ρ 1 →0 b > 0 for any ρ 0 > ρ 2 0 . Since b > b, we get ρ 2 0 < ρ 1 0 . Similarly, we derive from (21) that lim ρ 1 →+∞ b| ρ 0 =0 < 0 while lim ρ 1 →+∞ b| ρ 0 →+∞ > 0. Therefore, there exists ρ 3 0 > 0 such that lim ρ 1 →+∞ b > 0 for any ρ 0 > ρ 3 0 . Now notice from (20) and (21) the following property: 24) and (26). Since ρ 2 0 > 0 is such that lim ρ 1 →0 b| ρ 0 =ρ 2 0 = 0, we derive from (20): b| ρ 1 =ρ 0 =ρ 2 0 < 0. On the con-

Lemma 1 .

 1 Denote p = (1, p 1 ) , w = (w 0 , w 1 ) and Â(w, p) = [â ij (w i , p j )]. Then p = Â (w, p)w.The factor market clearing equation depends on the input coefficients from the private perspective. Lemma 2. Denote x = (1, x 1 ) , y = (y 0 , y 1 ) and A(w, p) = [a ij (w i , p j )]. Then A(w, p)y = x.

Proposition 4 .

 4 Under Assumptions 1-2, let 1 > β 10 β 01 β 11 β 00 and 1 >

Proposition 5 .

 5 Under Assumption 1, let β 10 β 01 β 11 β 00 > 1 and 1 >

1 →0b = 1 -β 10 β 01 β

 1101 It clearly appears from Proposition 2 that under the conditions of Lemma 5, b and b are monotone decreasing function of ρ 1 . We may now prove Proposition 4: In the first part of the proof we consider positive values of ρ 1 only. Notice that L'Hôpital's rule gives: lim ρ 1 →0 g(ρ 1 ) = β 11 δ+g -ρ 0 (1+ρ 0 ) β01 (16) Consider the capital intensity difference at the private level b. The previous result implies lim ρ right-hand-side of (17) is a monotone function of ρ 0 . Under Assumption 2 and β 10 β 01 /β 11 β 00 < 1, we derive lim ρ 1 →0 b| ρ 0 ≥0 > 0. It follows that there exists ρ 1 0 ∈ [-1, 0) such that lim ρ 1 →0 b > 0 for any ρ 0 > ρ the right-hand-side of (18) is a monotone function of ρ 0 . If the following condition holds β 10 β 01 β 00 β 11

  again that the right-hand-side of (21) is a monotone function of ρ 0 . If the following condition holds

b| ρ 1 =ρ 0 =ρ 2 0= 1

 21 function of ρ 0 since β 00 β 11 /β 10 β 01 < 1. Considering that ρ 2 0 ∈ (-1, 0) is defined as lim ρ 1 →0 b| ρ 0 =ρ 2 0 = 0, we derive from (20): -β 11

lim ρ 1 →+∞ b| ρ 0 =ρ 2 0

 12

See Benhabib and Farmer[START_REF] Benhabib | Indeterminacy and Sunspots in Macroeconomics[END_REF].

See Benhabib and Nishimura[START_REF] Benhabib | Indeterminacy and Sunspots with Constant Returns[END_REF] for Cobb-Douglas economies and Nishimura and Venditti[START_REF] Nishimura | Indeterminacy and the Role of Factor Substitutability[END_REF] for CES economies with symmetric elasticities of capital-labor substitution.

See Benhabib and Nishimura[START_REF] Benhabib | Indeterminacy and Sunspots with Constant Returns[END_REF].

See Nishimura and Venditti[START_REF] Nishimura | Indeterminacy and the Role of Factor Substitutability[END_REF].

trary, since ρ 3 0 > 0 is such that lim ρ 1 →+∞ b| ρ 0 =ρ 3 0 = 0, we derive from (21):

Therefore we get b| ρ 1 =ρ 0 > 0 for any ρ 0 ≥ ρ 3 0 . Considering Lemma 5, we then conclude the following results: c) for any given ρ 0 ∈ (ρ 2 0 , ρ 3 0 ), there exists ρ 2 1 ∈ (0, ρ 0 ] such that b > 0 for all ρ 1 ∈ [0, ρ 2 1 ); d) for any given ρ 0 > ρ 3 0 , b > 0 for all ρ 1 ≥ 0. Notice that since b > b, we get ρ 2 1 > ρ 1 1 . Therefore, we conclude from points a)-d) that local indeterminacy occurs in the following cases: i) for any given ρ 0 ∈ (ρ 2 0 , max{ρ 1 0 , ρ 3 0 }) and ρ 1 ∈ (ρ 1 , ρ1 ) with ρ 1 ∈ (ρ 1 , ρ 0 ) and ρ1 = ρ 2 1 ∈ (0, ρ 0 ]; ii) for any given ρ 0 > max{ρ 1 0 , ρ 3 0 } and ρ 1 > ρ 1 with ρ 1 = ρ 1 1 ∈ (0, ρ 0 ).


Amir Shahdin
email: amir.shahdin@isae.fr

Laurent Mezeix
email: laurent.mezeix@ensiacet.fr

Joseph Morlier
email: joseph.morlier@isae.fr

Yves Gourinat
email: yves.gourinat@isae.fr

Christophe Bouvet
email: bouvet@lgmt.ups-tlse.fr

Diagnosis of the Fabrication Process of a Sandwich Structure with Fiber Network Core by Vibration Testing

The aim of this paper is the fabrication and vibration testing of a sandwich structure including a new core material known as fibre network sandwich material. As fabrication norms for such a material do not exist as such, so the primary goal is to reproduce successfully fibre network sandwich specimens. Enhanced vibration testing diagnoses the quality of the fabrication process. These sandwich materials possess high damping values, making them suitable for vibro-acoustic applications where structural strength is of secondary importance e.g., internal panelling of a helicopter.

Introduction

The vibration characteristics of sandwich materials have drawn much attention recently. The dynamic parameters of a structure i.e. natural frequency, damping and mode shapes are determined with the help of vibration testing which provides the basis for rapid inexpensive dynamic characterization of composite structures [START_REF] Gibson | Modal Vibration Response Measurements for Characterization of Composite Materials and Structures[END_REF]. D.J Ewins gave a detailed overview of the vibration based methods [START_REF] Ewins | Modal Testing Theory and Practice[END_REF]. A wide amount of literature is present related to vibration testing of composite sandwich structures [START_REF] Hwu | Vibration Suppression of Composite Sandwich Beams[END_REF][START_REF] Hu | Free Vibration of Delaminated Composite Sandwich Beams[END_REF][START_REF] Nilsson | Prediction and Measurement of some Dynamic Properties of Sandwich Structures with Honeycomb and Foam Cores[END_REF][START_REF] Renji | Experimental Modal Densities of Honeycomb Sandwich Panels at High Frequencies[END_REF][START_REF] Renji | Loss Factors of Composite Honeycomb Sandwich Panels[END_REF]. The equations that explain the dynamic behavior of sandwich beams are described extensively in the literature and notably in the references [START_REF] Yan | Governing Equations for Vibrating Constrained-Layer Damping Sandwich Plates and Beams[END_REF][START_REF] Mead | A Comparison of some Equations for the Flexural Vibration of Damped Sandwich Beams[END_REF].

The importance of material damping in the design process has increased in recent years as the control of noise and vibration in high precision, high performance structures and machines has become more of a concern. At the same time, composites sandwich researchers have focused more attention on damping as a design variable [START_REF] Wang | Damping of Honeycomb Sandwich Beams[END_REF][START_REF] Li | Effects of thickness and Delamination on the Damping in Honeycomb-Foam Sandwich Beams[END_REF]. One way of increasing damping in sandwich materials is by putting a viscoelastic layer as core between the two laminates [START_REF] Hao | Vibration and Damping Analysis of a Sandwich Beam Containing a Viscoelastic Constraining Layer[END_REF][START_REF] Rao | Dynamic Analysis and Design of Laminated Composite Beams with Multiple Damping Layers[END_REF]. Jueng and Aref reported that sandwich structures with combined honeycomb-foam cores have higher damping than those with individual honeycomb or solid viscoelastic cores [START_REF] Jung | A Combined Honeycomb and Solid Viscoelastic Material for Structural Damping Application[END_REF].

These advancements have led to the need for developing materials possessing better damping characteristics. New materials like fiber network materials can be used as potential dampers and sound absorbers in specific applications like the inner paneling of a helicopter, where structural strength is not the primary requirement. Fiber network materials are made from natural materials (wool, cotton etc) as well as artificial ones (carbon, steel, glass etc) and are quickly becoming of widespread use as sound absorbers [START_REF] Garai | A Simple Empirical Model of Polyester Fiber Materials for Acoustic Applications[END_REF]. Bonded metal fibers network materials offer advantages for use as heat exchanger [START_REF] Golosnoy | Optimisation of Metallic Fiber Network Materials for Compact Heat Exchangers[END_REF] or insulation [START_REF] Zhang | Experimental and Theoretical Studies on High-Temperature Thermal Properties of Fibrous Insulation[END_REF]. These materials possess low relative density, high porosity and are cost-effective. Sandwich structures normally consist of two thin skins separated by a thick core. Traditional core materials are usually honeycomb, foam or balsa. Recently, a novel type of sandwich has been developed with bonded metallic fibers as core material [START_REF] Markaki | Mechanics of Thin Ultra-Light Stainless Steel Sandwich Sheet Material, Part I. Stiffness[END_REF][START_REF] Dean | Energy absorption during projectile perforation of lightweight panels with metallic fibre cores[END_REF][START_REF] Zhou | Mechanical Properties of Fibrous Core Sandwich Panels[END_REF]. This material presents attractive combination of properties like high specific stiffness, good damping capacity and energy absorption. Fiber network materials with carbon fibers have also been studied for using as core material [START_REF] Mezeix | A New Sandwich Structured Composite with Entangled Carbon Fibers as Core Material[END_REF]. Entangled cross-linked carbon fibers present many advantages as core materials i.e., Diagnosis of the fabrication process of a sandwich structure with fiber network core by vibration testing: Amir Shahdin open porosity, multifunctional material or the possibility to reeve electric or control cables on core material. In addition, L. Mezeix et al. studied the mechanical behavior of fiber network materials in compression [START_REF] Mezeix | Experimental Data and Modelling of Entangled Fibers During Compression[END_REF]. Mechanical testing has also been carried out on specimens made of wood fibers [START_REF] Castéra | Comportement Physico-mécanique des Matériaux Fibreux Cellulosiques considéré comme des Milieux Aléatoires[END_REF], glass fibers [START_REF] Baudequin | Identification des Mécanismes Physiques mis en jeu lors de la reprise d'épaisseur de la Laine de Verre[END_REF] and various matted fibers [START_REF] Poquillon | Experimental Data about Mechanical Behavior during Compression Tests for Various Matted Fibres[END_REF]. Although few works can be found in the scientific literature devoted to the mechanical testing of sandwich fiber network materials, and in fact no scientific literature can be found related to the vibration testing of sandwich fiber network materials or even simple fiber network materials.

Therefore, the main motivation of this work is to develop a process in order to fabricate sandwich fiber network specimens. Whereas, vibration testing is used to diagnose the quality of the fabrication process and to verify the potential damping capabilities of these sandwich specimens.

Fabrication Process

The core of the sandwich test specimens used in this article consists of carbon fibers that are made of a yarn of standard carbon filaments. For the cross-linking of carbon fibers epoxy resin is used. The skin materials are made of unidirectional pre-impregnated plies of type T700/M21. The upper and lower skins consist of four plies each with a lay-up of [0/90/90/0]. For the test specimens used in this article, approximately 900 g of fibers of 10mm length are cut manually. The carbon fibers are then treated in a solution of dichloromethane for 24 hours and are then cleaned for 2 hours in methanol. These uncoated carbons are then separated by a blow of compressed air. The mixture of resin and hardener is then sprayed on the separated carbon fibers by a spray paint gun. The fibers vaporized by the resin are then placed in the mold between the two skins of unidirectional composites. In order to polymerize the fiber network sandwich specimens, the mold is then heated in an autoclave up to 180°C for two hours. The dimensions of the vibration test specimens are 480x50x11 mm. The thickness of the fiber network core is 10 mm and that of each skin is 0.5 mm.

The fiber density of the fiber network core is chosen to be 150 kg/m 3 . First a small mold (510x65x11 mm) is used to fabricate a single vibration test specimen with a fiber core density of 150 kg/m 3 . Then a larger mold (510x250x11 mm) is used to produce four vibration test specimens at one time. In the first instance, four identical specimens are produced, in view to have the same composition as that of the single specimen fabricated from small mold. These four specimens shall be referred to as heavy specimens in this article. In the next step, four lighter specimens are fabricated having 20 g less resin than the heavier ones, keeping the same fiber core density of 150 kg/m 3 . The composition of these three types of test specimens is listed in Table 1.

Vibration Testing

The experimental equipment used to obtain the vibration test results discussed in this paper is shown in Figure 1. The experimental set-up is that of a free-free beam excited at its center, based on Oberst method [START_REF] Wojtowicki | New Approach for the Measurements of Damping Properties of Materials using Oberst Beam[END_REF]. The Oberst method states that a free-free beam excited at its center has the same dynamical behavior as that of a half length cantilever beam. The test specimen is placed at its center on a B&K force sensor (type 8200) which is then assembled on a shaker supplied by Prodera having a maximum force of 100 N. A fixation system is used to place the test specimens on the force sensor. The fixation is glued to the test specimens with a HBM X60 rapid adhesive. The response displacements are measured with the help of a non-contact and high precision Laser Vibrometer The center of the test specimens is excited at Point 17 as shown in Figure 2. Burst random excitation is used which is a broadband type excitation signal (0-2650 Hz). The signal is averaged 10 times for each measurement point. Hanning windows are used for both the output and the input signals. The linearity is checked and a high frequency resolution (∆f = 0.25Hz) for precise modal parameter estimation is used. Response is measured at 33 points that are symmetrically spaced in three rows along the length of the beam. The modal parameters are extracted by a frequency domain parameter estimation method (Polymax) integrated in the data acquisition system.

Results and Discussion

The aim of the vibration testing is to diagnose the variability of the fabrication process and to extract the global modal parameters i.e., natural frequencies and damping ratios. The modal parameters for the first four bending modes are studied. Firstly, vibration tests are carried out on the vibration test specimen fabricated from the small mold and then on the four light and heavy specimens fabricated from the large mold. The modal parameters extracted from 33 high quality frequency response functions with the help of Polymax algorithm integrated in the LMS data acquisition system, are presented in Table 2. The difference in weight between each of the four heavy and light specimens is due to the fact that with the currently available fabrication technology it is difficult to spray the resin equally on the fibers. This difference in weight is also evident in the natural frequencies of these specimens (Table 2). The heavy specimen 5 and light specimen 6 have comparatively high natural frequencies due to them being relatively heavier in weight than the rest.

Keeping in view the difficulty level of the fabrication process, a major aim of this work is to successfully reproduce multiple sandwich fiber network specimens having the same characteristics. For this reason, the intent is to reproduce four vibration test specimens identical to Specimen 1 (150 kg/m 3 fiber core density and 43 g of epoxy resin) in a large mold. The comparison of the modal parameters of Specimen 1 with those of the heavy specimens (Specimen 2-5) show that large sandwich specimens with fiber network core can be reproduced with a relative efficiency. The natural frequency of the first mode for Specimen 1 is 225 Hz (Table 2), where as for the heavy specimens (Specimens 2-5) the average natural frequency for the first mode is 211 Hz with a standard deviation of 18 Hz. The main reason for this dispersion is the uneven spray of resin on the fibers. So vibration tests have been used here as a diagnostic tool in order to verify the quality of the fabrication process.

On the other hand, the light specimens (Specimens 6-9) possess better damping characteristics because lesser amount of resin increases the friction between the carbon fibers in the Diagnosis of the fabrication process of a sandwich structure with fiber network core by vibration testing: Amir Shahdin sandwich core (weakening the cross-linking between the fibers), which leads to higher values of damping ratios (Table 2). The vibration tests results show that the heavy specimens (Specimens 1-5) being more dense are less dissipative in nature thus possessing lower damping characteristics whereas, the light specimens (Specimens 6-9) having higher damping values. This observation can be further explained by studying the comparison of the sum of frequency response functions (FRF) for Specimen 2 (Heavy) and Specimen 7 (Light) in Figure 3. The sum of the frequency response functions can be compared for all the vibration test specimens in this article, as same number of measurement points (33) have been chosen for all the specimens i.e. symmetricity has been respected. The resonance peaks for the first four bending modes for each specimen are also highlighted in Figure 3. It can be seen in Figure 3, that the resonance peaks of the heavy specimen are more pointed and possess higher amplitude than the light specimen. The resonance peaks of the light specimens are comparatively rounded specially after 800Hz. An interesting global approach is developed next in order to compare the average vibratory level of the specimens. The average vibratory level is calculated by taking the average of the amplitude (in dB) of the sum of the frequency response function for each specimen. The average vibratory level (AVL) is also compared for the two types of specimens, and it can be seen that the light specimens possess on average a 15 dB lower amplitude than the heavy ones.

From Figure 3, it can be inferred that the difference in the natural frequencies between the heavy and light specimens signifies a loss in rigidity, whereas the change in vibratory level signifies a reduction in the amplitude of vibration. The percentage change in frequency and the change in vibrational level between the four heavy and four light vibration test specimens (150kg/m 3 fiber core density) for the first four bending modes are calculated with the help of Equations ( 1) and ( 2) and the resulting values are laid out in Table 3.

Change in frequency

(∆f) = H L H f (k) f (k) f (k) - (1)
where f H (k) is the average frequency for the four heavy specimens (Specimens 2-5) for the k th mode and likewise f L (k) is the average frequency for the four light specimens (Specimens 6-9) for the k th mode

Change in vibratory level

(∆a) = H L a (k) a (k) - (2)
where a H (k) is the average amplitude in dB for the four heavy specimens for the k th mode and a L (k) is the average amplitude in dB for the four light specimens for the k th mode

Conclusion

The aim of this article is to fabricate a sandwich material that possesses high damping characteristics for specific applications like the inner paneling of a helicopter cabin. For this reason, a new type of sandwich material having fiber network material as core and unidirectional carbon preimpregnated plies as skins is fabricated. The advantage of the vibration testing is to diagnose the variability of the fabrication process and to verify experimentally the potentially good damping characteristics of the fiber network sandwich specimens. The vibration tests on these specimens showed that the lighter specimens possess five times more damping and on the average a 15 dB lesser vibratory level than the heavier ones. Despite the complexity of the fabrication process, large fiber network sandwich specimens are relatively successfully reproduced.

Composites20092ndECCOMAS

  Thematic Conference on the Mechanical Response of Composites 1-3 April 2009 Imperial College London, UK 1 Corresponding author Email: amir.shahdin@isae.fr, Phone no: + (33) 5 61 33 83 86, OFV-505 provided by Polytec. The shaker, force sensor and the laser vibrometer are manipulated with the help of a data acquisition system supplied by LMS Test Lab.

Figure 1 .Figure 2 .

 12 Figure 1. Experimental set-up for vibration testing Figure 2. Vibration test specimen specifying the location of the excitation and the measurement points

Figure 3

 3 Figure 3 Comparison of the Sum of Frequency Response Function (FRF) for a light (Specimen 3) and a heavy specimen (Specimen 7) having 150 kg/m 3 fiber core density.

Composites20092ndECCOMAS

  Thematic Conference on the Mechanical Response of Composites 1-3 April 2009 Imperial College London, UK 1 Corresponding author Email: amir.shahdin@isae.fr, Phone no: + (33) 5 61 33 83 86,

Table 1 .

 1 Composition of fiber network sandwich specimens for vibration tests fabricated from the small and large mold having 150 kg/m 3 fiber density

		Small Mold	Large Mold
	Density	150 kg/m 3	150 kg/m 3 (Light)	150 kg/m 3 (Heavy)
	Weight of Fibers	39 g	39 g	39 g
	Weight of Resin & Hardener	43 g	18 g	43 g
	Weight of Skin (Upper + Lower)	38 g	38 g	38 g
	Total Weight of Specimen	120 g	95 g	120 g

Table 2 .

 2 Modal parameters comparison for the light and heavy vibration test specimens fabricated from the small and large mold (150 kg/m 3 fiber core density)

	Type of	Specimen	Specimen	Damped Natural Frequencies		Damping Ratios (%)
	Specimen	No	Weight		(Hz)
			(g)
				1st	2nd	3rd	4th	1st	2nd	3rd	4th
				Mode	Mode	Mode	Mode	Mode	Mode	Mode	Mode
	Heavy (Small Mold)	1	120.0	225	622	1064	1536	0.331 0.386	1.17	1.24
		2	124.1	222	652	1069	1550	0.331 0.386 0.622 1.235
	Heavy	3	116.0	189	506	947	1493	0.464 0.697 1.204 1.263
	(Large Mold)	4	118.6	202	562	981	1428	0.556 0.641 1.033 1.124
		5	126.4	230	632	1107	1673	0.691 0.792 1.351 1.126
		6	98.0	160	349	701	1217	2.75	3.29	2.65	1.59
	Light	7	93.0	140	295	636	1160	3.31	2.18	3.98	1.25
	(Large Mold)	8	94.7	145	311	680	1179	3.63	2.98	2.53	0.93
		9	94.5	144	320	681	1171	3.62	2.59	2.28	1.10

Table 3 .

 3 Change in frequency and vibrational level for the light and heavy vibration test specimens for the first four modes

	Mode 1 Mode 2 Mode 3 Mode 4


L Montastruc
email: ludovic.montastruc@ensiacet.fr

T Liu

PRODUCTION OF LIPOPEPTIDES BY BIOREACTION SEPARATION COUPLING

Keywords: surfactin, adsorption, fermentation, integrated process

The lipopeptide Surfactin produced by Bacillus subtilis is one of the most widely studied biosurfactants because of its high biotechnological and therapeutical significance. Surfactin has several advantages over the chemically synthesized surfactants such as lower toxicity, higher biodegradability and biocompatibility. In view of Surfactin production, we propose a new approach integrating the fermentation process with microfiltration and adsorption. The adsorption column, designed as a fixed bed activated carbon reactor, recovers the Surfactin and eliminates the foam formation. Moreover, the adsorption step reduces the concentration in the effluent so that it does not exceed permissible limit.

Surfactin is a bacterial lipopeptide produced biologically by Bacillus subtilis. It has the advantages of biodegradability, low toxicity and biocompatibility. As a result, it has been found an attractive alternative or supplement to chemically synthesized surfactants that have a detrimental effect on the environment and successful application in areas such as bioremediation [START_REF] Mulligan | Surfactant-enhanced remediation of contaminated soil: A review[END_REF][START_REF] Mulligan | Environmental applications for biosurfactants[END_REF] and oil recovery [START_REF] Lin | Biosurfactants: Recent advances[END_REF][START_REF] Desai | Microbial production of surfactants and their commercial potential[END_REF]. However, biosurfactants are not widely available because of their high production cost, which result primarily from low strain productivities and high recovery expenses. Moreover, under the aerated conditions, surfactin accumulation generates foam that is apt to flow out of the reactor; which decreases the valid reactor volume and makes the process difficult to control. Process integration is an attractive approach to foam elimination and to produce valuable products of higher quality at reduced costs. Typical example is the fermentation process integration with liquid isolation of a specified reaction product [START_REF] Schügerl | Integrated bioprocesses[END_REF].

In this study, in view of surfactin production, we propose a new integrated process comprising fermentation and adsorption as shown in Fig. 1. A micro filter and a heat-exchanger are added for cell recovery and temperature control before the adsorption column. The permeate flow containing no cell enters the adsorption column whereas the retentate flow containing cells returns into the bioreactor. The adsorption column is designed as a fixed bed of activated carbon, which is applied for surfactin recovery and foam elimination.

The column is designed based on the adsorption experience obtained in our preliminary work [START_REF] Liu | Integrated Process for Production of Surfactin (Part 1): Adsorption Rate of Pure Surfactin onto Activated Carbon[END_REF][START_REF] Montastruc | Integrated Process for Production of Surfactin (Part 2): Equilibrium and kinetic study of Adsorption Surfactin onto Activated Carbon[END_REF]. Both adsorption equilibria and kinetics show that the activated carbon acts as a very effective adsorbent for surfactin adsorption. In view of using an integrated process, in this work the adsorption of surfactin from real medium solutions is studied in both batch and continuous fixed bed conditions. Fixed bed adsorption experiments were performed to study the adsorption dynamics and give accurate data for scale up. Main objective of the fixed bed adsorption is to reduce concentration in the effluent so that it does not exceed permissible limit. The most important criterion in the design of fixed bed adsorption systems is the prediction of column breakthrough or the shape of the adsorption wave front, which determine the operation life span of the bed and regeneration times. Some experiments were necessary to determine the adsorption model. The effects of parameters of fixed bed process, such as operation temperature, flow rate and adsorbent dose on the shape of breakthrough curves were investigated using the predicted model. Finally, more than 94 % of the surfactin adsorbed onto the fixed bed column was eluted by desorption with methanol indicating that an activated carbon fix bed could be used for surfactin recovery effectively in the integrated process. Moreover, a production kinetic model was determined in batch culture. Adding the breakthrough curve model, this study was the preliminary work of the global process optimization.

EXPERIMENTAL

Fermentation procedure

The surfactin was produced by Bacillus subtilis ATCC 21332. The strain selected could produce only one kind of lipopeptide -surfactin. The cultures from late exponential growth phase were harvested as the inoculum for surfactin fermentation experiments. Optical density of the culture was determined at 600 nm and culture was washed and concentrated 10 times by centrifugation. The seed cultures were inoculated into 1000 mL flasks containing 200 mL of Landy modified medium [START_REF] Liu | Integrated Process for Production of Surfactin (Part 1): Adsorption Rate of Pure Surfactin onto Activated Carbon[END_REF] in order to obtain 0.5 as initial optical density. Batch cultures were incubated at 37 °C and pH at ~7 with an agitation rate of 200 rpm. The cell growth and surfactin concentration were monitored with respect to time. After the fermentation was ended, the culture medium was centrifuged. The supernatant was collected for the following adsorption experiments.

Adsorption of surfactin from culture medium onto the activated carbon

Merck activated carbon (Ref. 1.02514. 1000) with spherical geometry was used. In our previous work, only the adsorption of pure surfactin from aqueous buffer solution was studied, and the sorption capacity of the adsorbents was obtained using the adsorption isotherm technique [START_REF] Liu | Integrated Process for Production of Surfactin (Part 1): Adsorption Rate of Pure Surfactin onto Activated Carbon[END_REF]. In this work, the surfactin was adsorbed directly from the culture medium in continuous fixed bed conditions. Continuous fixed bed adsorption was also studied to evaluate the column performance for surfactin adsorption from culture medium. Experiments were conducted in a 1.3 cm diameter and 30 cm length glass column packed with 0.9 mm particle-size of the activated carbon with of 0.9 mm diameter between two supporting layers of glass balls with 2 mm diameter. Nine grams of activated carbon were added to produce 110 mm length column. The influent culture medium was put in an overhead feed reservoir of 3.0 L capacity and kept agitation during the experiment process to make the culture medium homogeneous. The influent feed flowed vertically through the column from down top. The flow rate was supplied and controlled by a peristaltic pump. Sampling of effluent was done at the chosen time, and the surfactin concentrations in the samples were measured. Effects of initial adsorbate concentration (350 mg/L and 550 mg/L) were investigated on the performance of breakthrough for the adsorption of surfactin directly from the culture medium.

RESULTS AND DISCUSSION

Dynamics of cell growth and surfactin production

B. subtilis ATCC 21332 was grown in Landy modified medium [START_REF] Liu | Integrated Process for Production of Surfactin (Part 1): Adsorption Rate of Pure Surfactin onto Activated Carbon[END_REF]. The time courses of cell growth and surfactin production are shown in Fig. 2, which indicates that surfactin production growth is apparently associated with cell growth. After about 45 hours fermentation process, both cell growth and surfactin production reach the maximum and then level off. Generally, the surfactin production in such a fermentation process can reach up to 500-600 mg/L. The surfactin concentration of the culture medium used for adsorption studying was controlled to be C 0 = 550 mg/L by water dilution or vacuum distillation concentration.

To modeling the experimental point of the biomass and the surfactin production, we propose the equation:

                  dt dX dt dP X X X dt dX   max 1 ( 1 )
where α is the kinetic constant (h -1 ), X max the maximal biomass concentration, and γ the productivity (mg•L -1 •OD -1 ). The best fit of the above equations to the experimental data yielded α = 0.277 h -1 , X max = 16.72 OD unit and γ = 29.7 mg•L -1 •OD -1 .

Breakthrough curve analysis

In a fixed bed adsorption system, the adsorbent located close to inlet where maximum adsorption takes place initially. This adsorption zone moves further as time passes and then approaches the exit of the bed. When the adsorption zone has moved through the column, the concentration of the adsorbate at the exit becomes equal to the feed concentration. Adsorption involves interface accumulation or concentration of an adsorbate species at an adsorbent surface or interface. A plot of exit concentration as a function of lapse time or volume throughput reacted is known as breakthrough curve. The characteristic shape of the breakthrough curve depends on the inlet flow rates, concentration, adsorption equilibria and kinetics, and other properties such as column diameter and bed height. The area under the breakthrough curve gives the total quantity of adsorbate adsorbed for a given feed concentration.

The model used in the present work to fit the experimental data takes account of the effect of both intraparticle and external-film mass-transfer resistances. This model was used in different works in particular for big molecule like α-Amylase [START_REF] Bautista | Adsorption of α-Amylase in affixed bed: Operating Efficiency and kinetic modeling[END_REF]. Simultaneously, instantaneous local adsorption equilibrium on the pore surface was considered. Furthermore, the following assumptions were made to set up the model equations:

1) The system operates under isothermal conditions.

2) The equilibrium of adsorption is described by an isothermal equation in the form q=f(c).

3) Intraparticle mass transport is due to Fickian diffusion, and it is characterized by the pore diffusion coefficient, i.e. D p . Mass transfer across the boundary layer surrounding the solid particles is characterized by the external-film mass transfer coefficient k f .

4) The linear velocity of the liquid phase along the column, and the mass-transfer parameters are independent of the solute concentration in the bulk liquid phase. 5) Axial dispersion is considered to account for nonideal flux along the longitudinal axis of the column.

6) The macroporous adsorbent particles are spherical and homogeneous in size and density.

The surfactin diffusion coefficient with molecular dynamics simulation is estimate like the following procedure. Classical molecular dynamics is employed to estimate the diffusion coefficient.

Because the surfactin molecule is primarily composed of amino acids, we choose GAFF (generalized AMBER force field) parameter set to model the intermolecular and intramolecular interactions, as well as atomic charges. [9] The cubic simulation cell is setup with only one surfactin molecule (two carboxylic groups are deprotoned), as well as sufficient water molecules, plus 2 K+ cation to maintain electric neutrality. We use SPC (Single Point Charge) model for the water molecule and employ standard periodical boundary conditions in all three directions. And the dimension of simulation cell is larger than 6 nm, so that it's enough to encapsulate the surfactin and prevent it from escaping. For both van der Waals and Coulomb potential, a cutoff radius of 1 nm is used, thus each surfactin does not have interaction with its image, so actually it's in an infinite dilute solution.

The initial configuration is prepared targeting an arbitrary density of 1.0 g/cm 3 , and equilibrated to 0.1 MPa, desired temperatures respectively with NPT ensemble. Then a 400 ps (with a timestep of 2 fs) NVT simulation with standard Nose-Hoover thermostat is carried out to further equilibrate the system, after which 3200 ps production run is performed to estimate the diffusion coefficient. In fact, we first calculate mean square displacement and plot it against time, using the Einstein relation then the diffusion constant could be obtained. For a temperature equal 310 K, the diffusion value is equal to 0.428×10 -5 cm 2 /s. The axial dispersion coefficient is calculated with the correlation proposed by [START_REF] Wakao | Effect of fluid dispersion coefficients on particle-to-fluid mass transfer coefficients in packed beds. Correlation of Sherwood numbers[END_REF]:

   Re 5 . 0 20 Sc D D m L    ( 2 )
The external mass transfer coefficient is calculated with the correlation proposed by [START_REF] Dwivedi | Particle-Fluid Mass Transfer in Fixed and Fluidized Beds[END_REF]:

              386 . 0 82 . 0 3 2 Re 365 . 0 Re 765 . 0   Sc v k f ( 3 )
The D p value was determined with minimizing the quadratic difference between experimental and modeling points;

The adsorption performance of surfactin from culture medium was studied at two inlet surfactin concentrations, 350 and 550 mg/L. The original culture medium of 550 mg/L was diluted with water to obtain 350 mg/L. Fig. 3 shows the breakthrough curves at the two initial concentrations. The column height was 110 mm, the flow rate was 0.295 ml/min, and temperature was kept at 37 o C. The two breakthrough curves broke through nearly at the same contact time. Moreover, in the first stage of exhaustion, the two columns were exhausted nearly as fast as each other. However, in the second stage, the column got exhausted faster at a higher initial surfactin concentration. As a whole, an increase in initial surfactin concentration of inlet culture medium increased the adsorbed surfactin quantity at the same contact time on the adsorbent column and the efficiency of the adsorbent column as well, which should also be due to increase of the mass transfer of the surfactin to the adsorbent at a higher initial surfactin concentration. The D p value were respectively 1×10 -11 m 2 •s -1 and 1.5×10 -11 m 2 •s -1 for an initial concentration equal to 350 and 550 mg/L.

Recovery of surfactin by methanol elution

It has been shown that the adsorption of surfactin onto the activated carbon is an irreversible process, during which the adsorbed surfactin can not be desorbed with water solutions. Therefore, methanol was used as eluent to recover surfactin adsorbed in the activated carbon. The adsorbent column was operated for 30 h under the following conditions: column height = 110 mm, flow rate = 0.295 mL/min, T = 37 o C and C 0 = 550 mg/L. The total quantity of surfactin adsorbed, which is about 82.0 mg. Then, after washed with water, the adsorbent column was eluted by methanol with a flow rate of 0.295 ml/min at 25 o C. Fig. 4 shows the evolution of surfactin concentration in effluent as a function of the throughput volume of eluent. The area under the curve gives the total quantity of surfactin recovered in effluent methanol, which is about 77.0 mg. Therefore, more than 94 % of surfactin adsorbed onto the activated carbon could be recovered by using the adsorption and subsequent desorption processes.

CONCLUSION

Surfactin productivity of up to 600 mg/l could be obtained by B. subtilis ATCC 21332 in the Landy modified medium. Surfactin adsorption capacity from culture media confirms that the activated carbon could be used as a good adsorbent for recovery of surfactin in an integrated process. Finally, more than 88 % of the surfactin adsorbed was eluted by desorption with methanol indicating that an activated carbon fixed bed could be used for surfactin recovery effectively in the integrated process.

Fig. 1

 1 Fig.1 Schematic of the integrated process

Fig. 2

 2 Fig. 2 Time-course of cell growth and surfactin concentration during Bacillus subtilis ATCC 21332 fermentation (line = model, symbol =experimental point)

Fig. 3

 3 Fig. 3 Experimental (dots) and model (lines) breakthrough curve for adsorption of Surfactin [column height = 110 mm, flow rate = 0.295 ml/min, T = 37 o C, C 0 (mg/L): (■)350; (+)550].

Fig. 4

 4 Fig. 4 Evolution of surfactin concentration in effluent as a function of the throughput volume of eluent (column height = 110 mm, flow rate = 0.295 ml/min and T = 25 o C)


Lynda Aiche

Hugues Vergnes

Bernard Despax

Brigitte Caussat
email: brigitte.caussat@ensiacet.fr

Hubert Caquineau
email: caquineau@laplace.univ-tlse.fr

SILICON OXIDE DEPOSITION FROM COLD REMOTE PLASMA ON POLYETHYLENE POWDERS IN FLUIDIZED BED

Keywords:

In this work, a fluidized bed has been coupled to a far cold remote nitrogen/oxygen plasma to deposit silicon oxide SiO x from silane SiH 4 on the surface of a polyethylene powder. The influence on the characteristics of the silica-like deposit on powders, of several parameters such as the amount of oxygen injected in the plasma, the SiH 4 flow rate and the treatment duration, has been studied. First, the particles wettability has been measured by contact angle (Washburn method), knowing that PE is hydrophobic and that silicon oxide would increase its hydrophilicity. The film chemical composition has been analysed by FT-IR spectroscopy. Its organization on the powder surface and its morphology have been studied by FEG SEM. Results indicate that the wettability of powders increases with the whole parameters studied. Deposition seems to occur under the form of nodules; its continuity is not ascertained. This is the reason why the increase of wettability could be due either to a better surface coverage of the deposition or to an increase of its thickness. Additional analyses are in progress to better understand the correlations between the deposition parameters, the plasma features and the deposition characteristics.

Introduction

The plasma technique is a dry process which can modify the surface properties of thermosensitive materials without changing their intrinsic bulk properties [1].

When coupled with a fluidized bed, a nitrogen cold remote plasma technology allows to modify the surface of thermo-sensitive powders either by grafting new functions (for instance hydroxyl OH groups) [2][3][4], or by depositing various materials, such as silicon nitride Si 3 N 4 from silane SiH 4 and nitrogen [START_REF] Davidson | Fluidization[END_REF]. The fluidized bed is a very efficient gas-solid contactor generating intense thermal and mass transfers between gas and particles, and then very uniform treatments. A very good contact between reactive species of the plasma discharge and powder surface is obtained [4]. And the plasma reactivity is not destroyed by the fluidized bed [3]. The reactivity of nitrogen/oxygen cold plasma is used first to act on powders surface by creating active sites and then to dissociate the gaseous precursors, to form deposition [START_REF] Aiche | Interface Controlled Organic Thin Films[END_REF].

The plasma assistance allows to operate at low temperature (< 90 °C in our case).

This study concerns silicon oxide deposition on the surface of a high density polyethylene (PE) powder belonging to Geldart's group B, i.e. easy-to-fluidize. The reactive gases were nitrogen, oxygen and silane SiH 4. The influence of some key operating parameters such as the amount of oxygen injected in the plasma, the SiH 4 flow rate and the treatment duration, has been studied on the characteristics of the deposition. First, the particles wettability has been measured by contact angle (Washburn method), knowing that PE is hydrophobic and that a SiO x coating would increase its hydrophilicity. Its chemical composition has been analysed by Fourier Transform Infra-Red (FT-IR) spectroscopy. Its organization on the powder surface, and its morphology have been studied by Field Effect Gun Scanning Electron Microscopy (FEG SEM). First the experimental setup is described, before to present and discuss the main results obtained.

Experimental

Experiments were carried out in a fluidization column in glass, 50mm in diameter and 0.9m high as shown in fig 1 . High density polyethylene (PE) powders, 260 µm in Sauter diameter and 930 kg/m 3 in grain density, have been studied. These particles were fluidized by a mixture of nitrogen and oxygen gases (respectively N50 and Alpha quality, Air Liquide). Gas flow rates were regulated by mass flow controllers. The particles were supported by a porous plate designed in our lab. The total pressure into the bed was maintained at 10 Torr by means of a vacuum pump. We ensured that the leak rate was less than 100mTorr/min.The bed pressure drop was measured by a MKS vacuum gauge. Elutriated particles were captured by a particle trap.

The gas flow mixture (N 2 and O 2 ) was excited by an electrodeless discharge by means of microwaves generator (2450MHz) that could deliver a transmitted power up to 300W. The quartz discharge tube (inner diameter 9 mm) was connected to the bottom of the fluidized bed reactor. The microwave cavity was positioned at 40cm from the reactor inlet.

Silane SiH 4 (max: 2.5 vol. % in nitrogen) was introduced directly in the fluidized bed through an annular injector located at 2.5 cm above the porous plate. We fixed this percentage in such a way that the calculated theoretical deposition thickness (corresponding to 100 % conversion of silane and assuming that the deposition was continuous and uniform around each grain) is in the range between 50 and 200 nm for the conditions tested here.

The PE powder mass treated in this reactor was always 51g. Compared to similar works of the literature [1,2], this represents an important amount of powders, for which the fixed bed height is of the order of the bed diameter.

The microwaves generator power was maintained at 300W. The total gas flow rate was fixed at 800 sccm (cm 3 /min STP) and the flow rate crossing the plasma was of 640 sccm.

In order to improve the silicon oxide film adhesion, the PE powder was always pre-treated by 0.3 vol. % of oxygen in nitrogen for 1hour before deposition. The pre-treatment conditions were determined after primary study based on our previous work [START_REF] Aiche | Interface Controlled Organic Thin Films[END_REF]. Besides, the deposit duration was fixed to 2hours. Where η l (Pa.s) , ρ l (kg.m -3 ) and σ (mN/m) are respectively the viscosity, density and total surface tension of the chosen liquid. The water contact angle of the non treated PE powder is of 90°.

The attenuated total multiple reflection (ATR) technique was used to analyse the chemical modifications produced by plasma treatment. A Nicolet 5700 spectrometer was used to obtain the IR spectra of the treated powders. A diamond prism was used and the incident angle of the IR radiation was 45°. One hundred scans with a resolution of 4 cm -1 were obtained and averaged per sample.

The morphology of the treated powders surfaces was analyzed using a Jeol JSM 6700F FEG-SEM microscope. The samples were secured onto copper mounts using silver paint and gold coating in a Polar on high resolution sputter coater to obtain enough contrast in the FEG-SEM micrographs. The energy of the electron beam was 20 kV.

Optical Emission Spectroscopy (OES) measurements have been performed to study the plasma characteristics just below the porous plate, using a GetSpec 3648 set-up.

Results and discussion

PE powders fluidization

First, the choice to work under a total pressure of 10 Torr results from a compromise between the concentration of the active species responsible for creating active sites on powder surfaces and also for the dissociation of SiH 4 and the fluidization quality. The concentration of the active species increases when the pressure decreases and the fluidization quality is improved when the pressure increases.

Then, the minimum fluidization velocity at reduced pressure (10Torr) was deduced from the variation of the pressure drop through the fluidized bed versus the gas velocity. Harrison and Richardson method was adopted [START_REF] Davidson | Fluidization[END_REF].According to experimental data, the minimum fluidization velocity (U mf ) was equal to 5.3 cm/s. This value agreed well with predicted value by Llop et al. correlation [START_REF] Llop | Fluidization at vacuum conditions[END_REF]. We chose to work at a fluidization ratio U/U mf of 10 since it provides a convenient bed hydrodynamics and ensures an important active species concentration.

The granulometry of powders was lowly modified by the plasma treatment, as verified by laser granulometry measurements.

Effect of the oxygen flow rate

The influence of the oxygen flow rate on the contact angle with water for the treated PE powders is shown in figure 2. The increase of the oxygen percentage in the discharge enhances the powders wettability since the contact angle regularly decreases with the oxygen percentage, to reach 0° at 3 sccm of O 2 . In fact, the plasma treatment (only considering N 2 and O 2 ) could create new chemical functions on the powder surface, which may change the surface chemical composition and roughness, affecting the polymer wettability [START_REF] Svorcık | [END_REF]. Indeed, Arpagaus et al. [11] show in their works that the wettability effect is strongly influenced by the plasma gas composition and the degree of wettability increases with higher O 2 content in an O 2 /Ar mixture. The intensity of oxygen incorporation into the surface increases with the amount of O 2 in the plasma gas but levels off at a value higher than 30 vol. % O 2 [11].

In a previous work about surface treatment of these PE powders by a nitrogen cold remote plasma [START_REF] Aiche | Interface Controlled Organic Thin Films[END_REF], we have shown that these new functions reduce the contact angle to 65° by adding 1% vol. oxygen in nitrogen in the cold plasma under 300W and microwave positioned at 20 cm far from the reactor. Consequently, it can be inferred that the reduction of the contact angle to 0° is not only due to these new functions grafted on the powders surface but mainly to the deposited material.

The chemical composition of the deposition has been analysed by FTIR-ATR as shown on Figure .7. Whatever the conditions tested, for all runs presented in this article, the spectra are very similar. The absorption at 1070 cm -1 and 820 cm-1 is due to stretching vibrational and bending mode of Si-O-Si group that mainly characterizes the deposition. The presence of carbonyl groups is also evident from the observation of the spectra since the deformation CH x absorbs in the range 1400-1500 cm -1 . FEG-SEM analyses have confirmed the presence of the deposition on the powders, as shown below. The PE surface before deposition is quite smooth. The deposition is relatively rough; it seems to occur under the form of SiO x nodules. Its continuity on the PE surface is not ascertained. As a consequence, the increase of wettability of the powders due to the increase of O 2 flow rate could be due either to a better surface coverage or to an increase of the deposition thickness or even to a modification of the deposit chemical composition with more oxygen deposited. Some additional characterizations are in progress to elucidate these points. The EOS investigations have revealed that in the deposition conditions without powders, the introduction of small amounts of oxygen strongly increases the concentration of the produced active species N( 4 S). For more details, it is interesting to refer to the study of Boudam et al. [12]. Higher oxygen flow rate in the discharge promotes the production of silicon oxide by the combination of oxygen atoms with silicon coming from SiH 4 decomposition. The produced N( 4 S) dissociates the O 2 and SiH 4 gaseous precursors and creates active sites on the powders surface.

It can be seen on figure 5 that the surface modifications induced by the deposition are not permanent. The water contact angle increases rapidly to reach a plateau at proximally 60° after a few days when 3 sccm of O 2 were added in the discharge, and after one day for 0.15 sccm. The increase of the coated PE powders wettability to 60° confirms our assumption regarding the wettability causes. In fact, when the hydrophilic character is induced by grafted functions, the PE powders become hydrophobic after a few days [4]. But in our case, the coated PE powders wettability stabilises at 60°. This indicates that water contact angle reduction to 0° is due to deposition. However, wettability evolution could be first the result of losing of the coated film by friction during handling. A second reason could be assigned to carbon contamination by the ambient atmosphere which would increase its hydrophobicity. In this part, the oxygen flow rate was maintained at 3 sccm, which gave the best wettability, as previously shown. Figure 6 shows the evolution of the contact angle with water for various silane flow rates. the deposition thickness are enhanced by the presence of more silicon atoms in the bed for the conditions tested. By considering an error of ±3° on experimental values on the water contact angles, the general trend is that increasing the coating duration decreases the contact angle, and so increases the powder wettability. By increasing the run duration, the theoretical deposition thickness also increases, since the amount of silicon injected is higher. So this parameter could explain the better wettability obtained here.

Effect of deposition duration

Conclusion

This work has demonstrated the feasibility to deposit silicon oxide on the surface of a polyethylene powder, using a far cold remote nitrogen/oxygen plasma and silane in a fluidized bed. This deposition occurs under the form of nodules and its continuity is not ascertained. It drastically increases the powder wettability. This hydrophilic character of the deposition is exalted by the oxygen concentration injected in the discharge, the SiH 4 flow rate and the treatment duration. Additional characterizations are in progress to understand if these parameters increase either the continuity of deposition, or its thickness or modify its chemical composition. The surface properties induced by the deposition are not permanent, probably due to carbon contamination.

Figure 1 :

 1 Figure 1: Experimental set up

Figure 2 .

 2 Figure 2.Variation of the water contact angle versus oxygen flow rate, 4sccm of SiH 4 diluted in 156 sccm of N 2 and 640 sccm of total plasma gas.

Figure 3 .

 3 Figure 3. Normalized FTIR-ATR spectra of the PE powders before and after deposition.

Figure 4 .

 4 Figure 4. FEG-SEM micrographs of the PE powders surface (a) before and (b) after plasma post discharge coating under 4 sccm of SiH 4 ,0.63 sccm of O 2 in 640 sccm of N 2

Figure 5 .

 5 Figure 5. Variation of the water contact angle versus storing time, 4sccm of SIH 4 diluted in 156 sccm of N2 and 640 sccm of total Plasma gas

Figure 6 .

 6 Figure 6. Variation of the contact angle versus SiH 4 flow rate, 3sccm of O 2 added in 640 sccm of total plasma gas.

Figure 7 Figure 7 .

 77 Figure 7 presents the influence of the deposition duration on the contact angle of the PE powders.