Olivier Blight
email: olivier.blight@imbe.fr

Jérôme Orgeas

Aline Le Menn

Angélique Quilichini

Blight

Orgeas Olivier

Jérôme Le Menn

Aline Quilichini

Interaction between a threatened endemic plant (Anchusa crispa) and the invasive Argentine ant (Linepithema humile)

Keywords: invasive ant, endemic plant, seed dispersal, conservation

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Interaction between a threatened endemic plant (Anchusa crispa) and the invasive Argentine ant (Linepithema humile)

Introduction

Seeddispersal is one of the most ecologically significant plant-animal mutualisms. This mutualism influences seedling recruitment, population dynamics, species distributions, plantcommunity composition and gene flow [START_REF] Nathan | Spatial patterns of seed dispersal, their determinants and consequences for recruitment[END_REF]. Due to their ubiquitous occurrence in terrestrial ecosystems, ants are one of the most important animal groups involved in seed dynamics [START_REF] Lengyel | Convergent evolution of seed dispersal by ants, and phylogeny and biogeography in flowering plants: A global survey[END_REF]. They contribute significantly to maintaining the structure and diversity of different natural communities [START_REF] Lengyel | Ants sow the seeds of global diversification in flowering plants[END_REF].

Elimination of native ants following invasion and particularly by the Argentine ant (Linepithema humile) is widely reported to lead indirectly to the near-complete collapse of seed dispersal services [START_REF] Christian | Consequences of a biological invasion reveal the importance of mutualism for plant communities[END_REF][START_REF] Carney | Invasive Argentine ants (Linepithema humile) do not replace native ants as seed dispersers of Dendromecon rigida (Papaveraceae) in California, USA[END_REF]. The Argentine ant is seen as an imperfect substitute for the full range of native ants that disperse elaiosome-bearing seeds (Rodriguez-Cabal et al. 2009b). Homogenization of native ant fauna following its invasion further diminishes seed dispersal services for native plant species [START_REF] Lach | Invasive Ants : Unwanted Partners in Ant-Plant Interactions ?[END_REF], and in some cases can favour the spread of highly invasive plants [START_REF] Rowles | New mutualism for old: Indirect disruption and direct facilitation of seed dispersal following Argentine ant invasion[END_REF].

As in other parts of its introduced range, replacement of native ants by L. humile in Southern

Europe has led to changes in the relationships between ants and plants, including myrmecochory (Gómez et al. 2003;[START_REF] Gómez | Can the Argentine ant (Linepithema humile Mayr) replace native ants in myrmecochory?[END_REF]. Argentine ants have been present and affected the local ecosystem of Corsica, a French Mediterranean island, for more than 60 years [START_REF] Blight | Competitive dominance in the organisation of Mediterranean ant communities[END_REF]. They spread mainly on the coasts, invading modified but also natural habitats including sites that serve as a habitat for an endemic myrmecochorous plant, Anchusa crispa Viv. [START_REF] Blight | Where and how Argentine ant (Linepithema humile) spreads in Corsica?[END_REF]. Anchusa crispa is a Mediterranean endemic species occurring on the islands of Corsica (France) and Sardinia (Italy). It occurs in isolated populations in coastal habitats. Anchusa crispa has been listed under the Berne Convention since 1979 and has been on the European Community list of protected species since 1992. This species is classified as endangered by International Union for Conservation of Nature.

In the past 20 years, the Argentine ant has significantly increased its population density in Corsica [START_REF] Blight | Where and how Argentine ant (Linepithema humile) spreads in Corsica?[END_REF], this study therefore aimed to evaluate its effects on the population of the rare and threatened endemic myrmecochorous plant, A. crispa, investigating the direct and indirect effects of the Argentine ant invasion on patterns of seed dispersal. Specifically, our study addressed the potential of L. humile to disperse elaisome-bearing seeds of A. crispa and its effects on (1) the composition of ant communities removing seeds, (2) the number of seed removals, (3) seed preference, (4) the distance of seed dispersion and (5) seed germination.

Materials and Methods

Experimental sites

Anchusa crispa, a short-lived (up to 4 years with 2-3 flowering years) hemicryptophyte, grows on coastal dunes and back-dunes, on sandy soils in open herbaceous vegetation. The fruits comprise four fused carpels that develop into nutlets which bear an elaiosome.

In Corsica, three populations occur on the west coast and two on the east coast. The total number of plants in each population in Corsica varies from less than 10 individuals to several hundreds.

Given the low number of A. crispa populations and there small size, we could performed our experiments on only the two biggest sites: one uninvaded, on the West coast (Portigliolo) and one on the East coast (Cannella) that is invaded by the Argentine ant [START_REF] Blight | Where and how Argentine ant (Linepithema humile) spreads in Corsica?[END_REF]). Both populations of A. crispa covered approximatively 80m 2 . All the populations are located near a river mouth. The station on the West coast is located near a river in the Gulf of Propriano. This site is fenced off from car access (Portigliolo). The second population, on the East coast (Cannella), is approximately 45km away and located on a dune sloping down from a restaurant and a camp site.

Ant assemblages

To determine local ant richness and species occurrence, we placed at both sites 20 plastic pitfall traps (35 mm in diameter and 70 mm deep) partly filled with a non-attractive mixture (ethyleneglycol) as a capture and preservation fluid, at one-metre interval along two transects of 10 meters set in parallel. The two transects were separated by 6m. After 48 hours, all pitfall samples were collected and the ant species were sorted and identified to the species level. Ant assemblages were evaluated in June (optimum of fructification) 2008 and 2009, two weeks before the experimental tests were conducted.

Offering tests

We determined the behavioural reaction of ants to mature seeds of A. crispa in the field by directly offering seeds to workers at both sites in June 2009. By using tweezers, we placed seeds on the ground next to randomly selected solitary workers that prospected for food. We offered 80 seeds at Portigliolo to the three most abundant species (28 seeds to both Messor minor and Tapinoma sp and 24 seeds to Aphaenogaster spinosa) and 80 seeds to the Argentine ant, the only species present at Cannella. Each ant's response was categorized following the [START_REF] Culver | Myrmecochory in Viola: dynamics of seed-ant interactions in some West Virginia species[END_REF] classification : (i) antennate and examine: the ant touched the seed with antennae and then continued to forage (ii) pick-up attempt: the ant touched the seed with antennae, inspected it and then released it without removing it and (iii) remove: the ant held the seed between its mandibles and removed it. When possible, we also recorded the number of seeds transported over less than 10 cm, between 10 cm and 50 cm, between 50 cm and 1 m and beyond 1 m. Seeds were collected directly from the plant and offered to ants at the sites where they were collected 24 hours after their collection. Differences in proportion of ant-seed interactions and in proportion of seeds dispersed over less than 50 cm were compared using Fisher's exact test.

Seed preference and removal events

To test seed attractiveness and seed transport rates at the community level, we placed in a 5.5 cm (diameter) plastic petri dish one seed of each of three plants, A. crispa, Silene sericea and Lotus cytisoides at the uninvaded site (Portigliolo) and A. crispa, S. sericea and Malcolmia ramosissima at the invaded site (Cannella), where there was no L. cytisoides. This test was conducted in June 2008 and June 2009. Anchusa crispa seeds are the only seeds bearing an elaiosome and are larger than the seeds of the three other species (19 mg for A. crispa, < 10 mg for S. sericea, L. cytisoides and M. ramosissima) that were dominant at their sites. We included non-elaisome bearing seed species to compare attractiveness of A. crispa seeds to the two dominant plant species in each site. We placed two petri dishes separated by one meter at onemetre intervals along the same two transects used for the ant sampling, one petri dish on each side of the transects. We recorded the seeds remaining on each petri dish every 30 min for two hours and then every hour until the end of the test. Observations started at 6:00 a.m. and finished at 11:00 a.m. Ants were not active after 11:00 a.m. when temperatures reach more than 40°C on the soil surface. We conducted this experiment over three consecutive days on the same two transects. A total of 120 seeds for each of the three plants on both sites were deposited. Seeds were collected directly on plants. Differences in proportions of seeds dispersed between plant species were tested using Fisher's exact test.

Seed germination

To test for an effect of Argentine ant workers on seed germination, we collected 112 seeds and placed them in petri dishes containing cotton moistened with distilled water. Before the beginning of the experiment, 55 seeds were deposited for 72h in a foraging arena connected to an artificial nest containing an Argentine ant colony. Workers were then allowed to remove seeds to the nest. The remaining 57 seeds were placed in the same container but without any ants. The dishes were placed in a single randomised block. Seeds were germinated at 18±2°C, under day-night conditions, and distilled water was added as needed. Seed germination was recorded every two days during two months, after which no further germination occurred.

Results

Ant assemblages

At the uninvaded site (Portigliolo), in 2008, we caught seven native species (Table 1). Messor minor, Tapinoma sp and Aphaenogaster spinosa were the three most dominant species occurring in 60%, 80% and 50% of the traps respectively. In 2009, we caught five species.

Tapinoma sp and A. spinosa were the most abundant species, occurring in 75% and 55% of the traps respectively. Although we did not catch M. minor workers in 2009 during the 48h trapping session, they were present a few metres away from the A. crispa population. Tapinoma sp was identified as Tapinoma nigerrimum but according to a recent review of this genus, it is probably Tapinoma magnum (Seifert et al. 2017).

At the invaded site, in both years we collected only one species, the Argentine ant. In 2008

Argentine ants occurred in 100% of the pitfall traps whereas in 2009 they were caught in 70% of the pitfall traps (Table 1). The difference in Argentine ant occurrence between the two years may be explained by the flooding of this study site one week before experimentation started.

Offering tests

The rates of seed removal at Portigliolo (Messor minor (n=28), Tapinoma sp (n=28) and

Aphaenogaster spinosa (n=24)) and Cannella (L. humile) were similar, respectively 46/80 (57.5 %) and 41/80 (51 %) (df = 1, P = 0.21, Fisher's exact test). All A. crispa seeds offered to the seed harvester M. minor were systematically removed, taken to the ant's nest more than 1 metre away (range 1 m -3 m) and deposited in the nest (n = 28). Tapinoma sp and the Argentine ant behaved similarly with seeds. Removal of seeds represented 50% and 51.3% of seed interactions for Tapinoma sp and the Argentine ant respectively (df = 1, P = 0.55, Fisher's exact test), but only 29% of A. spinosa seed interactions (Fig. 1). All seeds transported by A. spinosa workers were carried directly to the nest more than one metre away.

Moreover, both Tapinoma sp and L. humile transported seeds over short distances, dropping them along foraging trails. Tapinoma sp workers transported 38% (5/13) of seeds over less than 10 cm, 38% (5/13) between 10 cm and 50 cm and 23% (3/13) between 50 cm and 1 m (Table 2). Linepithema humile transported 41% (14/34) of seeds over less than 10 cm, 32% (11/34) between 10 cm and 50 cm and the remaining 26% (9/34) were transported over distances ranging between 50 cm and 1 m (Table 2). Proportions of seeds dispersed by Tapinoma sp and L. humile over less than 50 cm were similar: 77% (10/13) and 74% (25/34) respectively (df = 1, P = 0.56, Fisher's exact test).

Seed preference and removal events

At the uninvaded site, the proportion of A. crispa seeds dispersed did not differ significantly between years, respectively 28/120 (23%) in 2008 and 19/120 (16%) in 2009 (Fig. 2) (df = 1,

P = 0.

Seed germination

We recorded no significant difference in the number of germinated seeds between the group of seeds that were in contact with Argentine ants for 72h and the control (df = 1, P = 0.19, Chisquare test). At the end of the test, 51% of the seeds that were in contact with Argentine ant workers and 37% of the seeds in the control germinated.

Discussion

The results of our study indicate that the Argentine ant is attracted to, and able to remove, the elaisome-bearing seeds of A. crispa, without affecting seed germination. Argentine ants behaved similarly to the presumed main native seed disperser Tapinoma sp with respect to their inherent seed-removing ability and to the distances of transport. However, at the community level, they collected and dispersed three times more seeds than did all the native ants together. Although their inherent ability to remove seeds does not seem to exceed that of native ants, the proportion of dispersed seeds was higher when Argentine ants were present in 2008. The high densities reached by Argentine ant colonies may explain this result, increasing the likelihood of seed/worker contacts by a factor of nearly five. In 2009, when the Argentine ant occurrence decreased, we recorded similar rates of seed removal between the two sites. Similar results were reported for Solenopsis invicta, which was found to be superior to native species in seed removal at the community level, but not in their inherent ability to exploit resources (Stuble et al. 2010). The presence of a "handle" on A. crispa seeds is also probably an important factor, facilitating their transport by Argentine ant workers despite their weight (20mg mass) (see Fig. 3). By attaching an artificial handle to seeds, [START_REF] Gómez | Ant behaviour and seed morphology: A missing link of myrmecochory[END_REF] demonstrated that this structure allowed small ant species to displace seeds that they had initially been unable to transport.

Comparisons of seed dispersion rates between

The ratio between seed mass and Argentine ant body size usually explains the lack of dispersal of large seeds and the dispersal distances observed in invaded sites [START_REF] Christian | Consequences of a biological invasion reveal the importance of mutualism for plant communities[END_REF]. Dispersal distance generally depends on ant body size, large ants being likely to displace larger seeds and over longer distances than smaller ants (Ness et al. 2004). According to the classification of ants and seeds used by Ness et al. (2004), ants with body length under 5 mm being classified as "small", both Tapinoma sp and the Argentine ant can be considered small ants and M. minor a "large" ant. and A. crispa seeds can be considered "large".

The fact that L. humile and Tapinoma sp, disperses seeds over short distances may not imperil A. crispa populations. If it is of no value to the majority of myrmecochores (e.g. [START_REF] Kjellsson | Seed fate in an ant-dispersed sedge, Carex pilulifera L.: recruitment and seedling survival in tests of models for spatial dispersion[END_REF], seed dispersal over short distances may provide benefits to A. crispa. Indeed, the progeny produced by related individuals shows greater fitness than the progeny produced by unrelated individuals, through outbreeding depression [START_REF] Quilichini | Evidence for local outbreeding depression in the Mediterranean island endemic Anchusa crispa Viv. (Boraginaceae)[END_REF]. Moreover, the fact that both Tapinoma sp and the Argentine ant loses seeds on route to nests may not be a limiting factor for seed germination. It removes direct competition from the mother plant and under the action of wind or rain, abandoned seeds are rapidly covered by a fine layer of sand that may facilitate their germination. The highest germination rates were observed when seeds remained within the top two centimetres of sand [START_REF] Quilichini | Seed dispersal and germination patterns ia a rare Mediterranean island endemic (Anchusa crispa Viv., Boraginaceae)[END_REF]. This is confirmed by our observation of several plants growing along main Argentine ant trails.

It is commonly argued that invasive ants disrupt seed dispersal, leading to a near-complete collapse of dispersal services and a decrease in seedling recruitment [START_REF] Lach | Invasive Ants : Unwanted Partners in Ant-Plant Interactions ?[END_REF]. Reducing predation and competition from related plants, minimizing the effects of fire, and depositing seeds under suitable conditions are the usually benefits that plants receive from native ants [START_REF] Giladi | Choosing benefits or partners: A review of the evidence for the evolution of myrmecochory[END_REF], and which Argentine ants have usually failed to provide (Ness & Bronstein, 2004;Rodriguez-Cabal, Stuble, Nuñez, & Sanders, 2009). However, invasive ants do not systematically interact to the detriment of native plants. In some cases, they even facilitate native species. For instance, Imported Fire ant has been described as responsible for more than half of all removals facilitating dispersal of elaiosome-bearing plant species (Stuble et al. 2010).

Generally, exotic species facilitate native myrmecochores when they replace native ants, fulfil their functional role and provide seeds with a means of escape from enemies or competitors [START_REF] Rodriguez | Can invasive species facilitate native species? Evidence of how, when, and why these impacts occur[END_REF]. Here, in addition to seed removal there is no direct negative effect on seed germination. Furthermore, the Argentine ant may decrease seed predation by displacing granivorous insects like the native ant M. minor. A previous study demonstrated that the presence of the Argentine ant could prevent seed predation by local Messor species [START_REF] Oliveras | A shift in seed harvesting by ants following Argentine ant invasion[END_REF]). Similarly, Solenopsis invicta creates heterogeneity in seed survival by reducing seed predation by arthropods in the vicinity of their nest [START_REF] Chandler | Invasive ants generate heterogeneity in seed survival[END_REF]. Among seed predators, harvester ants are considered primary seed consumers [START_REF] Beattie | Ant-plant interactions[END_REF]. They may decrease seed germination by direct predation [START_REF] Schöning | Seed predation of the tussock-grass Stipa tenacissima L. by ants (Messor spp.) in southeastern Spain: The adaptive value of trypanocarpy[END_REF]) although they could play a dispersal role when they drop seeds in suitable conditions for germination in midtransport or in refuse piles [START_REF] Retana | Dual role of harvesting ants as seed predators and dispersers of a non-myrmechorous Mediterranean perennial herb[END_REF][START_REF] Bulot | Refuse pile turnover by harvester ants (Hymenoptera: Formicidae) increases seed density and seedling species richness in dry grasslands[END_REF]). However, this positive effect may be very limited. For example, ant-dispersed diaspores accounted for only 0.1% of seeds harvested in grasslands by Messor barbarus [START_REF] Detrain | Seed drops and caches by the harvester ant Messor barbarus: Do they contribute to seed dispersal in Mediterranean grasslands?[END_REF]. In the time of the experimentation, we never noted adult plants or seedlings in the immediate vicinity of M. minor nests (e.g. refuse piles), suggesting the species likely negative impact on A. crispa seeds.

Although, the absence of replication due to the very low number and small size of A. crispa populations in Corsica may limit our conclusions, the results obtained from the different experimentations suggest a functional replacement of local ants in A. crispa seed dispersal by the Argentine ant. In the line of this hypothesis, long-term surveys of A. crispa populations showed an increase in the number of individuals over 10 years at the invaded site, Cannella [START_REF] Bosc | Gestion conservatoire des sites du Golfe du Valinco (Site N2000 n°FR9400594[END_REF]. From 2007 onwards, the number of plants is higher at Cannella than at Portigliolo, which was initially the biggest population in Corsica [START_REF] Bosc | Gestion conservatoire des sites du Golfe du Valinco (Site N2000 n°FR9400594[END_REF].

This study has important implications for the management of this rare and threatened endemic plant. Even if the Argentine ant may not be a threat to A. crispa survival, the invaded population needs to be regularly monitored, for example every 5 years. Indeed, the fact that A. crispa seed dispersal relies on one species can make the resilience of this service lower than in the uninvaded site (concept known as the "insurance hypothesis", Yachi and Loreau 1999), especially because invasive ant populations can vary substantially over time and experience dramatic population declines and local extinctions [START_REF] Lester | Booms, busts and population collapses in invasive ants[END_REF].

  A. crispa, an elaiosome-bearing seed species, and three non-elaiosome-bearing seed species confirmed Argentine ants' preference for elaisosome-bearing seeds. Both in 2008 and 2009, A. crispa seeds were by far the most dispersed seeds by Argentine ants. At the uninvaded site, in 2008, the numbers of A. crispa and L. cytisoides seeds dispersed did not differ significantly, whereas in 2009 it was mainly A. crispa seeds that were dispersed. This drastic decrease of seed dispersion rates of nonelaiosome-bearing seed species (22% in 2008 and 2.5% in 2009 of L. cytisoides seeds removed, and 12% and 3.3% seeds removed for S. sericea) between years at the uninvaded site may be explained by the low activity of the harvester ant M. minor during the three days of experimentation in 2009. At the opposite, A. crispa seed removal rates were similar between years (23% in 2008 and 16% in 2009), probably because Tapinoma sp continued to disperse them.

Figure 1

 1 Figure 1 Proportion of ant-seed interactions for the three most abundant native species Messor

Figure 2

 2 Figure 2 Percentage of Anchusa crispa seeds dispersed in both invaded and uninvaded sites, in

Figure 3

 3 Figure 3 An Argentine worker transporting a seed of Anchusa crispa by the handle.

  Fig. 1

Table 2

 2 Number of seeds of all species transported within five distances categories by Tapinoma sp (n=13), Linepithema humile (n=34), Messor minor (n=28), and Aphaenogaster spinosa (n=7).

		Distances of seed removal (cm)
		< 10	10 -50	51 -100	> 100
	Aphaenogaster spinosa	0 (0%)	0 (0%)	0 (0%)	7 (100%)
	Messor minor	0 (0%)	0 (0%)	0 (0%)	28 (100%)
	Tapinoma sp	5 (38%)	5 (38%)	3 (23%)	0 (0%)
	Linepithema humile	14 (41%)	11 (32%)	9 (26%)	0 (0%)

Acknowledgements

This work was supported by a grant from the "Office de l'Environnement de la Corse" and the "Direction Régionale de l'Environnement". Our thanks to M. Sweetko for English language revision.


V Perzynski

Peyre

C Kern

R Aquino

E Dubois

R Perzynski

V Peyre

Dispersions of Magnetic Nanoparticles in the Mixture Ethyleneglycol-Choline Chloride : the Role of Solvent Association

Keywords:

Dispersions of magnetic nanoparticles in the mixture

ethyleneglycol-choline chloride: The role of solvent association

Introduction

Dispersions of nanoparticles (NPs) in ionic liquids (ILs) have drawn recently huge attention. The electrochemical and conducting properties of the ILs can indeed be combined with specific (electronic, magnetic, optical...) properties of the nanoparticles to obtain multiple stimuli responsive materials with possible applications in electrochemistry, synthesis, catalysis, separation fields [START_REF] He | Nanoparticles in ionic liquids: interactions and organization[END_REF][START_REF] He | Ionic liquid and nanoparticle hybrid systems: Emerging applications[END_REF].

However, one serious drawback of such solvents is their price, toxicity and availability. For this reason, deep eutectic solvents (DES) and related complex solvents can be interesting alternatives. These liquids are made of a quaternary ammonium salt (choline chloride in our case) mixed with a molecular hydrogen bond donor (HBD). Since the molar ratio of ammonium salt in the final liquid is huge, the new solvent is closer to an ionic liquid than to an electrolyte solution. Indeed, the physicochemical properties of DESs such as negligible vapor pressure, thermostability, wide electrochemical potential windows, tunability are similar to those of the traditionally used ionic liquids. DESs can also have the ILs' drawbacks such as high viscosity [START_REF] Zhang | Deep eutectic solvents: syntheses, properties and applications[END_REF]. Since the interest for DESs is more recent than that for standard ILs, data about both the DESs and dispersions of particles in DESs are still scarce [START_REF] Abo-Hamad | Potential applications of deep eutectic solvents in nanotechnology[END_REF][START_REF] Mjalli | Effect of nano-particles on the rheological properties of Reline[END_REF]. The complexity of the solvent, made of three components (the cation, the anion and the HBD) makes any prediction about colloidal stability in DESs impossible in the actual state of our knowledge. For this reason and due to the interesting properties of DESs such as cost and availability there is a huge need for experimental data about dispersions of NPs in DESs.

In this work, we use a solvent mixture made of choline chloride (ChCl), a widely used, cheap, bio-sourced component, mixed with ethyleneglycol (EG) as HBD. Thanks to its composition, it is closely related to DES and is sometimes classified among them although it is not an eutectic mixture on the strict basis of melting temperatures. This solvent will be named ChEG hereafter. Though somehow toxic, EG is chosen since it can form with ChCl a liquid of low viscosity at room temperature (around 35 mPa.s at 20 • C for ChCl:EG (1:3 in mol:mol) compared to more than 1000 mPa.s for the widely used ChCl:urea (1:2) [START_REF] Yadav | Densities and Viscosities of (Choline Chloride + Urea) Deep Eutectic Solvent and Its Aqueous Mixtures in the Temperature Range 293[END_REF]). Based on the analysis of the conductivity and viscosity, it was shown [START_REF] Abbott | Application of Hole Theory to Define Ionic Liquids by their Transport Properties[END_REF] that ChCl: EG mixtures switch from a dilute salt solution behavior to an ionic liquid behavior for ChCl:EG composition about 1:4 (20 mol% = 36 vol% of ChCl). For this reason, a composition slightly higher (1:3 -25 mol% = 43 vol% of ChCl) is used in this work.

A dispersion of NPs in ChEG could be obtained by direct synthesis in the complex solvent. This leads to NPs with surface properties different from the synthesis in water and to enhanced catalytic properties [START_REF] Chen | Ionothermal synthesis of Fe3o4 magnetic nanoparticles as efficient heterogeneous Fenton-like catalysts for degradation of organic pollutants with H2o2[END_REF][START_REF] Huang | Ionothermal synthesis of Cu-doped Fe3o4 magnetic nanoparticles with enhanced peroxidase-like activity for organic wastewater treatment[END_REF]. However, the nature of the surface is not wery well understood and controlled . Morever, it can be difficult to eliminate the side-products of the NPs synthesized in ChEG.

Here, we chose a NPs synthesis in an aqueous medium which is easier, cheaper and already well known. Therefore, a transfer to ChEG after a synthesis in water appears more advantageous, all the more so since the interface NPs-solvent can be controlled and adapted during this step. The chosen nanoparticles are iron oxide maghemite particles (diameter ≈10nm), synthesized by coprecipitation in alkaline aqueous medium [START_REF] Massart | Preparation of aqueous magnetic liquids in alkaline and acidic media, I.E.E.E[END_REF][START_REF] Mamusa | Microstructure of colloidal dispersions in the ionic liquid ethylammonium nitrate: influence of the nature of the nanoparticles' counterion[END_REF][START_REF] Lucas | Surface charge density of maghemite nanoparticles: Role of electrostatics in the proton exchange[END_REF]. Their surface state can be tuned to monitor the surface charge by changing the pH, or by adsorption of various ligands such as citrate or polyacrylate. Such particles have already been successfully dispersed in an ionic liquid, ethylammonium nitrate [START_REF] Mamusa | Microstructure of colloidal dispersions in the ionic liquid ethylammonium nitrate: influence of the nature of the nanoparticles' counterion[END_REF][START_REF] Mamusa | Tuning the colloidal stability in ionic liquids by controlling the nanoparticles/liquid interface[END_REF][START_REF] Mamusa | Concentrated assemblies of magnetic nanoparticles in ionic liquids[END_REF][START_REF] Guibert | Controlling nanoparticles dispersion in ionic liquids by tuning the pH[END_REF]. The particles being magnetic, the observed stable dispersion is indeed a ferrofluid (FF). The best colloidal stability was obtained for particles coated with citrate or poly(acrylate), using Na + as a counterion [START_REF] Mamusa | Microstructure of colloidal dispersions in the ionic liquid ethylammonium nitrate: influence of the nature of the nanoparticles' counterion[END_REF][START_REF] Guibert | Controlling nanoparticles dispersion in ionic liquids by tuning the pH[END_REF]. The stability was ensured by the formation of layers of solvent ions around the particles, providing a steric hindrance to particles approach [START_REF] Mamusa | Concentrated assemblies of magnetic nanoparticles in ionic liquids[END_REF]. The determining effect of small amounts of water was also evidenced, leading to enhanced flocculation by destructuration of these protective layers [START_REF] Mamusa | Concentrated assemblies of magnetic nanoparticles in ionic liquids[END_REF].

In the present work, we transfer similar citrate-coated NPs (Na + counterions) from water to ChEG. In the first part, we detail the samples and techniques used. We then analyse the solvent -mixture ChCl-EG (1:3) -by measuring viscosity, density and refractive index in the temperature range 20-45 • C, as a function of the water content. These parameters are indeed necessary for the analysis of the obtained colloidal dispersions in ChEG. A model to describe the results is proposed. This solvent is subsequently used to disperse maghemite nanoparticles. We assess the colloidal stability by means of visual observations, optical microscopy, small angle X-ray scattering (SAXS), magnetization measurements, viscosimetry, and dynamic light scattering. Finally, we discuss the results and conclude on the possible influence of a local viscosity different from the macroscopic one to understand the data.

Materials and methods

The following products are purchased and used as received: choline chloride (Sigma Aldrich); ethyleneglycol (Normapur, VWR); nitric acid (69.5% water solution, Carlo Erba); hydrochloric acid (37% water solution, AnalaR Normapur, VWR); citric acid (H 3 Cit, Rectapur, Prolabo); iron(II) chloride (AnalaR Normapur, VWR); iron(III) chloride (Prolabo); iron(III) nitrate (Technical, VWR); trisodium citrate (Na 3 Cit, Merck); acetone (technical, VWR); diethyl ether (99.8% pur., AnalaR Normapur).

Water content is analyzed using coulometric Karl Fisher titration (Schott TitroLine KF Trace). Density is measured thanks to an Anton Paar DSA 5000M, with precision 10 -6 g/mL and temperature accuracy better than 10 -3 • C. Viscosity of ChEG and ChEG/water mixtures is determined using an Anton Paar Automated MicroViscometer (AMVn) instrument by using the falling ball model in a 1.8 mm (resp. 3.0 mm) capillary with a 1.5 mm (resp 2.5 mm) falling ball. The instrument is first calibrated with reference standard oils N14 (resp. N44) from Cannon Instrument Company provided by Anton Paar (η =71.29 mPa• s (resp. [START_REF]CRC Handbook of chemistry and physics[END_REF].68 mPa• s), ρ = 0.8249 g/mL (resp. 0.8092 g/mL) at 25 • C)). Dynamic light scattering is performed on a Vasco instrument from Cordouan Technologies, operating at λ = 656 nm, with a temperature control better than 0.1 • C. The instrument uses a very thin film (around 200 µm) and a backscattering detection at θ = 135 • , which avoids multiple scattering and allows measurements in optically absorbing samples such as ferrofluids. The corresponding scattering vectors Q values are Q = 2.3 10 -3 Å -1 in water and 2.5 10 -3 Å -1 in ChEG. The index of refraction is measured with a refractometer Arago from Cordouan Technologies at 656 nm, with a temperature control better than 0.1 • C. The samples are observed on the micron scale with an optical microscope (Olympus BX51 with lens 40x/0.6 and camera x5). Small angle X-rays scattering (SAXS) is performed at the SWING beam line of synchrotron Soleil (France). Two different configurations are used, giving access to the Q-range 2.10 -3 Å -1 < Q < 0.4 Å -1 . The beam energy of 7 keV (corresponding to λ= 1.77 Å) is chosen to avoid X-ray absorption by iron. Standard correction procedures are applied for sample volume, empty cell signal subtraction and detector efficiency to obtain the scattered intensity in absolute scale (cm -1 ). Such data reduction is done using the software Foxtrot R . Magnetization is registered using a home-made Vibrating Sample Magnetizer (VSM [START_REF] Foner | Versatile and Sensitive Vibrating?Sample Magnetometer[END_REF][START_REF]Magnetic Fluids and Applications Handbook[END_REF]). The stability under magnetic field is checked from the absence of diffraction pattern when the sample is illuminated by a He-Ne laser under the field of an electromagnet. Above the threshold of stability, a diffracted line appears in the direction perpendicular to the magnetic field due to the formation of concentrated phase needles resulting from a phase separation [START_REF] Bacri | Ionic ferrofluids: a crossing of chemistry and physics[END_REF].

Choline Chloride-Ethyleneglycol (ChEG)

The complex solvent chosen here, based on choline chloride (ChCl) and ethyleneglycol (EG) is prepared by simple mixture of the two components at 60 • C in closed bottles in order to avoid water uptake. No liquid mixture could be observed for a molar ratio ChCl:EG of composition (1:1)(50mol%) even at high temperature and crystals appeared upon cooling for the ratio (1:2)(33mol%). For this reason and according to Abbott's criterion to obtain an ionic liquid behaviour [START_REF] Abbott | Application of Hole Theory to Define Ionic Liquids by their Transport Properties[END_REF], we chose to work with a molar ratio (1:3). The properties of both components and resulting solvent are given in Table 1. The amount of water in the prepared solvent, determined from Karl Fisher titration, is 0.2 w%.

Only few data are available in literature (see Table 1), with scattered values. Our density is close to the results of [START_REF] Abbott | Application of Hole Theory to Define Ionic Liquids by their Transport Properties[END_REF]. However, the viscosity we obtain is much higher. Since the water contents can be of importance for the viscosity, we investigate the density and viscosity of the ChEG, in the temperature range 20-45 • C (Figure 1 added water. Unless a huge amount of water is present in literature samples, this cannot be the reason for the observed viscosity discrepancies 1 . The evolution of the viscosity with the water content of Fig. 2 shows the importance of controlling this parameter. Indeed these solvents are hygroscopic. We left a vial open to the ambient air and recorded the mass increase, attributed to atmospheric water uptake, for several days. The water uptake is very low (+0.24 w%) during the first four hours. It is thus possible to work quickly with short opening of the containers for the necessary handling of ChEG. In the long run, the water uptake is +13 w% after 3 weeks. Thus the solvent must be 1 Another reason for the low viscosity values found in the literature could be a degradation of the solvent, for example due to a heating at too high temperature, as this solvent presents a thermal decomposition above 100 • C [START_REF] Abbas | Synthesis and characterization of choline chloride based binary mixtures[END_REF] stored in appropriate conditions to avoid water contamination. The subsequent elimination of water by freeze-drying for instance is not possible, since EG is eliminated as well.

Dependence of the density on the water-content as a

function of temperature Molar volume of the solvent (here, wet ChEG) V slvt can be calculated from experimental density values ρ for each temperature according to :

V slvt = M slvt /ρ (1)
(M slvt is the molar mass of the solvent)

The quantity of water is taken into account in M slvt through:

M slvt = (1 -x)M ChEG + xM H 2 O ( 2
)
where x is the molar fraction of water deduced from Karl-Fisher titration, M ChEG and M H 2 O are, respectively, the molar mass of the pure ChEG (table 1) and water. V slvt vs water content x is plotted in supporting information (Fig. SI-1), for temperatures between T =20 • C and 45 • C. The plots are linear and can be approximated for each T by :

V slvt = V * ChEG + x(V ∞ H 2 O -V * ChEG ) (3)
where V ∞ H 2 O is the partial molar volume of water at infinite dilution and V * ChEG is the molar volume of pure (dry) ChEG, at the given temperature T . The y-intercept is thus V * ChEG at T . From the slopes of the curves, one can calculate V ∞ H 2 O as a function of T . Table 2 reports these V * ChEG and V ∞ H 2 O as well as that of the molar volume of pure water V * H 2 O calculated from [START_REF] Tanaka | Recommended table for the density of water between 0 • C and 40 • C based on recent experimental reports[END_REF], in the temperature range explored. V ∞ H 2 O is smaller by 1 cm 3 compared to V * H 2 O at 20 • C and stays small whatever the temperature. 3

The same decrease was observed for water dissolved in pure EG [START_REF] Egorov | Volumetric properties of the water-ethylene glycol mixtures in the temperature range 278333.15 k at atmospheric pressure[END_REF] and can be attributed to a more efficient packing of water in the mixture with less H bonds around H 2 O that in bulk water. The expansion of V slvt with temperature can be quantified with the volumetric isobaric thermal expansion coefficient, defined as α P = (1/V slvt )(∂V slvt /∂T ) P . The coefficient α P values are between 5.27 10 -4 and 5.36 10 -4 K -1 in the temperature range explored whatever the water content. These values are close to those determined for the mixture ChCl/EG (1:2) (5-6 10 -4 K -1 ) [28] and a bit smaller than the one of EG (6.4 10 -4 K -1 [START_REF] Nain | Ultrasonic and viscometric studies of molecular interactions in binary mixtures of formamide with ethanol, 1-propanol, 1,2-ethanediol and 1,2-propanediol at different temperatures[END_REF]). This indicates that ChEG is a slightly more associated liquid than pure EG.

Dependence of viscosity on the water-content as a function of temperature

The temperature-dependence of macroscopic viscosities η of ChCl:EG (1:3) mixtures (Fig. 2) can be described with Eyring's model [START_REF] Eyring | Viscosity, Plasticity, and Diffusion as Examples of Absolute Reaction Rates[END_REF] for each water content as:

η = hN A /V slvt exp(∆G a /RT ) (4)
with h, the Planck constant, N A , the Avogadro number and ∆G a the activation Gibbs energy for the viscous flow. From equation 4, ∆G a is obtained for each x and T using the experimental η and V slvt . In a first step, let us analyse ∆G a at 298 K as a function of the water molar fraction x. We obtain a linear function, which can be described with ∆G a = -22,9x + 21,5 (kJ/mol). The y-intercept (21.5 kJ/mol) corresponds to ∆G a value for pure (dry) ChEG. The presence of water lowers the activation Gibbs energy for the flow, i.e., the energy barrier to overcome for the fluid to flow is smaller. This reduction of ∆G a can originate either from a smaller enthalpy contribution (intermolecular interactions in the fluid are smaller) or from an entropic contribution (more configurations are accessible) during the flow process according to Eyring's hole theory for viscosity.

In a second step, the analysis of ∆G a as a function of T for each water content gives the activation flow enthalpy and entropy from ∆G a = ∆H a -T ∆S a . When a molecule is on its way to change location (producing a flow), it must overcome an energy barrier made of two contributions. First, it must fight against the loss of interaction energy with its neighbours of the initial place : this is quantified as ∆H a . Second, since it is not linked any more to its neighbours, it gains rotation entropy. This 3) with different water contents (in weight percent). 0% was obtained using extrapolated values of molar volumes and viscosities to null water content. Values for pure EG are from [START_REF] Nain | Ultrasonic and viscometric studies of molecular interactions in binary mixtures of formamide with ethanol, 1-propanol, 1,2-ethanediol and 1,2-propanediol at different temperatures[END_REF]. For the ferrofluid, the molar volume is that of the driest ChEG (0.2% water) with the local viscosity discussed in the text.

is ∆S a . The flow will be easier if its activation energy ∆G a is low, e.g., if ∆H a and -T ∆S a are low (high ∆S a ). These contributions can be determined according to equation ( 4) supposing ∆H a and ∆S a are independent of T. Rln[ηV slvt /hN A ] = ∆G a /T plotted vs (1/T ) (Eyring plot, figure 3) is a straight line which gives access to ∆H a (slope) and ∆S a (y-intercept) (Table 3). This is applied for each water content as well as for the extrapolated "pure dry solvent" using V * ChEG and viscosity data linearly extrapolated to zero water content (data in SI). All ∆H a values are scattered around the mean value 29.5 ± 0.5 kJ/mol, close to the enthalpy of activation of pure EG (30.0 kJ/mol) [START_REF] Nain | Ultrasonic and viscometric studies of molecular interactions in binary mixtures of formamide with ethanol, 1-propanol, 1,2-ethanediol and 1,2-propanediol at different temperatures[END_REF]. This implies that, for the viscous flow, the formation of an activated species in the solvent made of ChCl and EG is probably related to the same kind of intermolecular forces as in pure EG. The presence of some water does not deeply modify this interaction energy. It is coherent with a view of the cohesion of ChEG due to H bonds between all three components (EG, ChCl and water).

∆S a values are scattered around 28.5 ± 1.5 J mol -1 K -1 (or -T ∆S a = -8.5 ± 0.6 kJ mol -1 at 298K) for all water contents. The entropic contribution -T ∆S a to ∆G a is negative and is about one third of ∆H a as it is also observed for pure EG (-T ∆S a = -10.7 kJ mol -1 at 298K [START_REF] Nain | Ultrasonic and viscometric studies of molecular interactions in binary mixtures of formamide with ethanol, 1-propanol, 1,2-ethanediol and 1,2-propanediol at different temperatures[END_REF]) . The entropy term is high and lowers the ∆G a value. However, the solvent, i.e. the mixture of EG and ChCl, is still a structured solvent that needs to locally destructurate in order to flow, as pure EG. A more detailed analysis of the contributions of -T ∆S a and ∆H a to ∆G a with the water content in ChEG is delicate since a small experimental imprecision on the slope determination in Figure 3 (i.e., ∆H a ) has a huge impact on the y-intercept ∆S a .

This set of data on the ChCl:EG (1:3) mixture provides a good picture of this new solvent, scarcely described in the literature, here studied within several conditions of water uptake. Its stability in the long run for T< 30

Dispersions in ChEG

The dispersions are produced by transfer of the NPs from water, solvent used for their synthesis, to ChEG. When a black liquid is obtained as seen by eye, it has to be analyzed further to evaluate the quality of the dispersion and its nanostructure. Micrometric structure is analyzed by optical microscopy. If the dispersion is homogeneous at this scale, small-angle X-ray scattering (SAXS), magnetization measurements and dynamic light scattering (DLS) are used.

Preparation of the dispersions

Iron oxide nanoparticles are synthesized by a routine process of coprecipitation of ferric and ferrous iron ions in alkaline aqueous medium [START_REF] Massart | Preparation of aqueous magnetic liquids in alkaline and acidic media, I.E.E.E[END_REF][START_REF] Mamusa | Microstructure of colloidal dispersions in the ionic liquid ethylammonium nitrate: influence of the nature of the nanoparticles' counterion[END_REF][START_REF] Lucas | Surface charge density of maghemite nanoparticles: Role of electrostatics in the proton exchange[END_REF]. It produces magnetite (Fe 3 O 4 ) nanoparticles then fully oxidized to maghemmite (γFe 2 O 3 ) which is chemically stable. Final dispersed nanoparticles with a diameter of the order of 10 nm are obtained in acidic aqueous medium (pH = 1.5). At this stage, the surface charge is positive (Fe-OH + 2 ) and counterbalanced by NO - 3 anions. The volume fraction of the NPs in the aqueous stock solution is Φ v = 2.25v%, as deduced from the determination of the iron content by Flame Atomic Absorption Spectroscopy and using a density of 5g/cm 3 for maghemite. Before their transfer to ChEG, the surface of the particles is covered by citrate ions, using the procedure described in [START_REF] Filomeno | Tuning the Solid/Liquid Interface in Ionic Colloidal Dispersions: Influence on Their Structure and Thermodiffusive Properties[END_REF]. Briefly, citric acid (H 3 Cit) is added to the acidic dispersion which leads to flocculation. HNO 3 is then removed by repeated additions/ removals of H 3 Cit at 0.020 mol/L. The pH is then increased between 7.0 and 7.5 by addition of NaOH. The final particles are thus covered with citrate ions, Na + being the positively charged counterions. This dispersion of citrate-coated NPs will be referred to as "aqueous dispersion" from now on.

If the particles are to be transferred in the ChEG, the aqueous dispersion is first flocculated by addition of acetone, washed 3 times by mixtures of acetone:water (9:1) and once by ether. After the removal of the ether, the desired amount of ChEG is added, and the dispersion is gently heated to remove any residual organic solvent.The dispersion is then centrifuged 10 minutes at 4000 rpm to get rid of the remaining few undispersed particles.

Structure by optical microscopy and Small Angle X-Ray Scattering (SAXS)

Two samples are compared, in water and in ChEG. As the aqueous systems have been extensively studied in several conditions and their colloidal stability is well controlled (see for example [START_REF] Mériguet | Magnetic fluids with tunable interparticle interaction : monitoring the under-field local structure[END_REF] and references therein), the dispersion in water is used as a reference. On the nanoscale, the structure is determined by SAXS. The scattered intensity I is recorded as a function of the scattering vector Q. For very dilute NPs dispersion, I(Q) is characteristic of the shape of the scattering objects over the whole Q range (form factor). For more concentrated dispersions, the nanoparticles undergo colloidal interactions. The intensity at small Qs is then representative of the structure factor S(Q) of the dispersion while at high Qs, it is still dominated by the form factor. A dilute aqueous sample is used for the determination of the form factor. It is obtained by dilution of the initial aqueous dispersion in HNO 3 10 -2 mol L -1 . Figure SI-2 plots the scattered intensity versus the scattering vector Q in absolute units. Figure 4 shows the corresponding Porod plot I×Q 4 , normalized by the volume fraction φ v and the contrast ∆ρ 2 , versus Q. The volume fraction is taken from the invariant of diffusion : φ v is 0.061 vol% for the reference sample in water and 1.13 vol% for the sample in ChEG. This representation I×Q 4 enhances the high Qs region and allows the comparison of the form factors. The curves in water and in ChEG are very similar, indicating that the same particles are present in both solvents. At such a low concentration of NPs in water, we can assume that the interparticle interactions are negligible. We adjust the curve I(Q) with a form factor of polydisperse spheres and a lognormal distribution. The scattering length densities ρ necessary to calculate the contrast ∆ρ2 are known (NPs : 40.6 10 10 cm -2 ; water 9.3 10 10 cm -2 ; ChEG 10.2 10 10 cm -2 ) therefore the only parameters are the median diameter d S AXS and the polydispersity σ S AXS . We use the software provided by the NIST 2 to model the experimental curves [START_REF] Kline | Reduction and analysis of SANS and USANS data using IGOR Pro[END_REF]. The fits are performed in the I×Q 4 representation (Figure 4). Good fits are obtained in water for d S AXS = 6.0 nm, σ = 0.43 and in ChEG for d S AXS = 6.0 nm, σ= 0.40. These results confirm that the size distribution is little affected by the solvent transfer. The quality of the fit is poorer at small Q values, indicating the presence of some aggregates in both cases (Figure SI-2).

The colloidal stability in ChEG means that some interparticular repulsions are present to counterbalance the van der Waals attraction. Information can be gained on the interactions from the scattered intensity at small Qs. The structure factor of the ChEG dispersion is calculated using the diffractogram in water (i.e., at very low concentration) as form factor :

S (Q) = [I ChEG (Q)/φ v,ChEG ]/[I water (Q)/φ v,water ]
. Indeed, we know from previous studies that interparticular interactions in water are negligeable at this φ [START_REF] Dubois | Liquid-gas transitions in charged colloidal dispersions: Small-angle neutron scattering coupled with phase diagrams of magnetic fluids[END_REF]. It is reported on figure 5. It presents an initial value S(0) lower that 1 and no structure peak at larger Qs, which is indicative of short range repulsions. The order of magnitude of the repulsion is similar to what was observed with the same kind of particles (maghemite covered with citrate, Na + counterions) dispersed in the ionic liquid ethylammonium nitrate [START_REF] Mamusa | Microstructure of colloidal dispersions in the ionic liquid ethylammonium nitrate: influence of the nature of the nanoparticles' counterion[END_REF].

Magnetization measurements

As the NPs are magnetic, the analysis of the magnetization curve of the dispersions can give information on the size distribution. However, the stability under field is checked first: both samples are stable up to 1350 Oe= 108 kA m -1 at room temperature. Above this threshold, the destabilization occurs in a few ten minutes. Therefore, the magnetization versus magnetic field is determined using a sequence in high fields as quick as possible in order not to destabilize the system during the measurement. The curves, plotted in Figure 6, can be fitted with the Langevin's equation coupled with a lognormal distribution of NP diameters [START_REF]Magnetic Fluids and Applications Handbook[END_REF]. We get here 314 kA/m for the magnetization of the material, d o = 7.1 nm and σ= 0.38 for the sample in water and d o = 6.8 nm and σ= 0.35 for the sample in ChEG. This distribution in ChEG is thus very close to the initial distribution in water. It means that only few large particles are removed during the transfer in ChEG, as was already concluded from the SAXS measurements.

Dynamic Light Scattering (DLS)

Much more accessible than SAXS, DLS is a laboratory experiment which enables studies with temperature, as well as aging and reversibility exploration. The light scattered by the sample is recorded versus time and an autocorrelation function is calculated. It measures how fast a NPs forgets its initial position thus giving access to a diffusion coefficient, which can be interpreted using appropriate models. Here, the measured intensity autocorrelation function g 2 (t) is converted into the field autocorrelation g 1 (t) function using : g 1 (t) = [g 2 (t)] 1/2 -offset. The function g 1 (t) of the studied samples can be fitted with a stretched exponential g 1 (t) = A × exp[-(t/τ) β ] and an exponent β close to 1, however not perfectly, especially at long times. Henceforward, we choose (i) to fit the curves for times corresponding to g 1 (t) > 0.1, without taking into account the tail which is less reliable, (ii) to use a mono-exponential, i.e., β= 1, in order to reduce the number of fitting parameters to only one. This corresponds to the simplest analysis, with a monodisperse population. The characteristic time τ is related to the translational diffusion coefficient D t = (τQ 2 ) -1 . 4 (and Figure 7), the diffusion coefficient D t increases with temperature, in a reproducible manner, even for 6 months old samples for which no evolution is seen, and going back to its value at 25 • C after a cycle 25 When interparticle interaction is negligible, the translational diffusion coefficient D t is related to the hydrodynamic diameter d H by the Stokes Einstein's equation

D t = k B T 3πη 0 d H (5)
where k B is the Boltzmann constant, T the absolute temperature, η 0 the local solvent shear viscosity. Since D t is obtained by DLS as an intensity average, the corresponding hydrodynamic diameter obtained through equation 5 is a mean value defined as d H =< d 6 > / < d 5 > [START_REF]Neutrons, X-rays and light: scattering methods applied to soft condensed matter[END_REF]. It can be compared to the value calculated from the lognormal distribution obtained from SAXS, d calc H . In the case of the aqueous FF-sample, using for η 0 the macroscopic viscosity of water in Eq. 5, we obtain d calc H =17.0 nm in reasonable agreement with the DLS experimental results d H =25 nm (see Table 4). The DLS value is indeed somehow higher due to the presence of small aggregates (see SAXS section).

However in the case of the ChEG FF-sample, this agreement fails if η 0 is assimilated to the ChEG macroscopic viscosity in Eq. 5. The apparent diameter d app H is then found much smaller than d calc H . Moreover, the size is unexpectedly temperature dependent.

Several arguments can be put forward to explain these observations. (i) A much lower polydispersity in the ChEG-FF sample could be an explanation, however not consistent with the SAXS results. (ii) Interparticle interactions could modify the apparent diffusion coefficient in the ChEG FF-sample, however the observed size decrease would correspond to a strong repulsion which is not compatible with SAXS data. (iii) A bad evaluation of the refractive index could be put forward, however the refractive indexes of liquids usually range roughly between 1.3 and 1.5 and their variation cannot explain such a large difference. (iv) The viscosity is not correct. Indeed, viscosity highly depends on the solvent composition. However a macroscopic water pollution cannot explain the results as a huge amount of water (more than 20% ) would be necessary to induce such a large decrease of the macroscopic viscosity.

The only possibility in this ChEG-FF sample, is thus a lower local viscosity around the NPs than the macroscopic ChEG viscosity. A direct comparison of the diffusion coefficients is presented in Figure 7. It plots the temperature-dependence of the experimental diffusion coefficients D ChEG t,exp and compares them with calculated values, using the sample in water at 25 4). We observe that this local viscosity is much lower than the macroscopic one and presents a less pronounced temperature dependance.

Analysis of the temperature-dependence and origin of the local viscosity in the ChEG-FF sample

The analysis with the model of Eyring of the temperature dependence of the local viscosity, considering that the molar volume of the solvent is unaffected by the presence of the particles, gives contributions to the flow Gibbs energy very different from those of pure ChEG (Figure 3 and Table 3). If at 298K the Gibbs energy value ∆G a is similar in the ChEG-FF and in the complex solvent (around 19 kJ/mol), its splitting into enthalpy ∆H a and entropy ∆S a is completely modified. Indeed, ∆H a is lowered from 30 to 10 kJ/mol while -T ∆S a changed sign from -8.5 kJ/mol to + 9 kJ/mol. This implies that the particles deeply modify the solvent organization around them, both from the solvent-solvent interaction point of view (a decrease of ∆H a means that a lower number of interactions between solvents molecules must be modified for the fluid to flow) and from the entropic point of view (higher -T ∆S a means that the movement of a solvent entity implies a loss of accessible configurations).

Ionic liquids are known to be strongly organized in hydrophilic and hydrophobic domains [START_REF] Hayes | Amphiphilicity determines nanostructure in protic ionic liquids[END_REF] and the DES made of choline chloride-urea was recently shown to be highly structured through H-bonds implying all its three components (Cl-, Cholinium, urea) [START_REF] Hammond | Liquid structure of the choline chloride-urea deep eutectic solvent (reline) from neutron diffraction and atomistic modelling[END_REF]. By analogy with ChCl-urea, ChCl-EG can be assumed structured. In such a structured solvent, the nanoparticles can preferentially accommodate one of the solvent components and disturb the network. In this case, the viscosity experienced by the particles is not the macroscopic viscosity of the solvent, but the local viscosity of the surrounding fluid in that domain. In our case, this local viscosity is much lower than the macroscopic one, our results suggesting that the less viscous fluid, ethyleneglycol, directly surrounds the particles.

Conclusion

The density and viscosity of a mixture of choline chloride and ethylene glycol (1:3) a complex solvent close to DESs, were carefully determined from 20 to 45 • C and for water contents between 0.2 and 4.6%. The analysis of the data using the model of Eyring indicates that this highly structured solvent needs to destructurate to flow. This is quantified by the entropic contribution that is more important than the enthalpic one in the activation Gibbs energy for viscous flow. In a second part, we demonstrate that maghemite nanoparticles can be dispersed in this complex solvent thanks to a transfer from water. They are well dispersed, interacting through weak repulsions sufficient to stabilize the dispersions in zero magnetic field and under field up to 1350 Oe. This threshold could be increased using smaller particles and reduced polydispersities. These results are similar to those previously obtained in an ionic liquid, EAN and confirm that the mixture ChEG studied here behaves like an ionic liquid. Indeed, a behaviour of concentrated electrolyte solution would not allow the dispersion of NPs which would flocculate due to the screening of electrostatic interactions. The local viscosity experienced by the NPs in ChEG dispersions as well as its temperature variation indicates that the particles locally modify the solvent around the NPs. This could be a local modification of organization or of composition. The results can probably be extended to other oxide nanoparticles, with the same kind of surface groups (hydroxides).

To go further, quantitative macroscopic viscosity measurements, difficult as samples are black and hygroscopic, would be interesting to compare to microscopic viscosities. Also parallel SAXS measurements as a function of temperature would enable to get a clear picture of the local behavior of the samples.

Figure 1 :

 1 Figure 1: Density versus temperature for the solvent ChCl:EG (1:3) with different water contents (in weight percent).

Figure 2 :

 2 Figure 2: Viscosity versus temperature for the solvent ChCl:EG (1:3) with different water contents (in weight percent).

  73.035 73.230 73.425 73.620 73.817 V ∞ H 2 O 16.994 17.083 17.174 17.266 17.361 17.487 V * H 2 O 18.032 18.053 18.079 18.108 18.141 18.178 Table 2: Molar volume of pure ChEG V *ChEG (cm 3 /mol) , partial molar volume of water at infinite dilution in the ChEG-water mixture V ∞ H2O , and molar volume of pure water V * H2O for various temperature T ( • C). Assuming an ideal mixture, V *ChEG is consistent with a molar volume of 72.0+/-1.5 cm 3 /mol at 20 • C which can be estimated from the pure component volumes with an imprecise density of choline chloride.

Figure 3 :

 3 Figure 3: Eyring's plot for the solvent ChCl:EG (1:3) with different water contents (in weight percent). 0% was obtained using extrapolated values of molar volumes and viscosities to null water content. Values for pure EG are from[START_REF] Nain | Ultrasonic and viscometric studies of molecular interactions in binary mixtures of formamide with ethanol, 1-propanol, 1,2-ethanediol and 1,2-propanediol at different temperatures[END_REF]. For the ferrofluid, the molar volume is that of the driest ChEG (0.2% water) with the local viscosity discussed in the text.

Figure 4 :

 4 Figure 4: Porod (I×Q 4 ) representation of SAXS data, in absolute unit. ( red crosses) Bare particles in aqueous HNO 3 (pH = 2) : Φ v = 0.061v% (purple triangles) Citrate-coated particles in ChEG : Φ v = 1.13v%. Fits were obtained with a lognormal distribution for the sizes of the particles. See text for details.

Figure 5 :

 5 Figure 5: Structure factor S(Q) of the dispersion in ChEG at Φ v = 1.13v% , using the diffractogramme in water as form factor

Figure 6 :

 6 Figure 6: Normalized magnetization versus magnetic field for the samples in water (squares) and ChEG (circles).

Figure 7 :

 7 Figure 7: Diffusion coefficients D t from DLS measurements. Points : experimental values D exp t,ChEG in ChEG at various T. Dashed line : calculated values D calc t,ChEG using the measured viscosities of ChEG. Solid line : calculated values D calc t,EG using the viscosity of pure EG .

1 .

 1 Hydrodynamic NP diameter and local viscosity in the ChEG-FF sample

Table 1 :

 1 and 2, data in SI) for various quantities of Properties of the individual components of ChEG and of ChEG at T=25

	Properties	Ethylene	Choline	ChCl:EG
		glycol (EG)	Chloride (ChCl) 1:3 (ChEG)
	M (g/mol)	62.07	139.63	81.46
	Melting T ( • C)	-13 [19]	247 [20]	-8 [21]
	Boiling T ( • C)	197.3 [19]	decomposition
	Density	1.10988 [22]	1.1 [20]	1.1367 [21]
	(g/cm 3 )	1.11004 [23]	(for 70w%	1.118 [7]
			in water)	1.115167*
	Viscosity	17.13 [24]	-	26 [21]
	(10 -3 Pa/s)			19 [7]
				31.6*
	Refractive	1.4318 (20 • C)	-	1.456*
	index	[19]
	Molar volume	55.92 [22]	127 ± 5 [20]	71.66 [21]
	(cm 3 /mol)			72.86 [7]
				72.53*

• C except when another temperature is specified. * = this work

Table 3 :

 3 Flow activation Gibbs energy, enthaly and entropy derived from viscosity for the ChEG solvent with 0.2% water, for EG and for the ferrofluid in ChEG. * this work regular checking of the samples over several months.

  • C -45 • C -25 • C.

	Solvent	Water ChEG ChEG ChEG
	T ( • C)	25	25	35	45
	D t,exp	1.92	1.43	1.65	1.97
	(m 2 s -1 )	10 -11 10 -12	10 -12	10 -12
	η macro	0.89	31.6	21.6	14.5
	(mPa.s)
	d app H (nm)	25.5	9.6	12.4	16.8
	η local from		12	10.7	9.2
	DLS (mPa.s)

Table 4 :

 4 Experimental diffusion coefficients D t,exp , macroscopic viscosities η macro , corresponding hydrodynamic diameter d H . Viscosities η local are determined from DLS for the samples with the hypothesis that d H = 25.5 nm for all T in ChEG.

NIST : National Institute of Standards and Technologyhttps://www.ncnr.nist.gov/programs/sans/data/red anal.html

We thank Laurent Michot for fruitful discussions and Javier Perez, our local contact on SWING beamline at SOLEIL for his help. We thank ANR TE-FLIC (grant number ANR-12-PRGE-0011-01) for financial support.

Supporting Information


Alain Dervieux
email: alain.dervieux@inria.fr

Eléonore Gauci
email: eleonore.gauci@inria.fr

Loic Frazza

Anca Belme
email: belme@dalembert.upmc.fr

Alexandre Carabias
email: alexandre.carabias@lemma-ing.com

Adrien Loseille
email: adrien.loseille@inria.fr

Frédéric Alauzet
email: frederic.alauzet@inria.fr

Mesh adaptation for k-exact CFD approximations

Keywords: computational fluid dynamics, mesh adaptation, adjoint state

Introduction

The purpose of mesh adaptation research is, thanks to an improved accuracy, to be able to compute new phenomena and also to master the numerical uncertain-ties which have been up to now unsufficiently controlled. An important strategy is to minimize the approximation error with respect to the mesh. A central question is then to find a good representation of the approximation error. A family of approximations plays a particular role in Computational Fluid Dynamics. The k-exact approximations provide a zero error when the exact solution is a polynomial of degree k. They involve finite elements like continuous and discontinuous Galerkin and ENO finite-volume approximations. Typically, k-exact approximations have a truncation error of order k + 1.

The purpose of this paper is to adapt mesh using error estimates for a few k-exact approximations in CFD.

We focus on methods which prescribe an anisotropic mesh under the form of a parametrization of it by a Riemannian metric. A Riemannian metric is a continuous symmetric matrix field defined on the computational domain Ω , for example in two dimensions:

M : Ω ⊂ R 2 → R 2 2 x → M (x) = R(x) t 1 ∆ ξ (x) 2 0 0 1 ∆ η(x) 2

R(x).

Notation R holds for a rotation for prescribing mesh stretching directions and ∆ ξ , ∆ η for prescribing mesh size in these directions. A mesh obeying these prescriptions is called a unit mesh for M . We observe that the very complex and discrete thing which is a mesh is replaced by a continuous function to be found as the minimum of a numerical error. Then we have to organise a process Metric → Mesh and discrete solution → Error → New metric which can be thought of as either a pure discrete process, or the discretization of a continuous process. Let us recall why metrics are particularly adapted to 1-exact approximations. These approximations involve most second-order methods based on continuous P 1 finite-element approximation, namely Galerkin, SUPG, Residual distribution and vertex-centered MUSCL approximations. First, the P 1 -interpolation error plays a central role in error estimates. Second, this interpolation error can be converted in terms of the mesh metric. We recall, following [START_REF] Loseille | Continuous mesh framework. Part I: well-posed continuous interpolation error[END_REF][START_REF] Loseille | Continuous mesh framework. Part II: validations and applications[END_REF], the main features of the continuous metric-based analysis initiated in several papers like [START_REF] Dompierre | Anisotropic mesh adaptation: towards a solver and user independent CFD[END_REF][START_REF] Castro-Díaz | Anisotropic unstructured mesh adaptation for flow simulations[END_REF][START_REF] Agouzal | Adaptive generation of quasi-optimal tetrahedral meshes[END_REF]. For a function u defined on the computational domain, we use the continuous interpolation error uπ M u instead of the discrete interpolation error u -Π M u :

u -π M u = |tr(M -1 2 |H u |M -1 2 )| ; |u -π M u| ≈ const.|u -Π M u|, (1)
where H u is the Hessian of u and M also denotes a unit mesh for metric M . We consider minimizing:

j(M ) = u -π M u L 1 (Ω h ) , (2)
and we define as optimal metric the one which minimizes the right-hand side under the constraint of a total number of vertices equal to a parameter N. After solving analytically this optimization problem, we get -solely using the fact that H is a positive symmetric matrix-the unique optimal (M L 1 (x)) x∈Ω as:

M L 1 = N Ω (det(H u )) 1 4 -1 (det(H u )) -1 4 H u . (3)
Knowing the continuous function u, we can derive the continuous optimal metric. In practice, u is solution of a PDE and the whole process of computing u and then deriving the optimal metric is operated by a fixed point iteration involving the generation of a mesh according to the metric, the solution of the PDE on the mesh and the building of a discrete metric.

In this paper, we discuss three types of functional j and the extension to higherorder approximations. For most cases, we propose to evaluate the method by measuring the mesh-adaptive convergence order α, defined by:

error = O(N -α dim ), (4)
where N is the total number of nodes and dim the dimension of computational domain, and α = k + 1 for a k-exact approximation.

2 Features-, goal-, norm-oriented formulations

These formulations are presented for the continuous case and applied to the secondorder accurate particular case of P 1 approximations on triangles and tetrahedra.

Feature-based (FB) adaptation

The continuous feature-based anisotropic method (2)-( 3) is generally defined by replacing the local interpolation error by the application of the recovered Hessian of the solution times a local mesh size defined by the continuous metric, see [START_REF] Alauzet | High order sonic boom modeling by adaptive methods[END_REF][START_REF] Loseille | Adaptive anisotropic simulations in aerodynamics[END_REF][START_REF] Castro-Díaz | Anisotropic unstructured mesh adaptation for flow simulations[END_REF][START_REF] Dompierre | Anisotropic mesh adaptation: towards a solver and user independent CFD[END_REF][START_REF] Vasilevski | Error bounds for controllable adaptive algorithms based on a hessian recovery[END_REF][START_REF] Huang | Metric tensors for anisotropic mesh generation[END_REF][START_REF] Frey | Anisotropic mesh adaptation for CFD computations[END_REF].

A typical example is the prediction of the sonic boom signature of a supersonic aircraft (see [START_REF] Loseille | Adaptive anisotropic simulations in aerodynamics[END_REF] for specific features). Let us consider the C25D geometry of the workshop [START_REF]2nd AIAA Sonic Boom Prediction Workshop[END_REF]. We use the Mach number M as sensor, i.e. j(M ) = Mπ M M L 1 (Ω h ) . Cuts of mesh and solution are depicted in Figure 1. The FB approach is particularly attractive due to its simplicity and its ability in taking into account physical aspects through the choice of the sensors. However, for systems, the choice of sensors is extremely sensitive.

The goal-oriented (GO) formulation

The GO mesh adaptation focuses on deriving the optimal mesh for computing a prescribed scalar quantity of interest (QoI). Many papers deals with a posteriori goal-based error formulation to drive adaptivity, using adjoint formulations or gradients, e.g. [START_REF] Ferro | Anisotropic mesh adaptation for crack propagation induced by a thermal shock in 2D[END_REF][START_REF] Yano | An optimization framework for anisotropic simplex mesh adaptation: application to aerodynamics flows[END_REF]. We investigate a priori based GO formulations for steady and unsteady problems. Loseille et al. [START_REF] Loseille | Anisotropic Norm-Oriented Mesh Adaptation for Compressible Flows[END_REF] derived the goal-based error estimate in a steady context for Euler flows, showing that the QoI error estimate is expressed as a weighted interpolation error on solution flow fields. This leads to an optimal metric computed as a sum of Hessians of Euler fluxes weighted by gradient components of the adjoint state and permits to focus on the capture of important features with respect to the chosen functional, such as sonic boom print at ground, see Figure 2.

Numerical corrector and norm-oriented (NO) formulation

Given a discrete problem, a mesh (of metric) M h and the discrete solution W h computed with the mesh, we call "numerical corrector" a discrete field W h such that the sum W h + W h is a significantly more accurate approximation of the exact solution than W h . In other words, W h is a good approximation of the error. Clearly, W h is useful for estimating the error, for correcting it, and for building a norm-oriented mesh adaptation algorithm.

A trivial way to compute W h could be to first compute an extremely accurate and extremely costly W h/2 k (k large) solution computed on a mesh M h/2 k obtained by dividing k times the elements of M h , and finally to put W h = W h/2 k -W h . But the interesting feature of a numerical corrector should be that its computational cost is not much higher than the computational cost of W h . We describe now a corrector evaluation of low computational cost relying on the application of a Defect Correction principle and working on the initial mesh M h :

Ψ h (W h +W h,DC ) ≈ -R h/2→h Ψ h/2 (R h→h/2 W h ) ; W h,DC = W h,DC -(π h W h -W h ),
where π h W h -W h is a recovery-based evaluation of the interpolation error (see [START_REF] Loseille | Anisotropic Norm-Oriented Mesh Adaptation for Compressible Flows[END_REF] for details). The notation R h/2→h holds for the transfer (extension by linear interpolation) operator from the twice finer mesh M h/2 to the initial mesh M h , while R h→h/2 holds for the transfer operator from the initial mesh M h to the twice finer mesh M h/2 . The finer-mesh residual Ψ h/2 (W h ) can be assembled by defining the sub-elements of M h/2 only locally around any vertex of M h . Applications of this method to the Navier-Stokes model can be found in [START_REF] Frazza | Nonlinear corrector for RANS equations[END_REF]. We present an application with the Euler model used for sonic boom prediction. We consider again the C25 geometry. The important input is the pressure signature at one-body length below the aircraft. Figure 3 depicts the pressure signal and the local error bar, from the non-linear corrector, for a tailored mesh (mesh aligned with the Mach cone) and for adapted meshes. The taylored mesh calculation may seem converged but the corrector remains large. The right-hand side shows a more coherent convergence, with a much smaller corrector.

We can now introduce the NO formulation. We base it on the L 2 -norm of approximation error. It consists in the minimization of the following expression with respect to the mesh M :

j(M ) = ||W -W M || 2 L 2 (Ω ) with Ψ M (W M ) = 0. ( 5
)
Fig. 3 Flow around a lowboom C25 geometry: Pressure levels (red) and non-linear corrector (black) error intervals for the pressure (z = 0, y = -C). Top: on an adhoc tailored mesh. Bottom: on a self-adaptive mesh (right). On right figure, the good convergence is indicated by much smaller intervals.

Introducing g = W -W M , we get a formulation similar to a GO formulation j(M ) = (g,W -W M ). The central idea of NO is to replace g by the numerical corrector W h as defined in this section. The rest of the NO process follows the GO algorithm with g = W h . The whole NO adaptation algorithm finally writes:

Step 1: solve state equation for W

Step 2: solve corrector equation for W h

Step 3: solve adjoint equation for W *

Step 4: evaluate optimal metric as a function of W and W *

Step 5: generate unit mesh for M opt,norm and go to Step 1. In order to give an idea of how this NO works, we consider as benchmark a test case from [START_REF] Formaggia | Anisotropic a priori error estimates for elliptic problems[END_REF] featuring a 2D boundary layer (Figure 4). The Laplace equation is solved with a RHS inducing the boundary layer depicted in the figure. FB and NO mesh-adaptive methods are compared by displaying the convergence curve related to Criterion (4). In abscissae the number of nodes used for computing the discrete solution u h is shown, and in ordinates the L 2 -norm of the aproximation error uu h which is measured from the known analytic solution. When the FB method is applied, a tremendous improvement of the error is obtained with 128 vertices, then a uniform element division and further FB adaptation are applied in alternance. While the element division is applied, the error is as expected divided by 4. In contrast, for 512, 1024, 2048,... vertices (abscissae in the figure), the effect of FB adaptation is to increase the error, and the second-order convergence is lost. On the contrary, with this test case, each NO mesh-adaptation phase improves (even slightly) the error norm, producing an asymptotic numerical convergence of order two.

Estimates for k-exact approximations

Due to their error size and characteristics (dispersion, dissipation), second-order accurate approximations are unable to compute many phenomena. For smooth contexts, high-order methods bring crucial improvements.

Higher-order (HO) estimates

Main existing HO schemes satisfy the so-called k-exactness property expressing the fact that if the exact solution is a polynomial of order k then the approximation scheme will give the exact solution as answer. The assembly of these schemes involves a step of polynomial reconstruction (e.g., ENO schemes), or of polynomial interpolation (e.g., Continuous/discontinuous Galerkin). The main part of the error can then be expressed in terms of a (k + 1)-th term of a Taylor series where the spatial increment is related with local mesh size. We want to stress that this is the key of an easy extension of metric-based adaptation to HO schemes. We illustrate this with the computation of 2D Euler flows. Considering a triangulation of the computational domain and its dual cells C i built with triangle medians, the exact solution W of Euler equations verifies (omitting initial conditions):

B(W,V 0 ) = 0, ∀ V 0 ∈ V 0 = {V 0 constant by cell}, with B(W,V 0 ) = T 0 ∑ i E i (W,V 0 )dt + T 0 ∂Γ F Γ (W ) • nV 0 dΓ dt E i (W,V 0 ) = C i V 0 ∂ π 0 W ∂t dΩ + 1 2 ∑ j ∂C i ∩∂C j V 0 (F (W )| ∂C i + F (W )| ∂C j ) • ndσ .
Here we denote by π 0 the operator replacing a function by its mean on each cell, F the Euler fluxes, and the second sum is taken over the cells j neighboring cell i. Let us define a quadratic Central-ENO scheme [START_REF] Barth | Higher order solution of the Euler equations on unstructured grids using quadratic reconstruction[END_REF][START_REF] Ivan | High-order solution-adaptive central essentially nonoscillatory (CENO) method for viscous flows[END_REF]. The computational cost of this scheme is rather large but acceptable for 2D calculations (its extension to 3D is even more computationally expensive). This scheme is based on a quadratic reconstruction on any integration cell C i using the means of the variable on cells around C i . Let us denote by R 0 2 the global reconstruction operator mapping the constant-by-cell discrete field into its quadratic-by-cell reconstruction. The CENO scheme writes in short:

Find W 0 constant by cell s.t. B(R 0 2 W 0 ,V 0 ) = 0, ∀ V 0 constant by cell.
A representative functional of goal-oriented error is:

δ j = (g, R 0 2 π 0 W -R 0 2 W 0 ).
Lemma 1 [START_REF] Carabias | Anisotropic goal-oriented error analysis for a third-order accurate CENO Euler discretization[END_REF]: Introducing the adjoint state W * 0 ∈ V 0 , solution of:

∂ B ∂W (R 0 2 W 0 )(R 0 2 V 0 ,W * 0 ) = (g, R 0 2 V 0 ), ∀ V 0 ∈ V 0 . (6)
we have the following equivalence:

(g, R 0 2 π 0 W -R 0 2 W 0 ) ≈ ∂ B ∂W (W )(R 0 2 π 0 W -W,W * 0 ). ( 7
)
This estimate is typical of a k-exact variational scheme and permits to express the error as a Taylor term of rank k + 1, rank 3 in our case, with respect to directional mesh size δ x, which we replace by the power 3/2 of a quadratic term:

δ j sup δ x T(δ x) 3 ≈ sup δ x | H|(δ x) 2 3 2 ∀δ x ∈ R 2 . (8)
In [START_REF] Carabias | Anisotropic goal-oriented error analysis for a third-order accurate CENO Euler discretization[END_REF] we fit the second-order tensor H to the third-order tensor T by least-squares, and the optimal metric is computed in a similar way to the second-order accurate case. An a priori better option for accounting higher-order interpolation error, not tested here, is to apply the strategy of [START_REF] Coulaud | Very High Order Anisotropic Metric-Based Mesh Adaptation in 3D. 25th International Meshing Roundtable[END_REF].

Remark: The a priori estimate of Lemma 1 is inspired by the a posteriori estimates of Barth and Larson [START_REF] Barth | A-posteriori error estimation for higher order godunov finite volume methods on unstructured meshes[END_REF] in which the authors explain that the analysis extends to many k-exact approximations of high-order. This is also true for the present a priori analysis. In particular, the analogous estimate for a k-exact discontinuous P k -Galerkin approximation writes:

(g, Π k W -W k ) ≈ ∂ B ∂W (W )(Π k W -W k ,W * k ), (9)
where W k and W * k are the DG k discrete state and adjoint and Π k the elementwise interpolation of degree k.

High-order accurate unsteady mesh adaptation

We illustrate the use of the above estimate ( 6)-( 8) with an application to an unsteady flow. For many propagation phenomena, the discretisation grid (space and time) necessary for a complete representation is very heavy. We consider here an acoustic wave propagation based on the Euler equations.

In order to apply an unsteady mesh adaptation, we adopt the so-called Global-Fixed-Point algorithm [START_REF] Belme | Time Accurate Anisotropic Goal-Oriented Mesh Adaptation for Unsteady Flows[END_REF]. In short, the time interval is divided in n adap subintervals. After a computation of state and adjoint on the whole time interval, an optimal space-time metric is evaluated as a set of spatial metrics for each of the time sub-intervals.

In Figure 5, the propagation of a noise from a road (bottom left) to a balcony (near top, right) around an anti-noise wall (middle of bottom) is computed. The functional is the pressure integral on an interval of the balcony. Since a few wavelengths are emitted at the noise source, the mesh adaptation process will concentrate on the part of the wave train which will hit the balcony. This dramatically reduces the region of the computational domain which needs to be refined. With 30980 vertices (mean of the 20 meshes used over the time interval) the resolution is about 10 points per half wave and would require 5 millions vertices if the mesh were a uniform mesh of same maximal fineness. As for Criterion (4), we have measured for this case α = 2.45, which is not satisfactory with respect to the theoretical order of approximation, which is 3, but already carries an important improvement with respect to analogous adaptation based on a second-order finite-volume approximation, see [START_REF] Carabias | Analyse et adaptation de maillage pour des schémas non-oscillatoires d'ordre élevé[END_REF].

Fig. 1

 1 Fig. 1 Lowboom C25 computation with a feature-based mesh adaptation: Cut plane x = 30 (left) and on symmetry plane (right).

Fig. 2 A

 2 Fig.2A typical comparison of feature-based (left) and goal-oriented (right) mesh adaptation for the computation of sonic boom. On left the whole flow is captured, on right, focus is put on the shock structures influencing the boom path at bottom.

Fig. 4

 4 Fig. 4 Elliptic test case of a 2D boundary layer. A comparison between uniform refinement ("FMG"), feature/Hessian-based, and norm-oriented mesh adaptation methods: error |uu h | L 2 in terms of number of vertices.

Fig. 5

 5 Fig. 5 Goal Oriented unsteady calculation of nonlinear acoustics propagation with third-order goal-oriented adaptation. Pressure at three different time levels and the corresponding meshes.

Dervieux-Gauci-Frazza-Belme-Carabias-Loseille-Alauzet

Conclusions

The k-exact analysis described in this note allows us to express errors in terms of interpolation errors. This holds for various k-exact approximations like FVM, FEM, DG, ENO. This also holds for three types of adaptation strategies, namely the feature-based, the goal-oriented, and the norm-oriented. Applications with P 1 -Galerkin and P 2 -CENO approximations are demonstrated. This method can be complemented with a special treatment of singularities, [25], and combined with a FMG process, [23].

Acknowledgements This work has been supported by French National Research Agency (ANR) through project MAIDESC n o ANR-13-MONU-0010. This work was granted access to the HPC resources of CINES under the allocations 2017-A0022A05067 and 2017-A0022A06386 made by GENCI (Grand Equipement National de Calcul Intensif).