Philippe Stempflé
email: philippe.stempfle@ens2m.fr

Xavier Bourrat

Olivier Pantalé

Richard Kouitat Njiwa

Jean-Philippe Jehl

Anne Domatti

Evelyne Lopez

Multiscale Structure of Nacre Biomaterial: Thermomechanical Behavior and Wear Processes

Keywords: Nacre, Organic matrix, Multiscale analysis, Frictional dissipated power

Sheet nacre is a hybrid biocomposite with a multiscale structure, including nanograins of CaCO 3 (97% wt.% -40 nm in size) and two organic matrices: (i) the interlamellar mainly composed of β-chitin and proteins, and (ii) the intracrystalline composed by silk-fibroin-like proteins. This material is currently contemplated for the manufacture of small prostheses (e.g. rachis and dorsal vertebra prostheses) which are subjected to micro-slip or fretting motion. In this work, the tribological behaviour of nacre is studied by varying the frictional dissipated power from few nW to several hundreds mW, in order to assess the various responses of the different nacre's components, independently. Results reveal various dissipative mechanisms vs. dissipated frictional power: organic thin film lubrication, tablet's elastoplastic deformations, stick-slip phenomenon and/or multiscale wear processes, including various thermo-mechanical processes (i.e., mineral phase transformation, organics melting and friction-induced nanoshocks process on a large range). All these mechanisms are controlled by the multiscale and anisotropy of its structure -and especially by its both matrices and respective orientation vs. the sliding direction.

INTRODUCTION

Sheet nacre (the pearly internal layer of molluscan shells) is a biocomposite displaying an organomineral multiscale structure [START_REF] Lee | Mother-of-Pearl: An Engineering Gem in Remarkable Natural Material Surfaces and Their Engineering Potential[END_REF][START_REF] Meyers | Biological materials, Structure and mechanical properties[END_REF][START_REF] Levi-Kalisman | Structure of the nacreous organic matrix of a bivalve mollusc shell examined in the hydrated state using cryo-TEM[END_REF][START_REF] Li | Nanoscale structure and mechanical characterization of a natural nanocomposite material : the shell of red abalone[END_REF][START_REF] Rousseau | Multiscale structure of sheet nacre[END_REF][START_REF] Li | Unveiling the formation Mechanism of Pseudo-Single-Crystal Aragonite Platelets in Nacre[END_REF]. It exhibits high strength and toughness though it is primarily composed of a hard and very brittle ceramic -so-called aragonite (97%wt.) [START_REF] Espinosa | Merger of structure and material in nacre and bone -persepectives on de novo biomimetic materials[END_REF][START_REF] Denkena | Ductile and brittle material removal mechanisms in natural nacre -a model for novel implant materials[END_REF]. The well-known nacre's superior mechanical properties lies in its hierarchical structure (Fig. 1) meaning that particular structures exist at distinct length scales. Thus, Dutta et al [START_REF] Dutta | Optimal overlap length in staggered architecture composites under dynamic loading conditions[END_REF] first reported the importance of length scales in maximizing both load-transfer capability and resistance to crack initiation in composites both under static and impact rates of loading.

At the first hierarchical level (Fig. 1a), sheet nacre is usually described as a layered brick & mortar architecture [START_REF] Levi-Kalisman | Structure of the nacreous organic matrix of a bivalve mollusc shell examined in the hydrated state using cryo-TEM[END_REF][START_REF] Espinosa | Merger of structure and material in nacre and bone -persepectives on de novo biomimetic materials[END_REF]: bricks corresponds to aragonite tablets (1-5 µm long and 500 nm thick) while the mortar appears as an interlamellar organic matrix made of highly cross-linked protein of polysacharride and b-chitin (30 nm thick) [START_REF] Meyers | Biological materials, Structure and mechanical properties[END_REF]. Although this organic matrix constitutes a few percent of the total weight of the shell (< 3%wt.) it has a significant influence on the overall mechanical properties of nacre, as demonstrated by many authors from years (eg. [START_REF] Meyers | Biological materials, Structure and mechanical properties[END_REF][START_REF] Vincent | Structural Biomaterials[END_REF][START_REF] Kaplan | Mollusc shell structures: novel design strategies for synthetic materials[END_REF][START_REF] Wang | Deformation mechanisms in nacre[END_REF][START_REF] Dutta | Crack tortuosity in nacreous layer -Topological dependence and biomimetic design guideline[END_REF]). Indeed, interlamellar layers of polymers act as energy absorbers which limit the propagation of cracks and finally avoid the formation of critical failure within the biocomposite [START_REF] Dutta | Crack tortuosity in nacreous layer -Topological dependence and biomimetic design guideline[END_REF].

However, recent works [START_REF] Li | Nanoscale structure and mechanical characterization of a natural nanocomposite material : the shell of red abalone[END_REF][START_REF] Rousseau | Multiscale structure of sheet nacre[END_REF][START_REF] Li | Unveiling the formation Mechanism of Pseudo-Single-Crystal Aragonite Platelets in Nacre[END_REF][START_REF] Ph | Mechanical properties of the elemental nanocomponents of nacre structure[END_REF]] demonstrated a second hierarchical level which takes place within the tablet (Fig. 1b). The latter is indeed not purely mineral but is itself an organomineral composite -Fig. 1c andd constituted by aragonite biocrystals (40 nm in size, Fig. 1e) drowned within a 4 nm thick intracrystalline organic matrix -constituted by silk-like proteins -organized as a tridimensional structure forming the micro-sized flat tablets (thickness about 500 nm, Fig. 1b) [START_REF] Peireira-Mouriès | Soluble silk-like organic matrix in the nacreous layer of the bivalve Pinctada maxima-a new insight in the biomineralization field[END_REF]. Hence, these tablets are further able to dissipate energy by elastic-plastic deformation leading to an overall increase in fracture toughness [START_REF] Ph | Mechanical properties of the elemental nanocomponents of nacre structure[END_REF][START_REF] Xu | Deformation strengthening of biopolymer in nacre[END_REF].

Because its hierarchical structure is a great source of inspiration in the design of new kinds of composites, sheet nacre has been largely studied from decades by researchers and engineers [START_REF] Dutta | Crack tortuosity in nacreous layer -Topological dependence and biomimetic design guideline[END_REF][START_REF] Munch | Tough bio-inspired hybrid materials[END_REF][START_REF] Luz | Biomimetic design of materials and biomaterials inspired by the structure of nacre[END_REF][START_REF] Zhu | A novel biomimetic material duplicating the structure and mechanics of natural nacre[END_REF][START_REF] Tang | Nanostructured artificial nacre[END_REF]. In particular, this material is currently being studied in two main research fields: (i) the creation of new organic /inorganic bio-inspired hybrid materials [START_REF] Dutta | Crack tortuosity in nacreous layer -Topological dependence and biomimetic design guideline[END_REF][START_REF] Munch | Tough bio-inspired hybrid materials[END_REF][START_REF] Luz | Biomimetic design of materials and biomaterials inspired by the structure of nacre[END_REF][START_REF] Zhu | A novel biomimetic material duplicating the structure and mechanics of natural nacre[END_REF] and (ii) the design of small prostheses [START_REF] Tang | Nanostructured artificial nacre[END_REF][START_REF] Atlan | Interface between bone and nacre implants in sheep[END_REF][START_REF] Lopez | Demonstration of the capacity of nacre to induce bone formation by human osteoblasts maintained in vitro[END_REF][START_REF] Westbroek | A marriage of bone and nacre[END_REF]. In this framework, classical approaches consist to strive to produce synthetic composites that display desired mechanical properties by mimicking the nacre structures at different scales [START_REF] Luz | Biomimetic design of materials and biomaterials inspired by the structure of nacre[END_REF]. On the one hand, investigations conducted by Dutta et al. [START_REF] Dutta | Crack tortuosity in nacreous layer -Topological dependence and biomimetic design guideline[END_REF] provided guidelines for designing light-weight and tough adhesively bonded structures and staggered architecture composites. On the other hand, Zhu et al [START_REF] Zhu | A novel biomimetic material duplicating the structure and mechanics of natural nacre[END_REF] created a nacre-like material at the microscale which is composed of PMMA tablets layered as a columnar arrangement where interactions between layers are only controlled by the tribological aspects at the interfaces [START_REF] Yu-Min Lin | Interfacial shear strength in abalone nacre[END_REF]. At a lower scale, Tang et al [START_REF] Tang | Nanostructured artificial nacre[END_REF] created an interesting organic/inorganic nanolayered structure of polyelectrolytes and clays. Because this nanolayered material exhibits tensile strength and Young's modulus very close to the bone ones, it seems to be a good candidate for creating medical implants.

Another way followed by researchers is to looking at the possibility of using nacre for reconstructing human bodies [START_REF] Atlan | Interface between bone and nacre implants in sheep[END_REF][START_REF] Lopez | Demonstration of the capacity of nacre to induce bone formation by human osteoblasts maintained in vitro[END_REF][START_REF] Westbroek | A marriage of bone and nacre[END_REF]. Owing to its natural biocompatibility with human, nacre became an obvious candidate as a possible alternative to existing implant materials [START_REF] Denkena | Ductile and brittle material removal mechanisms in natural nacre -a model for novel implant materials[END_REF][START_REF] Westbroek | A marriage of bone and nacre[END_REF]. Indeed, current implants display a lot of drawbacks which are mainly connected to tribological behavior such as unsatisfactory life time or wear particlesinduced inflammatory reactions, for instance. Thus, Atlan et al [START_REF] Atlan | Interface between bone and nacre implants in sheep[END_REF] first tested nacre as a bone implant and showed that nacre implant triggers and encourages osteogenic -or bone forming -activity [START_REF] Lopez | Demonstration of the capacity of nacre to induce bone formation by human osteoblasts maintained in vitro[END_REF].

Taking into account the natural biocompatibility of nacre with bone as well as its tremendous durability connected to its great mechanical properties, this materials could definitely be considered as a potential substitute for defective bone [START_REF] Atlan | Interface between bone and nacre implants in sheep[END_REF][START_REF] Lopez | Demonstration of the capacity of nacre to induce bone formation by human osteoblasts maintained in vitro[END_REF][START_REF] Westbroek | A marriage of bone and nacre[END_REF] by direct implant or by creating some new organic/inorganic bioinspired hybrid materials [START_REF] Tang | Nanostructured artificial nacre[END_REF]. However, these challenging applications clearly need some knowledge about the frictional and wear behavior of this material [START_REF] Ph | Tribological behaviour of nacre -Influence of the environment on the elementary wear processes[END_REF][START_REF] Ph | Friction-induced sheet nacre fracture : effects of nanoshocks on cracks location[END_REF][START_REF] Ph | Bourrat Thermal-induced wear mechanisms of sheet nacre in dry friction[END_REF][START_REF] Ph | Evaluation of the real contact area in three-body dry friction by micro-thermal analysis[END_REF][START_REF] Ph | Nanotribology of nacre: Anisotropic dissipation in a multiscale hybrid material[END_REF][START_REF] Tian | Biological coupling anti-wear properties of three typical molluscan shells -Scapharca subcrenata, Rapana venosa, Acanthochiton rubrolineatus[END_REF][START_REF] Jia | Microstructure and friction-wear characteristics of bivalve shells[END_REF].

Recent tribological results, obtained in dry friction [START_REF] Ph | Tribological behaviour of nacre -Influence of the environment on the elementary wear processes[END_REF][START_REF] Ph | Friction-induced sheet nacre fracture : effects of nanoshocks on cracks location[END_REF][START_REF] Ph | Bourrat Thermal-induced wear mechanisms of sheet nacre in dry friction[END_REF][START_REF] Ph | Evaluation of the real contact area in three-body dry friction by micro-thermal analysis[END_REF][START_REF] Ph | Nanotribology of nacre: Anisotropic dissipation in a multiscale hybrid material[END_REF][START_REF] Tian | Biological coupling anti-wear properties of three typical molluscan shells -Scapharca subcrenata, Rapana venosa, Acanthochiton rubrolineatus[END_REF][START_REF] Jia | Microstructure and friction-wear characteristics of bivalve shells[END_REF] and in liquid medium [START_REF] Ph | Tribological behaviour of nacre -Influence of the environment on the elementary wear processes[END_REF], reveal that the wear mechanisms are strongly controlled: (i) by the environment itself, especially in liquid medium [START_REF] Ph | Tribological behaviour of nacre -Influence of the environment on the elementary wear processes[END_REF] through various physico-chemical interactions, occurring inside of the water-soluble intracrystalline organic phase [START_REF] Peireira-Mouriès | Soluble silk-like organic matrix in the nacreous layer of the bivalve Pinctada maxima-a new insight in the biomineralization field[END_REF]; and (ii) by the amount of energy dissipated during tribological tests in dry friction [e.g., 28]. Thus, as a function of the amount of frictional power dissipated within the multi-asperity tribocontact, nacre can be submitted to peculiar thermal [START_REF] Ph | Bourrat Thermal-induced wear mechanisms of sheet nacre in dry friction[END_REF] and mechanical [START_REF] Ph | Tribological behaviour of nacre -Influence of the environment on the elementary wear processes[END_REF][START_REF] Ph | Friction-induced sheet nacre fracture : effects of nanoshocks on cracks location[END_REF][START_REF] Tian | Biological coupling anti-wear properties of three typical molluscan shells -Scapharca subcrenata, Rapana venosa, Acanthochiton rubrolineatus[END_REF][START_REF] Jia | Microstructure and friction-wear characteristics of bivalve shells[END_REF] damage processes that occur at various orders of magnitude involving its double composite structure. This way, various multiscale wear mechanisms can be observed, leading to various wear rate levels, controlled by the multiscale structure of nacre itself [START_REF] Ph | Tribological behaviour of nacre -Influence of the environment on the elementary wear processes[END_REF][START_REF] Ph | Friction-induced sheet nacre fracture : effects of nanoshocks on cracks location[END_REF][START_REF] Ph | Bourrat Thermal-induced wear mechanisms of sheet nacre in dry friction[END_REF][START_REF] Ph | Nanotribology of nacre: Anisotropic dissipation in a multiscale hybrid material[END_REF]. Besides, organic matrices are always directly involved in the friction-induced energy dissipation mechanisms and, in the subsequent damage mechanisms. So, the way the organic matrices control the wear mechanisms can be successively studied by varying the amount of frictional power, which is dissipated within the contact -that is, by changing the size of the real contact area (i.e, the contact pressure) and the sliding velocity within a large range.

The aim of this work is to study each wear mechanism independently -as a function of its own frictional dissipated power -by using multiscale tribological tests. Thus, frictional dissipated powers range from few nanowatts (which is sufficient for inducing self-heating elastic deformation and/or long recovery-time viscoelastic deformation of the organomineral tablets [START_REF] Ph | Nanotribology of nacre: Anisotropic dissipation in a multiscale hybrid material[END_REF]) to several hundreds milliwatts (which enable phase changes of the mineral phase [START_REF] Ph | Bourrat Thermal-induced wear mechanisms of sheet nacre in dry friction[END_REF][START_REF] Ph | Evaluation of the real contact area in three-body dry friction by micro-thermal analysis[END_REF][START_REF] Bourrat | Nacre biocrystal thermal behavior[END_REF][START_REF] Huang | Nanoscale structural and mechanical characterization of heat treated nacre[END_REF]).

EXPERIMENTAL PART

Sheet nacre samples

Samples are made of sheet nacre extracted from giant oyster Pinctada maxima [START_REF] Rousseau | Multiscale structure of sheet nacre[END_REF][START_REF] Ph | Nanotribology of nacre: Anisotropic dissipation in a multiscale hybrid material[END_REF]. They are polished more or less parallel to the aragonite tablets orientation in a so-called face-on configuration (size: 5 µm, thickness: 400 nm, Ra: 14.5 ± 0.6 nm). Note that a residual angle (about 8° vs. the horizontal) is generally observed by AFM. The average size of the nanograins assessed by AFM image analysis is 38 ± 21 nm (Fig. 1e). Mechanical properties of samples -and their components -are compiled in Table 1. Some of them have been measured by combining acoustic microscopy and spherical nanoindentation, as described in §2.2. The others were compiled from our previous works which used spherical and sharp nanoindentation tests coupled with homogenization theory [START_REF] Ph | Mechanical properties of the elemental nanocomponents of nacre structure[END_REF][START_REF] Ph | Friction-induced sheet nacre fracture : effects of nanoshocks on cracks location[END_REF].

Acoustic microscopy (AM) is a non-destructive method based on the use of ultrasonic waves [START_REF] Briggs | Acoustic Microscopy[END_REF][START_REF] Gr | Acoustic Microscopy: Fundamentals and Applications[END_REF].

Mechanical properties of heterogeneous or multi-phased materials can be determined from the knowledge of the acoustic wave propagation velocity [START_REF] Robert | Characterisation of TiN and carbon-doped chromium thin film coatings by acoustic microscopy[END_REF]. The acoustic waves are highly focused by an acoustic lens (482 MHz, see Fig. 2a) in order to obtain a good spatial resolution attaining a few micrometers. Two main operating modes are used [START_REF] Comte | Microprobe-type measurement of Young's modulus and Poisson coefficient by means of depth sensing indentation and acoustic microscopy[END_REF] : 2) is a qualitative assessment which has a capacity for providing tomographic information related to stress fields associated to subsurface material discontinuities and damage. Using constant defocussing, acoustic images display qualitative information on the material structure or surface topography (Fig. 2b). When the lens is focused below the sample surface, buried structures appear, as shown in Fig. 2c for sheet nacre. This technique allows a clear distinction of tablets and interlamellar matrix from the different mechanical properties of the both mineral and organic phases.

• (i) the Imaging Mode (Fig.
• (ii) the Acoustic Signature Mode (Fig. 3) is a quantitative mode where elastic parameters such as Young's modulus and Poisson ratio local modulus can be measured with microprobe resolution. As shown in Fig. 3a, the so-called V(z) curve -V refers to the video or envelope-detected signal that is used to modulate the brightness of the picture, and z refers to the amount by which the specimen surface is displaced from the focal plane of the lens [START_REF] Briggs | Acoustic Microscopy[END_REF] -exhibits a strong pseudo periodicity that can be connected to specific surface acoustic wave velocity. Using signal processing based on a Fourier transform analysis (FFT, Fig. 3b) this period is accurately extracted to calculate the velocity of the surface wave responsible for these oscillations. For composite such as sheet nacre -which is composed of mineral and organic phases -two different signals can be obviously assessed. These signals can be easily deconvoluted because the propagation velocity of acoustic waves is very different in the two phases.

From Acoustic Signature assessments, assuming linear elastic behaviour, the model links the standard elastic parameters such as the Young's modulus and the Poisson's ratio to the density and two specific velocities [START_REF] Robert | Characterisation of TiN and carbon-doped chromium thin film coatings by acoustic microscopy[END_REF]:

where, V L is the velocity of longitudinal waves, which create compression and traction stresses along the wave propagation direction; and V T is the velocity of transverse waves whose induced shear strains are perpendicular to the wave propagation direction; and r is the density. However, V T is not directly assessed in practice. Instead Rayleigh wave velocity, V R is measured locally in addition to V L from the FFT acoustic signatures (Fig. 3). Equation 3 is used to get V R from V L and V T in order to finally compute E and n using equations (1) and (2) :

Various assessments need to be done at various defocusing distances. Defocusing is actually the process of decreasing the separation of the lens and specimen relative to the focal distance [START_REF] Briggs | Acoustic Microscopy[END_REF]. V T is finally computed using Eq. 3 formulated as :

Only one root is physically acceptable in term of mechanical properties. However, it is worth of mentioning that Acoustic Microscopy fails to provide the whole elastic properties because the lack of equations versus the number of unknowns [START_REF] Comte | Microprobe-type measurement of Young's modulus and Poisson coefficient by means of depth sensing indentation and acoustic microscopy[END_REF]. Indeed, one of the elastic properties -generally the Poisson's ratio -needs to be known beforehand by means of another technique. Comte et al [START_REF] Comte | Microprobe-type measurement of Young's modulus and Poisson coefficient by means of depth sensing indentation and acoustic microscopy[END_REF] demonstrated that nanoindentation can be combined with acoutic microscopy because the material volume which is involved during the test is in the same order. Thus, knowing the Poisson ratio evaluated by nanoindentation (Table 1), the Young's modulus can accurately be deduced from V R using the following relationships [START_REF] Robert | Characterisation of TiN and carbon-doped chromium thin film coatings by acoustic microscopy[END_REF]: with In practice, the systematic error was evaluated at 1.07% by using reference samples. Hence, the standard deviation of the Young's modulus is evaluated at 5%. Three measurements have been done for each sample with five defocusing distances per point. As mentioned above, the resolution is evaluated at 3 µm owing to the coupling liquid (distilled water). Experimental results give V L = 2400 Hz and V R = 3250 Hz leading to E tablet = 58 ± 5% GPa and E matrix = 13± 5% GPa, assuming the nacre density equal to 1.18 [START_REF] Meyers | Biological materials, Structure and mechanical properties[END_REF][START_REF] Vincent | Structural Biomaterials[END_REF].

For the organomineral tablets, Young's modulus is in good agreement with the one previously measured using spherical nanoindentation [START_REF] Ph | Mechanical properties of the elemental nanocomponents of nacre structure[END_REF] (see Table 1). This is not the case for the interlamellar matrix since its Young's modulus is twice as that evaluated previously by spherical nanoindentation [START_REF] Ph | Friction-induced sheet nacre fracture : effects of nanoshocks on cracks location[END_REF]. This difference is clearly connected to the presence of porosity within the interlamellar organic matrix. Indeed, spherical nanoindentation needs to use a structural model in order to extract the mechanical properties of each phase of a composite. The structural model used in our previous paper [START_REF] Ph | Friction-induced sheet nacre fracture : effects of nanoshocks on cracks location[END_REF] is pictured in Fig. 4a. It considered the organic matrix as a continuous phase, in agreement with SEM analysis of Rousseau et al [START_REF] Rousseau | Multiscale structure of sheet nacre[END_REF] which clearly demonstrated that interfaces between tablets of Pinctada maxima's sheet nacre were only organic without any mineral bridges. Using this structural model, the Young's modulus assessed at high loads spherical nanoindentation corresponds to the transversal Young's modulus (E T ) of a stacking of layers made up of pure matrix and reinforced layers (mineral and organic). Its value is connected to the ones of the tablets E f and the interlamellar matrix E m by the following relationship which take into account the volume fraction of the tablets V f , considering nacre as a composite of tablets and interlamellar organic matrix [START_REF] Ph | Friction-induced sheet nacre fracture : effects of nanoshocks on cracks location[END_REF]:

Using this relationship, spherical nanoindentation leads to a Young's modulus of the continuous matrix equal to 6.3 GPa as reported in Table 1 from [START_REF] Ph | Friction-induced sheet nacre fracture : effects of nanoshocks on cracks location[END_REF] to be compared at 13 GPa obtained using acoustic microscopy.

As suggested in Fig. 2c, a new structural model of sheet nacre -which takes into account the matrix porosity -is needed to validate the AM's mechanical properties by using spherical nanoindentation. But how much porosity does the interlamellar matrix hold in? As shown in Fig. 5a, the void amount has been estimated at minima by discriminating the interlamellar matrix and its porosity in different transversal cuts observed in darkfield TEM. The minimum value of crossed porosity is about 35% ± 5.6%. Beside, other SEM preparations reveal a honeycomb structure in the interlamellar area under the tablets (Fig. 5b). Porosity is here estimated by stereology at 59% ± 3%. Thus, one can consider that our Pinctada maxima sheet nacre is constituted by an interlamellar matrix (6.47% wt.) which displays a porosity of about 47% ± 25%. We will see in §3 that this heterogeneity will strongly affect the wear mechanisms.

Knowing the amount of porosity within the interlamellar matrix, its mechanical properties can be now reevaluated by using the previous spherical nanoindentation results applied but with a structural model which takes into account the porosity of the matrix (Fig. 4b). Thus, it appears that the previous matrix Young's modulus (ie 6.3 GPa) is no longer that of a continuous matrix but the one of the composite layer which is constituted by organic matrix and voids. Assuming the Young's modulus of the pores equal to zero and the above ratio of porosity, the real Young's modulus of the interlamellar matrix is now re-evaluated at 13.5 GPa, which is in good agreement with both the value assessed by acoustic microscopy, and the one reported by Xu et al -ie., 11 ± 3 GPa [START_REF] Xu | Elastic modulus of biopolymer matrix in nacre measured using coupled atomic force microscopy bending and inverse finite element technique[END_REF].

Thermal properties of sheet nacre -Scanning Thermal Microscopy

The local thermal properties of the samples, and especially the relationship between the frictional dissipated power and the contact temperature are assessed with a SThM (TA Instruments µTA 2990 with a TA Instruments controller TA 5300, New Castle, DE), which is an analytical system that combines the high resolution visualization and positioning methods of scanning probe microscopy with the technology of thermal analysis [START_REF] Ph | Bourrat Thermal-induced wear mechanisms of sheet nacre in dry friction[END_REF]. The standard AFM probe is replaced by a thermal probe made from a Wollaston wire (5 µm diameter platinum -10% rhodium wire enclosed in a silver sheath), which allows the acquisition of the surface contact temperature, and simultaneously acts as a highly localized heater. The vertical deflection of the assembly is monitored by a light pointing technique. Spring constant is 10 N.m -1 . The constant current setpoint and the z-setpoint are 1 mA and 50 V, respectively. The scanning rate of the probe is 100 µm.s -1 .

The latter is sufficiently weak so that the heat transfer regime stays in the steady state conduction as in the high power tribological tests. The spatial resolution and the thermal sensitivity are about 100 nm and 1°C, respectively. Calibration of the SThM and procedure are detailed in [START_REF] Ph | Evaluation of the real contact area in three-body dry friction by micro-thermal analysis[END_REF].

Multi-scale tribological tests

Macro-tribological tests (1 -630 mW)

The experimental device is a pin-on-disc tribotester CSM Instruments (Peseux Switzerland) [START_REF] Ph | Evaluation of the real contact area in three-body dry friction by micro-thermal analysis[END_REF]. High power tribological tests are carried out at ambient air and room temperature in dry conditions by repeated friction of a 3.5 mm square shaped pin of nacre against the surface of a polished disc of nacre (AE 44 mm).

The total sliding distance is about 100 m. Frictional dissipated power varies within the range of 1 to 630 mW by changing the normal load from 1 to 15 N (with a sliding velocity fixed at v = 10 mm/s) or the sliding velocity from 1 to 150 mm/s -1 (with a normal load fixed at F n = 10 N), respectively. According to the mechanical and thermal properties of samples, the situation during the macrotribological tests closely approximates the steady state thermal conduction [START_REF] Ph | Evaluation of the real contact area in three-body dry friction by micro-thermal analysis[END_REF][START_REF] Zhang | Nano/Microscale Heat Transfer[END_REF].

Wear rates are determined at the end of tests by measuring the weight of the material lost during the tests (precision:10 -4 g). Any phase transformation of the wear debris is assessed using X-Ray diffraction. However, wear debris generated in dry friction are generally constituted by agglomeration of elemental nanodebris [START_REF] Ph | Tribological behaviour of nacre -Influence of the environment on the elementary wear processes[END_REF]. In order to assess the real size of the elemental wear debris -leading to the actual wear mechanisms occurring at the nanoscale -before any agglomeration process, wear particle size distribution measurements were conducted by using Photon Cross-correlation Spectroscopy (PCCS, NanoPhox, Sympatec, Germany [START_REF] Wang | A comparison study on the measurement of nanoparticles[END_REF]) in addition to traditional SEM and AFM observations. This technique extends the well-established principle of Photon Correlation Spectroscopy [START_REF] Theodore | Nanotechnology -Basic Calculations for engineers and scientists[END_REF] to highly dilute suspensions. Prior any analysis wear debris are dispersed in distilled water with an ultrasonic bath. Then, the technique involves passing a laser beam into the suspension and measuring the Doppler shift of the frequency of dynamic light scatter at an angle with respect to the incident beam. This shift is related to particle velocity, which in turn, is inversely related to the particles size. Photon Cross-correlation Spectroscopy (PCCS) allows the measurement of elemental wear debris in the size ranges of about 1 nm to some µm [START_REF] Wang | A comparison study on the measurement of nanoparticles[END_REF].

Multi-asperity nanotribological tests (2 nW -0.1 mW)

The experimental device is constituted by a ball-on-disc nanotribometer CSM Instruments (Switzerland) [START_REF] Ph | Nanotribology of nacre: Anisotropic dissipation in a multiscale hybrid material[END_REF]. A pin is mounted on a stiff lever, designed as a frictionless force transducer (K x = 265.1 N.m -1 ; K z = 152.2 N.m -1 ). The friction force is determined during the test by measuring the deflection of the elastic arm (low load range down to 50 µN). The ball (Si 3 N 4 -AE1.5 mm) is loaded onto a flat nacre sample with a precisely known force using closed loop. The load and friction resolutions are about 1 µN [START_REF] Ph | Multi-asperity nanotribological behaviour of single-crystal silicon: Crystallography-induced anisotropy in friction and wear[END_REF]. Tribological tests are carried out in linear reciprocating mode at room temperature (22°C) under ambient air (RH 35%).

The sliding distance is about 10 m. The frictional dissipated power varies in the range of 2 nW to 0.1 mW by changing the normal load from 10 to 80 mN (at v = 1 mm/s) and the sliding velocity from 10 -3 to 28 mm/s (at 40 mN), respectively. The thermal contribution is here greatly limited and clearly too low for damaging the matrices [START_REF] Ph | Nanotribology of nacre: Anisotropic dissipation in a multiscale hybrid material[END_REF][START_REF] Zhang | Nano/Microscale Heat Transfer[END_REF].

At this scale classical post mortem wear analysis -like profilometry or AFM assessment -is not really suitable because (i) the error in wear assessment owing to the elastic recovery is no longer negligible and (ii) the presence of nanometric tribolayer within the contact cannot be taken into account when the contact is opened afterward [START_REF] Ph | In situ running-in wear assessment in multi-asperity nanotribology[END_REF]. In other words, post mortem assessment can only be used when the wear level is much more significant than the deformation one. Hence, an in situ wear assessment is carried out by using a real-time depth measuring sensor on the nanotribometer [START_REF] Ph | Multi-asperity nanotribological behaviour of single-crystal silicon: Crystallography-induced anisotropy in friction and wear[END_REF]. The depth range varies from 20 nm to 100 µm with a resolution of about 20 nm. Tribological assessments are compiled using a triboscopic approach giving simultaneously, and for each cycle: (i) the power map (e.g., Fig. 17a) plotting the evolution of the power dissipated by friction along the friction track (this power is the product of the friction force by the sliding velocity ie, P f =F t .v) ; (ii) the wear map (e.g., Fig. 17b) which reveals any-time dependent wear process and/or potential build-up of a tribolayer within the contact. This map is computed from an in situ depth map by taking into account the tilting and the initial deformation of the samples, as reported previously [START_REF] Ph | In situ running-in wear assessment in multi-asperity nanotribology[END_REF].

Finally, an in situ wear rate assessment can be directly extracted from this wear map as a classical profile analysis (e.g., Fig. 17c) by using a topographical software (http://gwyddion.net/). The complete in situ wear assessment procedure has been described previously [START_REF] Ph | In situ running-in wear assessment in multi-asperity nanotribology[END_REF].

Numerical simulations of wear process

In addition to experimental multiscale tribological tests, Finite Element (FE) simulations are carried out in order to understand elemental wear mechanisms that occur during the tribological tests, especially in the range which is not investigated experimentally (in between 0.1 to 1 mW). As shown in Fig. 6, FE model simulates the effect of a rigid impactor on nacre sample with and without any shock occurrence. The associated damage mechanism will be analyzed and correlated to the results of tribological tests -as the assessment of wear debris, for instance. Numerical simulations need the mechanical properties -assessed using acoustic microscopy and nanoindentation -reported in Table 1. Besides, the numerical model will be able to simulate the influence of the matrix's porosity by keeping the matrix continuous while its Young's modulus will be changed from 6.3 to 12.6 GPa.

In practice, Abaqus Explicit FEM code [START_REF]ABAQUS -ABAQUS Standard User Manual, V. 6.8[END_REF] is used to solve the problem. The choice of an explicit code is connected to the presence of severe non-linearities induced both by the large deformations and the behavior laws that are used [START_REF]ABAQUS -ABAQUS Standard User Manual, V. 6.8[END_REF]. The mesh of each component has already been described in details [START_REF] Ph | Friction-induced sheet nacre fracture : effects of nanoshocks on cracks location[END_REF]. We just point out that a ductile damage law is coupled with the erosion algorithm of the Abaqus Explicit code [START_REF]ABAQUS -ABAQUS Standard User Manual, V. 6.8[END_REF] to simulate the wear process. Thus, when the ductile damage criterion is reached, the corresponding element is considered instantly completely damaged, and is removed from the FE model. The ductile damage law is based on the maximal equivalent plastic strain value (ε max ) of the interlamellar organic phase (10%), which is extracted from the work of Ji and Gao [START_REF] Ji | Mechanical properties of nanostructure of biological materials[END_REF]. The plastic deformation of the organomineral tablets is assumed to be controlled by the intracrystalline organic matrix surrounding the nanometric biocrystals [START_REF] Ph | Mechanical properties of the elemental nanocomponents of nacre structure[END_REF].

Therefore, the maximum strain value of the tablets is estimated from the volume fraction of the organic matrix, within the tablet, assessed by AFM image analysis -about 12% in 2D corresponding to 4.2% in 3D [START_REF] Ph | Mechanical properties of the elemental nanocomponents of nacre structure[END_REF]. Thus, the maximum strain value ε max of tablets can be estimated to be about 12% of the one of the interlamellar organic matrix (ε max = 10%, see Table1) because both presented models are bi-dimensional ones.

RESULTS AND DISCUSSION

Evolution of the tribological behavior for the high power tribological tests (P f > 1 mW)

Figure 7 shows the variations of the dissipated power by friction -P f = µF n v -as a function of the sliding distance for various normal loads (2 N to 15 N). After a running-in period, the deduced friction coefficient is about 0.44 ± 0.02. This value also appears quite insensitive to any variation of the sliding speed in the range 1 to 150 mm/s. Thus, surprisingly the friction coefficient stays constant when the frictional dissipated power varies from 1 to 630 mW.

In contrast, wear rate is very sensitive to the frictional dissipated power, as reported in Fig. 8, which displays the effect of any change of normal load (Fig. 8a) and sliding velocity (Fig. 8b), respectively. Thus, various abrupt changes are observed when the frictional dissipated power exceeds 22 mW, 53 mW and 213 mW, respectively, whereas in the same time, the friction coefficient does not change and stays constant at 0.44±0.02. This peculiar behavior is probably due to the presence of a tribolayer covering the friction track, which controls the frictional behavior as a negative feedback loop [START_REF] Ph | Nanomechanical behaviour of the 3 rd body generated in dry friction -Feedback effect of the 3 rd body and influence of the surrounding environment on the tribology of graphite[END_REF]. However, the various rises of the wear rate can be studied independently, by considering the various thermomechanical effects that are likely to occur within the contact.

Thermal -induced wear process at the highest frictional powers

Wear debris generated under various levels of frictional dissipated powers are assessed using X-Ray diffraction, and compared to the original spectrum of aragonite in Fig. 9. The intensity of the calcite peak (expected at 34.28° in Fig. 9) -i.e the other crystallographic form of the CaCO 3 -gradually increases when the frictional dissipated power increases from 10 to 103 mW. Below 22 mW, the proportion of calcite in the wear debris is similar as the one that is assessed in the Pinctada maxima original spectrum. It is worth of noting that the irreversible transformation of calcium carbonate from aragonite to calcite only occurs when the temperature reaches 470°C [START_REF] Bourrat | Nacre biocrystal thermal behavior[END_REF][START_REF] Huang | Nanoscale structural and mechanical characterization of heat treated nacre[END_REF]. Thus, these results show that high temperatures can be briefly reached, at the level of the microscopic asperities, when the frictional dissipated powers exceed 50 mW (Fig. 9), leading to a phase transformation of the mineral part of the wear debris. Besides, cathodoluminescence assessments carried out on the tribolayer covering the friction track, reveal that this latter is also completely transformed to calcite [START_REF] Ph | Evaluation of the real contact area in three-body dry friction by micro-thermal analysis[END_REF].

However, since thermal-induced structural changes of the mineral are observed when the frictional dissipated power is higher than 22 mW, the temperature that drives the melting of the organic phases should be reached for the lower dissipated powers -somewhere in-between 1 and 20 mW. An accurate value of this latter can be assessed using a Scanning Thermal Microscopy [START_REF] Ph | Evaluation of the real contact area in three-body dry friction by micro-thermal analysis[END_REF][START_REF] Zhang | Nano/Microscale Heat Transfer[END_REF]. Practically, µDTA curve (Fig. 10a) reveals the relationship between the dissipated power and the contact temperature, and µTMA curve (Fig. 10b) simultaneously displays the change of the vertical displacement of the cantilever, as a function of the contact temperature. Whereas the continuous rise of the sensor vs. the contact temperature corresponds to the thermal expansion of the nacre sample (Fig. 10b), sensor jumps which are observed beyond 225°C clearly reveal that the tablets get unstuck when the organic phase is degraded in the slot of temperature 200°C to 350°C [START_REF] Yu-Min Lin | Interfacial shear strength in abalone nacre[END_REF][START_REF] Ph | Evaluation of the real contact area in three-body dry friction by micro-thermal analysis[END_REF]. Besides, the amplitude of the sensor's jumps of 250 to 500 nm appears very close to the value of the tablet's thickness. AFM views (Fig. 10b, inset), carried out before and after heating, confirm the matrix degradation as soon as the contact temperature exceeds 225°C. Finally, µDTA plot (Fig. 10a) provides an accurate value of the minimum thermal power that is necessary for melting the organic phase (about 17 mW).

Thus, these various X-Ray and SThM observations enable us to explain the tribological results (Fig. 7) and the variations of the wear rate vs. the frictional dissipated power (Fig. 8), by comparing the thermal response of sheet nacre (Fig 10a ) with the thermal power which is really dissipated by friction during the tribological tests (Fig. 7). Assuming this latter being around 98% of the frictional power, macrotribological results reveal that:

• The first abrupt change of the wear rate (around 22 mW in Fig. 8) can be attributed to the degradation of the surface organic matrices (T c 282°C at 24.5 mW in Fig. 10a), when the frictional dissipated powers are greater than 17 mW. Indeed, this friction-induced contact temperature rises up to the melting point of the organic matrices, as observed in Fig. 10b.

• The aragonite-calcite phase transformation (expected around 470°C [START_REF] Bourrat | Nacre biocrystal thermal behavior[END_REF]) is clearly observed within the contact when the frictional dissipated power is higher than 49 mW. This value is extrapolated from the SThM curves at T c 508°C and is in good agreement with the calcite peak observed in Fig. 9 • Finally, the avalanche effect observed beyond 213 mW (Fig. 8b) could be attributed to the propagation of the heat flow within the subsurface when the steady state regime is reached. So, the matrix can be degraded in depth, leading to an increased rate of the wear. Thus, for this range of dissipated powers (P f > 17 mW), wear of sheet nacre is clearly induced and controlled by thermal effects -combining melting of the organic matrices and friction-induced phase transformation of the mineral phase. Besides, for lower power levels (P f < 17 mW ), the frictional-induced thermal component is not sufficient for melting the organic matrices (T c < 210°C). Thus, another kind of wear mechanism must occur in this range of frictional dissipated powers as documented in the following.

Shock-induced wear process, at medium frictional powers

Typical AFM views of the friction track (Fig. 11a) can be seen after tribological tests conducted in the range of 1 to 17 mW. Nacre tablets are clearly crumbled; the edge lines are rough and let appear the nanograins constituting biocrystals. As a result, wear debris are very heterogeneous in size because they are constituted by single biocrystals, agglomerates, pieces of tablets and tablets as observed by PCCS (Fig. 11b). Obviously, this wear mechanism involves both organic matrices, ie. intracrystalline and interlamellar.

In order to explain how these matrices work together in this peculiar wear mechanism, numerical simulation using FEM models is carried out by varying the sliding velocity of the slider (so-called impactor) in the range of 10 down to 1 mm/s. These velocities enable us to extrapolate results to the lowest dissipated powers in order to make the connection with the next nanotribological tests. As mentioned above, a ductile damage law is coupled with the erosion algorithm of the Abaqus Explicit code to simulate the fracture inside both materials [START_REF]ABAQUS -ABAQUS Standard User Manual, V. 6.8[END_REF]. Thus, when the ductile damage criterion is reached, the corresponding element is considered instantly completely damaged, and is removed from the FE model. The intercrystalline matrix is first considered as continuous without any porosity.

Figure 12 shows simulation for two impactor velocities -i.e., two dissipated powers -as observed at the same time increment (4.10 -2 s). Although both matrices are clearly solicited, major part of the fracture is located inside of the aragonite tablet whatever the impactor velocity (Fig. 12). This is the main consequence of the dynamic effects, and the presence of repeated shocks. Figure 13 illustrates the shock-induced wear mechanims leading to the formation of elemental wear debris, the size of which is very close to the initial biocrystals one, as observed from a typical tribolayer AFM view (Fig. 13). Besides, stress distribution shows that during the impact, the ratio between the equivalent stress and the yield stress ( y ), inside of the biological interlamellar organic matrix, is lower than the one inside of the tablet. The fracture propagation inside the interlamellar organic matrix is clearly limited: tablet's debonding does not occur. Hence, fracture inside the tablet leads to the formation of pieces instead of whole tablet.

However these numerical results seems to disagree with the experimental PCCS assessments (Fig. 11) which reveals a multimodale distribution rather as a mononodale one, as it could be suggested by simulations.

Henceforth, new numerical simulations have been carried out by introducing the presence of porosity within the interlamellar matrix. This porosity is taken into account by keeping the matrix continuous while its

Young's modulus is increased from 6.3 to 12.6 GPa. As shown in Fig 14a, crack propragation progressively moves outside the tablet when porosity ratio increases to finally occur inside the organic matrix leading to the formation of larger wear debris. Thus, the multinodale distribution of wear debris, see in Fig. 11B, can be satisfactorily understood by keeping in mind the large standard deviation that has been determined for the interlamellar matrix porosity -ie. 47± 25%.

As a result, the complete damage mechanism involves here the fracture of both matrices, explaining the experimental high friction coefficient generally observed in this range of frictional powers (µ = 0.44 ± 0.02) [START_REF] Dutta | Optimal overlap length in staggered architecture composites under dynamic loading conditions[END_REF]. In addition, these simulations enable us to satisfactorily explain the formation of nanometric wear debris -as observed in Fig. 11 and 13 -which have a size very close to that of the initial nanograins constituting the tablets (Fig. 1e vs. Fig 13). In addition, simulations reveal that nanodebris can be directly generated without subsequent tablet's crushing (Fig. 13). This mechanism is mainly due to the high velocity-induced shocks which occur within the tablets. This fracture behavior is also observed when the impactor's velocity is decreased by a factor 10 (i.e., 1 mm/s, Fig. 12b), so this friction-induced nanoshocks mechanism is likely to be the driving force that controls the wear mechanism of nacre on a large range. Is this mechanism still licit at the lower power tribological tests as observed in multi-asperity nanotribological tests?

Evolution of the tribological behaviour for the low power tribological tests (1 nW < P f < 0.1 mW)

In the range of low dissipated powers, there are two different tribological regimes (Fig. 15a); this is in contrast to what is observed in the previous high power tribological tests, where only one single high level frictional regime is observed (µ = 0.44), controlled by a tribolayer of heterogeneous wear debris. First, a high frictional regime is very close to the one observed in our macrotribological experiments ( §3.1). Second, a low frictional regime is only observed on a very smooth surface. The transition to the upper regime clearly occurs when the slider crosses cracks with a critical depth higher than the thickness of a tablet (more than 500 nm), as shown in Fig. 15b. So, any significant change in the roughness can involve this transition from low to high frictional regime. Which component is likely to control this transition?

Nanotribological tests involving the high frictional regime

As shown in Fig. 15a, the high frictional regime displays a high level coefficient of friction (0.35 ± 0.15), which is very unstable and close to the one observed in the high power macrotribological experiments (0.44 ± 0.02). Obviously, these high friction values are connected to a high wear rate, which is assessed using a phase-shifting interferometric profilometer (20 nm/cycle/N). Despite of the high frictional regime, there is no thermal-induced phase transformation here as revealed by the Raman spectrum carried out within the friction track after sliding (Fig 16). Since classical post mortem analysis is here useless, an in situ analysis should help to go further.

Figure 17 displays the typical in situ tribological behavior as observed in the high frictional regime. The average frictional power dissipated during this test is around 21 µW. As shown in the Fig. 17a, the power map is very unstable along the friction track and also in time. This is because a heterogeneous tribolayer is continuously built-up and removed within the contact, as reported in the wear map (Fig. 17b). In order to estimate the amount of wear debris trapped within the contact, the tribolayer's thickness is computed from a profile analysis (Fig. 17c), carried out on the respective wear map (Fig. 17b). Its thickness evolves with the number of cycles but, it increases and even overshoots several micrometers. Thus, this thickness is always higher than a single tablet's thickness (about 500 nm), revealing that several layers of tablets are probably involved in the wear process. Besides, successive jumps and drops observed on the wear profile are always in the same order of magnitude as the tablet's thickness, suggesting some tablet's movements within the contact: hence, pieces of tablets are directly involved by the wear mechanism, as suggested by the SEM observations (Fig. 17d). However, the tablet pieces that are initially detached, are finally crushed within the contact which produces the non-cohesive tribolayer (Fig. 17b ande); this process maintains itself continuously. Hence, the crushing of the tablets -leading to nanodebris -seems to occur after the tablets' debonding. So, the interlamellar matrix is broken before the intracrystalline one, this would explain the high level of friction which is observed here.

To better understand this process, the previous FEM model is used again, but the initial impactor velocity is still reduced by a factor 10 in Fig. 18a, and finally set to zero in Fig. 18b in order to remove the shocks occurrence. For the sake of comparison with the above simulation (Fig. 12), the interlamellar matrix is again considered without any pores. As shown in the Fig. 18, the damage mechanism changes and the location of fracture progressively moves from the tablet to the interlamellar matrix when the impactor velocity is reduced. As a result, this latter is completely fractured, while the tablet remains intact after deformation and becomes free. In absence of shock -i.e., in a quasi-static compression or shearing test (Fig. 18b) -high shear region and crack propagation may only be located inside of the interlamellar organic matrix. Besides, timehistory evolution of the stress reveals that this mechanism needs more energy than that of the nanoshocks (Fig. 13) because it involves the interlamellar matrix, which is 4 times stiffer than the intracrystalline one (see Table 1). Nanoshocks mechanism (Fig. 18a) involves stress peaks which are very sharp whereas interlamellar damage mechanism rather involves a continuous peeling of the matrix (Fig. 18b). Once in the contact, the detached tablets are completely crushed into homogeneous wear debris, as reported on various SEM views (Fig 17e). As a result, the amount of the wear debris trapped within the contact is also very important -and so the wear rate. In presence of porosity, the average size of debris is of course much more dispersed as shown above.

Consequently, the main wear mechanism in the nanotribological high frictional regime involves the fracturing of both organic matrices as a continuous mechanism. The high frictional energy is mainly dissipated by fracturing the interlamellar matrix, explaining the high level friction coefficient recorded during the test. The continuous tribolayer -which is initially constituted by huge debris and pieces of fractured tablets -is progressively built up by crushing the fractured tablets into homogeneous nanodebris.

So, this mechanism also involves the damage of the intracrystalline matrix, but as a secondary process.

Nanotribological tests involving the low frictional regime (down to µW)

Typical in situ tribological behavior is reported in Fig 19 (the average dissipated power is around 5.5 µW). In contrast to what is observed in the previous high frictional regime, power map (Fig. 19a) is here very smooth along the friction track and with time. The dissipated power is reduced by a factor 4 with respect to the high frictional regime (around 21µW). This difference is very close to the ratio between the strengths of the organic matrices (see Table 1 and [START_REF] Ph | Mechanical properties of the elemental nanocomponents of nacre structure[END_REF]). Thus, a change of wear mechanism -that would involve the fracturing of intracrystalline instead of interlamellar matrix -is surely expected. Wear map (Fig. 19b) is also quite homogeneous but reveals that a cohesive tribolayer is now continuously built-up during the test. It is never removed during the tests, in contrast to what is observed for the high frictional regime. The cohesion ability of tribolayers being generally linked to the size of the elemental wear debris that constitute them, AFM image analysis reveals that their size is very close to the one of the initial nanograins constituting the biocrystals (Fig. 19e vs. Fig. 1e). Wear profile extracted from wear map (Fig. 19c) also confirms the presence of a very thin tribolayer whose thickness is seldom over the initial tablet's thickness (above 500 nm). That means here that only one layer of tablets is involved in the wear process. This result is also suggested in typical SEM pictures (Fig. 19d). Thus, in that case, the wear mechanism would be mainly a fracturing within the tablet, involving the intracrystalline organic matrix only (mainly composed by silk-fibroin-like proteins).

That would explain the connection between the reduction of the frictional dissipated power and the difference between the matrices' stiffness, as mentioned above. Thus, the main wear mechanism clearly involves the detachment of nanograins of the biocrystals from the top row of tablets by breaking of the intracrystalline organic matrix. Wear debris -close to the nanograins (Fig. 19e) -are directly generated, without any debonding the tablets as previously observed in the high frictional regime. Hence, tablets's debonding mechanism could be likely the driving force controlling the transition from the low to the high regime (Fig. 15b). Let us check this assumption out.

Transition from low to high frictional regime as the gap between nano-and macro-scale

As reported in Fig. 15b, the transition between the low frictional regime and the high frictional one is quite irreversible and, is only observed in the face-on configuration -ie when the tablets are oriented more or less parallel to the direction of sliding (see Fig. 18b). In contrast, when tablets are no longer flat but upright on the section, and parallel to the direction of sliding -so-called edge-on configuration -the low frictional regime is always observed in a large range of dissipated powers [START_REF] Ph | Nanotribology of nacre: Anisotropic dissipation in a multiscale hybrid material[END_REF]. So, this transition only seems to be controlled by the shear resistance of the interlamellar matrix. Referring to recent results carried out on faceon configuration of sheet nacre [START_REF] Ph | Nanotribology of nacre: Anisotropic dissipation in a multiscale hybrid material[END_REF], friction and wear are very sensitive to the track smoothness. As long as the crack is still below a critical size (close to the tablets' thickness), fracturing mainly occurs within the tablets involving the intracrystalline matrix only (Fig. 19c). But as soon as the crack's thickness overshoots the critical size (Fig. 17c), the fracturing location moves from the intracrystalline matrix to the interlamellar one. As a result, the frictional dissipation power suddenly increases because the two matrices are now involved, but the influence of the intracrystalline is almost imperceptible, completely hidden by the interlamellar one.

So, the high frictional regime clearly makes the connection between the low and high dissipated power process that is, between the nanotribological and the macrotribological tests: as same coefficients of friction, and wear rate which depends on the friction dissipated power. The transition is only observed as soon as the interlamellar matrix is broken. Thus, this frictional power-induced transition is clearly the key connecting tribological process from nano-to macroscale, which is controlled by the shear resistance of the "interlamellar" matrix (polysaccharides in between the tablets).

Tribological tests involving the ultralow frictional regime (down to nW)

As shown in Fig. 20a, the friction coefficient drops when the frictional dissipated power decreases until several nW -by decreasing the sliding velocity. Wear rate is too low to control the dissipated frictional power, as mentioned above. So, any variation of the sliding velocity involves a change of the friction coefficient. However, an anisotropic effect is clearly observed when the orientation of the sample is switched from face-on to edge-on configuration. In the face-on configuration, stick-slip phenomenon -due to the elastic recovery of the structure -is clearly observed at very low sliding speeds (Fig. 20b), locking the friction coefficient drop around 0.09 whatever the frictional dissipated power down to the nW. The blue curve (in Fig. 20b) also reveals a low dissipative wear mechanism that is clearly coupled with the stick-slip phenomenon and probably controlled by the elastic properties of the organomineral tablets -ie. the intracrystalline matrix itself.

In contrast, same tribological tests, which are carried out in the edge-on configuration, reveal that the friction coefficient continuously drops when the sliding velocity decreases. In addition, no stick-slip phenomenon is observed in that configuration. Thus, for the same sliding speed, the friction coefficient is much lower in the edge-on configuration than in the face-on one because elastic dissipation and wear process are here replaced by an irreversible deformation that could be connected to the plasticity of the organomineral tablets (Fig. 20c). Hence, the dissipative frictional power is much lower, and so the wear rate. The reduction of the friction coefficient is probably due to the transfer of the organic component -as a lubricant and protective film -on the friction interface, as suggested by Jia et al [START_REF] Jia | Microstructure and friction-wear characteristics of bivalve shells[END_REF] using EDX analysis of the friction track.

Since the edge-on configuration seems to be the best one for any sheet nacre tribological applications, friction tests have been finally carried out at 70 µW on over 5000 cycles in order to check the lifespan (Fig. 21). As expected, the initial contact involves a vertical displacement of about 80 nm under the slider. This deformation could result of some plastic deformations within the vertical organomineral tablets as mentioned above for the short tribological tests -ie., 100 cycles. However, Fig. 21 suggests that this initial "plastic"

displacement is not at all permanent since the wear profile increases over the first thousands cycles until the complete surface recovery. This means that the contact is rather submitted to very long recovery-time viscoelastic deformations within the tablets [START_REF] Xu | Deformation strengthening of biopolymer in nacre[END_REF][START_REF] Meyers | Mechanical strength of abalone nacre: Role of the soft organic layer[END_REF] -ie., several hundreds cycles. As a result of the viscoelastic contact deformation the friction coefficient strongly oscillates during this period. But, as soon as the wear profile reaches its initial value -ie., z = 0 nm -the friction coefficient decreases abruptly on over 1000 cycles because of an increasing tribolayer-induced lubricant that is progressivelly built within the contact [START_REF] Jia | Microstructure and friction-wear characteristics of bivalve shells[END_REF]. Finally, the coefficient of friction stays low and stable until the end once the film's thickness reaches an uniform value -which is about 60 nm. Thus, the edge-on parallel configuration seems clearly the best way to succeed in the nacre-like bioprostetic design displaying a high lifespan.

CONCLUSION

Multiscale tribological behavior of face-on sheet nacre has been studied by varying the frictional power which is dissipated within the contact. Results reveal various dissipative mechanisms controlled by the two organic matrices on various scales:

• When the frictional dissipated power is higher than 17 mW, wear of sheet nacre is mainly controlled by thermal effects -combining melting of the organic matrices and friction-induced phase transformation of the mineral phase. These two mechanisms start from 17 mW and 49 mW, respectively. Beyond, the wear rate dramatically increases as a so-called avalanche effect due to the propagation of the heat flow within the subsurfaces, damaging matrices in depth.

• When the frictional dissipated power is in between 1 µW and 17 mW, wear is mainly controlled by a friction-induced nanoshock process involving the fracture of the intracrystalline matrix only. However, this process can be preceded (or not) by the fracture of the interlamellar one increasing (or not) the frictional dissipated power by a factor 4 or more. The interlamellar damage occurrence clearly controls the transition from low to high frictional regime and then, the wear rate on a large range. This irreversible transition is completely controlled by the shear resistance of the interlamellar matrix, and obviously depends on the orientation of the sheet nacre (face-on vs. edge-on configuration)

• When the friction dissipated power is decreased down to 1 nW, the tribological behavior is completely controlled by the orientation of the samples (face-on vs. edge-on), and so by the organic matrices. In face-on configuration, stick-slip occurrence simultaneously controls friction and wear behaviors. In contrast, in edge-on configuration, an interesting lubricant effect -probably due to an organic tribolayer generated by the organic interlamellar phase -is observed in association with some viscoelastic deformations of the organomineral tablets. This orientation leads to the better tribological behaviour displaying a low and stable friction coefficient along thousands cycles without any detectable wear ;

  [START_REF] Huang | Nanoscale structural and mechanical characterization of heat treated nacre[END_REF]. So, this phase transformation well corresponds to the second change of the slope of the wear rate, that is observed in Fig 8b around 53 mW.

Figure 1 :

 1 Figure 1 : Multiscale structure of nacre a) the brick & mortar architecture ; b) enlargement on the edge of a

Figure 2 :Figure 3 :Figure 4 :

 234 Figure 2 : a) Acoustic microscope ; b) Acoustic imaging of face-on sheet nacre sample (482 MHz, at z = 0

Figure 6 :Figure 7 :

 67 Figure 6: Finite element model simulating the wear mechanims

Figure 8 :Figure 12 :Figure 13 :

 81213 Figure 8: Variation of the wear rate with a) the normal load; b) the sliding velocity

Figure 15 :Figure 16 :Figure 21 :

 151621 Figure 15: a) Evolution of the coefficient of friction with the normal load in high and low frictional

Table 1 :

 1 Mechanical properties of the Pinctada maxima sheet nacre and its components

ACKNOWLEDGEMENTS

This work was supported by grants from the Agence Nationale de la Recherche Française under the awards NanoBioCarbonate ANR 10 INTB 90901 Organic mediation on nanostructured biomaterials, biomimetic example of otoliths and freshwater pearls. Authors are grateful to Mr. Philippe Violle from Sympatec GmbH for lending the PCCS Nanophox equipement which allowed us the multiscale wear debris assessement.


L.-A Duflot

B Tamadazte

N Andreff

A Krupa

Wavelet-based visual servoing using OCT images

This paper deals with the development of an Optical Coherence Tomography (OCT) based visual servoing. The proposed control law uses the wavelet coefficients of the OCT images as the signal control inputs instead of the conventional geometric visual features (points, lines, moments, etc.). An important contribution is the determination of the interaction matrix that links the variation of the wavelet coefficients to the OCT probe (respectively to the robotic platform) spatial velocity. This interaction matrix, required in the visual control law, is obtained from time-derivation of the wavelet coefficients. This work is carried out in a medical context which consists of automatically moving a biological sample in such a way to go back to the position of a sample region that corresponds to a previous optical biopsy (OCT image). For instance, the proposed methodology makes it possible to follow accurately the progress of a pathological tissue between an optical biopsy and a former one. The developed method was experimentally validated using an OCT imaging system placed in an eye-tohand configuration viewing the robotic platform sample holder. The obtained results demonstrated the feasibility of this type of visual servoing approach and promising performances in terms of convergence and accuracy.

I. INTRODUCTION

A. Motivation

The wavelet transform is a mathematical tool introduced in 1910 by Alfred Haar but has grown in popularity only in the late 1980s with the arrival of the early work of Yves Meyer [START_REF] Meyer | Ondelettes, fonctions splines et analyses graduées[END_REF]. The rise of digital image processing and the need of compression in order to process, encode, transmit, and save images with increasing resolutions have led to the development of advanced and efficient compression methods. The wavelet framework was particularly used for elaborating the JPEG2000 compression [START_REF] Christopoulos | The jpeg2000 still image coding system: an overview[END_REF]. In fact, wavelet multiscale decomposition of an image is obtained by repeatedly smoothing and subsampling an image signal while also storing the detail information lost at each stage of this process. Such decomposition is not only useful for defining increasingly coarse approximations of images, but also to obtain sparse representations which means that most coefficients describing the detail information lost in each transformation step are typically close to zero. Therefore, the coarse approximation coefficients of an image can be considered as signal control inputs in the formulation of a robotic visual L.-A. Duflot Andreff are with FEMTO-ST, AS2M, Univ. Bourgogne Franche-Comté, Univ. de Franche-Comté/CNRS/ENSMM, 25000 Besanc ¸on, France brahim.tamadazte@femto-st.frnicolas.andreff@femto-st.fr control law. A 6-degrees-of-freedom (DOF) visual servoing approach based on the low-pass approximation of an image obtained from a discrete wavelet transform was proposed in [START_REF] Ourak | Waveletsbased 6 dof visual servoing[END_REF] using a conventional projective camera as a sensor. In the same manner, shearlet transform (considered as the extension of the wavelets one) was applied in the context of medical image-guided tasks, where a 6 DOF ultrasound (US) based visual servoing method was proposed for automatically positioning an ultrasound probe actuated by a robotic arm [START_REF] Duflot | Shearlet-based vs. photometric-based visual servoing for robot-assisted medical applications[END_REF]. These methods have demonstrated very interesting behavior during the different experimental validations of the respective control laws. These preliminary investigations already demonstrated the feasibility and potential benefits of considering wavelet-based multiscale representations in a visual servoing scheme, in particular with respect to unfavorable conditions (noise, partial image occlusion, illumination changes). In fact, using multiscale wavelet decomposition could serve as an interesting compromise between purely geometric feature-based approaches [START_REF] Chaumette | Visual servo control. i. basic approaches[END_REF] and image-intensitybased (photometric) ones [START_REF] Collewet | Photometric visual servoing[END_REF], [START_REF] Tamadazte | A direct visual servoing scheme for automatic nanopositioning[END_REF]. This is especially valuable in case of less textured images such as medical images: US, OCT, in which the detection, extraction and tracking of visual features over time is very challenging.

Our work takes place in an ambitious project dealing with the development of a microrobotic-based nasal endoscopy system embedding a fiber-based OCT system. The medical aim is to perform optical biopsy of the olfactory mucosa in order to prevent an eventual smell deficiency which can be directly related to the appearance of neurodegenerative diseases. In this paper we present one important aspect of this project that concerns the development of a visual control law in which the robotic sensor is an OCT imaging system. The objective is to automatically align with a robotic platform a tissue sample observed by the OCT system in such a way to reach a desired OCT image previously captured at a early stage of the pathology for examination purpose. OCT is an emerging technological imaging device used to obtain high resolution cross-sectional tomographic images. First developed in 1991 [START_REF] Xu | Real-time compressive sensing spectral domain optical coherence tomography[END_REF], this device, which allows reaching a micrometric resolution and millimetric penetration depth, was tested on retina and on the coronary artery. Actually, OCT is more and more often used for medical mini-and noninvasive applications, in particular for ophthalmology and dermatology. This device uses low-coherence interferometry with near-infrared light in order to produce cross-section images of a scattering medium. Nevertheless, the acquisition of 3D volumes remains slow (nearly one minute for a centimetric size area) due to mechanical scanning as well as the size of the acquired data.

B. Contributions

A CCD camera is rigidly attached to our OCT probe. That is why, one previous work was already using a wavelet decomposition of the OCT scan to control the in-plane movements [START_REF] Ourak | Partitioned camera-oct based 6 dof visual servoing for automatic repetitive optical biopsies[END_REF]. The image captured by the CCD camera was used to control the other DOF. Our goal now is to control all the DOF of the robot based on OCT information. Given that volume acquisition remains too slow to apply a visual servoing scheme, this paper will propose a direct visual servoing using only OCT cross-sectional image as visual information. All the DOF can theoretically be controlled with the work presented here. Nevertheless, since rotations are difficult to manage with our robot with such resolutions, only 3 DOF will be controlled experimentally. They correspond to the two in-plane (xz) and the out-of-plane y translations of the OCT image plane (see Fig. 3 for the definition of the x, y and z axes). The wavelet decomposition is not the same as the one used in [START_REF] Ourak | Partitioned camera-oct based 6 dof visual servoing for automatic repetitive optical biopsies[END_REF] and use more coefficient as signal input in order to upgrade the controller. The work presented in [START_REF] Yu | Calibration and integration of b-mode optical coherence tomography for assistive control in robotic microsurgery[END_REF] is also making a 6 DOF visual servoing using OCT probe with a geometric feature-based approach.

C. Plan

In the sequel, Section II first recalls the general framework of the wavelet transform and then presents our contribution on the design of a visual servoing approach that considers the wavelet coefficients as visual features (signal control inputs). Section III shows the experimental validation of the proposed wavelet-based visual servoing approach using a Fourier-Domain OCT (FD-OCT) imaging system as a robotic sensor. This section also provides a discussion on the performances of the developed method.

II. FRAMEWORK

A. Wavelet Signal Decomposition

Real-world data, like signal or images, are composed by smooth (homogeneous) regions and non-smooth ones (e.g., edges). These edges often represent the most interesting information for signal processing. In contrast to Fourier transforms, which only yield information about which frequencies are present in a signal but not their locations, wavelet transforms allow a simultaneous and efficient analysis of the behavior of a signal in both the time (in 1D case) or the space (in 2D case) and the frequency domains.

In fact, a wavelet is a rapidly decaying wave-like oscillation, centered in 0 and existing for a given location, contrary to sine functions in the Fourier transform. A wavelet system is constructed by shifting and scaling a wavelet function ψ in the Hilbert space L 2 (R) of square integrable functions. Then the generating wavelet ψ shifted by k and scaled by j is described by

ψ j,k (x) = 2 j/2 ψ(2 j x -k) (1)
In the late 1980s, pioneers like Ingrid Daubechies [START_REF] Daubechies | Orthonormal bases of compactly supported wavelets[END_REF] and Stéphane Mallat [START_REF] Mallat | A theory for multiresolution signal decomposition: the wavelet representation[END_REF] introduced discrete wavelet transforms more adapted for implementation. In this paper, we opt for the Daubechies wavelets included within the MATLAB WAVELET TOOLBOX.

Note that the basic formula of a continuous wavelet transform of a function f ∈ L 2 (R) is defined by

(W ψ f )(j, k) = f, ψ j,k = R ψ( x -k2 -j 2 -j )f (x)dx (2)
Thus, the wavelet transform allows extracting different levels of details coefficients B 1 , B 2 and B 3 and smooth approximation coefficients A 3 of a given signal S.

In practice, a discrete wavelet transform is performed by applying successively high-pass and low-pass filters to a signal S as represented in Fig. 1. Therefore, in the case of 2D signal (i.e., image), we consider low-pass and high-pass filtering along each direction leading to 3 types of detail coefficients: horizontal, vertical and diagonal ones and approximation coefficients. In other words, the details coefficient previously introduced by B are divided in three types: horizontal h, vertical v and diagonal d coefficients (see Fig. 2).

B. Control Law

The objective of a visual servoing scheme is to control the motion of a robotic structure in order to allow that a set of visual features s (s ∈ R k ), defining a robot pose r(t) ∈ SE(3) i.e., s = s r(t) , reaches a set of desired features s * (s * ∈ R k ) expressing the desired robot pose r * ∈ SE(3). In other words, the aim is to minimize the visual error given by e = s r(t) -s *

For this purpose, it is needed to establish the link between the velocity twist vector, which in our case contains only the translation components v = (v x v y v z ) of the OCT system frame, and the variations of s by ṡ = L s v where L s ∈ R k×3 is called the interaction matrix.

In order to obtain a decrease of the error e such as ė = -λe, with λ being a positive gain, the classical controller usually considered in the literature is given by [

5] v = -λ L s + s(t) -s * (4)
where L s + is the Moore-Penrose pseudo-inverse of an estimation of the interaction matrix. We propose therefore to use this control law to control the 3 DOF robotic platform in the OCT system frame.

In this work, the visual information s used in the control loop consists of the wavelet coefficients computed by the discrete Daubechies wavelet transform W ψ (.) on a square image I of size (N × N ) as

    A 3 h 3 v 3 d 3 h 2 v 2 d 2 h 1 v 1 d 1     = W ψ (I) (5)
More precisely, we take into account only one quarter of the coefficients which are B 2 , B 3 and A 3 such as

B m = [d m ; h m ; v m ] with index m = {2, 3} (6)
The resulting visual feature vector is therefore built as

s =[B 2 (1, 1), ..., B 2 (i, j), ..., B 2 ( 3N / 4 , N / 4 ), B 3 (1, 1), ..., B 3 (i, j), ..., B 3 ( 3N / 8 , N / 8 ), A 3 (1, 1), ..., A 3 (i, j), ..., A 3 ( N / 8 , N / 8 )] (7)
As example, Fig. 2 shows the Daubechies wavelet decomposition of an OCT image (a xz image slice) of a LEGO toy element. The first level of details B 1 corresponds to the higher frequency assimilated to noise and is therefore not kept to compute the visual features. As N = 512 in our test case, then the size of s is equal to 256 × 256 = 65536 coefficients. with

L B2 (i, j) =[∇ x (B 2 ) (i,j) , ∇ y (B 2 ) (i,j) , ∇ z (B 2 ) (i,j) ] L B3 (i, j) =[∇ x (B 3 ) (i,j) , ∇ y (B 3 ) (i,j) , ∇ z (B 3 ) (i,j) ] L A3 (i, j) =[∇ x (A 3 ) (i,j) , ∇ y (A 3 ) (i,j) , ∇ z (A 3 ) (i,j) ] (9)
The functions ∇ x (.), ∇ y (.) and ∇ z (.) give the components along x, y and z of the wavelet transform of the 3D gradient of the considered image.

They are calculated from

∇ x (.) =W ψ (∇I x ) ∇ y (.) =W ψ (∇I y ) ∇ z (.) =W ψ (∇I z ) (10)
where ∇I x = ∂I/∂x, ∇I y = ∂I/∂y, ∇I z = ∂I/∂z are the value of the image gradient for the point of coordinates (x, y, z) with a derivative filter as the Sobel one and W ψ (.) is given by [START_REF] Chaumette | Visual servo control. i. basic approaches[END_REF]. Whereas the computation of ∇I x and ∇I z uses the OCT slice at the desired position, the computation of ∇I y needs two parallel slices on each side of the desired slice to be calculated (see Fig. 3).

III. EXPERIMENTAL VALIDATION A. OCT Device

As mentioned in the introduction, OCT is an emerging technology for performing high-resolution, cross-sectional, tomographic imaging in real time. OCT system operation is based on a low-coherence interferometric technique, usually using near infrared light [START_REF] Xu | Real-time compressive sensing spectral domain optical coherence tomography[END_REF]. In fact, OCT images are obtained by the measure of the reverberate time delay and the intensity of the back-scattered light from the viewed object that gives data about the different scrambling layers of the tissue.

Fig. 4: Functioning of a Fourier-Domain OCT system

The OCT images (depending on the used lighting source wave-length) have micrometer resolution and millimeter penetration depth while US images offer a millimeter resolution with several centimeters of depth. Due to the numerous advantages of the OCT, it is considered "optical biopsy", in various medical fields, which can be a substitute of the conventional "physical biopsy". The OCT imaging system used is shown in Fig. 5. It consists of a Fourier-Domain Telesto-II OCT from THORLABS. It is characterized by 5.5µm and 7µm axial and lateral resolutions, respectively, and provides up to 3.5mm of depth. The Telesto-II allows a maximum field-of-view of 10×1 ×13.5mm 3 with a maximum A-Scan (1D optical core) acquisition rate of 76 kHz.

B. Robotic Setup

The proposed methods and materials were experimentally tested using a robot placed as a sample holder i.e., under the OCT imaging system, in a eye-to-hand configuration (Fig. 5). More precisely, the robotic platform is a 6 DOF 3PPSR parallel robot (a Space-FAB SF-3000 BS from Micros) characterized by a translation range of (t x , t y , t z ) max = (50, 100, 12.7) (mm) and a rotation range of (r x , r y , r z ) max = (10, 10, 10)( • ), a linear resolution of 0.2µm and an angular resolution of 0.0005 • ; and two computers communicating with an asynchronous TCP/IP protocol, a 3.2 GHz i5 core Intel CPU (with a Linux OS) for computer vision and control law implementation and a 2.33 GHz Xeon Intel CPU (with Windows OS) for robot inner control law implementation. Note that, with a non-optimized implementation of the method (essentially scripted using MATLAB), the control loop takes approximately 100 milliseconds. During the visual servoing process, we grab also the white light image of the CCD camera rigidly attached in a co-axial configuration with the OCT light path. Therefore, the CCD images are used to essentially verify the good functioning of the developed control law. In order to validate the visual servoing controller in which the signal control inputs are the wavelet details coefficients, we choose two scenarios: using an artificial sample (a LEGO piece) and a biological sample (part of a shrimp) (Fig. 6). Two tests are presented in detail in the following; "LEGO piece test 1" corresponds to subsection III-C and "shrimp sample test 1" corresponds to the subsection III-D. The first experiment was performed using a LEGO of size 1.6 × 0.8 × 1.6 cm 3 . First, the sample is placed at a desired position r * and moved to an arbitrary (initial) position r. So, the initial error is estimated (using the robot's high resolution encoders) to be ∆T initial = (-0.5, -0.5, 0.3) (mm). The image difference between the initial OCT image I oct and desired one I * oct is depicted in Fig. 7.(b). Note that, the displacement along the x and z axes are clearly visible on the OCT slice, contrary to the one along y axis, this is due to the LEGO structure (Fig. 7.(b)). However, this difference along y axis can be shown in Fig. 8.

(b).

The first experiment achievement is shown in Fig. 7, where Fig. 7.(a) is the initial image (position), Fig. 7.(b) the initial difference image, and Fig. 7.(c) the final difference image at convergence. As can be noticed, this final difference image contains mostly noise which proves that final and desired images are completely superimposed demonstrating the good accuracy of the control law. This is also demonstrated by the CCD images sequence illustrated (Fig. 8). The final error is estimated (again using the robot's encoders) to ∆T f inal = (-0.0118, -0.0149, 0.0048) (mm). The Cartesian error decay in each DOF is recorded and plotted in Fig. 9 as well as the norm of the global error e = s-s * which is shown in Fig. 10. One can underline that all the errors converge towards zero, in approximately 80 iterations. However, it is important to remember that the control is performed on the OCT image (with an unfavorable signal-to-noise ratio); this does not generate exponential convergence of the Cartesian errors, unlike in case of a posebased visual servoing scheme.

D. Shrimp Experiment

Concerning the second experiment, the LEGO sample was replaced by a biological one, i.e., a part of a shrimp of size 2 × 1 × 0.5 cm 3 approximately. The scenario remains the same, the operator records the desired position and then moves the sample (using the robotic platform) to an arbitrary initial position. The visual servoing task should therefore automatically return the robot to the desired position. For this experiment, the initial error is ∆T initial = (-0.5, -0.5, 0.3) (mm). The image difference between the initial OCT image and the desired one is shown in Again, the error in each DOF for the second test decreases to zero as shown in Fig. 13. As can be highlighted, all DOF converge to the desired position similarly to the first test. Also, the norm of the error e = s-s * was recorded and plotted in Fig. 14 and its evolution towards zero confirms the good convergence behavior of the OCT-based control law.

E. Others Experiments

Table I presents the initial positioning errors and the final positioning ones (measured at convergence) of all the experiments done to validate our method. Two tests were performed using the LEGO piece and three using the shrimp sample. The mean positioning error considering all axes is 0.01mm and the standard deviation (STD) considering all axes is 0.02mm. This accuracy shows the precision of the presented framework. We can observe that the y axis (i.e. the out-of-plane axis) is slightly more than half as accurate as the others despite the fact that the interaction matrix was computed using only four slices adjacent to the slice considered as the desired image.

IV. CONCLUSION AND DISCUSSION

The main objective of this paper was the study of an OCT-based direct visual servoing scheme. In fact, instead of considering geometric visual features in the interaction matrix, we took into account the wavelet coefficients. Furthermore, the OCT imaging system that is usually used only for diagnosis purposes (in-situ optical biopsies acquisition) was employed here also as a robotic sensor. The proposed controller was experimentally tested using a highly accurate parallel robotic structure placed under the OCT system in the role of sample holder platform. Despite of the nature of the OCT images (unfavorable signal-to-noise ratio), the proposed approach demonstrated interesting behaviors, especially in term of accuracy. Because of this unfavorable signal-tonoise ratio, taking directly the pixels of the image in the visual servoing scheme fails to provide an effective visual servoing scheme. The work presented in [START_REF] Ourak | Partitioned camera-oct based 6 dof visual servoing for automatic repetitive optical biopsies[END_REF] was already using wavelet coefficients to overcome the limitations of a direct visual servoing based on pixel intensities. In the framework presented here, a larger amount of coefficients is used to compute the visual features of the control law and the interaction matrix is computed differently. With our experimental setup, we obtained an average final error positioning of 0.01mm which represents only 3% of the average initial displacement.

Further work will concern the extension of this investigation to a 6 DOF control law by taking into account the rotations. This haven't be done yet because of technical issues but the framework presented here contains all the theoretical notions to provide a 6 DOF direct visual servoing. More than five tests should also be proposed as well as a comparison with a pixel intensities based visual servoing. Another way of improvement will be the implementation of the whole proposed materials in a more efficient software framework, e.g., a C++ one which can improve consequently the dynamic of the controller. Finally, OCT allows to generate volumetric data. A visual servoing based on a C-scan could be a groundbreaking work for optical biopsies. Nevertheless, this process is time costly (near 1 minute for a 1cm 3 sample). One idea should be to combine some compressive sensing acquisition scheme (which uses often wavelet transforms) with this direct visual servoing framework.

Fig. 1 :

 1 Fig. 1: Diagram of a wavelet decomposition of a signal S.

Fig. 2 :

 2 Fig. 2: Wavelet transform of an OCT image of a LEGO toy element

Fig. 3 :

 3 Fig. 3: Diagram showing the operation of a typical OCT imaging system

Fig. 5 :

 5 Fig. 5: Photography of the experimental setup including the 6 DOF robotic platform as well as the OCT imaging system.

Fig. 6 :

 6 Fig. 6: Sample materials used in the experimental validation: LEGO and shrimp

Fig. 7 :

 7 Fig. 7: Test 1 (OCT): (a) desired OCT image I * oct , (b) initial difference image I * oct -I oct , (c) the final difference image I * oct -I oct .

Fig. 8 :Fig. 10 :

 810 Fig. 8: Test 1 (CCD camera): (a) desired white light image I * cam , (b) the initial difference image I * cam -I cam , and (c) I * cam -I cam acquired at convergence.

Fig. 11 .

 11 (b). Contrary to the first task, the displacement along the y axis is visible (Fig.11.(b)). As experimentally demonstrated in Fig.11.(c), which is completely gray, the visual servoing control law converges accurately towards the desired position. The final error (i.e., at convergence) is ∆T f inal = (-0.0074, 0.0035, -0.0091) (mm).

Fig. 11 :Fig. 12 :Fig. 14 :

 111214 Fig. 11: Task 2 (OCT): (a) desired OCT image I * oct , (b) initial difference image I * oct -I oct and (c) the final difference image I * oct -I oct at convergence.

  and A. Krupa are with Univ Rennes, Inria,

	CNRS,	IRISA,	Campus	universitaire	de	Beaulieu,	Rennes
	35042,	France

lesley-ann.duflot@inria.fr alexandre.krupa@inria.fr L.-A. Duflot, B. Tamadazte and N.

TABLE I :

 I Summary of the different performed tests

	Case	error (mm)	Tx	Ty	Tz
	LEGO piece	e initial	-0.5	-0.5	0.3
	test 1	e f inal	-0.0118 -0.0149	0.0048
	LEGO piece	e initial	-0.3	-0.6	0.5
	test 2	e f inal	-0.0002 -0.0020 -0.0111
	shrimp sample	e initial	-0.5	-0.5	0.3
	test 1	e f inal	-0.0074	0.0035	-0.0091
	shrimp sample	e initial	-0.5	-0.2	0.3
	test 2	e f inal	-0.0209	0.0677	-0.0087
	shrimp sample	e initial	0.2	-0.2	0.2
	test 3	e f inal	-0.0058	0.0054	-0.0070
		mean(|e f inal |)	0.0092	0.0187	0.0081
		STD(|e f inal |)	0.0077	0.0278	0.0024

ACKNOWLEDGMENT

This work is partially supported by DESWEEP project funded by the Région de Bretagne and from the project NEMRO (ANR-14-CE17-0013-001) funded by the ANR, France. It is also performed in the framework of the Labex ACTION (ANR-11-LABEX-01-01).