Edgar Alejandro Pachon-Rodriguez

Emmanuel Guillon

Geert Houvenaghel

Jean Colombani
email: jean.colombani@univ-lyon1.fr

Wet creep of hardened hydraulic cements -example of gypsum plaster and implication for hydrated Portland cement

Keywords: Humidity, B: microstructure, C: creep, D: admixture, dissolution

Gypsum plaster exhibits a dramatic creep when placed in a very humid environment. We have combined mechanical tests of wet bending creep of set plaster and holographic interferometry measurements of dissolution rate and diffusion coefficient to look for the origin of this wet creep. Both these experiments have been performed in absence and presence of various known anti-creep admixtures. It appears that the creep rate and dissolution rate are strongly correlated. This correlation has allowed to propose surface-driven pressure solution creep as mechanism of wet creep of gypsum plaster, i.e., the sequence dissolution in the grain boundary water/diffusion/precipitation at the grain surface. An order of magnitude analysis shows that this dissolutiondiffusion-recrystallization series can also contribute to the creep of hydrated Portland cement.

Introduction

Hydraulic cements are mineral powders that harden under water, and subsequently remain cohesive in presence of water. They constitute the main materials of the building industry, used under the form of pastes to enable molding in the desired shape. Portland cement and gypsum plaster are the most used hydraulic binders because of their availability, low cost and easy installation. Hardened Portland cement and its derivatives (mortar, concrete) has also the virtue to be load-bearing, and hydrated plaster of Paris to be light, isolating and fire resistant. One of their limitations is their longtime plastic strain, or creep, mainly indoor for gypsum plaster and outdoor for hardened Portland cement. For gypsum, this creep is strongly enhanced by the presence of humidity.

Being chemically and structurally much simpler than hydrated cement pastes, this study is devoted to the creep of set plaster of Paris, with the aim to identify the underlying mechanisms and to estimate their applicability to the cementitious materials. Indeed the elimination, or at least limitation, of this drawback requires the understanding of its microscopic origin. Few studies have been devoted to the investigation of the link between the microstructure and the mechanical properties of set plaster. Concerning the influence of water, the few existing studies have brought some clues for the interpretation of the stiffness and resistance of set plaster in humid or wet environments. But up to now, the creep in presence of water had not re-ceived any explanation. We have proposed recently that it derives from the dissolution of gypsum [START_REF] Pachon-Rodriguez | Pressure solution as origin of the humid creep of a mineral material[END_REF]. This finding has enabled to propose pressure solution creep as the mechanism of wet creep of set plaster. We detail here the experiments (mechanical tests and interferometric measurements) leading to this result and discuss its implications for hydrated Portland cement.

Mechanical properties of gypsum plaster in presence of water

Set plaster is constituted of intricate gypsum (CaSO 4 , 2 H 2 O) needles, roughly 20 µm long, obtained from the hydration of plaster of Paris (CaSO 4 , 1 2 H 2 O).

The cohesion of the material derives from the bonds between the needles and from the tenon and mortise joints between them [START_REF] Chen | A hierarchical study of the mechanical properties of gypsum[END_REF]. It has been postulated

about ten years ago that the bond between the needles is of the same nature as the bond between flocculated colloids [START_REF] Chappuis | A model for a better understanding of the cohesion of hardened hydraulic materials[END_REF]. Therefore the gypsum microcrystallites should be linked via a nanometric water film. The attraction between them should stem from van der Waals interactions between the facing charged faces and ionic correlations between the Debye layers developing in the water close to the surface, and the repulsion from the exclusion of the Debye layers. At the ends of the water layers, capillary forces develop at menisci and contribute to the cohesion between the needles. The balance between these forces determines the liquid film thickness. It has been added

a few years later that the presence of "bridging", i.e., solid, interfaces, beside these "non-bridging" liquid interfaces between needles, is necessary to obtain a more comprehensive interpretation of the set plaster properties [START_REF] Gartner | Cohesion and expansion in polycrystalline solids formed by hydration reactions -the case of gypsum plasters[END_REF].

This vision of the microstructure of the material enables to interpret several mechanical properties of wet or humid set plaster:

• it has been observed that Young's modulus of set plaster decreases when the relative humidity increases [START_REF] Sâadaoui | Internal friction study of water effect on set plaster[END_REF][START_REF] Badens | Relation between young's modulus of set plaster and complete wetting of grain boundaries by water[END_REF]. This can be interpreted by the fact that the equilibrium thickness of the water inter-needle layers increases with the relative humidity enhancement, from ∼ 1 nm in dry conditions to ∼ 10 nm in humid ones [START_REF] Chappuis | A model for a better understanding of the cohesion of hardened hydraulic materials[END_REF]. And a thicker layer induces a weaker bond between needles. The assumption has been made that this weakened connection can result in a reversible slip between some microcrystallites, increasing the elastic strain, thus decreasing Young's modulus. The bridging bonds deform only by elastic bending and guarantee that no irreversible strain occurs.

• The flexural strength, i.e., the bending failure stress of this brittle material, has also been seen as decreasing when the material is soaked in water [START_REF] Coquard | Water and solvent effects on the strength of plaster[END_REF] or when it adsorbs water [START_REF] Andrews | The effect of water contents on the strength of calcium sulfate plaster products[END_REF]. This feature can also be ascribed to the thickening of the water layers in a humid environment. Indeed the resulting slip between microcrystallites implies a lower contribution of the non-bridging bonds to the strength, so a lowered failure stress.

• It has also been observed that, if the hardness of a set plaster sample decreases after being plunged in water, the hardness recovers its initial value once the sample is dried [START_REF] Coquard | Water and solvent effects on the strength of plaster[END_REF]. The reversibility of the influence of water can again been interpreted by the fact that the mechanical resistance of the material reflects the values of the inter-needle water slab thickness, in equilibrium with the quantity of water available. If more water is available, the layer thickens and the cohesion diminishes, and vice-versa.

Wet creep of gypsum plaster

Beside these successes of the theory, one should mention that unfortunately it does not provide any mean to understand the increase of creep in a humid environment [START_REF] Sattler | Elastic and plastic deformations of plaster units under uniaxial compressive stress[END_REF][START_REF] Craker | Plastic deformation of gypsum[END_REF]. Indeed creep is a slow process occurring over periods of days or months, whereas the equilibration of the intercrystalline water layers is a quasi-instantaneous mechanism. Thereby the progressive plastic strain occurring during creep cannot be explained by a progressive increase of the slab dimension leading to a loss of cohesion.

We show here that the wet creep of set plaster is a consequence of a phenomenon called pressure solution creep (figure 1). When an external stress, or simply its own weight, is applied to a gypsum board, the gypsum needles are subject to local stresses. These stresses induce an increase of the chemical potential of the gypsum. So when water is present, in particular in the intercrystalline contacts, to recover chemical equilibrium, the chemical potential of the liquid increases also to equalize with the one of the solid. This leads to the enhancement of the solubility of gypsum, thereby to a dissolution of the solid in the liquid. Therefore concentration gradients appear along the water layers, which induce Fick diffusion of dissolved gypsum. When the sulfate and calcium ions reach areas without stress, their solubility recovers its initial value and they precipitate on the solid at rest. This dissolutiondiffusion-precipitation series continues as long as the local stress exists, and induces a transfer of matter from high stress to low stress regions. By this way a plastic strain occurs, which accommodates the applied stress, and the material creeps.

This phenomenon is well known in geology. It has been ascribed several This sequence induces a local transfer of matter, so a macroscopic plastic strain.

contributions to the upper crust evolution, for instance during non-seismic strain of active faults, or for the transformation of loose sediments into cohesive sedimentary rocks [START_REF] Gratier | The role of pressure solution creep in the ductility of the earths upper crust[END_REF]. Pressure solution creep has in particular be evidenced in wet gypsum particulates under uniaxial and hydrostatical load [START_REF] Demeer | Creep of wet gypsum aggregates under hydrostatic loading conditions[END_REF][START_REF] Demeer | Uniaxial compaction creep of wet gypsum aggregates[END_REF].

Numerous experiments of pressure solution creep in water have been performed in laboratory, with the final aim of understanding geological situations. Many of these studies use model systems or model configurations from which we can learn on the basic mechanisms and characteristic kinetics of pressure solution [START_REF] Gratier | The role of pressure solution creep in the ductility of the earths upper crust[END_REF]. But their specific features, generally devoted to geology, make their application to industrial materials difficult. For instance, the available works on pressure solution in gypsum study low porosity assemblies of non-cohesive quasi-spherical gypsum crystallites,. All of these characteristics differ from industrial gypsum, thereby making the application of these studies difficult to building materials. We have not found such a study in the materials science field. Therefore a protocol enabling to validate the existence of pressure solution in the creep of humid gypsum plaster is necessary.

Before going on, we would like to recall that it was formerly thought that the decrease of the mechanical strength of set plaster with moisture could be due to the dissolution of small gypsum crystals, precipitated at the end of the plaster setting and bridging the gypsum needles [START_REF] Murat | Corrélation texture cristallinepropriétés mécaniques de plâtres durcis. étude préliminaire[END_REF][START_REF] Lewry | The setting of gypsum plaster Part II The development of microstructure and strength[END_REF]. This phenomenon was sometimes misleadingly called "dissolution-recrystallization".

Pressure solution creep mentioned here also involves dissolution and recrystallization but in a totally different way: dissolution occurs at high stress regions of inter-needle contacts and recrystallization at low stress regions or at the needles surface. No microcrystal appears or disappears. This hypothesis of dissolving-recrystallizing microcrystals was unambiguously discarded by hardness and bending tests in various conditions of plaster setting and relative humidity [START_REF] Coquard | Water and solvent effects on the strength of plaster[END_REF].

To establish a protocol of validation of the presence of pressure solution creep, we benefit from the methodology of the works performed on the geological side. In these studies, the presence of pressure solution creep, and its exact nature, is usually determined by measuring the creep kinetics. The slowest step in the reaction-transport-recrystallization sequence limits the matter transfer rate, so drives the whole kinetics and determines the evolution laws.

The theoretical determination of the exact expression of ε(t), the strain evolution with time, requires to define precisely many parameters: order of the chemical reaction, reactive surface area, thickness of the liquid slab, locus of the precipitation sites, porosity, size distribution of the mineral grains, width of the solid contacts, . . . The final ε(t) curves have been found to be highly dependent on the above characteristics of the system and on their interplay. A particularly complete modelization of pressure solution creep in sandstone for instance can be found in Ref. [START_REF] Renard | Enhanced pressure solution creep rates induced by clay particles: Experimental evidence in salt aggregates[END_REF].

As no complete pressure solution modelization of a highly porous medium like set plaster exists, we have to make with first-order models, bringing at least trends of the strain evolution with time. We have chosen the acclaimed model of Raj [START_REF] Raj | Creep in polycrystalline aggregates by matter transport through a liquid phase[END_REF]. His modelization considers the creep of a unique stressed cubic mineral sample in a solvent present in channels at its surface. It states that :

• if the kinetics is driven by the mass transport (due in general to a low flow rate of diffusion in tiny channels), the strain rate writes:

dε/dt ∼ σD s/d 3 (1)
• if the kinetics is driven by the surface reaction (due to its slowness), the strain rate writes:

dε/dt ∼ σk s/d (2)
In these expressions, σ is the applied stress, D the diffusion coefficient of the dissolved species, k the reaction rate constant of the mineral, s its solubility and d the characteristic size of the constrained interface.

In geological laboratory experiments, the limiting step is estimated in varying the grain size d and determining if the strain rate scales as d -1

or d -3 . In our case, varying the gypsum crystallites size in set plaster is difficult. To identify the limiting stage, we have benefited from the existence of admixtures added to the plaster industrially to limit the humid creep.

These additives are efficient in slowing down the creep strain rate, but again, their mechanism of action is not understood yet. So the idea of our work is to find which factor of the above expressions, if any, these admixtures modify, to lower the strain rate, thus indicating the working mechanism of creep.

Experiments

In equations 1 and 2, the parameters the knowledge of which is needed are d, s, k, D and dε/dt.

Admixtures

The investigated anti-creep admixtures are a tartaric acid (C The acid mixture is made of 1/6 of tartaric acid and 5/6 of boric acid in weight.

Bend creep tests

For the bend tests, gypsum (from Mazan quarry, France) is ground and dehydrated to make plaster (CaSO 4 , 1 2 H 2 O). The resultant powder is mixed with water to make a paste with a water/plaster weight ratio of 0.8, bringing a convenient compromise between the fluidity of the paste and the porosity of the final product (57%). The mixture is cast in a parallelepipedic mold, placed in a closed vessel during 24 h to achieve complete hydration, dried, and stored in calcium sulfate saturated water until the test, to avoid dehydration [START_REF] Lewry | The setting of gypsum plaster Part II The development of microstructure and strength[END_REF]. The same protocol was also followed with water containing the various admixtures.

Standard bend tests have been performed to measure the creep strain rate of set plaster. The above-described samples were loaded in the middle on the top face and supported at their ends. The deflection was recorded with a Linear Variable Differential Transformer displacement sensor every 2 h during 15 days. During the tests, the beams were immersed in water -to study wet behavior-saturated with calcium sulfate -to avoid normal dissolution and be sure to observe, if any, pressure dissolution. The maximum load was chosen as 20% of the tensile strength, measured for each batch on one sample before the test, to remain outside the stress range where subcritical crack growth inside the samples is expected, risking to blur the results [START_REF] Sattler | Elastic and plastic deformations of plaster units under uniaxial compressive stress[END_REF][START_REF] Meille | Mechanisms of crack propagation in dry plaster[END_REF].

The force and displacement are converted in the stress σ and strain ε at the top face in the middle of the beam with the elastic approximation:

σ = 3P L 2wh 2 and ε = 6hδ L 2 (3)
In these expressions, P is the load, L the support span, w = 20 mm the width of the beam and h = 20 mm its height.

The zero-stress curve inside the beam may depart from the center of the beam, due to non-symmetry of the compressive and tensile strain mechanism, which may make these formulas non valid. But the determination of the exact stress field is not possible and would deserve a study for itself. MPa.

An example of ε(t) curves at various stresses for one admixture is shown in figure 2. The strain rate dε(t)/dt, needed for the test of the above-mentioned equations is obtained in derivating the experimental ε(t) curves numerically.

From it, the creep compliance rate (dε(t)/dt)/σ is computed. All strain-time data and curves for all admixtures and stresses can be found as Supplementary Material.

A parameter that may play a role in the elaboration of the samples is the concentration of admixture in the water used to manufacture the gypsum from the plaster powder. Several concentrations between 0.05% and 0.5% in weight have been tested for each additive. The creep results have been found to be independent on the concentration, except for the smallest concentrations ( 0.1%) where the anti-creep action is less efficient.

The strain obviously depends on the bending stress. To get an estimate of the evolution of the creep intensity with the applied stress, the value of the bend strain at one given time (namely 111h) has been drawn versus the bend stress in figure 3. It can be stated that the creep is roughly proportional to the applied stress in the range investigated in this study.

Contact size

The determination of the inter-needle contact size d is not a trivial task and for this we have performed scanning electron microscopy observations of the microstructure of the set plaster samples (figure 4). For all the admix-tures used, the microstructure remains of the acicular type. None of them reveal lenticular or columnar habit, sometimes encountered in natural gypsum, depending on the impurities adsorbing on specific crystalline planes.

The characteristic size of the microcrystals is similar at first order in all pictures and we have considered that the contact size d should also be similar among all samples. For the computations in next section, the average value of d = 1 µm has been chosen.

Nevertheless we can mention that gypsum plaster elaborated with Sequion reveals unexplained micrometer-size etch pits at the needles surface.

Whereas we have seen that, according to the SEM pictures, the influence of the additives on the set plaster microstructure is not significant, we have attempted to get a further evidence of this lack of influence in testing another protocol of elaboration of the samples. In this process, all solid samples are first elaborated from plaster and pure water, in the absence of any admixture. Subsequently, each sample is soaked during 12h in water containing a given concentration of additive, to make the molecules impregnate the gypsum crystallites network of the material. The creep bend tests are then performed as detailed in section 4.2. The interest of this procedure is to guarantee that all samples have strictly the same microstructure, having all been elaborated in pure water.

Figure 5 shows the creep curves for one admixture present in the impregnation water at various concentrations. As the applied stresses are almost similar, the difference between the various curves can be ascribed to the concentration of additive in the soaking water. The more concentrated in additive the impregnation solution, the more efficient the anti-creep effect. The origin of this evolution is to be searched in the variation of the dynamics of adsorption and diffusion in a porous medium of the admixture for the various concentrations. This result seems to indicate that the quantity of molecules inside the sample is not the same for the various concentrations, even with the long impregnation time we have chosen. The adsorption and diffusion dynamics may be different from one admixture to another, which makes the comparison between the results obtained with the various additives quite delicate. Therefore, we have abandoned this process and exclusively used the protocol presented in the previous section, where we have the certainty that the molecules are embedded in the samples.

Solubility

The solubility of gypsum in aqueous solutions of the various admixtures has been determined by induced coupled plasma atomic emission spectroscopy. Certainly due to the low concentration of additive here, no departure from the solubility of gypsum in pure water (2 g/L, 15 mmol/L) has been found, whatever the added product.

Dissolution rate constant

The gypsum-water interfacial reaction rate constant is also needed, being either the dissolution or precipitation rate constant in Raj's equation. De

Meer and Spiers identified precipitation as the driving mechanism, which can be expected in their low porosity system, where unstressed precipitation sites are rare [START_REF] Demeer | Uniaxial compaction creep of wet gypsum aggregates[END_REF]. But the dissolution and precipitation rate constants should be close. Indeed the rate of attachment and detachment of ions at a solid surface are strongly linked, and a change of the surface reactivity influence both [START_REF] Lasaga | A model for crystal dissolution[END_REF]. We have chosen to study the dissolution rate constant. But if a correlation between creep and dissolution is found and shows that the phenomenon is interface-driven, it is very likely that a correlation between creep and precipitation also exists.

The measurement of the dissolution rate constant is a delicate task. As we have shown in preceding studies, the usual solution chemistry methods provide dissolution rates blurred by mass transport phenomena (diffusion, convection) [START_REF] Colombani | Measurement of the pure dissolution rate constant of a mineral in water[END_REF][START_REF] Colombani | Dissolution measurement free from mass transport[END_REF]. As the possible effect of admixture on the dissolution rate constant may be tiny, the standard dissolution measurement techniques

were not appropriate and we have used an alternative methodology, holographic interferometry. This technique has been described in detail in Ref. [START_REF] Colombani | Holographic interferometry study of the dissolution and diffusion of gypsum in water[END_REF]. It presents two major advantages. First the experiment is performed in quiescent water, thereby avoiding any convective disturbance. Secondly the concentration is directly measured at the solid-liquid interface, whereas in standard methods it is measured in the flowing liquid far from the surface.

The dissolution rate constant k of gypsum of the same origin as in the bend tests -to allow comparison-in water containing the various admixtures has been measured by holographic interferometry and for all of them, k has been found to be modified by the admixture [START_REF] Pachon-Rodriguez | Pure dissolution kinetics of anhydrite and gypsum in inhibiting aqueous salt solutions[END_REF]. The results are summarized in table 1.

Diffusion coefficient

The holographic interferometry experiments have also the advantage to give access to the diffusion coefficient D of dissolved gypsum in water. Therefore D has been measured for gypsum in water containing the various admixtures and it has been seen that this coefficient is almost constant for all Admixture k (10 -6 mol m -2 s -1 ) D (10 admixtures, probably due to the low concentration of the products, as shown in table 1.

Results and discussion

Before testing our assumption of pressure solution creep using all the experiments described in the preceding section, we would like to focus first on the ε(t) curves. They constitute the first systematic study of the wet creep of gypsum plaster. By fitting the curves, we have noticed that all of them obey to a power law: ε(t) = At n , with n < 1. Thereby, we see that a consolidating mechanism is active during the wet creep, which slows down progressively the strain. The creep exponent n depends on the admixture with which the set plaster sample has been manufactured: 0.69 for pure water, 0.71 for boric/tartaric acid, 0.52 for Trilon, 0.34 for Sequion, 0.39 for Dequest and 0.38 for STMP. It is striking to state that it evolves from ∼2/3 to ∼1/3 from pure water to the most efficient anti-creep admixture. Figure 6 illustrates this evolution. We have not found yet the origin of this ability Indeed the ε(t) curves are non linear. We have chosen t 0 = 1 × 10 5 s, in the middle of the investigated time, but we have checked otherwise that the We would like to mention here the compressive creep tests performed by Hoxha et al. with natural gypsum rocks [START_REF] Hoxha | Deformation of natural gypsum rock: Mechanisms and questions[END_REF]. They mention that the duration of their experiments is too short (∼ 15 days) to evidence pressure solution.

Therefore they explain the expansion of their samples by a mechanism of reversible migration of water molecules, from the solid to the pore space. As shown here, the consequence of pressure solution creep may be observable even for such short period of time. With the material at hand, we are not able to explain why they do not observe precipitation-limited pressure solution creep like de Meer & Spiers with a system of similar porosity [START_REF] Demeer | Uniaxial compaction creep of wet gypsum aggregates[END_REF].

Now that the basic mechanism of the wet creep of gypsum plaster is elucidated, a detailed theoretical analysis of pressure solution in an as porous material as gypsum plaster would enable to make quantitative predictions about its kinetics. But knowing that the dissolution kinetics is at the basis of wet creep enables to try to find tools against this drawback of the material.

Implications for hydrated Portland cement

Despite being the manufactured material most used on earth, the structure and cohesion of hydrated Portland cement is not perfectly understood yet. Along a schematic view, a hydrated Portland cement paste can be considered as a floculated colloidal suspension of calcium-silicate-hydrate (C-S-H), i.e., (CaO) 1.7 (SiO 2 )(H 2 O) 1.8 [START_REF] Allen | Composition and density of nanoscale calcium-silicate-hydrate in cement[END_REF], nanometric particles, called gel, with partial crystallinity [START_REF] Pellenq | A realistic molecular model of cement hydrates[END_REF]. Ca(OH) 2 nanocrystallites, and other minor hydration products, are also present in the gel. Like in standard colloidal gels, the remarkable strength of this hydraulic cement stems from electrostatic and ionic correlation forces between the particles via the water layers between them [START_REF] Gmira | Microscopic physical basis of the poromechanical behavior of cement-based materials[END_REF]. Capillary forces contribute also in unsaturated materials, where water-air menisci are present.

It is generally admitted that C-S-H is found in hydrated pastes in 2 or 3 forms, of identical chemical composition, but clearly distinct organizations and densities [START_REF] Chappuis | A model for a better understanding of the cohesion of hardened hydraulic materials[END_REF][START_REF] Jennings | Colloid model of C-S-H and implications to the problem of creep and shrinkage[END_REF][START_REF] Vandamme | Nanogranular origin of concrete creep[END_REF]. The proportion of these phases varies with the water/cement ratio used to manufacture the cement paste, the drying, and the aging of the material. This structural heterogeneity induces a multi-scale porosity, also evolving with the just-mentioned parameters.

One major concern about cementitious materials is their aging, in the form of shrinkage, creep, fractures, . . . partly due to the slow drying of the products of cement hydration [START_REF] Bazant | Prediction of concrete creep and shrinkage: past, present and future[END_REF]. It has been shown that the concrete creep strain results exclusively from the irreversible strain of the cement hydration products [START_REF] Acker | Swelling, shrinkage and creep: a mechanical approach to cement hydration[END_REF] and the creep of hardened cement is still an active field of research [START_REF] Vandamme | Nanogranular origin of concrete creep[END_REF].

We discuss here only about wet creep, i.e., the plastic strain under small load of the water saturated material. In these conditions, capillary menisci inside the material are absent, which suppresses a major source of aging of the material. Therefore in this case the creep cannot derive from shrinkage induced by water loss, and corresponding models do not apply [START_REF] Bazant | Prediction of concrete creep and shrinkage: past, present and future[END_REF]. Recent assumptions of creep origin applying in this situation are: reorganization of high density phase globules, analog to dislocation migration in crystals [START_REF] Jennings | Colloid model of C-S-H and implications to the problem of creep and shrinkage[END_REF], reorganization of C-S-H particles, leading to an increase of the packing factor of the 3 phases, analog to granular matter flow [START_REF] Vandamme | Nanogranular origin of concrete creep[END_REF], better alignment of C-S-H sheets, analog to house-of-card collapse [START_REF] Acker | Swelling, shrinkage and creep: a mechanical approach to cement hydration[END_REF], . . .

All of these hypotheses rely on the sliding of C-S-H nanoparticles, or of C-S-H sheets, enabling reorganization [START_REF] Sanahuja | Creep of a C-S-H gel: a micromechanical approach[END_REF]. An alternative proposal, beside sliding, has also been made to explain the relative motion of C-S-H particles.

The possibility of the existence of the dissolution-diffusion-recrystallisation series, leading to a transfer of matter among the C-S-H particles, inducing plastic strain, has also been proposed [START_REF] Chappuis | A model for a better understanding of the cohesion of hardened hydraulic materials[END_REF]. Theoretical predictions of hydrating concrete creep have even been proposed [START_REF] Grasley | Constitutive modeling of the aging viscoelastic properties of portland cement paste[END_REF][START_REF] Suter | Constitutive model for aging thermoviscoelasticity of reacting concrete ii: results and discussion[END_REF]. But these analytic laws are phenomenological and postulate a priori the existence of a significant influence of the applied stress on the dissolution of the hardened material. We have shown here that this influence is significant and measurable in the case of wet gypsum and want to discuss here the case of hydrated cement.

Hydrated cement pastes share common characteristics with gypsum plaster: They both simultaneously shrink and harden during drying, they swell when immersed in water after setting, and they experience humid creep. The question of the existence of pressure solution in saturated hardened cement arises. As for gypsum, a direct observation is not possible for the moment and correlations have to be sought. Recent C-S-H nanoindentation measurements have shown a logarithmic creep [START_REF] Vandamme | Nanogranular origin of concrete creep[END_REF]. As no comprehensive model of pressure solution creep exists, it is unfortunately not possible to draw any conclusion about pressure solution from this lograrithmic evolution of the strain [START_REF] Gratier | The role of pressure solution creep in the ductility of the earths upper crust[END_REF]. The order of magnitude of the creep compliance rate in these experiments evolves from 10 -9 to 10 -10 % s -1 MPa -1 , for samples ages similar to ours (∼ 10 days). In our experiments with gypsum plaster, its values is about 10 -6 % s -1 MPa -1 (figure 8).

The crystallites in C-S-H are nanometric. The inter-particle contact length is therefore of the same order of magnitude or lower, so 3 orders of magnitude smaller than in set plaster, where it is micrometric (figure 4).

Therefore if pressure solution plays a role, the mass transport time between particles should be negligible and the phenomenon should be reaction-driven, as in the case of gypsum, and depend on the dissolution rate constant of the material via equation 2.

Making the first order assumption that the correlation in gypsum plaster mol m -2 s -1 [START_REF] Baur | Dissolution-precipitation behaviour of ettringite, monosulfate, and calcium silicate hydrate[END_REF]. This value is of the same order of magnitude as the value we think necessary to make pressure solution active during C-S-H creep.

Therefore this order of magnitude analysis brings pressure solution among the possible phenomena acting in the creep of C-S-H.

We would like to stress on the fact that our assumption is consistent with all the creep models mentioned above. The novelty does not lie in the geometry of the motion of the nanoparticles, but in the way of this motion, dissolution-recrystallisation instead of sliding. To strengthen this hypothesis, the measurement of the dissolution rate constant of C-S-H, and other hydrated phases, in aqueous solutions representative of the liquid present in the nanopores of hardened cement are highly desirable, essential to base on experimental evidence the contribution of pressure solution to concrete creep. Besides, if the role of dissolution is corroborated, its exact role on the creep strain will have to be specified. For instance, whether the dissolutiondiffusion-precipitation sequence results in a creep strain accomodating the stress like in gypsum plaster [START_REF] Chappuis | A model for a better understanding of the cohesion of hardened hydraulic materials[END_REF], or the dissolution induces a thinning of the crystallites inducing a viscoelastic strain progressively increasing with time [START_REF] Suter | Constitutive model for aging thermoviscoelasticity of reacting concrete ii: results and discussion[END_REF], will have to be clarified.

Conclusion

We Following the study of gypsum plaster, an order of magnitude analysis has shown that pressure solution may also contribute to the creep of hydrated Portland cement. Indeed, alternatively to the sliding between the hydrated phase nanoparticles, the dissolution-diffusion-recrystallization sequence was shown to be another mean of the reorganization of the nanoparticles during creep.

Figure 1 :

 1 Figure 1: Sketch of the consecutive steps of pressure solution creep in gypsum plaster: an external load creates a local compression stress between two gypsum needles, which induces dissolution, diffusion of the dissolved species, and recrystallisation in a non-stressed area.

Figure 2 :

 2 Figure 2: Evolution with time of the bending strain for the gypsum plaster samples manufactured with 0.15% Sequion in the preparation water. The applied stress, from the bottom to the top curve is 0.244 (red), 0.332 (orange), 0.356 (blue) and 0.386 (green)

Figure 3 :

 3 Figure 3: Bend strain ε after 111 h versus applied stress σ for the various admixtures: pure water (black circles), boric/tartaric acid (red upward triangles), Trilon (pink downward triangles), Dequest (blue leftward triangles), Sequion (light blue diamonds), STMP (green squares).

Figure 4 :

 4 Figure 4: Scanning electron microscope pictures of set plaster samples, pure and elaborated with the various admixtures. The dimension of the images is 12.7 × 11.0 µm 2 .

Figure 5 :

 5 Figure 5: Evolution with time of the bending strain for the pure gypsum plaster samples impregnated by Sequion. From the top to the bottom curve, the concentration of Sequion in the impregnation water is 0.50 g/L for a 0.406 MPa stress (red), 4.97 g/L for a 0.524 MPa stress (green) and 9.94 g/L for a 0.496 MPa (blue).

Figure 6 :

 6 Figure 6: Bend creep exponent of wet gypsum plaster manufactured with the various admixtures.

Figure 7 :

 7 Figure 7: Compliance rate for wet bending creep of gypsum plaster elaborated with a given additive at t 0 = 1 × 10 5 s, as a function of a coefficient proportional to the diffusion coefficient of dissolved gypsum in a solution of the same additive. Red upward-pointing triangles: tartaric/boric acid; black circles: pure water; pink downward-pointing triangles: Trilon; blue left-pointing triangles: Sequion; light blue diamonds: Dequest; green squares: STMP.

Figure 8 :

 8 Figure 8: Compliance rate for wet bending creep of gypsum plaster elaborated with a given additive at t 0 = 1 × 10 5 s, as a function of a coefficient proportional to the dissolution rate constant of gypsum in a solution of the same additive. The color code is the same as in figure 7. The black dashed line is a linear fit of the data.

  and hydrated cement pastes are similar, we should have ( ε/σ) C-S-H /( ε/σ) gypsum ∼ (ks/d) C-S-H /(ks/d) gypsum . Let us review the parameters in this equation.The solubilities of the hydrated phases of cement depend highly on the nature and structure of the phase and on the chemical environment. But the order of magnitude of the solubility of calcium has been measured as 10 mmol/L, comparable to the 15 mmol/L solubility of calcium in the case of gypsum[START_REF] Chen | Solubility and structure of calcium silicate hydrate[END_REF]. The creep compliance rate ε/σ of hardened cement mentioned above are 3 orders of magnitude lower than the one of gypsum plaster. The characteristic size d of the interface between C-S-H particles is also 3 orders of magnitude lower than in gypsum plaster. Therefore, a dissolution rate constant of k C-S-H ∼ 10 -6 k gypsum ∼ 10 -11 mol m -2 s -1 would enable to explain the change in the nanoparticle structure responsible for the creep observed in the experiments of Vandamme & Ulm [30]. Measurements with radiotracers of the dissolution rate of C-S-H suspensions in aqueous solutions have brought a value k C-S-H ∼ 3 × 10 -12

  have performed wet bending creep tests to measure the strain rate of set plaster manufactured with various anti-creep admixtures (boric/tartaric acid, Trilon, Sequion, Dequest, STMP). Besides we have carried out holographic interferometry experiments to measure the dissolution rate constant of gypsum in water containing these anti-creep admixtures, and the diffusion coefficient of dissolved gypsum in these solutions. A strong correlation has been found between the wet creep strain rate and the dissolution rate constant of the material. This clear dissolution-creep link has enabled to propose reaction-driven pressure solution creep as the underlying mechanism of the wet creep of gypsum plaster. This is the first time that this phenomenon is evidenced experimentally in an industrial material.

Table 1 :

 1 -10 m 2 s -1 ) Dissolution rate constant k and diffusion coefficient D of gypsum in water containing various admixtures, measured by holographic interferometry.

	Without	46	7.1
	Tartaric-boric acid	74	5.9
	Trilon	21	4.3
	Sequion	11	6.1
	Dequest	8.0	4.9
	STMP	3.3	6.5

Acknowledgements

We thank Elisabeth Charlaix, Ellis Gartner, François Renard et Dag Dysthe for fruitful discussions. This work was supported by Lafarge Centre de Recherche, Région Rhône-Alpes and CNES (french spatial agency).


Laure Petit

Catherine Barentin
email: barentin@lpmcn.univ-lyon1.fr

Jean Colombani

Christophe Ybert

Lydéric Bocquet

Size Dependence of Tracer Diffusion in a Laponite Colloidal Gel

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Soft glassy materials -as exemplified by concentrated emulsions, granular materials, colloidal glasses, etc. -are complex and disordered materials encountered in a wide range of every day life applications, from paints to cosmetics or food industry. 1,2 The description of these systems, the behavior of which is intermediate between solid and fluid, is however fundamentally challenging due to their intrinsic out-of-equilibrium nature, associated with the dynamical arrest in disordered state occurring at the jamming (or glass) transition. 3 As such, the understanding of the complex phenomenology shared by these diverse systems has attracted numerous theoretical and experimental works devoted in particular to the breakdown of ergodicity and the existence of long relaxation times leading to the so-called aging of the material, the identification of dynamical heterogeneities in the material between 'frozen' and fast relaxing regions, 4,5 or the appearance of a strongly heterogeneous material response under flow and deformation associated with shear-banding phenomena and flow cooperativity. [6][7][8] Among the various systems studied so far, aqueous suspension of Laponite clay platelets have been used extensively due to its industrial interest as gelling agent or oil drilling fluid, together with practical advantages associated with its transparency -which makes it well suited for light scattering or other optical techniques -and the fact that it exhibits a complete phenomenological picture, from aging to shear-thinning, thixotropy and shear-banding. Laponite suspensions in their glassy state were used for instance to study aging dynamics associated with slow structural relaxation due to cooperative reorganizations of particles 9,10 in soft glassy materials. They were used also to investigate the theoretical proposal of extending the well-known fluctuation-dissipation theorem formalism to describe out-of-equilibrium systems, through the incorporation of an effective temperature T eff. differing from the bath temperature. [11][12][13][14][15] One should however acknowledge that the origin of the dynamical arrest in Laponite suspension remains debated: whether the solidification of the material stems from the attraction between the negatively charged faces and positively charged edges of Laponite platelets, providing percolating aggregates as in gels, or from the repulsion between the electrical double layers of the Laponite platelets, as in glasses, has not received a general agreement. 10,[16][17][18] Overall, this points to the difficulties inherent to the description of these soft glassy materials and to the need for additional information on the local dynamical properties in such systems. This is the purpose of the present work where the Laponite suspension behavior and structure is probed through the measurement of the diffusive properties of colloidal tracer particles of various sizes.

The diffusive motion of particles in simple liquids is well described by the celebrated Stokes-Einstein relation:

D = kT ξ (1)
where ξ = 3πησ is the Stokes friction coefficient for a sphere with diameter σ ; η is the viscosity of the simple liquid considered as a continuum medium, k the Boltzman constant and T the absolute temperature. The Stokes expression for the friction coefficient holds in the case of a single particle diffusing in an infinite bulk liquid characterized by its viscosity η. When diffusion of the tracer occurs in an heterogeneous material -like porous media, gels or concentrated complex systems -, the diffusion is hindered by interactions, especially the long-range hydrodynamic interaction, of the tracer with the matrix structure. [19][20][21][22][23] While the structure of glassy and glass-forming materials are spatially disordered, 3 such systems have been shown to be dynamically correlated on long time scales. 4,25 In this context, diffusion of tracers has been used to probe the dynamics of these materials, like in colloidal systems, 5,14 polymer networks, 26,27 as well as in glass-forming (supercooled) liquids. 28 Accordingly, tracers are usually chosen with the same size as the material constituents, as e.g. for the above-mentioned studies: molecular size in Refs. 26,28 and colloidal sizes (from 100 nm to 1 µm) in Refs. 5,27 Although theoretical works suggest that the tracer characteristics dictate the way it couples to the soft glass and affect for instance the effective temperature it probes, 29 it appears that very little is known on the experimental side about how changing the tracer size modifies its diffusive motion in such systems. This is therefore the point on which we will more specifically focus in this study: we use a FRAP (Fluorescence Recovery After Photobleaching) technique to investigate the diffusive motion of tracers in a (colloidal) glass as a function of their size, covering probe sizes ranging from molecular (1 nm) up to 100 nm. This range of scale allows us to probe the microscopic length scales of the Laponite suspension (platelet size and inter-distance), typically of the order of ten nanometers.

The paper is organized as follows. First, we discuss all experimental details in section , presenting the dedicated FRAP setup for measuring diffusion coefficients, and performing careful validations for diffusivity measurements in Laponite suspensions , regarding in particular aging effects and the sensitivity to Laponite preparation protocol. Then, we present the experimental results, i.e. the tracers diffusion coefficients in Laponite as a function of the probe size and Laponite concentration and analyze them in terms of Laponite structure and hindered diffusion in porouslike medium before concluding.

Materials and methods

Sample preparation

The investigated soft glassy material is a suspension of Laponite, that consists of disks of ∼30 nm diameter, 1 nm thickness, and 2.53 g cm -3 density, dispersed in aqueous solution. 30,31 The faces of these synthetic clay platelets of Laponite, with formula Si 8 [Mg 5.5 Li 0.4 O 20 (OH) 4 ] 0.7-Na 0.7+ 0.7 , acquires a negative charge of roughly 1 e -1 .nm -2 (e charge of electron) upon solvation of Na + cations, while their edges are positively or negatively charged, depending on pH. [START_REF] Olphen | An introduction to clay colloid chemistry[END_REF][START_REF] Thompson | [END_REF][34] Solution are obtained by progressively adding the desired amount Laponite RD (Rockwood additives Ltd, used as received) to a dilution of fluorescent tracer particles in water, to obtain Laponite concentrations from 1 to 3.4%. The Laponite suspension self-buffers at pH 10, with an ionic strength of 2 to 8 mM 17 in the absence of any added salt. The obtained solution is stirred vigorously for 30 minutes to remove the initial turbidity and subsequently introduced in the optical cell and sealed with paraffin. The age of the Laponite system is calculated from this time on. A control vessel containing the same solution is kept to check the macroscopic gelation of the gel * . Typically the * The term "gelation" is used in this paper to describe the macroscopic dynamical arrest of the structure.

gelation time ranges from a few minutes to a few months, and decreases with increasing Laponite concentration (see below).

Note that the range of available Laponite concentration is quite narrow. For mass fraction of 1%, i.e., 10 g L -1 , or less, the gelation time is generally too long and the Laponite solution is damaged before gelation (due in particular to the long time chemical detioration due to CO 2 ). For mass fraction of 4 %, i.e., 42 g L -1 , or more, gelation occurs before the stirring has dispersed the powder, therefore an inhomogeneous and cloudy gel is obtained.

As fluorescent probe particles, we used either fluorescein molecules, the hydrodynamical diameter of which is about 1 nm, or commercial polystyren fluorescent nanobeads with diameters ranging from 25 to 100 nm. We mention that finding fluorescent tracers with size smaller than 100 nm, suitable for working in Laponite suspension is a difficult task, as emphasized in Ref. 35 The stability of the colloidal tracers dispersed in water buffered at pH 10 was systematically tested.

We only considered stables colloids with diffusion coefficients showing no significant variations on time scales comparable with Laponite suspension gelation time (see Section "Measurement of tracer diffusion in Laponite"). In a few situations (nanoH R fluorescent beads) the gelation of Laponite solution was observed to be prevented by the presence of the tracers, and those measurements were accordingly discarded. Altogether the different fluorescent probe particles used in this study are presented in Table 1. The reported sizes (1, 25, 37, 65 and 100 nm) have been determined through a FRAP measurement of their diffusive properties in pure water solutions. Initial colloidal solutions are diluted in ultrapure water (Milli-Q, Millipore) at concentration below 0.2% in volume which provided a satisfactory trade-off for FRAP measurements: low enough to remain bleachable without significant heating of the liquid, and high enough to lead to a good signal-to-noise ratio.

Finally, the combination of small-sized particles and weak density mismatch (1.05 for polystyren spheres) prevented the occurence of sedimentation processes in the course of the experiments.

FRAP measurement setup

Diffusion coefficient measurements were carried out by means of a modified version of the Fluorescence Recovery After Photobleaching (FRAP) technique which has been widely used in biophysics in the study, for instance, of protein diffusion in phospholipid bilayers. 36 The general principle of the measurement is that an intense laser beam is used to photobleach the fluorescence of tracers in a small region of the sample. A much weaker beam then monitors the recovery of the fluorescence signal as unbleached tracers diffuse into the bleached area. In order to gain accuracy and robustness against possible artefacts, our implementation of the FRAP technique differs from the classical one of Axelrod et al. 36 where only the overall fluorescence signal of the bleached area is recorded. Adapting a previously proposed configuration, 37 we image here the imprinted photobleaching pattern as a function of time.

More precisely (see Figure 1), the 488 nm ray (2 Watts power) of an Ar + laser (Spectra-Physics) is divided into a photobleaching beam and an illuminating by a 90%/10% beam-splitter.

The photobleaching beam passes through a cylindrical-spherical lens combination to generate a horizontal laser sheet having a vertical beam waist of 110 µm at the location of the sample cell. Illuminating beam is further attenuated by a half-wave plate combinated with a Glan-Taylor polarizer before passing through a spatial filter and being enlarged to provide uniform illumination at the scale of the observed sample region. The latter is monitored onto a cooled CCD Camera (CoolSnap HQ2, RoperScientific) through a microscope objective and lens tube (Plan Fluor x4, NA=0.13, Nikon) and an emission bandpass filter (545AF75, Omega). The whole setup is located on a vibration-damping table, while photobleaching and illuminating beam lines are controled by computer-driven shutters.

Depending on the fluorescent probe characteristics and concentration, and on the cell geometry, a photobleached layer is imprinted into the sample in a time ranging from 50 ms to 3 s (Figure 2a).

The evolution of the photobleached layer is then followed through the intensity profile I(z,t) along the vertical direction (perpendicular to bleached layer) extracted from camera images (Figure 2b).

To improve accuracy, remaining inhomogeneities in the illuminating beam are corrected using a The initial fluorescent intensity profile of the imprinted layer just after photobleaching appears perfectly described by a gaussian profile I(z, 0) = I 0 exp(-2z 2 /w 2 0 ) and therefore remains gaussian throughout its evolution (Figure 2b) governed by 1-dimensional diffusion laws:

I(z,t) = I 0 w 0 w(t) exp - 2z 2 w(t) 2 (2)
where w(t), the 1/e 2 profile waist, obeys the classical relationship (Figure 2c)

w(t) 2 /2 = 4Dt + w 2 0 /2, (3)
from which the diffusion coefficient of the fluorescent probe is obtained † .

Compared with usual implementations of FRAP technique, imaging the photobleached area presents several advantages. First, the actual initial condition immediately after photobleaching is actually measured, and does not need being modeled. Second, the presence of hydrodynamic flows is readily detected, avoiding artefacts in analysis of the fluorescence intensity evolution. This is particularly important for FRAP experiments in low viscosity systems where thermal heating during photobleaching can give rise to convective instabilities. This was especially true with colloidal probes, the quantum yield of which was, according to the manufacturers, at most below 50%, getting much lower when the colloids size increases. This effect manifested itself directly on global distorsions of the photobleached horizontal layer resulting from non-zero velocity field, as can be seen in the inset of Figure 2(a). Such a convective effect is also evidenced on the squared width evolution, w 2 (t), which departs from a linear time behavior (not shown) as a consequence of the existence of a hydrodynamical flux replacing the diffusional migration of the fluorescent colloids.

In the following, special care was taken to avoid these thermo-convective effects, first by mini-mizing the photobleaching time and power while keeping a good enough signal-to-noise ratio; and second by confining the liquid in narrow optical cells which helps stabilizing the system. Depending on the viscosity of the system, containers with sizes from 250 µm to 3 mm were used.

Overall, our setup provides a very good accuracy determining diffusion constants in water in the range (1 to 10 3 µm 2 /s) with standard deviations on measurements of few percents. The rather short gelation time measured for this low laponite concentration is probably due to the presence of surfactants conveyed by the added solution of fluorescent colloids.

Measurements of tracer diffusion in Laponite

The diffusion coefficients of the fluorescent tracers in the Laponite suspension were measured as a function of the time elapsed since cell filling. For all studied tracers and concentrations of Laponite, the time evolution of the diffusion coefficient presents first a decrease, due to the dynamical arrest process, before reaching a plateau after a "gelation time" t g (see Figure 3). The dynamical arrest process, responsible for the gelation and aging dynamics of Laponite solutions, has been the object of numerous studies: t g is known to increase when the Laponite concentration decreases, going here from less than a day for c ≥ 3 wt%, to a few days for c 2wt% and up Size dependence of tracer diffusion in a . . . to 20-60 days for c 1 wt%, values which are consistent with other studies 17,38,39 although the techniques of investigation are different. Indeed the reported values of t g 40 demonstrate appreciable disparities in gelation time determinations. This can be explained partly by differences in definition depending on the performed experiments (dynamic light scattering 17,38,40 or rheological measurements [39][40][41] ). Associated with this source of discrepancies, it is worth mentionning that the gelation process is also affected by small physico-chemical changes 17,39 and is therefore expected to be somewhat sensitive to the incorporation of tracers, even though at low concentration.

In the present paper, we therefore choose to concentrate on the behavior of the system past the initial evolution of the diffusion coefficient of the colloids, at times t t g . We report and discuss hereafter the dependency of the diffusion coefficient on the plateau D ∞ (see in Figure 3) as a function of the Laponite concentration and probe dimensions. We stress that D ∞ may also experience a slow evolution due to the ageing of the Laponite suspension. But this change occurs on timescales much greater than the ones investigated here and was not observed. In addition to gelation time, the phase diagram of the Laponite gel is known to be very sensitive to small changes in the experimental conditions. 10,17,42 Before moving to experimental results, we therefore checked first the influence of the detailed procedure of Laponite preparation on the previously defined diffusion coefficient of the probes D ∞ .

Independence of diffusion on experimental protocole

D D D D ∞ ∞ ∞ ∞ / / / / D D D D ∞ max ∞ max ∞ max ∞ max 0.4 0.6 0.
First of all, water readily adsorbs on Laponite clay and we therefore compared results obtained when the Laponite powder is either dried or not before its dispersion in water. Such a drying of the clay was achieved following the procedure used in Ref.: 10 the powder is heated in an oven at T = 400 K during four hours before being added to ultrapure water. In addition to water adsorption, filtration of the suspension also proved to be of primary importance when studying the light scattering signal from Laponite solutions. 43,44 We therefore also checked the effect of such a filtering on our probe diffusion measurements. When filtered, the suspension is pushed through a 0.45 µm filter to break up remaining aggregates.

Overall, a benchmark comparison was conducted on the probe diffusion properties for 4 different sample preparation procedures: (1) undried and unfiltered suspension, (2) undried and filtered suspension, (3) dried and unfiltered suspension, (4) dried and filtered suspension. In all cases a nominal laponite concentration of c n = 2.9 wt% was achieved by dispersing the desired amount of the operated clay in ultrapure water.

∞ measured with the procedure (2). The measured concentrations are normalized by the nominal sample concentration c n = 2.9 wt%.

Operating

Titrimetry ICPAES Titrimetry FRAP procedure concentration ratio diffusion ratio c (wt%) Table 2). This value is fully consistent with available results for water adsorption isotherms 45 showing that in standard conditions of temperature and humidity, Laponite adsorbs about 15% of water. The powder drying procedure indeed removes a significant part of the retained water but still shows a lower Laponite concentration (c < c n , protocol (3), Table 2) associated with a remaining 5% water content -presumably due in part to water re-adsorption outside the oven. Finally, the effect of suspension filtering on the final laponite content is assessed by comparing protocols (1,2) or (3,4) in Table 2; overall it shows that about 7% of the initial amount is lost in the process, a value again consistent with the litterature where the effect of different filter sizes on light scattering measurements have been reported. 43 Turning now to the diffusion properties of the same 37 nm colloidal probes in the differently prepared samples, it can be seen in Table 2 that D ∞ indeed varies with the preparation protocol.

c (wt%) c/c n D ∞ /D (2) ∞ (
However, as can be clearly seen in Figure 4, such variations are merely related to the changes in the actual Laponite concentration and do not depend on the filtering itself -and the associated removal of platelet aggregates to which light scattering is sensitive-: the higher the actual concentration c of the prepared suspension, the lower the diffusion coefficient.

Provided the Laponite concentration is properly evaluated, the so-defined probe diffusion property thus appears well-defined and insensitive the details of sample preparations. In the following, all samples are prepared in the simplest way (undried/unfiltered) and Laponite concentrations c are measured independently using titrimetry. After these different experimental validations, we can now turn to the examination of the colloidal probes' dynamics in Laponite glasses. ‡ More precisely, the concentration of Mg 2+ is determined by titrating with ethylenediamine tetra-acetic acid (EDTA). The end of the titration is determined by the disappearance of the Mg 2+ -eriochrome black T complex so by the suspension color change from red to blue.

Probe size dependence of diffusion in Laponite Diffusion coefficient of colloids in Laponite

We now examine the diffusive properties of colloidal tracer particles of various sizes, in order to probe Laponite suspension behavior and structure and capture some additional information on the local dynamical properties in this soft glassy material. Tracers characteristics were, for instance, theoretically predicted 29 to dictate the way the tracer couples to the soft glass thus affecting the effective temperature it probes. We therefore more specifically focus here on the role of the probed lengthscale, say σ in the Stokes-Einstein relationship in Eq. ( 1), on the measured diffusion coefficient D ∞ . As already noticed, in a simple liquid the product (Dσ ) should be constant (Eq. 1), whereas it is expected to vary in a glassy system if the size of the tracer approaches a characteristic length of the colloidal suspension. 27,28,35 We report in Figure 5 the experimental diffusive properties of probes with sizes ranging from about 1 nm up to 100 nm (see Table 1 First, two extreme behaviors are readily observed. On the small size side, fluorescein molecules, with an hydrodynamic diameter around 1 nm, appear to keep a constant diffusion coefficient throughout the explored Laponite concentrations, thereby being totally unaffected by the presence of the Laponite disks in the glassy medium. On the large size side, at the opposite, the 65 and 100 nm diameter colloidal tracers are completely frozen in the Laponite glass, even for the lowest clay concentration we could investigate: c ≈ 1%. Their diffusivity therefore dropped to an extremely low value, below the sensitivity of our technique, in line with the high macroscopic viscosity of the Laponite glass under these conditions. 16,44 Second, for intermediate probe sizes, corresponding here to 25 and 37 nm diameter colloids, the tracers diffusivity is lower than the one corresponding to the free solvent -water-dynamics, and gradually evolves toward a complete arrest ("gelled" medium) as the Laponite concentration increases. Comparing the evolution for the two different sizes, one evidences that the larger probe reaches the same D * ∞ (corresponding to the same reduction in diffusivity relative to the free solvent dynamics) for a lower Laponite concentration than the smaller one. To better emphasize this size sensitivity of the probe slowing down, let us stress that 25 and 65 nm diameter colloids differ in normalized diffusion coefficient by more than 2 decades while only differing by less than a factor 3 in size. This result is similar to the enhancement in rotational diffusion of small molecules measured in ref., 28 even though the investigated length scales are different in the two studies. As an illustration, we performed a test experiment using a Kisker nanosphere batch having a bimodal diameter distribution (with peaks given by the manufacturer at 26 and 75 nm). FRAP measurements performed in water (not shown) enabled to establish that the actual diameters were rather 25 and 65 nm. When dispersed in the same c = 2.4 wt% Laponite suspension, a complete separation in diffusion time scales is evident as shown in Figure 6. The effective photobleached width first increases due to 25 nm diameter colloids diffusion, with a dynamics agreeing with the one measured on monodisperse batch for a similar Laponite concentration (Figure 5). At larger times, only small colloids have homogeneized and the photobleached width is dominated by the 65 nm colloids signal. The latter appear completely frozen in the glass, with a diffusion dynamics orders of magnitude slower than expected from the simple size ratio, in agreement with the sieving effect already mentionned.

Hindered diffusion in the Laponite network

From the reported change of D * ∞ with size, it can be immediately inferred that a structural length around a few ten nanometers does exist in the Laponite glass. The exact structure of this material is still under discussion, but this characteristic length should be related in some manner to the average disk-to-disk distance d.

Consequently, the evolution of D *

∞ with the diameter σ of the probe can be interpreted in terms of a simple mechanism based on diffusion hindrance by the Laponite structure. Tiny probes essentially migrate in the inter-disks free solvent without feeling the platelets network, therefore showing an unaltered diffusion coefficient. As the probes become larger, they eventually experience motion hindrance by the surrounding Laponite disks that gets more and more pronounced as their size increases, in analogy to diffusion in a porous medium. Finally, when the probes are large enough that σ reaches a value of the order of the distance between Laponite particles, the hindrance is (-) theoretical prediction from Eq. 5 with C = 0.9. such that they appear embedded in the disks network therefore experiencing an extremely viscous environment and no detectable diffusion.

In this context, we thus expect that the measured decrease of diffusion coefficient D * ∞ with the increase of Laponite concentration c should be directly related to the concomitant decrease of the disk-to-disk characteristic distance d. From a purely geometrical point of view, and assuming a homogeneous spatial distribution of Laponite particles, this suggests d ∝ c -1/3 . More quantitatively, one expects d = (6/π) 1/3 (ρ lap v/ρ glass ) 1/3 c -1/3 with c Laponite mass concentration, ρ lap , ρ glass the density of the Laponite disks and of the Laponite solution § , and v the volume of a Laponite disk.

In Figure 7, all experimental data for the normalized probes diffusivity D * ∞ are plotted as a function of the ratio between the probes diameter and the above estimate of the mean disk-to-disk distance σ /d(c). Interestingly, these simple "geometrical confinement" considerations do quite satisfactorily account for the various data sets, that gather to form a continuous evolution -although a definite evidence of a true master curve would require more overlap among the various data sets.

In accordance with previous reasoning, the tracer diffusion is seen to become undetectable within § Due to the presence of water, the Laponite glass density only depends slightly on c. For c ≤ 3.5% wt, this dependence can be neglected as ρ glass and ρ water differ from less than 2%.

our experimental resolution when their size reaches the Laponite disk-to-disk distance: σ ∼ d(c).

Going further, the dependence of diffusion on geometrical parameters, σ and d(c), suggests that the diffusion hindrance can be interpreted in terms of the hydrodynamic interactions between the probe and the Laponite disks structure, and not by their physico-chemical interaction. Accordingly the mechanism of probe diffusion in the Laponite glass is similar to that of hindered diffusion in an open porous network. 19,20 Due to the long-range nature of hydrodynamic interaction, the description of diffusion processes in static (or dynamic) networks is in general a formidable task. We point for example to Refs. 22,23 for recent works on the diffusion of tracers in rod networks. In the present case, our measurements were performed after gelation occured and one may thus consider the laponite network as static for the tracer diffusion. Accordingly we do not consider the influence of ageing of the laponite structure at very long times: this should result in a small, long-time diffusion of the largest probes, which is not measured with the present experimental setup (see discussion below). Furthermore, the laponite network is made of an arrangement of disks and to our knowledge tracer diffusion in such a network has not been considered theoretically. Yet, we thus propose a simplified approach, aiming at rationalizing our experimental results.

In the limit where the probe size is small compared to the disk-to-disk distance, σ d, the long-ranged hydrodynamic interactions can be expanded in powers of σ /d, thus leading to a probe diffusion coefficient behaving as: 20,[START_REF] Happel | Low Reynolds number hydrodynamics[END_REF]

D D 0 = 1 -α σ d + β σ d 2 + . . . (4)
with α, β numerical coefficients (α > 0). In general, the expansion organizes itself as a multipole expansion of the hydrodynamic interactions between the probe and the structure.

Of course, such an expansion is expected to break down as the size of the probe reaches a fraction of the interparticle distance, σ ∼ d: this is the dense regime. For laponite disks with diameter 30nm, this corresponds also to the situation where the probe size is of the same order as the laponite disk size. In this limit, the probe's motion is strongly confined between the disks of the Laponite netword and the hydrodynamic interactions are strongly screened. But one may note that in this regime, probe diffusion should involve mainly motion of the beads parallel to the laponite disks, at a distance to the disk fixed by a fraction of d (since motion perpendicular to the disks is even more strongly hindered [START_REF] Happel | Low Reynolds number hydrodynamics[END_REF] ). Under a drastic simplification, we propose accordinly to reduce the probe diffusive dynamics to this elementary process, i.e. involving motion parallel to laponite disks.

The mobility of a sphere moving parallel to a plane has been considered by Brenner, [START_REF] Happel | Low Reynolds number hydrodynamics[END_REF] and more recently in Refs. [START_REF] Joly | Nanohydrodynamique au voisinage d'une surface solide : de la caractérisation expérimentale à l'équilibre aux conséquences sur la dynamique des systèmes chargés[END_REF][START_REF] Joly | [END_REF][49] in the context of diffusion in strongly confined geometry. An expression for the parallel diffusion coefficient between two walls separated by a distance d has been obtained in this geometry [START_REF] Happel | Low Reynolds number hydrodynamics[END_REF][START_REF] Joly | Nanohydrodynamique au voisinage d'une surface solide : de la caractérisation expérimentale à l'équilibre aux conséquences sur la dynamique des systèmes chargés[END_REF][START_REF] Joly | [END_REF][49] and reads

D D 0 = 1 2 1 -C σ d -1 (5)
with C a numerical constant (C = 9/8 for diffusion between two infinite rigid walls). This equation was shown to provide a good approximation for the diffusion processes down to strong confinement. 49 Here a limit is also provided by the condition σ < d/C. For larger d, a vanishing diffusion coefficient is expected. This expression should be merely considered as an interpolation between a strongly hindered regime, with σ ∼ d and D → 0, and the dilute regime, with σ d, in which the diffusion coefficients expands in σ /d along Eq.( 4).

Nonetheless, it provides a useful fit for the evolution of the diffusion coefficient with σ /d: as is shown in Figure 7, the experimental results show good agreement with the fitting formula, Eq.

(5), with C = 0.9.

Let us conclude this part by a few remarks:

First, as noted above, we have considered a "frozen" Laponite structure while in fact the diffusion motion of the probe inside the Laponite glass may be able to induce a local deformation of the structure. Indeed due to the soft nature of the Laponite glass, the clay platelets could be able to rotate or move a small distance to make the probe migration easier. For instance, the time needed for the intermediate 37 nm probe, diffusing in 2 wt% Laponite, to move over a clay particle diameter is about 10 -4 s. This is to compare with the characteristic time of the free diffusion of Laponite, i.e., the relaxation time of the rattling motion of a platelet in the cage formed by its neighbors, which has been evaluated by dynamic light scattering to be ∼10 -4 s. 10,40 Consequently, the Laponite network may reorganize locally during the passage of a nanosphere and potentially facilitate its diffusion.

Second, an interesting remark is that the simple relationship d(c) ∝ c -1/3 provides a sound estimate of the disk-to-disk distance characterizing the diffusive dynamics insider the Laponite glass structure. This promotes the idea that the spatial distribution of the platelets is homogeneous, although a definite proof of this homogeneous structure has not been reached up to now in the litterature. Using the distinction between a colloidal gel and a colloidal glass given in ref.

, 16 our result is in favor of a glass whose characteristic length scale is of the order of the inter-particle distance whereas the characteristic length scale of a gel would be expected to be much larger than the size of its constitutive colloids.

Finally, it is interesting to mention that the above evolution of D(σ ) can be alternatively interpreted by considering that the colloids migrate in an inhomogeneous medium showing a scaledependent effective viscosity η eff (σ ). This interpretation has been proposed in a different context for the scale dependent viscosity of polymers melts, 50 as well as in recent experiments on diffusion in crowded micellar solutions 24 .

In the present case, this viscosity can be estimated using a Stokes-Einstein type relation, thus defining a viscosity η eff (σ ) = kT /(3πσ D(σ )). The extreme values of η eff (σ ) are known: in the limit where the colloids are so small that they diffuse freely in the Laponite structure then lim σ →0

η eff (σ ) = η water ; in the opposite limit where the colloids are large compared to the Laponite structure, they feel the macroscopic viscosity of the Laponite glass and lim σ →∞ η eff (σ ) = η macro ( η water ). Typically, η macro is five to six orders of magnitude higher than the water viscosity. 16,44 For intermediate scales, the scale dependent viscosity will interpolate between these extreme values.

Size dependence of tracer diffusion in a . . .

Assuming η macro = ∞, Eq. 5 provides an empirical expression for the scale dependent viscosity, in the form η eff (σ ) = η water 2(1 -Cσ /d) -1 -1 , for σ < d/C, while η eff = ∞ for σ > d/C.

Conclusion

In the present work, we have used a modified Fluorescence Recovery After Photobleaching setup to measure the diffusion coefficient of fluorescent nanoprobes, of sizes ranging between 1 and 100 nm, in Laponite glasses of concentrations from 1 to 3.4 wt%. A change of diffusive behavior is evidenced for probe sizes around few ten nanometers, which corresponds to the typical distance between the disk of clay constituting the soft glass skeleton. Far below this value, the nanospheres migrate in the solvent showing little interaction with the Laponite, and their diffusion coefficient is the same as in pure solvent. Far above this value, the motion of the nanospheres is strongly hindered by the network of Laponite platelets and their diffusion coefficient is seen to vanish. In between, increasing the probe size leads to a gradual increase of the hindrance of diffusive motion by Laponite platelets from free solvent diffusion down to experimentally undetectable motion. The diffusion coefficient measured for various probe sizes and Laponite concentrations are shown to be well accounted for when considering only a dependency on probe size to disk-to-disk distance ratio. This simple dependence promotes the picture of a homogeneous glass structure characterized by a well defined interparticle distance, and indicates that the hindered diffusion originates in the hydrodynamic interaction between the probe and the Laponite glass structure. In particular, it

shows that in such situation the relationship between diffusivity and probe size results from the heterogeneous nature of the soft glass at the scale of the probe, which can be accounted for by a scale dependent viscosity in line with results for polymers.

Figure 2 :

 2 Figure 2: Evolution of a photobleached slab of solution. a) Typical image of the photobleached sheet (inset: sheet distorsion in presence of thermal convection); b) Fluorescence intensity profiles of fluorescein solution photobleached at t = 0, after various elapsed times (from 3 to 64 s); c) Time evolution of the squared width w 2 of the gaussian photobleached intensity profile during fluorescence recovery.

Figure 4 :

 4 Figure 4: Diffusion coefficient of 37 nm diameter colloids as a function of the measured concentration of Laponite elaborated with different operating procedures (see text). The nominal concentration of Laponite is c n = 2.9 wt% in the four cases.

Figure 5 :

 5 Figure 5: Evolution of the normalized diffusion coefficient, D * ∞ = D ∞ (c)/D 0 , of colloidal tracers in Laponite, as a function of the concentration of laponite for various tracer sizes: (♦) 1 nm fluorescein molecules; ( ) 25 nm diameter colloids; ( ) 37 nm diameter colloids; ( ) 65 nm diameter colloids; (•) 100 nm diameter colloids.

  ) in Laponite glasses of various concentrations c. This figure displays, for the various tracers, the evolution of the probe diffusion coefficient in the Laponite glass, normalized by its measured value in pure water (D * ∞ = D ∞ (c)/D 0 ), as a function of Laponite Size dependence of tracer diffusion in a . . . concentration c. Each data point gathers the results of 10 to 100 independent measurements on different samples, from which the average value and standard deviation is obtained.

Figure 6 :

 6 Figure 6: Non-monotonic evolution of the photobleached stripe width for a batch of bidisperse fluorescent colloids provided by Kisker R in a Laponite glass of concentration 2.4 wt%. The insert shows the short time evolution of the waist diameter.

Figure 7 :

 7 Figure 7: Normalized diffusion coefficient D * ∞ as a function of the ratio between the probe size σ and the disk-to-disk distance d(c) (see text): (♦) 1 nm fluorescein molecules; ( ) 25 nm diameter colloids; ( ) 37 nm diameter colloids; ( ) 65 nm diameter colloids; (•) 100 nm diameter colloids;(-) theoretical prediction from Eq. 5 with C = 0.9.

50

 50

Figure 8 :

 8 Figure 8: Graphic for TOC

Table 1 :

 1 Summary of fluorescent probe particles used. The actual size is deduced from FRAP measurements of the particles diffusion coefficient in water through Stokes-Einstein law. Colloids' manufacturers:(a) , FluoProbes R ; (b) Polyscience R ; (c) , Kisker R ;(d) , Molecular Probes R .

	FRAP measured nominal diffusion standard
	diameter	diameter coefficient	error
	σ (nm)	σ n (nm)	µm 2 s -1	µm 2 s -1
	1	/	427	5
	25 ± 0.5	28 (a)	17.5	0.3
	37 ± 2	50 (b)	11.7	0.6
	65 ± 7	70 (c)	6.7	0.7
	100 ± 2	100 (d)	4.3	0.1
	mirror		beam sampler	mirror
	BLEACHING
	BEAM	computer
	cylindrical
	lens
		camera
	spherical lens	OBSERVATION BEAM
		emission filter
		lens
	microscope objective
	cell
				laser
	spherical
	lens	LIGHTNING
	pinhole	BEAM
	microscope objective
		prism
	half-wave plate
		shutter
			mirror

shutter Figure 1: Fluorescence Recovery After Photobleaching setup (see text for details).

Table 2 :

 2 Influence of the sample preparation procedure on the actual laponite concentrations and on the diffusion coefficients D ∞ of 37 nm diameter colloids in the associated laponite gel. The diffusion coefficients are normalized by the diffusion coefficient D

  Spectroscopy (ICPAES). The principle of Laponite suspension titrimetry is described in details in:42 it consists in dissolving Laponite at low pH before analyzing the concentration of dissolved Mg 2+ by using complexiometric titrations ‡ . Starting with the effect of water absorption by

	Laponite powder, we note, comparing actual (c) and nominal (c n ) Laponite concentrations, that the
	undried clay retains 13% of water (protocol (1), table
	1) undried/unfiltered laponite	2.52	2.57	0.87	0.74
	(2) undried/filtered laponite	2.34	-	0.81	1
	(3) dried/unfiltered laponite	2.74	-	0.95	0.54
	(4) dried/filtered laponite	2.54	-	0.88	0.65

We first examine the actual Laponite concentration c resulting from the different sample preparations (see in Table

2

), as determined from titrimetry and Induced Coupled Plasma Atomic Emis-Size dependence of tracer diffusion in a . . . sion

† The gaussian amplitude evolution can also be used to deduce diffusivity although less accurately due mostly to residual bleaching during the recovery process.

Acknowledgements-We thank Jean-Louis Barrat for many interesting discussions and suggestions. We acknowledge support from the ANR, through the program 'ANR Blanche' (Sllocdyn).