F Démé

E Peuvrel-Disdier

B Vergnes
email: bruno.vergnes@mines-paristech.fr

Loss of crystalline structure and swelling kinetics of maize starch and flour granules in glycerol excess: the role of the envelope structure

Keywords: Maize flour, Starch, Glycerol, Swelling, Amylose, Crystallinity

    Loss of crystalline structure and swelling kinetics of maize starch and flour granules in glycerol excess: The role of the envelope structure Florian Démé, Edith Peuvrel-Disdier, Bruno Vergnes

Introduction

Starch gelatinization has been studied for a long time [START_REF] Olkku | Gelatinization of starch and wheat flour starcha review[END_REF][START_REF] Donovan | Phase transitions of starch water system[END_REF]Ratnayake and Jackson, 2008), essentially in water, but also more recently in other liquids, like glycerol [START_REF] Liu | Phase transitions of maize starches with different amylose contents in glycerol-water systems[END_REF], NMMO [START_REF] Koganti | Solvent effects on starche dissolution and gelatinization[END_REF][START_REF] Koganti | Effect of granule organization on the behavior of starches in the NMMO (N-methyl morpholine N-oxide) solvent system[END_REF] and ionic liquids (Liu and Budtova, 2013;[START_REF] Mateyawa | Effect of the ionic liquid 1-ethyl-3-methylimidazolium acetate on the phase transition of starch: dissolution or gelatinization?[END_REF]. Different methods were used to characterize gelatinization kinetics and mechanisms, like differential scanning calorimetry (DSC), optical and electron microscopy, birefringence, wide angle X-ray diffraction, light scattering, viscosity measurements. DSC is one of the most popular and was largely used to explore the phase transitions in starch/water systems (Wootton and Bamunuarachchi, 1979;[START_REF] Russell | Gelatinisation of starches with different amylose/amylopectin content. A study by differential scanning calorimetry[END_REF][START_REF] Zanoni | Naive model of starch gelatinization kinetics[END_REF][START_REF] Spigno | Gelatinization kinetics of rice starch studied by nonisothermal calorimetric technique: influence of extraction method, water concentration and heating rate[END_REF][START_REF] Sopade | Gelatinisation of starch in mixtures of sugars. II. Application of differential scanning calorimetry[END_REF]. Combinations of rheology and microscopy can also be used [START_REF] Tan | Combined rheological and optical investigation of maize, barley and wheat starch gelatinisation[END_REF].

It is reported that suspension viscosity increases during gelatinization, then reaches a peak and decreases [START_REF] Eliasson | Viscoelastic behavior during the gelatinization of starch. I. Comparison of wheat, maize, potato and waxy-barley starches[END_REF][START_REF] Yang | Complex viscosity-temperature master curve of cornstarch dispersion during gelatinization[END_REF][START_REF] Yu | Study of starch gelatinization in a flow field using simultaneous rheometric data collection and microscopic observations[END_REF]. By studying starches of different botanical origins or with various amylose/amylopectin ratios, it was possible to evidence the strong influence of this parameter [START_REF] Russell | Gelatinisation of starches with different amylose/amylopectin content. A study by differential scanning calorimetry[END_REF][START_REF] Tester | Swelling and gelatinization of cereal starches[END_REF][START_REF] Cooke | Loss of crystalline and molecular order during starch gelatinisation: origin of the enthalpic transition[END_REF][START_REF] Jenkins | Gelatinisation of starch: A combined SAXS/WAXS/DSC and SANS study[END_REF][START_REF] Fredriksson | The influence of amylose and amylopectin characteristics on gelatinization and retrogradation properties of different starches[END_REF][START_REF] Liu | Gelatinization of cornstarch with different amylose/amylopectin content[END_REF][START_REF] Chen | Phase transition of starch granules observed by microscope under shearless and shear conditions[END_REF][START_REF] Blazek | Pasting and swelling properties of wheat flour and starch in relation to amylose content[END_REF][START_REF] Liu | Phase transitions of maize starches with different amylose contents in glycerol-water systems[END_REF][START_REF] Chen | Internal structure and phase transitions of starch granules during gelatinization[END_REF]. It was shown that the amylose content is very influent on granule swelling in water excess. The granule size, the heating rate [START_REF] Patel | Effect of heating rate on starch granule morphology and size[END_REF] and the addition of fatty acids [START_REF] Blazek | Effect of monopalmitin on pasting properties of wheat starches with varying amylose content[END_REF] can also change granule swelling kinetics. Theses fatty acids do not have the same interactions with maize starches of different amylose content [START_REF] Chang | The physicochemical properties of swelled maize starch granules complexed with lauric acid[END_REF]. Some explanations of the differences in swelling kinetics regardless to the amylose content were proposed by Debet andGidley (2006, 2007). They evidenced the role of the granule envelope structure on the swelling behaviour. This structure is due to a network made of proteins and V-type crystallites, resulting from helical arrangements of amylose chains. The recent development of plasticized starch/thermoplastic blends [START_REF] St-Pierre | Processing and characterization of thermoplastic starch/polyethylene blends[END_REF][START_REF] Rodriguez-Gonzalez | High performance LDPE/thermoplastic starch blend: a sustainable alternative to pure polyethylene[END_REF][START_REF] Li | Morphology development and interfacial interactions in polycaprolactone / thermoplastic starch blends[END_REF], based on ternary starch/glycerol/water mixtures, recently led to the study of starch gelatinization in such media [START_REF] Forssell | Phase and glass transition behaviour of concentrated barley starch-glycerol-water mixtures, a model for thermoplastic starch[END_REF][START_REF] Nashed | The plasticization effect of glycerol and water on the gelatinisation of wheat starch[END_REF][START_REF] Tan | Investigation of the starch gelatinization phenomena in water-glycerol systems: application of modulated temperature differential scanning calorimetry[END_REF][START_REF] Li | The relationship between starch gelatinization and morphology control in melt-processed polymer blends with thermoplastic starch[END_REF][START_REF] Liu | Phase transitions of maize starches with different amylose contents in glycerol-water systems[END_REF]. If the gelatinization of starch is well documented, it is less the case for flours. However, from an economic point of view, flour is less expensive than starch to be used in blend with a thermoplastic. Consequently, it is also interesting to look at flours behavior. Therefore, in the present study, we propose to characterize the mechanism of starch and flour gelatinization in glycerol suspensions, and to evaluate the swelling kinetics of starch and flour granules with different amylose/amylopectin contents in pure glycerol.

Materials and methods

Four starchy products, kindly provided by ULICE (Riom, France), were investigated in this study: three maize flours (standard, waxy, high amylose) and a standard maize starch. SEM pictures of these materials are shown in Figure 1 and their main characteristics are given in Table 1. They differed either by the variety and/or the treatment, e.g. extraction of fats and proteins for the starch.

The three varieties of maize flour essentially differed by their respective ratio of amylose and amylopectin: the waxy maize contained less than 1 wt% amylose (dry matter), whereas standard maize contained about 30 wt%, and high amylose maize (or amylomaize) 70 wt%. The initial water content was also different for the various starchy products: it varied from 8.4 wt% (waxy flour) to 12.3 wt% (standard starch). This initial water content was not modified during trials in glycerol excess. Experiments at the same water content may have been of interest, but were not carried out in the present study. Storage in closed bags of the samples in a freezer allowed to avoid changes in water content. It was effectively checked that the storage did not modify the water content. The global gelatinization process in glycerol excess is assumed to be not affected by small variations of the water content. This is of course fully different from plasticization of starch at low glycerol content, where small variations of the water content have an important effect on the plasticization process and the product rheological behavior. The lipid content in flours was globally proportional to the amylose amount. The protein content was almost zero in standard starch whereas it was quite high in flours (around 9 wt%).

Glycerol was chosen as starchy phase plasticizer. Its melting temperature was 17°C, it is boiling at 290°C, but may begin to evaporate around 170°C. It had a density of 1.26 and a viscosity of 1 Pa.s at 25°C. Lauric acid was also used in the study. It is a mono-fatty acid, with a melting temperature around 45°C. It has no double bond and can create complexes with amylose.

Microscopic observations were performed on a Leitz microscope (Metallux 3), equipped with a CCD camera (JVC KY-F75U). Observations in polarized light were made with a hot stage (Linkam TMS 91) on a very dilute suspension of dry flour in glycerol, to estimate the temperature of loss of crystallinity (Maltese cross extinction) and to observe the swelling of the granules with the temperature.

Viscosity measurements under continuous shear of suspensions of granules in glycerol were carried out on a parallel plate rheometer (Stresstech © , Reologica). 40 mm diameter plates and a 1 mm gap were used. The initial volume fraction of flour or starch varied between 6.3 and 6.6 wt% (2 g of starch or flour suspended in 40 mL of glycerol). After a pre-shear at 25 s -1 to break eventual aggregates, the suspension was sheared at 16 s -1 , with a heating rate in the range 1 to 10°C/min. Calorimetric analyzes were performed on a differential scanning calorimeter (Perkin Elmer DSC7).

To avoid loss of material (water evaporation), high pressure sealed capsules were used.

Results and discussion

Gelatinization in glycerol excess

We first analyzed the temperatures of loss of crystallinity in excess of glycerol. The loss of crystallinity can be characterized by the loss of birefringence in optical microscopy and by the appearance of a melting endotherm in DSC experiments. For the loss of birefringence, a suspension of granules (flour or starch) in glycerol was observed in optical microscopy, under polarized light (Fig. 2). For DSC experiments, the mixture contained 40 wt% flour (or starch) and 60 wt% glycerol. Indeed, the quantity of starchy phase had to be significant in order to obtain a signal. Moreover, above 50% glycerol, its amount has almost no influence on the gelatinization temperature [START_REF] Van Soest | Influence of glycerol on melting of potato starch[END_REF][START_REF] Habeych | On the applicability of Flory-Huggins theory to ternary starch-water-solute systems[END_REF]. In both cases (microscopy and DSC), the heating rate was 10°C/min.

Values of temperature of loss of cristallinity, obtained by both microscopy and DSC are indicated in Table 2. Values for the beginning of gelatinization are similar for all products, although high amylose maize flour presents a higher melting temperature. In fact, both techniques do not detect the same phenomena. Temperature range of gelatinization observed by optical microscopy corresponds to the loss of birefringence due to the melting of crystallites. In comparison, melting endotherms measured by DSC are due to both loss of crystallinity and dissociation of double helices formed by amylose/lipid complexes [START_REF] Cooke | Loss of crystalline and molecular order during starch gelatinisation: origin of the enthalpic transition[END_REF]. In addition, melting endotherms of crystallites and amylose/lipid complexes (more stable) can be merged [START_REF] Jenkins | Gelatinisation of starch: A combined SAXS/WAXS/DSC and SANS study[END_REF][START_REF] Matveev | The relationship between thermodynamics and structural properties of low and high amylose maize starches[END_REF][START_REF] Liu | Gelatinization of cornstarch with different amylose/amylopectin content[END_REF], explaining that the melting temperature range seen by DSC is wider than the one seen by optical microscopy. The higher lipid content in high amylose flour explains its broader melting endotherm (35°C) relatively to waxy maize (20°C). In addition, the melting enthalpy increases with amylose content, while the degree of crystallinity is often lower for high amylose starches [START_REF] Tester | Starch composition, fine structure and architecture[END_REF]. In our case, the degree of crystallinity deduced from XRD experiments was 16.5 %, 21.8% and 24.8% for high amylose, standard and waxy flours, respectively. This highlights the role of amylose content on a more persistent structure, mainly due to more stable amylose/lipid complexes.

For each sample, we also observed an exotherm just before the melting endotherm (around 50°C in Fig. 3). This exotherm is linked to the formation of hydrogen bonds between starch and polyols such as glycerol [START_REF] Van Soest | Influence of glycerol on melting of potato starch[END_REF][START_REF] Smits | Interaction between dry starch and plasticisers glycerol or ethylene glycol, measured by differential scanning calorimetry and solid state NMR spectroscopy[END_REF][START_REF] Habeych | On the applicability of Flory-Huggins theory to ternary starch-water-solute systems[END_REF]. This peak is only observed at low water content in water/glycerol/starch mixtures. Indeed, too much water saturates the hydroxyl groups of starch chains, making it impossible to link starch and glycerol [START_REF] Habitante | Phase transitions of cassava starch dispersions prepared with glycerol solutions[END_REF]. As for gelatinization, the transition corresponding to this peak is irreversible: nothing is observed on cooling, or during a second scan (Fig. 3).

The location of these two peaks (endotherm and exotherm) changes when increasing the heating rate from 1°C/min to 5°C/min. Fig. 4 depicts the variation of the peak temperatures with the heating rate. The temperature of the endotherm peak increases slightly with the heating rate (from 113 to 126°C), whereas the exotherm one shows a stronger dependence (from 46 to 85°C). The melting temperature of the crystals should not depend on the heating rate. The small increase of the temperature of the melting endotherm with the heating rate can be interpreted as a little overheating although a 10°C/min heating rate is usually chosen to avoid this phenomenon. As mentioned above, the exotherm is due to another physical phenomenon. Assuming that it is linked to starch/glycerol interactions, the diffusion of glycerol into starch granules must be essential in these interactions and this phenomenon is thermally activated. In addition, some annealing effects on heating could not be excluded. These measurements evidence a different effect of the heating rate on the two physical phenomena.

To validate the physical interpretation for the exothermic peak, the starch suspension was left at 25°C for 7 days. Glycerol had thus time to diffuse into starch granules and form hydrogen bonds.

Accordingly, maturation of the mixture leads to the suppression of this exothermic peak (Fig. 5) and the gelatinization temperature was shifted to a lower value. [START_REF] Smits | Interaction between dry starch and plasticisers glycerol or ethylene glycol, measured by differential scanning calorimetry and solid state NMR spectroscopy[END_REF] similarly reported the disappearance of the exotherm on dehydrated amorphous amylopectin/glycerol mixtures after storage at 20°C for 7 days.

Microscopic observations of starch and flour swelling

The loss of crystallinity is a necessary but not sufficient condition to characterize the destruction of the granular structure. We also measured kinetics of swelling of flour and starch granules, as the swelling behavior depends on the starch structure.

Figure 2 shows the behavior of flour granules suspended in glycerol, with a heating rate of 5°C/min, observed by optical microscopy between crossed polarizers. The Maltese cross reflects the semicrystalline structure of starch granules at low temperature. Granules start to swell just before the loss of crystallinity. The gelatinization takes place over a temperature range of 10 to 20°C, at around 130°C, with a gradual disappearance of the Maltese cross.

In the case of standard and high amylose maize flours, swelling took place in two stages: a first stage where large granules (diameter ≈ 20 μm) swell just after gelatinization, and a second stage, beyond 150°C, where small granules (diameter ≈ 5 μm) started to swell whereas larger ones continued to swell. Waxy maize flour and standard maize starch did not exhibit this behavior.

Above a certain temperature, granules became less apparent: they are usually referred to as "ghosts" (Fig. 2d). Thermal disturbance of the solution above 170°C make the observation of the bursting of these granules unclear. Nevertheless, it occurred around 180°C and they fully disappeared at 200°C.

From these observations, it was possible to plot the evolution of the granule diameter as a function of temperature (Fig. 6). Measurements of the diameters were carried out on a tenth of granules of each type and size range, using an image analysis software. We observe an important increase of diameter above 130-140°C, i.e. after gelatinization.

Rheological measurements during starch and flour swelling

Stability tests over time were carried out at room temperature and at different shear rates to ensure that suspensions do not have a thixotropic behavior. Each suspension was first pre-sheared at 25 s -1 for 60 s to disperse aggregates of granules. It was then subjected to a constant shear rate of 16 s -1 while temperature was increased with time from 25 to 200°C, at a constant heating rate.

The variation of the viscosity of a suspension of standard maize flour in glycerol with temperature is shown in Fig. 7. Whereas glycerol viscosity regularly decreases with temperature increase, the flour suspension shows a different behavior. Initially, the viscosity of the suspension is very close to that of glycerol, due to the low volume fraction of the dispersed phase (6.5 vol %). However, after gelatinization, the swelling, first of the granules and then of "ghosts", results in a sharp increase in the volume fraction and therefore in the viscosity of the suspension. A two-stage process is observed in the rise in viscosity, corresponding to optical microscopy observations. The first viscosity increase should correspond to the swelling of large granules whereas the second step is attributed to the swelling of small granules. The drop in viscosity at the end of the measurement corresponds to the rupture of the "ghosts": the suspension is then transformed into a solution.

As we can see in Fig. 7, the viscosity evolution is affected by the heating rate. The first viscosity increase, corresponding to the swelling of large granules, is clearly influenced by the heating rate.

Swelling at a high heating rate begins at a higher temperature. This is in agreement with the fact that swelling of large granules occurs just after the gelatinization, whose temperature increases as the heating rate is increased (see Fig. 4). Differences of swelling kinetics with heating rate were also reported by [START_REF] Patel | Effect of heating rate on starch granule morphology and size[END_REF] in the case of water. They explained that granule swelling occurred at a higher temperature at higher heating rates, what is coherent with our observations. The rheological behavior of all flour or starch suspensions in glycerol was later measured at the same heating rate (5°C/min). Measurements for the different maize flour suspensions are compared in Fig. 7. This figure shows that the flour amylose content has a huge effect on the swelling properties of the granules. The swelling is delayed to higher temperatures and the swelling intensity (evidenced as the viscosity increase) decreases as the amylose content increases. Granules of high amylose maize flour swell regularly and homogeneously. Numerous studies of starch swelling in water reported similar results on the correlation between swelling properties and amylose content but with a single-step swelling kinetics [START_REF] Tester | Swelling and gelatinization of cereal starches[END_REF][START_REF] Sasaki | Effect of wheat starch structure on swelling power[END_REF][START_REF] Tester | Annealing of maize starch[END_REF][START_REF] Patel | Effect of heating rate on starch granule morphology and size[END_REF][START_REF] Chen | Phase transition of starch granules observed by microscope under shearless and shear conditions[END_REF][START_REF] Blazek | Pasting and swelling properties of wheat flour and starch in relation to amylose content[END_REF].

The reduced swelling of enriched amylose starches was attributed to the presence of amylose/lipid complexes in the granules. [START_REF] Debet | Three classes of starch granule swelling: Influence of surface proteins and lipids[END_REF] noticed that cleaning the surface of standard maize granules allowed a more rapid swelling, but did not change the kinetics for high amylose maize. They deduced that proteins and phospholipids, rather than amylose, could play a role.

However, other studies demonstrated that proteins and phospholipids are not sufficient to preserve the integrity of starch granules without amylose [START_REF] Debet | Three classes of starch granule swelling: Influence of surface proteins and lipids[END_REF]. The most probable explanation is that, during lixiviation, proteins present on the granule surface limit the diffusion of amylose chains. These amylose chains may complex and form a network with the phospholipids on the granule surface. Granule swelling would thus result from a competition between starch/glycerol interactions, leading to swelling, and the strength of the network created at the surface by proteins and amylose/lipid complexes. The fact that small granules present a reduced swelling after gelatinization compared to large ones could be related to the surface/volume ratio, which is more important for small granules.

The behaviors of standard maize flour and starch are depicted in Fig. 9. Contrarily to flour, starch swelling is very fast and occurs in one single step. As it contains very few proteins and lipids, the surface network cannot be built. Consequently, the behavior of flour and starch should be different in terms of transformation and processing, at least in processes with a large amount of plasticizer. In plasticized starch and in plasticized starch/thermoplastic blends, where the amount of glycerol is typically 33-36 wt% [START_REF] St-Pierre | Processing and characterization of thermoplastic starch/polyethylene blends[END_REF][START_REF] Li | Comparison of sorbitol and glycerol as plasticizers for thermoplastic starch in TPS/PLA blends[END_REF], it would be interesting to see if the difference in the envelope strength can still be detected in compounding conditions. We recently showed that the rheological behavior and the morphology of plasticized flour/thermoplastic blends were determined by the amylose/amylopectin ratio, the plasticizer content and the processing conditions, as for plasticized starch/thermoplastic blends (Deme et al., 2014). [START_REF] Becker | Relevance of amylose-lipid complexes to the behaviour of thermally processed starches[END_REF] studied the influence of a thermal treatment with glycerol stearate on various starches. They showed that enriched amylose starches exhibited a reduced swelling in water, contrarily to waxy maize starch which swelled even in cold water. In the present case, we investigated the effect of adding 0.2 g of lauric acid (corresponding to 10 wt% relatively to starch phase) in the standard maize starch suspension. Lauric acid is a fatty acid without insaturation, able to easily complex with amylose [START_REF] Tang | Analyses of complexes between lipids and wheat starch[END_REF]. We can see in Fig. 10 that the addition of lauric acid results in a two-step swelling for the standard maize starch granules, like for flour granules. Swelling is thus delayed in presence of fatty acid, confirming the crucial role of the formation of amylose/lipid complexes at the granule surface. As expected, in the case of waxy maize flour, the effect of lauric acid is insignificant on the granule swelling (represented by the viscosity increase), as shown in Fig. 11.

Measuring swelling kinetics of the granules by optical microscopy (Fig. 6) allows us to establish a relationship between the evolution of the suspension viscosity and the one of the granule volume fraction . The evolution of the volume fraction can be deduced from the evolution of the granule diameter. By introducing this volume fraction into a rheological law of suspension, it is possible to calculate the viscosity evolution with temperature. This calculation of the suspension viscosity from the knowledge of the granule size was performed although experimental conditions in which rheological and optical measurements were conducted were very different. In optical microscoscopy, the granule swelling was followed at rest and in a very dilute suspension. In rheometry, the granule swelling was followed under shear in a more concentrated medium (6.5 vol%). Anyway, we tried to evaluate the viscosity evolution by using a Krieger-Dougherty equation [START_REF] Krieger | A mechanism for non-Newtonian flow in suspensions of rigid spheres[END_REF]

: 0 () ( ) ( ) 1 a max T TT         (1)
where  0 (T) is the viscosity of the glycerol,  max is the maximum packing volume fraction and a is an exponent, function of the maximum packing fraction and of the intrinsic viscosity of the solid filler.  max and a values depend on type, shape and size polydispersity of the particles. For example,  max = 0.74 and  max = 0.64 are classical values for a suspension of rigid monodispersed spheres in face centered cubic and random class packing, respectively. a is often equal ( or close) to 2 (see for example [START_REF] Quemada | Modélisation Rhéologique Structurelle[END_REF]. Fig. 12 shows that, for the standard maize flour, the model provides a correct approximation of the viscosity change, despite a slight overestimation at low temperatures.

This prediction was obtained for  max = 0.52 and a = 3.

Similar calculations were performed for the other varieties of granules. The values of  max and a determined for the standard maize flour suspensions were used for the viscosity estimation of the other suspensions. Fig. 13 shows the comparison between experimental measurements and the evolution of the viscosity using the Krieger-Dougherty law. In each case, for the different varieties of flours and starch, the evolution of the solid volume fraction with temperature was calculated from the experimental measurements of the granule diameter kinetics. We can see that there is quite a good agreement between experimental measurements during the swelling step and the Krieger-Dougherty prediction. This confirms that the viscosity evolution is essentially related to the granule swelling and corresponding change in volume fraction.

Conclusion.

We investigated the loss of crystalline structure and swelling kinetics of various maize flours and starch in excess of glycerol. Loss of crystallinity was investigated by DSC and optical microscopy.

Swelling kinetics was followed by optical microscopy and via viscosity measurements.

The present work allowed us to highlight differences in temperature of loss of crystallinity for different varieties of flours in excess of glycerol. It appears that an increased amylose content makes the flours more difficult to gelatinize. This is more clearly seen in DSC measurements than by observations of loss of birefringence, because a significant portion of the starch structure comes not only from amylopectin crystallites but also from helical arrangements linked to amylose/lipid complexes. A different effect of the heating rate on the ordered structure and on interactions between glycerol and starch was evidenced.

We observed the same differences in swelling, because the swelling starts after gelatinization occurs. Granule swelling in water depends on the amylose content, but occurs in one step. In excess of glycerol, a two-step process is observed for flours containing amylose, which confirms that the ghost integrity is kept by the envelope strength.

Swelling mechanisms in glycerol are the same as those observed in water, but the envelope structure is more persistent. The strength of this envelope also comes from the formation of networks including proteins, lipids and amylose. This explains the greater difficulty to deconstruct granules of standard maize flour than standard maize starch, which contains much less fat and protein than the flour. When processed at high level of plasticizer, flour (containing amylose)

should be thus more difficult to destructure than starch. Tests in presence of lauric acid highlighted the role of lipids on the envelope strength.

These results can explain the existence of persistent "ghosts" in glycerol/flour mixtures, even at high temperature, without intensive shear or addition of water.

Figure captions

Fig. 1 .Fig. 2 .

 12 Fig. 1. SEM micrographies of the pristine granules. (a) standard maize flour, (b) waxy maize flour,

Fig. 7 .Fig. 8 .

 78 Fig. 7. Effect of heating rate on the viscosity evolution with temperature of a standard maize flour

Fig. 10 .

 10 Fig. 10. Influence of lauric acid on the viscosity evolution with temperature (10°C/min) of standard

Fig. 12 .

 12 Fig. 12. Viscosity evolution with temperature (10°C/min) of standard maize flour suspension in glycerol. Comparison between experiment () and model ( __ ).

Fig. 13 .

 13 Fig. 13. Viscosity evolution with temperature (10°C/min) of flour and starch suspensions in glycerol. Comparison between experiment (symbols) and model (lines). , __ : waxy maize

  Fig. 1. Demé et al.

Fig. 9 .

 9 Fig. 8. Demé et al.

Fig. 10 .

 10 Fig. 10. Demé et al.

  Fig. 11. Demé et al.

Fig. 12 .

 12 Fig. 12. Demé et al.

Fig. 13 .

 13 Fig. 13. Demé et al.

Table 1 .

 1 , __ : waxy maize flour, , __ : high amylose maize flour, , ----: standard maize starch. Starchy products density and composition (in weight % of dry matter).

		Standard	Standard	Waxy	High amylose
		maize starch	maize flour	maize flour	maize flour
	Raw density (g/L)	1470	1407	1449	1420
	Amylose (wt%)	30	30	1	70
	Water (wt%)	12.27	8.76	8.39	9.44
	Proteins (wt%)	0.4	8.33	9.16	9.8
	Lipids (wt%)	0.92	2.9	1.66	5.49
	Ashes at 900°C (wt%)	0.06	0.9	0.53	1.29
		Data provided by ULICE

Table 2 .

 2 Temperatures of loss of birefringence (microscopy) and gelatinization (DSC) of starch and flour granules in glycerol excess.

Fig. 3. Demé et al.

Acknowledgements

This work was carried out within the project CEREMAT, supported by Céréales Vallée cluster. We gratefully thank Fond Unique Interministériel (France) for financial support. The authors wish to thank Professor Jean-Marc Haudin (CEMEF, MINES ParisTech) for fruitful discussions on the

interpretation of the different effects of the heating rate on the shift of the exotherm and endotherm temperatures.


G Normand

E Peuvrel-Disdier

B Vergnes
email: bruno.vergnes@mines-paristech.fr

Matrix degradation during high speed extrusion of polypropylene/clay nanocomposites -influence on filler dispersion

Keywords: Twin screw extrusion, Nanocomposite, High speed, Degradation, Dispersion

We prepared polypropylene/organoclay nanocomposites by melt blending in a twin-screw extruder, exploring the domain of high screw speeds (up to 1100 rpm). The samples were characterized at both microscale (size of agglomerates) and nanoscale (level of exfoliation).

We show that, despite a satisfactory exfoliation, the polymer matrix suffered important thermomechanical degradation by chain scission. We propose a way to correct this degradation on the viscosity curves and we confirm that high screw speeds are not necessarily favorable to clay exfoliation, essentially because of the too high melt temperatures encountered during the process.

Introduction

Polymer/clay nanocomposites were first developed in the Toyota Research Laboratories by [START_REF] Usuki | Synthesis of nylon 6-clay hybrid[END_REF]. Since this pioneering work, it has been shown that mixing a small amount of clay (<5 wt%) with a polymer matrix could improve many properties such as mechanical [START_REF] Cho | Nylon 6 nanocomposites by melt compounding[END_REF][START_REF] Gloaguen | Plastic deformation behaviour of thermoplastic/clay nanocomposites[END_REF][START_REF] Hong | Preparation and mechanical properties of polypropylene/clay nanocomposites for automotive parts application[END_REF]Hedge et al., 2013), barrier [START_REF] Messersmith | Synthesis and barrier properties of poly(εcaprolactone)-layered silicate nanocomposites[END_REF][START_REF] Yano | Synthesis and properties of polyimide-clay hybrid films[END_REF] and flame retardancy properties [START_REF] Doh | Synthesis and properties of polystyrene-organoammonium montmorillonite hybrid[END_REF][START_REF] Gilman | Flammability properties of polymer-layered-silicate nanocomposites. Polypropylene and polystyrene nanocomposites[END_REF].

The interest of organoclay compared to other nanofillers is their important specific area.

Indeed, clay presents itself as aggregates, composed of tactoids which are made of several tenths of individual layers stacked together. The difficulty during processing consists in separating those different layers in order to reach the full reinforcing potential of the organoclay. However, the clay is naturally hydrophilic and thus difficult to disperse in a nonpolar matrix. Therefore, the clay usually undergoes a chemical treatment, allowing to increase the interlamellar distance, and to make it organophilic. This chemical modification is often enough to allow a good dispersion in polar matrices such as polyurethane [START_REF] Wang | Nanolayer reinforcement of elastomeric polyurethane[END_REF] and epoxy [START_REF] Massam | Clay nanolayer reinforcement of a glassy epoxy polymer[END_REF].

However, in non-polar matrices, a compatibilizer must be used to increase the chemical affinity between clay and matrix. For polypropylene (PP), the most common compatibilizer is polypropylene grafted with maleic anhydride (PP-g-MA), which has proven to allow a good dispersion of the clay with an exfoliated structure [START_REF] Kawasumi | Preparation and mechanical properties of polypropylene-clay hybrids[END_REF][START_REF] Lertwimolnun | Influence of compatibilizer and processing conditions on the dispersion of nanoclay in a polypropylene matrix[END_REF]. PP/organoclay nanocomposites appear as interesting materials because PP is affordable, presents good mechanical properties and an easy processability. In order to prepare such nanocomposites, the most interesting option is melt blending, using a twin-screw extruder. This method presents several advantages, as it is a continuous process, with high levels of mechanical energy, and flexibility: screw profile and processing conditions (screw speed, feed rate, barrel temperature) can be easily modified to optimize the clay dispersion state.

In order to characterize the dispersion state, microscopic observations and X-ray diffraction are commonly used on polymer/clay nanocomposites. Scanning electron microscopy allows the observation of the clay aggregates at microscale. X-ray diffraction helps to characterize the interlayer distance, and thus the intercalation state. Melt rheology in small amplitude oscillatory shear is also often used to characterize such materials, because it is very sensitive to the clay exfoliation level, and hence gives a good idea of the dispersion achieved at nanoscale during melt processing.

The influence of processing conditions such as screw speed [START_REF] Incarnato | Preparation and characterization of new melt compounded copolyamide nanocomposite[END_REF][START_REF] Lertwimolnun | Effect of processing conditions on the formation of polypropylene/organoclay nanocomposites in a twin screw extruder[END_REF][START_REF] Modesti | Thermal behaviour of compatibilised polypropylene nanocomposite: effect of processing conditions[END_REF][START_REF] Lin | Nylon 66/clay nanocomposite structure development in a twin screw extruder[END_REF][START_REF] Domenech | Influence of twin-screw processing conditions on structure and properties of polypropylene-organoclay nanocomposites[END_REF], feed rate [START_REF] Fasulo | Extrusion processing of TPO nanocomposites[END_REF][START_REF] Scatteia | Processing, rheology and structure of melt compounded PBT-clay nanocomposites having different chemical composition[END_REF][START_REF] Lertwimolnun | Influence of screw profile and extrusion conditions on the microstructure of polypropylene/organoclay nanocomposites[END_REF] and barrel temperature [START_REF] Modesti | Effect of processing conditions on morphology and mechanical properties of compatibilized polypropylene nanocomposites[END_REF][START_REF] Shah | Organoclay degradation in melt processed polyethylene nanocomposites[END_REF][START_REF] Domenech | Influence of twin-screw processing conditions on structure and properties of polypropylene-organoclay nanocomposites[END_REF] on the clay dispersion state has been widely studied in the literature. It has been shown that a low feed rate is beneficial for the dispersion, because a long residence time helps to achieve a good level of exfoliation [START_REF] Domenech | Influence of twin-screw processing conditions on structure and properties of polypropylene-organoclay nanocomposites[END_REF]. However, feed rate seems to have a limited impact on intercalation [START_REF] Lertwimolnun | Effect of processing conditions on the formation of polypropylene/organoclay nanocomposites in a twin screw extruder[END_REF]. It is also generally admitted that a low temperature is better for dispersion because it increases the level of shear stresses during melt processing [START_REF] Modesti | Effect of processing conditions on morphology and mechanical properties of compatibilized polypropylene nanocomposites[END_REF][START_REF] Domenech | Influence of twin-screw processing conditions on structure and properties of polypropylene-organoclay nanocomposites[END_REF]. It also prevents the degradation of the components, as both polypropylene, PP-g-MA and organoclay can be degraded if the temperatures reached during melt processing are too high [START_REF] Barbas | Evolution of dispersion along the extruder during the manufacture of polymer-organoclay nanocomposites[END_REF]. In some cases, the degradation of the clay chemical modifier can cause polymer chain scission, as with polystyrene [START_REF] Nassar | Melt intercalation in montmorillonite/polystyrene nanocomposites[END_REF], polyethylene [START_REF] Shah | Organoclay degradation in melt processed polyethylene nanocomposites[END_REF] and polyamide [START_REF] Fornes | Polymer matrix degradation and color formation in melt processed nylon 6/clay nanocomposites[END_REF] matrices, but it has not been reported yet with polypropylene.

It has also been shown that increasing screw speed helps to achieve a good level of exfoliation, because increasing shear stress helps the separation of the individual layers [START_REF] Cho | Nylon 6 nanocomposites by melt compounding[END_REF][START_REF] Lertwimolnun | Effect of processing conditions on the formation of polypropylene/organoclay nanocomposites in a twin screw extruder[END_REF]. However, [START_REF] Peltola | Effect of rotational speed of twin screw extruder on the microstructure and rheological and mechanical properties of nanoclay-reinforced polypropylene[END_REF] obtained good exfoliation levels regardless of the screw speed used, reporting limited impact of this parameter. Only a few studies have shown the impact of very high screw speeds (up to 1200 rpm). [START_REF] Fel | Residence time distribution in a high shear twin screw extruder[END_REF] reported these effects on the residence time distribution. [START_REF] Louizi | Contribution of high-shear processing to the compatibilization of (PP/EPR)/PE ternary blends[END_REF] noticed an improvement of the morphology and properties of nanocomposites made of PP/EPR and silica around 800 rpm, and a degradation of the matrix above. The main purpose of this paper is thus to explore the potentiality of modern extruders, and specifically of high screw speeds, for the preparation of PP/organoclay nanocomposites.

Materials and Methods

Materials

The polypropylene used (Moplen HP400R, LyondellBasell) is an injection grade isotactic homopolymer, with a MFI of 25 (2.16 kg/10 min) and a melting temperature of 164°C. The PP-g-MA (Epolene G-3015, Eastman) has a MFI of 40 (2.16 kg/10min) and a melting temperature of 162°C. The organoclay (OMMT) is a Cloisite 20 (C20), provided by BYK Additives (formerly Southern Clay Products). It is a Na+-montmorillonite chemically modified by dimethyl di(hydrogenated tallowalkyl) quaternary ammonium. It has a density of 1.77 g.cm -3 , and a cation exchange capacity of 95 meq/100g.

Samples preparation

Before compounding, OMMT and PP-g-MA were dried for 15 hours at 80°C under vacuum.

The PP/PP-g-MA/C20 composites were prepared with respective weight fractions of 85/10/5, using a laboratory scale co-rotating twin-screw extruder (ThermoFisher Rheomex PTW24). It had a diameter of 24 mm and a L/D ratio of 40. The extruder was fed with two gravimetric feeders: the first one (Brabender DDW-MD3-DDSR20-10) allowed to introduce the polymer pellets (PP + PP-g-MA) in barrel 1, and a second one (Brabender DDW-MD1-MT-1) allowed the introduction of the clay in barrel 3 (Fig. 1). The barrel temperature was set à 180°C. Five screw speeds were chosen: 100, 300, 600, 900 and 1100 rpm. The feed rates were set at 2.47 kg/h for the matrix (PP + PP-g-MA) and 0.13 kg/h for the clay, hence a total feed rate of 2.6 kg/h. The clay amount was systematically checked for each sample by weighing solid residuals after complete matrix calcination in an oven at 600°C. Extrudate temperatures were measured at the die exit for each extrusion condition using a manual thermocouple.

The extrudate was cooled down in water at die exit before being granulated. For the various characterizations, the nanocomposite pellets were melted and compressed in a hydraulic press (Carver M 3853-0) at a pressure of 20 MPa, during 8 minutes at 200°C. The samples were then disk-shaped, with a 25 mm diameter and a 1.5 mm thickness.

Microstructure characterization

In order to characterize the microstructure, the polished surface of the compressed samples was observed by scanning electron microscopy (SEM) with a Philips XL30 ESEM, under an acceleration voltage of 15 kV. The size and repartition of the clay agglomerates were characterized using an image analysis software (Visilog 6). It enables the calculation of the clay area ratio A r (expressed in %):

100 clay r total A A A = × (1)
where A clay is the area of the clay agglomerates and A total the total area analyzed. This parameter characterizes the clay dispersion state at the microscale. Due to the microscope resolution, agglomerates with an equivalent diameter under 5 μm were not considered. The total area analyzed was 25 mm 2 .

The samples were also observed via transmission electron microscopy (TEM), performed on the CCMA EM Core Facility (University of Nice Sophia Antipolis, France). We used a

Jeol JEM 1400, equipped with an Olympus MORADA SIS camera. Before observation, the disk shaped samples were cryofractured. Then, ultra-thin samples were prepared via a cryoultramicrotome (Leica FC6) at a temperature of -100°C. Local observations were then performed on the samples under an acceleration voltage of 100 kV.

The intercalation state was characterized via X-ray diffraction measurements. The diffractometer used was a PANalytical X'pert Pro MRD, with the Cu Kα radiation and a wavelength of λ = 1,540456 Å. A 2θ range between 1.5 and 10° was analyzed. The interlamellar distance d 001 of the clay was calculated using the Bragg's law:

001 2 sin n d λ θ = (2)
where n is the order of the reflexion and λ the radiation wavelength.

Small amplitude oscillatory shear rheology measurements were performed using a stresscontrolled Anton Paar MCR-302 rheometer. Frequency sweep measurements were carried out in the molten state at a temperature of 180°C, a strain of 1% (i.e. in the linear regime) and a gap of 1.25 mm under a nitrogen atmosphere. Before the measurements, the samples were stabilized at 180°C for 30 minutes. A Carreau-Yasuda model with yield stress was used to describe the complex viscosity η*(ω) dependence with the angular frequency ω [START_REF] Lertwimolnun | Effect of processing conditions on the formation of polypropylene/organoclay nanocomposites in a twin screw extruder[END_REF][START_REF] Vergnes | The use of apparent yield stress to characterize exfoliation in polymer nanocomposites[END_REF]:

1 0 0 ( ) 1 ( ) n a a σ η ω η λω ω -   = + +   (3)
where σ 0 is the apparent melt yield stress, η 0 is the zero-shear viscosity, λ is a characteristic relaxation time, a is the Yasuda parameter and n is the shear-thinning index.

The yield stress σ 0 characterizes the elasticity of the network formed by the clay lamellae.

It was shown that, at fixed clay loading, this parameter can be correlated to the exfoliation of the clay, and thus to the efficiency of the dispersion at nanoscale.

Results and Discussion

Microscopic observations

To characterize the clay dispersion state at microscale, the samples were observed via scanning electron microscopy (SEM). Fig. 2 shows the pictures obtained for the samples prepared at different screw speeds. As the screw speed increases, the number and size of clay agglomerates decreases. This means that the clay aggregates are sheared more severely when the screw speed increases, which explains the decreasing size. The number of clay aggregates decreases when the screw speed increases from 100 to 600 rpm, and afterwards the effect of screw speed is not so pronounced, as the visual aspect seems to remain the same.

Fig. 3 shows a quantitative analysis of the SEM pictures, with the evolution of the area ratio with screw speed. At the lowest screw speed (N = 100 rpm), the area ratio is rather high (0.6%), and decreases to a value below 0.1% at N = 600 rpm. Then, as the screw speed increases again, the area ratio shows a slight increase. It therefore appears that, at this scale, the optimal screw speed is around 600 rpm, and that higher screw speeds do not improve the state of microscopic dispersion.

TEM observations were also performed on the samples, and the pictures are presented in Fig. 4. At each screw speed, the clay lamellae have been exfoliated, at least partially, and we can observe single clay layers as well as aggregated ones. A few "large" tactoids could be observed for the screw speeds from 100 to 600 rpm, with a length between 500 nm and 1 μm.

It also appears that the number of layers per tactoids decreases between 100 and 900 rpm. At higher screw speeds, these tactoids seem smaller, and only a few could be observed, which suggests a better dispersion at nanoscale.

Intercalation state

The samples were characterized by X-ray diffraction. Fig. 5 shows the diffractograms of the five nanocomposites and pure clay for comparison. For each screw speed, the diffraction peak corresponding to the [001] plane of the clay is shifted to the smaller angles. This indicates that the interlamellar distance was increased during extrusion, through intercalation of the clay galleries by polymer chains. The corresponding interlamellar distances are presented on Fig.

6

. The temperatures of the melt at the die exit, measured during extrusion, are also indicated on the graph. For each screw speed, the calculated distance is comprised between 3.6 and 3.9 nm, which indicates an intercalation compared to pure clay (d 001 = 3.28 nm). However, the interlamellar distance decreases as the screw speed increases, which can indicate a degradation of the intercalation state at high screw speed.

A decrease of interlamellar distance with screw speed has been already observed by [START_REF] Domenech | Influence of twin-screw processing conditions on structure and properties of polypropylene-organoclay nanocomposites[END_REF], who obtained basal distances inferior to the one of pure clay for nanocomposites prepared at the highest screw speeds (between 500 and 900 rpm). They attributed this decline to the thermal degradation of the surfactant, due to the high temperatures caused by viscous dissipation. Similar results were reported by [START_REF] Dintcheva | Thermo-mechanical degradation of LDPE-based nanocomposites[END_REF] on polyethylene/OMMT nanocomposites. [START_REF] Lertwimolnun | Effect of processing conditions on the formation of polypropylene/organoclay nanocomposites in a twin screw extruder[END_REF] observed no effects of the screw speed on intercalation, but the screw speeds were rather low, comprised between 100 and 300 rpm. [START_REF] Barbas | Processing conditions effect on dispersion evolution in a twin-screw extruder: polypropylene-clay nanocomposites[END_REF] observed a slight increase of the interlamellar distance when the screw speed increased from 50 to 100 rpm, and no evolution at higher screw speeds. The assumption of thermal degradation of the surfactant at high screw speeds is realistic in our case, as the temperatures reached at 900 and 1100 rpm were close to 230°C, as indicated in Fig. 6. It has been reported in the literature that a significant part of the organomodifier is lost between 190 and 240°C [START_REF] Shah | Organoclay degradation in melt processed polyethylene nanocomposites[END_REF][START_REF] Cervantes-Uc | Thermal degradation of commercially available organoclays studied by TGA-FTIR[END_REF][START_REF] Cui | Effect of organoclay purity and degradation on nanocomposite performance, Part 1: surfactant degradation[END_REF].

Rheological measurements

Frequency sweep measurements have been performed on the nanocomposites and the matrix in the molten state (Fig. 7). The evolution of the complex viscosity shows the presence of an apparent yield stress at low frequency for the five nanocomposites at the contrary of the matrix. This indicates the presence of a percolated network formed by the clay platelets [START_REF] Cassagnau | Melt rheology of organoclay and fumed silica nanocomposites[END_REF][START_REF] Domenech | Formation of fractal-like structure in organoclay-based polypropylene nanocomposites[END_REF]. It confirms a good level of exfoliation whatever the screw speed used.

If the low frequency measurements are sensitive to clay exfoliation, the high frequency ones rather characterize the matrix behavior. However, the curves are crossing at high frequency, the viscosity increasing up to 300 rpm and decreasing above. For the sample extruded at 1100 rpm, the viscosity is quite similar to the one of the matrix. To clearly interpret these results, we repeated the extrusion tests with the matrix alone, under the same conditions as for the nanocomposites: five screw speeds N (from 100 to 1100 rpm), at a flow rate Q of 2.47 kg/h and a barrel temperature of 180°C. We then performed rheological characterization of the samples obtained, in the same conditions as for the nanocomposites.

Fig. 8 shows the evolution of the complex viscosity of the matrix with angular frequency for the different screw speeds. It is clear on this graph that the viscosity of the matrix decreases as the rotational speed increases, reflecting a thermomechanical degradation of the polymer. Indeed, it is well known that, under severe thermal and mechanical conditions, PP suffers chain scissions, reducing both molecular weight and viscosity [START_REF] Hinsken | Degradation of polyolefins during melt processing[END_REF]Gonzalez-Gonzalez et al., 1998;[START_REF] Canevarolo | Chain scission distribution function for polypropylene degradation during multiple extrusions[END_REF]. This explains the cross-over of the viscosity curves at high frequency.

It would be interesting to correct the complex viscosity curves of the nanocomposites in order to better analyze the data and correctly assess the rheological behavior. In the same way that one can perform a time/temperature superposition, we propose to use a shift factor a d to obtain a master curve by plotting η*(ω)/a d as a function of ω a d . Fig. 9 shows the mastercurve obtained, considering the curve at 100 rpm as a reference (a d = 1). It appears that the curves are very well superimposed, validating the suggested approach. Fig. 10 shows the change in shift factor with the screw speed. At the highest screw speed, it drops to a value of 0.5, meaning that the viscosity was divided by two in these processing conditions. If we consider only the PP and if we assume that the Newtonian viscosity changes with the molecular weight with a power 3.6 [START_REF] Leonardi | Détermination de la distribution des masses molaires d'homopolymères linéaires par spectrométrie mécanique[END_REF], we can estimate the order of magnitude of the molecular weight of the degraded products. Starting from 123 kg/mol for the virgin PP, we reach 103 kg/mol at 1100 rpm. These values are in agreement with the order of magnitude of molecular weight evolutions measured by size exclusion chromatography for thermomechanically degraded PP (Gonzalez-Gonzalez et al., 1998;[START_REF] Canevarolo | Chain scission distribution function for polypropylene degradation during multiple extrusions[END_REF].

If we assume that the degradation of the matrix is not modified by the presence of the clay, we can now apply these shift factors to the viscosity curves of the nanocomposites to correct the degradation of the matrix. In fact, we can also apply directly a time/degradation superposition to the viscosity curves of the nanocomposites, by assuming that the values at high frequency are perfectly superimposed. We then obtain values of the shift factors very close to the previous ones, what justifies the assumption that the degradation of the matrix is not strongly modified by the presence of the clay. Fig. 11 shows the "corrected" curves. They are now ranked as expected. It appears that the curve corresponding to a screw speed of 300 rpm shows the highest increase of viscosity at low frequency. Curves corresponding to screw speeds of 600, 900 and 1100 rpm are almost superimposed, slightly below the one at 300 rpm.

Discussion

The Carreau-Yasuda model with a yield stress was applied to the corrected complex viscosity curves. The yield stress increases rapidly from 240 to 510 Pa when the screw speed increases from 100 rpm to 300 rpm (Fig. 12). Above that speed, the yield stress stabilizes at a slightly lower value, around 400 Pa. This implies that beyond 300 rpm, the clay exfoliation is not improved. A decrease of the exfoliation at high screw speed was also observed by [START_REF] Barbas | Processing conditions effect on dispersion evolution in a twin-screw extruder: polypropylene-clay nanocomposites[END_REF] on similar nanocomposites. The authors attributed it to the decreasing matrix viscosity, which may have the effect of reducing the shear stresses and prevent optimal exfoliation. This hypothesis is plausible in our case, as a large decrease in matrix viscosity was observed at high screw speed, due to thermomechanical degradation. The decrease of clay intercalation at high screw speed (Fig. 6) could also limit the exfoliation and explain this evolution.

The area ratio and the melt yield stress allow to characterize the clay dispersion state at both microscale and nanoscale. Several authors [START_REF] Lertwimolnun | Influence of screw profile and extrusion conditions on the microstructure of polypropylene/organoclay nanocomposites[END_REF][START_REF] Médéric | Structural and rheological properties as a function of mixing energy for polymer/layered silicate nanocomposites[END_REF][START_REF] Domenech | Influence of twin-screw processing conditions on structure and properties of polypropylene-organoclay nanocomposites[END_REF][START_REF] Domenech | The importance of specific mechanical energy during twin screw extrusion of organoclay based polypropylene nanocomposites[END_REF] have proposed to correlate the efficiency of the dispersion at these two scales with the specific mechanical energy (SME). To do that, we have first to evaluate the energy provided to the nanocomposite after clay introduction.

Experimentally, we can only measure the global SME provided by the extruder and we cannot extract the part devoted to the nanocomposite after the clay introduction. For this reason, we can only calculate the SME provided to the nanocomposite by numerical modelling. We used for that the Ludovic © software, dedicated to the calculation of flow parameters in a twinscrew extruder and already largely used for understanding the compounding of nanocomposites (Lertwimolnun andVergnes, 2006, 2007;[START_REF] Bahloul | Reactive extrusion processing of polypropylene/TiO 2 nanocomposites by in situ synthesis of the nanofillers: Experiments and modeling[END_REF][START_REF] Domenech | Influence of twin-screw processing conditions on structure and properties of polypropylene-organoclay nanocomposites[END_REF][START_REF] Domenech | The importance of specific mechanical energy during twin screw extrusion of organoclay based polypropylene nanocomposites[END_REF]. Ludovic® is a 1D modelling software for twin screw extruders, which makes it possible to calculate the main flow parameters (temperature, pressure, shear rate, viscosity, residence time, specific energy, filling ratio, etc.) along the screws. A local one-dimensional approach is adopted. The screws are divided into sub-elements, each element corresponding to the volume occupied by the material around one screw. Computation of the various parameters is done separately for each type of screw element (partially or totally filled right-handed screw elements, left-handed screw elements and blocks of staggered kneading disks) and for the die components. The data for the simulation can be split into three main parts:

first, a description of the extrusion configuration, such as screw and barrel profiles and die geometry, is necessary. Second, the rheological and thermal properties of the polymer have to be defined. Third, processing conditions must be indicated (feed rate, screw speed, barrel temperatures, heat transfer coefficients).

Fig. 13 shows the evolution of the area ratio and the melt yield stress with the calculated SME. It appears that there is an energy threshold, between 400 and 600 kWh/t, above which both the exfoliation and the dispersion of clay aggregates is not improved. Domenech et al.

(2012) also observed, for similar nanocomposites, an energy threshold around 600 kWh/t, beyond which the melt yield stress and the area ratio did not change. Mederic et al. (2009) detected a critical threshold of 140 kWh/t, but for PA12/clay samples prepared in an internal mixer. It seems therefore that this threshold value may depend on several parameters, including the affinity between the filler and the matrix.

The extrudate temperatures are also indicated on Fig. 13. Several authors have reported that degradation of clay surfactant or the matrix can occur if the temperatures reached inside the extruder are too high [START_REF] Cui | Effect of organoclay purity and degradation on nanocomposite performance, Part 1: surfactant degradation[END_REF][START_REF] Domenech | Influence of twin-screw processing conditions on structure and properties of polypropylene-organoclay nanocomposites[END_REF][START_REF] Barbas | Evolution of dispersion along the extruder during the manufacture of polymer-organoclay nanocomposites[END_REF][START_REF] Gelfer | Relationships between structure and rheology in model nanocomposites of ethylene-vinyl-based copolymers and organoclays[END_REF]. We observe in Fig. 13 that the optimal level of nanometric dispersion is achieved between 200°C and 217°C. When the temperature reaches and exceeds 220°C, the yield stress stabilizes around 400 Pa and no longer increases. Above 400 kWh/t, higher energy levels, up to 1100 kWh/t, appear to result in a similar exfoliation. It seems thus that, beside SME, the temperature reached by the nanocomposite is a key element for the clay exfoliation. It is important to provide enough energy to enable nanoscale dispersion, but by simultaneously trying to keep the lowest possible temperature. However, it is difficult to decouple the effects of temperature and SME, as these two parameters are directly connected. Another potential hypothesis explaining the level-off of the yield stress is that a too high energy could induce a breakage of the clay sheets. We can see on the TEM observations (Fig. 4) that the size of the clay sheets seems to be lower for higher screw speeds, which could limit the interactions between exfoliated platelets, and thus result in a lower yield stress for an equivalent level of exfoliation.

Conclusion

This study has evaluated the impact of high speed extrusion on the dispersion state of an organoclay in a polypropylene matrix. It was first shown that all screw speeds, from 100 to 1100 rpm, lead to an exfoliated structure. The formation of a percolated network was confirmed by rheological measurements. However, it was demonstrated that high screw speeds, above 600 rpm, appear to damage the polymer matrix. The approach we proposed allowed us to analyze the complex viscosity curves by correcting the degradation of the matrix. We were then able to confirm that high screw speeds (above 600 rpm) do not result in a better exfoliation.

One of the reasons for this limitation can be the thermal degradation of the chemical modifier of the clay, which results in a collapse of the clay galleries, and in turn prevents an optimal exfoliation state. Another reason can be the mechanical breakage of the clay sheets, resulting in a lower melt yield stress for an equivalent exfoliation level. Finally, the degradation of the matrix can also lower the level of shear stress inside the extruder during melt processing, which in turn reduces the level of exfoliation.

Figure captions

Figure 1 :

 1 Figure 1: Screw profile.

Figure 2 :

 2 Figure 2: SEM images of the nanocomposites prepared at different screw speeds: a) 100 rpm,

Figure 3 :

 3 Figure 3: Evolution of the area ratio with the screw speed (T barrel = 180°C, Q = 2.6 kg/h).

Figure 4 :

 4 Figure 4: TEM images of the nanocomposites prepared at different screw speeds: a) 100 rpm,

Figure 5 :

 5 Figure 5: X-ray diffractograms of the nanocomposites prepared at different screw speeds.

Figure 6 :

 6 Figure 6: Interlamellar distance of the nanocomposites prepared at different screw speeds.

Figure 7 :

 7 Figure 7: Complex viscosity η* of the nanocomposites PP/PP-g-MA/C20 and the matrix as

Figure 8 :

 8 Figure 8: Evolution of the complex viscosity of the matrix with the screw speed (T = 180°C,

Figure 9 :

 9 Figure 9: Reduced complex viscosity as function of reduced frequency for the samples

Figure 11 :

 11 Figure 11: Shifted nanocomposites complex viscosity curves.

Figure 12 :

 12 Figure 12: Evolution of the melt yield stress with the screw speed (T barrel = 180°C, Q = 2.6

Figure 13 :

 13 Figure 13: Evolution of a) the area ratio and b) the melt yield stress with the SME provided to

Figure 1 :Figure 3 :Figure 4 :Figure 6 :

 1346 Figure 1: Screw profile

Figure 7 :Figure 8 :Figure 9 :

 789 Figure 7: Complex viscosity η* of the nanocomposites PP/PP-g-MA/C20 and the matrix as

Figure 11 :

 11 Figure10: Shift factor as function of screw speed.

Fig. 13 :

 13 Figure 12: Evolution of the melt yield stress with the screw speed (T barrel = 180°C, Q = 2.6

Acknowledgements

This study was carried out through the European project EVOLUTION, in the frame of the 7 th Framework Program (FP7/2007-2013), under grant agreement 314744. We are grateful to the EU for financial support. The authors greatly acknowledge the CCMA (Centre Commun de

Microscopie Appliquée, University of Nice Sophia-Antipolis, Microscopy and Imaging platform Côte d'Azur, MICA) and its personal, more particularly Sophie Pagnotta for the sample preparations and TEM observations.


Vivien Despax
email: vivien.despax@lmpt.univ-tours.fr

A common limit in large rank for Markov chains dened from representations of classical Lie algebras

From the datum of an integer partition and a classical Lie algebra, one can dene a Markov chain on an associated multiplicative graph. For each classical family A, C, B, D, we thus obtain a sequence of Markov chain which is indexed by the rank of the considered algebra. In this article we show that, for each type, the transition kernel of the Markov chain has a limit when the rank tends to innity. Moreover, the limit kernel does not depend on the considered type.

Introduction

In what follows, δ is a nonzero partition. For all r suciently large, δ is regarded as a dominant weight of g r = gl r , sp 2r , so 2r+1 or so 2r . For xed r, it is then possible to dene a graph G (δ,gr) reecting the multiplicative structure that arises when one decomposes the successive tensor powers V gr (δ) , V gr (δ) ⊗2 , V gr (δ) ⊗3 , . . . into irreducible components. Introducing then a stochastic matrix Π (δ,gr) on G (δ,gr) , we now have a Markov chain M (δ,gr) . The aim of our article is to study what happens when r tends to innity.

The Markov chain M (δ,gr) originally appears in [START_REF] O'connell | Conditioned random walks and the RSK correspondence[END_REF][START_REF] Lecouvey | Random walks in Weyl chambers and crystals[END_REF][START_REF] Lecouvey | Conditioned random walks from Kac-Moody root systems[END_REF] as a random walk on the weight lattice of g r conditioned to never exit the cone of dominant weights. In these articles, it is shown that the transition matrix of such conditioned process can be expressed in terms of specializations of the corresponding normalized Weyl characters. The determination of the harmonic functions on the graph G (δ,gr) is a closely connected subject which is studied in [LT]. It is then natural to ask what happens when r tends to innity.

Roughly speaking, we prove in this article that the four classical families A = (gl r ) r , C = (sp 2r ) r , B = (so 2r+1 ) r , D = (so 2r ) r give rise to the same limit Markov chain. Our result intersects with a similar phenomenon observed by Nikitin and Vershik in [NV] for some graded graphs and their Pascalized versions. Nevertheless, the graphs that we consider in this article cannot be seen in general as Pascalized versions of simpler graded graphs.

To give sense to this statement, we start by observing that the sequence of multiplicative graphs G (δ,gr) r admits a natural limit. This is a consequence of the stabilization of the tensor multiplicities for large rank. Next, we prove that the sequence of stochastic matrices Π (δ,gr) r is convergent. To accomplish this, we establish the result in type A by showing that the specializations of the normalized Schur functions used to dene Π (δ,gl r ) converge when r tends to innity. This mainly follows from the Weyl character formula. By using the notion of Kashiwara-Nakashima tableaux, we show the convergence to the same limit in type C, B, D. This stabilization in large rank and the existence of a common limit when this rank tends to innity illustrates a general phenomenon in the combinatorics of classical root systems. For example each tensor product multiplicity c λ δ,κ (g r ) appearing in the decomposition

V gr (δ) ⊗ V gr (κ) = λ V gr (λ) c λ δ,κ (gr)
stabilizes in large rank and the common limit when r tends to innity is the Littlewood-Richardson coecients c λ δ,κ (see for example [L]). The article is organized as follows. Section 2 introduces the required material and notation on representation theory. Section 3 describes the Markov chain M (δ,gr) on nite rank multiplicative graph. Section 4 is concerned with the limit Markov chains. We rst dene the limit graph ( 4.1) and then study the asymptotic behaviour of Π (δ,gr) r ( 4.2). We start with the type A case ( 4.2.1) and pursue with types C, B, D ( 4.2.2). Our main theorem is then stated in 4.2.3. Finally in Section 5 we link our results to the generalized Pitman transform dened in [BBO] and to the study of the extremal harmonic functions on multiplicative graphs achieved in [LT].

Background on representation theory and combinatorics

In this article, we only consider complex Lie algebras and their nite-dimensional representations. Let X = (g r ) r be one of the four classical families A = (gl r ) r , C = (sp 2r ) r , B = (so 2r+1 ) r or D = (so 2r ) r . Let us x a positive integer r for the whole section. We introduce some objects dened from g r . Nevertheless, we do not include this parameter in our notation for the moment.

Root system and weight lattice

The Cartan subalgebra of g r consisting of the diagonal matrices is denoted by h.

There exists an Euclidean space (E, . , . ) endowed with an orthonormal basis (ε 1 , ε 2 , . . . , ε r ) such that the root system R of g r (with respect to h) is identied to the following nite subset of E:

R =          {± (ε i -ε j ) : 1 ≤ i < j ≤ r} if g r = gl r {±2ε i : 1 ≤ i ≤ r} ∪ {± (ε i ± ε j ) : 1 ≤ i < j ≤ r} if g r = sp 2r {±ε i : 1 ≤ i ≤ r} ∪ {± (ε i ± ε j ) : 1 ≤ i < j ≤ r} if g r = so 2r+1 {± (ε i ± ε j ) : 1 ≤ i < j ≤ r} if g r = so 2r .
Denote by F the subspace of E spanned by R. The Weyl group W of R is regarded here as the subgroup of O (F ) generated by the orthogonal reections with xed points the {α} ⊥ , α ∈ R. Let I be {1, 2, . . . , r -1} if g r = gl r and {1, 2, . . . , r} in the other cases. We introduce the simple system S = (α i ) i∈I such that

α i = ε i -ε i+1 if 1 ≤ i < r and α r =      2ε r if g r = sp 2r ε r if g r = so 2r+1 ε r-1 + ε r if g r = so 2r which yields the partition R = R + (-R + ) where R + =          {ε i -ε j : 1 ≤ i < j ≤ r} if g r = gl r {2ε i : 1 ≤ i ≤ r} ∪ {ε i ± ε j : 1 ≤ i < j ≤ r} if g r = sp 2r {ε i : 1 ≤ i ≤ r} ∪ {ε i ± ε j : 1 ≤ i < j ≤ r} if g r = so 2r+1 {ε i ± ε j : 1 ≤ i < j ≤ r} if g r = so 2r .
The family of fundamental weights Ω = (ω i ) i∈I associated to S is such that

       ω i = i j=1 ε j , i ∈ I if g r = gl r , sp 2r ω i = i j=1 ε j , 1 ≤ i < r, ω r = 1 2 r j=1 ε j if g r = so 2r+1 ω i = i j=1 ε j , 1 ≤ i < r -1, ω r-1 = 1 2 r-1 j=1 ε j -1 2 ε r , ω r = 1 2 r j=1 ε j if g r = so 2r . If we set ω ∨ i = 2 α i ,α i ω i , then the family Ω ∨ = (ω ∨ i ) i∈I is such that ω ∨ i , α j = δ i j i, j ∈ I.
Notice that ω ∨ i = ω i when X = A. Introduce the weight lattice of g r

L =      r i=1 Zε i if g r = gl r r i=1 Zω i = r i=1 Zε i if g r = sp 2r r i=1 Zω i = r i=1 Zε i + Z 1 2 r i=1 ε i if g r = so 2r+1 , so 2r and set P = r i=1 Z + ε i if g r = gl r L if g r = sp 2r , so 2r+1 , so 2r .
The weights of the (polynomial if g r = gl r ) representations of g r are identied with the elements of P : if V is a representation of g r , then it admits a weight space decomposition

V = ω∈P V ω = ω∈P (V )
V ω where

V ω = {v ∈ V : hv = ω (h) v for any h in h} and

P (V ) = {ω ∈ P : V ω = {0}} .

Irreducible representations and characters

Introduce the set of dominant weights of g r

P + =          { r i=1 λ i ε i ∈ P : λ 1 ≥ λ 2 ≥ . . . ≥ λ r } if g r = gl r r i=1 Z + ω i = { r i=1 λ i ε i ∈ P : λ 1 ≥ λ 2 ≥ . . . ≥ λ r ≥ 0} if g r = sp 2r r i=1 Z + ω i = { r i=1 λ i ε i ∈ P : λ 1 ≥ λ 2 ≥ . . . ≥ λ r ≥ 0} if g r = so 2r+1 r i=1 Z + ω i = { r i=1 λ i ε i ∈ P : λ 1 ≥ λ 2 ≥ . . . ≥ λ r-1 ≥ |λ r |} if g r = so 2r
.

The set P + parametrizes the irreducible (polynomial if g r = gl r ) representations of g r and every representation V of g r decomposes into irreducible components appearing with multiplicities

V = λ∈P + V (λ) ⊕m V λ
where V (λ) is the irreducible representation of g r corresponding to λ. In particular, for any δ in P + , the tensor powers V (δ) ⊗n (n > 0) and the tensor products

V (λ) ⊗ V (δ) (λ ∈ P + ) admit such decomposition: V (δ) ⊗n = µ∈P + V (µ) ⊕fn µ V (λ) ⊗ V (δ) = µ∈P + V (µ) ⊕m λ µ .
Given a list of variables (x i ) 1≤i≤r and ω in L, we set

x ω = r i=1 x ω,ε i i . For λ in P + , the Weyl character of the irreducible representation V (λ) is s λ (x) = ω∈P K λ ω x ω = ω∈P (λ) K λ ω x ω where K λ ω = dim V (λ) ω and P (λ) = P (V (λ)).
Recall the Weyl character formula:

s λ (x) = a λ+ρ (x) a ρ (x)
where

a ω (x) = σ∈W (σ) x σω ω ∈ L
being the sign character of W and ρ = i∈I ω i . The dominator a ρ (x) can be expressed as

a ρ (x) = x ρ α∈R + 1 -x -α .

Partitions

Denition 2.3.1 Any weakly decreasing sequence of nonnegative integers with a nite number of nonzero terms is called a partition.

Each partition is identied to its Young diagram. Given a partition λ, we denote by (λ) the number of its parts, that is the number of its nonzero terms. We denote by |λ| the sum of its terms, that is its number of boxes. The set of partitions with at most parts is denoted by P . The sequence (P ) ∈N is increasing and the set of all partitions is P ∞ = ∈N P . If λ = (λ 1 , λ 2 , . . . , λ , 0, 0, . . .) is in P with ≤ r, then we identify λ with the element λ 1 ε 1 + λ 2 ε 2 + . . . + λ ε of P + . According to this, one has then P r = P + if g r = gl r , sp 2r and P r ⊂ P + if g r = so 2r+1 , so 2r .

If δ and λ are in P r , it can be shown that we have in fact

V (δ) ⊗n = µ∈Pr V (µ) ⊕fn µ V (λ) ⊗ V (δ) = µ∈Pr V (µ) ⊕m λ µ .
(1)

We can be even more precise by showing that we can restrict the above direct sums to µ in P r such that |µ| ≤ n |δ| and |µ| ≤ |λ| + |δ|, respectively. Given λ in P r , the representation V (λ) appears with a positive multiplicity in the decomposition of V ( ) ⊗|λ| into irreducible. One deduces in particular

dim V (λ) ≤ (dim V ( )) |λ| =          r |λ| if g r = gl r (2r) |λ| if g r = sp 2r (2r + 1) |λ| if g r = so 2r+1 (2r) |λ| if g r = so 2r .
One also observes a stabilization and independence phenomenon of the tensor multiplicities in large rank. We record it now for later use in the following proposition.

Proposition 2.3.2 Let δ, λ be in P r . Let n be a positive integer.

1. If δ, n are such that r ≥ n (δ), then one has

f (δ,gr) n µ = f (δ,g r+1 ) n µ µ ∈ P r .
2. If δ, λ are such that r ≥ (λ) + (δ), then one has

m (δ,gr) λ µ = m (δ,g r+1 ) λ µ µ ∈ P r . If δ, λ, µ are such that r ≥ (λ) + (δ) and |µ| = |λ| + |δ|, then one has m (δ,gr) λ µ = m (δ,gl r ) λ µ .

Brief review on the Kashiwara-Nakashima tableaux

Let λ be in P r and set = (λ): λ = (λ 1 , λ 2 , . . . , λ , 0, 0, . . .). To the irreducible g rmodule V (λ) is associated its crystal graph B(λ), a combinatorial object encoding many informations on V (λ). Here, we only mention the features and properties that are useful for our purposes and refer the reader to the articles [K], [KN] and the book [HK] for a complete exposition.

The graph B (λ) is nite and connected, its number of vertices being equal to the dimension of V (λ). It is also a colored and oriented graph, the arrows being labelled by the simple roots S = (α i ) i∈I . There exists a weight graduation wt : B (λ) → P such wt (B (λ)) = P (λ) and

v α i -→ v =⇒ wt (v) -wt (v ) = α i v, v ∈ B (λ) , i ∈ I.
There is a unique source vertex v 0 and one has wt (v 0 ) = λ.

When g r = gl r , we have P + = P r and the crystal graph B (λ) admits a realization in terms of semistandard Young tableaux of shape λ over the alphabet A = {1 < . . . < r}, that is the llings of λ with letters from the totally ordered set A such that the rows are weakly increasing as we move to the right and such that the columns are strictly increasing as we go down.

When g r = sp 2r , so 2r+1 or so 2r , there exists a similar parametrization by analogous objects, the so-called Kashiwara-Nakashima g r -tableaux. A Kashiwara-Nakashima g r -tableau with shape λ is a lling of λ by letters of the ordered alphabet

A =      1 < 2 < . . . < r < r < . . . < 2 < 1 if g r = sp 2r 1 < 2 < . . . < r < 0 < r < . . . < 2 < 1 if g r = so 2r+1 1 < 2 < . . . < r -1 < r, r < r -1 < . . . < 2 < 1 if g r = so 2r
according to combinatorial rules depending on g r . The set of Kashiwara-Nakashima g r -tableaux with shape λ contains then all the ordinary semistandard tableaux with shape λ along with some generalizations that we do not detail here. One can show that the weight of a Kashiwara-Nakashima g r -tableau T of shape λ is then the element wt (T ) of P such that the coordinate on ε i is the number of occurrences of the letter i minus the number of occurrences of the letter ī. The source vertex v 0 is identied to

T 0 = 1 . . . . . . . . . 1 (λ 1 boxes) 2 . . . . . . 2 (λ 2 boxes) . . . . . .

(λ boxes)

.

Given two adjacent vertices

T α i -→ T in B (λ),
they only dier by one letter, the letter in T being replaced in T by its immediate successor in A. The Weyl character corresponding to λ can be computed with these tableaux:

s λ (x) = T ∈B(λ)
x wt(T ) .

(2)

3 Markov chains on the nite rank multiplicative graphs

In this subsection we x a nonzero partition δ. We now introduce the multiplicative graph on which the Markov chains we consider are dened.

Finite rank multiplicative graphs

Let r be an integer such that r ≥ (δ). Dene

V (δ,gr) = n>0 V (δ,gr)
n where

V (δ,gr) n = (µ, n) ∈ P r × N * : f (δ,gr) n µ > 0 n > 0.
Given (µ, n + 1) in P r × N * , one has

(µ, n + 1) ∈ V (δ,gr) n+1 ⇐⇒ ∃λ ∈ P r (λ, n) ∈ V (δ,gr) n m (δ,gr) λ µ > 0.
Let us introduce a set of weighted arrows E (δ,gr) on V (δ,gr) : for (λ, n) in V (δ,gr) n and (µ, n + 1) in V (δ,gr) n+1 , we set

(λ, n) m (δ,gr ) λ µ ----→ (µ, n + 1)
if and only if the multiplicity m (δ,gr) λ µ is positive.

Denition 3.1.1 The graph G (δ,gr) = V (δ,gr) , E (δ,gr) is called the multiplicative graph for (δ, g r ).

Example This condition is not sucient since , 2 is not a vertex of the multiplicative graph for ( , so 7 ). Beside this, there are adjacent vertices with the same number of boxes and some weights are greater than 1: in the multiplicative graph for ( , so 7 ), one has the weighted arrow

): , 1 , 2    , 2    ↓ ↓ , 3 , 3    , 3    3. Let (λ, n) be in P r × N * . If
, 2 2 - → , 3 .
In particular this last multiplicative graph is not the Pascalized version of a braching graph of type A.

Markov chains on the nite rank multiplicative graphs

Let r be an integer such that r ≥ (δ). In this paragraph, we introduce the normalized Weyl characters and then use them to dene a stochastic matrix on the adjacent vertices of G (δ,gr) . Let λ be in P r . We have

x -λ s gr λ (x) = ω∈P gr (λ) K gr λ ω x -(λ-ω) .
If ω is in P gr (λ), then there exists T in B gr (λ) such that wt (T ) = ω and there is a path in B gr (λ) starting at T 0 and ending at T . Then λ -ω = wt (T 0 ) -wt (T ) must be a linear combination of the simple roots S = (α i ) i∈I with nonnegative integer coecients: more precisely, the coecient of α i in wt (T 0 ) -wt (T ) is equal to the number of browsed arrows α i -→ in any path from T 0 to T . Using the family

Ω ∨ = (ω ∨ i ) i∈I , one can also write λ -ω = i∈I λ -ω, ω ∨ i α i .
Dene a family of variables (y i ) i∈I by setting

y i = x -α i . If we introduce y [ω] = i∈I y ω,ω ∨ i i
for ω in L, then we can write ω∈P gr (λ)

K gr λ ω x -(λ-ω) = ω∈P gr (λ)
K gr λ ω y [λ-ω] .

Thus S gr λ (y) = ω∈P gr (λ) K gr λ ω y [λ-ω] is a polynomial in (y i ) i∈I with nonnegative integer coecients. Given a nite sequence of positive reals θ = (θ i ) i∈I and ω in L, the real number obtained by letting y i = θ i in y [ω] (respectively S gr λ (y)), is denoted by θ [ω] (respectively S gr λ (θ)). When b is a positive real, we write S gr λ (b, b, . . . , b)

for the real number obtained by replacing each y i by b in S gr λ (y). In other words:

S gr λ (b, b, . . . , b) = ω∈P gr (λ) K gr λ ω b λ-ω,ρ ∨ (3)
where ρ ∨ = i∈I ω ∨ i . For any nite sequence of positive reals θ = (θ i ) i∈I , we dene a matrix Π (δ,gr) θ on G (δ,gr) : for any (λ, n) , (µ, n + 1) in P r × N * such that

(λ, n) m (δ,gr ) λ µ ----→ (µ, n + 1) , set Π (δ,gr) θ ((λ, n) , (µ, n + 1)) = m (δ,gr) λ µ S gr µ (θ) S gr λ (θ) S gr δ (θ)
θ [λ+δ-µ] .

It is a stochastic matrix because (1) implies

s gr λ (x) s gr δ (x) = µ∈Pr m (δ,gr) λ µ s gr µ (x) λ ∈ P r .
For every xed parameter θ, we thus obtain a Markov chain M (δ,gr) θ associated to the pair G (δ,gr) , Π (δ,gr) θ .

Markov chains on the innite rank multiplicative graphs

Let us x once for all a nonzero partition δ. Let X = (g r ) r be A, C, B or D.

A common stochastic matrix for the innite rank multiplicative graphs

Let θ = (θ i ) i≥1 be a sequence of positive reals and set

θ [r] = (θ i ) i∈I = (θ 1 , θ 2 , . . . , θ r-1 ) if X = A (θ 1 , θ 2 , . . . , θ r ) if X = C, B, D
for any positive integer r.

Our aim is now, under a suitable hypothesis on θ, to establish the pointwise convergence of the sequence of matrices Π (δ,gr)

θ [r] r
on the set P ∞ × N * and also identify the limit.

We start by showing that the sequence S gr λ θ [r] r is convergent for any λ in P ∞ . Our method consists in showing this result for X = A and then use it in the other cases.

Case

X = A Let λ be in P ∞ and set = (λ): λ = (λ 1 , λ 2 , . . . , λ , 0, 0, . . .). Assume X = A.
Our rst ingredient is the following proposition. Our proof is directly inspired by the proof of the dimension formula given in [FH].

Proposition 4.2.1 Let b be a positive real. One has

S gl r λ (b, b, . . . , b) = 1≤i<j≤ 1 -b λ i -λ j +j-i 1 -b j-i 1≤i≤ <j≤r 1 -b λ i +j-i 1 -b j-i r > .
Proof. In this proof, we omit to include the parameter gl r in our notation. Let r be an integer such that r > . Let z be a variable. For any

ω 0 in L = r i=1 Zε i , intro- duce now the morphism φ ω 0 : C [L] → C [[z]] dened by φ ω 0 (x ω ) = exp ( ω, ω 0 z) for any ω in L.
Firstly, we have

S λ (b, b, . . . , b) = ω∈P (λ) K λ ω exp ( λ -ω, ρ ln b) since ρ ∨ = ρ when X = A. Observe that S λ (b, b, . . . , b) is obtained by letting z = -ln b in φ ρ x -λ s λ (x) .
Secondly, the Weyl character formula gives

x -λ s λ (x) = x -λ a λ+ρ (x) a ρ (x) .
For any ω 0 , ω in L, observe that we have

φ ω 0 (a ω (x)) = σ∈W (σ) φ ω 0 (x σω ) = σ∈W (σ) exp ( σω, ω 0 z) = σ∈W σ -1 exp ω, σ -1 ω 0 z = φ ω (a ω 0 (x)) .
Then

φ ρ x -λ a λ+ρ (x) = φ ρ x -λ φ λ+ρ (a ρ (x)) = exp ( -λ, ρ z) φ λ+ρ   x ρ α∈R + 1 -x -α   = exp ( ρ, ρ z) α∈R + (1 -exp (-α, λ + ρ z))
and

φ ρ (a ρ (x)) = exp ( ρ, ρ z) α∈R + (1 -exp (-α, ρ z)) .
This shows

S λ (b, b, . . . , b) = α∈R + 1 -b λ+ρ,α 1 -b ρ,α .
As X = A, we have

R + = {ε i -ε j : 1 ≤ i < j ≤ r}
and ρ = r i=1 (r -i) ε i . Reminding that λ j = 0 for any integer j such that j > , the result follows.

Lemma 4.2.2 For any real t in (0, 1) and for any positive integers a, the innite product n>0

1-t a+n 1-t n is convergent. Proof. We have indeed ln 1-t a+n 1-t n ∼ n→∞ (1 -t a ) t n .
Theorem 4.2.3 Assume θ is a bounded sequence of positive reals such that sup θ < 1. The sequence S gl r λ θ [r] r is convergent. The limit is denoted by S A λ (θ) in the sequel.

Proof. Any semistandard tableau on the alphabet {1 < 2 < . . . < r} is also a semistandard tableau on the alphabet {1 < 2 < . . . < r < r + 1}, with same weight. Then Identity (2) implies that the sequence S gl r λ θ [r] r is weakly increasing.

Recall that, for any positive integer r such that r ≥ , the polynomial S gl r λ (y) has nonnegative integer coecients. We have thus Proposition 4.2.4 We have

0 ≤ S gr λ θ [r] -S gl r λ θ [r] ≤      b r-+1 (2r) |λ| if g r = sp 2r b r-(2r + 1) |λ| if g r = so 2r+1 b r-+2 (2r) |λ| if g r = so 2r r ≥ .
Proof. Suppose X = C, the arguments are similar in the other cases. Let r be such that r ≥ . We have

S gr λ θ [r] = T ∈B gr (λ) θ [λ-wt(T )] [r]
.

Since B gr (λ) contains B gl r (λ) and since the simple roots α 1 , α 2 , . . . , α r-1 are the same for g r and gl r , the left inequality is clear. Now let T be in B gr (λ) \ B gl r (λ): there is at least one letter of T which is in r, r -1, . . . , 1 . A path in B gr (λ) starting at T 0 and ending at T must browse arrows α -→, . . . , αr -→, each of them at least one time, in order to change one letter into an element of r, r -1, . . . , 1 . So, looking at the weight graduation, we have to substract at least one time each of the simple roots α , . . . , α r in order to get wt (T ) from wt (T 0 ). Since b < 1, one can write then

S gr λ θ [r] -S gl r λ θ [r] = r i= θ i T ∈B gr (λ)\B gl r (λ) θ [λ-wt(T)-r i= α i] [r] ≤ b r-+1 T ∈B gr (λ)\B gl r (λ) 1 ≤ b r-+1 dim V gr (λ) ≤ b r-+1 (2r) |λ| .
Corollary 4.2.5 We have

S gr λ θ [r] ---→ r→∞ S A λ (θ) . Proof. As b = sup θ < 1, we have b r r |λ| ---→ r→∞ 0
and the result is then a straightforward consequence of the previous proposition. ---→ r→∞ 0.

Main result

Proof. Suppose X = B, the arguments are similar in the other cases. Let us x a positive integer such that λ + δ -µ is in i=1 Zε i . As the family

(ε 1 -ε 2 , . . . , ε -1 -ε , ε
) is a basis of i=1 Zε i , there exists integers a 1 , . . . , a such that

λ + δ -µ = -1 i=1 a i (ε i -ε i+1 ) + a ε .
Now observe that a = |λ| + |δ| -|µ|. Recall that ε i -ε i+1 = α i for any integer i such that 1 ≤ i < and ε = r i= α i for any integer r such that r ≥ . We deduce

θ [λ+δ-µ] [r] = -1 i=1 θ a i i r i= θ i a r ≥ . If a = |λ| + |δ| -|µ| = 0, then θ [λ+δ-µ] [r] = -1 i=1 θ a i i r ≥ . If a = |λ| + |δ| -|µ| > 0, then the assumption sup θ < 1 implies θ [λ+δ-µ] [r] ≤ r i= θ i r ≥ and r i= θ i ---→ r→∞ 0.
Now, we state our main result for which X can be A, C, B or D.

Theorem 4.2.7 For any (λ, n) , (µ, n + 1) in P ∞ × N * such that 5 Further results

(λ, n) m (δ,X) λ µ ----→ (µ, n + 1) , one has Π (δ,gr) θ [r] ((λ, n) , (µ, n + 1)) ---→ r→∞ m (δ,A) λ µ S A µ (θ) S A λ (θ)S A δ (θ) θ [λ+δ-µ] if |µ| = |λ| + |δ| 0 if |µ| < |λ| + |δ|

Connection with the generalized Pitman transform

It was shown in [START_REF] Lecouvey | Conditioned random walks from Kac-Moody root systems[END_REF] that the Markov chains dened in 4 coincide with some random Littelmann paths conditioned to stay in Weyl chambers and can also be obtained from non conditioned such random paths by applying the generalized Pitman transform introduced in [BBO]. For gl r , this generalized Pitman transform can also be described from the Schensted insertion algorithm on semistandard tableaux. Similar algorithms on Kashiwara-Nakashima tableaux also exist for the other classical types in nite rank based on Kashiwara crystal basis theory.

The generalized Pitman transform essentially associates to each concatenation of Littelmann paths for the representation V (δ) considered its corresponding highest weight path which entirely lies in the Weyl chamber. This construction cannot be directly generalized in innite rank because the relevant representations to consider are not then of highest weight. Nevertheless the crystal basis construction can be adapted together with the insertion algorithms on tableaux (see [L]). It is interesting to observe that the same kind of stabilization phenomenons then appear in large rank: the generalized Pitman transforms admit a limit in large rank and the four limits so obtained in each classical types are essentially the same.

Extremal harmonic functions on the limit multiplicative graphs

The extremal harmonic functions on the multiplicative graphs G (δ,gr) with g r of type X have been characterized in [LT] in terms of the functions S λ (θ [r] ) introduced in 3.2. It follows from our main Theorem 4.2.7 that the extremal harmonic functions on the limit multiplicative graphs G (δ,X) only depend on the partition δ considered and not on the type X considered. It thus suces to consider the extremal harmonic functions on G (δ,A) which can be expressed in terms of Schur functions by rening the work of Kerov and Vershik on the Young lattice (see [LT]).

  b = sup θ. Since b < 1, Proposition 4.2.1 and Lemma 4.2.2 show that the sequence S gl r λ (b, b . . . , b) r is bounded and so is S gl r 2 Case X = C, B, or D Let λ be in P ∞ and set = (λ). Assume X = C, B, or D. Assume θ is bounded and sup θ < 1. Set b = sup θ.

  Proof. Proposition 2.3.2 and Corollary 4.2.5 show the convergence of the sequences m |µ| = |λ| + |δ|, Proposition 2.3.2 also mentions that we have m

  The multiplicative graph for ( , sp 2r ) is the Pascalized graph of the previous one: starting from the multiplicative graph for ( , gl r ), we complete the level n + 1 by reecting the level n -1 with respect to the level n, with the corresponding arrows. Each arrow has weight 1. Then (λ, n) in P r × N is a vertex of this graph if and only if |λ| ≤ n and |λ| = n mod 2. Below are represented some levels of the multiplicative graph for( , sp 4

3.1.2 1. The multiplicative graph for ( , gl r ) is the Young lattice of partitions with at most r parts: (λ, n) in P r × N * is a vertex of this graph if and only if |λ| = n. The second component of a vertex is then useless. Each arrow has weight 1. Below are represented some levels of the multiplicative graph for ( , gl 2 ): ↓ 2.

  it is a vertex of the multiplicative graph for ( , so 2r+1 ), then it must satisfy |λ| ≤ | | × n = 2n and |λ| = 0 mod 2.

Innite rank multiplicative graphs

For xed (µ, n) in P ∞ × N * , the rst point of Proposition 2.3.2 shows that the sequence of integers f (δ,gr) n µ r eventually becomes constant. Denote by f (δ,X)

is positive exactly means that, for all r suciently large, (µ, n) is an element of V (δ,gr) n . This naturally leads us to dene a limit set of vertices by setting

For xed λ, µ in P ∞ , the second point of Proposition 2.3.2 shows that the sequence of integers m (δ,gr) λ µ r eventually becomes constant. Denote by m (δ,X) λ µ this stabilized multiplicity. Let us introduce a set of weighted arrows E (δ,X) on V (δ,X) : for

if and only if the stabilized multiplicity m (δ,X) X) is called the multiplicative graph for (δ, X).