The TouchBox: open-source haptic device for finger-based interaction

S P * M F † I C M S T , Z U A , Z , S
The TouchBox is a low-cost DIY human-computer interface providing fast, wide-band and high-dynamics vibrotactile feedback, tracking the position of up to two finger-pads in contact with its top surface, and measuring their contact areas as well as the applied normal and lateral ( D) forces. Finger-pressing is a common gesture while interacting with everyday objects, including digital devices. Also, while one or more fingers are in contact with an object (as the result of finger-pressing), another common gesture is that of pushing or sliding the finger(s) over the touched surface, giving rise to shear strain. The proposed interface addresses a simplified version of such scenario. Applications range from using the interface as calibrated measurement device to advanced human-machine interaction. The decoupling of normal and lateral forces represented the main mechanical challenge, however this was an important prerequisite as commercial D force sensors with comparable performance usually cost around -USD, while our low-cost solution uses separate unidirectional MLCs currently priced about USD each. To achieve that, a key step was to rely on precise CNC-milled aluminum components, interconnected by various elements affording controlled deformation of the resulting structure. Since the MLC sensors are made out of single pieces of aluminum with screw holes at both ends, they were used also as building components: The complete structure is resting on the larger MLC measuring normal force, while the four smaller MLCs sensing lateral forces provide support to the sides. Four spring steel wires hold the normal force plate in place. While the wires can withhold vertical forces, they afford horizontal sliding of the force plate due to lateral forces. Attached to each of the four smaller MLCs is a side plate, each using two headless screws that minimize the gap to the force plate while avoiding direct contact with it. In this way, the side MLCs are only engaged when lateral force components are oriented towards them. The Plexiglas top panel is suspended over the force plate by means of rubberfoam spacers fixed with double sided tape, so that vibratory energy is not dissipated through the metal structure. A software application collects sensor data via the interface's USB connections, analyzes and visualizes them in a graphical interface. The images taken from the camera are processed to compute the position of up to two finger-pads in x-y coordinates relative to the center of the top panel and their contact areas in mm 2 . Tridimensional force data along the x, y and z axes (see Fig. ) are instead calculated within the application and presented numerically as values in N. The accuracy of contact area and force measurements were tested and calibrated, while the characterization of the actuators' response is planned and will follow the same procedure used for the previous design (see the appendix of [ ]). The measured quantities are represented graphically as various interactive overlays over a square which symbolizes the top panel of the interface. The computed data are also made available externally by the application for other uses.

The

Copyright is held by authors A few examples are currently provided with the interface that generate vibrotactile and auditory feedback controlled by the sensors data, namely: interactive synthesis physics-based models [ ] of crushing, scraping and friction, and multiple sine wave oscillators creating beatings. Auditory and vibrotactile stimuli can be set separately, or conversely a same signal can be routed to both the actuators and external audio equipment such as headphones.

  TouchBox is conceived as an open hardware/software project (CC BY-SA . ): An open-access data repository linked to GitHub stores DIY instructions, mechanical and electronic specifications and schematics, D models, software for analyzing sensors data, and for generating interactive vibrotactile and auditory feedback. A former version of the device -measuring only normal force and contact area of a single finger -was recently used in a psychophysical experiment that tested vibrotactile sensitivity thresholds under active touch conditions, bringing results of high interest for the design of haptic interfaces [ ].

Figure :

 : Figure : The interface prototype with overlays showing a finger, relative normal and lateral forces, and contact area.

A

This research is pursued as part of project HAPTEEV (Haptic technology and evaluation for digital musical interfaces), funded by the Swiss National Science Foundation.


Manon Kania
email: manon.kania@insa-lyon.fr

Mathieu Gautier
email: mathieu.gautier@insa-lyon.fr

Anne Imig

Philippe Michel

Rémy Gourdon
email: remy.gourdon@insa-lyon.fr

Comparative characterization of surface sludge deposits from fourteen French vertical flow constructed wetlands sewage treatment plants using biological, chemical and thermal indices

Keywords: Sludge deposits, Constructed wetland, indices, Characterization, Organic matter, Ageing

Highlights

 Particulate organic matter (POM) in sludge deposits (SD) exhibits specific properties.

 Young deposits (≤1 y) and mature de-posits (>3 y) identified as 2 distinct groups.

 Additional pretreatments affected young-age sludge deposits characteristics.

 State of maturation of POM in SD inter-mediate between peat and wet meadows

Introduction

French Vertical Flow Constructed Wetlands (VFCWs) have become in the recent years in France the sewage treatment system most widely used in rural communities of ≤2000 Population Equivalent [START_REF] Bellin | L'épuration végétalisée s'ouvre à de nouveaux marchés[END_REF]. This rapid development may be explained by the high level of performance of these systems, their good social acceptability and landscape integration, their low energy requirements, easy operation, and low maintenance (Liu et al., 2015a;[START_REF] Semeraro | A constructed treatment wetland as an opportunity to enhance biodiversity and ecosystem services[END_REF][START_REF] Wu | Development of constructed wetlands in performance intensifications for wastewater treatment: a nitrogen and organic matter targeted review[END_REF].

The so-called "French VFCW system" comprises two stages of filters where the first stage is fed with unsettled raw wastewater. This design results in the formation of a sludge layer at the surface of the first filter which has mainly been considered until recently through its hydraulic aspects and its negative consequences with regards to the potential risk of clogging [START_REF] Hua | Influence of clogging and resting processes on flow patterns in vertical flow constructed wetlands[END_REF][START_REF] Knowles | Clogging in subsurface-flow treatment wetlands: Occurrence and contributing factors[END_REF][START_REF] Nivala | Clogging in subsurface-flow treatment wetlands: measurement, modeling and management[END_REF][START_REF] Yang | Study of the spatial and temporal distribution of accumulated solids in an experimental vertical-flow constructed wetland system[END_REF]). Yet, the sludge layer may contribute positively to the treatment efficiency. For example, suspended solids removal, water retention and the biodegradation or retention of pollutants may be improved [START_REF] Chazarenc | Influence of surface layer on hydrology and biology of gravel bed vertical flow constructed wetlands[END_REF][START_REF] Kim | Physical-chemical characterization of sludge and granular materials from a vertical flow constructed wetland for municipal wastewater treatment[END_REF][START_REF] Molle | French vertical flow constructed wetlands: a need of a better understanding of the role of the deposit layer[END_REF]. On the other hand; the retention of most of feed organic matter within the sludge layer may limit the efficiency of subsequent denitrification [START_REF] Lee | Nitrogen removal in constructed wetland systems[END_REF][START_REF] Van Oostrom | Denitrification in constructed wastewater wetlands receiving high concentrations of nitrate[END_REF]. The effective roles of the sludge layer in the system may depend on its structural, compositional and reactional properties which are controlled by numerous factors, such as the composition of the influent wastewater, the operating conditions of the plant, or other local environmental factors [START_REF] Fu | Organic matter transplant improved purification performance of newly built constructed wetlands[END_REF].

Although extensive work has already been published on similar issues in the fields of natural wetlands [START_REF] Grasset | Can Soil Organic Carbon Fractions Be Used as Functional Indicators of Wetlands?[END_REF]Rouwane et al., 2017), anthropic sediment [START_REF] Badin | Répartition et influence de la matière organique et des microorganismes sur l'agrégation et le relargage de polluants dans des sédiments (Distribution and influence of organic matter and microorganisms on the aggregation and release of pollutants in sediments)[END_REF][START_REF] Huot | Formation, fonctionnement et évolution d'un Technosol (Formation, functionning and evolution of a Technosol)[END_REF], activated sludge [START_REF] Alvarenga | Sewage sludge, compost and other representative organic wastes as agricultural soil amendments: Benefits versus limiting factors[END_REF][START_REF] Lachassagne | Devenir de micropolluants présents dans les boues d'épuration, du traitement à l'épandage agricole : Application aux micropolluants métalliques (Cd, Cu) et organiques (médicaments) issus du traitement biologique conventionnel d'effluents urbains ou hospitaliers (Becoming of micropollutants in sewage sludge, from treatment to agricultural application: Application to metallic (Cd, Cu) and organic micropollutants (drugs) resulting from the conventional biological treatment of urban or hospital effluents)[END_REF], or sludge drying reed beds (SDRB) [START_REF] Collard | Comparison of three different wastewater sludge and their respective drying processes: Solar, thermal and reed beds -Impact on organic matter characteristics[END_REF][START_REF] Nielsen | Sludge treatment reed bed facilities--organic load and operation problems[END_REF][START_REF] Nielsen | Sludge quality after 10-20 years of treatment in reed bed systems[END_REF]Nielsen & Larsen, 2016;[START_REF] Nielsen | Sludge treatment and drying reed bed systems in Denmark[END_REF][START_REF] Vincent | Mechanical and hydraulic properties of sludge deposit on sludge drying reed beds (SDRBs): influence of sludge characteristics and loading rates[END_REF], very few studies have focused on VFCWs' surface sludge deposits [START_REF] Bois | Indicateurs de Colmatage de FPR à EV. Etude comparative de 4 stations de traitement après 10 ans de fonctionnement[END_REF][START_REF] Chazarenc | Influence of surface layer on hydrology and biology of gravel bed vertical flow constructed wetlands[END_REF][START_REF] Kim | Physical-chemical characterization of sludge and granular materials from a vertical flow constructed wetland for municipal wastewater treatment[END_REF][START_REF] Kim | Effect of aging on phosphorus speciation in surface deposit of a vertical flow constructed wetland[END_REF]. [START_REF] Molle | French vertical flow constructed wetlands: a need of a better understanding of the role of the deposit layer[END_REF] and [START_REF] Perron | Caractérisation des boues d'épuration municipales[END_REF] reported the influence of the VFCW plant design and operating conditions, such as organic and hydraulic loads or feed and rest periodicity. [START_REF] Kim | Physical-chemical characterization of sludge and granular materials from a vertical flow constructed wetland for municipal wastewater treatment[END_REF], reported that although mainly organic, sludge deposits contained significant amounts of inorganic constituents originating from the local geological background or generated by possible operations of pretreatment of the influent such as phosphate precipitation by FeCl3 addition. [START_REF] Kim | Physical-chemical characterization of sludge and granular materials from a vertical flow constructed wetland for municipal wastewater treatment[END_REF] and Kania et al. (2018Kania et al. ( , 2018a) ) investigated the extent of organic matter evolution within the sludge deposits, such as mineralization. Finally, the authors of this study reported structural properties data of surface sludge deposits in a recent article [START_REF] Kania | Caractérisation des dépôts de surface des filtres plantés de roseaux à écoulement vertical[END_REF]. In the specific context of sludge drying reed bed systems, few studies investigated the mineralization of the sludge and the organic matter stabilization [START_REF] Masciandaro | Sewage sludge and waterworks sludge stabilization in sludge treatment reed bed systems[END_REF][START_REF] Nielsen | Mineralization of hazardous organic compounds in a sludge reed bed and sludge storage[END_REF][START_REF] Nielsen | Stabilisation and mineralisation of sludge in reed bed systems after 10-20 years of operation[END_REF][START_REF] Peruzzi | Stabilization process in reed bed systems for sludge treatment[END_REF][START_REF] Peruzzi | Organic matter stabilization in reed bed systems: Danish and Italian examples[END_REF].

The objective of the present study was to contribute to a better description of the properties of sludge deposits using chemical, biological and thermochemical methods. Samples from fourteen VFCW sewage treatment plants in operation in France were analyzed to assess whether specific typologies would be observed and, if so, identify the dominant factors of influence, especially with regards to the particulate organic matter (POM).

Materials and methods

Sites of investigation

Fourteen French VFCW sewage treatment plants located in the south-eastern part of France, were selected for the study. The treatment plants were all equipped with a preliminary gridding stage set between 3 and 30 mm to remove coarse solids and debris. Four of them were also equipped with a sand trap (Table 1). Three typical types of VFCW process designs were represented in the selected plants:

-Process type I, usually referred to as the "classical French system", where gridded, unsettled raw wastewater was percolated through two successive stages of vertical flow filters.

-Process type II, where an additional stage of aerobic biological trickling filter was used to treat gridded wastewater prior to the two successive stages of vertical flow filters, -Process type III, comprising the same operations as in type II plus the introduction of a FeCl3 solution into the outlet of the aerobic trickling filter in order to precipitate phosphorus prior to infiltration through the two successive stages of filters. These processes were further divided into two sub-categories according to the level of saturation of the first-stage filter as shown in Table 1. Full drainage (unsaturated conditions) was implemented to ensure good aeration in all process type I plants ("classical French system") and in some plants of process types II and III patented as AZOE ® and AZOE-P ® respectively.

Partially saturated conditions were implemented in the other plants of types II and III (patented as AZOE-N ® and AZOE-NP ® ) in the objective to allow aeration of the upper unsaturated zone for nitrification, and anoxic conditions in the lower flooded zone for denitrification. More details were given in a previous article [START_REF] Kania | Caractérisation des dépôts de surface des filtres plantés de roseaux à écoulement vertical[END_REF][START_REF] Kim | Performance evaluation of partially saturated vertical-flow constructed wetland with trickling filter and chemical precipitation for domestic and winery wastewaters treatment[END_REF].

Sampling and preparation of the sludge deposits samples

Sludge deposits were sampled in winter time between January and March 2015 from most wetlands, COR was sampled in November 2015 and RAM and PRE in February 2017 due to their more recent dates of implementation of these wetlands. The annual load of dry solids varied from one constructed wetland to another depending on its nominal design and its actual operating conditions. It was estimated between 20 and 35 kg of dry solids per year and m² of first stage filter cells. At the time of the field sampling operations, the sludge layer had not yet been removed from any of the wetlands and it has not been removed since. The lower part of the layer therefore contained deposits from the very beginning of the plant operation, whereas the upper part contained fresh deposits. The entire depth of the sludge layer was sampled on each site in order to obtain average samples representative of the mean composition of the sludge deposits over the whole period of operation of the VFCW plants.

All samples were taken following the same protocol from the stage filters. A few kilograms of deposits were taken by shoveling the entire depth of the sludge layer from 5 to 10 spots (depending on the filter size) at the surface of the cell under resting period for 0-1 week.

The sampling spots were randomly located at a distance of at least 5 m one from each other and at least 2 m from the feed pipes. The samples taken from each spot were treated on site to extract manually reeds rhizomes, and possible traces of gravels. They were then carefully homogenized by quarting and mixing together with shovels. The representative samples from each plant thereby constituted were then taken to the laboratory within a maximum of 4 h. They were transported to the lab by car placed in a non-heated trunk at a maximal temperature of 10°C due to the winter season. There, an aliquot fraction of each sample was used to determine water content and the organic content of the dry solids. The rest was dried in thin layers at 35°C for one day, finely grinded and stored at 4°C.

Analytical methods

General analytical parameters

Upon reception in the laboratory, a few grams of fresh samples from each plant were dried in triplicates in an oven at 105°C until they reached a constant mass in order to determine their humidity. Dry solids were then burnt in an oven at 550°C for four hours to determine their organic matter contents. The rest of each sample was spread in thin layers of 1-2 cm on a laboratory bench and air-dried for 1 day at 35°C, then grinded down to less than 1 mm to obtain "dry powdered samples" which were stored at 4°C and used for all the other analyses.

Elemental analyses were done by alkaline fusion (LiBO2) followed by acid digestion (HNO3) and subsequent analyses of the solutions by inductively coupled plasma atomic emission spectrometry (ICP-AES) (IRIS Advantage ERS, Thermo Scientific). The following elements were analyzed : silicon (Si), aluminum (Al), iron (Fe), manganese (Mn), magnesium (Mg), calcium (Ca), sodium (Na), potassium (K), titanium (Ti) and phosphorus (P). Trace metals chromium (Cr), copper (Cu), nickel (Ni), zinc (Zn), cadmium (Cd) and lead (Pb) were analyzed using a Sciex Perkin Elmer ELAN 5000a. ICP-mass spectrometer.

Organic carbon contents were determined by burning at 1400°C 1g of dry powdered samples, previously de-carbonated with acid and under high temperature, and by infrared absorption with a Carbon -Sulfur EMIA 320V2 (HORIBA).Total nitrogen contents were measured by burning 1g of powdered samples at 1050°C under a flow of helium and oxygen mixture in microanalyzer designed and developed at UMR 5280 Institute of Analytical Sciences which operates according to the Dumas method.

Thermogravimetric and differential thermal analysis (TGA/DSC)

Thermogravimetric analyses (TGA) were carried out using a METTLER TOLEDO TGA analyzer whose temperature precision was ±0.5°C and microbalance sensitivity less than ±0.1mg. A dry mass of ±20 mg of each sample was heated from 35°C to 900°C at a rate of 10°C/min, under a flow of air of 50 mL per minute. The analyses were done in duplicates and blank experiments were carried under the same conditions as the assays but without any sample.

The mass loss due to thermal degradation of the sample was recorded as a function of the temperature. Heat fluxes were also recorded during the process by differential scanning calorimetry (DSC).

RTGA index was calculated from experimental data as the ratio of the mass loss recorded between 200 and 400°C (m1) to that recorded between 400 and 600°C (m2), as proposed by several authors [START_REF] Stevenson | Humus chemistry: genesis, composition, reactions[END_REF][START_REF] Fernandez | Use of thermal analysis techniques (TG-DSC) for the characterization of diverse organic municipal waste streams to predict biological stability prior to land application[END_REF]. In each given range of temperature, the unit mass heat release, also called "energy density" E (in J.mg -1 ), was also calculated from the ratio between the surface area of the corresponding peak in DSC profiles over the mass loss recorded in the same range of temperatures (E 1 and E 2 ). The R DSC index was finally calculated as the ratio of E2/E1 [START_REF] Achour | Caractérisation de la Matière Organique dans les Ordures Ménagères -Recherche d'Indicateurs de Stabilité (Characterization of Organic Matter in Household Rubbish -Search for Stability Indicators)[END_REF].

Determination of humification rate

The contents in humic-like and fulvic-like substances (referred to as HULIS and FULIS, respectively) were analyzed to evaluate the humification degree of the organic matter in the sludge deposits [START_REF] Albrecht | Co-Compostage de Boues de station d'épuration et de Déchets Verts (Co-Compostage of sludge from wastewater treatment plant and waste Green)[END_REF]. The protocol for the HULIS and FULIS extraction was adapted from [START_REF] Serra-Wittling | Stability and maturity of a green waste and biowaste compost assessed on the basis of a molecular study using spectroscopy, thermal analysis, thermodesorption and thermochemolysis[END_REF]. HULIS and FULIS were extracted by suspending for 2 h under constant agitation 1 g of powdered samples in 50 mL of a 0.1 M sodium hydroxide aqueous solution. The suspensions were then centrifuged at 10 000 g for 15 min and vacuum filtered at 0.45 µm using Sartorius cellulose acetate filters. The solutions, containing HULIS and FULIS fractions were analyzed for total organic carbon (TOC) using a Shimadzu TOC-L analyzer, then acidified to pH 1.5 with 1 M sulfuric acid aqueous solution and left at rest overnight at +4°C to allow precipitation of HULIS fraction. FULIS fraction was collected in solution by centrifugation at 6000 g for 15 min followed by filtration at 0.45µm. The solution was analyzed for TOC using the same protocol as for the HULIS + FULIS extracts.

The proportion of HULIS + FULIS in the organic matter was taken as an indicator of the degree of humification of the sludge organic matter and calculated as follows:

Humification rate = 100 x [HULIS]+[ FULIS] TOC (1)

Biodegradation under aerobic conditions

Respirometric assays were conducted to evaluate the biodegradability of the sludge deposits considered here as an indicator of stability of their organic matter. A given mass of each powdered sample, calculated to contain 75.5 mg of organic matter, was introduced into 500 mL BOD flasks containing 10 mL of a mineral medium made of an aqueous solution of 28.25 g.L -1 KH2PO4 ;146.08 g.L -1 K2HPO4; ;3.66 g.L -1 CaCl2 ; 28.64 g.L -1 NH4Cl ; 3.06 g.L -1 MgSO4, 0.7 g.L -1 FeSO4 and 0.4 g.L -1 ZnSO4 (Liu et al., 2015b). The flasks were then inoculated with 10 mL of a seed suspension prepared from biological sludge obtained from a municipal wastewater activated sludge treatment plant located in Villeurbanne (France) and cultivated in the laboratory prior to the beginning of the assays. They were finally spiked with ca. 0.2 mL of a 5 g.L -1 allylthio-urea aqueous solution used as a nitrification inhibitor, hermetically closed by manometric caps to monitor pressure variations, and incubated at 30±1°C in the dark. Carbon dioxide formed from biodegradation was trapped by sodium hydroxide pellets placed in a specific compartment in the headspace of the test flasks. The recorded pressure therefore decreased proportionally to the oxygen consumed and was subsequently converted into BOD values expressed in mg O2 per g of sludge TOC.

Biodegradability was finally calculated according to eqn. ( 2) from the ratio between the cumulated consumption of O2 (g) over 14 days of incubation (BOD14) and the theoretical oxygen demand (ThBOD) of 2667 mg O2 per g of TOC (corresponding to a consumption of 32 g O2 for a complete bio-oxidation of 12 g C based on a stoichiometry of 1 mole of O2 for 1 mole of C assuming that carbon in the sludge organic constituents was at an average degree of oxidation of zero).

Biodegradability (%) =

BOD 14 (gO 2 per g TOC) 2. 667x 100

(2)

Results and discussion

Organic content and elemental composition

Organic content

Table 2 shows the average contents in organic matter and organic carbon analyzed in the surface deposits samples. Organic matter (OM) contents ranged between 50.3 and 86.2% of total dry weight (% w/ w). Regardless of their wetland of origin, the surface deposits were therefore in all cases mostly organic material. Fig. 1 illustrates the plot of organic matter contents of the surface deposits vs. the number of years of operation of the VFCW sewage treatment plants of origin. The graph showed that deposits from young systems with newly established thin sludge layers of about 1-3 cm contained larger concentrations of organic matter than those from older systems of ≥3 years with a sludge layer of about 10-25 cm thick. This observation suggested that hydrolysis and / or mineralization of POM progressively reduced the organic matter content of the deposits within the sludge layer. However, although COR, RAM and PRE samples were all taken from young plants (1 year and less), COR sample was less organic than those from RAM and PRE.

Time was therefore not the only factor of influence. Other analytical results discussed below have to be considered to explain this observation.

Elemental compositions

The contents in major elements were quite similar in all the samples, except for MAT which exhibited high Mg, Na, K and Mn concentrations (Table 2), probably due to atypical local conditions as discussed in the sections below. Phosphorus (P) contents in the sludge deposits from plants with FeCl3 injection (2.0; 1.6; 2.5 and 2.0% w/w in VER, ORB, COR and MAT, respectively), were two-fold higher than in the plants without any additional treatment where P concentrations ranged between 0.5 and 1.2% w/w of dry matter. This result confirmed the good efficiency of FeCl3 injection to precipitate phosphorus. The samples from plants with a trickling filter also revealed slightly higher P contents (1.2%; 1.9% and 1.5% in NEU, BAY and MIN samples respectively) than those with no additional treatment at all. This observation could be attributed to the assimilation of phosphorus within the biofilm in the trickling filter, which after detachment would be subsequently retained within the surface sludge layer and induce a relative increase in P contents [START_REF] Andersson | Characterization of bacterial biofilms for wastewater treatment[END_REF]. Total Nitrogen (TN) contents ranged between 1.5 and 4.8% w/w of dry mass in all the samples. These results were close to those reported in sludges from lagoons sewage treatment plants, but lower than those found in suspended solids from activated sludge systems [START_REF] Vincent | Les lits de séchage de filtres plantés de roseaux pour le traitement de boues activées et les matières de vidange (Sludge drying reed beds for treatment of sludge activated and septage)[END_REF]. COR sample exhibited a TN concentration of 4.1% w/w of dry mass, which was about double of the concentrations measured in the samples of similar age RAM and PRE (1.5 and 2.1% w/w of dry mass, respectively). This observation may be explained by the interactions (complexation, coagulation, precipitation) of soluble N species with FeCl3 used in COR plant. Other authors (Boucher et al., 1999;[START_REF] Hamdani | Caractérisation et traitement par coagulation-décantation d'un effluent de laiterie[END_REF][START_REF] Poon | The use of ferric chloride and anionic polymer in the chemically assisted primary sedimentation process[END_REF] reported the effect of FeCl3 on the removal of TN from dairy effluents. This effect was observable here in young-age plants but not in older plants (Table 2),

showing that other ageing effects became predominant over the years of operation.

Table 2 showed that the contents in trace metals were in the same order of magnitude in all the samples. The accumulation of trace metals and other micropollutants is a potential environmental concern when considering land application or nutrients recovery from the dredged sludge [START_REF] Fang | Comparative characterization of sewage sludge compost and soil: Heavy metal leaching characteristics[END_REF][START_REF] Gautier | Investigation of heavy metals leaching from VFCWs sludge deposit to evaluate potential risks associated to land application[END_REF][START_REF] Kim | pH and Eh effects on phosphorus fate in constructed wetland's sludge surface deposit[END_REF][START_REF] Peccia | We Should Expect More out of Our Sewage Sludge[END_REF][START_REF] Peysson | Determination of 136 pharmaceuticals and hormones in sewage sludge using quick, easy, cheap, effective, rugged and safe extraction followed by analysis with liquid chromatography-time-of-flight-mass spectrometry[END_REF]. Total contents in Cd, Cr, Ni, Pb, Cu and Zn in the sludge deposits analyzed here ranged in most cases between ca. 0.9 and 1.6, 35 and 74, 18 and 37, 32 and 69, 196 and 425, 430 and 953 mg.kg DM -1 respectively. These concentrations were in the ranges classically reported in activated sludge, respectively 1-10 for Cd, 10-100 for Cr, Ni and Pb and 100-1000 mg.kg DM -1 for Zn and Cu [START_REF] Lachassagne | Devenir de micropolluants présents dans les boues d'épuration, du traitement à l'épandage agricole : Application aux micropolluants métalliques (Cd, Cu) et organiques (médicaments) issus du traitement biologique conventionnel d'effluents urbains ou hospitaliers (Becoming of micropollutants in sewage sludge, from treatment to agricultural application: Application to metallic (Cd, Cu) and organic micropollutants (drugs) resulting from the conventional biological treatment of urban or hospital effluents)[END_REF]. A few samples revealed however relatively high contents in some elements. Sludge deposits from SABAG plant exhibited a high copper concentration (1162 mg.kg DM -1 ), samples from VER a high zinc concentration (1218.5 mg.kg DM -1 ) and samples from BAY a high lead contents (121.5 mg.kg DM -1 ). These observations may be related to the specific nature of anthropic discharges (domestic, industrial or both) to the sewage water in the water catchments of the respective constructed wetlands [START_REF] Chipasa | Accumulation and fate of selected heavy metals in a biological wastewater treatment system[END_REF][START_REF] Ustun | Occurrence and removal of metals in urban wastewater treatment plants[END_REF][START_REF] Vriens | Quantification of Element Fluxes in Wastewaters: A Nationwide Survey in Switzerland[END_REF]. Table 2 showed the concentrations of trace metals analyzed in the 14 sludge deposit samples. As an informative reference, these concentrations may be compared to the upper limit concentrations of a series of trace metals VFCW plants, suggesting a progressive accumulation of trace metals in the sludge deposits over the years of operation, which may be explained in part by their sorption onto POM and/or mineral fractions of the sludge deposits. The implementation of additional treatment did not induce observable differences of total metal concentration in the samples from mature plants.

The average concentrations were 1201±386 mg.kg DM -1 in samples from classical VFCW (no additional treatments), 1115±237 mg.kg DM -1 in samples from AZOE ® treatment plants, and 1529±450 mg.kg DM -1 in AZOE-NP ® treatment plants. Regarding sludge deposits from the young plants, the effect of FeCl3 addition (co-precipitation with and/or sorption onto flocculated mineral and organic phases) may explain here again the observed difference between the samples from COR (ca 1711 mg.kg DM -1 ) and RAM and PRE (721 and 833 mg.kg DM -1 respectively) as already discussed by El Samrani et al. ( 2008) and [START_REF] Fu | Removal of heavy metal ions from wastewaters: a review[END_REF]. The situation in the sludge layer in the VFCW seems in many aspects comparable with the sludge layer building up in SDRB systems. In sludges from reed bed systems, very similar contents were indeed reported for TOC and TN [START_REF] Masciandaro | Sewage sludge and waterworks sludge stabilization in sludge treatment reed bed systems[END_REF][START_REF] Nielsen | Stabilisation and mineralisation of sludge in reed bed systems after 10-20 years of operation[END_REF][START_REF] Peruzzi | Organic matter stabilization in reed bed systems: Danish and Italian examples[END_REF], and trace metals [START_REF] Caicedo | Comparison of heavy metal content in two sludge drying reed beds of different age[END_REF][START_REF] Matamoros | Musk fragrances, DEHP and heavy metals in a 20 years old sludge treatment reed bed system[END_REF][START_REF] Nielsen | Sludge quality after 10-20 years of treatment in reed bed systems[END_REF].

Biodegradability of sludge deposits' organic matter

Fig. 2 shows the results of cumulated O2 consumption measured over 14 days of incubation at 30°C of the sludge deposits samples. It can be seen that the samples from the mature plants exhibited respiration levels ranging between 200 and 500 mgO2.TOC -1 in 14 days of incubation, which may be considered as low values according to the scale used for compost [START_REF] Wood | Review of Stability Testing[END_REF] and with respect to the theoretical maximum of 2667 mg O2 per g of TOC.

These results therefore revealed the poor biodegradability and/or bioavailability of the organic matter in these samples. The samples from the youngest three plants PRE, RAM and COR exhibited drastically higher respiration levels (1729, 1387 and 2577 mg O2 per g of TOC respectively), indicating that their organic matter was readily biodegradable and bioavailable in the conditions of the assays. [START_REF] Vincent | Les lits de séchage de filtres plantés de roseaux pour le traitement de boues activées et les matières de vidange (Sludge drying reed beds for treatment of sludge activated and septage)[END_REF] reported that fresh organic matter from raw wastewater was responsible for the major part of O2 consumption in "classical French" VFCWs. Results obtained from the analysis of sludge deposits from VFCW sewage treatment plants of increasing ages therefore showed that both the organic contents of the sludge deposits and the biodegradability of their organic matter decreased over time, suggesting the occurrence of both mineralization and humification processes. Similar phenomena were reported in other environmental media [START_REF] Albrecht | Co-Compostage de Boues de station d'épuration et de Déchets Verts (Co-Compostage of sludge from wastewater treatment plant and waste Green)[END_REF][START_REF] Francou | Stabilisation de la matière organique au cours du compostage de déchets urbains (Stabilization of organic matter in the urban waste composting)[END_REF].

Indicators of maturation of particulate organic matter

TOC/TN ratio

TOC/TN ratio is a parameter classically used to monitor the progress of biodegradation in the processes of organic waste composting [START_REF] Iglesias Jiménez | Evaluation of city refuse compost maturity: a review Biological wastes[END_REF][START_REF] El Ouaqoudi | Evaluation of lignocelullose compost stability and maturity using spectroscopic (FTIR) and thermal (TGA/TDA) analysis[END_REF]. The ratio normally decreases over time since biodegradation releases carbon (in the form of CO2) whereas nitrogen remains within the waste material during composting operations. Compost is considered mature when the ratio falls below 15 [START_REF] El Ouaqoudi | Evaluation of lignocelullose compost stability and maturity using spectroscopic (FTIR) and thermal (TGA/TDA) analysis[END_REF]. Grimaud (1996) also reported that secondary sludge from conventional domestic wastewater treatment would exhibit a TOC/TN ratio between 6 and 12, whereas primary sludge would be above 12.

Based on these studies, the ratio was tentatively used here as an indicator of the evolution of organic matter in the sludge deposits. Table 2 showed that low values between 5.5 and 7.2 were measured in sludge deposits taken from VFCW plants of three years and more, whereas samples from the young plants RAM and PRE exhibited much higher values of 24.7 and 15.6 respectively. COR sample however, although also originating from a young plant, revealed a low TOC/TN value of 5.9. The ratio TOC/TN ratio was however considered nonrelevant as an indicator of POM evolution in VFCW. As a matter of fact, unlike composting where N is not exported from waste material, the conditions of evolution of surface sludge deposits in VFCW allowed both the release of N in the form of soluble and leachable species and/or the accumulation of N species from wastewater via complexation and/or adsorption.

Other authors [START_REF] Brewer | Maturity and Stability Evaluation of Composted Yard Trimmings[END_REF][START_REF] Huang | Transformation of organic matter during co-composting of pig manure with sawdust[END_REF]) also reported that TOC/TN ratio was not sufficient or not always relevant to discriminate the level of maturity of OM. Other indicators widely used in published studies, such as the humification rate, were therefore considered.

Humification rate

The contents in HULIS and FULIS fractions analyzed in the different samples, expressed as % w/w of organic carbon, and the calculated humification rate are shown in Fig. 3. The humification rate of almost all the samples from mature plants were found to be relatively high and very similar, ranging from 27.9±0.5 to 33.8±0.9% w/w of sludge TOC. This indicator was therefore consistent with the results discussed above relative to the biostability of their organic matter. [START_REF] Masciandaro | Sewage sludge and waterworks sludge stabilization in sludge treatment reed bed systems[END_REF] obtained very similar values (humification rate between 23.7 and 30.4%) in sludge treatment reed bed systems and concluded to the effectiveness in stabilizing sludge organic matter. One sample however, also collected from a mature plant, MAT, exhibited on this indicator a much higher value than all other samples, as high as ca. 46.4±2.8%, suggesting that the organic matter in this sample was particularly humified, although it did not exhibit the smallest biodegradability in Fig. 2. The particular properties of this sample are discussed below. Samples from the young plants RAM and PRE showed lower values of humification rates, 12.5±0.6 and 10.3±2.9% respectively, which were relatively close to level reported for activated sludge (16.5% in [START_REF] Riffaldi | Humic Substances in Sewage Sludges[END_REF] and 10.9% in [START_REF] Li | Compositional and functional features of humic acid-like fractions from vermicomposting of sewage sludge and cow dung[END_REF]). COR sample however exhibited a humification rate of ca 20.1±1.6% w/w of TOC which was comprised between the levels measured in the other two young plants RAM and PRE and in mature plants. This observation was attributed to the transformation of the organic matter through the trickling filter and the detachment of biofilm fragments which may favor the formation of humic-like substances as reported by other authors [START_REF] Maeng | Reduction of organic matter in drinking water using a hybrid system combined with a rock biofilter and membrane in developing countries[END_REF][START_REF] Zhao | Removal and transformation of organic matters in domestic wastewater during lab-scale chemically enhanced primary treatment and a trickling filter treatment[END_REF].

Thermal analyses

TGA and DSC profiles obtained from thermal analyses of the sludge deposits samples are shown in Fig. 4 in the range of temperatures between 200° and 600°C known to induce thermal degradation of organic compounds of natural origins [START_REF] Torres-Climent | Chemical, Thermal and Spectroscopic Methods to Assess Biodegradation of Winery-Distillery Wastes during Composting[END_REF]. The first part of the graphs, from 200°C to 400°C, was considered to correspond to the degradation of relatively labile organic compounds, such as hemicelluloses, cellulose and some aliphatic compounds [START_REF] Chen | Thermal characteristics and kinetics of refining and chemicals wastewater, lignite and their blends during combustion[END_REF][START_REF] Gomez | An evaluation of stability by thermogravimetric analysis of digestate obtained from different biowastes[END_REF][START_REF] Serra-Wittling | Stability and maturity of a green waste and biowaste compost assessed on the basis of a molecular study using spectroscopy, thermal analysis, thermodesorption and thermochemolysis[END_REF][START_REF] El Ouaqoudi | Evaluation of lignocelullose compost stability and maturity using spectroscopic (FTIR) and thermal (TGA/TDA) analysis[END_REF]. The second part between 400 and 600°C was considered associated to more complex and stable organic compounds such as lignin and other high molecular weight poly-aromatic structures such as humic and fulvic substances [START_REF] Chen | Thermal characteristics and kinetics of refining and chemicals wastewater, lignite and their blends during combustion[END_REF][START_REF] Cuypers | Amorphous and condensed organic matter domains: the effect of persulfate oxidation on the composition of soil/sediment organic matter[END_REF][START_REF] Melis | Thermal analysis for the evaluation of the organic matter evolution during municipal solid waste aerobic composting process[END_REF][START_REF] El Ouaqoudi | Evaluation of lignocelullose compost stability and maturity using spectroscopic (FTIR) and thermal (TGA/TDA) analysis[END_REF]. For all samples, except RAM and PRE, TGA profiles showed that around 1/4 to 1/3 of the initial dry mass (26.2 to 35.1% w/w of dry mass) was degraded between 200 and 400°C.

The mass losses between 400 and 600°C were in the same order of magnitude as from 200 to 400°C, ranging between 19.8 and 32.4% w/w of dry mass. RAM and PRE samples, taken from the youngest VFCW plants, exhibited different patterns, with about 2/3 of initial dry mass degraded between 200 and 400°C (66.4 and 66.1% w/w of dry mass respectively), and only around 16% w/w of initial dry mass degraded between 400 and 600°C. Similar results were reported for activated sludge by [START_REF] Magdziarz | Thermogravimetric study of biomass, sewage sludge and coal combustion[END_REF], with around 50% degradation of initial dry mass between 200 and 400°C and 20% w/w of dry mass above 400°C. MAT sample exhibited an atypical profile. Although between 200 and 400°C the mass loss was 32.0% w/w of dry mass as in almost all samples, above 400°C the mass loss was the lowest of all samples (11.1%).

The thermal degradation processes were logically exothermic since the analyses were made in the presence of air (DSC profiles in Fig. 4). The heat released per unit mass of organic matter oxidized in the assays was related to the energy of inter-atomic bounds and the degree of oxidation of carbon in the organic constituents. It can be observed that the ratio E 1 between 200 and 400°C was much smaller than E2 between 400 and 600°C (Table 3). The thermal degradation was therefore much less exothermic per unit mass of organic constituents below 400°C than above 400°C. It was therefore concluded that the POM constituents degraded below 400°C were composed of molecules such as carbohydrates with relatively weak inter-atomic bounds such as C-O, C-C and C-N (activation energy of 350-412 kJ.mol -1 ) [START_REF] Leinweber | Differential thermal analysis, thermogravimetry and insource pyrolysis-mass spectrometry studies on the formation of soil organic matter[END_REF], and a relatively high degree of oxidation of carbon indicative of the oxygenation index of the molecules [START_REF] Barré | The energetic and chemical signatures of persistent soil organic matter[END_REF][START_REF] Soucémarianadin | Is Rock-Eval 6 thermal analysis a good indicator of soil organic carbon lability? -A method-comparison study in forest soils[END_REF]. The organic constituents degraded above 400°C were considered to contain more C=C bounds of higher energy (activation energy of 520-840 kJ.mol -1 ) such as in humic-like substances, lignin and lignocellulosic complexes [START_REF] Magdziarz | Thermogravimetric study of biomass, sewage sludge and coal combustion[END_REF], with carbon at a lower degree of oxidation, such as in (poly)-aromatic compounds [START_REF] Leinweber | Differential thermal analysis, thermogravimetry and insource pyrolysis-mass spectrometry studies on the formation of soil organic matter[END_REF]. The thermal indices RTGA and RDSC were calculated and the results given in Table 3. For all samples except those from the youngest plants PRE and RAM, the RTGA index was between 0.8 and 1.1, indicating the presence of relatively stable organic constituents. For comparison, compost is considered mature for an RTGA index above 0.8 [START_REF] Serra-Wittling | Stability and maturity of a green waste and biowaste compost assessed on the basis of a molecular study using spectroscopy, thermal analysis, thermodesorption and thermochemolysis[END_REF][START_REF] El Ouaqoudi | Evaluation of lignocelullose compost stability and maturity using spectroscopic (FTIR) and thermal (TGA/TDA) analysis[END_REF]. RTGA indexes of PRE and RAM samples were 0.2 and 0.3 respectively, confirming the previous discussions related to the predominance of labile organic compounds in these samples and the lower humification rate. RDSC values were found between 1.6 and 3.2. Samples from the young plants RAM and PRE showed a higher index than those from the mature plants, confirming their lower stability [START_REF] Achour | Caractérisation de la Matière Organique dans les Ordures Ménagères -Recherche d'Indicateurs de Stabilité (Characterization of Organic Matter in Household Rubbish -Search for Stability Indicators)[END_REF].

COR sample exhibited again a particular profile, with RTGA and RDSC indexes closer to the group of mature plants than young plants. These results therefore confirmed the effects of the additional treatments, notably the trickling filter, on the characteristics of sludge deposits in young-age VFCW plants.

Finally, the TGA/DSC analysis allowed to highlight the particular behavior of MAT sample which did not feature a clear bimodal pattern but showed instead a peak around 320°C followed by a shoulder at 445°C. The particularities of MAT sample are further discussed below.

3.5. Multi-parameter cross-analysis of sludge deposits characteristics

Inter-comparison of the 14 sludge deposits

A correlation analysis was conducted on the different parameters of OM maturity and stability used in this study. Results gathered in Table 4 showed that OM content was relatively correlated with aerobic biodegradation and RDSC, and anti-correlated with the humification rate.

A high organic content was typical of young-age VFCW plants, where fresh organic matter was readily biodegradable and poorly humified. RDSC index was anti-correlated with the humification rate and RTGA was not significantly correlated to any other indexes. To allow a relevant and discriminant graphical representation of the different SD samples with respect to their characteristics, the most discriminant parameter within each type of analyses was selected (namely biological, thermochemical, and chemical analyses). Based on the results shown in the previous sections, biodegradability and humification were selected as the biological and chemical parameters. Based on the statistical correlation assessment, RTGA index was selected as the thermochemical parameter because it did not show significant correlations with the first two parameters according to Table 4, and was therefore considered to provide independent additional information. The corresponding 3-D representation shown in Fig. 5 clearly revealed distinct typologies of samples. A first group was identified with the samples from the young-age VFCW plants RAM, PRE and COR. COR sample was positioned slightly apart from RAM and PRE samples, underlining the influence of the additional treatments on the characteristics of the sludge deposits within the young-age of the VFCW plants. A second group was clearly formed with the samples from the mature plants. This group was very homogeneous, revealing the very strong similarities in the characteristics of the sludge deposits of the VFCW plants of three years of operation and more, regardless of the implementation or not of additional treatments and other local conditions. Sludge samples from mature VFCW plants exhibited a humification rate between 20 and 40%, an OM biodegradation rate between 0 and 35%, and a RTGA between 0.75 and 1. Finally, MAT sample was excluded from both previous groups in the graphical representation of Fig. 5. Its overall characteristics were therefore distinct from either groups of young-age and mature VFCW plants. MAT plant was actually an atypical VFCW plant as compared to the others. For example, a predominance of baby's wipes was observed at the surface of the first filter when the sludge deposits were sampled. These wipes are made of about 60% polyester and 40% cellulose for most of them and affect the composition of the sludge deposits. Fig. 5 therefore allowed to clearly identify a third group of samples whose characteristics revealed a particular situation.

Comparison to other samples at different levels of POM evolution

Although some debates still remain on the subject, it is usually considered in the literature that ageing of natural particulate organic matter (POM) occurs via (i) the hydrolysis and mineralization of at least part of the most readily biodegradable constituents and (ii) the rearrangement of the remaining constituents into complex supra-structures usually called humus and usually considered as made of humic-like and fulvic-like substances. These phenomena are considered to increase (i) the proportion of HULIS+FULIS within the particulate organic matter and (ii) the proportion of HULIS vs. FULVIC within humus fraction.

To assess the extents to which these processes occurred in the sludge deposits, Fig. 6 illustrates the plot of the ratio FULIS concentration/HULIS concentration of the samples as a function of the overall proportion of HULIS+FULIS within the total organic carbon of the samples. Data reported by [START_REF] Grasset | Can Soil Organic Carbon Fractions Be Used as Functional Indicators of Wetlands?[END_REF] for different types of samples from natural systems were included in Fig. 6, along with other data calculated for activated sludges using the results of other studies [START_REF] Li | Compositional and functional features of humic acid-like fractions from vermicomposting of sewage sludge and cow dung[END_REF][START_REF] Riffaldi | Humic Substances in Sewage Sludges[END_REF]. Fig. 6 illustrated the maturation pattern of natural organic matter from the early stages of evolution (fresh OM) in the top left part to the mature state in the bottom right of the graph. The VFCW sludge deposits samples appeared as two major groups in Fig. 6. The samples from young VFCW plants (RAM, PRE and to a lesser extent COR) constituted a first group situated at an earlier stage of maturation from the second group made of the samples from the mature VFCW plants. COR sample was actually positioned between the young and mature groups, revealing the effects of the additional treatments discussed previously. Altogether, the spots showing the VFCW sludge deposits exhibited a specific state of evolution of the organic matter as compared to the other samples. Their position in the graph revealed a specific typology of samples, despite the differences existing amongst them. Most sludge deposits samples were positioned between the spot of digested sludge samples and peatlands and the spot of wet meadows. On the contrary, RAM, PRE and COR samples were positioned close to the area of peatlands and digested sludge samples.

Conclusions

Biological, chemical and thermochemical methods were applied to characterize surface sludge deposits sampled from 14 French VFCW sewage treatment plants. Results clearly revealed the existences of two groups of samples. The dominant factor of distinction between the two groups was the age of the VFCW plant of origin. The first group corresponded to sludge deposits taken from VFCW plants which were still in their first year of operation at the time the samples were collected. These samples exhibited a high content of organic matter of high biodegradability, high thermal lability, and low humification. Although the number of samples in this category (3) was low to permit a straightforward generalization, it was observed that the implementation of additional treatments in one of the three VFCW plants of this group modified the characteristics of its sludge deposits in terms of humification rate and thermal behavior. This observation showed that the presence of additional treatments modified the characteristics of the sludge deposits into a signature close to that of samples from mature systems. The second group gathered samples from 10 distinct French VFCW plants in a remarkably homogenous manner. The plants of origin were three years of age or more. Their sludge deposits were characterized by a lower organic content, lower biodegradability and thermal lability, and higher humification than those of the first group. This common set of properties was favorable to a relative accumulation of trace metals, which were analyzed at higher concentrations in this group of samples. However, the concentrations measured were found compatible with a land application practice should the sludge have to be dredged out from the filters for maintenance.

One last sample, although originating from a mature VFCW plant, revealed quite different properties as compared to the other ten samples of the group. This observation could not be explained by the implementation of additional treatments. Rather, it revealed that the sample was identified as "atypical", suggesting a particular situation in the VFCW plant of origin.

This study therefore showed that the organic matter captured within the surface sludge layer in French VFCW sewage treatment plants was submitted to a relatively rapid evolution process over the very first year of operation of the plants. After one to three years of operation, a certain equilibrium was established, resulting from the balance between the flow of fresh POM continuously brought to the surface and the evolution of the rest of the deposits within the depth of the layer. This equilibrium state was found to be remarkably similar in all the plants of 3 years and more, regardless of the implementation of additional treatments, local conditions, and time. Surface sludge deposits therefore appeared as integrative tracers of the evolution of the system.

Fig. 1 .

 1 Fig. 1. Organic matter contents of the surface deposits versus operating time of their VFCW plants of origin. Blue dots: Classical French system with no additional treatment. Green dots: additional treatment on trickling filter only. Red dots: trickling filter and FeCl 3 addition.

  considered in French regulation for land application of sludge (decree N° 97-1133 of 8/01/1998), expressed in mg of element per kg of DM: Cd and Hg 10 DM; Cr and Cu 1000, Ni 200; Pb 800; Zn 3000; Cr+Cu+Ni+Zn 4000. It was observed that the concentrations increased in time from around 780 mg.kg -1 of DM in young plants RAM and PRE to ≥1200 in the older "classical"

Fig. 2 .

 2 Fig. 2. Aerobic biodegradation measured over 14 days of incubation at 30°C on the surface sludge deposits sampled from French VFCW sewage treatment plants of different types. Classical system with no additional treatment (blue bars); additional treatment on trickling filter only (green); trickling filter and FeCl3 addition (red).

Fig. 3 .

 3 Fig. 3. Distribution of organic carbon in sludge deposits (SD) organic fractions, and humification rate measured in the 14 VFCWs sludge deposits. Classical system with no additional treatment (blue); additional treatment on trickling filter only (green); trickling filter and FeCl3 addition (red).

Fig. 4 .

 4 Fig. 4. Mass loss (a) and differential scanning calorimetry (b) profiles recorded from surface sludge deposits from 14 French VFCW sewage treatment plants. Classical system with no additional treatment (blue); additional treatment on trickling filter only (green); trickling filter and FeCl3 addition (red).

Fig. 5 .

 5 Fig. 5. Differentiation of sludge samples according to their indices related to the stability and maturity of the OM. Each sample is represented with a circle whose center is positioned according to the biodegradation rate and humification rate of the samples, and diameter is proportional to their RTGA indices. Classical system with no additional treatment (blue); additional treatment on trickling filter only (green); trickling filter and FeCl3 addition (red).

Fig. 6 .

 6 Fig.6. CFULIS/CHULIS against CHULIS+FULIS/TOC for natural wetlands (aquatic wetlands, peatlands and wet meadows), activated sludges and the 14 sludge deposits of this study.

Table 1

 1 Characteristics of the 14 French VFCW sewage treatment plants selected for the study.

	1
	VFCW plants	Process type	Gridding / sieving	Sand trap	Process line stages Trickling filter injection FeCl3	Saturation of 1 st stage filter	Number of years of operation at sampling date	Nominal capacity (Population Equivalent)	Average hydraulic load (m 3 / y)
	BAG1

Table 2

 2 General composition of sludge deposits samples.

	118
			OM b	TOC c	Si d	Ca d	Al e	Fe d	Mg e	Na f	K d	Mn e	Ti e	P d	TN g	TOC/TN	Cr e	Cu d	Ni d	Zn e	Cd f	Pb d	∑metals h
		VFCW labels a
			% w/w of DM					% w/w of DM										mg.kg DM -1
		BAG1	59.0	24.7	7.7	2.6	2.1	1.0	0.3	0.2	0.5	0.02	0.2	1.2	4.0	6.2	49.8	338.0	23.2	615.6	1.0	43.7
		BAG2	57.4	24.2	13.3	2.9	2.0	0.9	0.3	0.2	0.7	0.01	0.2	0.8	3.3	7.2	41.4	363.8	21.7	810.2	1.2	37.2
		SABAG	60.3	21.0	11.6	2.6	1.8	1.4	0.3	0.2	0.6	0.04	0.2	1.1	3.6	5.8	46.3	1162.0	23.1	557.4	1.0	33.4
		VIL	60.9	22.7	10.5	2.4	2.4	1.2	0.3	0.1	0.5	0.01	0.3	1.2	3.4	6.7	46.5	425.2	18.0	430.0	0.6	31.9	952
		BAG3	50.3	18.4	13.2	3.5	2.4	1.4	0.3	0.3	0.7	0.02	0.2	1.0	3.1	5.9	59.3	282.3	25.1	481.4	1.0	34.7	884
		PRE	86.2	36.4	2.1	2.5	0.4	0.2	0.1	0.1	0.2	<D.L.	0.1	0.5	1.5	24.7	19.3	216.9	7.7	463.1	0.4	13.6	721
		RAM	84.3	32.3	3.4	3.2	0.6	0.3	0.2	0.1	0.2	<D.L.	0.1	0.7	2.1	15.6	18.4	304.7	10.0	481.0	0.6	18.1	833
		BAY	70.5	28.5	3.3	4.7	1.5	0.5	0.3	0.1	0.3	0.01	0.2	1.9	4.5	6.4	34.5	232.9	20.3	968.9	1.5	121.5
		MIN	73.5	28.2	5.0	3.5	1.3	0.6	0.4	0.2	0.4	0.02	0.3	1.4	4.8	5.9	51.9	195.9	30.3	731.6	0.9	32.2
		NEU	58.6	21.7	9.6	3.7	2.3	1.2	0.4	0.3	0.7	0.04	0.2	1.1	3.6	6.0	51.8	223.9	26.9	573.9	1.2	45.4	923
		VER	62.0	23.0	5.7	3.8	2.1	3.5	0.3	0.1	0.4	0.02	0.2	2.0	3.8	6.1	57.4	622.5	30.5	1218.5	1.6	64.2
		ORB	58.2	22.7	7.3	3.8	2.0	3.5	0.3	0.1	0.5	0.03	0.2	1.6	3.6	6.2	74.0	379.6	36.7	952.9	1.3	48.9
		COR	72.0	24.7	2.8	4.6	0.8	5.2	0.3	0.2	0.5	0.02	0.2	2.5	4.1	5.9	51.4	691.4	21.0	908.8	1.0	37.2
		MAT	50.6	14.0	11.2	1.6	3.2	5.9	0.4	0.4	1.2	0.5	0.3	2.0	2.5	5.5	57.6	201.7	23.4	744.7	1.1	68.7
	119	OM: Organic Matter; DM: Dry Matter; DL = Detection Limit
		a See Table 1

e Relative uncertainty of 10% f Relative uncertainty of 15% g Relative uncertainty of 0.3% h ∑metals = [Cr] + [Cu] + [Ni] + [Zn] + [Cd] + [Pb].

Table 3 R

 3 TGA , E 1 , E 2 and R DSC values for 14 VFCW sludge deposits.

		RTGA	E1	J.mg -1	E2	RDSC
	BAG1	0.95	8.6		20.7	2.4
	BAG2	0.75	10.5		16.7	1.6
	SABAG	0.83	9.2		15.4	1.7
	VIL	0.77	8.9		19.1	.2
	BAG3	0.76	10.2		21.2	2.1
	PRE	0.24	8.9		24.1	2.9
	RAM	0.28	7.9		24.6	3.2
	BAY	0.92	6.6		16.3	2.5
	MIN	1.05	6.0		17.3	2.4
	NEU	0.85	8.0		17.7	2.2
	VER	0.76	11.7		22.0	1.9
	ORB	0.76	9.5		15.6	1.7
	COR	0.79	8.5		18.7	2.2
	MAT	0.35	13.6		18.7	1.4

Table 4

 4 Correlation coefficients (and P-value) between the parameters of biological, chemical and

	thermal characterization. Correlation coefficients shown in bold were considered statistically significant,
	with P<0.05.
		OM	Humification rate	Aerobic biodegradation	RTGA	RDSC
	OM	1	-0.834 (2.10 -4 )	0.623 (0.02)	-0.363	0.828 (3.10 -4 )
	Humification rate		1	-0.702 (0.01)	0.427	-0.750 (2.10 -3 )
	Aerobic biodegradation			1	-0.467	0.436
	RTGA				1	-0.253
	RDSC					1

b Analytical results from the replicates of each sample did not exceed ±2.5%

Acknowledgments

The authors would like to thank the technicians of DEEP laboratory Richard Poncet and Hervé Perier-Camby for their contributions, the SARM (Service d'Analyse des Roches et des Mineraux -CNRS -CRPG) and the ISA (Institut des Sciences Analytiques) for the elementary analyses. They are grateful to the reviewers for their contribution to the improvement of this manuscript, but also to SCIRPE and CIFRE (Conventions Industrielles de Formation pour la REcherche) for funding this research project.


Damien F G Minenna
email: author.damien.minenna@univ-amu.fr

Yves Elskens

Fabrice Doveil

Frédéric André

Universality of the Abraham-Minkowski dilemma for photon momenta beyond dielectric materials

come

Electromagnetic momenta for wave-particle systems in vacuum waveguides -Universality of the Abraham-Minkowski dilemma beyond dielectric materials

Whenever light is slowed down, for any cause, two different formulas give its momentum. For dielectrics, the coexistence of those momenta was the heart of the century-old Abraham-Minkowski dilemma, recently resolved. We demonstrate that this framework extends to momentum exchange in wave-particle interaction; in particular to Langmuir waves for Landau damping and to vacuum waveguides of electron tubes (metallic slow-wave structures). Focussing on the latter, we show that the dilemma resolution is not limited to discriminating between kinematic and canonical momenta but also involves a non-negligible flowing momentum from Maxwell's electromagnetic stress. The existence of two momenta in materials, plasmas, and waveguides, for which light velocity modification has entirely different origin, points to the universality of the Abraham-Minkowski dilemma.

The question "what is the momentum of light propagating through glass?" seems simple enough to appear on a high-school quiz [1,2]. Nevertheless, it took physicists more than a hundred years to answer it in a vigorous debate confronting two rival theories and called the Abraham-Minkowski controversy. For Minkowski [3], respectively Abraham [4], the momentum carried by light is proportional (p Mink = n ω/c), respectively inversely proportional (p Abra = ω/(nc)), to the refractive index n of dielectric and magnetic materials (see Table I). The problematic is the same for classical fields or photons. Despite many theoretical and experimental works [START_REF] Jones | Proc. R. Soc. Lond. A[END_REF][START_REF] Jones | Proc. R. Soc. Lond. A[END_REF][START_REF] Brevik | [END_REF][8], it took a century until recent contributions [9][10][11] proposed a resolution of the controversy, arguing that both theories are correct. While this resolution is largely accepted, the dilemma is still actively debated in materials [16,17].

Mathematically speaking, the notion of momentum admits two definitions. The linear (kinematic) momentum of a particle (i) is its mass times its velocity (p kin,i = m qi , linked to forces), while the conjugate (canonical, in Hamiltonian formalism) momentum is the derivative of the Lagrangian (p can,i = ∂ qi L, linked to action). In newtonian mechanics, there is no difference between both momenta, but this is no more the case in quantum physics or in electrodynamics. In that respect, the canonical momentum [START_REF] Jackson | Classical Electrodynamics[END_REF] of a charged particle (i) immersed in a magnetic field is p can,i = p kin,i + eA(q i ), with its electric charge e, and the vector potential A such that B = rot A. The discrepancy between both momenta comes from the involvement of the field. Therefore, we should also expect discrepancies between momenta of light. This is precisely where the controversy originates: the canonical momentum of light is Minkowski's expression p Mink while Abraham's expression p Abra is kinematic. According to Refs. [10,11], in dielectric and magnetic media, both momenta are linked via the total momentum p matter can + p Mink = p matter kin + p Abra , with the canonical Table I. Established expressions [8,12] for the momentum of light and its density, in dielectric and magnetic materials. A single photon has wavenumber k = ω/v φ = nω/c, with angular frequency ω, phase velocity v φ , and refractive index n = c/v φ . Classical electrodynamics of media calls on the electric field E, the magnetic field H, the electric displacement D = E, the magnetic induction B = µH, the permittivity = r 0, the magnetic permeability µ = µrµ0, and the speed of light in vacuum c = ( 0µ0)

-1/2 . The group index of a dispersive medium is ng = c/vg, with group velocity vg = ∂ k ω. Since the group index is ng = n + ω ∂ωn, one can assume ng ≈ n, for low frequencies or if n is independent of the angular frequency as in non-dispersive media. Consequently, for photons, there is an n 2 ratio between both momenta, while defining n = √ rµr, and v φ = ( µ) -1/2 (for homogeneous, isotropic, non-dispersive, space-time independent media), leads to the same n 2 ratio for classical fields. In dispersive media, an nng ratio is admitted for photons [13] and sometimes inferred [START_REF] Jones | Proc. R. Soc. Lond. A[END_REF]8,14,15] for electromagnetic fields though no broad consensus has emerged.

Momentum of

Momentum density a single photon a of classical fields

Minkowski's p Mink = k = n ω c g Mink = D × B Abraham's p Abra = k nng = ng ω c g Abra = E × H c 2 a Some authors set k = ω/c with p Mink = n k, p Abra = k/ng.
p matter can and kinematic p matter kin momenta of matter. At first glance, the dilemma seems to arise when light simply propagates through a medium, suggesting that the effect originates from the permittivity and/or the magnetic permeability. However, it does not. The key mechanism in the Abraham-Minkowski dilemma is the modification of the wave/photon velocity compared with the speed of light in vacuum, or equivalently the modification of the wavenumber, due to various causes. While the dilemma was only investigated in matter, this letter presents two systems with starkly different causes for light speed modification. In both cases, we obtain Abraham's and Minkowski's momenta. The first system considers electromagnetic fields propagating inside the dispersive metallic waveguide of a vacuum electron tube. The second system considers Langmuir waves within the Landau damping (or amplification) context in plasmas. Our main goal is to demonstrate the universality of the Abraham-Minkowski dilemma beyond materials, so we give priority to the first system since forces and energies in vacuum are well established. In addition, we have simulations to assess the extention of the dilemma to waveguides with an algorithm benchmarked against experimental measurements (see below). Evidences that the dilemma occurs in plasmas (using an analogous treatment) are raised at the end of this letter. The resolution of the dilemma for both systems goes beyond the resolution for materials since it involves a flowing momentum from Maxwell's electromagnetic stress.

Before proceeding, we should stress that, for both plasmas or waveguides, we describe the wave-particle interaction with the less popular N -body (many body or molecular dynamics) description. Compared to kinetic (e.g. Vlasovian) description, it involves a huge number of degrees of freedom that seem difficult to compute, but this approach was proven feasible [START_REF] Elskens | Microscopic dynamics of plasmas and chaos[END_REF][START_REF] Minenna | [END_REF][21] for both cases we investigate, and provides easily the two momenta of light.

Regardless of the cause, the light velocity modification leads to a refractive index n = c/v φ , with the phase velocity v φ and, for dispersive systems (n depending on the frequency), a group index n g = c/v g , with the group velocity v g . In matter, both indices (and velocities) are naturally linked to the permittivity and the magnetic permeability. In vacuum waveguides, like resonant cavities of slow-wave structures, it is the device geometry that determines both velocities. Indeed, their dispersion relations can be calculated [START_REF] Jackson | Classical Electrodynamics[END_REF] from sourceless Maxwell equations with the metallic wall boundary conditions. For instance, the helical slow-wave structure of a traveling-wave tube [START_REF] Gilmour | Principles of traveling wave tubes[END_REF] (a typical electron tube used for wave amplification, similar to free-electron lasers) can slow down the phase velocity by a factor 1/7. Large refractive indices (e.g. n ≈ 7) are frequent in waveguides, making the difference between Abraham's and Minkowski's momenta more dramatic. The effects of this deceleration on wave propagation are the same as in matter, but it is easier to investigate the dilemma in waveguides since we can then start from Maxwell's equations in vacuum with sources. To complete the system, we consider an electron beam propagating through the waveguide in resonance with the waves (|v| v φ ). For a moderate beam intensity, the particles do not modify the dispersion relation of the waveguide and the momentum exchange does not change the refractive index. Deriving Abraham's and Minkowski's expressions in a waveguide requires a proper definition of the refractive and group indices in the device. This is done using a decomposition [START_REF] Kuznetsov | [END_REF]24,[START_REF] Minenna | DIMOHA: an advanced algorithm for complete traveling-wave tube simulations[END_REF] separating dependence on space (r) and time (t) for the electric E and magnetic H fields,

E(r, t) = k V k (t)E k (r) , (1)
H(r, t) = k iI k (t)H k (r) , (2)
where the fields (with wavenumber k) obey the Maxwell equations in vacuum with sources. The imaginary i factor in Eq. ( 2) simplifies the connection below between the evolution equations obtained from Maxwell and Hamilton formalism. A sum instead of an integral over all k's is chosen for convenience. More accurately, a sum over all possible propagation bands is necessary, but, for the sake of simplicity, only the dominant mode is considered. Using the Weyl gauge, the electric field is E = -Ȧ, with the electromagnetic vector potential A = k iI k A k . Since we deal with wave propagation in vacuum, the same coefficients I k as for the magnetic field are used, though this would no longer hold in matter. Using this field decomposition, Helmholtz' equations (sourceless Maxwell equations) read rot

E k = -iµ 0 ω k H k , and rot H k = i 0 ω k E k , with eigenfrequencies ω k . We normalize eigenfields E k , H k , by Ω k = V 0 |E k (r)| 2 dV = V µ 0 |H k (r)| 2 dV . (3)
With this decomposition, the Maxwell-Ampère and Maxwell-Faraday evolution equations read

Vk (t) = ∂ t V k = -I k (t) ω k + ı e qı • E * k (q ı ) Ω k , (4)
İk (t) = ∂ t I k = V k (t) ω k , (5)
respectively. We then re-express the well-known electromagnetic energy [START_REF] Jackson | Classical Electrodynamics[END_REF] (Hamiltonian of fields) in vacuum

1 2 V ( 0 |E| 2 + µ 0 |H| 2 ) dV as [24] H em = k 1 2 V k (t)V * k (t) + I k (t)I * k (t) Ω k . (6)
In this representation, V k and I k are canonical variables, with V k the "conjugate momenta" and I k the "generalized coordinates". The normalisation (3) ensures their Poisson brackets are canonical. The dynamics of charged particles (i) in electromagnetic fields is dictated by the Lorentz force i mq i = -e Ȧ(q i ) -e( qi • ∇)A(q i ) + i e qi × µ 0 H, leading to the well-known particles energy [START_REF] Jackson | Classical Electrodynamics[END_REF][START_REF] Landau | Electrodynamics of Continuous Media[END_REF] (a.k.a. coupling Hamiltonian) H cpl = i (2m) -1 |p can,i -eA(q i )| 2 . Relativistic expressions leave unchanged the field momentum. From the total energy (self-consistent Hamiltonian) H tot = H em +H cpl , the Hamilton evolution equations are relations (4)-( 5

) if -iω k A k = E *
k , in agreement with the Weyl gauge, and on setting Ω k = ω k .

The Lagrangian of the system is obtained with the Legendre transformation of the total Hamiltonian, as

L tot = i p can,i • qi + k V k İk -H tot .
This Lagrangian is invariant under space translations q i → q i + δ r and I k → I k e -ik•δr . Hence, the infinitesimal variation of the Lagrangian (a.k.a. the generalised force on the system)

∂L tot ∂δ r = dp tot dt = i ṗcan,i + k ik d dt V * k I k , (7)
must vanish. Applying Noether's theorem [START_REF] Minenna | [END_REF] leads to a first formulation of the conserved total momentum in the waveguide

p tot = i p can,i + k iV * k I k k , (8)
as the sum of canonical momenta of the particles and the fields, with Minkowski's momentum in the waveguide

p wg Mink = k iV * k I k k . (9)
In addition, the derivative of Eq. ( 3) enables the calculation of the group velocity

v g = ∂ k ω k , with ω k = Ω k .
Using the derivative of Helmholtz' equations, the iden-

tity B • (∇ × A) = ∇ • (A × B) + A • (∇ × B)
, for arbitrary A, B, and the divergence theorem, we obtain

v g = (1/ω k ) V E * k (r) × H k (r) dV + c.c
., with c.c. denoting complex conjugate. A simple rearrangement, with ω k = kv φ = kc/n and v g = c/n g , leads to the relation for the wavevector

k = nn g c 2 V E * k (r) × H k (r) dV + c.c. , (10)
in the monochromatic (a.k.a. continuous waveform, CW) regime where group and refractive indices are considered as constant. Then, Eq. ( 9) yields the Minkowski momentum density

g wg Mink (r, t) = nn g c 2 E × H , (11)
for dispersive waveguides in the monochromatic regime (see the difference with materials in Table I).

On the other hand, Abraham's momentum is found exactly as in vacuum, using the classic reformulation [START_REF] Jackson | Classical Electrodynamics[END_REF][START_REF] Griffiths | Introduction to Electrodynamics[END_REF] of the Lorentz force density

f = ∇ • σ -∂ t (E × H)/c 2 , ( 12
)
obtained from Maxwell equations in vacuum with sources, with the Maxwell stress tensor σ, and the Abraham momentum density

g wg Abra (r, t) = 1 c 2 E × H , (13)
as for materials (see Table I). For waveguides, we use decomposition (1)-( 2) [START_REF] Minenna | [END_REF] and Eq. ( 10) for the wavevector to re-express Eq. ( 13) as the Abraham momentum

p wg Abra = k 1 nn g iV * k I k k . (14)
The last item needed to resolve the Abraham-Minkowski dilemma for waveguides is the link between both momenta, Eqs ( 9) and ( 14). This is done by comparing the two balances of forces of the system: a first one from the Noether approach Eq. ( 7) i ṗcan,i + ṗMink = 0, and a second one from the Lorentz force Eq. (12) i ṗkin,i + ṗAbra -V ∇ • σ dV = 0. Comparing both leads to a second formulation for the conserved total momentum

p tot = i p can,i + p Mink = i p kin,i + p Abra + p flow , (15)
with the sum running on particles (i), which is the crucial relation needed to solve the dilemma. In our field representation, this gives the flowing momentum

p flow = k 1 - 1 nn g iV * k I k k + i k eiI k A k (q i ) , (16)
as the difference ensuring Eq. (15).

Most importantly, the use of the Lorentz force introduces the time derivative [START_REF] Griffiths | Introduction to Electrodynamics[END_REF] ṗflow = -V ∇•σ dV of the flowing momentum leaving volume V . As soon as amplification or attenuation occurs, the system is not spatially homogeneous since the fields exert a stress and thereby the force balance needs to be completed by the Maxwell stress. Indeed the time derivative of Eq. ( 16) coincides with the integral of the divergence of Maxwell's stress tensor [START_REF] Jackson | Classical Electrodynamics[END_REF][START_REF] Griffiths | Introduction to Electrodynamics[END_REF]

∇ • σ = 0 (∇ • E)E + µ o (∇ × H) × H + 0 (∇ × E) × E.
The key difference of Eq. ( 15) with the resolution of the dilemma in materials [10,11] is the inclusion of the flowing momentum which becomes crucial for waveguidebased amplifiers (while it vanishes for mere propagation in a passive medium). Figure 1 shows an example of momentum exchange in a traveling-wave tube with a large nn g . The algorithm used is based on an N -body approach [START_REF] Minenna | [END_REF]24] with a decomposition similar to Eqs (1)-( 2), specialized for the traveling-wave tube geometry and combined with a degrees-of-freedom reduction. It has shown an excellent agreement with real tube measurements [START_REF] Minenna | Proceedings of the 18 th IEEE International Vacuum Electronics Conference (IVEC)[END_REF][START_REF] Minenna | Proceedings of the 20 th IEEE International Vacuum Electronics Conference (IVEC)[END_REF][START_REF] Minenna | DIMOHA: an advanced algorithm for complete traveling-wave tube simulations[END_REF]. For Figure 1, we removed specific tube features (waveguide losses, industrial adjustments, imperfect adaptations) to limit the interaction only to the momentum exchange (15). In helix traveling-wave tubes, fields can be assumed almost longitudinal, leading to only a small difference between canonical and kinematic momenta. Since Abraham's momentum is much smaller than Minkowski's, and because, in typical electron tubes, per cavity (viz. waveguide cell) of a traveling-wave tube, in monochromatic regime [START_REF] Minenna | [END_REF], from an algorithm [START_REF] Minenna | Proceedings of the 18 th IEEE International Vacuum Electronics Conference (IVEC)[END_REF][START_REF] Minenna | Proceedings of the 20 th IEEE International Vacuum Electronics Conference (IVEC)[END_REF][START_REF] Minenna | DIMOHA: an advanced algorithm for complete traveling-wave tube simulations[END_REF] used to design industrial traveling-wave tubes. To ensure conservation of the total momentum, we adapted the simulation to consider only the wave-particle exchange in the system and to discard any losses or external interactions. The electron velocity is set slightly faster than the phase velocity (given by the structure geometry) to compel the wave amplification from the beginning. After 175 cavities, nonlinear trapping [START_REF] Elskens | Microscopic dynamics of plasmas and chaos[END_REF] of particles provokes the decline of the wave momentum. a) Conservation of i pcan,i + p Mink from Eq. ( 15) per cavity. Solid line: canonical momentum lost by particles (plotted positively). Dashed curve: Minkowski's momentum Eq. ( 9). b) Conservation of i p kin,i + p Abra + p flow from Eq. (15) per cavity. Solid line: kinematic momentum lost by particles (plotted positively). Dashed curve: Abraham's momentum Eq. ( 14). Dash-dot curve: flowing momentum Eq. ( 16) leaving cavities. The conservation of total momentum (15) is observed as the particle momentum loss (plotted positively) coincides with the wave momentum increase. The ratio between Minkowski's and Abraham's momenta is nng = 62.2, with n = 7.2 and ng = 8.6, in agreement with the model parameters, through the whole device.

the difference between canonical and kinematic momenta of electrons can be really small, we immediately see that the flowing momentum can obviously not be ignored.

Note that, with these new forms for Abraham's and Minkowski's expressions, we can oversimplify the quantization by enforcing iV * k I k → Nk , in Eq. ( 9) and Eq. ( 14), to quantize classical fields in terms of photon counting operators Nk . Though crude, this procedure reflects on the debate on photon momentum. Using this transform, Minkowski's momentum, respectively Abraham's, for a single photon becomes the familiar p Mink = k, respectively p Abra = k/(nn g ) (see Table I).

The dilemma does not only extend to vacuum waveguides but is relevant to every wave-particle system (with v φ = c) involving momentum exchange. Arguably the most famous one is Landau damping [START_REF] Landau | [END_REF]32] (and Landau growth, a.k.a. bump-on-tail instability, with the same origin [21]) occurring in plasma physics (beam-plasma system). Here, the propagation medium of the Langmuir waves is a plasma and the dispersion relation derives from the (bulk) electron velocity and position distribution functions. The interaction between waves and (beam) electrons is the key to Landau damping. To investigate the latter, the N -body Hamiltonian approach has shown [START_REF] Elskens | Microscopic dynamics of plasmas and chaos[END_REF]33] that the associated dynamics conserves a total wave-particle momentum. The conservation of this total momentum implies a non-linear synchronization between the particles and the wave, leading to the physical result that Landau damping or growth is linked to the slope of the particle velocity distribution function f (v i ) near the phase velocity of the wave. Although it may seem surprising to relate Landau damping and vacuum waveguides, this connection was already fruitfully made decades ago [34,35]. Beside the transition to chaos [36], Landau damping (more precisely the nonlinear synchronisation between an electron beam and waves) has been accurately investigated [37] using a traveling-wave tube. In both cases, the physics is globally the same, based on momentum exchange, and only differs in the origin of the dispersion relation for the wave propagation.

Because the wave-particle interactions for travelingwave tubes and for Landau damping are analogous [START_REF] Elskens | Microscopic dynamics of plasmas and chaos[END_REF]37], a similar spectral decomposition to Eqs (1)-( 2), applies to Langmuir waves in plasma. However, there are some critical differences. In particular, Langmuir waves can be considered as electrostatic, which implies equality of the canonical and kinematic momenta of particles (p can,i = p kin,i ). Yet, the Abraham-Minkowski dilemma still applies, thanks to the crucial flowing momentum: A complete demonstration that Langmuir waves have two momenta will be the subject of a forthcoming publication. Indeed, conservation of the total momentum, as in Eq. ( 8), has already been demonstrated experimentally [37] and theoretically [START_REF] Elskens | Microscopic dynamics of plasmas and chaos[END_REF]21] (without identifying Minkowski). On the other hand, when expressing the electric force -e∇ϕ, with the electrostatic potential ϕ (proportional to the square of the wave intensity), Abraham's momentum arises complemented with the flowing momentum to ensure momentum balance as well, in agreement with Eq. (15).

In many ways, the Abraham-Minkowski controversy looks like a modern version, involving electrodynamics, of the vis viva controversy [38] (during the 17 th and 18 th centuries, both "linear momentum" (mv) and "kinetic energy" (mv 2 ) were discussed as the same concept of "force" as it was called). The problem originates from the difference between definitions one uses. For either physical system, one obtains two different equations for the momentum conservation: one from the Hamilton properties leading to Minkowski's (canonical) expression, and one from the force balance leading to Abraham's (kinematic). This leads to the conclu-sion that the Abraham-Minkowski dilemma arises in vacuum waveguides of electron tubes and in Landau damping/amplification.

Finding two momenta of light not only in dielectrics and magnetic materials, but also in waveguides and in plasmas, implies that the difference between Abraham's and Minkowski's is a universal issue, relevant to many chapters of physics. Investigating momentum exchange in wave-particle systems requires taking notice of the dilemma, because it results from the same fundamental processes: we expect the debate to arise in every domains where momentum exchange occurs with slowed down light. It is worth noting that simulations presented in Figure 1 were made with an algorithm [START_REF] Minenna | Proceedings of the 18 th IEEE International Vacuum Electronics Conference (IVEC)[END_REF][START_REF] Minenna | Proceedings of the 20 th IEEE International Vacuum Electronics Conference (IVEC)[END_REF][START_REF] Minenna | DIMOHA: an advanced algorithm for complete traveling-wave tube simulations[END_REF] used for industrial applications. Consequently, the distinction between Abraham's and Minkowski's approaches and considering the Maxwell stress, is relevant to actual devices.

Unlike for materials, solving the Abraham-Minkowski dilemma for waveguides and plasmas implies taking account of the flowing momentum (which cannot be ascribed to a supporting material medium), meaning this dilemma also highlights a difference about the impact of the electromagnetic stress on the expression of the light momenta.

Figure 1 .

 1 Figure1. Momenta balance (time averaged per period 2π/ω k ) per cavity (viz. waveguide cell) of a traveling-wave tube, in monochromatic regime[START_REF] Minenna | [END_REF], from an algorithm[START_REF] Minenna | Proceedings of the 18 th IEEE International Vacuum Electronics Conference (IVEC)[END_REF][START_REF] Minenna | Proceedings of the 20 th IEEE International Vacuum Electronics Conference (IVEC)[END_REF][START_REF] Minenna | DIMOHA: an advanced algorithm for complete traveling-wave tube simulations[END_REF] used to design industrial traveling-wave tubes. To ensure conservation of the total momentum, we adapted the simulation to consider only the wave-particle exchange in the system and to discard any losses or external interactions. The electron velocity is set slightly faster than the phase velocity (given by the structure geometry) to compel the wave amplification from the beginning. After 175 cavities, nonlinear trapping[START_REF] Elskens | Microscopic dynamics of plasmas and chaos[END_REF] of particles provokes the decline of the wave momentum. a) Conservation of i pcan,i + p Mink from Eq. (15) per cavity. Solid line: canonical momentum lost by particles (plotted positively). Dashed curve: Minkowski's momentum Eq. (9). b) Conservation of i p kin,i + p Abra + p flow from Eq. (15) per cavity. Solid line: kinematic momentum lost by particles (plotted positively). Dashed curve: Abraham's momentum Eq. (14). Dash-dot curve: flowing momentum Eq. (16) leaving cavities. The conservation of total momentum (15) is observed as the particle momentum loss (plotted positively) coincides with the wave momentum increase. The ratio between Minkowski's and Abraham's momenta is nng = 62.2, with n = 7.2 and ng = 8.6, in agreement with the model parameters, through the whole device.

The authors gratefully thank Olivier Agullo, Didier Bénisti, Yann Camenen, Caroline Champenois, Lénaïc Couëdel, Nicolas Dubuit, Dominique F. Escande and David Zarzoso for their critical reading of the manuscript, and F. Javier Artola, Thomas Durt, André Nicolet, Valentin Pigeon, Alexandre Poyé, Brian Stout, Gérard Tayeb and Frédéric Zolla for fruitful discussions.