Françoise Vernier
email: francoise.vernier@cemagref.fr

Paul Bordenave
email: paul.bordenave@cemagref.fr

Marie Chavent
email: chavent@sm.u-bordeaux2.fr

Odile Leccia

Kevin Petit

Modelling scenarios of agriculture changes on freshwater uses and water quality at a large watershed scale -the case of the Charente watershed (France)

Keywords: AGRICULTURE, SCENARIOS, WATER RESOURCES, SWAT MODEL, INDICATORS, PESTICIDES. 1. Vernier

The Charente River Basin (10 000 km2) is linked to the Pertuis Charentais Sea by a large tidal influence. The agriculture impacts strongly the quantity and the quality of freshwaters and influences the salinity and the quality of coastal waters needed by the oyster culture. In the context of 2 projects dealing with ICZM (EU Spicosa project, Respireau), we focused on the environmental assessment of spatialised scenarios of changes in agriculture. Different methods of clustering and data analysis were used to define a relevant typology of spatial units on the area. Combined with an analysis of agricultural practices based on surveys (irrigation, fertilizers and pesticides), the agricultural activities can be described in a simplified way, to be linked with hydrological models regarding the water quantity and quality. Scenarios are built using surveys, expertise and discussion with stakeholders. Measures for agriculture changes are either considered in the agro-environmental programs or defined by stakeholders. Their effectiveness on the reduction of nutrients and pesticides fluxes is assessed when applied on the whole area or within sub-basins, depending on the farming system and vulnerability. Assessment was carried out using the spatially distributed model SWAT. The effectiveness of each scenario is considered in terms of relative reduction of water uptake for irrigation, nitrates, pesticides, suspended matter compared to the current situation.

Introduction

Agricultural policies have recently experienced major reformulations and have become more spatialised. Defining policy priorities requires appropriate tools (indicators, models) with relevant results about ecological and social features of agricultural practices (CEC, 2001). Concerning water resources, a major policy issue is currently the implementation of WFD (Water Framework Directive) in European countries. It stresses that an assessment is required to implement efficient measurement programs to preserve or restore the good ecological status of water bodies (to be reached in 2015). Compliance with this WFD implies a reduction in the impact of agricultural pressure on the environment and the implementation of measures designed to reduce pollution of surface and ground waters from agriculture. The river basin has been designated as an appropriate level for an integrated water management. In this approach, both point and non-point sources (NPS) of pollution are subject to control. Pollution from NPS typically includes nutrients and pesticides applied to the arable land surface. The European Parliament recently approved a new EU pesticides legislation (to be implemented in 2011). Member States must adopt National Action Plans for reducing "risks and impacts" of pesticide use on human health and the environment, including timetables and targets for use reduction. They must take measures to protect the aquatic environment from the impact of pesticides. These are to include "buffer zones" around water bodies and "safeguard zones" for groundwater used for drinkable water. In France, the program "Ecophyto 2018" aims to a specific reduction target of 50% for pesticides. The funding available for these mitigation measures must be used in the most effective way. As the characteristics of agricultural production may differ across a catchment, policy instruments can lead to extremely different results in terms of environmental effectiveness and consequently in terms of implementation costs. The aim of this paper is to propose a methodology for characterizing the agricultural activities and their possible evolution (scenarios) at the scale of a large river basin, in a distributed way and for evaluating the environmental impact of the chosen scenarios using modelling and indicators. The goal is to provide operational results for the science-policy discussions, in the context of stakeholders or local managers groups. The work from which results are presented here was conducted in the Charente river basin (10 000 km2) and Pertuis Charentais Sea, in the context of two projects dealing with ICZM. The European project SPICOSA (Science and Policy Integration for Coastal System Assessment) proposed a system approach framework and integrated assessment platforms, in order to consider the ecological, social and economical dimensions of coastal systems management (Prou and al, 2009). The Respireau project (French National Program Liteau) is centred on the discussion process between scientists and stakeholders and deals with the water quality issue on the same area.

2.

Material and methods

The Charente river basin

Agriculture activity covers about 60% of this area and about 11% of the cultivated area is irrigated. The annual water supply to human activities is 4 millions m3, with 34% dedicated to drinking water and 57% to irrigation which are therefore the two main human activities of concern in the freshwater issue. This river basin shows a risk of failing the objectives of the WFD towards the good ecological status (52% of the water bodies), due to agriculture diffuse pollution (nitrates, suspended matter and pesticides) and water shortage recurrent events. In addition, these failures of the freshwater management system have impacts on the marine waters (salinity) and the coastal ecosystems.

Spatial analysis

Different zoning methods could reveal these new entities by making it possible to identify units of systems with the same functioning at a selected level. First we chose the WFD elementary spatial units to analyse all variables at this scale. The WFD spatial unit (water bodies) is the elementary partition of aquatic environments selected for the water status assessment (140 hydrological units). A special focus has been done on some representative sub-basins (Né for both quality and quantity issues, Boutonne for intensive irrigation, Arnoult for pesticides). We also took in account reference spatial units (RSU) which are homogeneous parts of the river basin for selected variables. Some variables concerning structural sensitivity (slope, soil and drainage) are needed to assess the potential risk of agricultural pesticide or nutrients transfers towards surface waters. On the other hand, data about soil occupation and human practices, also needed, are available at the administrative entities scale. The territorial limits of environmental zoning do not fit with the French administrative limits, so that we have to use specific methods to take in account the different levels of organization (Daalgard and al, 2003). Aggregation and breaking up tools allowed recovering the agricultural data at RSU scale. (Vernier and al., 2010).

Clustering and typologies

When all the values for the chosen variables are calculated at the scale of spatial entities, data analysis tools can be used for clustering and defining relevant groups of spatial units for both geographic and thematic fields. Classical clustering methods (Parti-decla method) were applied to build a typology of RSUs. A new method using geographic constraints was tested: DIVCLUS-T is a descendant hierarchical clustering algorithm based on a monothetic bipartitional approach allowing the dendrogram of the hierarchy to be read as a Vernier F., Bordenave P., Chavent M., Leccia O., Petit K. / Modelling scenarios of agriculture changes on freshwater uses and water quality at a large watershed scale -the case of the Charente watershed (France) decision tree. It was applied to RSUs described by agricultural and environmental variables, in order to take their geographical contiguity into account in the monothetic clustering process [START_REF] Chavent | Monothetic Divisive Clustering with Geographical Constraints[END_REF]. This typology was built to describe the Charente watershed and the agricultural activities in a simple way. So we can describe the agricultural activities (SAU, irrigated SAU, technical parameters and uptake of water in a spatialised way or in an aggregated way. We also built a typology of farming systems using statistical data from Agricultural Census and Common Agricultural Policy data. This typology can be used to generalize the data about agricultural practices (irrigation, fertilizers and pesticides) at the large river basin scale.

Scenarios

Besides the consultation of a participant group regarding the scenarios of evolution of the system (local managers) (Prou and al, 2009), we performed specific surveys among local authorities and professionals (experts, advisers, farmers) about the evolution of Agriculture in the next ten years. 2020 is relevant for two main purposes in terms of public policies: WFD specific programs ( 2015) and agro-environmental measures (MAET, Ecophyto 2018). These surveys allow proposing to stakeholders, different options for the evolution of agriculture, regarding the potential impact on the water resources. The scenarios are built following a participative approach. They are intended to function as combinations of future trends for climate, soil occupation (increase, decrease or change) and management options.

The current situation of agriculture on the area is defined as the baseline scenario "scenario 0". The parameters for this scenario 0 are defined by regional references, the previous work and the surveys made on the area. Each other scenario to be simulated is an occurrence of this first one. For each scenario, a calculation of all variables used has to be done for each sub-basin. Therefore, one global scenario on the Charente river basin comes in different local scenarios for each spatial unit (RSU) used in the models.

Effectiveness assessment -agro-hydrological modelling

The effectiveness of scenarios to reduce environmental impacts from agriculture is assessed using the semi-distributed river basin Soil and Water Assessment Tool (SWAT) model. (Arnold andal., 2005, Neitsch andal., 2001). The SWAT model was chosen for its ability to simulate nutrients and pesticide concentrations at the watershed scale, its worldwide use and its potential to simulate agricultural management practices. The model integrates all relevant ecological, hydrological and in-stream processes at the sub basin scale. The RSus of the SWAT model are called Hydrological Response Units (HRUs) in which the vertical, lateral and sub-lateral flows of water and nutrients are calculated. Some specific calculations [START_REF] Lescot | A decision support to choose between changes of agricultural practices; A spatially distributed Cost-Effectiveness assessment framework[END_REF] are made to define relevant spatial land use entities in order to build the HRUs while intersecting data about soils and climate (4648 HRUs on the studied area). The management practices are defined at the HRU level by specific management operations (beginning and end of growing season, timing of cultural operations, timing and amount of fertilizers and pesticides spraying, irrigation management). Nutrients and pesticides fluxes from HRUs are routed to the 140 subbasin outlets. Several changes in the source code of the SWAT model were introduced for equations determining the FILTER function. In urban zones, we took herbicide spraying into account. Attention was paid to implementing an estimator for point losses such as those related to the cleaning of dressing/spraying equipment and to improve the representation of physical processes in filter strips. (Veerle and al, 2008). Once the model is modified and calibrated, it can be used to simulate the chosen scenarios and evaluate their effectiveness. Before calibration and validation of SWAT, manual sensitivity analysis was carried out using a method similar to the one proposed by [START_REF] Ullrich | Application of the Soil and Water Assessment Tool (SWAT) to predict the impact of alternative management practices on water quality and quantity[END_REF]. The most sensitive parameters were CN2, USLE-P and Filter and these parameters were calibrated by comparison with measurement data. For point source pollution, we followed recommendations given by Gevaert (Veerle and al, 2008).

General approach (assessment)

The links between the different methods used are summarized in figure1. The definition of the typologies of farming systems and practices per RSU, the definition of scenarios and the discussion about their effectiveness and environmental impacts (in progress) include a participative process. Some scenarios can be simulated using the SWAT model (daily time scale for a 22 year period) and some others not. The results are about the scenarios which have been currently simulated. New iterations taking in account stakeholders' remarks are to be done.

Preliminary results

Characterization of the agriculture in the river basin

Eight classes of RSUs (140 hydrological units) and five main farming systems were defined on the area using the clustering methods detailed previously.

For each RSU, crop areas and crops areas are defined, as well as the type of watering uptake (figure 2). Using these two typologies, a crop management sequence including fertilization, pesticides and irrigation practices (see the example of maize in figure 1) can be allocated for each main crop and each sub-basin of the area. These values are used as an entry to the management files of the models for the scenario 0. Regarding the crop rotations, surveys allowed defining the main crop rotations for irrigated and non-irrigated crop area, depending on the type of soil and of the farming systems (table 1). Using these data, we calculated at the RSU scale some agro-environmental indicators combining pressure from agriculture and sensitivity (soils, slopes, drainage). These indicators are useful for setting the different parameters of scenarios.

Scenarios

The main options defined with the stakeholders about the evolution of Agriculture in the next ten year are presented in table 2. The reference values for crop rotations and crop management sequences defined for the baseline scenario (scenario 0) are adapted and spatialized at the RSU scale for each simulated scenario. The values of the areas of irrigated and non irrigated crops are recalculated in a spatialised way for each sub-basin, and also the percentage of water uptakes from different sources (river, reservoirs and groundwater).

Irrigation scenarios

Considering the options chosen by the stakeholders, our expertise on the area and the constraints from the simulation model, the simulated scenarios are (currently):

-IR0: scenario without irrigation on the area. It comes from the baseline scenario (business as usual) but without any irrigation. This "virtual" scenario is used as a calibrating one, -IR1: scenario which fits with less authorized uptakes; irrigation is forbidden when the river flow reaches a threshold of 12 m3 s -1 , -IR2: scenario without any threshold for the uptake in the river; it is also a calibrating scenario, all the water available in the river could be used for irrigation uptakes.

The SWAT model calculates the daily volume needed for the irrigation of the irrigated crop area on the sub-basin and then the type of water uptake (from the river, from the water table or from reservoirs) is defined in some parameters with the values coming from the previous steps (see paragraph 3.1).

Scenarios for pesticides assessment

Some mitigation measures were tested:

Vernier F., Bordenave P., Chavent M., Leccia O., Petit K. / Modelling scenarios of agriculture changes on freshwater uses and water quality at a large watershed scale -the case of the Charente watershed (France)

-RBS: setting riparian buffers all along river courses, except when soil occupation is grasslands or meadows ; the parameters chosen are 5, 10 or 20 meters wide (RBS5, RBS10 and RBS 20), -CC: catch crop implemented between winter crops and spring crops, -RP: decrease in the quantity of pesticides applied (RP10, decrease of 10%, RP20 decrease of 20%); stakeholders thought these values are the more realistic, -LR-R: diversification in crop rotations with increasing the areas of winter rape, -LR-P: diversification in crop rotations with increasing the areas of pea (protein plants).

Most of these scenarios were evocated by the stakeholders. They are used in the current mitigation measures proposed on the area. Several others scenarios are going to be simulated and proposed for discussions and feedback to stakeholders.

Results of simulations

The results presented in the tables 3,4 and 5 consist in annual means for the 12 last years of the 22-year-period simulation based on a daily calculation of all variables. Some indicators can be calculated for each scenario (crop yields). These indicators can be used to calculate a deficit of crop production due to some strategies (economic impact assessment). The comparison of hydrological effectiveness is carried out and considered in terms of relative reductions in a particular pollutant for scenarios regarding nutrients and pesticides.

with Average concentration over the ten last years of hydrological simulation

Baseline concentration

Effectiveness is computed at the outlet of each HRU, at the outlet of each sub basin and at the Charente river basin outlet. The figures 3a and 3b show an example of graphic output of the modelling of the baseline scenario (scenario 0) and of the RBS5 scenario. This type of output is produce for all the simulated scenarios in order to be discussed with stakeholders.

    -1 s μgl C -1 -1 s μgl (%) 0 μgl C Effectiveness = 1 -C             -

Figure 3a and 3b: spatial restitution of the simulation results for two scenarios, the baseline scenario and the RBS5 scenario (riparian buffers)

The results from table 3 show that irrigation impacts strongly the discharge of the Charente River even if the irrigated area is only 10% of the total cultivated area. Without policy constraints on water uptake for irrigation, crop production is higher but the river flow during the dry period decreases dramatically and doesn't allow a good functioning of the ecosystem; problems could occur for the uptake of drinkable water. These values are given for the Charente water course and more difficulties appear in upstream water bodies. The results presented in table 4 assert that irrigation influences the other factors of water quality. The IR2 scenario without constraints on irrigation water uptakes increases the concentrations in pesticides and suspended matter; the same scenario decreases the nitrate concentrations due to a better use of fertilization on the plots. The IR1 scenario which fits with less authorized water uptakes for irrigation is efficient mainly regarding the river flow (biodiversity).

As described in previous studies [START_REF] Lescot | A decision support to choose between changes of agricultural practices; A spatially distributed Cost-Effectiveness assessment framework[END_REF], it is shown in table 5 that the riparian buffers (RBS scenarios) are effective for reducing the pesticides concentrations in the river course. These measures are applied on a small cultivated area and consequently the costs to implement them are not very high. Despite of this efficiency, these scenarios are not effective enough to decrease the concentration under a value of 0,5 µg/l which is considered as a standard (for all molecules) in current policies. These scenarios should be combined to other mitigations measures regarding the amount of pesticides applied on the plots. The RP scenarios (RP10, RP20) have a good effectiveness (20 to 30%) for reducing the pesticides concentrations. In these scenarios, a part of the herbicide spraying is replaced by mechanical weeding (zero herbicide) and not by other molecules. Another explanation is that herbicides are mainly found in the rivers, more than fungicides and insecticides. So the main cultivated area (cereals, maize and rapeseed) is concerned by this strategy. The catch crop scenario (CC) is also interesting because the action is effective regarding both nitrates and pesticides whereas RBS scenarios are less effective for reducing nitrates (particularly in well-drained areas).

The response of the sub basins to the pesticide pressure from agriculture is very different depending on their location in the river basin and on the natural and human conditions (figure 3a). Consequently, the effectiveness of each scenario comes in different ranges depending on the HRUs and sub basins (figure 3b).

Conclusions and Recommendations

The methodology used, based on several methods of diagnosis, spatial analysis, clustering and distributed modelling was tested at the scale of a large river basin, which is a relevant scale for integrated water management, seems effective for evaluating "ex-ante" the potential impacts of the possible evolution of Agriculture on the area. The typologies of RSUs and agricultural practices, then the results given by the simulations provide relevant output indicators for the science-policy discussions and more objectivity in the debate between stakeholders. The spatial resolution used for the restitution (maps) provides additional information for the management of the local areas or the implementation of mitigation measures on the area. The long-term time scale used for the simulations allows freeing from the short term variability like annual climate variations, annual variation of agricultural practices and therefore a better interpretation of water quality monitoring (mainly for pesticides). Another interest of this methodology is to provide a multicriteria evaluation (nitrates, pesticides, suspended matter and discharge) of each chosen scenario.

The participatory process (discussion with stakeholders) can be taken in account with possible iterations for each scenario, to test several assumptions. The preliminary results show some scenarios could be effective for reducing the pesticides concentrations in the rivers but it will be necessary to implement the mitigation measures in a relevant way depending on the sub basin (farming systems, natural conditions) and on a large cultivated area. The most effective scenarios (RBS scenarios) should be implemented in the largest range of farms but the concerned cultivated area is quite small (5 to 10 meters along the river). A perspective of the method could be a coupling with an economical model to assess the cost of implementation of the most effective scenarios, depending on the farming systems and the "on contract" cultivated area (CE cost effectiveness methods).

Figure 1 :

 1 Figure 1: General approach

Figure 2 :

 2 Figure 2: Types of farming systems and water uptake on the area

  , Bordenave P.,Chavent M., Leccia O., Petit K. / Modelling scenarios of agriculture changes on freshwater uses and water quality at a large watershed scale -the case of the Charente watershed(France)

Fig

  Fig.3a. Baseline scenario -pesticides concentration

Table 1 :

 1 crop rotations depending on soil and farming system (two types of soils among 7 defined on the area) Vernier F., Bordenave P., Chavent M., Leccia O., Petit K. / Modelling scenarios of agriculture changes on freshwater uses and water quality at a large watershed scale -the case of the Charente watershed(France)

	Type of soil	Crop rotation : Non irrigated	Irrigated
	Doucins (silty soils)	maize / wheat / wheat or barley	Maize
		rapeseed-wheat-sunflower-wheat
	Marshland soils,	Maize	Maize
	valleys, terraces	rapeseed-wheat-sunflower-wheat	Maize-wheat-barley

Table 2 :

 2 main options from the stakeholders(if simulated in bold)

	Soil occupation by	increase for non	Decrease for	Stability of increase
	agriculture	irrigated crop area	irrigated crop area	for total arable land
	Crop rotation	diversification	more spring or winter	More "catch crops"
			crops (less single crop	implemented
			farming)
	Types of crops	More industrial crops	Increase for wheat,	Decrease for
		(hemp)	sunflower and	irrigated maize and
			leguminous plants	rapeseed
	Water uptake	More dams	Less authorised	Specific varieties of
			uptake	crops (less water
				demand)
	Nutrients	More organic farming
		systems and
		grasslands
	Pesticides	-10 à -20% of	-30 % pesticides	More organic farming
		pesticides applied	applied	systems and
				grasslands
				Riparian buffers

Table 3 :

 3 Impact of irrigation strategies on the river flow (for 3 chosen scenarios)

		Number of days (June to October) with a daily discharge threshold of
		1 m3/s	6 m3/s	8 m3/s	10 m3/s	12 m3/s	14 m3/s	16 m3/s
	S0	0	0	0	0	3	4	6
	IR1	0	2	3	4	10	11	12
	IR2	13	18	21	23	26	28	30

Table 4 :

 4 Impact of irrigation strategies on the nitrates, pesticides and suspended matter (for 3 chosen scenarios)

		ET (mm)	Irrigation rate (mm)	SM (T ha -1 )	Erosion (kg ha -1 an -1 )	N03-N (T an -1 ) *	Pesticides (µ l -1 )	Crops yields TMS ha -1
	S0	365	0	23600	530	35100	1.06	6.9
	IR1	371	140	25000	580	33400	1.10	9.0
	IR2	410	160	30000	290	25200	1.15	10.1

ET evapotranspiration ; SM suspended matter in the river ; Erosion at the HRU scale ; Pesticides sum of the molecules ; crop yields in tons of dry matter per hectare

Table 5 :

 5 effectiveness of scenarios to reduce the river concentrations at the outlet

	Effectiveness	RBS 5	RBS10	RBS20	CC	RP10	RP20	LR-R LR-P
	(%)
	pesticides	25	41	59	16	23	30	-12	9
	reduction
	suspended
	matter	13	21	31	-1	4	4	5	1


Doru Constantin
email: constantin@lps.u-psud.fr

Patrick Davidson

Éric Freyssingeas

Anders Madsen

Slow dynamics of a colloidal lamellar phase

Keywords: numbers: 82.70.Dd, 87.15.Ya, 61.05.cf a)

We used X-ray photon correlation spectroscopy to study the dynamics in the lamellar phase of a platelet suspension, as a function of the particle concentration. We measured the collective diffusion coefficient along the director of the phase, over length scales down to the inter-particle distance, and quantified the hydrodynamic interaction between the particles. This interaction sets in with increasing concentration and can be described qualitatively by a simplified model. No change in the microscopic structure or dynamics is observed at the transition between the fluid and the gel-like lamellar phases.

I. INTRODUCTION

In order to reach a thorough understanding of colloidal suspensions, it is crucial to probe their dynamics at length scales comparable to the inter-particle distance. Thus, a considerable amount of experimental and theoretical work has been concerned with the dynamics of dense colloidal suspensions [START_REF] Pusey | Colloidal suspensions[END_REF] . Space-and time-resolved experiments were mainly performed using dynamic light scattering (DLS) on micron-size spherical particles, both in the isotropic [START_REF] Segrè | Viscosity and structural relaxation in suspensions of hard-sphere colloids[END_REF] and the crystalline 3 phases.

DLS was also used to study dilute isotropic suspensions of elongated particles [START_REF] Graf | Dynamic light scattering by aqueous solutions of rodlike fd-virus particles[END_REF] or solutions of disk-shaped particles close to the sol-gel transition [START_REF] Kroon | Dynamic light scattering studies on the sol-gel transition of a suspension of anisotropic colloidal particles[END_REF] , but using this technique for the study of concentrated and/or ordered solutions becomes challenging, due to the required particle sizes. This is nevertheless a regime where the system should exhibit rich dynamics, influenced by the (short-or long-range) order and by the hydrodynamic coupling, the latter being quantified by the hydrodynamic function.

Theoretical efforts aiming to calculate this function have focussed mainly on suspensions of spherical particles [START_REF] Nägele | Long-time dynamics of charged colloidal suspensions: hydrodynamic interaction effects[END_REF] ; they have been largely validated by the experimental results (see below). For anisotropic particles, on the other hand, the calculations are much more involved and they were mostly restricted to solutions of slender rods [START_REF] Batchelor | The stress generated in a non-dilute suspension of elongated particles by pure straining motion[END_REF][START_REF] Shaqfeh | The hydrodynamic stress in a suspension of rods[END_REF] . We are not aware of any analytical results for the hydrodynamic function of plate-like particles. Experimental measurements at high scattering vector (corresponding to the interparticle distance) are also lacking, due to the intrinsic wavelength limitation of DLS.

In recent years, the accessible distance range increased significantly via the use of X-ray Photon Correlation Spectroscopy (XPCS), which is fundamentally similar to DLS but uses X-ray radiation as a probe. Although this technique is only effective on systems with high scattering contrast and slow relaxation rates, it has already been used to study concentrated hard sphere solutions [START_REF] Robert | Structure and dynamics of electrostatically interacting magnetic nanoparticles in suspension[END_REF][START_REF] Banchio | Many-body hydrodynamic in-teractions in charge-stabilized suspensions[END_REF][START_REF] Robert | Dynamics in dense suspensions of charge-stabilized colloidal particles[END_REF] , aging suspensions [START_REF] Bandyopadhyay | Evolution of particle-scale dynamics in an aging clay suspension[END_REF][START_REF] Wandersman | Heterogeneous dynamics and ageing in a dense ferro-glass[END_REF] , particles dispersed in complex fluids [START_REF] Caronna | Dynamics of nanoparticles in a supercooled liquid[END_REF][START_REF] Guo | Nanoparticle motion within glassy polymer melts[END_REF] , and interface dynamics [START_REF] Gutt | Observation of heterodyne mixing in surface x-ray photon correlation spectroscopy experiments[END_REF][START_REF] Gutt | Partially wetting thin liquid films: Structure and dynamics studied with coherent x rays[END_REF] . Some XPCS studies were performed on mesophases. For instance, the collective diffusion coefficient of a colloidal nematic phase was determined over a wide range of the wave vector q. The analysis showed that the dynamics of the system displays strong slowing down over length scales larger than the interparticle distance [START_REF] Poulos | Short-range dynamics of a nematic liquid-crystalline phase[END_REF] . The relaxation of capillary surface waves has also been measured [START_REF] Madsen | Viscosity of a liquid crystal near the Nematic-Smectic A phase transition[END_REF] . Smectic phases -which have a certain degree of positional concentrated in the vicinity of the Bragg peaks, leading to high count rates. That is why some of the first XPCS results were obtained on such systems, in particular under the form of free-standing [20][21][22] films. The bulk dynamics of a lamellar lyotropic phase of surfactant were also studied [START_REF] Constantin | Dynamics of bulk fluctuations in a lamellar phase studied by coherent x-ray scattering[END_REF] . The results, confirmed by DLS measurements, could be interpreted in terms of the continuum elastic theory of smectics 24,25 .

In this paper we present results obtained on lamellar lyotropic phases composed of large inorganic colloids; in this system, the slow relaxation rates and the high scattering contrast greatly extend the accessible q-range. We measure the collective diffusion coefficient D(q z ), where ẑ is the direction of the lamellar director, for a fairly wide range of scattering vectors along the lamellar director, q z (covering at least the first Bragg peak of the phase). We then obtain the hydrodynamic function H(q z ). The large aspect ratio of the particles and their lamellar order enable us to describe the dynamics using a simplified analytical model, which is in semi-quantitative agreement with the data. As a function of the concentration, the hydrodynamic coupling goes from very weak to extremely strong.

II. EXPERIMENTAL

We used concentrated suspensions of phosphatoantimonate (H 3 Sb 3 P 2 O 14 ) particles, with a typical thickness of 1 nm and at least 300 nm wide [START_REF] Gabriel | Swollen liquid-crystalline lamellar phase based on extended solid-like sheets[END_REF] , synthesized as in reference 27. The lateral size distribution of the particles was assessed using a scanning electron microscope equipped with a field emission gun (SEM-FEG). In a typical experiment, a colloidal suspension was diluted down to a volume fraction of 2.5 10 -4 by adding distilled water. Then, a drop of suspension was spin-coated at ∼ 1000 rpm onto a silicon wafer. Close inspection of the samples, by SEM-FEG at 10 kV and by optical microscopy, revealed the presence of plate-like particles that sometimes had a clear hexagonal shape. Their size distribution is extremely broad and ranges from a few hundred nanometers to more than five microns.

The batch solutions were diluted, either with pure water or with a 50:50 (wt%) glycerol/water mixture to reach the desired volume concentration φ. The room-temperature phase diagram was determined by visual inspection of the vials, in natural light and between crossed polarizers. The birefringent lamellar phase has a gel-like texture above φ = 1.8% (does not flow under its own weight, i.e. it has a zero-frequency elastic modulus) and is fluid with a concentration φ I < 0.1%. These phase boundaries are compatible with the phase diagram presented in Figure 3 of reference 26, for the case of no added salt (ionic strength below 10 -4 mol/l).

The lamellar samples were prepared by aspiration in round glass capillaries, 1 mm wide and with a wall thickness of 10 µm. By scanning the capillaries in the beam, it is easy to find aligned domains (usually, with the director normal to the capillary walls).

The small-angle X-ray scattering (SAXS) and XPCS measurements were performed at the TROIKA beam line ID10A of the ESRF with an X-ray energy of 8 keV (λ = 1.55 Å) selected by a single-bounce Si(111) monochromator, in the uniform filling mode of the storage ring. A (partially) coherent beam is obtained by inserting a 10 µm pinhole aperture a few centimeters upstream of the sample.

We used a 2D Maxipix detector consisting of 256 × 256 square pixels (55 µm in size) and with acquisition rates of up to 1 kHz. For XPCS, a few thousand images were recorded and the intensity autocorrelation functions were calculated by ensemble averaging [START_REF] Fluerasu | Slow dynamics and aging in colloidal gels studied by x-ray photon correlation spectroscopy[END_REF] over pixels with the same absolute value of the scattering vector (see Figure 1). The SAXS pattern is obtained as the average intensity over all the images in each run. Since we are only concerned with the q z dependence of the measured quantities, in the following we will denote throughout q z by q.

The relaxation of concentration fluctuations with a given wave vector q is reflected in the field correlation function at that value of the scattering vector:

g 1 (q, t) = E * (0)E(t) / E * (0)E(0) .
As we will see below, our data is well described by a single exponential relaxation:

g 1 (q, t) = exp[-Ω(q)t].
Experimentally, we measure the normalized intensity correlation function g 2 (q, t), related to the field correlation function by the Siegert relation:

g 2 (q, t) = 1 + b(q) |g 1 (q, t)| 2 = 1 + b(q) exp[-2Ω(q)t] (1)
where the contrast factor b is a few percent. Fitting the experimental data to Equation

(1), as illustrated in Figure 2, yields the relaxation rates Ω(q). We will further define a scale-dependent diffusion constant, D(q) = Ω(q)/q 2 . In the following, we will work with the (static) structure factor S(q) and with the diffusion constant D(q). For the sample with φ = 2.1 vol% (used as an example in Figures 1, 2 and 3) these parameters are displayed in

III. RESULTS AND DISCUSSION

The structure factor and fluctuations of the smectic phase have been studied in detail over the last decades by many authors (see Ref. 29 for an in-depth review). The deformation free energy of the phase consists essentially of two terms, proportional to the compression modulus (along the smectic director) and to the bending stiffness of the layers. Both the amplitude and the relaxation of the deformation eigenmodes (corresponding to a unique phonon) can be described in a relatively straightforward (although algebraically cumbersome) fashion.

However, if one is interested in the dynamic structure factor, i.e. in the collective behavior of the system at a given wave vector, the calculations become extremely involved, since all the phonons contribute in non-trivial ways. In order to make the analytical treatment as easy as possible, so that the underlying physics is not obscured by the mathematical formalism, some simplifying assumptions must be made.

In this work, we are dealing with rather dilute phases composed of very stiff and very large platelets. We will therefore assume that the compression modulus is low and the bending stiffness is large, so that the most important fluctuations are those affecting the spacing between platelets along the director ẑ and we treat the system as effectively onedimensional, within the framework of a damped harmonic chain model. (materialized by a dotted line) is the high-q value of the diffusion constant, corresponding to the range where the structure factor saturates to 1.

A. Statics

For this model, the static structure factor has an analytical solution, given by Refs. 30 and 31:

I(q, 0) = S(q) = sinh q 2 σ 2 2 cosh q 2 σ 2 2 -cos (qd) (2)
with σ the typical fluctuation amplitude, defined by σ 2 = k B T /α as a function of the strength of the harmonic potential, α.

Clearly, the harmonic model is not very accurate at high q, as one can see in Figure 5 for the fits to the low-concentration data. The interaction potential is probably stiffer, leading to pronounced second-and even third-order peaks, which are better described by a hard-particle model [START_REF] Wertheim | Analytic solution of the Percus-Yevick equation[END_REF] (plotted as dashed lines). In the following we will nevertheless use the harmonic approximation, which describes the first peak rather well and which is more tractable as far as the dynamics are concerned (see Section III B).

From the analysis of the structure factor data in Figure 5 using the model (2) we extract the repeat distance d and the fluctuation amplitude σ. In Figure 6 we present both the dilution law d(φ) and the ratio σ/d as a function of the volume fraction φ. Three important conclusions can be drawn from this data:

• The platelet thickness δ, given by the slope of the dilution law, is 7.4 ± 1.2 Å, much smaller than the 10 Å value given in the literature [START_REF] Gabriel | Swollen liquid-crystalline lamellar phase based on extended solid-like sheets[END_REF] . This discrepancy could be due to imperfect exfoliation of the layers, leading to a lower effective concentration in the lamellar phase.

• There is no systematic difference in d and σ between the systems formulated in pure Curves shifted vertically in steps of 1. The solid lines are fits to the harmonic chain model (2). For the two lowest concentrations, we also present as dashed lines the fit with the hard-rod model (see the text for more details).

water (at room temperature) and those with 50 % glycerol (at 253 K). Since the phase boundaries are also the same, we conclude that the static structure of the phase is not affected by the presence of glycerol. At the same time, there is no discernible difference between the two synthesis batches.

• The ratio σ/d = 0.2 ± 0.05 remains almost constant as φ (and hence d) vary by almost an order of magnitude. given in the literature [START_REF] Gabriel | Swollen liquid-crystalline lamellar phase based on extended solid-like sheets[END_REF] . Bottom: the ratio between the fluctuation amplitude σ (see Equation 2) and d. The data is shown for all the available samples, both in pure water at 293 K and in a 50:50 glycerol/water mixture at 253 K.

B. Dynamics

To investigate the dynamics of the system we consider a damped harmonic chain, following the notations of Geisel [START_REF] Geisel | Interacting brownian particles and correlations in superionic conductors[END_REF] . The chain consists of N particles at positions x j , and we define the displacements u j = x j -x 0 j with respect to the reference positions, which obey x 0 j+1 -x 0 j = d, where d is the average interparticle distance. For definiteness, we assume periodic boundary conditions: u N +1 = u 1 and only consider odd values for N.

We start by considering the case of hydrodynamically uncoupled particles, i.e. the energy dissipation for each particle only depends on its velocity with respect to the surrounding fluid, and not on the position or velocity of the other particles.

The equations of motion are:

mẍ j + mγ ẋj + ∂V ∂x j = F j (t) (3a) V = α 2 k (x k+1 -x k -d) 2 (3b) F j (t)F i (0) = 2mγk B T δ ij δ(t) (3c)
where m is the particle mass, mγ is a damping coefficient (to be discussed in detail further on) and F j (t) is the random force acting on particle j at time t.

The quantity of interest is the time-dependent correlation function (depending also on the wave vector q):

I(q, t) = 1 N N l,k=1 e iq(x l (t)-x k (0)) (4)
which reduces for t = 0 to the static structure factor (2).

The treatment is quite standard: the displacements u j are developed over the basis of normal modes (phonons). The amplitudes and time relaxation rates of the phonons obtained from (3) are used to express I(q, t) by developing the right-hand side in Eq. ( 4).

In the following, we restrict ourselves to the overdamped case, γ ≫ α/m = ω 0 . In this limit, the inertial term in Eq. (3a) can be neglected and a unique relaxation rate is associated to each phonon 34 . The resulting expression for I(q, t) is unwieldy, so we use the cumulant expansion [START_REF] Koppel | Analysis of macromolecular polydispersity in intensity correlation spectroscopy: The method of cumulants[END_REF] . The first cumulant, defined as K 1 (q) ≡ -1 I(q,0) ∂ ∂t I(q, t) t=0 represents an average relaxation rate for a given wave vector. Our experimental data is well described by a single exponential decay, I(q, t) = I(q, 0) e -Ω(q)t , in which case the first cumulant is just the relaxation rate: K 1 (q) = Ω(q), an assumption we will make throughout the analysis below.

In the uncoupled case, one has simply:

K 1 (q) = D 0 S(q) q 2 (5) with D 0 = σ 2 ω 2 0 γ = k B T mγ
The relaxation rate obeys a diffusive law, with a diffusion constant D(q) = D 0 /S(q)

which contains explicitly the well-known 1/S(q) dependence ("de Gennes narrowing" [START_REF] De Gennes | Liquid dynamics and inelastic scattering of neutrons[END_REF] ).

D 0 is the value of D(q) at high wave vectors, where S(q) reaches 1, so that one should have D(q) • S(q)/D 0 = 1 at all q. In other words, only the thermodynamic forces (related to S(q)) are taken into account, and the prefactor D 0 is simply proportional to the mobility of an isolated particle, (mγ) -1 . For a thin circular plate of radius R moving normal to its plane at low Reynolds number in a fluid with viscosity η, one has 37 : mγ = 16ηR. FIG. 7. Experimental data for the rescaled hydrodynamic function H(q)/H(∞) (symbols) and the static structure factor S(q) (solid lines) at various concentrations φ indicated alongside the curves.

The left and right panel are plotted against a common y-axis. For clarity, the curves are shifted vertically in steps of 3 units.

However, one must also account for the hydrodynamic interactions, which lead to a more complex, and generally scale-dependent, form for the dissipation. Their effect is quantified in terms of the hydrodynamic function, H(q), defined as 6 :

H(q) = D(q) D 0 S(q) (6)
In dilute solutions, H(q) → 1 at high q values (where S(q) saturates at 1) but is in general different from unity below this range. We emphasize that in the simplified model discussed above, H(q) = 1 identically (from equations ( 5) and ( 6).) at moderate concentrations. In three-dimensional systems they have been studied quite extensively, both for spherical [START_REF] Nägele | Long-time dynamics of charged colloidal suspensions: hydrodynamic interaction effects[END_REF] and for elongated [START_REF] Batchelor | The stress generated in a non-dilute suspension of elongated particles by pure straining motion[END_REF][START_REF] Shaqfeh | The hydrodynamic stress in a suspension of rods[END_REF] particles. In recent years, the twoand one-dimensional cases have also been considered, in particular in the context of confined spherical particles [38][START_REF] Cui | Anomalous hydrodynamic interaction in a quasi-two-dimensional suspension[END_REF][START_REF] Beatus | Phonons in a one-dimensional microfluidic crystal[END_REF] . In contrast, the phase under investigation consists of very anisotropic platelets and the one-dimensional character is given by the orientational order of the phase, rather than by confinement; hence, the models cited above are probably not adapted. Since the distance between platelets along the smectic director (face to face) is much lower than their lateral extension, we only consider hydrodynamic coupling along this direction, neglecting the effect of the in-plane neighbors (edge to edge).

The simplest model that accounts for these features is the well-known Stefan equation [START_REF] Bird | Dynamics of Polymeric Liquids[END_REF] describing the squeezing flow between two parallel plates at low Reynolds number (in the lubrication approximation). The viscous force acting on the plates is:

F = - 3πηR 4 ḣ 4h 3 (7)
with η the viscosity of the fluid, R the radius of the (circular) plates and h the gap width.

In this case, the dissipation is no longer proportional to the velocity of an individual particle with respect to the surrounding fluid, but rather to the velocity difference between neighboring particles:

F visc j = -ζ [2 ẋj -( ẋj-1 + ẋj+1 )] (8)
where ζ = 3πηR 4 4d 3

For this reason, the dissipation associated to the homogeneous (q = 0) mode vanishes: the particles can move together, at any velocity and maintaining the spacing d. The consequence is an infinite relaxation rate. To remove this artificial divergence, we also preserve an individual friction term (formally identical to the second term on the lhs in Equation (3a).)

In this case, however, the quantitative value for mγ is different from the free case, since the interaction of each particle with the fluid is "screened" by its neighbors. A simple yet fairly realistic model is that of a cylindrical stack of platelets: ζ corresponds then to the relative motion of particles within the stack, while mγ is associated with a solid-like translation of the cylinder along its axis, the dissipation taking place in the surrounding medium (with an effective viscosity η eff which is in general different from that of the solvent). Using wellknown formulas for the longitudinal mobility of a cylinder [START_REF] Van Bruggen | Long-time translational self-diffusion in isotropic dispersions of colloidal rods[END_REF] and considering an aspect ratio (length L to diameter 2R) of the order of 10, one has: D = k B T πη eff L , amounting to a friction coefficient per particle mγ = k B T D d L = πη eff δ/φ. With the ingredients above, and after some algebra (detailed in the Appendix) one obtains for H(q) an explicit formula:

H(q)= 2 √ 1 + 2β N -1 p=0 ′ 1 - p N cos(qdp) exp - q 2 σ 2 2 p × ×z p , with z = 1 + 1/β -1 + 2β/β ( 9
)
where N is the number of particles and β = 2ζ/(mγ) is the ratio of the collective dissipation described by (8) to the individual particle dissipation; it provides a quantitative measure of the hydrodynamic coupling. The primed sum symbol indicates that the p = 0 term should be halved. In the high-q limit, H(q) → H(∞) = (2β + 1) -1/2 so that the corresponding diffusion constant is:

D ∞ = D 0 √ 2β + 1 = k B T mγ 4ζ/(mγ) + 1 (10)
As shown in the Appendix, for high values of the coupling constant β the rescaled hydrodynamic function H(q)/H(∞) approaches the static structure factor S(q); in other words, D(q) becomes constant since the hydrodynamic effects are much stronger than the thermodynamic ones. This tendency is indeed visible in Figure 4, where the "dips" present at the peak positions for low concentrations flatten out [START_REF]At high φ, D(q) still exhibits a low-angle increase, which is not captured by our model and leads to an H(q) that is systematically higher than S(q) below the peak position in Figure 7[END_REF] . We would then expect that, as the concentration increases, the hydrodynamic function evolves from a constant value to the limiting shape S(q). This is indeed observed in Figure 7, where there is a clear difference between the data at φ = 0.8 % and those at higher concentration. Indeed, at φ = 1.1 % H(q) is already similar to S(q) and the similarity becomes clearer above this φ value. More points within 0.8 % ≤ φ ≤ 1.1 % would be needed to resolve the onset of coupling.

The similarity between S(q) and H(q) is very useful, since the static structure factor can be measured much more precisely and to higher q values than the hydrodynamic function.

We therefore rescale H(q), bringing it as close as possible to S(q) (the data in Figure 7 has already been rescaled). This operation also sets the value of D ∞ without the need of measuring D(q) to very high q.

To summarize, the high-q value of the diffusion coefficient in the two regimes is given by: 14

D ∞ = D 0 = k B T 16η eff R uncoupled (11a) D ∞ = k B T πη eff d 1 3 R d 4 η η eff + 1 coupled (11b)
where we considered that the dissipation by squeezing flow between the particles involved the solvent viscosity η = 50 mPa s (for a 50:50 wt % mixture of glycerol in water at 253 K 44 ) while for the dissipation of the individual particles one needs to use the effective viscosity of the medium, η eff , taken as independent of the concentration in the investigated range.

The smectic repeat distance d is given in Figure 6, so the only adjustable parameters are η eff and the platelet radius R. Good agreement with the experimental data is obtained with the values: η eff = 3.2η and R = 3.5 µm (see Figure 8). Indeed, at the lowest concentration φ = 0.8 % one has D ∞ = D 0 given by Eq. (11a) (dashed line in Figure 8), while at higher concentration the data is well described by the dependence (11b), plotted as solid line. The coupling occurs for φ between 0.8 and 1.1 %, in agreement with the interpretation proposed above for the hydrodynamic function. It is noteworthy that no significant change in the microscopic dynamics can be detected at the transition between the fluid and the gel-like lamellar phases.

The rather large value of the platelet radius obtained above is not surprising, since the largest particles dominate both the scattering signal (with a contribution proportional to the square of the particle volume) and the correlation function, contributing the slowest relaxation. Furthermore, the coupling in Equation (11b) goes as the fourth power of the radius. The size distribution being very wide (see Section II), all experimental results should be severely skewed towards the large particles.

IV. CONCLUSION

We measured the static and dynamic properties of a lamellar phase composed of rigid platelets and quantified the hydrodynamic coupling between nearest neighbors (along the director of the phase). The coupling is almost absent at low concentrations, where the dissipation occurs at the level of the isolated particle, but it quickly becomes dominant at higher concentrations. The hydrodynamic function of the phase H(q z ) is relatively well described by an analytical one-dimensional model.

This result is noteworthy insofar the hydrodynamic function -although indispensable for understanding the relaxation at the particle size-is generally difficult to calculate, even for spherical colloids. To our knowledge, no explicit models have been proposed for anisotropic particles. Somewhat surprisingly, the order of the particles (which should further complicate the study) allows us in the present case to use a very simplified approach.

The high-q value of the diffusion coefficient D ∞ decreases rapidly with the concentration in the fluid lamellar phase, but it remains practically constant across the sol/gel transition and well into the gel-like regime. No sharp variation is observed at the transition, in either the dynamic or the static parameters (in particular, we do not detect any spatial inhomogeneities), showing that this transition involves longer length-and/or time-scales than probed in our experiment.

FIG. 1 .

 1 FIG.1. SAXS scattering pattern for the sample with φ = 2.1 vol%; the lamellar director is roughly horizontal. The feature at the center of the image is the first Bragg sheet. The shadow of the beamstop is visible at the left edge. The averaging was done within the range delimited by the two dashed lines, on circular arcs corresponding to a given scattering vector q. One arc is shown as solid line, for q = 14.5 10 -3 Å-1 .

Figure 3 .

 3 Figure 3. The diffusion constant is shown for all φ values in Figure 4.

FIG. 2 .

 2 FIG.2. Normalized intensity correlation function g 2 (q, t) for the sample with φ = 2.1 vol% (see Figure1), shifted vertically in steps of 0.3. The different symbols correspond to different values of the scattering vector q. The solid lines are fits to Equation (1).

FIG. 3 .

 3 FIG.3. Structure factor S(q) and diffusion constant D(q) for the sample with φ = 2.1 vol%. D ∞

5 FIG. 4 .

 54 FIG.4. Scale-dependent diffusion constant D(q) for different φ values. The "dips" visible in the curves with φ = 0.8 and 1.1 vol% (and, to a lesser degree, for φ = 1.4 vol%) occur at the position of the peak in the static structure factor S(q).

5 FIG. 5 .

 55 FIG.5. Structure factor S(q) for different concentrations φ of the lamellar phase (diamonds).

FIG. 6 .

 6 FIG. 6. Top: dilution law. The smectic repeat distance d is plotted against the volume fraction φ in log-log representation. The solid line represents the d ∼ φ -1 variation expected for a onedimensional system. The resulting platelet thickness, 7.4 ± 1.2 Å, is clearly below the 10 Å value

FIG. 8 .

 8 FIG.8. High-q value of the diffusion constant, D ∞ , as a function of the volume fraction φ of platelets in a 50:50 glycerol/water mixture at 253 K. The phase diagram is also indicated: the (fluid or gel-like) lamellar phase at high concentration and an isotropic (I) phase at very low concentration, with a rather wide coexistence range.

Acknowledgements

We are deeply indebted to Patrick Batail, Franck Camerel, and Jean-Christophe Gabriel for providing us with the samples of H 3 Sb 3 P 2 O 14 colloidal suspensions and for helpful discussions. The ESRF is acknowledged for the provision of beam time (experiment SC-2690).

Appendix: Coupled hydrodynamics

In this Appendix we detail the calculations of the hydrodynamic function for the harmonic model (3), with or without the coupling term (8).

The boundary conditions specified in section III B impose the eigenvector basis:

such that -π/d < q n < π/d (restriction to the first Brillouin zone). The normal modes (phonons) are given by f n (x l ) = exp(iq n dl) = exp(2iπln/N) and the individual displacements are expressed as:

The amplitude coefficients obey:

Normal mode expansion of Equation (3a) with the additional dissipative term (8) yields the relaxation rate of the phonons:

where we remind that β = 2ζ/(mγ) is the ratio of the collective dissipation to the individual particle dissipation.

The time-dependent correlation function is then obtained by simple substitution in (4):

and the first cumulant is obtained according to (5), yielding for H(q):

where the coefficients C(p, β) = 1

N n cos(qndp)

1+β [1-cos(qnd)] can be reduced (e.g. by going to the continuum limit and performing a complex integral over the unit circle) to: C(p, β) = z p √ 2β+1 , finally yielding equation ( 9) above. For the uncoupled case β = 0 this result is greatly simplified, since the sum over the modes in Equation ( 9) becomes: 1 N n cos(q n dp) = δ p,0 and thus H(q, β = 0) = 1, leading to the second equality in Equation (5).

The strongly coupled form lim β→∞ H(q) = S(q) √ 2β + 1 is obtained by noting that in this limit z approaches 1 from below, so that the damping factor exp -( q 2 σ 2 2 -ln z)p is dominated by the first term in the exponent. One can therefore drop the z p factor in Eq. (A.5) and retrieve an obvious expansion of the static structure factor S(q). Of course, for this to hold the scattering vector must also be above a certain value, otherwise the first term in the exponent might be too small. We checked that for reasonable numerical values the condition is fulfilled for all accessible values of q.