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.

A promising approach was launched by Dan Sperber [START_REF] Sperber | Explaining Culture: A Naturalistic Approach[END_REF] in the mid-nineties: cultural epidemiology. It is theoretically well developed, but still empirically not well tested due to the challenge in setting up concrete experiments. We aim to further test the cultural epidemiology framework with case studies, and advance the debate on a common ontology for cognitive and social sciences.

We use the MemeTracker dataset [START_REF] Leskovec | Meme-tracking and the Dynamics of the News Cycle[END_REF] to track the evolution of quotes in blogspace: quotes should not change, but authors often unconsciously transform them nonetheless when writing a blog-post.

By tracking word substitutions when quotes are copied, we characterize one of the ways in which these pieces of information are transformed as they go from blog to brain and back to blog, and so on.

Smartphones and tablets let us gather very large amounts of data under controlled and filterable conditions [START_REF] Miller | The smartphone psychology manifesto[END_REF].

We will develop a smartphone game using the Science en Poche tools to accurately measure how transformations are introduced and/or constraints are followed when people repeat short sentences, in various conditions.

The insight on transformations will provide the micro level for multi-agent models of epidemiology of representations.

A key claim from the theory that is to be tested is the emergence of stable cultural attractors as fixed points of the whole dynamical system. Most of the questions in this poster also relate to the notions of interpretation, meaning (or making sense), and social interaction, which are still all left out of the discussion. So does this approach properly address the social-cognitive communication question? Can we improve?

Smartphones are a new kind of microscope [START_REF] Anthony G Greenwald | There is nothing so theoretical as a good method[END_REF]. Can we leverage it better to ask questions relevant to both cognitive and social sciences?

Introduction Experiment 1 -

 1 psycholinguistics on quotes from the Internet Experiment 2 -real-life chinese whispers game-experiment Recent years have seen several attempts to bring the cognitive and social sciences together. Social cognition or Cognitive economy are examples of such initiatives. But getting the cognitive and social levels of explanation to interoperate so they can participate in common questions requires a common ontology which does not yet exist.

  The ontology consists of mental representations (those of cognitive science), and their physical counterpart, public representations (those of social science). Mental representations live in the mind, are produced as public representations, and are interpreted by other minds to form new -different-mental representations. Cognitive modules determine how and why representations are interpreted and/or produced.

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  It was a very strange situation » « It was a very dynamic situation » log(f (daughter)) -log(f (mother)) > How word frequency and number of phonemes of a word vary upon substitution How word frequency and number of phonemes relate to susceptibility to substitution Wider questions Poster built with Inkscape. Except where otherwise noted, content on this poster is licensed under CC-BY-SA-4.0 Image built from work created and shared by the Android Open Source Project (design pages of the developer website), used according to terms described in the Creative Commons 2.5 Attribution License.


Preliminary approach for crypt detection in Inflammatory Bowel Disease Bassem Ben Cheikh 1,2,3, * , Philippe Bertheau 4 , Daniel Racoceanu 1,2,3 1 Sorbonne Universités, UPMC Univ Paris 06, UMR 7371, U1146, Lab. d'Imagerie Biomédicale, F-75013, Paris, France. Abstract -Crypt architecture is one of the most significant histological features used for the examination of colorectal biopsy specimens enabling clinical decisions in the investigation of Inflammatory Bowel Diseases. However, the architecture modelling remains a challenging problem leading to variability in reporting and subjectivity in pathological examination. In this context, intestinal gland detection represent a necessary step before a clinical study of their architecture. This work presents a graph-based technique describing spatial relationships over sparse structures for crypt detection using morphological mesh filtering operators.

Index Terms -Image Processing, Medical Informatics, Microscopy.

I. INTRODUCTION

Inflammatory Bowel Diseases (IBDs) are characterized by chronic inflammation of the gastrointestinal tract, principally the colon and small intestine. The routine diagnosis of an IBD is based on biopsy; where a tissue is removed from the suspected organ and examined by the pathologist under a microscope. A biopsy of an affected colon tissue may show abnormalities in its histological structure. The morphological features of the intestinal glands are significant indicators for the severity of the disease. In order to introduce these morphological features, gland detection is a necessary step ahead. For this purpose, most of the existing methods use replicated classifications of color [1], texture [2], or graph [3] features. In this work, we present a new graph-based technique and morphological mesh filtering transformations for intestinal gland detection. Section 2 presents an overview of our method and section 3 presents the results and the limitations of this approach.

II. MATERIALS AND METHODS

One of the most distinctive properties of a gland is that they usually exhibit a closed shape structure surrounded by a layer of epithelial cell nuclei. Compared to the connective tissue, the inner area of an intestinal gland doesn't contain nuclei, being composed of cytoplasm, goblet cells, Based on these observations, we propose a novel graph approach to distinguish the structure of an intestinal gland from the rest of the tissue. For this purpose, a biopsy tissue image is first decomposed into sets of points, associated to different tissue components. Furthermore, we use their distribution characteristics in order to determine the locations of gland structures.

II.1. Image decomposition

One of the most widely used routine-stains in histopathology is the Hematoxylin and Eosin (H&E). In a typical image of a H&E-stained colorectal tissue, there are mainly three colour groups of pixels. The nucleus pixels have dark purple colour, whereas the cytoplasm and stroma pixels have varying degrees of pink, and lumen pixels are white. Hence, the pixels of an image are quantized into three clusters with the k-means algorithm.

II.2. Node identification

In order to set up nodes to lead the graph reconstruction, we need first to translate the color class information to node information. For this purpose, a grid is placed on the resulting segmented image. For each grid entry we calculate the ratio of pixels belonging to a color class to the size of the grid entry. If this ratio is higher than a threshold then a node is set within the grid entry.

  2 CNRS, UMR 7371, Laboratoire d'Imagerie Biomédicale, F-75013, Paris, France. 3 INSERM, U1146, Laboratoire d'Imagerie Biomédicale, F-75013, Paris, France. 4 Saint-Louis Hospital, Department of Pathology, APHP, and University Paris Diderot USPC, Paris, France * bassem.bencheikh@lib.upmc.fr.

  Figure 1: (a.1) connective tissue, (a.2) crypt and (a.3) blood vessel. (b) Red: lumen. Yellow: goblet cells. Blue: epithelial cell nuclei. Green: immune system cell nuclei. Pink: cytoplasm. Purple: stroma.


J Grynberg

E Gouillart

M.-H Chopinet

M J Toplis

Importance of the atmosphere on the mechanisms and kinetics of reactions between silica and solid sodium carbonate

Using thermogravimetric analysis, the reaction kinetics between powders of silica (SiO2) and sodium carbonate (Na2CO3) are investigated below the melting point of sodium carbonate, under different atmospheres. These experiments show that the reaction kinetics critically depend on the partial pressure of CO2 in the surrounding atmosphere. Under a flow of nitrogen, the reaction rate is constant with time and follows an Arrhenian law, with a weak influence of the grain sizes of the two reactants. Under a flow of carbon dioxide, the reaction rate is found to be much slower, to have non-linear time dependence and to have a weaker temperature dependence. The influence of grainsize is also found to be more significant in this case. Two different reaction mechanisms between silica and sodium carbonate may account for these results: (i) sodium carbonate dissociates at a rate depending on the partial pressure of CO2, releasing sodium to the vapor, that may then react with the surface of silica grains; or (ii) silica and sodium carbonate grains react at granular contacts between individual grains. A quantitative equation is proposed that accounts for the influence of temperature, pCO 2 and grain size on the reaction rate between silica and sodium carbonate.

I. INTRODUCTION

The system sodium carbonate -silica is one of the simplest possible from which a silicate melt may be produced at temperatures less than 1000 • C. In addition, grains of silica and sodium carbonate are among the principal constituents used in the window-or bottle-glass industry.

The reaction kinetics between sodium carbonate and silica powders have been extensively studied [START_REF] Howarth | A study of the fundamental reactions in the formation of sodalime-silica glasses[END_REF][START_REF] Howarth | The rate of reaction between silica and sodium carbonate at different temperatures and the process of vitrification[END_REF][START_REF] Pole | Kinetics of solid-phase reactions of certain carbonates with mullite, silica and alumina[END_REF][START_REF] Gibsont | Reactions in solid state: 111, reaction between sodium carbonate and quartz[END_REF][START_REF] Wilburn | The application of differential thermal analysis and thermogravimetric analysis to the study of reactions between glass-making materials -part 1. the sodium carbonate -silica system[END_REF][START_REF] Ott | Kinetics and mechanism of the reaction between sodium carbonate and silica[END_REF][START_REF] Hrma | Reaction between Sodium Carbonate and Silica Sand at 874 • C < T < 1022[END_REF], using various techniques such as thermogravimetric analysis [START_REF] Howarth | The rate of reaction between silica and sodium carbonate at different temperatures and the process of vitrification[END_REF][START_REF] Wilburn | The application of differential thermal analysis and thermogravimetric analysis to the study of reactions between glass-making materials -part 1. the sodium carbonate -silica system[END_REF], in-situ X-ray diffraction [START_REF] Michael | Kinetics of glass batch reactions[END_REF], or in-situ 23 Na NMR [START_REF] Aled R Jones | Tracing the reactive melting of glassforming silicate batches by in situ 23Na NMR[END_REF]. Reactions are known to be accelerated above the melting point of sodium carbonate at 865 • C, when the molten carbonate rapidly wets silica grains [START_REF] Hrma | Reaction between Sodium Carbonate and Silica Sand at 874 • C < T < 1022[END_REF]. However, reactions may also occur below the melting point of sodium carbonate, either to form crystalline Na 2 SiO 3 or Na 2 Si 2 O 5 , or even a silicate melt given that the eutectic of the SiO 2 -Na 2 O binary is at 785 • C [START_REF] Fc Kracek | The system sodium oxide-silica[END_REF][START_REF] Hrma | Batch melting reactions[END_REF]. Quantification of the mineralogy and spatial distribution of reaction products is of primary interest, as these factors will influence the evolution of the system at higher temperatures.

There is general agreement [START_REF] Hrma | Reaction between Sodium Carbonate and Silica Sand at 874 • C < T < 1022[END_REF][START_REF] Aled R Jones | Tracing the reactive melting of glassforming silicate batches by in situ 23Na NMR[END_REF][START_REF] Kim | Decomposition of Na2CO3 by interaction with SiO2 in mold flux of steel continuous casting[END_REF] that reaction between silica and solid sodium carbonate first produces crystalline Na 2 SiO 3 :

Na 2 CO 3 + SiO 2 → Na 2 SiO 3 + CO 2 .
(1)

However, different authors used different grain sizes, atmospheres or bulk composition, making results on reaction kinetics difficult to compare: published reaction rates at 800 • C [START_REF] Howarth | The rate of reaction between silica and sodium carbonate at different temperatures and the process of vitrification[END_REF][START_REF] Michael | Kinetics of glass batch reactions[END_REF][START_REF] Nandi | Reaction of silica with Na2CO3 and with the eutectic and double carbonate of Na2CO3 and CaCO3[END_REF] differ by one order of magnitude. Furthermore, activation energies ranging from 100 kJ.mol -1 [START_REF] Ott | Kinetics and mechanism of the reaction between sodium carbonate and silica[END_REF] to 490 kJ.mol -1 [START_REF] Nandi | Reaction of silica with Na2CO3 and with the eutectic and double carbonate of Na2CO3 and CaCO3[END_REF] have been described. Finally, previous studies [START_REF] Gibsont | Reactions in solid state: 111, reaction between sodium carbonate and quartz[END_REF][START_REF] Hrma | Batch melting reactions[END_REF][START_REF] Moriya | On the rate of solid reaction in the Na$ 2$CO$ 3$-SiO$ 2$ system[END_REF]] noticed that reaction rates typically decrease when the sample size increases; this effect was attributed to differences in the partial pressure of CO 2 but the literature shows no example of controlled experiments that tested this hypothesis. In this work, the reaction kinetics of SiO 2 -Na 2 CO 3 are revisited, with emphasis being placed on the effect of CO 2 partial pressure. Isothermal thermogravimetric analyses (TGA) are used to measure reaction kinetics as a function of time, for two different atmospheres. Our main result is the identification of two different reaction mechanisms between SiO 2 and Na 2 CO 3 . In the first, decomposition of sodium carbonate leads to the production of a sodium-containing vapor that reacts with the surface of silica grains that are in the vicinity, but not necessarily adjacent. In the second, reaction occurs at physical contacts between adjacent grains of sodium carbonate and silica. The nature of the dominant mechanism is found to depend on several parameters, in particular the partial pressure of CO 2 and the temperature, but also on grain size.

II. EXPERIMENTAL METHODS

a. Raw materials -Roncevaux silica sand (99.8% silica, 0.05%Al 2 O 3 , 0.03%CaO, 0.03%Fe 2 O 3 , 0.01%K 2 O 0.02%TiO 2 ) and Solvay sodium carbonate have been used as starting products. We use industrial-grade (instead of reagent-grade) raw materials because we aim at characterizing reaction kinetics relevant for industrial processes (in particular, with grain sizes close to that used in industry). Sodium carbonate grains have an important micro-porosity. Grains of both carbonate and silica were sieved in three different size bins: 100-160 µm (small -S), 200-250 µm (medium -M) and 400-500 µm (coarse -C). The average size of silica grains used in the float glass industry is close to our medium bin-size. Raw materials were dried at 110 • C for several hours before the experiments. All experiments use the same relative weight proportions of the two reactants: 64% SiO 2 and 36% Na 2 CO 3 (corresponding to a final glass composition of 75% SiO 2 and 25% Na 2 O). Compared to industrial soda-lime silica glass, this amounts to replacing the fraction of calcium carbonate by sodium carbonate. Mixtures were made from these sieved materials, each mixture denoted by two letters, the first corresponding to the size of silica grains and the second to that of sodium carbonate grains. For example, M-S corresponds to medium silica grains and small sodium carbonate grains.

b. Thermal analyses -Two different thermogravimetric instruments have been used: a Netzsch STA 449C apparatus equipped with an electronic microbalance, operating under a flow of nitrogen, and a Netzsch STA 409C apparatus equipped with a mechanical balance, that operates under a flow of CO 2 . A flow rate of 10 mL/min was used. We choose to perform isothermal thermogravimetric experiments, since the parameters governing the kinetics of reaction (1) would change with time for a nonisothermal treatment. In all experiments, the sample was heated from room temperature to the temperature of interest at a rate of 10K/min. Samples were then held at constant temperature for one hour in the experiments under nitrogen and some experiments under CO 2 , and for 24 hours for the remaining experiments under CO 2 . Separate blank runs were used for baseline correction. A sample mass of 100 mg was used for all experiments, with the exception of three experiments, for which a much larger sample mass of 1.4 g was used in order to assess the influence of the size of the system, as in [START_REF] Gibsont | Reactions in solid state: 111, reaction between sodium carbonate and quartz[END_REF][START_REF] Moriya | On the rate of solid reaction in the Na$ 2$CO$ 3$-SiO$ 2$ system[END_REF].

c. Static melting and observations -Complementary experiments on 5-g mixtures Na 2 CO 3 -SiO 2 were performed, in order to observe the mineralogy and spatial distribution of reaction products after quenching to room temperature. Samples were characterized using either a binocular microscope or with scanning electron microscopy (SEM). For the acquisition of SEM images the samples were first impregnated with epoxy resin, cut vertically, and polished using a water-free lubricant.

III. RESULTS

A. Nitrogen atmosphere

Measured weight loss is assumed to correspond to that of CO 2 according to Eq. (1). Hence, the fraction X of reacted sodium carbonate may be directly deduced from the thermogravimetric results. The variation of X as a function of time is shown in Fig. 1 for experiments performed under nitrogen, for four different pairs of grain sizes (S-M, M-S, M-M, and M-C), and at three different temperatures (750 • C, 800 • C and 820 • C).

For most experiments, a linear variation of X with time is observed, corresponding to a constant reaction rate:

X = αt.
(2)

820 • C MM-N 2 MC-N 2 MS-N 2 SM-N 2 (c)
0.00090 0.00095 0.00100 However, we observe that at 750 • C (and to a lesser extent at 800 • C) the reaction rate is somewhat faster during the earliest stage of the experiment, before reaching a linear regime. Hence, we define the reaction rate α as the linear regression of data obtained between 15 and 60 min. Reaction rates are shown vs. 1/T in Fig. 1 (d). For all grains sizes, an Arrhenian dependence of the reaction rate is observed

1/T (K -1 ) 10 -3 10 -2 v (% NC .s -1 ) MM-N 2 MC-N 2 MS-N 2 SM-N 2 MM-CO 2 MM-CO 2 -24h (
α(T ) = A exp - E 1 RT . (3)
The value of the activation energy is similar for all sizes, E 1 490 ± 30kJ.mol -1 , and comparable to that measured by [START_REF] Nandi | Reaction of silica with Na2CO3 and with the eutectic and double carbonate of Na2CO3 and CaCO3[END_REF] (443 kJ.mol -1 ).

The effect of grain size is small, but at fixed temperature experiments performed with smaller grains have a reaction rate that is systematically higher than experiments with larger grains. For the different mixtures, the relative order of the reaction rates is the same for all temperatures. However, the ratio between the fastest and the slowest reaction rates at fixed temperature is less than 2, while the surface area of grains is multiplied by a factor of 16 between small and coarse grains. Therefore, the evolution of the reaction rate is clearly not proportional to the specific surface area of grains of either species.

B. CO2 atmosphere

Additional TGA experiments were performed under CO 2 . For the M-M mixture, thermal treatments for one hour at 750, 800 and 840 • C, and for 24-hours at 800 and 820 • C were made (see Fig. 2 (a)). 24-hour experiments were also performed at 820 • C for mixtures M-S and S-M (see Fig. 2 (b)).

Comparison of the final extents of reaction after 1 or 24 h shows that heating under a CO 2 atmosphere leads to far less reaction than under nitrogen. To quantitatively compare reaction kinetics under the two atmospheres, a reaction rate was also calculated for the CO 2 experiments as a linear growth rate between 15 and 60 minutes . The reaction rates for the CO 2 experiments defined in this way are shown in Fig. 1 (d): compared to the nitrogen atmosphere, the reaction is slower by more than one order of magnitude. Furthermore, an Arrhenian fit gives a much lower activation energy, E 2 162kJ.mol -1 .

However, the time-dependence of reacted fraction X under CO 2 is not linear, but can be fitted by a function proportional to the square root of time

X = C √ t. (4)
The coefficient C depends on both temperature and grain size. At constant temperature, the relative ordering of reaction rates as a function of grain size is not the same as that observed under a flow of nitrogen. For example, under CO 2 small sand grains (S-M) lead to a slower reaction rate than medium grains, whereas the opposite is observed under N 2 (Fig. 1 (c)). Under CO 2 , the fastest rate is observed for small sodium carbonate grains. At long times (typically after 10 hours), the evolution of X levels-off and is no longer fitted by a square-root function of time, but rather by a (slow) linear evolution.

Overall, given that changing the atmosphere from N 2 to CO 2 leads to very different reactions rates, and that the variation of X as a function of time is different, it appears reasonable to suspect that different reaction mechanisms might be at play in each case.

C. Static melting, air atmosphere

Two static-melting experiments were performed in air atmosphere, using 5 g of the M-M batch. For the first experiment, a lid partially covered the top of the crucible, whereas no lid was used in the second experiment. The samples were heated at 820 • C for four hours in two separate experiments. The weight loss was measured after quenching the samples. For the sample with a lid, a reacted fraction X of 38% was computed, whereas the reacted fraction was 83% for the sample without a lid. For comparison, in the TGA experiments the reacted fraction after 4 hours at 820 • C is 18% under CO 2 flow (and predicted to be 100% under N 2 , from the extrapolation of Fig. 1). Therefore, the reaction rate in air seems to lie in-between those measured under for CO 2 and N 2 atmospheres, with a faster reaction when no lid is present.

SEM images of polished sections of the sample with a lid are shown in Fig. 3 (c-d). Most quartz grains are surrounded by a porous reaction layer, this layer appearing wider close to Na 2 CO 3 grains. Using Raman spectroscopy, we observe that this reaction layer is composed of crystalline Na 2 SiO 3 and Na 2 Si 2 O 5 . On the SEM images, we also observe that the quartz grains have sintered together in many places.

IV. REACTION MECHANISMS

It has been suggested [START_REF] Ott | Kinetics and mechanism of the reaction between sodium carbonate and silica[END_REF] that reaction (1) is controlled by diffusion of alkalis at the surface of silica grains, and later by diffusion through the reacted shell. However, this reaction mechanism alone cannot explain the huge difference between the behavior observed under N 2 and CO 2 atmospheres, in particular the very different timedependence of reaction extent.

In order to relate the observed differences to reaction mechanisms, we have performed two simple experiments, that aim to highlight to what extent grain-grain contacts are important. In the first experiment, a 5-g SiO 2 -Na 2 CO 3 sample was held for 5 min at 750 • C in an air atmosphere. After the treatment, sodium carbonate grains are surrounded by a shell of quartz grains that adhere to the central sodium carbonate grain (Fig. 3 (a)), indicating that reaction has taken place primarily at the contacts between silica and sodium carbonate.

In the second experiment, sodium carbonate and silica were put in physically-separated alumina crucibles, with the silica-containing crucible nested inside the sodium carbonate-containing crucible (inset of Fig. 3 (b)). A lid partially covered the surface of the outer crucible. This sample was held for four hours at 820 • C. After quenching, it was observed that the upper layers of sand grains have sintered together (see Fig. 3 (b)). We attribute the mechanical cohesion of these sand grains to a layer of reaction product. Although the reaction layer was too thin to determine its composition using Raman spectroscopy, we believe that it is composed of sodium silicates. In the absence of a physical contact between SiO 2 and Na 2 CO 3 , the origin of the reaction product is implied to be the decomposition of Na 2 CO 3 and the subsequent diffusive transport of a sodium-containing gas, that reacts with the surface of silica grains. This reaction mechanism is discussed in more details in the next section. Surface diffusion along the alumina crucible can be excluded as the primary reaction mechanism, since the sintered grains are found on the upper layers of the sand crucible, and not along its walls. Recent work using in-situ tomography [START_REF] Gouillart | In-situ synchrotron microtomography reveals multiple reaction pathways during soda-lime glass synthesis[END_REF] observed that secondary sodium silicates cover a large fraction of the available quartz surface, and not only those regions where quartz and sodium carbonate were in direct contact, providing independent evidence for vapor-phase transport of sodium.

The thermal decomposition of solid sodium carbonate in the absence of any other chemical species has been studied by Preston and Turner [START_REF] Preston | The decomposition of sodium carbonate[END_REF], using thermogravimetric measurements in dry air as well as in an atmosphere containing water vapor. In their experiments, less than 1% of sodium carbonate was found to have decomposed during a 1-hour thermal treatment at 800 • C in dry air, while 5% of sodium carbonate decomposed in an atmosphere of water vapor after 1 hour at 815 • C. Using the FactSage Software, we computed the equilibrium constants of the two reactions

Na 2 CO 3 → 2Na (g) + 1 2 O 2(g) + CO 2(g) (5
)

Na 2 CO 3 + H 2 O (g) → 2NaOH (g) + CO 2(g) , (6)
to be respectively 4.10 -22 and 10 -11 at 820 • C. Such weak values explain the slow decomposition rate observed by Preston and Turner. However, when silica is present, the sodium-bearing gases produced by reaction ( 5) or ( 6) may be removed from the vapor to produce Na 2 SiO 3 :

SiO 2 + 2Na + 1 2 O 2 → Na 2 SiO 3 , (7)
or in the presence of water in the vapor

SiO 2 + 2NaOH → Na 2 SiO 3 + H 2 O. ( 8
)
In this case, reaction ( 5) or ( 6) are continually displaced to the right, according to Le Chatelier's principle. The decomposition of Na 2 CO 3 can thus be accelerated, resulting in the equivalent reaction (1). This mechanism may be used to rationalize the results obtained under a flow of N 2 . Most importantly, the activity of sodium-bearing gases in ( 5) or ( 6) decreases when p CO2 increases. If the partial pressure of CO 2 is weak (e.g. under a nitrogen atmosphere), the partial pressure of gaseous Na or NaOH at thermodynamic equilibrium is higher than for the case of an atmosphere rich in CO 2 . We therefore propose the following expression for the reaction rate:

v(p CO2 , T, s) = A p CO2 (1 + (s)) exp - E 1 RT , (9)
where s is a geometrical parameter accounting for the grain sizes. The factor 1 + (s) represent the weak dependency of the reaction rate on grain sizes. Since the CO 2 partial pressure can depend on the location inside the sample -under nitrogen flow, one expects p CO2 to be smaller close to the sample surface than at the bottom of the crucible -, the reaction rate of Eq. ( 9) is local and can vary from one place to another. Our SEM observations confirm that the progress of the reaction always seems to be faster close to the surface of the sample. In addition, three supplementary TGA experiments at 750, 800 and 840 • C under N 2 flow, with medium grain sizes M-M and with a larger sample of 1.4 g, show (Fig. 4) that for the two higher temperatures the reaction rate can be fitted with the same activation energy as for the smaller 100mg samples, but that the reaction rate is significantly smaller for the larger samples. This difference can be attributed to the increased difficulty to evacuate CO 2 for larger samples, a mechanism that had already been suggested in [START_REF] Hrma | Batch melting reactions[END_REF]. At 750 • C, the reaction rate is comparable to that of smaller samples, probably because of a combination of the vapor-phase and contact reaction mechanisms.

Under CO 2 flow, the partial pressure of CO 2 is close to 1 everywhere, hence the equilibrium partial pressure of gaseous Na or NaOH is very weak. We deduce that a second reaction mechanism must be at play, that does not depend much on the CO 2 partial pressure. The reaction at intergranular contacts identified in Fig. 3 (a) is probably the second mechanism. It corresponds to reaction (1) that is thermodynamically favored, with an equilibrium constant of 10 3 at 700 • C. However, it is kinetically limited by the contact between the two species, explaining why the reaction rate is much slower. With this in mind, we propose the following expression for the reaction rate:

v(T, s, t) = C(s) √ t exp - E 2 RT . ( 10
)
The value of the geometrical factor C(s) is discussed in the next section. Since v does not depend on the atmosphere in Eq. ( 10), we expect that v is constant within the sample, contrary to the case where p CO2 is low.

V. DISCUSSION

A. Effective kinetics

For the vapor-phase reaction (favored by an N 2 atmosphere), it may appear surprising that the reaction rate is constant up to a large value of the reacted fraction (up to 75 %). This observation implies that the same quantity of Na 2 O and SiO 2 react per unit time. The porous microstructure of the product layer shown in Fig. 3 (d) may be responsible for this behavior if it consists of open porosity, allowing the gaseous phase to always access the surface of unreacted portions of silica grains. The size of the pores observed in Fig. 3 (d) (a few microns) is indeed much larger than the mean free path of gaseous sodium at 1 bar (¡ 100 nm).

In contrast, the reaction kinetics under CO 2 appear to be diffusion-limited. Different diffusion mechanisms can be invoked to explain the √ t scaling for the reaction rate. It is likely that the appearance of the reaction layer creates a barrier through which the sodium has to diffuse, resulting in reaction kinetics that slow down with time. The departure from the square-root dependence observed after 10 hours may correspond to the complete reaction of a fraction of the silica grains.

In our TGA experiments, we have considered extreme cases for the CO 2 partial pressure: p CO2 1 and p CO2 = 1. A different reaction mechanism is favored for each of these two cases. In most glass melting experiments, an intermediate value of the CO 2 partial pressure may lead to a combination of the two reaction mechanisms. We therefore propose that the local effective kinetics are 0.00090 0.00095 0.00100

1/T (K -1 ) 10 -4 10 -3 10 -2 10 -1 v (% NC .s -1 ) MM-N 2 MC-N 2 MS-N 2 SM-N 2 MM-N 2 -1,4g Turner MM-CO 2 MM-CO 2 -24h FIG. 4.
Arrhenian plot of our experimental results, and comparison with Turner's reaction rates. In addition to the same experimental results as in Fig. 1, we have represented the results obtained by Turner [START_REF] Howarth | The rate of reaction between silica and sodium carbonate at different temperatures and the process of vitrification[END_REF] for fine powders (blue triangles). The latter results are well fitted by two Arrhenian curves corresponding to the vapor-phase and the contact reaction mechanisms respectively. In green circles, we have represented the reaction rates obtained for larger samples of 1.4 g.

described by the sum of ( 9) and ( 10):

v(p CO2 , T, s, t) = A p CO2 (1 + (s)) exp - E 1 RT + C(s) √ t exp - E 2 RT . (11)
Because of the different dependence on time and p CO2 , the relative importance of the two mechanisms can vary in space and time. For example, during the initial stages of an experiment, reaction at physical contacts is favored by the large number of initial contacts, explaining why the reaction rate is larger at short times before reaching a constant rate at 750 • C under N 2 atmosphere (Fig. 1). Similarly, in a large crucible in a neutral atmosphere, it is expected that the vapor-transport mechanism will be more efficient close to the surface, where the concentration of CO 2 is low, whereas the contact reaction may be more efficient in the bulk, where the CO 2 partial pressure is higher.

B. Comparison with results from the literature

The identification of two distinct reaction mechanisms sheds new light on results from the literature, in particular the experiments of Turner [START_REF] Howarth | The rate of reaction between silica and sodium carbonate at different temperatures and the process of vitrification[END_REF]. Turner and collaborators performed TGA experiments in air for the same composition as that studied here (25%Na 2 O -75%SiO 2 ), using SiO 2 and Na 2 CO 3 powders sieved between 80 and 100 µm. The reaction rates of [START_REF] Howarth | The rate of reaction between silica and sodium carbonate at different temperatures and the process of vitrification[END_REF], measured during 15 and 60 minutes for different temperatures are reported on Fig. 4. Data at temperatures below 775 activation energy consistent with that of the contact reaction. In contrast, the two data points corresponding to temperatures above 775 • C are fitted with an activation energy consistent with that of the vapor-phase reaction (dashed lines in Fig. 4). The temperature of 775 • could therefore be interpreted as the transition from a domain where the contact mechanism dominates to one where vapor transport dominates. The transition temperature is higher for Turner's experiments than in ours because the former used finer powders, for which the contact reaction is favored. The difference in particle size also explains why the value of reaction rate is higher in Turner's experiments, especially in the contact reaction regime.

The more recent results of Dolan and Misture [START_REF] Michael | Kinetics of glass batch reactions[END_REF] are also in good agreement with our interpretation. [START_REF] Michael | Kinetics of glass batch reactions[END_REF] observed an activation energy of 157kJ.mol -1 , with particles of 100 µm, between 600 and 800 • C, consistent with the roles of small particle size and low temperature that are found here to favor the mechanism involving direct grain contacts. On the other hand, experiments performed under partial vacuum by Pole and Taylor [START_REF] Pole | Kinetics of solid-phase reactions of certain carbonates with mullite, silica and alumina[END_REF] and Nandi and Mukerji [START_REF] Nandi | Reaction of silica with Na2CO3 and with the eutectic and double carbonate of Na2CO3 and CaCO3[END_REF] may be fitted by activation energies of 486 and 444 kJ.mol -1 , in good agreement with the vapor-phase mechanism favored at low total pressure and low partial pressure of CO 2 .

C. Model for the effective reaction surface

When the contact reaction is the prevailing mechanism (Fig. 2), reaction rate is found to depend significantly on grain sizes. The values of the prefactor C(s) in Eq. ( 4) are given in Table I and in this subsection, we propose a simple preliminary model to rationalize values of C.

Denoting the radius of silica and sodium carbonate grains as r SiO2 and r Na2CO3 respectively, and assuming that the reaction rate is proportional to the fraction of silica surface in contact with sodium carbonate, the reaction rate can be estimated from

C(r SiO2 , r Na2CO3 ) ∝ n(r SiO2 )4πr 2
SiO2 Ω(r SiO2 , r Na2CO3 ), [START_REF] Kim | Decomposition of Na2CO3 by interaction with SiO2 in mold flux of steel continuous casting[END_REF] where n(r SiO2 ) is the number of quartz grains per unit volume, n(r SiO2 )4πr 2 SiO2 is the quartz surface per unit volume, and Ω(r SiO2 , r Na2CO3 ) is the surface fraction of a quartz grain that reacts with sodium carbonate. With For the contact reaction, the fraction of reactive surface of a silica grain is modeled as the total solid angle that Na2CO3 grain(s) occupy on the silica surface. The reaction surface is delineated in black on the quartz surface. For a given composition, small Na2CO3 grains (left) are more numerous, but each grain occupies a smaller solid angle dΩ on the quartz surface than for medium grains (right). (b) Reaction rate C vs. total reaction surface given by Eq. ( 13). The red square corresponds to the correction of Eq. [START_REF] Preston | The decomposition of sodium carbonate[END_REF].

n(r SiO2 ) ∝ r -3 SiO2 , we obtain

C(r SiO2 , r Na2CO3 ) ∝ r -1 SiO2 Ω(r SiO2 , r Na2CO3 ). ( 13
)
For a given contact between one grain of SiO 2 and one grain of Na 2 CO 3 , the reaction surface may be modeled by the solid angle occupied by Na 2 CO 3 on the SiO 2 surface (see Fig. 5 (a)). For one SiO 2 grain, Ω(r SiO2 , r Na2CO3 ) is the sum of the solid angles occupied by all its Na 2 CO 3 neighbors: Ω(r SiO2 , r Na2CO3 ) = i dΩ i .

In order to compute Ω, one therefore needs to know (i) the average number Z S -NC of sodium carbonate neighbors of each silica grain, and (ii) the solid angle dΩ occupied by a grain of size r 1 on a grain of size r 2 : Ω(r SiO2 , r Na2CO3 ) = dΩZ S -NC .

(

) 14
For frictionless grains, the average number of touching neighbors Z of a grain is 6. However, large grains will tend to have a larger number of neighbors than small grains. Concerning the solid angle, for a given silica grain the solid angle will be larger as the size of the surrounding carbonate grains increases. Analytical models [START_REF] Danisch | Model of random packings of different size balls[END_REF] have been proposed to describe the geometry of polydisperse granular media. The analytical formula proposed in [START_REF] Danisch | Model of random packings of different size balls[END_REF] are used here (as are detailed in the Appendix).

The resulting values for the number of contacts Z and the surface fraction Ω are given in Table I. Calculations for silica grains of class corresponding to our medium-class, mixed with small and medium Na 2 CO 3 grains shows that the number of contacts is much greater for small grains, but that the solid angle occupied by an individual small grain is also smaller than for a medium grain (see Fig. 5 (a)). For these two experiments (M-M and M-S), the ratio of contact surface areas (Eq. ( 13)) is approximately that of experimentally determined reaction rates between these two mixtures, as illustrated in Fig. 5 (b). For small silica grains (S-M), Z S -NC is less than one, and a fraction of the silica grains does not have a single sodium carbonate neighbor. However, the silica surface per unit volume is more important, and the solid angle dΩ is large. In Fig. 5, it can be seen that the predicted reaction surface per unit volume for the experiment S-M (blue point) is larger than that expected based on the measured rate and the data for the M-M and M-S mixture. One possible explanation for this discrepancy is that the whole solid angle dΩ is not available for the reaction when the quartz grains are smaller than the Na 2 CO 3 grains, since the reaction occurs at the surface of the quartz grains. With this in mind, when the quartz grains are smaller than carbonate grains we propose to replace dΩ by the equivalent solid angle for Na 2 CO 3 grains of the same size as the quartz grains. Making this assumption we find that the experiment S-M verifies relation [START_REF] Nandi | Reaction of silica with Na2CO3 and with the eutectic and double carbonate of Na2CO3 and CaCO3[END_REF] as well, as shown in Fig. 5 (red square). In light of these results, the following model is proposed to calculate the geometric factor C:

C(r SiO2 , r Na2CO3 ) ∝ r -1 SiO2 Zd Ω(r SiO2 , r Na2CO3 ), (15)
d Ω(r SiO2 , r Na2CO3 ) = min (dΩ(r SiO2 , r Na2CO3 ), dΩ(r SiO2 , r SiO2 )) .

While this model provides a promising approach to quan-tification of reaction rates, experiments with other grain sizes are needed to test the geometric model. It would also be interesting to investigate the extent of the reaction surface between silica and molten sodium carbonate (above the melting point of sodium carbonate): on the one hand, molten sodium carbonate may spread on the quartz surface and increase the reaction surface; but on the other hand, capillary forces may lead to molten sodium carbonate wetting preferentially some quartz grains at the expense of other quartz grains initially in contact with solid sodium carbonate.

D. Conclusions

In conclusion, it seems appropriate to quote W.E.S. Turner, who stated in 1933 [START_REF] Howarth | The rate of reaction between silica and sodium carbonate at different temperatures and the process of vitrification[END_REF] "Despite several earlier investigations on the subject of the reaction between sodium carbonate and silica [...], it seemed to us that much remained yet to be learnt about the early stages of the reaction, particularly below the melting point of sodium carbonate". Nevertheless, our results as well as most previous results from the literature can be explained by a combination of the two different reaction mechanisms identified in this article. In summary, when CO 2 can be efficiently removed from a decarbonating mixture, the partial pressure of sodium is large enough so that a vapor-phase reaction dominates. On the other hand, sodium carbonate and silica also react at the contacts between the two species. In this case, the reaction rate is dominated by diffusion through the reacted layer, leading to a reacted fraction proportional to √ t. In most cases, a combination of the two mechanisms will be observed.

Whereas the mechanism involving vapor transport is sensitive to local variations in p CO2 , the other mechanism is found to be sensitive to the local geometry of grains, and the effective contact surface area between grains of different composition. A model for the effective contact surface area has been proposed, that is shown to account well for our experimental results.

We gratefully acknowledge the help of Catheline Cazako, Nathalie Ferruaud, and Samuel Pierre for the experiments, and fruitful discussions with Cyril Condolf, Pierre Jop, Sophie Papin and William Woelffel.

APPENDIX: COMPUTATION OF REACTION SURFACES

For the geometric model of reaction surfaces, we have used the formula derived in Danish et al. [START_REF] Danisch | Model of random packings of different size balls[END_REF] for the geometry of a bidisperse packing of grains. The solid angle Ω ij that a ball of radius R j occupies on a ball of radius R i can be computed from the geometry only as

Ω ij /2π = arcsin R j R i +R j 0 sin θdθ = 1-1 - R j R i + R j 2 .
(17) Noting x the fraction of grains of species 1, the average fraction of occupied surface of a grain of species i is then

Ω occ i = xΩ i1 + (1 -x)Ω i2 . (18)
Noting Z the average coordination number in the packing (Z = 6 for frictionless grains), Z i the average coordination number of grains of species i, and Z ij the average number of neighbors j for a grain i, Danish et al. derive the following relations:

Z 11 = Z 2 1 x Z , Z 12 = Z 1 Z 2 (1 -x) Z , Z 21 = Z 1 Z 2 x Z , Z 22 = Z 2 2 (1 -x) Z ; (19)
Z 1 = Z x + (1 -x) Ω occ 1 Ω occ 2 , Z 2 = Z x Ω occ 2 Ω occ 1 + (1 -x) , (20)
from which contacts statistics of Table 1 are derived.

FIG. 2 .

 2 FIG. 2. Evolution of the reacted fraction X of sodium carbonate vs. time, under CO2 atmosphere. (a) Thermal treatments of 24 h at 800 and 820 • , for the medium grain size. (b) Thermal treatments at 820 • C and for different grain sizes.

FIG. 3 .

 3 FIG. 3. (a) Grains of quartz (transparent) sticking to a central grain of sodium carbonate (opaque white) after a 4-min thermal treatment at 750 • C. A solid-state reaction between the two species is evidenced by the sintering of the grain. The reaction must take place primarily at the contacts between grains, since only quartz grains in contact with sodium carbonate grains sintered. (b) Contact-free experiment. Inset: sketch of the experiment. A crucible filled with quartz grains (yellow) was placed inside a larger crucible filled with sodium carbonate grains (red). Main frame: after 4 hours at 820 • C, the quartz grains have sintered together. (c) SEM micrograph of a quenched SiO2 -Na2CO3 mixture heated for four hours at 820 • C. A porous reaction layer is visible around the quartz grains. (d) Detail of the same micrograph as c).

FIG. 5 .

 5 FIG. 5. (a)Sketch of a granular packing. For the contact reaction, the fraction of reactive surface of a silica grain is modeled as the total solid angle that Na2CO3 grain(s) occupy on the silica surface. The reaction surface is delineated in black on the quartz surface. For a given composition, small Na2CO3 grains (left) are more numerous, but each grain occupies a smaller solid angle dΩ on the quartz surface than for medium grains (right). (b) Reaction rate C vs. total reaction surface given by Eq. (13). The red square corresponds to the correction of Eq. (16).

TABLE I .

 I • C are well fitted by an Arrhenian temperature dependence with an Correlating reaction rates and geometry: for the different CO2 experiments, reaction rate prefactor C, computed number of contacts ZS -NC between silica and sodium carbonate, and fraction of a sand grain available for reaction with sodium carbonate dΩZS -NC.

		M-M M-S S-M
	C	1.21 1.96 0.80
	ZS -NC	2.15 7.26 0.76
	dΩZS -NC 0.29 0.42 0.19