Ludovic Bellot-Gurlet

Philippe Dillmann

Delphine Neff

From archaeological sites to nanoscale: the quest of tailored analytical strategy and modelling

Last developments of nano probes allows scientists of Cultural Heritage to assess a new kind of information that is crucial in the different topics concerned by the field: determining ancient manufacturing processes, studying use and provenance of ancient artefacts, revealing the degradation processes and developing adapted cleaning and conservation treatments. Nevertheless to be useful, these nanoscale approaches must be integrated in a tailored multi-step analysis. The final aim of these approaches will be to reach understanding and/or reliable modelling of the behaviours of the ancient systems. A first part of this chapter will review the issues in Cultural Heritage and the nature of the physico-chemical data that can be collected on the systems. In a second part, a selection of examples dealing with nano characterisation in Cultural Heritage will allow us to present several up to date techniques and methodologies employed in Cultural Heritage science. Then, the third part of the chapter will review some of the different modelling attempts that where already made in the domain of Cultural Heritage, and that were based on the use of physico-chemical descriptions at different scales. The challenge for the next future will be, for different kind of materials and environments, to propose multiscale models from nano to functional scale. Some key steps to face these challenges bridging the gap between multiscale descriptive characterisation and numerical modelling are reviewed here.

Introduction

The different chapters of this book demonstrated on the one hand the pertinence of looking at physico-chemical systems of Cultural Heritage at the nanometre scale, on the other hand the fact that, despite the absence of measurement tools to observe them at nanometre scale, ancient craftsmen's mastered the control of nanoparticules and nano-structuration for some marvel of ancient technical realisation 1 Bellot-Gurlet L., Dillmann Ph., Neff D. (2016). From archaeological sites to nanoscale: the quest of tailored analytical strategy and modelling, in : "Nanoscience and Cultural Heritage",Dillmann Ph., (metallic lustres, ceramics, etc). For studying such nano-structures or understanding the properties originating from the nano-scale, the development of nanoprobes or nano-investigation techniques is called to increase in heritage studies. Nevertheless, an investigation at the nano-scale alone does not allow an expert of heritage science to understand the system under investigation and to answer to the scientific question asked. Therefore this is only by integrating this nanometer scale approach in a tailored multi-step and multi-scale scientific and analytical methodology that the "nanometer scale will make sense".

Thus, today challenges and breakthrough are double:

• to develop the use of accurate and efficient nanoprobes or nanometre scale investigation techniques in multistep methodologies, • to bridge the gap between the different scales, concerning the understanding of Cultural Heritage systems and/or the modelling of their behaviour. The first aspect is, still today, linked by a limited access to the nanometre scale investigation methods. This is chiefly due to financial reasons limiting the widespread of devices and associated analytical know-how. Moreover most of the time experiments at nanoscale are costly and time consuming. This is an additional reason to integrate them in a step to step analytical strategy, beginning with reliable macroscopic investigation methods and a careful selection of the representative sample at a given scale. Then, at the end of the process, only the most representative sample should be submitted to nanometre observation, avoiding waste of time and means. To that purpose, and to help scientist of Cultural Heritage to develop right analytical strategies, the first part of this chapter will present a brief review of the various aspects of physico-chemical issues, at different scales of investigation, linked to the systems of Cultural Heritage.

To select the most adapted nanoscale investigation method it is necessary, for the conservation scientist or the scientist practicing archaeometry who will implement it in close collaboration with the specialist of the method, to have in mind the potentiality of the different available methods and the nature of the information they can deliver. This is the aim of the second part of this chapter that will give a tentative overview, through selected and concrete examples, of the cutting edge techniques used in the field of heritage but also in some related domains of research that have comparable scientific and analytical concerns.

The second aspect, aiming to bridge the gap between the different scales, may seem to be trivial but is not. This issue is not restricted to Cultural Heritage materials, but it is a crucial issue in other domains of science dealing also with heterogeneous materials in which a nanometre scale process can be (or not) at the origin of a macroscopic behaviour at the functional scale (environmental science, material science, etc.). The synergy of different potential mechanisms and processes happening at different scales renders the interpretation and the evaluation of the dominant parameter or mechanism (if there is one) very difficult. To reach that goal, we believe that a very important, challenging and promising research direction is linked to the multiscale modelling of such systems. This modelling, even by being only descriptive (and not predictive) will allow one to better understand the role of the different parameters and their respective influence on the macroscopic behaviour. In the most favourable case they could be able to predict the future of Cultural Heritage systems, being a wonderful tool for our societies that have the duty to transmit the Cultural Heritage and its knowledge to the next generations. These aspects will be detailed in the third and last part of the present chapter, reviewing the modelling attempts in the domain of Cultural Heritage research and pointing the challenges for the forthcoming researches that will try to develop multiscale models.

Issues dealing with "Heritage systems"

Listing or reviewing all the studies dealing with "material aspects" and Heritage is not straightforward. According to the review or book aim, author background or specialities, it will focus on, or highlight, various aspects of the numerous facets of heritage systems or studies (for examples: [START_REF] Pollard | TEM observations of goethite dehydratation: allpication to archaeological samples[END_REF][START_REF] Colombini | Organic materials in art and archaeology[END_REF][START_REF] Casadio | Molecular analysis for art, archaeometry and conservation[END_REF]; Physical techniques in the study of Art, Archaeology and Cultural Heritage 2006 ;[START_REF] Edwards | Analytical archaeometry: selected topics[END_REF][START_REF] Madariaga | Analytical chemistry in the field of cultural heritage[END_REF][START_REF] Dillmann | Circulation et provenance des matériaux dans les sociétés anciennes[END_REF]. For issues dealing with analytical studies, then focusing on objects or materials, one could try to depict the Heritage problematics through a simplified object life pathway (Fig. 24). All objects start from some raw material, which could be manufactured only by shaping, grinding (or any transformations without chemical changes), or through techniques/processes, which will transform the constituents to obtain some new materials. Societies and their knowledge could be studied through their ability of knowing their environment/resources, the ways for exploiting resources, the abilities for procurement and distribution of goods, and the techniques to manufacture objects or materials for specific uses or purposes, all these aspects describing a technical (and even sometimes technological) level and relations between societies and their natural environment. During history a wide dynamic exists, according to places, cultures, their interactions and evolutions. Many studies in the field of Cultural Heritage aim to reconstruct the history of some materials' use, some techniques or some technical gesture. This especially thanks to the information that analytical methods deliver to identify raw materials, their geographic origin or reveal ancient techniques. All these clues are linked to physico-chemical features of the constituent of the materials/artefacts.

After being manufactured objects were used, and according to some use wear or residues the human activities related to objects could be clarified (as stone tools [START_REF] Astruc | Multi-scale tribological analysis of the technique of manufacture of an obsidian bracelet from Aşıklı Höyük (Aceramic Neolithic, Central Anatolia)[END_REF] or ceramics contents [START_REF] Evershed | ORGANIC RESIDUE ANALYSIS IN ARCHAEOLOGY: THE ARCHAEOLOGICAL BIOMARKER REVOLUTION*[END_REF]). An important aspect is also the recycling practices that can be investigated by analytical approaches.

Fig. 24. Illustration of the life cycle of objects

Before being transmitted to the present, objects and masterpieces could remain "in use" (as buildings, mural or canvas paintings, etc.) or buried by constituting the archaeological sites. Along time, with multisecular duration, degradation mechanisms modified the objects leading sometimes to their complete change (according to their nature and their environment). For heritage studies, first the effects of these degradations need to be understood for being able to differentiate them from information related to previous object life. Second, if this testimony of past societies is decided to be conserved for future generations, one must understand the running degradation mechanisms in order to be able to diagnose the stability of the object and recommend some conservation policies, strategies or treatments. On the other hand, in order to be presented to a public, an object should be "easily" understood and thus some adapted restoration treatments need to be developed and undertaken in compliance with existing guidelines (as UNESCO, International Council of Museums-Committee for Conservation (ICOM-CC)6 and the International Centre for the Study of the Preservation and Restoration of Cultural Property7 ).

Dealing with "heritage systems", requires to apprehend all the scales/systems that constitute it: from a small artefact/object, to a building and a whole site and its environment. Just taking into account the system studied, problems are multiscaled, with interactions between the scales or parts of objects/systems. Addressing these challenges requires the development of adapted analytical strategies and concepts to make the links between on one hand, measures and scales, and on the other hand, studied systems and problematics. Such challenges are the daily work of scientists and Cultural Heritage experts.

Thus, for each of these stages of the life (and vanishing) of heritage systems, and for their protection and conservation, different issues arise, linked to historical, cultural or conservation purposes. These issues can be expressed in terms of scientific concepts or questions. Then, to answer these questions, specific physicochemical parameters must be measured on the systems. These measurements must be made at different scales, from the nanometre to the functional scale, in order to determine the synergies between these scales and pertinent (dominant) processes. These scientific questions and parameters are summed up in Table 1.

Concerning the manufacturing processes (from transformation of raw materials to secondary and final elaboration of artefacts), the key issues are linked to the determining of the raw materials (nature and origin) and the nature (temperature, chemical conditions, duration, efficiency, etc.) of the processes.

The first issue linked to historical aspects is the determination of supply network of goods and raw materials in ancient societies. This is linked to the crucial question of provenance and circulation of materials and artefacts. Determining the chemical compositions of the trace elements [START_REF] Leroy | The medieval iron market in Ariège (France). Multidisciplinary analytical approach and multivariate analyses[END_REF]) and/or the isotopic composition [START_REF] Brauns | Provenance of Iron Age iron in southern Germany: a new approach[END_REF]) can approach this. Because ancient systems and artefacts are per nature heterogeneous at different scales, and because these trace elements and isotopes are heterogeneously distributed in the material, investigation at different scales must here also be implemented.

The raw material selection is also involved in the manufacturing processes, as technologies are linked to the composition of the materials, but besides to the properties and reactivity of the constituents. This information can be reached by knowing the chemical composition (mainly of major or minor elements) but also and chiefly the nature of the phases in relation to their thermodynamic properties. The sizes of the different precipitates and aggregates in the systems are also crucial parameters to be determined. Lastly, linked to the duration and the efficiencies of the processes, transport phenomena, depending on temperatures, nature of media, phases and involved species are key parameters and can be understood. Transport mechanisms could be assessed by measuring the porosity of the system if they are porous (gaseous state transport), by the sizes of particles (implying active surfaces available) or by the nature of the phases in which solid state transport could happen (mostly at high temperatures).

If one considers now the use and the functions of the artefacts (tools, utensils, weapons, buildings, etc.) other questions must be considered. Beyond that shapes they are on the one hand linked to the aesthetical aspect of the materials (surface state, colour, …), sometimes due to surface treatments, and on the other hand to the mechanical resistance and behaviour . To understand the first of these aspects, the nature (chemical and physical/optical) of the former surface layer (called limit of the initial surface by the conservation scientists [START_REF] Bertholon | The original surface of corroded metallic archaeological objects: characterization and location[END_REF]) must be investigated. This implies to determine local composition and structural changes (including sizes of nanophases) at the location of the initial surface. The mechanical behaviour can be estimated with the nature of the material (composition) but also by its texture down to the micro and nano-structure (nature and organisation of phases and precipitates, linked to the move of dislocation for example). As far as degradation processes are involved, the challenges are here to determine the nature of the processes and the degradation rates linked to the stability of the systems in regards to their environments. This requires to assess to the thermodynamic properties of the phases but also their chemical and sometimes electrochemical reactivity and transport inside the alteration layers. These scientific questions need to determine here again the chemical and structural nature of the alteration layers at different scales, but also the chemical and physical properties (as reactivity, conductivity, hardness, density, etc.). Transport phenomena of species, the most of the time in porosities saturated in hydrous fluids and sometimes in solid state (vacancies, ions) in nanolayers, must also be considered.

Lastly, a crucial issue dealing with heritage is the setting up and the use of adapted cleaning, restoration and protection treatments. This topic needs to consider on the one hand the efficiency and durability of treatments but, also on the other hand the influence of this treatment on the integrity of the heritage system (modification of appearance, reversibility). This implies to characterise the nature (including chemical composition, structure and thermodynamic properties) of the new formed phases after interaction with the treatments, or the treatment product itself, but also to locate these compounds in the system and especially in the degradation system. Here again, in addition to the reactivity, the transport in the system (characterisation of porosities) is crucial. Lastly, the evolution under different environmental conditions of the new-formed phases (chemistry and structure) or the products used are key parameters to determine. Because the treated objects must be stable for the longest period, considering this behaviour requires a fine understanding of their durability properties over periods not mandatory for everyday life contexts.

Analytical techniques

From macro to nano scales

Analyses of Cultural Heritage materials and systems need specific protocols adapted to each context. The constraints on the sampling as well as the fragility of the artefacts can orientate the drawing of the analytical procedure. For that, the analytical set-up can be adapted to the potential various sample dimensions (from millimetre to decametre). As said before, characteristic of the samples are various and numerous and deals with the determination of their physical properties (among them: hardness, porosity, morphology of particles, conductivity), and their chemical properties mainly connected to their chemical composition and the nature and crystallinity of phases in presence. The choice of the most appropriate analytical technique to assess them is crucial to get the desired information.

Most of the studies dealing with the nanoscale, namely the characteristics and properties of the material at the nanoscale, start with a multiscale analysis. Indeed, apart the difficulty to access to nanoprobe equipment (due to their high cost, then rare occurrence, especially in Cultural Heritage laboratories), answering scientific issues concerned by nanoscale properties often starts at first by surveys and studies at lower scales (i.e. macro and microscopic scales). This process needs a first analytical chain composed of classical devices and setups used in material sciences and described in details in previous publications [START_REF] Edwards | Analytical archaeometry: selected topics[END_REF]. It is not in the scope of this chapter to describe it exhaustively. Yet it is important to insist on the fact that macro to microscale characterization prior to nanoscale analyses is unavoidable to ensure satisfying data acquisition and treatment. Actually the nanoscale analyses could be realized to the detriment of the representativeness regarding the totality and complexity of the studied artefacts. That is why the characterisation at the lower magnifications is required. First observations can be realized thanks to optical microscopy, especially when samples are prepared on sections which enlighten their heterogeneities. This provides information on the morphology and texture of the sample and allows one to locate the zones of interest for further analyses, by screening large surfaces and various samples. Additional observations can be obtained thanks to Scanning Electronic Microscopy (SEM). This analytical tool is the ultimate multiscale device thanks to its wide range of magnification. Particularly Field Emission Gun (FEG) SEM capable to work down to few nanometers resolution is a preferred technique to correlate information between micro and nanoscales.

In the following paragraph the different instruments working at the nanoscale are briefly presented, through selected examples. The aim is to illustrate the possibilities offered by these nanoscale techniques. Selected presented studies are mostly dealing with Cultural Heritage, and the authors hope that this overview will help new actors of the Cultural Heritage to enlarge the spectra of the techniques they can potentially use.

Analyses at nanometer scale

High Resolution X-ray Diffraction (HR-XRD) analyses can provide information on the sizes of nanocristallites present in the sampled materials, by using the Scherrer formula. For example Bayle [START_REF] Bayle | Déchloruration des objets archéologiques ferreux par le processus de stabilisation subcritique -Caractérisations physico-chimiques des systèmes transformés[END_REF] measured the crystallite sizes of akaganeite powders by XRD under synchrotron radiation to determine the effect of a dechlorination treatment on the progressive transformation of this chlorinated phase present in the corrosion layers of archaeological iron artefacts. Helary et al. [START_REF] Hélary | Contribution de la diffraction de rayons X sous incidence rasante à l'étude de céramiques lustrées[END_REF]) measured the crystallite size of silver particles to understand the technology used in the medieval period for the production of metallic lustre ceramics. Another example, Ungár et al. [START_REF] Ungár | Revealing the powdering methods of black makeup in Ancient Egypt by fitting microstructure based Fourier coefficients to the whole x-ray diffraction profiles of galena[END_REF] studied by XRD under synchrotron radiation the crystallite size of black make-up of Ancient Egypt. Last a recent study used XRD under synchrotron to determine particle sizes on a range from the micrometric to the nanometric scale on various ancient materials like Roman terra sigillata, Neolithic flint (Fig. 2) or Roman iron nails [START_REF] Dejoie | Complementary use of monochromatic and white-beam X-ray micro-diffraction for the investigation of ancient materials[END_REF]. This was achieved thanks to a micrometric and monochromatic beam combined to the exploitation of Laue diffraction. For surface analyses, Atomic Force Microscope (AFM) is one of the techniques used by few studies (Crina Anca Sandu et al. 2011), mainly dealing with the understanding of coatings like gilding or the assessment of protection properties for coatings protective treatments. This technique presents a lateral resolution, depending on the apex chosen for the surface probing, of about 10 nanometers. For example Ambrosi et al. [START_REF] Ambrosi | Colloidal Particles of Ca(OH)2: Properties and Applications to Restoration of Frescoes[END_REF]) studied the efficiency of the solvation of Ca(OH) 2 in various organic solvent by AFM. The Ca(OH) 2 is used to reinforce the wall paintings and the carbonic stones in the field of Cultural Heritage. The interest of AFM is to observe the dispersions of the nanoparticles in their solvents and to measure the sizes of the aggregates formed (Fig. 26).

Dal Bianco et al. [START_REF] Dal Bianco | Sol-gel silica coatings for the protection of cultural heritage glass[END_REF] worked on mosaics recovered by a golden leaf. Initially the golden leaf was coated by a thin overlaying glass layer called cartellina but its weathering leave the golden leaf tessera unprotected. That is why they studied a protection treatment of this type of artefacts thanks to various sol-gel silica coatings. This latter was applied on model sample and the roughness of their surfaces was studied thanks to AFM. In the same way Manoudis et al. [START_REF] Manoudis | Superhydrophobic films for the protection of outdoor cultural heritage assets[END_REF]) studied the effect of siloxane based coatings on the enhancement of hydrophobic properties of stones. AFM was a complementary technique showing that the addition of SiO 2 nanoparticles in the coat-ings conducts to the formation of nano-aggregates on the stone surface that impacts its colour. This consequence conducts the authors to recommend against adding nanoparticles in the coatings to respect the aesthetical aspect of the stone. Concerning the direct studies at the nanometric scale up to now the most used technique is the Transmission Electronic Microscope (TEM). This technique provides imaging with a resolution of less than 0.1 nanometer when a suitable preparation of the sample is realised. The electron beam is focused on the sample and images are recorded in transmission. In addition the electronic diffraction can provide structural analyses on the compounds observed. Moreover the interaction between the atoms of the sample and the electrons provokes the emission of X-rays providing qualitative and semi-quantitative information about the elementary composition of the samples thanks to the Energy Dispersive Spectroscopy (EDS). Last, the Electron Energy Loss Spectroscopy (EELS), although more delicate to interpret, can be used to provide information on the speciation of chemical elements present in the samples.

Complementary to AFM studies, the TEM imaging is used to determine the aggregates size used for the consolidation of stones [START_REF] Baglioni | Colloid and Materials Science for the Conservation of Cultural Heritage: Cleaning, Consolidation, and Deacidification[END_REF]. For example Chelazzi et al. [START_REF] Chelazzi | Hydroxide nanoparticles for cultural heritage: Consolidation and protection of wall paintings and carbonate materials[END_REF]) and chapter 3.3 in this book, determined the aggregate size of Ca(OH) 2 , Mg(OH) 2 or Ba(OH) 2 dispersed in propanol. Yet, the control of the particle sizes and the polydispersity should be of the same range than the porosity of the treated surfaces. TEM is involved in this study to tailor these particles by varying the synthesis parameters. One other interesting example of TEM concerns the study of pigments sampled in ornamented prehistoric caves. In their studies Pomiès et al. and Chalmin et al. (Pomiès et al. 1998;[START_REF] Chalmin | Les blasons de Lascaux[END_REF][START_REF] Chalmin | Minerals discovered in paleolithic black pigments by transmission electron microscopy and micro-X-ray absorption near-edge structure[END_REF]) have shown that many information can be obtained from a very low amount of pigments collected on the wall and deposited on a copper grid (Fig. 27).

Particularly in the case of the Lascaux cave different mineral phases have been identified depending on the colour: hematite (Fe 2 O 3 ) for the red, goethite (FeOOH) for the yellow and mix of manganese oxides (MnOOH, MnO 2 , Mn 2 O 3 , Mn 3 O 4 ) for the black pigments. Moreover the particle morphology indicates their crystallinity degree. Specifically for the hematite various class of low crystallinity or high crystallinity pigments have been established. Furthermore TEM imaging can document the pigment heating treatment as it revealed that the presence of porosity on hematite pigments is due to the deshydratation of a goethite (FeOOH) when heated to obtain hematite [START_REF] Pollard | TEM observations of goethite dehydratation: allpication to archaeological samples[END_REF]). These nano-scale experiments provide indication of the mastering of heating treatment, highlighting the use of firing techniques for pigment preparation to produce red hematite from yellow goethite. Moreover TEM-EDS analyses indicate the presence of various minor elements that allows one to discriminate several pigments of the same mineral class, showing the possible correlation between different paintings panels and the paintings pots excavated in the cave. A last example of TEM is given by Leon et al. [START_REF] Leon | Evolution of terra sigillata technology from Italy to Gaul through a multi-technique approach[END_REF] with the observation of samples of terra sigillata, ceramics of the Roman period, which provides information about the technological transfer between Italy and Gaul. Indeed thanks to the complementarity with microXRD under synchrotron radiation, Raman spectroscopy, SEM and TEM, it has been evidenced that the technology of production has evolved between Italy where the production started and Gaul where this production was implemented. In this study TEM delivered information on the ceramic production process, particularly by showing the shape of the particles originated from the clay used for the production. In this case the shape and composition of particles were related to the firing temperature and the materials chosen to produce the terra sigillata.

Thanks to the development of synchrotron facilities, specific experiments are developed that allow one to obtain nanometer range beam [START_REF] Sakdinawat | Nanoscale X-ray imaging[END_REF]). Among them another technique is getting involved in the characterisation of Cultural Heritage artefacts, the Scanning Transmission X-ray Microscopy (STXM). This microscopy is provided under synchrotron light source and consists in scanning a thin sample by a tuneable light to obtain a chemical image (or map). The energy chosen depends on the elemental composition of the sample. Through the scanning in energy around the absorption edge of an element (K, L depending on the element), the XANES (X-ray Absorption Near Edge Structure) spectra can be reconstituted. The principal advantage of this technique is to deliver hyperspectral images that allow to preserve the information about the location on the samples of the extracted XANES spectra. [START_REF] Chalmin | Minerals discovered in paleolithic black pigments by transmission electron microscopy and micro-X-ray absorption near-edge structure[END_REF] XANES provide structural data about the crystalline environment/speciation of a chemical element. Thus, it can help to determine the crystalline or amorphous nature of a phase as well as the oxidation degree of an element. STXM delivers a beam size of 25x25 nm² and is applied on thin samples of various thicknesses, depending on the composition of the sample and the chemical element that is studied. STXM is mostly used in material, geosciences or Natural Heritage as for example on fossils [START_REF] Bernard | Multiscale characterization of pyritized plant tissues in blueschist facies metamorphic rocks[END_REF][START_REF] Galvez | Chemical and structural imaging of fossilized tissues at the nanoscale and assessment of their taphonomy[END_REF]). One of application on archaeological samples is given by Leon et al. [START_REF] Leon | Interfacial layer on archaeological mild steel corroded in carbonated anoxic environments studied with coupled micro and nano probes[END_REF]. The aim of the study is to understand the corrosion mechanisms of ancient nails corroded in an anoxic soil. This study showed that an iron oxide layer of few hundred nanometers has precipitated at the interface with the metal. The sample of 100 nm thickness has been prepared by Focused Ion Beam (FIB-see below). As the whole corrosion layer of several hundred micrometer thickness is composed of an iron +II carbonate this interfacial oxide layer presents a mixed iron +II/+III oxidation degree (Fig. 5). This specificity could confer to this thin layer specific conductive property that could play a major role in the control of the iron corrosion kinetics.

Synchrotron facilities also afford high resolution tomographic setups [START_REF] Sakdinawat | Nanoscale X-ray imaging[END_REF]. For organic or light materials the spatial resolution of a voxel can reach 150 nm. For example fossils of arthropods embryo dated from 520 million year old can now be studied in 3D, avoiding any invasive sampling. Their analyses reveal the presence of mouth, teeth and gut on a few hundred micrometer sample size. Some examples of X-ray fluorescence synchrotron nanotomography (SR-TOMO) are also provided by the planetary sciences that analysed stardust [START_REF] Silversmit | X-ray Fluorescence Nanotomography on Cometary Matter from Comet 81P/Wild2 Returned by Stardust[END_REF]). This technique enables to analyse nondestructively the chemical composition of small samples. Back to laboratory, high spatial resolution facilities are nowadays developed. The Focussed Ion Beam (FIB) is one of them. This technique is based on the sample sputtering by a fine focused ion beam, generally gallium, allowing an efficient thin sample preparation for TEM or SEM. An additional interest is the coupling to a Scanning Electron Microscope (SEM). The specific interest of such dual-beam Focused Ion Beam Scanning Electron Microscopy (FIB-SEM) is to collect images during the milling, so that a volume of the sample can be 3D-imaged with a few tenth nanometers resolution. The first study in Cultural Heritage using the combination of FIB and SEM is illustrated by Giachi et al. (Giachi et al.). The authors observed archaeological waterlogged wood impregnated with colophony (Fig. 29). The milling of local surfaces allowed to observe with a high resolution the wood cell walls, showing that they were totally filled confirming the deposition of the colophony at least on the surface of the cells. A second example of preparation of sample thanks to FIB is given by Villanueva-Amadoz et al. [START_REF] Villanueva-Amadoz | Focused ion beam nano-sectioning and imaging: a new method in characterisation of palaeopalynological remains[END_REF]. In their study these authors worked on the characterisation of palaeopalynological remains. Slice spores and pollen grains were sectioned by FIB-SEM for reconstructing their ornamentation and wall-ultrastructure, providing information on their internal structure. [START_REF] King | Forming Cohesive Calcium Oxalate Layers on Marble Surfaces for Stone Conservation[END_REF]) used 3D reconstruction by FIB-SEM to investigate the formation of cohesive calcium oxalate layers on marble surfaces for the conservation of stone. They used FIB-SEM complementary to SEM analyses to determine the 3D porosity network at a submicrometric scale showing that nanopores are partially connected inside a treated marble cube. Various spectroscopies can now reach nanometric scales. In the UV-Visible spectroscopy with a synchrotron light source the analysed volume can be of a few hundred micrometers. [START_REF] Thoury | Synchrotron UV-Visible Multispectral Luminescence Microimaging of Historical Samples[END_REF]) illustrated this technique on samples of ancient varnish of a musical instrument and on painting pigments containing ZnO. The authors obtained mapping of the organic materials present in the varnish at the submicrometer scale. They also visualized the heterogeneities of the semiconductor pigment material. Tip Enhanced Raman Spectroscopy (TERS) thanks to the Near-field Scanning Optical Microscopy (NSOM) allows now the acquisition of Raman spectra with a nanometric resolution [START_REF] Fleischer | Near-field scanning optical microscopy nanoprobes[END_REF]. Few examples mostly related to the analyses of dyes are available in the literature. [START_REF] Kurouski | Tip-Enhanced Raman Spectroscopy (TERS) for in Situ Identification of Indigo and Iron Gall Ink on Paper[END_REF] analysed in situ indigo and iron gall ink on historical documents of the 19 th century. These techniques allowed the authors to identify pigments even at low concentration on the papers. Last, X-ray photoelectron spectroscopy (XPS) is a sensitive surface technique that provides chemical information on the few first hundred nanometers in the depth of materials but the lateral resolution is in the micrometric range. Nowadays, thanks to the coupling with synchrotron, X-ray photoelectron emission spectroscopy (XPEEM) can deliver lateral resolution of about 10 nm. Among the few examples available in the Cultural Heritage field, De [START_REF] Stasio | The multidisciplinarity of spectromicroscopy: from geomicrobiology to archaeology[END_REF]) tested the surface of Roman silver coins. and conclude on the presence of fluorine that can avoid the tarnishing of the studied silver coins.

The principle of Secondary Ion Mass Spectrometry (SIMS) has been developed for various facilities that are able to deliver submicronic beams. Among them is the ToF-SIMS (Time of Flight SIMS) that can provide a 200 nm lateral resolution in specific acquisition modes. Complementary to this technique the nano-SIMS provides a beam of about 50 nm of lateral resolution. Both ToF and nano-SIMS allow one to analyse the chemical and isotopic composition of thin surface of the samples. They also work in mapping mode preserving the information on the location of the analyses. The principle of these techniques is to sputter ions on the sample surface, collect and analyse the masses of the ejected secondary ions. The mass/charge ratio is analysed thanks to a mass spectrometer that delivers the elemental and isotopic composition of the samples. ToF-SIMS (Fig. 30) has been used to analyse painting pigments [START_REF] Richardin | Identification of Different Copper Green Pigments in Renaissance Paintings by Cluster-TOF-SIMS Imaging Analysis[END_REF] or human skin remains [START_REF] Cersoy | Cluster TOF-SIMS imaging of human skin remains: analysis of a South-Andean mummy sample[END_REF]. Concerning nano-SIMS, this technique is mostly used in the geoscience field (see for example [START_REF] Hoppe | NanoSIMS: technical aspects and applications in cosmochemistry and biological geochemistry[END_REF]). One study dealing with the understanding of iron corrosion mechanisms focused on an archaeological nail of the 16 th century. In order to determine the transport properties inside the corrosion layer, it was immersed in deuterated water during 3 months [START_REF] Leon | Interfacial layer on archaeological mild steel corroded in carbonated anoxic environments studied with coupled micro and nano probes[END_REF]). The nail was cut and prepared on transverse section. Thanks to the deuterium mapping at the interface with the metal, the authors showed that a micrometric interfacial layer was not porous, blocking the deuterated water penetration. Fig. 31 sums up in a simplified way the scales reached by the quoted nanotechniques and the nature of information they bring. It has to be stressed that, in this simplified view, the techniques at the micrometer are not taken into account, nor the possible requested sample preparation treatment (thin film, etc.) that can render the use of a given technique very tricky. Nevertheless, this short review demonstrates clearly that nanoscale techniques can provide unique information about the chemical composition and structure of ancient materials. Unfortunately, almost all of these techniques imply the sampling of heritage artefacts, not always allowed regarding conservation policies. Nevertheless, the constant improvement and development of new probes and micro-sampling solutions will facilitate their use on fragile or rare materials and thus improve their access for Heritage Science samples.

From description at nanoscale to multiscale modelling: a new challenge for Cultural Heritage

The first parts of the present chapter and the other chapters of this book (réf), show important examples of nano and, more generally, multiscale descriptions and analytical methods of systems linked to Cultural Heritage and ancient materials. These studies are more and more developed and tailored to assess crucial information on dimensioning parameters and to propose pertinent chemical mechanisms or processes. At the contrary very few attempts have been made, using these parameters, to model the behaviour of "heritage systems". A model can have different aims. A first one is, for example, a descriptive approach, allowing one to better understand the influence of each parameter on the global behaviour of the system. This, naturally conducts to an explanatory stage where the past and present behaviour laws of the system are understood. Lastly a predictive model can be build, in order to predict the behaviour of the system for the forthcoming years and centu-ries. Actually these different sorts of models can be combined to describe the complexity of a given system, as for example "multi-agent" approaches. They also can be based on analytical and empirical or phenomenological considerations.

Each of these approaches can concern the so-called: "heritage systems", which behaviours have to be modelled and predicted to propose scenarii for the past evolution of the societies and their environments (historical purpose), or the becoming of human achievements (conservation purpose). These studied systems could be ancient societies or human groups, archaeological landscapes and sites, ancient materials and of course, artefacts and structures that have to be protected and conserved for the future generations.

Models have been used for a long time, for example to understand the properties of materials during their manufacture, as for example glass viscosity [START_REF] Bingham | Roman blue-green bottle glass: chemical-optical analysis and high temperature viscosity modelling[END_REF], or thermodynamic phase predominance domains in metallurgy or corrosion. Nevertheless, one of the most exciting challenge for the modelling of "heritage systems" is to link properties and behaviours at different scales. For example, the modelling of the corrosion behaviour of a heritage artefact made of metal needs to understand the nanometer scale properties of the layers formed on the metallic surfaces, but also the climatic evolution of the area where the artefact is stored or exposed. The need of a multiscale approach is actually not a specificity of conservation science or Cultural Heritage approaches, but is a demand in several domains and fields of science and engineering [START_REF] Yang | An ontological conceptualization of multiscale models[END_REF]. In chemical engineering, because of the need to go from a laboratory scaled production to the industrial one, this multiscale approach is relatively popular for several ten years as detailed by Vlachos [START_REF] Vlachos | A Review of Multiscale Analysis: Examples from Systems Biology, Materials Engineering, and Other Fluid-Surface Interacting Systems[END_REF]. In ecology and environmental sciences, some efforts have been made to perform multi-scale approaches to evaluate structures of landscapes [START_REF] Burnett | A multi-scale segmentation/object relationship modelling methodology for landscape analysis[END_REF]. For "Heritage systems" as buildings, multiscale modelling combining micro and macro approaches allows to understand ancient building techniques in order to better diagnose the seismic risks [START_REF] Mele | Modelling and analysis of a basilica under earthquake loading[END_REF][START_REF] Abruzzese | Mechanical behavior of leaning masonry Huzhu Pagoda[END_REF], or understanding the influence of use of some materials (wooden beams reinforcement) for resistance to earthquakes [START_REF] Kouris | Detailed and simplified non-linear models for timber-framed masonry structures[END_REF]. In the same area of Cultural Heritage, multiscale data are often used for 3D modelling [START_REF] Remondino | 3D Modeling of Complex and Detailed Cultural Heritage Using Multi-resolution Data[END_REF][START_REF] Luca | Methods, formalisms and tools for the semantic-based surveying and representation of architectural heritage[END_REF], but also for the modelling of large and complex archaeological areas [START_REF] Guidi | A Multi-Resolution Methodology for the 3D Modeling of Large and Complex Archeological Areas[END_REF]). Nevertheless, these approaches do not include as large scales gap than the ones that must be involved for other domain of Heritage science as conservation or even multidisciplinary Archaeological sciences. For example, to model the use of resources or materials by ancient societies, to apprehend the qualities of raw materials or artefacts and consequently their technical and economical values, it is necessary to perform analytical studies (referring to Archaeometry or Archaeological science) to assess the nanometric and microscopic composition or structure of the archaeological materials found during excavations (see other chapters). This data can then be included in wider databases, containing information on the different archaeological sites and on the link between the different sites. This holistic analysis of these different kinds of data, collected at different scales (from the nanometre for the chemical composition of the artefacts, to the kilometre scale for the archaeological sites), would allow one, in the future, to model the technico-economical developments of ancient societies, based on the use of the data collected on "Heritage systems". Another crucial issue is linked to Cultural Heritage conservation. The aim here, is to make reliable diagnosis to help the restoration decision [START_REF] Kim | An experience curve-based decision support model for prioritizing restoration needs of cultural heritage[END_REF][START_REF] Cacciotti | A diagnostic ontological model for damages to historical constructions[END_REF]. This implies a multidisciplinary interpretation and a comprehensive knowledge based system in which modelling and predicting alteration processes are crucial steps. Here again, both nanometre (molecular interactions) and kilometre (climate) scales must be understood and integrated in a comprehensive multiscale interactive system. Let us develop in the following the different concepts linked with the prediction of the degradation of materials of Cultural Heritage, which is one of the most important challenges for the protection and transmission of Cultural Heritage.

Dealing with corrosion, several holistic multiscale models have been developed linking the behaviour of a given structure to the one of the surfaces involved in the corrosion processes [START_REF] Cole | Multiscale modelling of the corrosion of metals under atmospheric corrosion[END_REF][START_REF] Cole | Designing molecular protection: new paradigm for developing corrosion resistant materials uniting high throughput studies, multiscale modelling and self-repair[END_REF]. Inspired from these approaches, models dealing with different kinds of alteration processes could try to describe and predict the behaviour of materials encountered in Cultural Heritage.

To that purpose we must identify the processes involved at the different ranges of scales. Inspired by the ones proposed by [START_REF] Cole | Holistic model for atmospheric corrosion Part 1 -Theoretical framework for production, transportation and deposition of marine salts[END_REF] for atmospheric corrosion, it is possible to propose seven scales to model processes (Fig.

32):

• Macro: linked to gross meteorological conditions (polar, tropical continental, etc.). • Meso: meteorological parameters concerning regions of about 100 km 2 (marine, urban, rural, etc.).

• Local: immediate vicinity of the artefact, the building or the site.

• Micro: absolute proximity of the surface material.

• Surface: physical response of the surface (deposition an retention of pollutants, condensation, evaporation), radiation (sun, lights).

• Micron: interaction within interfaces in the material (material/electrolyte), charge and mass transfer and transport phenomena (in electrolyte).

• Nanometer: chemical reactions, charge and mass transfers, transport phenomena (in the solid). Different sub-models could then be defined, linked to each of their scale. Concerning the macro and meso-stage, to our knowledge, no model was directly developed in the field of conservation science, except the one developed by Tidblad et al. [START_REF] Tidblad | Atmospheric corrosion of heritage metallic artefacts: processes and prevention[END_REF] to predict the effect of climatic change on corrosion of metals. Nevertheless, already existing ones could be used in a forthcoming holistic approach. For example some modules developed by [START_REF] Cole | Holistic model for atmospheric corrosion Part 1 -Theoretical framework for production, transportation and deposition of marine salts[END_REF]) can be used for metals but probably also for a wide range of other materials. Up to now, models already developed in the domain of conservation of Cultural Heritage deals only with a single stage and the largest scale is the one corresponding to local conditions.

For example, de la Fuente et al. At the "micro" scale, the 3D models on buildings, as the one developed by Stefani et al. [START_REF] Stefani | Developing a toolkit for mapping and displaying stone alteration on a web-based documentation platform[END_REF]) on the Chambord Castle, can be helpful to locate the different degradation patterns by efficient alteration mapping and to link them to local environmental variations. This is the first step to build models at the "micro" and "surface" stages of the Cole's scale. Then, the local variations can be linked to modelled variation of outdoor or indoor atmospheres. Considering in-door atmosphere of buildings and museums, some attempt has been made to model the evolution of heat and moisture using meteorological data, building properties and users behaviour and a combination of a hydrothermal simulation and finite element approach [START_REF] Huijbregts | Modelling of heat and moisture induced strain to assess the impact of present and historical indoor climate conditions on mechanical degradation of a wooden cabinet[END_REF]. Another example linked to that scale of interest: the modelling of the temperature and humidity distribution by computational fluid dynamic based on finite volumes methods in the Hall of the Domus Aurea in Rome [START_REF] Albero | {CFD} modeling for the conservation of the Gilded Vault Hall in the Domus Aurea[END_REF]). At the "surface" levels, three dimensional computational fluid dynamics model were used to understand the influence of thermohygrometric parameters and airflow patterns responsible for salt recrystallization in the stones of the Crypt of the Lecce Cathedral in South Italy (D' Agostino et al. 2014). At the "micron" scale, several models dealing with local chemical processes have been developed. For example, to understand the reaction of sulphur dioxide from the atmosphere with calcareous porous stones (found for example in statuary) the models considered reactivity and diffusion of species to give a behaviour law and a quantitative description of the processes by 3D simulation [START_REF] Giavarini | A non-linear model of sulphation of porous stones: Numerical simulations and preliminary laboratory assessments[END_REF]. Parameters as crucial as diffusion coefficient of some species were collected and are used to serve modelling of processes at this stage. For example this was performed in a study on bronze corrosion in sulphate medium in order to understand the influence of tin content on corrosion processes [START_REF] Muller | α-CuSn bronzes in sulphate medium: Influence of the tin content on corrosion processes[END_REF]). An analytical model of the long term corrosion of iron reinforcement embedded in hydraulic binders using parameters directly measured on archaeological artefacts was proposed by Chitty et al. (Chitty et al. 2008a;Chitty et al. 2008b). This model considers the kinetic of oxygen diffusion through the system and its consumptions at the metal/corrosion product interface. Also dealing with corrosion of iron from Cultural Heritage buildings, Monnier et al. [START_REF] Monnier | Corrosion of iron from heritage buildings: proposal for degradation indexes based on rust layer composition and electrochemical reactivity[END_REF]) proposed a phenomenological model based on the establishment of a reactivity index of the constitutive phases of the corrosion layers formed on the artefacts. Several models involving the nanoscale have been proposed to describe long term corrosion of iron. They are constructed by taking into account a limiting step, for example the presence of a corrosion layer or nanopores hindering transport at this level [START_REF] Hoerlé | Advances in Understanding Atmospheric Corrosion of Iron II -Mechanistic Modelling of Wet-Dry Cycles[END_REF][START_REF] Bataillon | Corrosion modelling of iron based alloy in nuclear waste repository[END_REF]. Another phenomenological model for long term prediction at "micron" and "nano" scales was also developed for glasses in the context of the deep storage of nuclear wastes [START_REF] Verney-Carron | Long-term modeling of alterationtransport coupling: Application to a fractured Roman glass[END_REF]). This latter one involves nanometer scale phenomenology to understand and predict the formation of a protective gel at the surface of the glass depending on the solution of the water of the surrounding medium (clay). These three latter models were developed after analytical and parametric studies on archaeological and heritage metallic/glassy artefacts despite their application domains are not directly linked to the conservation of Cultural Heritage. Thus, they can perfectly be transposed back to "heritage systems" with the advantage to directly address "realistic" materials and conditions by directly exploiting the formalisms of the developed models.

This short review clearly shows that, by combining all these different scales, from global climatic changes to nano properties of the heritage systems, it would be possible in the next future, at least for several kinds of these systems, to pro-pose such an holistic model. This new challenge could be a wonderful mean to federate research teams around a global interdisciplinary project for conservation of Cultural Heritage.

Nevertheless, these kinds of holistic models could only be achievable by using consolidated databases containing heterogeneous data. Thus, one of the next forthcoming challenge, dealing with all sort of data (from climatic change to nanoscale measurements), is to construct perennial and interoperable databases. Several research teams are now on that path but this objective can only be achievable with a coordination at a national or European level for example.

Last but not least, any model has to be validated. Considering predictive models aiming to predict the behaviour of heritage systems on several 10 years or even centuries, this validation could be a real breakthrough. One solution could be a retroactive approach, considering the behaviour of the system in the past, studying "analogues", for benchmarking. These challenge linked with long term prediction is also one of the most exciting one for researches in the future.

This multistep and multiscale approach for predictive modelling will have to face several challenges to reach the goal. First it has to be stressed that the link between the fine characterisation of mechanisms and the properties of the system and the numerical modelling must constantly be preserved by regular come and go between all the concerned disciplines. This could be implemented in the future in the frame of global interdisciplinary projects dealing with one kind of material (metal, stone, etc.) in a given environment. This kind of project have a sense at the European scale and must gather all the specialists from laboratories skilled in fine and multiscale characterisation to teams specialised in numerical modelling. We believe that it is only in that frame that the synergies between all disciplines will be found. Another crucial aspect, we can think that in the future these new model for predicting the evolution of heritage systems with environment and climate can be considered as a new kind of immaterial Cultural Heritage, in the same way that the digitalised museums or libraries collections. This new "digitalised" Cultural Heritage resource, allowing future generation of scientist to develop and use predictive models will be based on the mastering of international and multi-source databases. This other aspect constitutes also an exciting challenge for future generations that could be faced at the European scale by gathering research and engineering teams using European facilities for data storage and treatment.

Use••

  Aesthetical aspect (surface state, layer, color) • Mechanical behaviour • Provenance and diffusion networks • Nature and properties of surface layer • Evolution of surface treatments • Optical behaviour • Study of use wear • Local composition and structural changes at the (limit of the original) surface • Nature and size of local precipitates • Chemical (trace elements) and isotopic (major elements) local and bulk composition Degradation • Nature of the processes and interaction with environments • Degradation rate • Stability of the system • Thermodynamic properties of the phases/compounds • Chemical reactivity of the layers/products • Transport of species in the layers (liquid/solid state) • Chemical nature of the phases/compounds (composition and structure) Physical behaviour (conductivity, density, hardness, local young modulus etc) • Porosity (variability, size and nature) Chemical reactivity of the phases of the system with the treatment • Transport of the treatment in the system • Stability/evolution of the products used for the treatment • Nature of the new formed phases (composition and structure) • Localisation of the new formed phase in the system • Porosities, mechanical properties • Chemical and structural evolution of the phases in function of the environmental stress (relative humidity, light, temperature, composition)

Fig. 25 .

 25 Fig. 25. Histogram of the apparent size of the coherently diffracting domains obtained after refinement of the powder data, showing a slight increase after the heating process of natural flints (Dejoie et al. 2015)

Fig. 26 .

 26 Fig. 26. Atomic Force image taken in a liquid phase (1 -propanol) of the Ca(OH)2 nanoparticles obtained at 60°C: image dimensions 3.3 µm x 3.3 µm,[START_REF] Ambrosi | Colloidal Particles of Ca(OH)2: Properties and Applications to Restoration of Frescoes[END_REF]

Fig. 27 .

 27 Fig. 27. (a) Black "crayon" of Combe Saunière, the only faceted block Marcillaud J.G. (b) G17C20 block of Combe Saunière: TEM micrograph of todorokite crystals identified by electron microdiffraction pattern ([001] zone axis single crystal),[START_REF] Chalmin | Minerals discovered in paleolithic black pigments by transmission electron microscopy and micro-X-ray absorption near-edge structure[END_REF]

Fig. 28 .

 28 Fig. 28. STXM investigations of the sample thin film GL10-03: (a) SDV component maps of the Fe(III), Fe(II) and Fe(0) species obtained with the siderite, maghemite and metal reference spectra. Grey scales for Fe(II), Fe(III) and Fe(0) maps: equivalent thickness (nm), for the residual map: optical density. (b) Overlay of the color-coded Fe(0), Fe(II) and Fe(III) component maps (c) C K-edge difference map (290-280 eV) of the thin film.[START_REF] Leon | Interfacial layer on archaeological mild steel corroded in carbonated anoxic environments studied with coupled micro and nano probes[END_REF]

Fig. 29 .

 29 Fig. 29. FIB/SEM image of the cell wall of pine treated with colophony, Giachi et al.

Fig. 30 .

 30 Fig. 30. High spatial resolution ion images of the skin cross section. (5a): Three color overlay between ion images (HYP: hydroxyproline fragment in red (R), leucine or isoleucine fragment in green (G) and calcium ion in blue (B)). (5b): hypodermis area, two color overlay between ion images (protein backbone in red (R) and fatty acids in blue (B)). Field of view 200x200 mm2; 512x512 pixels, pixel size 390 nm.[START_REF] Cersoy | Cluster TOF-SIMS imaging of human skin remains: analysis of a South-Andean mummy sample[END_REF]

Fig. 31 .

 31 Fig. 31. Summary of the nanoscale techniques used in Cultural Heritage (SR-TOMO : X-ray tomography under synchrotron radiation, AFM : Atomic Force Microscopy, FIB-SEM: Focused Ion Beam Scanning Electron Microscopy, HR XRD: High Resolution X-ray Diffraction, STXM: Scanning Transmission X-ray Microscopy, TEM: Transmission Electron Microscopy, TERS: Tip Enhanced Raman Spectroscopy, ToF-SIMS: Time of Flight Secondary Ion Mass Spectrometry, XAS: X-ray Absorption Spectroscopy, XPS: X-ray Photoelectron Spectroscopy.

  (de la Fuente et al. 2011), aims to assess and predict the effect of different pollutants on materials and object of the Cultural Heritage. Depending on the variation of concentration of different pollutants (SO 2 , NO 2 , O 3 and PM10-Particulate Matter with size bellow 10µm) and the alteration rate observed on different kinds of materials (bronze, limestone, …), dose response functions were empirically established. They allowed one to propose maps corresponding to different scenarios of pollutions and damages.

Fig. 32 .

 32 Fig. 32. Framework for an holistic model of alteration of Cultural Heritage (inspired from Cole et al. 2003)

Table 1 .

 1 Topic, historical, cultural and conservation issues, scientific questions and associated physico-chemical parameters to be determined at different scale on the Cultural Heritage systems Chemical composition: major/minor/trace elements • Isotopic ratio of some elements • Valence/oxidation state/local structure • Nature of the phases (structure) • Size of the particles, precipitates and aggregates • Nature and size of porosities

	Physico-chemical information to be gathered
			•
	Scientific questions		• Composition, texture and structure of the	bulk material or coatings/surfaces • Thermodynamic properties of phases,	Partitioning of elements during proc-	esses • Reactivity linked to several parameters	(temperature, oxido-reduction,…) • Transport of species (solid and gaseous	states)
	Historical, cultural and	conservation issues	• Initial raw materials used	• Provenance and diffusion	networks	• Nature of the processes
				Manufacturing
	Topic

http://www.icom-cc.org/

http://www.iccrom.org/


Adrian Daerr
email: adrian.daerr@univ-paris-diderot.fr

Juliette Courson

Margaux Abello

Wladimir Toutain

Bruno Andreotti

The charmed string: self-supporting loops through air drag

The string shooter experiment uses counter-rotating pulleys to propel a closed string forward. Its steady state exhibits a transition from a gravity dominated regime at low velocity towards a high velocity regime where the string takes the form of a selfsupporting loop. Here we show that this loop of light string is not suspended in the air due to inertia, but through the hydrodynamic drag exerted by the surrounding fluid, namely air. We investigate this drag experimentally and theoretically for a smooth long cylinder moving along its axis. We then derive the equations describing the shape of the string loop in the limit of vanishing string radius. The solutions present a critical point, analogous to a hydraulic jump, separating a supercritical zone where the wave velocity is smaller than the rope velocity, from a subcritical zone where waves propagate faster than the rope velocity. This property could be leveraged to create a white hole analogue similar to what has been demonstrated using surface waves on a flowing fluid. Loop solutions that are regular at the critical point are derived, discussed and compared to the experiment. In the general case, however, the critical point turns out to be the locus of a sharp turn of the string, which is modelled theoretically as a discontinuity. The hydrodynamic regularisation of this geometrical singularity, which involves non-local and added mass effects, is discussed based on dimensional analysis.

Introduction

In the broad field of fluid structure interactions, the study of flexible slender objects in axial flow has been motivated by many different engineering problems from nuclear reactors to aeronautics [START_REF] Païdoussis | Fluid-structure interactions: Volume 2: Slender structures and axial flow[END_REF], and also by its importance in living organisms, for example for the propulsion of swimmers [START_REF] Gazzola | Gait and speed selection in slender inertial swimmers[END_REF]. In the textile industry, ring spinning, air-jet weft insertion and demand for high weaving speed have triggered research on the dynamics of light fibres. Here we show that the hydrodynamic drag of air is a crucial ingredient in the dynamics of the string shooter, a physics toy in which a closed loop of flexible string is longitudinally entrained. This toy was popularised online by [START_REF] Yeany | String shooter video[END_REF], and its understanding was one of the problems of 2019's International Physicists' Tournament [START_REF] Collomb | International physicists' tournament[END_REF].

We also show that string dynamics formally corresponds to 1D hydrodynamics, with pressure in the Navier-Stokes equations replaced by string tension. As opposed to the analogous incompressible flow through a flexible pipe [START_REF] Doaré | The flow-induced instability of long hanging pipes[END_REF][START_REF] Païdoussis | Fluid-structure interactions: Volume 2: Slender structures and axial flow[END_REF] however, the geometrical inversion -the solid boundary is moving in a surrounding liquid (air) at rest, as opposed to fluid flowing inside a guiding tube -leads to very different dynamics of the string shooter because momentum is transferred to the environment. This motivates a particular focus on the hydrodynamic drag experienced by the string. 2 Preliminary observations

Experimental set-up

A string loop is entrained by rubber wheels mounted on two identical DC motors rotating at the same velocity but in opposite directions (figure 1). The wheels impose the linear velocity V and the angle at which the string is ejected. The transition from a pendent to a self-supporting shape is observed only for strings of low linear mass. We have used a three-ply cotton string, with a very low effective elasticity, of radius R = 5.5 • 10 -4 m and of density ρ s = 300 kg m -3 . Introducing its cross-sectional area A s = πR 2 , the linear mass density is ρ s A s = 2.88 • 10 -4 kg m -1 as directly measured using chemistry scales with a resolution better than one percent. The length of the string, the ejection angle of the string at the shooter and the velocity V are our three control parameters.

Evidence for the hydrodynamic lift of the string

Figure 1 shows low and high velocity steady state shapes of the string, which are directly related to the fact that it makes a loop. Indeed, a very long, unclosed string entrained by the same two wheels traces a long inverted catenary trajectory, as expected [START_REF] Biggins | Growth and shape of a chain fountain[END_REF]. The role of air becomes clear when performing the experiment at reduced pressure (figure 1c): despite high kinetic energy that would propel a free string many meters up and away (here V 2 /g 23 m), the looped string hangs down as for low velocities at atmospheric pressure. Further evidence for the importance of air drag is the possibility of replacing the wheel entrainment (figure 1e) by an aerodynamic entrainment (figure 1f).

The string can be safely considered as inextensible: in our experiments the maximum elongation is of order ρ s A s V 2 /EA s = 10 -3 , based on the string's measured stiffness EA s = 35 N, and tensions of order ρ s A s V 2 = 0.03 N for V = 10 m s -1 . In the steady state, the continuity equation applied to any piece of inextensible string implies that the velocity is tangent everywhere to the string and is constant in modulus. As a consequence, it can be written as V t, where V is imposed by the motor. Consider first a simple model in which one would neglect the hydrodynamic drag exerted by the air on the string. Then, the string would be a conservative system whose energy is minimal at equilibrium. The kinetic energy, 1 2 ρ s A s V 2 , does not depend on the shape of the string. As a consequence, such a model predicts that the string shape should always be the one minimising the potential energy, leading to a solution independent of the driving velocity (Walton & Mackenzie, 1854, p. 66) -in contrast to the observations reported in figure 1.

A minimal string shooter model

A minimal model

The minimal model for the self-sustained string must therefore involve inertia, gravity, string tension and the drag by the surrounding fluid. From dimensional analysis, the drag force per unit length exerted on a string moving along its tangent takes the form: -αρ f R|v|v, where α is a dimensionless 'skin' friction, ρ f the fluid density and v the material velocity. Assuming inextensibility, the equation of motion reads:

ρ s A s dv dt = ρ s A s g + ∂(T t) ∂s -αρ f R|v|v (1)
where s is the curvilinear coordinate and t the unit tangent vector (figure 1d). Importantly, the acceleration is the total time derivative of the velocity:

d dt = ∂ ∂t + v • t ∂ ∂s .
In the steady state, the continuity equation implies v = V t, resulting in the following force/inertia balance:

ρ s A s g + ∂ ∂s T -ρ s A s V 2 t -αρ f RV 2 t = 0 (2)
The effect of inertia is to shift the tension to negative values by a constant ρ s A s V 2 . We will call T -ρ s A s V 2 the effective tension which, contrary to the true tension T , can be negative. We introduce the dimensionless number comparing tension to inertial forces as:

T = T ρ s A s V 2 (3)
As T /ρ s A s is the velocity of transverse waves, the Mach number is simply related to the rescaled tension as T -1/2 . The 'Mach 1' critical point therefore coincides with T = 1 and separates a supersonic regime at T < 1 from a subsonic regime at T > 1.

The transsonic regime T ∼ 1 is discussed at the end of this article.

How can the hydrodynamic drag lift the string?

Integrating equation ( 2) over the closed loop, one sees that the weight ρ s A s g must be balanced by the discontinuity of the effective tension (T -ρ s A s V 2 )t across the driving point. Indeed, the hydrodynamic drag is constant and tangent to the string and has therefore a null resultant. The tension is controlled by inextensibility in a way analogous to pressure in incompressible hydrodynamics: its discontinuity is a consequence 4) with ∆ and S from the raw shape, blue triangles: equation ( 4) with ∆ and S of the numerical best fit of the string steady state shape by the theory. The green squares are independent measurements on free falling string segments. The grey diamonds are measurements on 2-strand acryl yarn reported by [START_REF] Uno | A study on an air-jet loom with substreams added[END_REF]. The dashed line curve is the parameter-free prediction in the hydrodynamically smooth regime (equation 7) for an infinite straight string. The solid line is the best fit in the hydrodynamically rough regime (r 0 = 180 µm). (b) Notations for the drag calculation. The velocity field is assumed to be axisymmetric.

of hydrodynamic drag, and would be zero in its absence (see section 3.4). The torque of the weight, evaluated at the driving point, is equal to -ρ s A s g∆ where ∆ is the horizontal distance between the centre of mass of the string and the driving point. The self sustained state is therefore characterised by a non vanishing torque due to weight (figure 1d). The effective tension on the other hand has a null lever arm at the driving point and therefore induces no torque. The hydrodynamic torque, obtained by integrating the drag moment along the string, is proportional to the surface area S of the loop, and reads 2Sαρ f RV 2 . Here we have used Green's theorem to relate the circulation to the area spanned by the loop:

| t × r ds| = (xdz -zdx) = 2 S dxdz = 2 S.
In the steady state, the hydrodynamic torque balances the torque due to the weight. It is therefore the hydrodynamic drag, exerted by the air on the string, which leads to a self-supporting loop. The horizontal displacement of the centre of mass ∆ increases with V but must remain below /4, by geometrical constraint. One accordingly predicts that the loop becomes slim at high speed, its area decreasing as S ∝ V -2 . Under the assumption that further dynamical mechanisms may be safely ignored, the torque balance can be used to estimate the hydrodynamic drag from the string shape, namely from the measurement of ∆ and S:

α = ρ s A s g 2ρ f RV 2 ∆ S (4)

Estimating the turbulent skin drag on the string

The effective drag coefficient deduced experimentally from equation ( 4) can be compared to theoretical expectations, based on well established phenomenological laws. Con-

The charmed string: self-supporting loops through air drag sider an infinite cylinder moving along its axis at constant velocity V (figure 2b). This is a turbulent boundary layer problem in axisymmetric geometry. In the steady state the momentum flux across any cylinder of radius r > R must be the same, so that the shear stress σ rz decreases as r -1 . Using the Prandtl mixing length approach [START_REF] Schlichting | Grenzschicht-Theorie[END_REF][START_REF] Van Driest | On turbulent flow near a wall[END_REF][START_REF] Flack | Review of hydraulic roughness scales in the fully rough regime[END_REF], the turbulent Reynolds stress outside the surface (viscous or rough) sublayer then reads:

σ rz ρ f κ 2 (r -R) 2 ∂u z ∂r ∂u z ∂r = - αρ f V 2 R 2πr (5)
where κ 0.41 is the von-Kármán constant. The momentum balance integrates into:

u z = V - α 2π V κ log 4R r 0 √ r - √ R √ r + √ R (6)
where the integration constant r 0 is identified as the hydrodynamic roughness in the limit r 0 R. In the smooth hydrodynamic regime, one expects a roughness r 0

1 8 2π α η f ρ f V
where the factor 1/8 is a phenomenological constant determined experimentally in the case of quasi-bidimensional turbulent boundary layers [START_REF] Schlichting | Grenzschicht-Theorie[END_REF], chap 17: logarithmic overlap law ). When the viscous sublayer is smaller than the corrugations of the wire, in the hydrodynamically rough regime, one rather expects r 0 to be a fraction of the geometrical roughness. The velocity must vanish at infinity, which leads to the relation:

α = 2π κ 2 log 2 4R r 0 (7)
The velocity tends to 0, far from the string, as 1/ √ r. As a consequence, the momentum per unit length stored in the air, 2πρ f ru z dr is infinite in the steady state (the integral diverges as r 3/2 ). This means that added mass effects, discussed later, would diverge for an asymptotically long and straight string: they must be limited by the finite size or the finite radius of curvature, which makes their quantitative modelling difficult.

Figure 2 shows that the measured values of α, using equation ( 4), are almost constant, only slightly decreasing with increasing Reynolds number. The average value α 0.167 corresponds to a hydrodynamic roughness of r 0 180 µm R/3, more than ten times larger than the viscous sublayer. This value is consistent with rough surface structures whose size is of the order of the string radius R, such as those formed by the three plied strands of our cotton yarn. The skin friction coefficient α was also independently measured in a free fall experiment using string segments attached to an excess weight. This drag is slightly lower, possibly due to the absence of counter-moving string. Measurements of longitudinal air-drag of very long cylinders are scarce in the literature. The air-drag of textile fibres is exploited in air-jet looms, but studies mostly consider short segments in non-homogeneous air-flow. As a notable exception [START_REF] Selwood | The axial air-drag of monofilaments[END_REF], using an elegant set-up to drag 20 m of nylon fibre through still air, reports a drag coefficient of 0.093 at the highest Reynolds number R = 60, well below our values as expected for a much smoother filament. Measurements on long ( /R > 4000) rough two-filament yarn by [START_REF] Uno | A study on an air-jet loom with substreams added[END_REF] on the other hand are in accordance with our data (figure 2a) and confirm that plied yarn has a larger effective roughness than mono-filaments.

Predicted shape

We express the curvilinear coordinate s = SV 2 /g along the string in terms of its dimensionless counterpart S using V 2 /g as the characteristic length. The equation (2), 1.1602, 1.43546, 2.18369, 4.21761, 9.74637), keeping the ejection angle and string length constant. The shapes have been superimposed at the singularity, whereas experimentally the leftmost point (corresponding to the shooter) is fixed. Lengths are in units of V 2 0 /g.

projected onto the tangent and the normal vectors, yields:

dT dS = sin θ + A (8a) dθ dS = cos θ T -1 (8b)
where the dimensionless drag coefficient scales as V 2 :

A = α ρ f RV 2 ρ s A s g (9)
Equation (8a) integrates to T T 0 + AS + Z, where Z is the altitude rescaled by V 2 /g and T 0 the tension immediately after the driving point.

Equation ( 8b) is singular at the critical point T = 1, except if the latter is located at the point whose tangent is vertical (θ = 3π/2). In this case, equations (8) admit around the critical point an asymptotic solution of the form:

T ∼ 1 + (A -1)S (10) θ(s) ∼ 3π/2 -CS 1/(A-1) (11)
The charmed string: self-supporting loops through air drag 5.29 m s -1 , same experiment as in figure 3). Seven deformations were tracked on both upper (red) and lower (green) branches, and their trajectories superimposed by taking t = 0 upon leaving the propelling wheels. Wave-fronts travelling in the direction of the string movement on the upper branch and in the reverse direction on the lower branch slow down critically as they approach the singularity at s = 0. The black lines are calculated from the tension shown in the right graph. (b) The string tension, corresponding to the best shape fit of upper and lower parts by our model (inset), increases along the string in the direction of string motion. T = 1 defines the singular point, which defines the origin of the curvilinear coordinate s in both graphs.

This solution is used as a starting point, parameterised by A and C, for numerical integration of equations ( 8). Figure 3a shows the family of solutions for a fixed value of the drag A. Loop shapes near critical drag (blue example in figure 3) are well reproduced, letting A be an adjustable parameter. Figure 3b shows solutions that differ in velocity V , but share the same ejection angle and string length. The succession of shapes shows the transition from gravity dominated shapes to lifted states.

Wave propagation

The theory predicts that all upper branches move at supersonic speeds with respect to transverse waves (T < 1), while on the contrary lower branches are subsonic (T > 1). This produces a white hole analogue similar to what can be achieved using surface waves on a flowing fluid [START_REF] Schützhold | Gravity wave analogues of black holes[END_REF][START_REF] Rousseaux | Horizon effects with surface waves on moving water[END_REF]. As opposed to finite time singularities that occur in the motion of open-ended strings [START_REF] Mcmillen | Whip waves[END_REF][START_REF] Brun | The surprising dynamics of a chain on a pulley: lift off and snapping[END_REF], the transsonic singularity in the propelled loop is stationary. To test its existence experimentally, we take advantage of the fact that the knot which ties the string into a loop produces small deformations of the loop shape at each passage through the propelling wheels. We tracked the position of seven deformations on both upper and lower branches in one given experiment, and find that these transverse perturbations exhibit a critical slowing down as they travel along the string (figure 4a). This is a signature of the singularity: transverse waves propagate at a speed ± T /ρ s A s relative to the moving string. Waves that propagate upstream, that is in the direction opposite to the string movement, therefore appear to move at a speed V -T /ρ s A s in the laboratory reference frame. This speed is positive (moving with the string) on the upper branch (where T /ρ s A s = √ T V < V ) and negative on the lower branch ( T /ρ s A s > V ), so that the perturbations travel towards the singularity on either side of it. The best shape fit this experiment by equations (8) yields a tension curve T (s) (figure 4b) from which we deduce trajectories ds/dt = V (1 -√ T ) that closely match the observations (figure 4a): this confirms the existence of the singularity and validates the minimal model, immediately above the threshold.

4 Conclusive remarks: beyond the minimal model 4.1 Departure from the pear shape When the speed is increased above the threshold, the shape becomes widest in the middle and tapers at both ends in the manner of a spindle. The spindle becomes slimmer with increasing velocity (figure 5a), in accordance with the minimal model and in particular the torque balance of equation ( 4). At high speed, however, the string exhibits an angular point at the singularity, which does not correspond to any of the smooth pear-shaped solutions of figure 3. An example is shown in figure 5b,c. The upper and lower branches are still well fitted by equations (8), on condition that both the angle and the tension are allowed to be discontinuous across the singularity. The discontinuity vanishes near the onset of the self-supporting loop regime (figure 5a), where the string has a vertical tangent at the singular point and the shape is well described by a pear-shaped solution.

Dynamical mechanisms in the vicinity of the critical point

As the upper and lower branches are still well described by the minimal model outside the class of smooth 'pear' solutions, some further dynamical mechanism must show up and dominate only in the vicinity of the singularity, and produce the small slightly asymmetric bend of finite curvature that can be observed in figure 5c. Placing an obstacle (a solid plate) near the tip causes important changes in this curved upper region around the critical point. This directly shows that non-local effects are present. Momentum is transferred across the loop via the entrained fluid, especially between counter-moving upper and lower branches near the end where the shape tapers. As boundary layers develop along a straight string, the drag force diminishes [START_REF] Gould | Air drag on synthetic-fibre textile monofilaments and yarns in axial flow at speeds of up to 100 metres per second[END_REF]. In contrast we expect the drag coefficient α to increase when the string is curved, explaining the extra-dissipation associated with the sudden change of tension around the corner.

Added mass effect and hydrodynamic torque

As air flows along the moving curved string, there must be an added mass effect i.e. a hydrodynamic force component normal to the string that can be estimated using a simple momentum balance argument. Let δ denote the effective thickness of the air layer that flows at the speed V of the string and that changes direction when the string is curved. The momentum flux of this layer is then ρ f R+δ R+r 0 2πru 2 z dr 8πρ f Rδ/ log 2 (4R/r 0 ), and the centripetal force required for a direction change is proportional to this flux times curvature. The reaction on the string can therefore be included by replacing ρ s A s with ρ s A s + ρ f A f , with A f = 8πδR/ log 2 (4R/r 0 ), in the inertial term of the momentum balance. We will not try here to model δ, which may depend on the whole shape of the string. As an upper-bound, one may consider the thickness of a boundary layer developing over a length ∼ , obeying δ log(δ/r 0 ) ∼ . As δ is at most ∼ 10 cm, the added mass effect can only increase the apparent density by 10 % of ρ s A s .

Additional effects come into play when we consider a string of finite diameter, for which we may notably have local torque N :

(ρ s A s + ρ f A f )V 2 ∂ ∂s t = ρ s A s g + ∂ ∂s (T t + N n) -αρ f RV 2 t (12) (ρ s I s + ρ f I f )V 2 ∂ 2 θ ∂s 2 = N (13)
where I s = πR 4 /4 is the string's moment of inertia and I f = 4πδ 3 R/3 log 2 (4R/r 0 ) the added mass effect, which is a first order description of the hydrodynamic moment exerted on the string. This time, the added mass effect is typically 10 3 larger than the solid contribution (prefactor 10 -7 vs 10 -10 in the dimensionless equations), and also larger than the moment resulting from the string's elastic bending stiffness [START_REF] De Langre | Flutter of long flexible cylinders in axial flow[END_REF]. The latter is estimated from the elasto-gravitary length a 3 cm -measured from the deflection of cantilevers made of string -to approximately -EI g 2 /ρ s AV 6 -g 3 a 3 /V 6 -3 • 10 -8 . Note that the signs of inertial and elastic stiffness are opposite, so the relative importance of inertial and elastic stiffness has a strong impact on the behaviour of the solution near the singularity. The torque contribution becomes of order 1 when the curvature and its derivatives change significantly over length scales smaller than 5 • 10 -3 V 2 /g 5 cm, compatible with the scale at which the string bends near the singularity (figure 5c). These examples of higher order hydrodynamic effects are by no means exhaustive: the dynamics of finite size bodies accelerating in flows is notoriously hard to model and includes History forces and drag corrections [START_REF] Mordant | Velocity measurement of a settling sphere[END_REF][START_REF] Calzavarini | Impact of trailing wake drag on the statistical properties and dynamics of finite-sized particle in turbulence[END_REF]. Understanding the string dynamics in the vicinity of the singularity will thus require a much more involved analysis of the hydrodynamics of the surrounding fluid.

Perspectives

The drag on cylinders in axial flow is dominated by the pressure contributions at the end caps for all but very elongated cylinders. Loops on the other hand experience only skin drag, making the experiment discussed here an ideal set-up for its study, including effects of finite size. A promising application in the textile industry may be the estimation of axial drag coefficients of fibres simply from the transition speed between gravitational and lifted states, and from the string's linear mass. Such a lowtech method may be of direct interest to textile producers wishing to adjust an air-jet loom to a new fibre type, without the need for a sophisticated and costly wind channel and drag measurement apparatus.

On a more fundamental level, the formal resemblance of string dynamics with 1D hydrodynamics, with tension playing the role of pressure, suggest a rich field of study both in steady and unsteady cases. The mathematical structure of the theoretical problem is particularly interesting, as the dynamical equations are easy to express in Lagrangian coordinates but the boundary conditions (obstacles for instance) are typically Eulerian. The creation of lift through axial drag forces, mediated by tension, is just a hint at more complex behaviour to be expected in unsteady situations. Of particular interest may be the interactions between moving filaments in turbulent flows, and propulsion systems involving several filaments (tentacles).

We thank Physique Expérimentale and the Physics Department of Université de Paris.

Figure 1 :

 1 Figure 1: A string loop entrained by two counter-rotating wheels at speed V exhibits a transition from (a) a gravity dominated regime at low velocity towards (b) a high velocity regime where the string takes the form of a self-supporting loop. (c) Reducing the pressure from atmospheric pressure to 20 kPa while keeping the high velocity reverts the string back to its lowered configuration. Images (a-c) were processed for readability by subtracting the background and enhancing contrast. Each image is 6.25 cm large. (d) Notations used in the text. The string is entrained by either (e) counter-rotating wheels or (f) by blowing air along a small portion of it.

Figure 2 :

 2 Figure 2: (a) The friction coefficient α decreases slightly as a function of the Reynolds number based on the string radius, R = ρ f RV /η f . Red circles: equation (4) with ∆ and S from the raw shape, blue triangles: equation (4) with ∆ and S of the numerical best fit of the string steady state shape by the theory. The green squares are independent measurements on free falling string segments. The grey diamonds are measurements on 2-strand acryl yarn reported by[START_REF] Uno | A study on an air-jet loom with substreams added[END_REF]. The dashed line curve is the parameter-free prediction in the hydrodynamically smooth regime (equation 7) for an infinite straight string. The solid line is the best fit in the hydrodynamically rough regime (r 0 = 180 µm). (b) Notations for the drag calculation. The velocity field is assumed to be axisymmetric.

Figure 3 :

 3 Figure 3: A typical experimental curve observed just above the threshold is shown in blue on both panels, obtained for V 0 = 5.29 m s -1 . (a) Solutions to equations (8) for various initial conditions and A = 1.44. Red are possible upper branches of the steady string loop, green possible lower branches. Note the up → down, red → green symmetry. In a given experiment (blue) the upper and lower parts of the loop each match one of the solutions. Note that the propelling system is on the left. Lengths are in units of V 2 /g. (b) Change of the selected shape as the drag parameter A increases (from top to bottom: 1.00798, 1.02168, 1.05893,1.1602, 1.43546, 2.18369, 4.21761, 9.74637), keeping the ejection angle and string length constant. The shapes have been superimposed at the singularity, whereas experimentally the leftmost point (corresponding to the shooter) is fixed. Lengths are in units of V 2 0 /g.

Figure 4 :

 4 Figure 4: (a) (a) Position of transverse wave-fronts on a string above the threshold (V 0 =5.29 m s -1 , same experiment as in figure3). Seven deformations were tracked on both upper (red) and lower (green) branches, and their trajectories superimposed by taking t = 0 upon leaving the propelling wheels. Wave-fronts travelling in the direction of the string movement on the upper branch and in the reverse direction on the lower branch slow down critically as they approach the singularity at s = 0. The black lines are calculated from the tension shown in the right graph. (b) The string tension, corresponding to the best shape fit of upper and lower parts by our model (inset), increases along the string in the direction of string motion. T = 1 defines the singular point, which defines the origin of the curvilinear coordinate s in both graphs.

Figure 5 :

 5 Figure 5: (a) Aspect ratio (blue squares) and angle discontinuity at the singular point (yellow circles) of the string loop at different velocities. Note the factor 2 in L/2W added to get a common scale. (b) Experimental loop shape (black, = 3.4 m, V = 9.53 m s -1 ) with superposed best fits by the simplified model (equations 8) of the upper (red) and lower (green) parts. (c) Detailed view of the vicinity of the singularity at the top right, showing that the apparent angle discontinuity is regularised in the experiment.