S Caré

E Hervé

Application of a n-Phase Model to the Diffusion Coefficient of Chloride in Mortar

Keywords: mortar, chlorides, diffusion, effective medium theory, homogenization technique, modeling

The determination of the chloride diffusion coefficient of a concrete is needed to help the prediction of the service life of concrete structure. In this paper, we propose first a critical review of models for chloride diffusion coefficients already used in literature at different scales and then we develop an analytical model, which takes into account the characteristics of the different phases of concrete. These materials are treated as a three-phase composite, consisting of a cement continuous phase, of an aggregates dispersed phase and of an interface transition zone. Chloride diffusion coefficient using an n-layered inclusion-based micromechanical modeling is predicted. The details of calculations are summarized hereafter and experimental measurements obtained on mortars are compared with predicted results.

Nomenclature c

volume fraction of ions (kg/m 3 ). C A volume fraction of aggregates (-). C I volume fraction of ITZ (-). D diffusion coefficient tensor in composite media (m 2 /s). D A diffusion coefficient of ions in aggregates (m 2 /s). D B diffusion coefficient of ions in bulk cement paste (m 2 /s). D i diffusion coefficient of ions in phase i (m 2 /s). D P diffusion coefficient of ions in cement paste (m 2 /s). D 0 diffusion coefficient of ions in water (m 2 /s). D eff effective diffusion coefficient of ions in porous media (m 2 /s). D eff-i predicted diffusion coefficient according a model i (m 2 /s). D eff max upper bound of the effective diffusion coefficient (m 2 /s). D eff min lower bound of the effective diffusion coefficient (m 2 /s). D 1 P estimation of the diffusion coefficient in cement paste (m 2 /s). i integer (-). I second-order unit tensor (-). j flux vector of ions (kg/m 2 s). k integer (-). n integer (-). R i radius of sphere i (m). t I thickness of ITZ (m).

Introduction

Chlorides, which are dissolved in the environment around a concrete structure can penetrate into it. Reinforcing steel in this structure begins to corrode when the chloride volume fraction at its level is high enough. The time to reach this critical volume fraction depends on the diffusion coefficient of chloride. This parameter is almost always determined experimentally for each concrete. The object of this study is to predict the value of this parameter, by considering the concrete properties due to its proportioning. The parameters, which have an effect on the chloride diffusion coefficient are the water to cement ratio (W/C) and the aggregates [START_REF] Atkinson | The diffusion of ions through water-saturated cement[END_REF]. The second section of this paper is a review of the determination of chloride diffusion coefficient at different scales. This review show that it is necessary to consider the concrete or the mortar as a three phase composite to match experimental observations. The third section introduces a new analytical model, using 'n theoretical layers' around each inclusion (aggregate) for the diffusion coefficient. The fourth section of this paper shows the validation of this model, after experiments on various mortar specimens.

Review of Models on Effective Chloride Diffusion Coefficient

DEFINITIONS

For models of diffusion coefficient, which consider the structural properties of concrete, some definitions are to be given about the scales, which are taken into account. Cementitious materials are considered as saturated, so that the only process of chloride is diffusion. As a rule, three scales are considered (Figure 1):

-The microscopic scale (10 -9 m) takes into account the pore features. For cement-based materials, the transfer process is mainly the diffusion of ions through water-filled pores (Garboczy, 1990). These pores can be characterized by different parameters: porosity , tortuosity β. -The mesoscopic scale (10 -6 -10 -3 m) corresponds to a 'theoretically homogeneous' material including cement paste and aggregates. At this scale, the chloride diffusion coefficient depends on the characteristics of the cement paste and of the aggregates. More precisely, concrete may be considered as a composite material with three phases (Figure 1). The first one, related to the aggregates, is a dispersed phase with homogeneous properties. The aggregates can be porous (calcareous aggregates) or not (siliceous aggregates). The second phase, ITZ (interfacial transition zone), is an interface, with a gradient of porosity, and volume fractions gradients of anhydrous and hydrated cement compounds. The ITZ microstructure is different from that of the bulk cement paste due to the 'packing' constraints imposed by the aggregate surface [START_REF] Garboczi | Analytical formulas for interfacial transition zone properties[END_REF]. The last phase is the bulk cement paste, a continuous phase also named matrix. -The macroscopic scale (order of magnitude: 10 -2 m) corresponds to the size of specimens used for determining the effective chloride diffusion coefficient D eff (m 2 /s) [START_REF] Francy | Measuring chloride diffusion coefficients from non-steady from non-steady state diffusion tests[END_REF][START_REF] François | Mesure du coefficient de diffusion des ions chlorures: comparaison régime permanent-régime transitoire[END_REF]. This coefficient is obtained from the steady-state regime according to the first Fick's law:

j = D eff ---→ gradc,
(1) where j represents the flux vector of free chloride ions (kg/m 2 s), c is the volume fraction (kg/m 3 ) of free chloride ions. When inside the pore systems of concrete, chloride ions are partitioned between free ions that diffuse toward region with lower chloride concentration and bound chloride. c is related to chloride-free ions, which do not interact with hydrated cement paste. The exchange between these different kind of ions is assumed to be instantaneous.

The steady-state regime is characterized by a linear concentration profile in the liquid phase between the upstream and the downstream cells. In this case, it is worth noticing that there is a local equilibrium between bound and free ions. It may be considered that there is no interaction between free chloride and solids phase. The second Fick's law reads then: div j = 0, (2)

USUAL MODELS

The usual models for chloride diffusion coefficient are as follows:

2.2.1. Empirical Modeling at Macroscopic Scale Previous data show that the diffusion coefficient in cement paste is well correlated with the water to cement ratio. According to [START_REF] Atkinson | The diffusion of ions through water-saturated cement[END_REF], in cement paste, the relationship between the chloride diffusion coefficient and the water to cement ratio, is approximately in an exponential one. According to [START_REF] Hobbs | Aggregate influence on chloride ion diffusion into concrete[END_REF], the diffusion coefficient in concrete depends on the water to cement ratio, but the study of this dependence shows that the correlation is weaker for concrete than for cement pastes. These results show that the effect of aggregates must be taken into account.

Microscopic Scale Models

As a rule, the effective diffusion coefficient in porous media D eff is lower than that of the same ions in water D 0 . According to [START_REF] Garboczi | Permeability, diffusivity and microstructural parameters: a critical review[END_REF], Van Barkel and Heetjes (1974) and [START_REF] Dormieux | Modélisation macroscopique du transport diffusif, apport des méthodes de changement d'espace[END_REF] D eff is given by

D eff = D 0 β = D 0 2 3 - , ( 3
)
where β = 2/(3 -) is the 'tortuosity' of the pore system. This result is obtained by an homogenization technique (Mori Tanaka's approach [START_REF] Dormieux | Modélisation macroscopique du transport diffusif, apport des méthodes de changement d'espace[END_REF] using Maxwell's model) and is an estimation of the effective diffusion coefficient of a composite material where the solid grains are considered as spherical inclusions in a fluid phase. Furthermore, another multiscale modeling technique may be used for the determination of the effective diffusion coefficient. [START_REF] Xu | Multiscale structures to describe porous media. Part II. Transport properties and application to test materials[END_REF] determine the diffusion coefficient by using the multiscale percolation system concept. In this case, the 'modeled material' is made of several elementary networks having mesh size proportional to the diameter of the real pores. The distribution of pore size is determined by mercury intrusion porosimetry.

These two kinds of models [START_REF] Xu | Multiscale structures to describe porous media. Part II. Transport properties and application to test materials[END_REF][START_REF] Dormieux | Modélisation macroscopique du transport diffusif, apport des méthodes de changement d'espace[END_REF] are applicable on any cement-based materials (cement paste, mortar, concrete). Nevertheless there is a poor agreement between the value predicted by these models and the experimental diffusion coefficients [START_REF] Daian | Evaluation des propriétés de transfert dans les matériaux cimentaires. Etude critique des modèles[END_REF]. As a rule, the predicted values are higher than the experimental ones.

Mesoscopic Scale Models

At this scale, the model materials is a two-phase or three-phase composite. The models take into account the volume fractions of the components (aggregates, bulk cement matrix and interfacial transition zone ITZ) and their diffusive properties. 2.2.3.1. Two-Phase Models. In this case, the material (concrete, mortar) is a twophase composite, one phase is the homogeneous cement paste and the other phase includes aggregates. The volume fraction of the aggregates is C A and that of the cement paste is (1 -C A ). The diffusion coefficients are D A for aggregate phase and D P for cement paste phase. Various models are given in literature. Hobbs (1997) proposed two models, which are based on the following assumptions: the decrease in chloride volume fraction and the flux (mass of chloride ions carried across a unit area) are the same in cement paste as in aggregates. The first assumption gives an upper bound for the effective diffusion coefficient (D eff max ) and the second one a lower bound (D eff min ). These two models are similar to the Voigt and Reuss models used to bound the elastic modulus of two-phase materials [START_REF] Hashin | A varationnal approach to the theory of the elastic behaviour of multi-phases materials[END_REF]. The effective diffusion coefficient D eff is a function of D P , D A and C A . The lower and upper bounds are estimated by

D eff min = 1 C A /D A + (1 -C A )/D P , D eff max = (1 -C A )D P + C A D A . ( 4
)
When aggregates are not porous (D A = 0), the effective diffusion coefficient

D eff min = 0 and D eff max is equal to D eff max = (1 -C A )D P . ( 5
)
Nevertheless, both relations given in Equation ( 4) correspond to a particular morphology: the two phases (aggregates and cement paste), have configurations in series or in parallel.

In the case of porous aggregates (D A = 0), an estimation of the effective diffusion coefficient D eff , which takes into account the matrix-aggregate configuration in the studied materials (aggregates considered as inclusions in the cement paste matrix) is given by [START_REF] Xi | Modeling chloride penetration in saturated concrete[END_REF] according to [START_REF] Christensen | Mechanics of Composite Materials[END_REF], and is equal to

D eff = D P 2D P (1 -C A ) + D A (1 + 2C A ) D P (2 + C A ) + D A (1 -C A ) (6)
For non-porous aggregates (D A = 0), Equation ( 6) with D A = 0 gives an estimation of D eff :

D eff = D P 2(1 -C A ) 2 + C A . ( 7
)
It is worth noticing that model in Equation ( 7) is always Maxwell's model and may be obtained from Equation (3) by replacing the porosity by the volume fraction of cement paste (1 -C A ).

2.2.3.2. Three-Phase Models. These models take into account the presence of an interface between bulk cement paste and aggregate (ITZ). [START_REF] Garboczi | Multiscale analytical/numerical theory of the diffusivity of concrete[END_REF] proposes analytical and numerical models. By applying homogenization technique to a dilute material (named dilute limit), [START_REF] Garboczi | Permeability, diffusivity and microstructural parameters: a critical review[END_REF] determines the diffusion coefficient D eff as an analytical function of the diffusion coefficient in the bulk cement matrix (D B ), of the aggregates size distribution and of the geometrical and physical properties of ITZ. But, this approach is applicable only for very low aggregate volume fractions (C A 1). By using a differential effective medium theory, the result of the dilute limit is used to obtain the effective diffusion coefficient whatever the aggregate volume fraction. For this approach, a numerical resolution is necessary. Furthermore the same author [START_REF] Garboczi | Multiscale analytical/numerical theory of the diffusivity of concrete[END_REF], with another numerical approach, puts the properties computed at one scale, micrometers for instance, into a model which is built at a higher scale, such as millimeters. The model proposed in [START_REF] Garboczi | Multiscale analytical/numerical theory of the diffusivity of concrete[END_REF] combines microstructure models for the cement paste surrounding a single aggregate and for a representative volume of concrete.

Furthermore, Jaiswal et al. (1998) determine the diffusion coefficient by using non-linear statistical analysis and physically based approximate models. An analytical formulation of the composite diffusion coefficient is based on Bruggeman's (1935) effective medium theory with percolation effects by [START_REF] Mclachlan | Electrical resistivity of composites[END_REF]. [START_REF] Jaiswal | Impact of the interfacial transition zone on the chloride permeability of concrete[END_REF] show that the ITZ must be included in models to match experimental observations, although the ITZ cannot be directly measured and its existence is often questioned.

SYNTHESIS OF THE REVIEW

Several models have been published to predict the diffusion coefficient. Three scales are considered for these models. At the macroscopic scale, empirical models propose the diffusion coefficient to be a function of the water to cement ratio in the material. At the microscopic scale, analytical models propose the diffusion coefficient to be a function of the microstructure parameters. Nevertheless there is a poor agreement between the predicted value of the diffusion coefficient and the experimental one [START_REF] Daian | Evaluation des propriétés de transfert dans les matériaux cimentaires. Etude critique des modèles[END_REF]. At the mesoscopic scale, numerical and analytical models are proposed to determine the diffusion coefficient of mortar as a two-phase or a three-phase composite. Few analytical models taking into account the presence of ITZ are proposed, although various works show that the ITZ must be included in models [START_REF] Jaiswal | Impact of the interfacial transition zone on the chloride permeability of concrete[END_REF].

The aim of this work is to propose a new analytical model for chloride diffusion coefficient in mortar, which takes into account the characteristics of the material considered as a three-phase composite. In order to predict the effective chloride diffusion coefficients in concrete, the (n + 1)-phase model, already used in elasticity, viscoelasticity, elastoplasticity [START_REF] Hervé | Thermal and thermoelastic behaviour of multiply coated inclusion-reinforced composites[END_REF]Zaoui, 1993, 1995) and in the case of thermal and thermoelastic behaviors [START_REF] Hervé | Thermal and thermoelastic behaviour of multiply coated inclusion-reinforced composites[END_REF], is extended to the case of ionic diffusion.

Model for Diffusion Coefficient in a n-Phase Material -Application to Mortar

This section is split into two parts. In the first one, we show how we can, in order to predict the diffusion coefficient in a n-phase material, directly use the (n + 1)-phase model [START_REF] Hervé | Thermal and thermoelastic behaviour of multiply coated inclusion-reinforced composites[END_REF] originally proposed in the case of thermal and thermoelastic behaviors. The (n + 1)-phase model is used to find an estimation of the effective diffusion coefficient of an n-layered-reinforced material. In the second part the (n + 1)-phase model is finally used to determine the overall diffusion of chlorides in concrete where the first phase is the aggregate, the second one the interfacial transition zone with homogeneous properties and the third one the bulk cement paste. Two cases are considered: diffusion can or cannot take place in aggregates.

THE (n + 1)-PHASE MODEL

To derive the ionic concentration field in an infinite medium constituted of an n-layered spherical inclusion, embedded in a matrix subjected to a uniform concentration rise at infinity (as partially done in [START_REF] Garboczi | The effect of the interfacial transition zone on concrete properties: the dilute limit[END_REF]), the ion volume fraction c (kg/m 3 ) compared to a base concentration is defined and as in [START_REF] Hervé | N-layered inclusion-based micromechanical modelling[END_REF] the following infinite medium, which has n isotropic phases, is considered. Phase 1 constitutes the central core and phase (i) lies within the shell limited by the two concentric spheres with the radii R i-1 and R i . Let (D = D i I) be the diffusion tensor of phase (i) (here i [1, n + 1], R 0 = 0 and R n+1 → ∞) (Figure 2), in which I is the second-order unit tensor (each phase is isotropic). The spherical coordinate system (r, θ, ϕ) has its origin at the common center of the above-mentioned spheres The axial symmetry is around x 3 -axis (x 3 = r cos θ). At infinity, the imposed volume fraction is given by

c |r→∞ = αx 3 , ( 8
)
where x k are Cartesian coordinates and α is a scalar. Each phase obeys the first Fick's law of ionic diffusion:

j = -D ---→ grad c, ( 9
)
where j is the flux vector in each phase.

There is no source or sink of ions and under steady state conditions we can use Equation ( 2).

Both Equations ( 9) and ( 2) lead to the governing equation ( c (k) = 0) for c (k) in each phase k. The general solution of this governing equation which respects the imposed remote condition ( 8) is given in phase (k) by

c (k) = A k r + B k r 2 cos θ, ( 10
)
where A k and B k are constants, B 1 = 0 to avoid singularity at the origin and A n+1 = α accounting for the imposed remote condition (it is worth noticing that in the case of a remote flux-type condition j = -γ e 3 , A n+1 = γ /D n+1 ).

The components of the corresponding flux vector are:

j (i) r = -D i A i -2 B i r 3 cos θ, j (i) θ = +D i A i + B i r 3 sin θ, j (i) ϕ = 0. ( 11
)
From the two conditions describing the continuity of j r and c at the interface r = R k between phases (k) and (k + 1) [START_REF] Hervé | Thermal and thermoelastic behaviour of multiply coated inclusion-reinforced composites[END_REF] all the coefficients A k and B k (k ∈ [1, n]) can be expressed in terms of the diffusion coefficient of each phase, their volume fraction and A n+1 . From these results the third component of the average ion volume fraction gradient is calculated in the whole n-layered inclusion:

c ,3 = 1 V n V n c ,3 dV = A n+1 + B n+1 R 3 n . ( 12
)
V n is the volume of this n-layered inclusion. Correspondingly, the third component of the average flux vector of ions in the whole n-layered inclusion reads:

j 3 = -D n+1 A n+1 + B n+1 R 3 n . ( 13
)
These solutions can be used if a true n-phase inclusion is considered in a welldefined matrix whose diffusion coefficient D n+1 is known. It can also be used if D n+1 (named then D eff ) denotes the effective diffusion coefficient of a composite material according to the (n + 1)-phase model.

To determine the effective diffusion coefficient of an isotropic composite material according to the '(n+1)-phase model', one of the two self-consistent conditions can be used. They both lead to relation B n+1 = 0. The first possible condition is to impose the third component of the average concentration gradient in the whole n-layered inclusion to be the same as the third component of the macroscopic concentration gradient imposed to the composite medium at infinite ( c ,3 = α). The second possible condition is to write that the third component of the average ion flux vector in the whole n-layered inclusion is the same as the third component of the ion flux vector imposed to the composite medium at infinity j 3 = -γ .

Setting B n+1 = 0 provides the effective diffusion coefficient [START_REF] Hervé | Thermal and thermoelastic behaviour of multiply coated inclusion-reinforced composites[END_REF] given hereafter by a recursive algorithm:

D eff (i) = D i + D i (R 3 i-1 /R 3 i ) (D i /(D eff (i-1) -D i )) + (1/3)((R 3 i -R 3 i-1 )/R 3 i ) , ( 14
)
where D eff (1) = D 1 .

APPLICATION TO THE DIFFUSION COEFFICIENT OF CHLORIDE IN CONCRETE

These results are applied to the problem of diffusion of chlorides in concrete.

As in [START_REF] Garboczi | The effect of the interfacial transition zone on concrete properties: the dilute limit[END_REF], a four-phase model is used. Phase 1 denotes the aggregates (considered as spherical inclusions) with the volume fraction C A and the diffusion coefficient D A . Phase 2 denotes the interfacial transition zone with the volume fraction C I and the diffusion coefficient D I . Phase 3 denotes the bulk cement paste with the diffusion coefficient D B . To find the effective diffusion coefficient in concrete we use Equation ( 14) and we proceed in three steps:

(a) We consider an homogeneous material made of aggregates:

D eff (1) = D A . ( 15
) (b) We consider a two-phase material made of inclusions (aggregates) embedded in a matrix (ITZ):

D eff (2) = D 1 + D 1 (R 3 1 /R 3 2 ) (D I /(D A -D I ) + (1/3)/((R 3 2 -R 3 1 )/R 3 2 ) , ( 16
)
where

R 3 2 -R 3 1 R 3 2 = C I C A + C I , R 3 1 R 3 2 = C A C A + C I . ( 17
)
(c) We consider coated inclusions (aggregates surrounded by the ITZ) embedded in a matrix (bulk cement paste):

D eff (3) = D B + D B (R 3 2 /R 3 3 ) (D B /(D eff (2) -D B )) + (1/3)((R 3 3 -R 3 2 )/R 3 3 ) , ( 18
)
where

R 3 3 -R 3 2 R 3 3 = 1 -C A -C I , R 3 2 R 3 3 = C A + C I . ( 19
)
Finally, substituting for D eff 2 from Equation ( 16) into Equation ( 18) and taking Equation ( 15) into account yields

D eff D B = N D , ( 20
)
where

N = 6D B D I (1 -C A )(C A + C I ) + C I (1 + 2C A + 2C I )(D I -D B ) × × (2D I + D A ) + 3D A [C I D B + C A D I (1 + 2C A + 2C I )],
and

D = 3D B D I (2 + C A )(C A + C I ) + C I (1 -C A -C I )(D I -D B ) × × (2D I + D A ) + 3D A [C I D B + C A D I (1 -C A -C I )].
The relationship (20) allows us to investigate materials with porous aggregates and with ITZ. Furthermore from this relation, several cases of concrete may be considered too. Indeed, this investigation makes it possible to identify the relevant parameters, which affect chloride transport in concrete with porous or non-porous aggregates and with or without ITZ.

When there is no ITZ (C I = 0 or D I = D B = D P ) and when aggregates are porous (D A = 0), relationship ( 20) is equivalent to Equation ( 6). When the diffusion coefficient of the aggregates is D A = 0 and when ITZ is absent (C I = 0 or D I = D B = D P ), relationship ( 20) is equivalent to Equation ( 7). In this case, D P is the diffusion of cement paste (D B ). Furthermore, when the diffusion coefficient of the aggregates vanishes (D A = 0), the relationship (20) reads:

D eff = D B 6D B (1 -C A )(C A + C I ) + 2C I (D I -D B )(1 + 2C A + 2C I ) 3D B (2 + C A )(C A + C I ) + 2C I (1 -C A -C I )(D I -D B ) .
(21)

Validation of the Model

MATERIALS AND EXPERIMENTAL PROCEDURES

The model of the diffusion coefficient of chloride in concrete given in Equation ( 21) is validated by experiments on specimens. The materials tested are one cement paste (P) and five mortars (M1C, M2C, M2M, M1F, M2F) with non-porous aggregates (D A = 0). Table I gives their compositions. Three different types of aggregates are used: a fine siliceous sand (0.315/1 mm) labeled 'F', a medium siliceous sand (0.315/2 mm) labeled 'M' and a coarse siliceous sand (2/4 mm) labeled 'C'. These compositions make it easy to study the influence of the interfacial transition zones on chloride diffusion process. The water to cement ratio (W/C = 0.45) and the type of cement (coarse cement) are chosen so that the microstructure of the (2003). Chloride-effective diffusion coefficient D eff is determined from non-steadystate diffusion tests by applying Fick's second law after determining for each material its total chloride profile and its chloride binding isotherm (Francy and François, 1996;[START_REF] François | Mesure du coefficient de diffusion des ions chlorures: comparaison régime permanent-régime transitoire[END_REF]. The microstructure of these materials is observed by scanning electron microscopy (SEM) and mercury intrusion porosimetry (MIP) is used to characterize the pore structure.

RESULTS

Interfacial Transition Zone Characteristics

The results obtained by MIP for the total porosity are summarized in Table II. For mortars, the finer is the sand used, the higher is the total porosity. The presence of aggregates changes the pore structure too [START_REF] Caré | Influence of aggregates on chloride diffusion coefficient into mortar[END_REF]. First, in the presence of aggregates, cement paste contains more fine pores. Second, pores appear with a size larger than what is observed in cement paste without aggregates and may be attributed to ITZ. These data agree with previously published results [START_REF] Bentur | A review of the work of the Rilem TC 159-ETC, Engineering of the interfacial transition zone in cementitious composites[END_REF]: a more porous ITZ is observed when bulk cement paste is more dense (due to a lower value of the water to cement ratio). Furthermore, SEM observations [START_REF] Caré | Influence of aggregates on chloride diffusion coefficient into mortar[END_REF] show that aggregates are surrounded by an interface between aggregate and bulk cement paste (ITZ). The thickness (t I ) of ITZ is measured by SEM and is about 30 μm, depends on the type of cement and not on the aggregate size distribution. The ITZ volume fraction C I is evaluated according to [START_REF] Garboczi | Analytical formulas for interfacial transition zone properties[END_REF]:

C I = (1 -C A ) -(1 -C A )e V (t I ). ( 22
)
The function e V (t I ) depends on aggregate volume fraction, on aggregate size distribution and on ITZ thickness [START_REF] Garboczi | Analytical formulas for interfacial transition zone properties[END_REF]. Equation ( 22) takes into account possible overlaps of interfacial transition zones. Table II shows the ITZ volume fraction C I for the studied mortars.

Effective Chloride Diffusion Coefficient D eff

Table III presents the experimental results concerning D eff , the diffusion coefficients for the studied materials. The errors on the experimental values of diffusion coefficients D eff are estimated to be 30% [START_REF] Caré | Influence of aggregates on chloride diffusion coefficient into mortar[END_REF]. It appears that the diffusion coefficient D eff depends on the aggregate volume fraction and on the aggregate size distribution.

The influence of aggregate properties can be evaluated by the estimation of the diffusion coefficient in the cement paste (named D 1 P ) of the mortars. To determine D 1 P , some authors [START_REF] Delagrave | Influence of the interfacial zone on the chloride diffusivity of mortars[END_REF] used the upper value D eff max as an estimation of the effective diffusion coefficient in Equation ( 5). But this equation does not take into account the matrix-aggregate configuration in the studied materials, as shown previously, so D 1 P is estimated by reversing Equation (7). D 1 P is given in Table III. The diffusion coefficient D 1 P of the cement paste of the mortars is higher than that of the cement paste P. The influence of aggregates on the diffusion coefficient D 1 P may be attributed to the presence of the porous ITZ in the materials.

So, it is interesting to compare the diffusion coefficient for interfacial transition zone D I with that for bulk cement paste D B . The diffusion coefficient for ITZ D I is evaluated for the five mortars, by reversing Equation ( 21), where C A , C I are determined from the mixture proportions. The diffusion coefficient D B is assumed to be always equal to that of plain cement paste P (without aggregate), although the assumption concerning a constant value for D B is not always exactly valid because the presence of aggregates changes the porosity of cement paste even far from the interfacial zone. The obtained values of ITZ diffusion coefficient D I are given in Table III. The ratio D I /D B appears to decrease with increasing ITZ volume fraction C I . This result may be explained by the assumption on the diffusion coefficient D B which may be overestimated for mortar with high ITZ volume fraction (their bulk cement paste is more dense than the plain cement paste P).

This analysis confirms that the cement paste in mortar includes two phases with different diffusive properties. Experimental results show that the effective diffusion coefficient depends on the aggregate properties (volume fraction and size) and cannot be modeled, at macroscopic scale, only by the water to cement ratio as shown by Hobbs (1990).

If microscopic scale model (Eq. ( 3) with D 0 = 1.5 × 10 -9 m 2 /s and given in Table II) is considered, the predicted values (named D eff-1 ) are higher than the experimental ones. This result shows that the microscopic model is not valid according to [START_REF] Daian | Evaluation des propriétés de transfert dans les matériaux cimentaires. Etude critique des modèles[END_REF].

If mesoscopic scale model is considered, the mortars may be considered as a two-phase composite or a three-phase composite. In a first step, the mortars are considered as a two-phase composite without ITZ. In this case, the diffusion coefficient in the cement paste is assumed to be equal to that of cement paste P (without aggregate). The predicted values of D eff (named D eff-2 ) are given in Table IV. The difference between the predicted values D eff-2 and the experimental coefficients is higher than the experimental uncertainties. So, this result confirms that the mortars have to be considered as a three-phase composite. The effective diffusion coefficient have to be estimated from the value of the diffusion coefficient in the interfacial transition zone D I and that of the bulk cement paste D B . The diffusion coefficient of the bulk cement paste is assumed to be that of the cement paste P without aggregate. As the diffusion coefficient of ITZ D I cannot be directly measured [START_REF] Jaiswal | Impact of the interfacial transition zone on the chloride permeability of concrete[END_REF], D I is evaluated from the result of the mortar M2C given in Table III. An only value for D I /D B is considered, although this ratio given in Table III is not the same for all mortars. When D I = 16.2D B , the coefficients for the mortars M1C, M2M, M1F and M2F (named D eff-3 ) are estimated without fitting. Calculated coefficients are presented in Table IV. The difference between the test results and the predicted results is small, suggesting that the proposed model is suitable for the effective diffusion coefficient because they are less than the uncertainties on the values (30%).

These comparisons suggest that the model given by Equation ( 21) is the best one to estimate the effective diffusion coefficient of mortar. So, a procedure to estimate the diffusion coefficient D eff of chloride in mortar can be as follows. When this diffusion coefficient is known for the related cement paste (without aggregate) and for a reference mortar, it can be estimated for any other mortars, with any aggregate volume fraction and any aggregate size distribution.

Conclusions

A new model based on 'n-layered inclusion' is established and used to predict the effective diffusion coefficient of chlorides in mortar or concrete. The material is composed of three phases: the first phase is aggregates, the second one the inter-facial transition zone and the third one the bulk cement paste. In this case, the model used is called a four-phase model. This model is validated by experiments on specimens made of plain cement paste and of mortar. The following conclusions can be drawn. The relevant factors that affect chloride transport in mortar are the aggregates volume fraction C A and the volume fraction of interfacial transition zone, ITZ. The effective diffusion coefficients in the bulk cement paste and in ITZ depend on the mixture proportions of cement paste (water to cement ratio).

According to this model, the effective diffusion coefficient of chloride depends on the mixture parameters which are:

-The volume fractions C A , for aggregates and C I , for ITZ. The latter parameter C I depends on the aggregate volume fraction, on the size distribution and on this ITZ thickness (the thickness is controlled by the cement type). -The diffusion coefficients D B , for bulk cement paste and D I , for ITZ can be determined directly and after fitting the results.

So the main advantage of the proposed model is its simplicity for predicting the effective diffusion coefficient of mortar. It means that it is easily applicable. But, for practical applications, it has to be improved because the thickness of interfacial transition zone must be known for any cement-aggregate couple. A further validation with porous aggregates will be of interest.

Figure 1 .

 1 Figure 1. Scales used for studying diffusion problems. At macroscopic scale, the effective diffusion coefficient D eff is determined. At mesoscopic scale, D eff = f [C(i), D(i)], where C(i) = volume fraction and D(i) = diffusion coefficient of the different phases (1 = aggregate, 2 = interfacial transition zone and 3 = bulk cement paste). At microscopic scale, D eff = f [D 0 , , β], where D 0 = diffusion coefficient of ions in water and = porosity and β = tortuosity of pore space.

Figure 2 .

 2 Figure 2. A n-layered spherical inclusion in an infinite medium.

Table I .

 I Composition of the tested cement paste and mortars

		Mixture
		P	M1C	M2C	M2M	M1F	M2F
	W/C	0.45	0.45	0.45	0.45	0.45	0.45
	Aggregate	/	C	C	M	F	F
	C A	0	0.25	0.5	0.5	0.25	0.5
	interfacial transition zone differs significantly from that of the bulk matrix. These
	mortars compositions studied are far from standard composition but they allow to
	validate model.
	The experimental procedures and the results obtained are developed in Caré

Table II .

 II Data on the microstructure of the tested materials

		Mixture
		P	M1C	M2C	M2M	M1F	M2F
	(%)	22.6	15.1	11	12.2	17.9	13
	C I (%)	/	1.8	3.6	10.3	8	15.4

Table III .

 III Experimental values ofThe modeling of the effective diffusion coefficient is considered at different scales. Predicted values of the effective diffusion coefficient are given in TableIVaccording to different models at microscopic and mesoscopic scales.

	4.3. MODELING OF THE EFFECTIVE DIFFUSION COEFFICIENT
	Table IV. Experimental (1) and predicted values (2, 3, 4) of D eff according to Equations (3), (7) and (21)	D eff and fitted values of D 1 P (Eq. (7)) and D I /D B (Eq. (21)) Mixture	Mixture M1C M2C M2M M1F M2F	P M1C M2C M2M M1F M2F (1): D eff (m 2 /s) from experiments 5.4 × 10 -12 (2): D eff-1 from Equation (3) 1.6 × 10 -10 4.8 × 10 -12 1.1 × 10 -10 7.4 × 10 -12 8.1 × 10 -12 1.3 × 10 -10 1.9 × 10 -10 9.5 × 10 -12 1.6 × 10 -10	D eff (m 2 /s) 5.65 × 10 -12 5.4 × 10 -12 4.8 × 10 -12 7.4 × 10 -12 8.1 × 10 -12 9.5 × 10 -12 Deviation (%) [(1) -(2)]/(1) × 100 -2800 -2300 -1600 -2200 -1600	D 1 P (m 2 /s) / 8.1 × 10 -12 1.2 × 10 -11 (3): D eff-2 from Equation (7) 3.8 × 10 -12 2.3 × 10 -12 1.85 × 10 -11 1.2 × 10 -11 2.4 × 10 -11 2.3 × 10 -12 3.8 × 10 -12 2.3 × 10 -12	D I /D B / 19 16.2 13 Deviation (%) [(1) -(3)]/(1) × 100 +30 +53 16.3 12.8 +70 +53 +76	(4): D eff-3 from Equation (21) 5.2 × 10 -12 4.8 × 10 -12 8.1 × 10 -12 8.5 × 10 -12 1.1 × 10 -11	Deviation (%) [(1) -(4)]/(1) × 100 +4 0 +0.1 -14.7 -16.4


Jing P Li

Philippe Cassagnau

Eliane Espuche

Pierre Alcouffe

V Eronique Bounor-Legar

Specific Properties of In Situ Ruthenium-Catalyzed Polyamide 12/Polydimethylsiloxane Compatibilized Blend

Keywords: extrusion, gas permeability, gas permeation, hydrosilylation, polyamide, polyamide 12, polydimethylsiloxanes, polymer blend, polysiloxane, surface free energy

The main objective of this work focused on the chemical modification of polyamide 12 (PA12) properties through the reaction with a hydride-terminated polydimethylsiloxane (PDMS-SiH). The investigated PA12/PDMS-SiH blend was compatibilized by ruthenium derivative catalyzed hydrosilylation reaction in molten state. This original route enhanced interfacial adhesion and avoid PDMS-SiH leaching phenomenon between the two immiscible phases. More specifically, the size of PDMS-SiH domains in the blend decreased from around 4 lm to 800 nm and from 30 to 1 lm after compatibilization with 10 and 20 wt % PDMS-SiH, respectively. For the best compatibilized PA12/PDMS-SiH blend, the introduction of PDMS lowered the surface free energy and the PA12-based blend turned from hydrophilic to hydrophobic behavior, as evidenced by the water contact angle measurements. Gas permeability and CO 2 /H 2 and CO 2 /He gas selectivity were also improved with the increase in PDMS content. Besides, the mechanical properties were enhanced with 13% increase in Young's modulus after in situ compatibilization with 15 wt % PDMS-SiH. Thermal stability was also improved after compatibilization as the initial degradation temperature of reactive blends obviously increased compared with nonreactive ones.

INTRODUCTION Polydimethylsiloxanes (PDMSs) are of particular interest as they have a wide range of commercial applications resulting from their special properties, [START_REF] Clarson | Siloxane polymers[END_REF][START_REF] Scott | [END_REF] such as high chemical and thermal stability, low toxicity, hydrophobicity, high gas permeability, and so on. Due to these specific properties, the introduction of PDMS to other kinds of polymers was widely studied to improve their related properties especially hydrophobicity and gas permeability. To realize such improvements, synthesis of PDMS-based copolymers are widely reported. Ho et al. 3 synthesized poly(dimethylsiloxane-urethane-urea) (PDMS-PUU) block copolymers based on aminopropyl endcapped dimethylsiloxane [H 2 N(CH 2 ) 3 (Si(-CH 3 ) 2 O) n Si(CH 3 ) 2 (CH 2 ) 3 ANH 2 ], isophorone diisocyanate [5isocyanato-1-(isocyanatomethyl)-1,3,3-trimethylcyclohexane] and 1,4-benzenedimethanol by a two-step polymerization. The surface properties of various segmented block copolymers had been studied using dynamic contact angle (CA) analysis. It was shown that the surfaces reorganized by a mechanism in which the hard block urethane-urea domains migrated through the soft block silicone to the polymerwater interface, and this copolymer established development of nontoxic, antifouling coatings for use in marine environments. Miyata and coworkers 4 also synthesized different kinds of membranes, consisting in ethanolpermselective PDMS and water-permselective poly(methylmethacrylate) (PMMA), using PDMS macro-azoinitiator initiating the addition reaction between vinyl-ester terminated PDMS and MMA in solution. The copolymers were prepared into membranes to investigate the relationship between their microphase separation and their permselectivity for aqueous ethanol solutions during pervaporation. They found that the PMMA-g-PDMS membranes changed from water-to ethanol-permselective at a PDMS content of 35 mol %.

Syntheses of such PDMS containing copolymers are useful but not widely used solution to combine them to other kinds of polymers due to the high cost and complex synthesis steps. In recent years, polymer blending process [START_REF] Parameswaranpillai | Polymer Blends: State of the Art, New Challenges, and Opportunities[END_REF][START_REF] Paul | Control of Phase Structure in Polymer Blends[END_REF] seems to be more potential route to realize thermoplastic modification based on PDMS. However, for most of such polymer blends, the immiscible problem is obvious, due to the low surface tension and low viscosity of PDMS. [START_REF] Clarson | Siloxane polymers[END_REF] To obtain stable materials, there are two common routes for polymer blends compatibilization, one of them is adding amphiphilic copolymer which can work as interfacial agent to promote interface stability. As represented by Xu et al., [START_REF] Xu | [END_REF] a proportion of PE-b-PDMS diblock copolymer achieving through the esterification reactions between monohydroxy-terminated poly(dimethylsiloxane) (PDMS-OH) and the corresponding carboxyl terminated polyethylene (PE-COOH) in the presence of tetrabutyl titanate was used as a compatibilizer in the blends of highdensity polyethylene (HDPE) and silicone oil. 1 wt % of the copolymer promoted the dispersion of silicone oil in HDPE going from more than 5 lm silicone oil dispersed domains to no obvious phase segregation approaching by SEM observation and improved the mechanical properties of HDPE/ PDMS blends. Besides, there has been recent interest in reactive blending. During the process, polymers have reactivity with each other and can react to achieve in situ compatibilization. [START_REF] Baker | Reactive Polymer Blending[END_REF] This method is effective to control morphology and to design new materials without previous compatibilizer synthesis. For example, as reported by Zhou and Osby, [START_REF] Zhou | [END_REF] they obtained polycarbonate (PC)/PDMS compatibilized blends by using hydroxyl-terminated PDMS (PDMS-OH) which reacted with PC through twin-screw extrusion at 280 8C. The newly formed PC-PDMS copolymer provides a compatibilization effect for the stable submicron blend morphology in an otherwise immiscible PC/PDMS blend system. Silicone material was found to be well dispersed in the PC major phase forming small spherical domains with the domain size of about 0.2-0.9 lm. Such PC/PDMS blend was proved to have excellent low temperature impact toughness.

Our work was inspired by our earlier study conducted by Bonnet et al. 10 They developed a new and original reactive blending between Ethylene-vinyl acetate copolymer (EVA) and PDMS based on Ru 3 (CO) 12 catalyzed EVA carbonyl hydrosilylation by SiAH groups of hydride-terminated PDMS (PDMS-SiH). The occurrence of the hydrosilylation reaction at the EVA/polysiloxane interface promoted a homogenization of the blend. The method was also successfully applied to polyester polybutylene terephthalate (PBT) and polymethylhydrosiloxane (PMHS) 11 ; it formed stable crosslinking network after reactive blending. The PBT/PMHS blend showed slight improvement in hydrophobicity compared with pure PBT. More recently, the Ru 3 (CO) 12 catalyzed reactive blending was extended from polyester to polyamide which was never been reported. 12 Through model study, it confirmed the possibility and efficiency of ruthenium catalyzed amide hydrosilylation reaction, concentration of Nsilylated compounds can reach 70 mol % after 2 h reaction at 100 8C. Then the amide hydrosilylation reaction was extended to the reactive blending of polyamide 12 (PA12) with PDMS-SiH under molten processing conditions. The occurrence of such reaction at the interface between PA12 and PDMS-SiH was proved, N-silylated copolymers formed at the interface and worked as compatibilizer. Therefore, a structured PA12/PDMS blend was investigated by rheology and electronic microscopy at different scales. Specifically, the dispersion of PDMS domains decreased from 3 to 4 mm to around 0.8 mm in diameter forming submicronic morphology. In the meantime, PDMS-SiH oxidation reaction was partially observed under such reactive conditions and led to a second PDMS gel-based dispersion with a diameter of approximately 20-30 nm. Therefore, we proved that ruthenium-catalyzed amide hydrosilylation provided an original, efficient, and controllable route for PA12/PDMS compatibilization. As mentioned before, the introduction of PDMS to other kinds of thermoplastics were aimed to lead properties modification. The main objective of this part of work focused on the hydrophobicity, gas permeability, and mechanical properties modification of PA12 with different proportions of PDMS and investigated related impact factor and mechanism.

EXPERIMENTAL

Materials

Polyamide 12 (PA12) was supplied by Arkema (AESNO TL RILSAN). The melting point is 170 8C, number average molar mass M n is 26,000 g mol 21 , weight average molar mass M w is 47,000 g mol 21 , and the density is 1.01 g cm 23 at room temperature.

Hydride-terminated PDMS (PDMS-SiH) with a molar mass of 726 g mol 21 and triruthenium dodecacarbonyl [Ru 3 (CO) 12 ] were commercial products from ABCR. Moreover, viscosity ratio of PDMS-SiH and PA12 (k PDMS-SiH/PA12 )i s33 10 27 , and the molar ratio of functional groups [SiH]/[ACONH] is 6.7 3 10 22 .

Processing of PA12/PDMS-SiH Blends PA12 pellets were dried in vacuum at 80 8C for 24 h. Melt reactive processing of PA12 and PDMS-SiH was carried out in a Haake Plasticorder intensive batch mixer equipped with a Rheomix 600 internal mixer. The temperature of the mixer chamber was set at 170 8C and the rotation speed was 50 rpm. The resistant torque and temperature were monitored during whole process. In a typical experiment, dried PA12 was added in the mixer chamber and first mixed for 3 min to melt the polymer until the torque curve reached a plateau. Then, the catalyst and PDMS-SiH mixture with a predetermined composition (Table 1) was added in the molten PA12 with a syringe. Furthermore, nonreactive PA12/PDMS-SiH blend and pure PA12 were prepared with the same processing as references.

Samples for mechanical measurement were prepared using a 15 g-capacity DSM microextruder (Midi 2000 Heerlen, NL) with corotating screws (length/diameter (L/D) ratio 5 18). PA12 pellets were added in the extruder first, then PDMS-SiH and catalyst mixture were added with a syringe from the top gap between the screw and chamber surface. The blends were processed at 200 8C for 5 min under a 100-rpm speed and injected in a 10 cm 3 mold at 80 8C to obtain dumbbellshaped specimens (ISO 527-Type 5A: dumbbell-shaped, 25 3 4 3 2mm 3 ). In addition, we confirmed by SEM that the processing methods (Haake plasticorder and micro-extruder) have no effect on the final morphologies.

Methods of Characterization

The morphology of the polymer blend was first characterized by scanning electron microscopy (SEM) using a FEI QUANTA 250 microscope with an accelerating voltage of 10 KV. The samples were prepared in two different ways, either fractured in liquid nitrogen or a smooth flat surface of these was obtained by cryo ultramicrotomy at 2155 8C with a UC7 LEICA ultramicrotome. Then the surfaces were coated with a gold/palladium thin layer of 10 nm. The morphology was also examined by transmission electron microscopy (TEM). Ultrathin sections of about 80 nm from samples, taken in triplicate throughout the whole materials, were cut by cryo ultramicrotomy at 2155 8C and observed using a PHILIPS CM120 microscope operated at an accelerating voltage of 120 KV. Differential scanning calorimetry (DSC) measurements were performed on a Q20 (TA instrument) from 30 to 200 8C for PA12 and PA12/PDMS-SiH blends. The samples were kept for 3 min at 200 8C to erase the thermal history before being cooled and then heated at a rate of 10 8Cm i n 21 under Helium flow 40 mL min 21 . The crystallinity was calculated using the melting enthalpy for a 100% crystalline yield to PA12 of 95 J g 21 . 13 Thermogravimetric analysis (TGA) measurements of PA12 and PA12/PDMS-SiH blends were performed on a TA Q500 at a heating rate of 10 8Cm i n 21 . The samples were heated from 30 to 700 8C at a rate of 10 8Cm i n 21 under helium flow.

Rheological measurements of the samples were carried out using a Rheometer ARES. The samples were first molded as disk (d 5 25 mm, h 5 1 mm). The samples were heated at 200 8C for 5 min until equilibrium was reached and the gap between the two plates was adjusted to 1 mm. Then the samples were equilibrated for 2 min before starting the test. Nitrogen gas was used to prevent thermal oxidation of the samples. The linear complex shear modulus (G*(x) 5 G 0 (x) 1 jG 00 (x), j 2 521) was measured at 200 8C.

Uniaxial tensile tests of PA12/PDMS-SiH blends were performed at room temperature in a Shimadzu universal testing machine model AG-10 KN, speed of 30 mm min 21 until fracture. For calculation of the elasticity modulus, speed of 2m mm i n 21 was used, using segment mode between 0.05 and 0.25% of strain (among elastic deformation). The tabulated results are average of at least five measurements.

The water uptake measurements were carried out after 12day long immersion of test samples (hot pressed film, thickness: around 0.5 mm) in distilled water, at room temperature. It was checked that this immersion time was high enough to reach water sorption equilibrium for all studied samples.

Surface free energy of modified PA12 was determined at room temperature with the sessile drop method using a Dataphysics Digidrop CA meter equipped with a CDD2/3 camera. From CA, measurements performed with water c l 572:8mNm as probe liquids on films with smooth surface; polar and dispersive components of surface free energy were determined by using Owens-Wendt theory. 14 The tabulated results are average of at least five measurements on different parts of each sample.

Gas permeation experiments were carried out for He, H 2 , and CO 2 at 20 8C under an upstream pressure equal to 3 bars. The permeation cell consisted of two compartments separated by the studied membrane. A desorption step under secondary vacuum was performed before each permeation experiment. The gas permeability coefficient (P)w a s calculated from the gas flux measured in the steady state (J) according to the following equation:

P5 LJ Dp ( 1
)
where L is the thickness of the film and Dp is the pressure difference between the two cell compartments. The permeability coefficient, P, expressed in barrier unit (1 barrer 5 10 210 cm 3 (STP) cm cm 22 s 21 cm Hg

RESULTS AND DISCUSSION

Processability and Morphological Behavior

In our previous work, the reactive compatibilization of PA12/PDMS-SiH has been successfully proved in molten conditions under shear. 12 The formation of N-silylated polyamide and polyamine at the interface, as illustrated in the Figure 1, allowed to increase the interfacial adhesion between the two immiscible polymers, and finally promoted the dispersion of PDMS-SiH into the PA12 matrix. In the Figure 1, the evolution of the system morphology can be associated to the variation in torque during polymer blending. It is clear that after the introduction of PDMS-SiH to molten PA12, the torque value decreases due to the PDMS-SiH lubrication. Then it increases to around 17 Nm for both in nonreactive 15 and reactive system. As reported by Scott and Macosko as well as Cassagnau and Fenouillot, 16,17 the mixing torque variation is primarily due to the change in blend rheology caused by a morphological transformation in such blend. For nonreactive blend, the torque variation keeps steady after the slight increase and we can observe from SEM analysis [Fig. 2(a)], for such nonreactive blend, PDMS-SiH domains with a diameter around 3 lm dispersed in the PA12 matrix. This means the occurrence of weak mixing in such immiscible blend despite the extremely low viscosity ratio due to the low molar mass (726 g mol 21 ) of PDMS-SiH. 18 On the contrary hand, the torque value of PA12/ PDMS-SiH reactive blend does not reach a steady plateau as for the nonreactive one but it increases rapidly from 17 to 53 Nm before decreasing again as shown in Figure 1. The torque variation corresponds to the variation in the apparent viscosity of the bulk material and consequently to some relevant microstructure development. 13,17,19 Comparing the morphology (Fig. 2), it is clear that in presence of catalyst, new formed copolymers such as N-silylated polyamide and Nsilylated polyamine act as compatibilizer. The mixing became effective and the morphology changed drastically until the torque reached a plateau. Then the torque decreased and finally compatibilized stable blend with a decrease in the PDMS-SiH domain size (around 0.8 lm) and narrow domain size distribution of PDMS-SiH in PA12 matrix is obtained. In addition, during the process, a second PDMS gel based dispersion with a diameter of approximately 20-30 nm is also formed due to self-oxidation 12 as shown in Figure 2(d).

As PDMS has a lot of unique properties, theoretically the more PDMS is introduced, the more the modification of organic polymer will be obvious as shown by Mackenzie et al., 20 who found that the organically modified silicates take on a more flexible nature as the PDMS concentration was increased. But, as mentioned before, the PDMS-SiH acting as a lubricant, the shear rate was located in the PDMS-SiH fluid which coated the wall of mixing chamber. When the PDMS-SiH concentration is not high, effective mixing can be carried out. Along with the PDMS-SiH amount increasing, lubrication is more pronounced and there is only a part of PDMS which can be mixed into molten PA12. Even though the reaction is efficient, the formed compatibilizers were not in high concentration enough to promote the dispersion of whole PDMS as reported in some literatures. 21 Thus, the whole addition of PDMS-SiH resulted in a poor mixing, and the blends remain totally separated at the macroscale (the PDMS-SiH leads to an oily surface). In our processing conditions, it was possible to increase PDMS-SiH content up to 20 wt %, but in a specific route. At first, we added 10 wt % PDMS-SiH and catalyst mixture in the molten PA12. In such conditions, efficient reactive blending was observed. When the torque has reached a steady plateau, another 10 wt % of pure PDMS-SiH (without catalyst) was introduced. The morphology of PA12/PDMS-SiH (20%)-reactive blend [Fig. 2(c)] proved the efficient blending, PDMS-SiH domains dispersed in the matrix with a diameter around 1 lm and a narrow distribution. In addition, through simply comparing 10 and 20 wt % PDMS-based blends morphology, we found a higher concentration of PDMS-SiH phase in the one with 20 wt % PDMS-SiH [Fig. 2(c)]. It should be mentioned that the sample is not oily like the one with poor mixing. Thereby, it was proved to be an efficient way to increase PDMS-SiH content by introducing it step-by-step.

The study of torque variation and related morphology confirmed the reactive compatibilization between PA12 and PDMS-SiH catalyzed by ruthenium catalyst. The in situ formed copolymers promote the dispersion of PDMS-SiH in PA12 matrix. Stable PA12/PDMS-SiH blends with a reduction in domains size and a narrow size distribution of PDMS-SiH phase was observed. Specifically, the size of PDMS domains of reactive blends with 10 and 20 wt % contents decreases to 0.8 and 1 lm, respectively. As the polymers compatibilization impacts the properties of the blend, 22 the rheological, thermal, mechanical, and gas permeability properties were investigated to understand the morphology-property relationships of such PA12/PDMS-SiH blends.

Rheological Behavior

The rheological behavior of two-phase polymer blends is affected by the flow-induced changes in morphology. Compatibilizer leads to smaller domain size and narrower size distribution of the dispersed phase. In physical compatibilized blends (compatibilization through adding premade copolymers), the rheological properties are influenced by the amount, molar mass, and the architecture of the compatibilizer. 23 Figure 3(a,b) shows typical behavior of the dynamic modulus for the pure matrix and the compatibilized blends.

Results show that the two modulus of blends increase at low frequency, the storage modulus increases more rapidly than the loss one. The addition of PDMS-SiH in PA12 matrix significantly affected the frequency dependence of G 0 and G 00 especially for compatibilized PA12/PDMS-SiH blend with higher concentration (20 wt %) of PDMS-SiH. It means the blends have additional relaxation process. 24,25 As reported in the literature, this additional relaxation could be attributed to the shape-relaxation of drops 26 and the interfacial viscoelasticity of compatibilizer 27 in the blend.

But in this case only the shape relaxation cannot explain alone this viscoelastic behavior, there is also shear thinning behavior and yield stress behaviors at low frequency in Figure 3(c). Actually, this behavior is also attributed to the blend morphology at the nano-scale [Fig. 2(d)], to be more exact, nanostructure of PDMS (gel particles around 20-30 nm) formed through oxidation reaction and presented a second gel based nanostructure. Such nano particles dispersed in the blend and their amount increased with the increase in PDMS-SiH content. The effect of them to viscoelastic behavior is similar to nano-fillers, some of them can induce a decrease in interfacial tension and stabilized the blend. Both sharp relaxation of PDMS droplets and effect of PDMS nano gel particles should both be taken into account, but are still difficult to quantify for a complete understanding.

Thermal Properties: TGA Blend compatibilization plays an important role in the degradation and thermal behavior of the blends. 28,29 The comparison of thermal stability of PA12, PA12/PDMS-SiH nonreactive, and reactive is shown in Figure 4. The initial degradation temperature shifts toward higher temperature for the Ru 3 (CO) 12 catalyzed reactive blend in comparison with the noncatalyzed blend, which indicated that compatibilized blend is relatively more stable than the nonreactive one. More precisely, the weight loss at around 419 8C for nonreactive blend coming from the degradation of PDMS phase was no more observed for the reactive one. This indicated the improvement of PDMS phase stability in the blend, because the formed PA12-PDMS-SiH copolymer by hydrosilylation reaction enhances the interaction between phases.

Furthermore, as described before, a part of PDMS-SiH changes to gel-like nano particles due to oxidation reaction in the presence of ruthenium catalyst also leading to enhance the thermal stability. Our results are in agreement with observations of other researchers, 28,30 such as Giri et al. 28 They found that the initial degradation temperature improved with the increase in in situ formed ethylenemethylacrylate (EMA)-g-PDMS copolymers concentration until reaching a maximum value (i.e., 332 8C for 0 wt %, 415 8C for 10 wt %, 416 8C for 12 wt %, and 415 8C for 14 wt % EMA copolymer). Such increase is also found in PA12/PDMS-SiH (20%) reactive blend, confirming that compatibilization also promotes the thermal stability even when the PDMS content increased. The residues of PA12/PDMS-SiH reactive blends at 700 8C are also more important than the ones for pure PA12 and PA12/PDMS-SiH nonreactive blend. As shown in Figure 4, the residue at 700 8C increases from 0.29 to 0.65 wt % for PA12/PDMS-SiH (10 wt %) nonreactive and reactive blends respectively and kept increasing with the increase in PDMS content (0.82 wt % for the one containing 20 wt % PDMS). Satyanarayana and Alper 31 proved that in the presence of rhodium catalyst, PDMS formed crosslinking network which exhibited both higher thermal stability and residue.

Surface Free Energy PDMS exhibits hydrophobic behavior and poor water wettability. The introduction of PDMS can modify the hydrophilic character of PA12. Such modification was proved by the change in surface free energy.

In this work, we used the Owens-Wendt theory which is the most general one used to determine the surface free energy 14 basing on CA results of water and diiodomethane. PA12 and PA12-PDMS-SiH compatibilized blends show different wettability depending on the different polarity of the two solvents. The static water and diiodomethane CA measurement results are shown in Figures 5 and6 and Table 2. For pure PA12, the water CA is around 80 6 h u sl e s st h a n9 0 8;t h i sl a s tC Av a l u e being often considered to determine the limit between wettability and nonwettability. 32 The PA12-PDMS-SiH compatibilized blends are both hydrophobic since the corresponding water CA are both larger than 908 and increased from 100 6 2.18 to 106 6 1.88 with PDMS content increasing from 10 to 20 wt %. For diiodomethane, an increase in CA is also observed as PDMS is introduced in the PA12 matrix. Specifically, the diiodomethane CA increased from 45 6 2.28 for pure PA12 to 63 6 1.38 and 68 6 1.58 for compatibilized blends with 10 and 20 wt % PDMS content, respectively. It has to be noted that CA measurement was not carried out on nonreactive blend, since its surface appears oily caused by PDMS migration. Similar result was achieved by Zhang et al., 33 The surface free energy at room temperature [34][35][36] was derived as follows (Table 2) The achievement of hydrophobicity due to PDMS introduction was also published in some literatures. 38,39 For example, Chen et al. 39 prepared UV-curable PDMS-containing polyurethane (PU) oligomer (UV-PDMS-PU) where such UV-PDMS-PU system consisted in 1% (w/w) photo-initiator was coated either on PET or polyamide textile surface and then cured by UV-radiation. They found the washing stability of textile was improved due to the micro-phase separation of PDMS on the surface and improve its hydrophobicity. This mechanism is also suitable for our PA12-PDMS-SiH compatibilized blends, since SEM results confirm the micromorphology of the samples that PDMS with a diameter around 0.8 to 1 lm dispersed in PA12 matrix. This microstructure provides hydrophobicity to PA12-PDMS-SiH compatibilized blend.

Gas Permeability PDMS as mentioned in the Introduction section is a polymer well known to have the highest gas permeability and diffusivity among all of the polymers manufactured on an industrial scale. 40,41 Therefore, an increase in gas permeability is expected for the PA12-based materials with introduction of PDMS. The evolution of CO 2 permeability coefficients as a function of the blend composition is presented in Figure 7.

As expected, the permeability increased as the PDMS content increased. However, significant differences were observed between the reactive and nonreactive PA12-PDMS-SiH blends. Higher permeability values were obtained for the reactive system. To discuss more deeply, the impact of the type of blends (noncompatibilized versus compatibilized) on the gas transport properties, we also indicated the permeability values calculated from Maxwell law in the Figure 7.

The Maxwell model developed in 1954, [START_REF] Maxwell | A Treatise on Electricity and Magnetism[END_REF] originally used for electrical conductivity of particulate composites, can be adapted to gas permeability of mixed materials membranes as eq 2: where P is the effective permeability of the blend, P m is the permeability of the continuous phase (PA12 matrix), / is the volume fraction of the dispersed phase, known as loading. k d is the permeability ratio P d /P m , where P d is the permeability of the dispersed phase (PDMS).

P P m 5
The permeability coefficient values determined on both systems (reactive and nonreactive blends) are lying below the theoretical curve. Maxwell law was developed for ideal binary systems, meaning that it neither takes into account an eventual modification of the matrix due to the presence of the dispersed phase nor potential interfacial effects. [START_REF] Gain | [END_REF] The former factor can be important especially for semicrystalline polymer. Actually, the polymer crystalline regions are considered to be impermeable to small molecules such as gases and thus change in crystallinity degree can lead to a modification of the final permeability. However, in our case through the DSC analysis, we found no notable changes in PA12 crystallinity after the introduction of PDMS. The PA12 matrix still maintained its individual property and kept the crystallinity around 60%.

For the reactive PA12-PDMS-SiH system, in situ compatibilization took place leading to the coexistence of micrometer and nanometer sized PDMS rich domains dispersed within the PA12 matrix. One could expect a high gas diffusion rate in such materials due to the formation of more interconnected permeable phases. However, it was shown that PA12-PDMS-SiH interfacial interactions were reinforced in these systems. Moreover, the presence of crosslinked PDMS rich nanoparticles was also evidenced. These last phenomena have a detrimental effect on gas diffusion and could explain the slightly lower values of permeability obtained with respect to the theoretical ones. For PA12-PDMS-SiH nonreactive blend, it can be noticed that Pco 2 is 1.3 barrer. This value is lower than one of the reactive PA12-PDMS-SiH with the same composition (1.6 barrer) and lower than the theoretical value (1.75 barrer). This is due to that without compatibilization, PDMS is easy to leach out and enrich on the blend surface driving by the significant difference between PA12 and PDMS. Therefore, there are only a part of PDMS actually dispersed in PA12.

The gas transport study was enlarged to H 2 and He for the reactive PA12-PDMS-SiH blend, and the relative permeability (the ratio of blend permeability to neat PA12 permeability) was reported for each gas and each blend composition in Figure 8. The relative permeability of H 2 and He is very close to each other, since these two gases have similar kinetic diameters of 2.34 and 2.65 Å, [START_REF] Gain | [END_REF] respectively. Furthermore, gas permeability increases as a function of PDMS content. However, the CO 2 relative permeability values are always higher than those measured for He and H 2 . This result can be assigned to the high solubility (S)o fC O 2 in PDMS resulting in a higher P, as in a fickian transport mechanism with D the diffusion coefficient.

P5SD

According to the literature, 44,45 a penetrant's relative permeability in PDMS can be largely determined by its relative solubility. Specifically, the solubility of CO 2 in PDMS is much higher than the one of H 2 [1.29 and 0.05 cm 3 STP /(cm 3 atm), respectively]. Different gas permeability coefficients of CO 2 , He, and H 2 provides a potential application as gas selectivities for PA12-PDMS-SiH blends.

Mechanical Behavior

The mechanical properties of pure PA12, PA12-PDMS-SiHnonreactive, and PA12-PDMS-SiH-reactive with different PDMS contents were carefully analyzed and shown in Figure 9 and Table 3 (10 and 15 wt %). First, the Young's modulus, elongation at break, and tensile stress decreases (50% elongation at break, 21% tensile stress, and 6% Young's modulus decrease) as PDMS is introduced without compatibilization comparing with pure PA12. As there is no change in the crystalline behavior of PA12 matrix, it is obvious that the decrease in mechanical properties is due to a weak interfacial adhesion between PDMS and PA12. This had been proved by a lot of researchers, 46,47 such as Prakashan et al. 47 who found that PP/PDMS blend in the absence of compatibilizer was easy to cause debonding at the elastomer-matrix interface. On the contrary, PA12-PDMS-SiH blend after compatibilization shows higher Young's modulus (890 MPa) than pure PA12 (750 MPa) and PA12-PDMS-SiH nonreactive blend (700 MPa). The yield stress almost kept the same value before and after introduction of PDMS. The increase in Young's modulus is related to the in situ formed copolymers promoting the compatibilization and enhancing the interfacial adhesion. In addition, a part of PDMS changed from liquid to gel through oxidation reaction in the presence of Ru 3 (CO) 12 forming a second nano-size gel based dispersion, which also promotes the mechanical properties comparing with nonreactive blend. The decrease in elongation at break after introduction of PDMS especially for nonreactive blend is due to the defects causing by the significant difference of viscosity between PA12 and PDMS.

Through increasing PDMS content in PA12-PDMS-SiH compatibilized blend, the mechanical properties decreases slightly (5% for the Young's modulus and 7% for both the elongation at break and tensile stress) compared with PA12-PDMS-SiH (10 wt %)-reactive blend. This behavior had been discussed in some publications as Bremner et al. 48 who found that polyurethanes/PDMS blend existed an optimal level of PDMS concentration around 1.5-2%, leading to an enhancement of the mechanical properties. Over this concentration, phase separation of PDMS became significant leading to a decrease in the mechanical properties. Such behavior can be expected in our studied blend series, explaining the decrease in mechanical properties of PA12-PDMS-SiH (15 wt %)-reactive blend, because the size of PDMS domain increased with PDMS content increase which had been confirmed by SEM study.

CONCLUSIONS

Ruthenium-catalyzed hydrosilylation provides an original and efficient route to compatibilize PA12 and PDMS-SiH blend. As a result, dispersion of PDMS in PA12 matrix is promoted with increasing interfacial adhesion between the two immiscible phases compared with nonreactive blend. The introduction of PDMS phase to PA12 with a well and stable dispersion can significantly modify some properties of PA12. Through characterization, it is clear that PA12 was modified from hydrophilic to hydrophobic since related water CA increased from 808 to over 1008. Gas permeability, especially for CO 2 also improved comparing with pure PA12 due to the introduction of PDMS phase. In addition, the thermal stability of PA12-PDMS-SiH blends was enhanced after compatibilization with a similar initial degradation temperature as PA12 but an increase in solid residue due to the formation of N-silylated copolymers and PDMS self-crosslinking via oxidation. Besides, the mechanical properties are enhanced with 13% increase in Young's modulus after in situ compatibilization with 15 wt % PDMS-SiH due to compatibilization and formation of PDMS second gel-based nanostructure.

21 5 7 .5 3

 73 10 218 Nm 3 mm 22 s 21 Pa 21 5 3.348 3 10 216 mol mm 22 s 21 Pa 21 ) was calculated from the slope of the straight line in the steady state. The precision on P values was better than 5%.

FIGURE 1 FIGURE 2

 12 FIGURE 1 Scheme of reaction between PA12 and PDMS-SiH and variation in the torque between PA12 and PDMS-SiH (10 wt %) under shear in the internal mixer at 170 8C: reactive blending (solid line) and nonreactive blend (dotted line). [Color figure can be viewed at wileyonlinelibrary.com]

FIGURE 3 FIGURE 4

 34 FIGURE 3 Rheological behaviors as a function of frequency for PA12, PA12-PDMS-SiH (10%), PA12-PDMS-SiH (20%) reactive blends: (a) variation in storage modulus, (b) variation in loss modulus, and (c) complex viscosity. [Color figure can be viewed at wileyonlinelibrary.com]

  who found a hydrophilic behavior for a phenoxy based polymer (water CA 748)w h i c hc h a n g e dt o hydrophobic with a water CA 948 after the introduction of 1.7 wt %P D M Sc o p o l y m e r .T h e yp r o v e dt h a tt h ec h a n g ei ns u r f a c e property is due to the surface enrichment of the low surface energy component such as PDMS to minimize the overall surface free energy of the binary macromolecular system.

  : pure PA12 38.0 mN m 21 , 37 PA12-PDMS-SiH (10 wt %)-reactive blend 26.9 mN m 21, and PA12-PDMS-SiH (20 wt %)-reactive blend 24.5 mN m 21 . Besides, there is also a decrease in both dispersive component (c s d ) and polar component (c s p ) of the surface energy. c s d decreased from 33.0 to 26.1 mN m 21 (with 10 wt % PDMS) and to 24.3 mN m 21 (with 20 wt % PDMS). c s p decreased from 5 mN m 21 for pure PA12 to 0.8 mN m 21 (with 10 wt % PDMS) and to 0.2 mN m 21 (with 20 wt % PDMS), respectively. It is clear that the introduction of PDMS significantly decreased the surface free energy of PA12.

FIGURE 5

 5 FIGURE 5 Water CA of pressed films: (a) pure PA12, (b) PA12-PDMS-SiH (10%)-reactive, (c) PA12-PDMS-SiH (20%)-reactive.

FIGURE 6

 6 FIGURE 6 Diiodomethane CA of pressed films: (d) pure PA12, (e) PA12-PDMS-SiH (10%)-reactive, (f) PA12-PDMS-SiH (20%)reactive.

  112/k d 21 ðÞ = k d 12 ðÞ 12/k d 21 ðÞ = k d 12 ðÞ(2)

FIGURE 8

 8 FIGURE 8 Evolution of the relative permeability (CO 2 , He, and H 2 ) as a function of PDMS-SiH content. [Color figure can be viewed at wileyonlinelibrary.com]

FIGURE 9

 9 FIGURE 9 Typical stress-strain curves of PA12 and PA12-PDMS-SiH blends and zoom of elastic deformation. [Color figure can be viewed at wileyonlinelibrary.com]

TABLE 1

 1 Composition of the PA12/PDMS-SiH Polymer Blends

		Sample	Designation	m(PA12):m(PDMS-SiH):m(Catalyst)
	Internal mixer	PA12-PDMS-SiH (10%)-nonreactive	PA12:PDMS-SiH	90:10
		PA12-PDMS-SiH (10%)-reactive	PA12:PDMS-SiH:catalyst	90:10:1
		PA12-PDMS-SiH (20%)-reactive	PA12:PDMS-SiH:catalyst	80:20:1
	Micro-extruder	PA12-PDMS-SiH (10%)-nonreactive	PA12:PDMS-SiH	90:10
		PA12-PDMS-SiH (10%)-reactive	PA12:PDMS-SiH:catalyst	90:10:1
		PA12-PDMS-SiH (10%)-reactive	PA12:PDMS-SiH:catalyst	85:15:1

TABLE 2

 2 Surface Properties of PA12 and PA12/PDMS Blends After Compatibilization Evolution of the CO 2 permeability coefficient as a function of PDMS content in PA12/PDMS blends.

		h water (8)	h diiodomethane (8)	c s (mN/m)	c s	d (mN/m)	c s	p (mN/m)
	PA12-pure	80 6 4.5	45 6 2.2	38.0 6 1.3	33.0 6 0.1	5.0 6 1.2
	PA12-PDMS-SiH (10%)-reactive	100 6 2.1	63 6 1.3	26.9 6 0.5	26.1 6 0.3	0.8 6 0.2
	PA12-PDMS-SiH (20%)-reactive	106 6 1.8	68 6 1.5	24.5 6 0.4	24.3 6 0.5	0.2 6 0.1

([trif]) is representative of Pco 2 of PA12-PDMS-SiH(10%)-nonreactive blend, ([squf]) is representative of Pco 2 of PA12/PDMS-reactive blend. The dotted line is Maxwell theoretical value tendency of Pco 2 . [Color figure can be viewed at wileyonlinelibrary.com]

TABLE 3

 3 Mechanical Properties of PA12 and PA12-PDMS-SiH Blends

		Young's	Yield	Elongation	Tensile
	Sample Name	Modulus (MPa)	Stress (MPa)	at Break (%)	Stress (MPa)
	PA12	750 6 14	52 6 0.3	160 6 47 0 6 0.6
	PA12-PDMS-SiH (10%)-nonreactive	700 6 25	53 6 1.4	80 6 10	55 6 1.1
	PA12-PDMS-SiH (10%)-reactive	890 6 20	51 6 0.8	130 6 10	70 6 1.1
	PA12-PDMS-SiH (10%)-reactive	850 6 21	50 6 1.1	120 6 96 5 6 1.3


Frédéric Roger

Ky Dang

come

Prediction of the weld shape in arc welding

A numerical modeling example in multiphysics coupling

A numerical model of the arc welding process is presented. It deals with all arc welding process with and without filler metal (Gas Metal Arc Welding and Gas Tungsten Arc Welding). We consider the three dimensional modelling of the stationary weld pool with a free deformable surface. All coupled equations are solved by the finite volume method. We consider heat transfer ands fluid flow in the weld pool and the geometry of the free surface subjected to arc pressure and drop impact in GMAW. The location of the energy supplied by drops in the weld pool is assumed to be a function of the Weber number. Comparison with macrography shows a really good agreement of the model. RÉSUMÉ. Un modèle numérique des procédés de soudage à l'arc est présenté. Il traite des procédés de soudage à l'arc avec et sans apport de matière (soudage MIG/MAG et TIG). Nous considérons la modélisation tridimensionnelle d'un bain de fusion stationnaire avec une surface libre déformable. Toutes les équations couplées sont résolues par la méthode des volumes finis. Nous considérons les transferts thermiques et l'écoulement dans le bain de fusion ainsi que la géométrie de la surface libre soumise à la pression d'arc et à l'impact des gouttes en soudage MIG/MAG. La position de l'énergie fournie par les gouttes dans le bain de fusion est supposée être une fonction du nombre de Weber. Des comparaisons avec des macrographies montrent un très bon accord du modèle. KEYWORDS: arc welding, weld pool, drop impact, multiphysics, finite volume method. MOTS-CLÉS : soudage à l'arc, bain de fusion, impact de gouttes, multiphysique, volumes finis.

Introduction

The weld geometry in arc welding introduces stress concentration. Thus it plays an important role in fatigue strength of welded assembly [LIEU 00]. Most of the time fracture is located in weld toe which can be assimilated to a notch with a small tip radius. The aim of the numerical model of the weld pool behaviour during welding is to predict the thermal field which is coupled with the fluid flow in the weld pool [ZAC 89], and the thermo-metallurgical and mechanical behaviour of the structure. This coupling allows to predict the residual stress field but not the shape of the weld after solidification. The purpose of this study is to be able to predict the geometry of the free surface and the fusion zone of the weld for Gas Tungsten Arc Welding and Gas Metal arc welding. In the GMAW process, the filler metal is taken into account with mass conservation, impact pressure of drops on the free surface and energy contribution of drops in the weld pool.

A three dimensional model has been developped to simulate heat transfer and fluid flow in the weld pool under the electric arc. To reduce computing time we consider a stationary weld pool relatively to the welding electrode. This assumption is based on the regularity of the weld shape.

In our numerical model considers the weld surface is considered as deformable submitted to arc pressure and drop impact in GMAW. It is shown that the filler metal supply a significant ratio of the total welding energy. Watkins [WAT 89] propose an average ratio of 40% of the total welding power. This energy transfer is located at different depths in the weld pool. We consider an empirical formula relating the critical Weber number with the splashing depth of drops. Once this number is known, the location of the heat transfer is determined.

The model is divided in two parts. The first is devoted to the coupling between heat transfer ans fluid flow in the weld pool. The second part is devoted to the prediction of the free surface geometry of the weld pool. The fusion limit resulting from the first part is a boundary condition for the second part.

Modeling of the weld pool

Fluid flow of the molten metal has been recognised as an important part of the heat transfer during the welding process. However the mechanisms causing the fluid flow have not been clearly identified.

The driving forces for the convection in the weld pool are :

1. Electromagnetic force which are body forces resulting from interaction between the electric arc and the fluid flow ;

2. Buoyancy force driven by thermal gradient in the weld pool ; p g 3. Surface tension force induced by Marangoni effect driven by surface tension gradients resulting from thermal gradients ; 4. Arc drag force with arc plasma flow near the free surface.

In the model, all the three first effects are taken into account.

The calculation domain for solving the governing equations is shown in figure 1. The coordinate system (x,y,z) is linked to welding electrode and then move at the constant welding speed u 0 relatively to the workpiece. The coordinate system of the workpiece is ( , y', z).

Calculation domain

Weld bead

Governing equations

The governing equations for the temperature field and fluid flow are mass conservation, momentum conservation and energy conservation in the (x,y,z) coordinate system linked to the moving electrode. We consider a steady state of the fluid flow. The fluid is assumed viscous and incompressible. Laminar flow is assumed since the size of the pool is small. The thermophysical properties are assumed temperature dependant. Based on the above assumptions, the equations governing the weld pool can be expressed as follows :

Mass conservation :

u x + v y + w z = 0 [1]
where u, v, w mean x, y and z-component of liquid metal velocity vector respectively.

Energy conservation :

x C p u -u 0 T + y C p vT + z C p wT = x k T x + y k T y + z k T z + S [2]
where S is a volumic heat source. In the model S corresponds to the heat supplied by drops in the GMAW weld pool.

Momentum conservation :

u -u 0 u x +v u y +w u z = - p x + 2 u x 2 + 2 u y 2 + 2 u z 2 + F x [3] u -u 0 v x +v v y +w v z = - p y + 2 v x 2 + 2 v y 2 + 2 v z 2 + F y [4] u -u 0 w x +v w y +w w z = - p z + 2 w x 2 + 2 w y 2 + 2 w z 2 + F z -g T -T 0 [5]
where, Fx, Fy and Fz mean x, y and z component of the electromagnetic force vector F, respectively. is the thermal expansion coefficient.

We can noticed in these equations that the moving reference is take into account by the relative velocity term (u-u 0 ).

Heat transfer

Heat flux from arc is the major melting heat source. This heat is due to the electron flow and the convection from the plasma. In GMAW, the heat from overheated drops bring up to 40% of the total welding power.

The surface heat flux from the arc is assumed to have a gaussian distribution radially symmetrical around the electrode axis. Heat loss due to vaporization of the weld pool are take into account by a surface heat flux. p g At the top of the weld pool, the thermal coupling with the plasma takes place in a thin layer wich is the anode boundary layer in GTAW and the cathode boundary layer in GMAW. This layer connects the arc plasma with the weld pool. We assume that this zone is a disc with a radius a. Ouside this zone we assume convection and radiation with the surrounding air.

Based on the above assumptions the heat flux at the top of the workpiece can be expressed as follows :

In the plasma contact zone :

-T z = q 0 e -kr 2 -wH

[6]
where is the thermal conductivity, r is the radial distance relatively to the electrode axis, is the process efficiency, w is the vaporization rate (g.m -2 .s -1 ) and H is the specific enthalpy of vaporization (J.kg -1 ).

Around this zone :

-

T z = -T -T 0 - T 4 -T 0 4 [7]
In Gas Metal Arc welding, the melted electrode bring overheated drops. The depth of penetration in the weld pool depends on velocity before impact, drop diameter and surface tension. So we use the Weber number which is a ratio of kinetics energy of translation and surface free energy [ROG 00]. According to the value of the impact Weber number the impact is followed by a jet or a vortex in the weld pool. For high values (We>>1), kinetics energy is high and the drop impact is followed by a jet formation. For the low values (We<<1), the drop sinks and the energy is brought in depth. For mercury the critical number between the two modes is 8 [DAV 94].

We found a Weber number for steel electrode from 1,8 to 12,4 for welding current from 180 to 280 A [START_REF] Roger | Etude et modélisation de la formation d'un cordon de soudure à l'arc[END_REF]. It is near the critical number for mercury. We decide to take surface heat source, like arc heat flux, for low Weber value and volumic heat source in the weld pool for high Weber value and then high welding current.

Electromagnetic force field in the weld pool

Electromagnetic force induced convection in the weld pool. The interaction between the electrical current flow and the magnetic induction creates a body force (Laplace force).

The magnetic induction field in the workpiece is created by the current flow in the workpiece and the arc plasma [ROG 01]. As the magnetic flux field through the weld pool surface is continuous, it is necessary to consider the combined effect of plasma and current in the workpiece to estimate the total magnetic field. Indeed, numerical results show that both effects are of the same order of magnitude [ROG 01]. The figure 2 shows the total Fx electromagnetic force component in the x direction along the weld pool. This figure shows the magnitude of the plasma induced force and the force induced by the current flow in the weld pool.

Figure 2. Comparison between the force induced by the plasma and by the the current in the workpiece

Marangoni effect

Streaming in a liquid due to a thermally caused surface tension gradient is called Marangoni effect. According to Ishizaki et al [START_REF] Ishizaki | Penetration in arc welding and -convection in molten metal[END_REF], circulation in a weld pool may partly be due to a surface tension gradient resulting from the temperature gradient in the molten metal. They support this view by model experiments with paraffin and stearic acid.

Small concentrations of surface active element affect weld pool shape by altering surface tension gradient on the weld pool surface [START_REF] Heiple | Surface active element effects on the shape of GTA, Laser, and Electron beam welds[END_REF] and thereby changing the magnitude and/or direction of fluid flow in the weld pool. According to the sign of the surface tension gradient, the fluid flow direction can change.

In the absence of significant concentration of surface active element, the surface tension of molten metals and alloys decreases with increasing temperature.

The figure 3 [START_REF] Roger | Etude et modélisation de la formation d'un cordon de soudure à l'arc[END_REF] shows the fluid flow in a GTAW weld pool cross section for a negative and positive surface tension temperature coefficient.

Boundary conditions

Boundary conditions for the temperature field and the fluid flow are summarized in figure 4. The calculation domain is symmetric along the (x,z) plane so we represent a half workpiece.

Modeling of the deformable free surface

The weld free surface (x,y)=z(x,y) leans on the fusion line in front of weld pool and connects the solidified zone. We assume the solidified zone is regular so z/ x=0. These limits define the geometric boundary condition for the calculation of the surface. Figure 5 shows these boundary conditions. The geometric free surface equation can be deduced from the minimization of the free energy of the surface E t subjected to the constraint of mass conservation with the filler metal. For GTAW, this energy is defined as follows :

E x y g Pa dxdy f dxdy t S S 1 1 2 2 2 2 [8]
with E t : total free energy (J), : surface tension (N.m -1 ), Pa: arc pressure (N.m -2 ).

Without filler metal the constraint equation is defined as follows :

dxdy g dxdy S S 0 [9]
Using Euler-Lagrange's equation for the minimization of the free energy, we found the partial differential equation defined as follows :

P a + -g = - 1 + y 2 xx -2 x y xy + 1 + x 2 yy 1 + x 2 + y 2 3 / 2
[10] p g With x = / x, xy = 2 / x y ; is a Lagrangian multiplier that must be adjusted to respect the constraint [9].

In GMAW, we have a similar equation but we add the drop impact pressure P g . Lagrange multiplier must be adjusted to take into account the amount of filler metal from the melted electrode. The governing equation for the surface is defined as follows :

P a + P g + -g = - 1 + y 2 xx -2 x y xy + 1 + x 2 yy 1 + x 2 + y 2 3 / 2
[11]

Numerical considerations

Figure 6 summarized coupling and boundary conditions. Calculation modules are framed with continuous lines. Interactions between the welding electrode wire, the electric arc and the weld pool are take into account with boundary conditions. Governing equations for temperature field, fluid flow, electromagnetic force field and free surface geometry are solved by the finite volume method with a three-dimensionnal mesh. SIMPLER (Semi-Implicit Method for Pressure Linked Equations Revised) algorithm [PAT 80] is used to solve the coupling between continuity [1] and momentum equations [3,4,5]. An algorithm detects the weld pool zone location in the workpiece and solve the fluid flow only in this area. This algorithm reduces drastically the computing time to a few hours using a Personnal Computer.

Results and discussion

We present two applications of the model for GTAW and GMAW of medium thickness plates.

Modeling of the Gas Tungsten Arc Welding

Welding parameters are listed in table 1.

Modeling of the Gas Metal Arc Welding

Welding parameters are listed in table 2. We estimate the arc heat flux represent 45% of the welding power.This energy is distributed on a disc of radius 4mm.The drops energy is a assumed to be a volumic heat source distributed in a sphere with a diameter of 2mm (drops diameter predicted) at the depth of 3.5 mm. Figures 8 and9 show numerical results and comparison with a macrography. It is necessary to developp a specific model for drop interaction to have a predictive weld shape. Figure 8 shows clearly that the drops influence weld pool penetration and dig a crater in the plate.

Conclusion

We have developped a global model to take into account multiphysics coupling in weld bead formation. Convective heat transfer in the weld pool plays a major role for the shape of the fusion zone for GTAW. In GMAW, filler metal plays also an important role because drops bring important energy, material and momentum.

We choose the finite volume method to discretize the governing equations. This method is well adapted for conservation equations with multiphysics coupling.

We reduce drastically computing time considering only the weld pool zone for the fluid flow calculations.

For the prediction of the free surface geometry, we solve a minimization problem under constraint. GTAW and GMAW models shows good agreement with experimental macrography of the weld shape.

The global model permits to study the influence of operating parameters on the weld pool shape and then to optimize it. The calculation of the free surface model permits to calculate the wetting angle of the weld which have a great influence on the fatigue behaviour of the welding assembly.

Frédéric Roger *

 Roger -Ky Dang Van** * Unité de Mécanique, ENSTA Chemin de la Hunière F-91120 Palaiseau ** Laboratoire de Mécanique des Solides Ecole Polytechnique F-91120 Palaiseau ABSTRACT.

Figure 1 .

 1 Figure 1. Calculation domain moving with the welding electrode

Figure 3 .

 3 Figure 3. Fluid flow in the weld pool, (a) negative surface tension coefficient (-10 -5 N/m.K) (b) positive surface tension temperature coefficient (10 -5 N/m.K)

Figure 4 .

 4 Figure 4. Boundary conditions for velocity and temperature on a half workpiece

Figure 5 .

 5 Figure 5. Geometric Boundary conditions for the calculation of the free surface

Figure 6 .

 6 Figure 6. Coupling and boundary conditions for the global model

Figure 7

 7 Figure7shows a comparison between numerical simulation and a top view of the weld bead after solidification where the electric arc has been cut in stationary operating conditions. The simulation view shows stationary temperature field and fluid flow in the weld pool. The comparison is quite good as we found the top shape of weld pool.

Figure 7 .

 7 Figure 7. Comparison between simulation and experiment for a top view of the plate in stationnary operating conditions

Figure 8 .Figure 9 .

 89 Figure 8. Comparison between predicted shape and real shape in a macrography Figure 9. View of the 3D stationary weld shape with pool depression

Table 1 .

 1 Welding parameters for GTAW

	Steel plate Soldur 700 (C=0.068)	Plate thickness : 5 mm
	Arc Voltage : 11,5 V	Current : 240 A
	Welding speed : 3 mm/s	Electrode diameter : 2,4 mm
	Arc length : 2 mm

Table 2 .

 2 Welding parameters for GMAW

	Low carbon steel plate	Plate thickness : 6 mm
	Arc voltage : 25 V	Current : 240 A
	Welding speed : 430 mm/mn	Electrode diameter: 1,2 mm
	Free electrode length : 16 mm	Gas composition : Ar -2% CO 2