Keywords: cement paste, long term behavior, porosity, microstructure, TEM, diffusivity, modeling. MOTS-CLÉS : pâte de ciment, comportement à long terme, porosité, microstructure, MET, diffusivité, modélisation, homogénéisation

Experimental and modeling studies of the link between microstructure and effective diffusivity of cement pastes

This paper presents a review of complementary experimental and modeling studies of the link between microstructure and diffusivity of cement pastes. These studies were performed in the framework of the workgroup Concrete Engineering Barrier headed by Andra. Knowledge of the link between microstructure and diffusivity is needed to run properly the coupled chemical, transport and mechanical models used to assess the long term behavior of concrete in deep repository conditions. The first part of the paper is devoted to a description of the porous texture of ordinary Portland cement pastes, in particular obtained using Transmission Electronic Microscopy. The second and third parts are respectively dedicated to the presentations of an analytical method and a 3D numerical one, both of them aimed at linking microstructure and diffusivity of cement pastes. RÉSUMÉ. Cet article présente une revue d'études complémentaires d'expérimentations et de modélisations relatives à la relation entre la microstructure et la diffusivité des pâtes de ciment. Ces études ont été effectuées dans le cadre du Groupe de Travail Barrière Ouvragée en Béton piloté par l'Andra. La connaissance de la relation entre microstructure et diffusivité est nécessaire dans le cadre de l'utilisation de modèles couplés chimie-transport-mécanique pour la prédiction du comportement à long terme des bétons en condition de stockage. La première partie de cet article propose une description de la texture poreuse des pâtes de ciment Portland, en particulier sur la base de Microscopie Electronique en Transmission. Les deuxième et troisième parties de l'article sont respectivement consacrées aux présentations d'une méthode analytique d'homogénéisation et d'une méthode numérique 3D d'homogénéisation, toutes deux ayant pour objectif de relier microstructure et diffusivité des pâtes de ciment.

Introduction

In the context of a nuclear waste deep repository, concrete used as an engineered barrier would be leached by underground waters containing aggressive chemical species such as carbonates and sulfates [START_REF] Harris | The performance of cementitious barriers in repositories[END_REF][START_REF]Dossier 2005 argile, Synthèse : Evaluation de la faisabilité du stockage géologique en formation argileuse[END_REF]. These solicitations might induce evolutions of the mechanical and confinement properties of concrete in the long run. Models coupling chemistry, transport and mechanics are generally used to simulate the material degradations in order to evaluate their residual properties in repository conditions [START_REF] Adenot | Modelling of the corrosion of the cement paste by deionized water[END_REF][START_REF] Adenot | Modelling of the corrosion of the cement paste by deionized water[END_REF][START_REF] Peycelon | Long-Term Behaviour of Concrete: development of operational model to predict the evolution of its containment performance, Application to cemented waste packages[END_REF][START_REF] Bary | Simplified chemo-transfert modelling of carbonation and sulphate attack in saturated cement pastes[END_REF][START_REF] Maltais | Predicting the durability of Portland cement systems in aggressive environments-Laboratory validation[END_REF][START_REF] Planel | Long-term performance of cement paste during combined calcium leaching-sulfate attack: kinetics and size effect[END_REF]. Due to underground water solicitations, minerals would locally dissolve or precipitate in the microstructure inducing porosity changes and possibly cracking. To run properly the coupled models aiming at simulating cementitious material degradations, a precise and reliable knowledge of the relationship between their porosity changes and the evolution of their transport properties is required. In this framework, the purpose of this paper is to present a review of experimental and modeling studies performed in the framework of the workgroup Concrete Engineering Barrier headed by Andra and devoted to the problem of the link between total porosity and effective diffusivity of cement pastes submitted to porosity changes (opening or plugging). Modeling of the link between porosity and diffusivity of cement-based materials is a difficult topic because their porosity is multi-scale and complex. Consequently, it has not been yet completely characterized using experimental techniques and no porosity description precise enough is available at the moment to allow a direct assessment of the material transport properties. To avoid the problem of the multi-scale porosity description, homogenization and composite methods seem well adapted to link microstructure and diffusivity of cement based materials. Their goal is to calculate the property of a heterogeneous system according to the elementary properties of the components of the system and its microstructure [START_REF] Torquato | Random Heterogeneous Materials, Microstructure and Macroscopic Properties[END_REF].

The models presented in this study are based on hydrate assemblage descriptions considering a fixed scale. In this framework, the section 2 is devoted to the definition of an OPC paste hydrate assemblage, that allows first a good description of the microstructure and the porous texture of the system, and second to which homogenization and composite techniques can be applied. In this study, the definition of the hydrate assemblage of OPC pastes is based in particular on the description of these paste microstructures obtained using Scanning and Transmission Electronic Microscopy observations. A second part of the paper (sections 3) is dedicated to the presentations of an analytical composite model and a 3D numerical tool, both of them aimed at linking microstructure and diffusivity of cement pastes. The proposed methods allow the estimation of the effective diffusion coefficients of cement pastes at the scale of the Representative Elementary Volume (REV). The models are able to consider porosity opening (hydrolysis) as well as porosity plugging (carbonation, sulfate attack). Finally, in section 4, the comparison between the various experimental and modeling approaches presented in this paper leads to a discussion related to the phenomenology of the diffusive transport in OPC pastes.

Study of the microstructure of OPC pastes using microscopy

Introduction

This section proposes a description of the porous texture of OPC pastes in relation with their hydrate assemblage using Scanning Electron Microscopy (SEM) and Transmission Electron Microscopy (TEM). These two microscopy techniques are respectively appropriate for visualizing the porosity of cementitious materials in the micrometer and the nanometer scale ranges [START_REF] Scrivener | Backscattered electron imaging of cementitious microstructures: understanding and quantification[END_REF][START_REF] Groves | TEM studies of cement hydration[END_REF][START_REF] Maggion | Etude de l'évolution microtexturale des pâtes de silicate tricalcique hydrate[END_REF][START_REF] Richardson | Microstructure and microanalysis of ha rde ne d ordina ry Portland cement pastes[END_REF][START_REF] Viehland | Mesostructure of Calcium Silica Hydrate Gels in Portland Cement Paste: Short Range Ordering, Nanocrystallinity, and Local Compositional Order[END_REF][START_REF] Richardson | The nature of hydration products in hardened cement pastes[END_REF]. Observation of cement pastes using these techniques is helpful to describe the system at the scale where it is heterogeneous. It helps characterizing the system by defining all its homogeneous elementary components. On the basis of such description and categorization, it is possible to estimate some physical properties of the system as a function of the elementary component properties and of the microstructure (the way the components are structured in the space) by using homogenization or composite methods (see e.g. [START_REF] Torquato | Random Heterogeneous Materials, Microstructure and Macroscopic Properties[END_REF] or 3D numerical tool [START_REF] Garboczi | Computer Simulation of the Diffusivity of Cement-Based Materials[END_REF][START_REF] Bentz | Influence of silica fume on diffusivity in cement based materials, I. Experimental and computer modelling studies on cement pastes[END_REF].

SEM is a useful technique to visualize 2D representations of capillary porosity in cementitious systems from which it is possible in particular to get information on the scale range of this phase [START_REF] Scrivener | Backscattered electron imaging of cementitious microstructures: understanding and quantification[END_REF]. On the other hand, a number of investigations of the hydrates and the Calcium Silicate Hydrate (C-S-H) porosity in cement pastes have been performed using TEM [START_REF] Groves | TEM studies of cement hydration[END_REF] [START_REF] Richardson | Microstructure and microanalysis of ha rde ne d ordina ry Portland cement pastes[END_REF][START_REF] Richardson | The nature of hydration products in hardened cement pastes[END_REF]. In this study, TEM observations of C-S-H porosity are interpreted on the basis of the Jennings model that supposes the existence of two types of this hydrate in OPC pastes: the Low Density and the High Density C-S-H (LD and HD C-S-H) [START_REF] Jennings | A model for the microstructure of calcium silicate hydrate in cement paste[END_REF][START_REF] Tennis | A model for two types of calcium silicate hydrate in the microstructure of Portland cement pastes[END_REF]. Because LD C-S-H porosity is higher than that of HD C-S-H, it will be assumed that LD C-S-H diffusivity is higher than that of HD C-S-H (see sections 3 and 4). The presence of these various phases is important as they probably impact the diffusive transport in cement pastes at the macroscopic scale depending on which diffusive phase (HD C-S-H, LD C-S-H, capillary porosity) percolates in the system and controls the transport [START_REF] Béjaoui | Modeling of the link between microstructure and effective diffusivity of cement pastes using a simplified composite model[END_REF].

To examine the reliability of this description of the heterogeneous system, the purpose of the first part of the paper is to obtain qualitative information about capillary porosity using SEM and to identify LD and HD C-S-H using TEM in OPC pastes with various w/c. A C S paste is also studied for comparison as it allows getting information related to C-S-H in a much simpler system (without aluminate and ferrite phases) compared to OPC pastes. Two series of OPC cement pastes and a laboratory made C 3 S paste are studied. A finely ground C S (with a Blaine specific surface of 5830 cm 2 /g) was used to mix the C 3 S paste. The different mixes are presented in Table 1. Table 2 presents the estimation of the amounts of capillary porosity, LD and HD C-S-H quantities in the studied materials according to the Jennings and Tennis models [START_REF] Jennings | A model for the microstructure of calcium silicate hydrate in cement paste[END_REF][START_REF] Tennis | A model for two types of calcium silicate hydrate in the microstructure of Portland cement pastes[END_REF]. The values related to the other phases of the systems (portlandite, aluminates…) are not presented here. The hydration GHJUHH . RI WKH & 3 S paste was determined using backscattered electron images and image analysis as proposed in [START_REF] Feng | Estimation of the degree of hydration of blended cement pastes by a scanning electron microscope point-counting procedure[END_REF] and [START_REF] Scrivener | Backscattered electron imaging of cementitious microstructures: understanding and quantification[END_REF]. It was found to be equal to 0.856 with a 0.0168 standard deviation. For this measurement, 15 images with a 500x magnification were analyzed, what corresponds to a 0.65 mm 2 observed surface. As SEM observations were not performed exhaustively on the studied OPC pastes, their hydration degrees were estimated using Equation [1] which links the w/c and the maximal hydration degree of OPC pastes cured under water. This equation was obtained by fitting a number of experimental results given in the literature [START_REF] Baroghel-Bouny | Caractérisation des pâtes de ciment et des bétons[END_REF][START_REF] Lam | Degree of hydration and gel/space ratio of high-volume fly ash/cement systems[END_REF][START_REF] Carino | Curing of High-Performance Concrete : Phase I study[END_REF]) [START_REF] Igarashi | Analysis of cement pastes and mortars by a combination of backscatter-based SEM image analysis and calculations based on the Powers model[END_REF][START_REF] Pane | Investigation of blended cement hydration by isothermal calorimetry and thermal analysis[END_REF] It is specified that care must be taken regarding the mineralogical composition of the studied C 3 S paste determined using the Jennings and Tennis model. Indeed, this system could be out of the field of validity of the model because of the high fineness of the C 3 S used to mix it.

Table 2. Estimation of the amounts of capillary porosity, LD and HD C-S-H in the OPC and the C 3 S pastes according to the Jennings model

SEM observations: identification of capillary porosity and of LD and HD C-S-H at the micro-scale

Figures 1 and 2 present SEM backscattered electron images performed on flat polished sections of the 2-0.43 and the 3-0.385 samples. Various minerals are identified in the system according to a method given elsewhere [START_REF] Scrivener | Backscattered electron imaging of cementitious microstructures: understanding and quantification[END_REF] and using EDS analyses. Figure 1A shows the global repartition of the phases in the OPC system at the REV scale. To reinforce the notion of REV, it should be remembered that OPC pastes can be seen as being a homogeneous medium at the macroscopic scale, this homogeneous medium simply consisting of a juxtaposition of REVs. The REV scale is usually defined as being much bigger than the biggest heterogeneities of the system [START_REF] Torquato | Random Heterogeneous Materials, Microstructure and Macroscopic Properties[END_REF]. As shown on Figure 1 The system consists of an assemblage of minerals: anhydrous phases surrounded by inner C-S-H shells, portlandite, outer product zones consisting of C-S-H and aluminates phases in which remains capillary porosity (appearing in black on the image).

Figure 1B and Figure 2 show outer product zones respectively in t h e 2-0.43 OPC paste and in the 3-0.385 C 3 S paste. Capillary porosity appears to be mostly situated in the micrometer scale. In relation with the Jennings model, C-S-H appearing on Figure 2 can be divided into two families: the LD C-S-H that seem to appear as a jagged phase and the HD C-S-H that seem to consist as a more compact one. The annotated HD C-S-H in the center of Figure 2 forms a shell which could correspond to a Hadley grain [START_REF] Hadley | On the occurrence of hollow-shell hydration grains in hydrated cement paste[END_REF]. The potential presence of this HD C-S-H forming a Hadley grain in a zone of outer product of the C 3 S paste could be explained by some aspects of the hydration theory supported by [START_REF] Garrault | Atomic Force Microscopy Study of C-S-H Growth on C 3 S Surface during its Early Hydration[END_REF] A zone of compact C-S-H (HD C-S-H) would form at the beginning of the hydration on the surface of the grain. Then, the C 3 S grain would dissolve through this diffusive barrier leading to the formation of less compact C-S-H (LD C-S-H) and leading to a Hadley grain. Hadley grains also seem to be sometimes delimited by only LD C-S-H (see for example Figure 2 middle right). Concerning the 2-0.43 OPC paste, inner C-S-H probably correspond to HD C-S-H as proposed by [START_REF] Tennis | A model for two types of calcium silicate hydrate in the microstructure of Portland cement pastes[END_REF] It can be seen on Figure 1B that small inner C-S-H particles can also be found in the so called "outer product zone" of the 2-0.43 OPC paste.

To summarize, two types of C-S-H are identified in the 2-0.43 OPC paste and in the 3-0.385 C 3 S paste. According to the Jennings model, they can be distinguished as being HD and LD C-S-H. SEM microscopy gives some qualitative information about their localization in the microstructure. Inner C-S-H in the OPC paste seem to be composed of HD C-S-H. Small particles of HD C-S-H seem to be also present in the so-called "outer product zones". However, in terms of C-S-H, these latter seem to be mostly composed of LD C-S-H. An important number of Hadley grains can also be identified in the outer product zones of the 2-0.43 OPC paste (see for example Figure 1A middle left and Figure 1B) and in the ones of the 0.385 C 3 S paste (see Figure 2). As it will be shown later (section 3.4), together with the existence of two types of C-S-H, the presence of Hadley grains could also contribute to explain some aspects of the diffusive transport phenomenology in OPC pastes. Two types of C-S-H have been observed using SEM. However, because of its resolution, this technique is not adapted to examine their intrinsic porosities at the nanometer scale. To achieve this goal, C-S-H porosity is observed using TEM in the next section.

TEM observations and interpretations

Cement samples are prepared to obtain 30 micron thickness blades (stuck on optical glass blades). A micro-pincher (O' Medenbach) is used to drill 3 mm diameter cement discs. Cu slots for TEM are stuck by a fine layer of araldite on these discs. The unit "disc + slot" is then unstuck from the optical glass. After drying, the unit is subjected to ionic abrasion using a PIPS slimming ionic beam (Gatan) until the drilling of cement in the central zone of the slot. The edges of the hole are then ready for TEM observations. Carbon metallization is necessary to obtain the electronic conduction of the cement pastes. Observations are performed using a high resolution field emission gun TEM JEOL 2010F (0.18 nm structural resolution, KEVEX EDS system) and a high resolution thermo-electronic LaB 6 gun TEM JEOL 2000FX (0.28 nm structural resolution, TRACOR série II EDS system).

Figure 3 presents TEM images of zones identified as being HD C-S-H in various studied samples. C-S-H identification was confirmed by semi quantitative EDS analyses. Whatever the w/c (except for the sample 2-0.70 in which no HD C-S-H was found), the HD C-S-H seem to keep an invariant morphology characterized by a very finely divided porosity at the nanometer scale and a compact solid structure.

LD C-S-H are identified in the various studied samples as shown in Figure 4. Contrary to HD C-S-H, LD C-S-H present various types of morphologies. The porosity of LD C-S-H presented on the images b and c is situated in the scale of about ten nanometers. On the images a, d and g, the morphology of LD C-S-H corresponds to a structure defined by foliated sticks delimiting a larger porosity (ten to a hundred nanometer scale). Finally, LD C-S-H on the images e, f, h and i present at least a hundred nanometer scale porosity. On the images h and i, this porosity is delimited by compact and relatively fine sticks. On the images e and f, sticks are larger and seem to be themselves porous with a fine porosity delimited by C-S-H particles also having a foliated texture. It is specified here that C-S-H were found to be very sensitive to the electron beam. When too focused, it induced characteristic degradations creating more or less big circular porosity bubbles in the C-S-H and destroying their foliated-like texture. For example, a beginning of degradation of the foliated-like texture of large sticks of C-S-H can be observed on image f. As TEM corresponds to local investigations, it is not possible to perform a quantitative analysis of the various LD C-S-H volume fractions in the studied samples. However, the observations carried out seem to indicate that the LD C-S-H are all the more composed of larger pores when the w/c is high. These local porosity variations are in good agreement with the fact that C-S-H grey levels in backscattered electron images follow a continuous distribution rather than a discrete one [START_REF] Scrivener | Backscattered electron imaging of cementitious microstructures: understanding and quantification[END_REF]. That implies that there is a more or less continuous family of C-S-H ranging from high to low density. So, even if considering HD and LD C-S-H improves the accuracy of C-S-H description, it has to be mentioned that this model still remains a large simplification of real systems.

Summary

This section presented a study of the porous texture of OPC and C 3 S pastes using SEM and TEM. Capillary porosity appears to be mostly situated in the micrometer scale and a quite important fraction of this phase seems to be embedded in Hadley grains in OPC and C S pastes. samples 1-0.25 (a), 2-0.32 (b), 2-0.43 (c), 2-0.43 (d), 1-0.50 (e), 1-0.50 (f), 2-0.70 (g), 2-0.70 (h), 3-0.385 (i The microscopy observations indicate that C-S-H can be divided into two families, according to the Jennings model. HD C-S-H are present in the various OPC and C S paste samples (except for the sample 2-0.70 in which no HD C-S-H were found). They seem to keep an invariant morphology. The HD C-S-H seem to be localized in the inner C-S-H zones and in some Hadley grains situated in outer product zones. LD C-S-H are present in all the studied samples. They seem to be all the more composed of structures with larger pores when the w/c is high and they are principally localized in outer product zones. As stated before, the higher porosity of LD C-S-H, compared to HD C-S-H, may explain their higher self-diffusion coefficient [START_REF] Bary | Assessment of diffusive and mechanical properties of hardened cement pastes using a multi-coated sphere assemblage model[END_REF] and their lower elastic modulus [START_REF] Constantinides | The effect of two types of C-S-H on the elasticity of cementbased materials: Results from nanoindentation and micromechanical modeling[END_REF]. Such microstructural information on the elementary components contributing to diffusive transport (capillary porosity, LD and HD C-S-H) are of importance in order to model the link between microstructure and transport properties of cement pastes using a homogenization or a composite method, or a 3D numerical tool. They contribute to the knowledge needed to define realistic simplified description of the heterogeneous system on which these methods have to be based, as is shown in the next sections.

Modeling of the link between microstructure and diffusivity of cement pastes using a simplified composite model and a 3D numerical tool

Introduction

The degradations of concrete that would occur in a nuclear waste repository due to underground water solicitations are generally predicted on th e b a s i s o f experiments and using coupled chemo-transport and mechanical models [START_REF] Adenot | Modelling of the corrosion of the cement paste by deionized water[END_REF][START_REF] Adenot | Modelling of the corrosion of the cement paste by deionized water[END_REF][START_REF] Peycelon | Long-Term Behaviour of Concrete: development of operational model to predict the evolution of its containment performance, Application to cemented waste packages[END_REF][START_REF] Bary | Simplified chemo-transfert modelling of carbonation and sulphate attack in saturated cement pastes[END_REF][START_REF] Maltais | Predicting the durability of Portland cement systems in aggressive environments-Laboratory validation[END_REF][START_REF] Planel | Long-term performance of cement paste during combined calcium leaching-sulfate attack: kinetics and size effect[END_REF]. These tools require in particular the knowledge of the local evolution of the material diffusivity according to porosity changes induced by the dissolution or the precipitation of minerals in the system. Because of the complex and multi-scale porosity of cement pastes (see section 1), no experimental technique allows a complete characterization of cement paste porosity so that it could be possible to directly determine from it the material diffusivity. Thus, empirical porosity-diffusion laws are generally used in most contributions [START_REF] Mainguy | Modelling of leaching in pure cement paste and mortar[END_REF][START_REF] Maltais | Predicting the durability of Portland cement systems in aggressive environments-Laboratory validation[END_REF][START_REF] Planel | Long-term performance of cement paste during combined calcium leaching-sulfate attack: kinetics and size effect[END_REF]. However, in order to improve the reliability of the simulations, homogenization and composite methods (see e.g. [START_REF] Torquato | Random Heterogeneous Materials, Microstructure and Macroscopic Properties[END_REF] and 3D numerical tool [START_REF] Garboczi | Computer Simulation of the Diffusivity of Cement-Based Materials[END_REF][START_REF] Bentz | Influence of silica fume on diffusivity in cement based materials, I. Experimental and computer modelling studies on cement pastes[END_REF] seem well adapted to link microstructure and diffusivity of cement-based materials. Their goal is to calculate physical properties of a heterogeneous system according to the elementary properties of the components of the system and its microstructure. This kind of approach was applied to the problem of cement paste diffusivity by using a 3D numerical method [START_REF] Garboczi | Computer Simulation of the Diffusivity of Cement-Based Materials[END_REF][START_REF] Bentz | Influence of silica fume on diffusivity in cement based materials, I. Experimental and computer modelling studies on cement pastes[END_REF]. This model allows estimating OPC paste diffusivity as a function of the self-diffusion coefficients of capillary porosity and C-S-H on the basis of 3D numerical descriptions of the hydrated system obtained with the CEMHYD3D hydration code (Bentz, 2000a;[START_REF] Bentz | Influence of silica fume on diffusivity in cement based materials, I. Experimental and computer modelling studies on cement pastes[END_REF]. From a phenomenological point of view, the authors have found that for systems in which the amount of capillary porosity is lower (higher) than the percolation threshold value, the diffusion flow is principally controlled by the C-S-H porous matrix (capillary porosity). An analytical method based on the general effective media equation was developed elsewhere to achieve the same goal [START_REF] Oh | Prediction of diffusivity of concrete based on simple analytic equations[END_REF]. The model accounts for the diffusion through capillary porosity and C-S-H. The formulation also considers the geometrical percolation threshold of capillary porosity. The C-S-H matrix composition is considered as being homogeneous in these modeling methods what implies that its self-diffusion coefficient changes with the addition of silica fume [START_REF] Bentz | Influence of silica fume on diffusivity in cement based materials, I. Experimental and computer modelling studies on cement pastes[END_REF]) [START_REF] Oh | Prediction of diffusivity of concrete based on simple analytic equations[END_REF] and the type of cement [START_REF] Oh | Prediction of diffusivity of concrete based on simple analytic equations[END_REF] according to empirical evolutions. As shown in section 1, the C-S-H matrix in the case of OPC pastes can be described more precisely according to the Jennings model which considers two types of this hydrate (HD and LD C-S-H). On this basis, a multi-coated sphere assemblage formulation taking into account these two types of C-S-H and the capillary porosity was developed in order to estimate OPC paste diffusivity submitted to porosity opening (leaching) and plugging (carbonation) [START_REF] Bary | Assessment of diffusive and mechanical properties of hardened cement pastes using a multi-coated sphere assemblage model[END_REF].

The link between porosity and diffusivity of cementitious materials is studied in this section. In particular, the diffusion of chemical species dissolved in the pore water of saturated cement pastes submitted to porosity opening and plugging is considered. Cracking and aggregate impacts on cementitious material diffusivity [START_REF] Garboczi | Modeling the influence of the Interfacial Zone on the DC Electrical Conductivity of Mortar[END_REF][START_REF] Garboczi | Multiscale Analytical/Numerical theory of the diffusivity of concrete[END_REF][START_REF] Garboczi | New effective medium theory for the diffusivity or conductivity of a multi-scale concrete microstructure model[END_REF][START_REF] Tognazzi | Couplage fissuration-dégradation chimique dans les matériaux cimentaires : caractérisation et modélisation[END_REF]Bentz, 2000b) are not within the scope of this paper and convective transport is not considered here. Neither are electro-diffusion effects that can be described using the appropriate Nernst-Planck transport formulation [START_REF] Truc | Numerical simulation of multi-species diffusion[END_REF].

A simplified composite model is first briefly presented in this paper as it was already described in details elsewhere [START_REF] Béjaoui | Modeling of the link between microstructure and effective diffusivity of cement pastes using a simplified composite model[END_REF]. It allows the estimation of the effective diffusion coefficient of cement pastes at the scale of the Representative Elementary Volume (REV) and avoids the problem of the multi-scale porosity description by considering the system at the scale of the hydrate assemblage. The 3D microstructural features of the system are indirectly taken into account using a limited number of parameters. The model is able to account for porosity opening and filling-in. Second, the early developments related to a 3D numerical model are also presented. This 3D model also aims at linking microstructure and diffusivity of cementitious materials. Compared to the simplified one, it is based on a more detailed description of the heterogeneous system and directly accounts for its 3D properties. Some aspects of the 3D diffusive transport numerical resolution will be discussed in this section. Finally, the comparison of the results obtained using the two models allows one to highlight some important characteristics related to the phenomenology of the diffusive transport in OPC pastes. Before the presentations of the simplified analytical model in section 3.3 and the 3D numerical model in section 3.4., the next section presents the macroscopic experimental data of cement paste diffusivities on which are based the modeling tools.

Experimental data

The models presented in sections 3.3 and 3.4 are based on a measurement series of tritiated water (HTO) diffusion in OPC and BFS-PFA (blast furnace slag and pulverized fly ash) cement pastes characterized by various water-to-cement ratios (w/c) [START_REF] Richet | Amélioration de la fiabilité du modèle de dégradation chimique des matériaux à base de liant hydraulique -Détermination des coefficients de diffusion dans les zones dégradées : Avancement[END_REF], as presented in Figure 5. The effective diffusion coefficients were measured according to the widely used through-diffusion method performed using diffusion cells [START_REF] Richet | Etude de la migration des radioéléments dans les liants hydrauliques -Influence du vieillissement des liants sur les mécanismes et la cinétique des transferts[END_REF][START_REF] Tits | Diffusion of tritiated water and 22Na+ through non-degraded hardened cement pastes[END_REF]. Before measurements, materials were cured in saturated lime water incorporating sodium and potassium hydroxide, during 12 months for OPC pastes and 18 months for BFS-PFA cement pastes. More details related to experimental measurements are given elsewhere [START_REF] Béjaoui | Modeling of the link between microstructure and effective diffusivity of cement pastes using a simplified composite model[END_REF]. Whatever the w/c, OPC paste diffusivities are about one order of magnitude greater than BFS-PFA cement paste ones whereas their total porosities are slightly lower (Galle, 2001). Together with experimental data, modeling results obtained using the simplified model and the 3D numerical one are given in Figure 5. As it will be explained in the next sections, the modeling developments are focused on OPC pastes and the application of the modeling methods to BFS-PFA cement pastes will be the purpose of a subsequent paper.

Description of the Microtrans simplified composite model

Formalism of the Microtrans model and application to OPC pastes

The Microtrans model aims at simply describing and assessing the diffusivity of OPC pastes as a function of their microstructure (for details, see [START_REF] Béjaoui | Modeling of the link between microstructure and effective diffusivity of cement pastes using a simplified composite model[END_REF]. It has been shown in section 1 that OPC paste is a heterogeneous system consisting in particular of three diffusive phases: the capillary porosity and the LD and HD C-S-H. The self-diffusion coefficient of capillary porosity is obviously higher than the one of C-S-H, and because of its higher porosity, it is assumed that LD C-S-H diffusivity is higher than that of HD C-S-H. So, three diffusive rates are expected to occur in the various OPC pastes depending on which diffusive phase percolates in the system and controls the transport. It is assumed that these percolation phenomena are the major factor impacting the diffusivity of OPC pastes at the REV scale. The formalism of the model was developed in order to be able to capture simply this assumed phenomenology.

Figure 6 (left) presents a system composed of a matrix (phase 1) characterized by its volume fraction Q 1 and its self-diffusion coefficient D sd1 (for a given diffusing species) in which are distributed inclusions (phase 2) characterized by their volume fraction Q 2 and their self-diffusion coefficient D sd2 (for the same diffusing species). It is considered that D sd2 > D sd1 . The diffusivity De of the whole system can be assessed with a good accuracy by using the equivalent system presented in Figure 6 (right) in which the percolating fraction of phase 2 is placed in parallel with a mixed component composed of the non percolating fraction of phase 2 and phase 1. This description means that the diffusive transport is principally controlled by the percolating fraction of the most diffusing phase in the system. In the mixed component equivalent to a matrix in which are distributed non percolating inclusions of the most diffusive phase the transport is controlled by the less diffusive phase. The 3D disposition of these percolating diffusing components (single percolating phases or mixed components), is taken into account in the model via their tortuosity coefficients. These quantities account for the 3D arrangement of the diffusive components. The effective diffusion coefficient of the system can then be calculated using Equation [2]. In Equation [2], yQ 2 is the percolating volume fraction of phase 2, while T 2 (yQ 2 ) and T 12 [(1-y)Q 2 +Q 1 ] are respectively the tortuosity coefficients related to the percolating fraction of phase 2, and to the mixed diffusive component composed of phase 1 and the non percolating fraction of phase 2.

1 2 12 1 sd 1 2 sd 2 1 2 1 2 2 2 2 sd 2 Q Q ) y 1 ( T 1 D Q D Q ) y 1 ( Q Q ) y 1 ( Q Q ) y 1 ( ) yQ ( T 1 D yQ De [2]
The application of the method to OPC pastes leads to the equivalent systems presented in Figure 7. These systems evolve as a function of increasing w/c. For low w/c, HD C-S-H controls the transport by forming a matrix in which are embedded non percolating inclusions of LD C-S-H and capillary porosity (Figure 7a). With increasing w/c, the mixed component composed of LD C-S-H and capillary porosity becomes progressively percolating (Figure 7(b1)) until full percolation (Figure 7(b2)). Then, in turn capillary porosity starts percolating (Figure 7(c1)) until full percolation (Figure 7(c2)). In other respects, the conceptual model proposed in Figure 7 is supported by experimental results related to ice formation in OPC paste porosities obtained using Low Temperature Calorimetry [START_REF] Snyder | Suspended hydration and loss of freezable water in cement pastes exposed to 90% relative humidity[END_REF][START_REF] Bentz | Capillary Porosity Depercolation/Repercolation in Hydrating Cement Pastes via Low Temperature Calorimetry Measurements and CEMHYD3D Modeling[END_REF][START_REF] Bentz | Curing, Hydration, and Microstructure of Cement Paste[END_REF]. In particular, these results show that ice can form through various pathways percolated by capillary pores, open gel pores (LD-C-S-H) or dense gel pores (HD-C-S-H) depending on the paste w/c, hydration degree and curing conditions. The experimental underlining of these three percolation modes conforms with the basic concept adopted in the proposed model of diffusive transport.

The parameters of the model (volume fractions, percolation properties, selfdiffusion coefficients and tortuosity coefficients of the diffusive phases) were determined as follows. The mineralogical compositions (in volume fraction) of the studied OPC pastes (see Figure 5) were determined using the Jennings and Tennis model [START_REF] Jennings | A model for the microstructure of calcium silicate hydrate in cement paste[END_REF][START_REF] Tennis | A model for two types of calcium silicate hydrate in the microstructure of Portland cement pastes[END_REF]. A single percolation function of the diffusive components was chosen on the basis of OPC paste microstructure descriptions obtained with the CEMHYD3D code [START_REF] Bentz | Influence of silica fume on diffusivity in cement based materials, I. Experimental and computer modelling studies on cement pastes[END_REF]. The percolation threshold of the diffusive components is assumed to be equal to 15% (in volume fraction), and full percolation is assumed to be reached at 25%. [START_REF] Béjaoui | Modeling of the link between microstructure and effective diffusivity of cement pastes using a simplified composite model[END_REF] The problem related to the discrepancies of the various percolation function data proposed in the literature [START_REF] Navi | Three-dimensional characterization of the pore structure of a simulated cement paste[END_REF][START_REF] Garboczi | The effect of statistical fluctuation, finite size error, and digital resolution on the phase percolation and transport properties of the NIST cement hydration model[END_REF][START_REF] Snyder | Suspended hydration and loss of freezable water in cement pastes exposed to 90% relative humidity[END_REF][START_REF] Hu | Depercolation threshold of porosity in model cement: approach by morphological evolution during hydration[END_REF][START_REF] Ye | Percolation of capillary pores in hardening cement pastes[END_REF] is discussed in more detail in [START_REF] Béjaoui | Modeling of the link between microstructure and effective diffusivity of cement pastes using a simplified composite model[END_REF]. The HTO self-diffusion coefficient (D sd HTO ) of capillary porosity is here the one of water in water (2.2 10 -9 m 2 .s -1 ) [START_REF] Mills | Self-Diffusion in Electrolyte Solutions[END_REF]. The tortuosity coefficient of capillary porosity is identified on the basis of the diffusion results obtained for the more porous pastes (w/c > 0.45) (see Figure 5) according to an exponential evolution, so that experimental and modeling results are in good agreement. The mixed diffusive components including HD and/or LD C-S-H are regarded as forming a matrix as soon as they are percolating and their tortuosity coefficient (T HD C-S-H = T LD C-S-H ) is determined using the two phase composite sphere model of Hashin (see [START_REF] Béjaoui | Modeling of the link between microstructure and effective diffusivity of cement pastes using a simplified composite model[END_REF]. This model allows in particular calculating the tortuosity coefficient of a percolating solid matrix in which the concentration of non diffusive spherical inclusions is progressively increased. The D sd HTO of HD C-S-H and LD C-S-H are then adjusted using the macroscopic diffusion results obtained respectively for the lowest w/c (0.25 and 0.30) and the medium w/c values (0.35 to 0.42). D sd HTO of HD C-S-H and LD C-S-H were respectively found to be equal to 1.10 -12 and 9.10 -12 m 2 s -1 .

All the results are synthesized on Figure 8 on which the HTO diffusivities (corresponding to (D sd HTO )/T(Q)) of capillary porosity, LD and HD C-S-H are expressed as a function of their volume fractions. The diffusivities of the mixed components accounted for the model (see Figure 7) can be determined using the data presented in Figure 8.

Application of the model to the determination of porosity-diffusion evolutions

The Microtrans simplified composite model is now able to estimate the evolution of the effective diffusion coefficient of OPC pastes of various w/c submitted to porosity opening (due to hydrolysis) or plugging (due to carbonation for example). For porosity opening, a leaching inducing a progressive dissolution of portlandite, calcium aluminate phases and anhydrous cement (except for C 4 AF) as proposed in references [START_REF] Adenot | Modelling of the corrosion of the cement paste by deionized water[END_REF][START_REF] Matte | Durabilité des bétons à ultra hautes performances : Rôle de la matrice cimentaire[END_REF][START_REF] Mainguy | Modelling of leaching in pure cement paste and mortar[END_REF] is considered. The volume fraction of capillary porosity is progressively increased in the system due to these dissolutions. For porosity closing, it is supposed that capillary porosity is fully filled first, before the one of C-S-H. The filling-in of C-S-H porosity during carbonation is supported by the very low HTO effective diffusion coefficient (8.0 10 -14 m 2 /s) measured on a 0.38 w/c OPC paste carbonated under water at a pH equal to 11.5 (Le [START_REF] Bescop | Action de solutions carbonatées sur l'altération chimique des matériaux à base de liants hydrauliques[END_REF]. This low value seems to be only explainable by the closing of C-S-H porosity. It is specified anyway that the composite model could also treat other phenomenologies. For the present application, the self-diffusion coefficients of HD and LD C-S-H are varying here as a function of their porosity closing according to a simple Archie's law as explained more precisely in [START_REF] Béjaoui | Modeling of the link between microstructure and effective diffusivity of cement pastes using a simplified composite model[END_REF].

Figure 9 presents the porosity-diffusion evolutions obtained using the Microtrans model for three OPC pastes with various w/c. Total porosities of cement pastes are calculated using the Jennings model. For comparison, expressions given in other contributions are also plotted: on the basis of simulations performed with their 3D approach, Bentz et al., 2000 derived a modified Archie law linking the capillary porosity and the diffusivity of OPC pastes, and an exponential evolution was fitted on the basis of the experimental data given in Figure 5 by [START_REF] Mainguy | Modelling of leaching in pure cement paste and mortar[END_REF] It is specified that total porosities of the sound studied cement pastes were calculated using the Jennings and Tennis model [START_REF] Jennings | A model for the microstructure of calcium silicate hydrate in cement paste[END_REF][START_REF] Tennis | A model for two types of calcium silicate hydrate in the microstructure of Portland cement pastes[END_REF] by summing capillary porosity and C-S-H porosities. Porosities of sound LD and HD C-S-H are defined here as being equals to 0.20 and 0.31 in reference to the Jennings model [START_REF] Jennings | Colloid model of C-S-H and implications to the problem of creep and shrinkage[END_REF], and excluding pores smaller than 1 nm. The water in these small pores probably contributes negligibly to the overall transport and to the porosity measurements because it is strongly fixed on the solid surfaces [START_REF] Baroghel-Bouny | Caractérisation des pâtes de ciment et des bétons[END_REF]. The calculated values of sound paste total porosities are in very good agreement with total water porosities measured next to a 60°C oven-drying on OPC pastes made with the same cement and various water to cement ratios [START_REF] Gallé | Effect of drying on cement-based materials pore structure as identified by mercury intrusion porosimetry: A comparative study between oven-, vacuum-, and freeze-drying[END_REF]. It is also specified that capillary porosities (and not total porosities) were used to calculate HTO effective diffusivities using the expression of Bentz et al., 2000, even if, for comparison, the results are plotted as a function of total porosity in Figure 9.

Starting from the sound materials, the porosity-diffusion evolutions determined for the three OPC pastes using the Microtrans model (see Figure 9) can be interpreted considering the progressive percolation (de-percolation) phenomena of the diffusive phases related to the increase (decrease) of total porosity due to leaching (carbonation). Leaching induces a progressive percolation of capillary porosity until its full percolation in the 0.40 and 0.50 OPC pastes. As illustrated on Figure 7b, for the 0.25 paste, leaching first induces a progressive percolation of the mixed component composed of LD C-S-H and capillary porosity (this mixed component thus progressively controlling the transport). Then, subsequent leaching leads in turn to the progressive percolation of capillary porosity until full percolation. Concerning the filling of porosity, OPC paste diffusivities are essentially impacted by the plugging of C-S-H porosity, the ones of capillary porosity having very little influences because initially this phase is not percolating in the sound systems (except for the first few percents of capillary porosity plugging in the 0.50 OPC paste). [START_REF] Mainguy | Modelling of leaching in pure cement paste and mortar[END_REF] and [START_REF] Bentz | Influence of silica fume on diffusivity in cement based materials, I. Experimental and computer modelling studies on cement pastes[END_REF] For total porosities greater than 30%, the evolution determined from Bentz et al., 2000 indicates higher diffusivities than the ones issued from the Microtrans model and [START_REF] Mainguy | Modelling of leaching in pure cement paste and mortar[END_REF]. The diffusivity of the percolating capillary porosity in [START_REF] Bentz | Influence of silica fume on diffusivity in cement based materials, I. Experimental and computer modelling studies on cement pastes[END_REF] is expressed as a function of the square root of its volume fraction according to a modified Archie law. Differently, it is expressed according to exponential evolutions in the Microtrans model and in [START_REF] Mainguy | Modelling of leaching in pure cement paste and mortar[END_REF]. These discrepancies explain why the diffusivity calculated from Bentz expression increases more rapidly with the fraction of percolating capillary porosity than the ones issued from the other references (the three methods finally giving of course the value of HTO diffusivity in water for a system consisting of 100% capillary porosity). When considering porosity opening due to leaching, porosity-diffusion evolutions determined from the Microtrans model seem to be well appropriate as they give good results when implemented into a coupled chemo-transport model aiming at describing OPC paste leaching under pure water [START_REF] Béjaoui | Modeling of the link between microstructure and effective diffusivity of cement pastes using a simplified composite model[END_REF]. In these simulations, portlandite, aluminates and anhydrous phases were supposed to dissolve progressively. Simulated leaching rates and equivalent diffusivities of the heterogeneous leached zone were found to compare favourably with experimental results [START_REF] Béjaoui | Modeling of the link between microstructure and effective diffusivity of cement pastes using a simplified composite model[END_REF]. It is mentionned that the porosity-diffusion equation issued from the CEMHYD3D code has also been successfully implemented into a model simulating OPC paste leaching [START_REF] Marchand | Influence of calcium hydroxide dissolution on the transport properties of hydrated cement systems[END_REF]. A limitation of these simulations is that only portlandite was supposed to dissolve.

For total porosities lower than 10%, the diffusivities of OPC pastes determined from the two approaches issued from the literature are clearly higher than the ones issued obtained by the Microtrans model (see Figure 9), because they do not consider any plugging of C-S-H porosity. Indeed, the parameters related to these approaches were adjusted using only sound material diffusivity values.

A limitation of the composite model has to be mentioned here. Modeling studies performed using the CEMHYD3D code have shown that hydration and leaching can probably not be considered as being simply inverse processes regarding capillary porosity percolation phenomena [START_REF] Bentz | Modeling the Leaching of Calcium Hydroxide from Cement Paste: Effects on Pore Space Percolation and Diffusivity[END_REF]. Re-percolation of capillary porosity seems to occur at a smaller value than that at which the initial depercolation during hydration occurred (e.g. respectively 0.16 versus 0.18 for a 0.45 w/c OPC paste). In first approximation, this effect is not taken into account into the composite model. The same observation can be performed concerning the tortuosity coefficient of capillary porosity. It is identified using experimental results related to sound OPC pastes with various porosity (see section 3.3.1). Using it to describe the evolution of the tortuosity coefficient of capillary porosity due to leaching remains an approximation.

A composite model allowing the estimation of HTO diffusivity of cement pastes in the cases of porosity opening and filling was presented in this section. It is suggested that the model can be simply extended to assess the diffusivity of other diffusing species by using the formulation of the geometrical formation factor of porous materials as defined in [START_REF] Snyder | The relationship between the formation factor and the diffusion coefficient of porous materials saturated with concentrated electrolytes: Theoretical and experimental considerations[END_REF]. The porosity-diffusion laws obtained with the model have been successfully integrated into a simplified chemo-transport model aimed at describing the leaching of cementitious materials (not presented here, see [START_REF] Béjaoui | Modeling of the link between microstructure and effective diffusivity of cement pastes using a simplified composite model[END_REF]. However a number of microstructural parameters such as the tortuosity coefficients and the percolation functions of the diffusive phases were determined using inverse analysis or various models issued from the literature. A better knowledge of the values of these parameters would be necessary to reach a better validation level of the Microtrans model. Moreover, the application of the model to BFS-PFA cement pastes would first require being able to determine the mineralogical composition of these systems, in particular in terms of different types of C-S-H, which is not yet possible at the moment.

For the purpose of having a tool that takes into account intrinsically the microstructural parameters such as the tortuosity coefficients and the percolation functions of the diffusive phases, a numerical model was developed. It allows estimation of the effective diffusion coefficient of numerically reconstructed 3D microstructures of cement pastes. This tool is presented in the next section.

Modeling of the link between microstructure and diffusivity of cement pastes using a 3D numerical method

Description of the 3D model

The purpose of the 3D model is to rebuild an OPC paste microstructure in order to estimate its effective diffusivity by using a 3D numerical description. On the basis of the latter and attributing diffusive properties to the elementary components of the microstructure, the diffusivity of the whole system is calculated using a finite element method by simulating a simple through-diffusion measurement. As explained below, the 3D numerical model is closed to the one initially proposed by [START_REF] Garboczi | Computer Simulation of the Diffusivity of Cement-Based Materials[END_REF]. The proposed 3D model principally differentiates from Garboczi et al., one due to the implementation of two types of C-S-H (HD and LD C-S-H), the numerical resolution of the diffusive transport problem (finite element method instead of a 3D electrical conductor network) and the fact the 3D model does not simulate hydration but directly build an hydrated microstructure.

The OPC paste 3D microstructure is created by randomly distributing in the space spheres and multi-coated spheres representing the various minerals of the system. Multi-coated spheres are composed of an anhydrous core surrounded by a first shell of HD C-S-H and a second one of LD C-S-H. Simple spheres are homogeneous. They can be composed of the various minerals of the microstructure (anhydrous, LD C-S-H, HD C-S-H, portlandite, ettringite, hydrogarnet). Capillary porosity simply consists of the unfilled space. Due to its principle, the 3D model can be compared to other 3D hydration models that also use (growing) spheres to represent the various components of hydrating OPC pastes [START_REF] Jennings | Simulation of Microstructure Development During the Hydration of a Cement Compound[END_REF]) [START_REF] Navi | Three-dimensional characterization of the pore structure of a simulated cement paste[END_REF][START_REF] Ye | Three-dimensionnal microstructure analysis of umerically simulated cementitious materials[END_REF][START_REF] Ye | Percolation of capillary pores in hardening cement pastes[END_REF]. However, it should be well noted that the 3D model does not simulate the hydration of OPC pastes to obtain their 3D numerical descriptions. It requires the knowledge of the mineralogical composition of the hydrated system to be studied. In this section, the 3D model is applied to the 0.40 w/c OPC paste for which experimental and modeling results related to diffusivity were presented in the previous sections. The mineralogical composition of this paste was determined using the Jennings and Tennis model [START_REF] Jennings | A model for the microstructure of calcium silicate hydrate in cement paste[END_REF][START_REF] Tennis | A model for two types of calcium silicate hydrate in the microstructure of Portland cement pastes[END_REF], as stated before. In first approximation, the size distribution of the spheres and the multi-coated spheres used to represent the 0.40 OPC paste microstructure is supposed to follow the one of the anhydrous cement grains used to mix the paste. This rough approximation could be improved in the future on the base of SEM observations and image analysis for example. Other limitations of the description adopted in the framework of these early modeling developments will be given throughout this section and the next ones. As shown in Figure 10, the 0.40 OPC paste microstructure is thus created by randomly distributing in a 10 μm 3 cube, multi-coated spheres whose diameters are ranging from 29 μm to 13 μm, and homogeneous spheres whose diameters are ranging from 11 to 1 μm. The composition of the multi-coated spheres is determined with respect to the relative proportions of the phases in the hydrated system. Then, the number of simple homogeneous spheres of each of the various phases of the microstructure is calculated knowing the mineralogical composition of the paste. The distribution of the multi-coated spheres and the simple spheres is performed using an adapted version of the Genpartnew C++ procedure of the CEMHYD3D code (Bentz, 2000b). In this application, the 10 μm cube is discretized into 1 μm cubes. Spheres are composed of assemblages of elementary cubes (Bentz, 2000b). They are distributed in the system by decreasing size order. In this work, the Genpartnew procedure was modified so that it allows checking the maximum diameter of the sphere that it is possible to distribute in each voxel of the system. Thus, the spheres are only randomly distributed on voxels that can accommodate them and not on all the voxels of the system. This modification of the procedure avoids the distribution failures when the filling rates of space become important. It also lowers the time needed to perform the distribution.

The results obtained for the 0.40 w/c paste are illustrated in Figure 11. It shows the mesh of the sound paste together with the meshes of the progressively more and more leached paste according to the scenario given in section 3.3. The composition in volume fraction of the sound paste corresponds to 13.4% capillary porosity, 6.5% anhydrous, 28.8% HD C-S-H, 21.6% LD C-S-H, 15.3% portlandite, 8.0% ettringite and 6.4% hydrogarnet. Leaching of the paste is simply performed when creating the microstructures by progressively distributing smaller and smaller spheres of the phase(s) to dissolve (until full dissolution).

Percolation of the phases in the 3D model

The Burn3d C++ procedure of the CEMHYD3D code is used to assess the percolation properties of the various diffusive phases in the 0.40 w/c OPC paste for increasing leaching. Percolation is assumed to occur between two elementary cubes of the same phase when they share together a same face. The amounts of capillary porosity corresponding to the different leaching levels considered are equal to 13.4% (sound paste), 24.6% (partial leaching of portlandite), 28.8% (full leaching of portlandite), 39.3% (full leaching of portlandite and partial leaching of aluminates), 43.2% (full leaching of portlandite and aluminates) and 49.7% (full leaching of portlandite, aluminates and anhydrous).

It is represented in

fractions of capillary porosity, LD C-S-H and the mixed component composed of LD C-S-H and capillary porosity as a function of their total volume fractions for increasing leached states of the 0.40 OPC paste (volume fractions of capillary porosity and mixed component progressively increase due to leaching as explained in the previous section)

The percolation threshold of capillary porosity is found to be near from 13%, in a reasonable agreement with the value proposed in the framework of the simplified composite method (see section 3.3). It also compares favorably to the value of 16% determined on the basis of the CEMHYD3D code for a 0.45 w/c paste submitted to leaching [START_REF] Bentz | Modeling the Leaching of Calcium Hydroxide from Cement Paste: Effects on Pore Space Percolation and Diffusivity[END_REF]. In the sound system, capillary porosity does not percolate, and as indicated in Figure 12, 87.5% of LD C-S-H is percolating in the system. However, it is also indicated in Figure 12, that 100% of the mixed component composed of capillary porosity and LD C-S-H (representing a total volume fraction of 35%) is percolating in the system. As explained in section 3.3, this situation corresponds to the configuration b2 in Figure 7. The mixed component will thus control the transport in the sound OPC paste. More precisely, the LD C-S-H will control the transport in this mixed component and thus in the whole paste. When increased to 24.6% volume fraction due to portlandite dissolution, capillary porosity is almost fully percolating in the system. Thus, for this leaching level and higher ones, this phase will essentially control the transport. Using the 3D model, the percolation threshold corresponding to full percolation of capillary porosity is found to be near 25%, which corresponds to the value used in the simplified modeling approach presented in section 3.3. The percolating fraction of capillary porosity and mixed component decrease a little when considering full leaching of the OPC paste. Indeed, in this system, a small fraction of capillary porosity is created by the dissolution of anhydrous phase embedded in C-S-H. Thus, this fraction of capillary porosity remains not connected. Percolation results will be helpful to interpret the calculations devoted to the assessment of the 3D numerical sound and leached paste diffusivities in the next section.

Assessment of the 3D numerical paste diffusivities using a finite element code

The 3D descriptions of the 0.40 OPC pastes presented in section 3.4.1 are implemented in the Cast3m finite element code developed at CEA in order to perform diffusion calculations using these systems. The finite element calculation consists in the simulation of a trough-diffusion experiment. A concentration gradient is imposed between the two opposite faces of the studied system (upstream and downstream faces). The boundary conditions on these faces are thus imposed fixed concentrations (C 0 and null concentration respectively on the upstream and the downstream faces). On the other faces, it is imposed a linear evolution of the concentration starting from the value corresponding to the upstream face (C 0 ) and decreasing till the value corresponding to the downstream face (null concentration). These uniform flow boundary conditions correspond to the ones that undergoes an equivalent hom ogeneous REV in the steady -state period of a through-diffusion measurement. The steady-state diffusion problem simply reduces to Equation [3]:

0 ) c ( d a gr . D div sdi [3]
where sdi D and c are respectively the self-diffusion coefficients of the various elements composing the system and the concentration of the tracer.

Self-Diffusion coefficients are attributed to the elements of the mesh according to which component of the microstructure they belong (capillary porosity, HD C-S-H…). The flow (mol/s) crossing the downstream face of the system is calculated by numerically solving Equation [3] using the Cast3m finite element code. The homogenized effective diffusivity of the whole system (m 2 /s) is then calculated by dividing the flow by the imposed concentration gradient (C 0 /L in mol/m 4 , where L is the length of the structure). In order to check the isotropy of the sy stem , the numerical resolution is performed for each of the three directions of the space. The flows crossing the lateral faces of the system are also calculated. The tritiated water (HTO) self-diffusion coefficients attributed to C-S-H correspond t o t h e o n e s identified in the framework of the simplified composite method (e.g. 1.10 -12 and 9.10 -12 m 2 .s -1 respectively for HD and LD C-S-H, for details see 3.3). The selfdiffusion coefficient of capillary porosity is first set equal to the one of HTO in water: 2.2.10 -9 m 2 .s -1 at 25°C [START_REF] Mills | Self-Diffusion in Electrolyte Solutions[END_REF]. Second, another expression will be also used for this parameter as explained later in this section. For numerical simulation, self-diffusion coefficients of the non diffusive phases (portlandite, ettringite…) are set to a negligible value (1.10 -20 m 2 .s -1 ) compared to the ones of the diffusive phases. In Figure 13, the assessment of the 0.40 OPC paste diffusivities for the various studied leached states using the 3D numerical model are presented. Diffusivities shown in Figure 13 are the mean values over the three space directions. The maximal difference between a value related to a given direction and the mean value was found to be lower than 8%. That indicates that the system can roughly be considered as being isotropic. A way to improve the isotropy of the 3D system would be to perform calculations over larger systems while keeping the same size for the greatest heterogeneities and for the mesh elements. The drawback of such a solution is that the calculation time and the required memory would increase dramatically with the size of the system and more precisely with the number of elements composing the mesh.

Results presented in Figure 13 first indicate that the 0.40 OPC paste diffusivities are largely overestimated when assessed using the 3D model and setting a value of 2.2.10 -9 m 2 .s -1 for the HTO self-diffusion coefficient of capillary porosity. The progressive percolation of capillary porosity does not appear clearly on the curve for total porosities ranging between 26 and 37% volume fractions. Indeed, it is specified that even if capillary porosity does not percolate in the sound system as stated in the previous section, it seems to control the system HTO diffusivity that almost reaches 1.10 -10 m 2 .s -1 . This discrepancy between the percolation and the diffusivity results may be attributed partly to the finite element calculation method. Indeed, the diffusion calculation allows a transfer between two elements not only if they share together a same face, but also if they share together a same edge or a same node. This difference induces that the percolation threshold corresponding to the diffusion calculation is lower than the one corresponding to the geometrical percolation as defined and measured the Burn3d C++ procedure of CEMHYD3D code. So, for the sound 0.40 OPC paste, capillary porosity has dropped the percolation threshold related to the diffusion finite element calculation but it remains below the geometrical one defined by the Burn3d procedure. That probably explains why capillary porosity is found to control the transport in this system even if it is not geometrically percolating. As soon as the capillary porosity reaches the geometrical percolation threshold (corresponding to full percolation) defined by the Burn3d procedure, the discrepancy between the percolation definitions related to the Burn3d procedure and to the finite element calculation becomes of the second order. Indeed, in this case the two tools will consider that capillary porosity is fully percolating.

The diffusion calculation could be improved by considering a transfer between two elements only if they share together a same face and not in the case that they share together an edge or a node. This type of numerical calculation can be performed more efficiently by using the mixed-hybrid finite element (MHFE) method implemented in the Cast3m code [START_REF] Dabbene | Mixed-hybrid finite elements for transport of pollutants by underground water[END_REF]. The MHFE formulation is intrinsically equivalent to the Burn3d one regarding the mass transfer: the flows are transmitted from one element to the adjacent only through their common face. For our calculations, the use of this formulation would probably essentially have an influence on the assessed value of the sound 0.40 OPC paste diffusivity. It would probably be lower if determined using the MHFE formulation. The other assessed diffusivities would be probably less influenced by the use of this formulation as, in these systems, capillary porosity is always geometrically percolating. Moreover, it is also specified that the finite element method is not the most appropriate to perform calculations on heterogeneous systems with high contrast between the phase physical properties, because it is not locally conservative, contrary to MHFE [START_REF] Dabbene | Mixed-hybrid finite elements for transport of pollutants by underground water[END_REF]. 3D numerical assessment of OPC paste diffusivities using the MHFE method will be the purpose of a subsequent paper. Nevertheless, it is proposed in the next section a phenomenological assumption allowing an explanation of the overestimation of the 3D assessed diffusivities when using a self diffusion coefficient of capillary porosity equal to 2.2.10 -9 m 2 .s -1 .

The 3D disposition of capillary porosity in the system confers to this phase a given tortuosity coefficient. However, the previous obtained results show that this coefficient is too low to obtain accurate diffusivity results. So, as the tortuosity coefficient of capillary porosity due to its 3D disposition is too low, a correction is added to the self-diffusion coefficient of this phase in order to artificially increase its tortuosity coefficient. 3D diffusion calculations were thus performed again using a corrected expression of the self-diffusion coefficient of capillary porosity so that the evolution of the effective diffusivity of capillary porosity (at the scale of the 3D system) as a function of its volume fraction is the same as the one issued from the simplified analytical method presented in section 3.3 (see Figure 8). That achieved using the expression given in Equation [4] As shown on Figure 13, diffusivity results obtained using the simplified method and the 3D numerical one are in good agreement when using Equation [4]. That validates to a certain extent the consistency between the two methods when only considering the technical aspects of the up scaling problem. However, the latter comment must be balanced for the sound paste. Indeed, in this case, the simplified method does not account for any percolation of capillary porosity in the system, although, as stated before, the numerical diffusion calculation does consider at least that a fraction of capillary porosity is percolating. So, rigorously, the simplified method result should be compared to a MHFE calculation and not to a finite element one.

It is still not explained at this point why capillary porosity seems to have a particularly high tortuosity coefficient. It is shown in this section that the 3D microstructural description of OPC pastes used in this study is probably not accurate as it does not allow assessing the leached 0.40 OPC paste diffusivities when setting the HTO self-diffusion coefficient of capillary porosity equal to 2.2.10 m 2 .s -1 . That means that one or several microstructural parameters are not taken into account in the 3D model. It is specified that physical phenomena may also have an influence on diffusive transport as for example water adsorption, diffusion in the double-layer… However, for high amounts of capillary porosity corresponding to quite big pores, these phenomena probably have a low impact on diffusivity compared to the influence of the microstructural features of the system. The microstructural gap of the 3D model forces one to implement a variable evolution of the self-diffusion coefficient of capillary porosity what should normally not be the case when using such a 3D method. The implementation of a variable evolution of capillary porosity self-diffusion coefficient is in fact an artificial means to correct the 3D description. Comparing the experimental and the modeling approaches presented in this paper, the next section proposes a discussion related to the microstructural aspects not taken into account in the present version of the 3D model.

Comparison of the experimental and the modeling approaches -Discussion

It was shown in the previous section that the simplified composite model allows a quite accurate determination of the link between microstructure and diffusivity of OPC pastes, in particular when dealing with the impact of leaching. However, capillary porosity is found to have a particularly high tortuosity coefficient which cannot be explained only considering the simplified method. On the other hand, the microstructural description implemented in the 3D numerical tool exhibits a gap as it does not allow reproducing the high tortuosity coefficient of capillary porosity in a leached 0.40 OPC paste. The 3D model neither allows one reproducing the high tortuosity coefficient of capillary porosity in sound OPC pastes with high w/c. Indeed, the model overestimates the 0.60 and the 0.65 OPC pastes diffusivities by about one order of magnitude. In these systems, it is specified that capillary porosities (respectively equals to 25.0% and 27.7%) are fully percolating (according to the simplified method and the 3D numerical one). So, the adopted 3D method clearly does not allow a precise description of OPC paste microstructure. To improve the representativeness of the simulated microstructure, t h e i n c l u s i o n representing the various hydrates could be extended to various shapes [START_REF] Stora | Influence of inclusion shapes on the effective linear elastic properties of hardened cement pastes[END_REF]. Moreover, a better definition of the inclusion size distribution and a refinement of the mesh used for simulation could also probably contribute to improve the simulated microstructure. Another assumption potentially explaining at least partly the apparently high tortuosity coefficient of capillary porosity in the sound pastes could be the presence of Hadley grains in the microstructure, as identified in the first section of the paper. Considering that capillary porosity situated in the core of Hadley grains could be often completely surrounded by a shell of C-S-H (see for example Figure 2), the presence of Hadley grains in the microstructure would systematically generate a significant fraction of non percolating capillary porosity in the system. So, according to this assumption, only a fraction of capillary porosity not embedded in Hadley grains would actually directly contribute to diffusive transport. The system diffusivity would thus be lowered first because the percolating volume fraction of the most diffusive phase would decrease. Otherwise, it is found that the percolation threshold of capillary porosity is in good agreement with some direct experimental measurements [START_REF] Snyder | Suspended hydration and loss of freezable water in cement pastes exposed to 90% relative humidity[END_REF] when using the present version of the 3D model. Accounting for the presence of Hadley grains would increase the capillary porosity percolation threshold in the 3D model. However, this effect could be balanced by a more realistic finer discretization of the system which would induce on the contrary a decrease of the percolation threshold [START_REF] Garboczi | The effect of statistical fluctuation, finite size error, and digital resolution on the phase percolation and transport properties of the NIST cement hydration model[END_REF]. Implementing such features in the 3D model would decrease the system diffusivity first because of the lowering of the percolating fraction of capillary porosity as stated before, and second because the tortuosity coefficient of this percolating fraction of capillary porosity would probably increase due to the refinement of the description. That also means that the parameters related to capillary porosity in the simplified composite method would not be accurate. Indeed, accounting for the presence of Hadley grains would induce that a fraction of capillary porosity would always remain unconnected even for high volume fractions of this phase. Thus the percolation function would have to be changed. That would also induce a change in the tortuosity coefficient of capillary porosity accounted for in the simplified method. Indeed, as a smaller quantity of capillary porosity would percolate, its tortuosity coefficient would have to be lowered to obtain the same diffusivity result. Such a 3D microstructural description accounting for various shapes of inclusion, a better definition of the inclusion size distribution, a refinement of the discretization of the system and the presence of Hadley grains could allow one getting a good consistency between the simplified method and the 3D one (with setting .) Por . Cap ( D sdHTO = 2.2.10 m 2 .s -1 ). The same type of phenomenological reasons could potentially explain the apparently high tortuosity coefficient of capillary porosity in the leached 0.40 OPC paste. Moreover, it could be possible that a number of hydrates (portlandite, aluminate phases…) more or less embedded in C-S-H create themselves encapsulated (non percolating) capillary porosities when dissolving. This last assumption could be checked in the framework of further work, in particular using SEM observations of leached OPC pastes.

Conclusion

This article presents experimental and modeling studies of the link between microstructure and effective diffusivity of cement pastes in saturated conditions. The knowledge of this link is needed in order to run properly the coupled chemical, transport and mechanical models used to evaluate the long term behavior of concrete submitted to chemical attacks due to underground waters in nuclear waste deep repository conditions. As cement-based material porosity is mutli-scale and complex, it is not yet possible today to completely characterize experimentally its texture in order to directly assess the material transport properties. Considering these aspects, homogenization and composite methods, as well as 3D numerical tool seem well adapted to treat the problem. Their goal is to estimate properties of heterogeneous systems at the REV scale as a function of the various component properties and also considering the microstructure, that is, the way the components are disposed in the space.

A first part of the paper is dedicated to the study of OPC paste microstructure and porosity using Scanning and Transmission Electron Microscopies. Observation of cement pastes using these techniques contributes to characterize the system at the scale where it is heterogeneous by identifying all its homogeneous elementary components. It also allows one getting qualitative and quantitative information related to the microstructural properties of these components (shape, disposition in the space, scale range…).

As presented in the second part of this study, these information are taken into account in the development of a simplified analytical composite model and a 3D numerical one, both of them aiming at linking microstructure and effective diffusivity of OPC pastes. The simplified composite model allows one estimating the effective diffusivity of OPC pastes by considering the contributions of capillary porosity and LD and HD C-S-H via their self-diffusion coefficients. The description avoids the problem of the multi-scale description of OPC paste porosity by considering the system at the scale of the mineralogical assemblage. The model formalism accounts for the fact that the diffusive transport is principally controlled by the most diffusive phase percolating in the system. The parameters of the simplified method being determined using inverse analysis or various models issued from the literature, it is available to estimate the evolution of OPC paste diffusivities as a function of porosity changes due for example to leaching or carbonation. However, a direct experimental measurement of the microstructural parameter values would allow to reach a better validation of the simplified model.

In parallel, a 3D numerical model is developed in order to have a tool which solves a complete heterogeneous system, taken to be as realistic as possible, with a size close to the REV one. The OPC paste 3D microstructure is created by randomly distributing in a 10 6 μm 3 cube spheres and multi-coated spheres representing the various minerals of the system (anhydrous, LD C-S-H, HD C-S-H, portlandite, ettringite, hydrogarnet). Capillary porosity simply consists of the unfilled space. The diffusivity of the numerical OPC paste is assessed by attributing self-diffusion coefficients to the components of the system and performing a finite element calculation that simulates a through-diffusion measurement through the numerical microstructure. It is shown in the paper that the adopted 3D method does not allow a precise description of OPC paste microstructure. The 3D microstructural description could be improved by accounting for various shapes of inclusion, a better definition of the inclusion size distribution, a refinement of the discretization of the system and the presence of Hadley grains in the system. Moreover, it is shown that the mixedhybrid finite element formulation would be more adapted than the finite element method to perform 3D diffusion simulations as it should allow accounting more precisely for the percolation phenomena.

  Béjaoui* -Benoît Bary* -Serge Nitsche** Damien Chaudanson** -Cécile Blanc* * CEA Saclay, DPC, F-91191 Gif-sur-Yvette cedex syriac.bejaoui@.cea.fr ** CRMCN-CNRS, Campus de Luminy, F-13288 Marseille cedex 09

Figure 1 .

 1 Figure 1. SEM images (backscattered electrons) of the 2-0.43 OPC paste: A) representative area of the sample, B) detailed image of an Outer Product Zone

  Figures 1 and 2 present SEM backscattered electron images performed on flat polished sections of the 2-0.43 and the 3-0.385 samples. Various minerals are identified in the system according to a method given elsewhere[START_REF] Scrivener | Backscattered electron imaging of cementitious microstructures: understanding and quantification[END_REF] and using EDS analyses. Figure1Ashows the global repartition of the phases in the OPC system at the REV scale. To reinforce the notion of REV, it should be remembered that OPC pastes can be seen as being a homogeneous medium at the macroscopic scale, this homogeneous medium simply consisting of a juxtaposition of REVs. The REV scale is usually defined as being much bigger than the biggest heterogeneities of the system[START_REF] Torquato | Random Heterogeneous Materials, Microstructure and Macroscopic Properties[END_REF]. As shown on Figure1, their sizes are about 30 P LQ WKH SDVWH

Figure 2 .

 2 Figure 2. SEM image (backscattered electrons) of an Outer Product Zone in the C 3 S paste 3-0.385

Figure 3 .Figure 4 .

 34 Figure 3. TEM images of HD C-S-H, samples 1-0.25 (a), 1-0.50 (b), 3-0.385 (c)

Figure 5 .

 5 Figure 5. HTO effective diffusion coefficients of OPC and BFS-PFA cement pastesas a function of w/c together with modeling results obtained using the simplified composite model and the 3D numerical one (respectively presented in sections 3.3 and 3.4) -3D simulations are only applied to the case of the 0.40 w/c OPC paste

Figure 6 .

 6 Figure 6. Sketch of a system made up of two phases having different self-diffusion coefficients (D sd2 > D sd1 ) (left) and its equivalent system in the Microtrans model (right) (from[START_REF] Béjaoui | Modeling of the link between microstructure and effective diffusivity of cement pastes using a simplified composite model[END_REF]

Figure 7 .

 7 Figure 7. Disposition of the diffusive phases of OPC pastes for increasing w/c in theMicrotrans model (from[START_REF] Béjaoui | Modeling of the link between microstructure and effective diffusivity of cement pastes using a simplified composite model[END_REF]

Figure 8 .

 8 Figure 8. HTO effective diffusivities (D sd HTO /T(Q)) of the OPC paste diffusive phases (from[START_REF] Béjaoui | Modeling of the link between microstructure and effective diffusivity of cement pastes using a simplified composite model[END_REF]

Figure 9 .

 9 Figure 9. Porosity-Diffusion evolutions (HTO) determined for various OPC pastesusing the Microtrans simplified composite model (the squares on the three OPC curves correspond to the sound materials) and given in the references[START_REF] Mainguy | Modelling of leaching in pure cement paste and mortar[END_REF] and[START_REF] Bentz | Influence of silica fume on diffusivity in cement based materials, I. Experimental and computer modelling studies on cement pastes[END_REF]

Figure 10 .

 10 Figure 10. Composition of the 0.40 w/c OPC paste expressed in term of the number of multi-coated spheres and simple spheres contained in a 10 μm cube

Figure 11 .

 11 Figure 11. Representation of the 0.40 w/c OPC paste meshes for various leaching levels: a) sound paste, b) paste without portlandite, c) paste without portlandite and aluminate phases, d) paste without portlandite, aluminate phases and anhydrous phases (hydrate and anhydrous phases are represented using various grey-levels, capillary porosity appears in black)

  Figure 12. Percolation fractions of capillary porosity, LD C-S-H and the mixed

Figure 13 .

 13 Figure 13. Assessment of the HTO diffusivities of the 0.40 w/c OPC paste for various leaching states using the 3D numerical model

Table 1 .

 1 Characteristics of cementitious materials used for microscopy studies

		Cement	Age (years)	Cure	w/c	Denomination
	Series 1	CEM I 52.5 PM ES HTS	11.0	Lime water	0.25 0.50	1-0.25 1-0.50
	Series 2	CEM I 52.5 PM ES CP2	1.25	Lime water	0.32 0.43 0.70	2-0.32 2-0.43 2-0.70
	C 3 S	C 3 S	0.083	Lime water	0.385	3-0.385

  .

	0	239 .	0	.	745	tanh	3	.	( 62	w	/	c	0	.	) 095	[1]

Acknowledgements

The authors thank Andra for financial support.

Adenot F., Durabilité du béton: caractérisation et modélisation des processus physiques et chimiques de dégradation du ciment, Ph-D thesis, Université d'Orléans, France, 1992.


Q Guegan

J-N Foulc

Characterization and effectiveness of ER fluids: an attempt to unify methods and criteria

This paper concerns a proposal to define a way to evaluate the effectiveness of ER fluids. After recalling the basic parameters of the ER fluids: static yield stress τ y , current density j, and zero-field viscosity η 0 , we determine for a lot of ER suspensions the experimental values of two corresponding parameters E 1 = τ y / j and E 2  = τ y / η 0  , and thereafter of a global effectiveness parameter E n = E 1 E 2 . We also compare the results obtained in the different cases of micrometer-sized and nanometer-sized particles-based ER fluids.

Introduction

The fundamental property of the ER fluids [1] is their capacity to change reversibly from a natural liquid state to a solid-like state in the presence of an electric field E. In its liquid state (E = 0) they should have low viscosity in order to limit the viscous losses when the fluid flows. In the solid state (E ≠ 0) it must withstand a strong yield stress to be able to transmit large mechanical forces. In addition, to increase the energy efficiency, it is necessary to reduce the consumption of electric current passing through the fluid. Finally, taking into account that the practical electrical field ranges from 3 to 5 kV/mm (electric field strength can be regarded as a nearly constant value) it can be considered that the main features of ER fluids are: i) the static yield stress and the current density, and ii) the zero-field viscosity. Since the discovery of the electrorheological (ER) effect more than fifty years, significant advances including the synthesis of new efficient ER fluids have emerged. The first important step forward led to replace the water-activated particles by the intrinsic conductivity particles [2,3]. This has provided more thermally stable fluids. The second significant development has emerged more recently. The use of nanometer-sized particles (having specific physical properties) has significantly increased the value of the static yield stress of the ER fluids under electric field. An increase in excess of 20 times or more of the yield stress appears in this case. This is known as the giant ER effect [4].

Characterization and effectiveness of ER fluids

Experimental procedure

The steady rheological response of ER fluids is usually modelled by the following equation: where the measured values are: the shear stress τ, the shear rateγ& , the electric field E and the calculated parameters are: the (static or dynamic) yield stress τ y , the plastic viscosity η p and the apparent viscosity η app .

τ (E, γ& ) = τ y (E) + η p γ& = η app . γ& (1)
Unfortunately, the different parameters defined in Eq. ( 1) are more or less dependent on the experimental conditions. So, if we want to properly compare the electrical and rheological responses of ER fluids it is necessary to apply at the characterization stage a well established experimental procedure. The temperature of the ER fluids is an important parameter influencing the responses of the fluids (for the yield stress and current density particularly). Other specific experimental conditions are: application way of the electric field and mechanical stress to the fluid prior to the measurements, order, duration and magnitude of the application of the field and the shear rate (or shear stess) during experiment, type of experimental set-up (shear or squeeze flow), geometry of measurement cell (parallel plates, concentric cylinders …).

Thus it seems like a necessity to clearly specify the whole experimental conditions for the electrorheological measurements (ERM) and better still to define standards of ERM. In this case one may expect to compare more effectively the performances of the ER fluids.

Proposal of an effectiveness ER parameter E n

Previous studies have highlighted a strong correlation between the yield stress and the electric current density for an ER fluid at rest or in solid-like state. This result is based on the theory (the conduction model) and the experience [5,6]. Furthermore it may also be noted that the recent nanoparticles-based ER fluids shows most of the time a relatively high value of the zero-field viscosity. These remarks lead us to propose: i) a first efficiency parameter E 1 in connection with the static yield stress τ y and the corresponding current density j, at a given electric field strength:

E 1 = τ y / j (2)
ii) a second efficiency parameter E 2 including the static yield stress and the zero-field viscosity η 0 :

E 2  = τ y / η 0 (3)
iii) an ER fluids effectiveness parameter E n :

E n = E 1 E 2 (4)
For the micrometer-sized-based ER fluids the zero-field viscosity is still quite limited (typical values: η 0 < 0.2 Pa . s for a volume fraction φ ~ 25 %) whereas for the nanometer-sized-based fluids the zerofield viscosity is widely greater (η 0 > 2 Pa . s). In some cases, for the latter ER fluids, a gel of the suspension at rest appears due to the strong interaction between nanoparticles. In this situation, one can used, a zero-field viscosity for a very low value of shear rate (for example γ& = = = = 1 s -1 ).

Some experimental results and discussion

We have gathered below some experimental data from the literature and from our previous works. We note that the data used (see legends of Figs. 1 and2) correspond to a large variety of ER fluids (size and formulation of the dispersed phase, various host insulating liquid and volume fraction of the solid phase …). We note in figure 1, for E ranging from 3 to 5 kV/mm, a widely dispersion of the values of the static yield stress τ y and current density j (τ y and j varie over three order of magnitude) whereas a much closer values for the ratio τ y / j (one order of magnitude). These results confirm the correlation between the static yield stress and the current density for ER fluids subjected to a dc electric field both for the micrometer-sized and the nanometer-sized ER suspensions. Figure 2 shows the variation of the static yield stress τ y as a function of the zero-field viscosity η 0 for a constant field strength E. A rather surprising result appears: all the data seem to follow a master straight line with a linear law (τ y = k η 0 ) meaning that E 2 can be roughly considered as a constant value k (Eq. 3). Here we find for E = 4 kV/mm, E 2 = k ≅ 0.6 x10 4 s -1 . Noting that above E = 2 kV/mm the static yield stress of the ER fluids varies almost linearly with the applied field, we can consider in the first approximation that k is proportional to E: k = k'E with k' ≅ 0.15x10 4 (s . kV/mm) -1 . Obviously, the experimental data used is not enough to justify a universal law for the dependence of the static yield stress on the zero-field viscosity for the ER fluids but this should be deepened in future work.

Taking into account a constant value for E 2 (E 2 = k) for a given electric field strength we then obtain for the effectiveness parameter:

E n = k E 1 .
If further experiments confirm that the zero-field viscosity is more or less proportional to the static yield stress (for a same field), this will imply that the ideal ER fluid concept (i.e. a strong yield stress and a low zero-field viscosity) is illusory. This situation is probably the consequence of the dispersion forces (van der Waals) between manometer-sized particles in "contact" and more generally to the importance of the dielectric interfacial properties at the nanometric scales [START_REF] Lewis | Conference on Electrorheological Fluids and Magnetorheological Suspensions IOP[END_REF]. However future progress on material science should lead to provide a better stabilization of the ER nanosuspensions and to obtain a satisfactory stress/viscosity compromise.

Conclusion

In this paper we proposed a global parameter E n intended to compare the effectiveness of the ER fluids. Firstly, we have noted the importance to well define experimental conditions for the relevance of this comparison. Then, we have defined two basic efficiency parameters. The first one E 1 specifies the correlation between the ER effect (static yield stress) and the conduction properties of the suspension (current density). The second parameter E 2 shows other correlation based on the rheological responses of the fluid, under electric field (static yield stress) and without field (zero-field viscosity). It would appear that the zero-field viscosity rises with the static yield stress. This behaviour seems to exist for a large variety of ER fluids (among others, for the micrometer-sized and the nanometer-sized particles-based fluids) and may be explained by the size effect of the dispersed phase of the suspensions. Finally, an effectiveness ER parameter E n is proposed taking into account the main chatacteristics of the ER fluids: static yield stress and current density at a given field strength, zerofield viscosity, and temperature (in this regard E = 4 kV/mm and T = 25 °C may be considered as a typical standard values).

11thFigure 1 .

 1 Figure 1. (a) Static yield stress τ y , (b) current density j and (c) ratio τ y / j vs applied electric field for various ER fluids. Symbols: nanometer-sized particle-based ERF, [4], [7], Q. Quegan et al. unpublished, micrometer particles-based ERF,  [8].

Figure 2 .11th

 2 Figure 2. Static yield stress at E = 4 kV/mm vs zero-field viscosity for various ER fluids. Symbols: nanometer particles-based ERF,[9],[10],[11], Q.Guegan et al. unpublished, micrometer particles-based ERF[START_REF] Ishino | Progress in electrorheology[END_REF],[7]. Dots: experimental points. Solid line: linear fit.