L P A Thai

F Mousseau

E K Oikonomou

M Radiom

J.-F Berret

Effect of Nanoparticles on the Bulk Shear Viscosity of a Lung Surfactant Fluid

Keywords:

Inhaled nanoparticles (< 100 nm) reaching the deep lung region first interact with the pulmonary surfactant, a thin lipid film lining the alveolar epithelium. To date, most biophysical studies have focused on particle induced modifications of the film interfacial properties. In comparison, there is less work on the surfactant bulk properties, and on their changes upon particle exposure. Here we study the viscoelastic properties of a biomimetic pulmonary surfactant in the presence of various engineered nanoparticles. The microrheology technique used is based on the remote actuation of micron-sized wires via the application of a rotating magnetic field and on timelapse optical microscopy. It is found that particles strongly interacting with lipid vesicles, such as cationic silica (SiO2, 42 nm) and alumina (Al2O3, 40 nm) induce profound modifications of the surfactant flow properties, even at low concentrations. In particular, we find that silica causes fluidification, while alumina induces a liquid-to-soft solid transition. Both phenomena are described quantitatively and accounted for in the context of colloidal physics models. It is finally suggested that the structure and viscosity changes could impair the fluid reorganization and recirculation occurring during breathing.

I -Introduction

Following their interaction with living organisms, nanoparticles (NPs) first interact with extracellular body fluids (ECF) before they come in contact with cellular structures. ECFs are composed of the blood plasma, of liquids from interstitial compartments, including the lymph, the mucus and the pulmonary surfactant and from small amounts of transcellular liquids in the ocular and cerebrospinal cavities. Combined, these fluids represent 1/3 of the total body fluids in humans, i.e. around 15 L for a person of normal weight. Upon contact with ECFs, NPs are dispersed in a waterborne phase rich in electrolytes, lipids, proteins and carbohydrates with which they interact. This is the case for example when NPs are introduced into the respiratory and digestive systems or the blood compartment. Over the last 20 years, studies have focused on NP interactions with serum proteins from the blood. [START_REF] Falahati | A Health Concern Regarding the Protein Corona, Aggregation and Disaggregation[END_REF][START_REF] Moore | Nanoparticle Colloidal Stability in Cell Culture Media and Impact on Cellular Interactions[END_REF] When mixed with serum, NP interfaces are spontaneously covered with proteins, leading to the protein corona formation. [START_REF] Lundqvist | Nanoparticle Size and Surface Properties Determine the Protein Corona with Possible Implications for Biological Impacts[END_REF][START_REF] Walczyk | What the Cell "Sees" in Bionanoscience[END_REF] For NPs, it is assumed that the protein corona determines their new biological identity and eventually regulate their interaction and biodistribution in vivo.

In the context of nanoparticles interacting with the ECFs, the respiratory zone in the lungs represents an interesting environment. In this region, the alveoli form a froth-like structure and their interface with the air is lined with a thin layer (< 1µm) of pulmonary surfactant. Secreted by alveolar epithelial cells, pulmonary surfactant is an interstitial fluid that contains lipids and proteins in a 90:10 ratio, its overall weight concentration being approximately 40 g L -1 . Its role is to reduce the surface tension with the air, to prevent the alveoli collapse at the lowest lung volumes and facilitate their expansion during subsequent inspirations. [START_REF] Casals | Role of Lipid Ordered/Disordered Phase Coexistence in Pulmonary Surfactant Function[END_REF][START_REF] Lopez-Rodriguez | Structure-Function Relationships in Pulmonary Surfactant Membranes: From Biophysics to Therapy[END_REF] The proteins present in the surfactant film such as SP-A, SP-B, SP-C and SP-D play an important role in the structure and function of the alveoli, in particular with regard to pathogen elimination and to interfacial film stability. For particles not captured in the upper airways, surfactant represents the first physical barrier against small sized particles (< 100 nm). Pertaining to the interaction mechanisms with lipids, the picture emerging from current literature is based on the protein corona model described previously, leading to a description in terms of biomolecular or lipid coronas. For particles with hydrophilic surfaces, this corona is often depicted as a supported lipid bilayer (SLB) which consists in a single bilayer adsorbed at the particle surface via adhesive forces. [START_REF] Richter | Formation of Solid-Supported Lipid Bilayers: An Integrated View[END_REF][START_REF] Mousseau | Supported Pulmonary Surfactant Bilayers on Silica Nanoparticles: Formulation, Stability and Impact on Lung Epithelial Cells[END_REF][START_REF] Liu | Interfacing Zwitterionic Liposomes with Inorganic Nanomaterials: Surface Forces, Membrane Integrity, and Applications[END_REF][START_REF] Mornet | The Formation of Supported Lipid Bilayers on Silica Nanoparticles Revealed by Cryoelectron Microscopy[END_REF][START_REF] Michel | Experimental Aspects of Colloidal Interactions in Mixed Systems of Liposome and Inorganic Nanoparticle and Their Applications[END_REF] However, recent studies have shown that the SLB formation is not spontaneous (in contrast to the protein corona) and depends on various factors such as particle and membrane charges as well as particle size. [START_REF] Dasgupta | Membrane-Wrapping Contributions to Malaria Parasite Invasion of the Human Erythrocyte[END_REF][START_REF] Deserno | Adhesion and Wrapping in Colloid-Vesicle Complexes[END_REF][START_REF] Froehlich | The Role of Surface Charge in Cellular Uptake and Cytotoxicity of Medical Nanoparticles[END_REF] So far, the exact nature of this lipid corona is not well established.

In the alveolar spaces, inhaled NPs first cross the interfacial lipid monolayer and thereafter diffuse through a three-dimensional network of interconnected membranes towards the hypophase. [START_REF] Hidalgo | Pulmonary Surfactant and Nanocarriers: Toxicity versus Combined Nanomedical Applications[END_REF] With a lipid concentration estimated at 40 g L -1 , [START_REF] Hobi | Physiological Variables Affecting Surface Film Formation by Native Lamellar Body-Like Pulmonary Surfactant Particles[END_REF][START_REF] Lewis | Surfactant and the Adult Respiratory-Distress Syndrome[END_REF][START_REF] Numata | Anionic Pulmonary Surfactant Lipid Regulation of Innate Immunity[END_REF][START_REF] Notter | Lung Surfactant: Basic Science and Clinical Applications[END_REF] the hypophase is a dispersion where lipids are organized in multilamellar vesicles and tubular myelins. [START_REF] Lopez-Rodriguez | Structure-Function Relationships in Pulmonary Surfactant Membranes: From Biophysics to Therapy[END_REF][START_REF] Notter | Lung Surfactant: Basic Science and Clinical Applications[END_REF][START_REF] Gil | Morphological Study of Pressure-Volume Hysteresis in Rat Lungs Fixed by Vascular Perfusion[END_REF] To date, most biophysical studies have focused on the modification of the interfacial film properties. Using pulmonary surfactant mimetics, Schleh et al. have shown that nanosized titanium dioxide induces a surfactant dysfunction associated with a dose-dependent increase of adsorption and surface tension. [START_REF] Schleh | The Effect of Titanium Dioxide Nanoparticles on Pulmonary Surfactant Function and Ultrastructure[END_REF] Similar results were found using gold, silica, polymeric NPs with either synthetic or exogenous formulations. [START_REF] Beck-Broichsitter | Biophysical Investigation of Pulmonary Surfactant Surface Properties upon Contact with Polymeric Nanoparticles In Vitro[END_REF][START_REF] Pera | Coverage and Disruption of Phospholipid Membranes by Oxide Nanoparticles[END_REF][START_REF] Bakshi | Metal Nanoparticle Pollutants Interfere with Pulmonary Surfactant Function In Vitro[END_REF] Surface pressure-area isotherms of synthetic lipid mixtures also depict strong modifications of the lipid interfacial organization caused by hydrophobic as well as hydrophilic NPs. [START_REF] Harishchandra | Nanoparticle Interaction with Model Lung Surfactant Monolayers[END_REF][START_REF] Sachan | High-Resolution Investigation of Nanoparticle Interaction with a Model Pulmonary Surfactant Monolayer[END_REF][START_REF] Fan | Adverse Biophysical Effects of Hydroxyapatite Nanoparticles on Natural Pulmonary Surfactant[END_REF][START_REF] Kodama | Investigating the Effect of Particle Size on Pulmonary Surfactant Phase Behavior[END_REF][START_REF] Kondej | Effect of Clay Nanoparticles on Model Lung Surfactant: A Potential Marker of Hazard from Nanoaerosol Inhalation[END_REF] During compression and expansion, it is found that particles lead to a disruption of the lipid phase behavior and alter the surface tension hysteresis cycle. Transposed to the lung environment, repeated exposure is expected to cause perturbations of the lung physiology associated with particle retention, cellular oxidative stress generation and pro-inflammatory effects. [START_REF] Hidalgo | Pulmonary Surfactant and Nanocarriers: Toxicity versus Combined Nanomedical Applications[END_REF][START_REF] Xia | Pulmonary Diseases Induced by Ambient Ultrafine and Engineered Nanoparticles in Twenty-First Century[END_REF][START_REF] Lelieveld | The Contribution of Outdoor Air Pollution Sources to Premature Mortality on a Global Scale[END_REF] In comparison, there is much less work done on the surfactant bulk properties and on the effects of nanoparticles. Bulk rheology measurements were reported by King et al. [START_REF] King | Bulk Shear Viscosities of Endogenous and Exogenous Lung Surfactants[END_REF] and by Lu et al. [START_REF] Lu | Kinematic Viscosity of Therapeutic Pulmonary Surfactants with Added Polymers[END_REF] who found that the viscosity of native and exogenous surfactants determined at physiological concentrations were in the range 5 -50 mPa s, indicating a relatively low viscosity ECF. In a recent work, we investigated the viscosity of the biomimetic lung surfactant Curosurf® as a function of the volume fraction 𝜙 in vesicles and found that it obeys the well-known Krieger-Dougherty law for colloids, [START_REF] Krieger | A Mechanism for Non-Newtonian Flow in Suspensions of Rigid Spheres[END_REF][START_REF] Larson | The Structure and Rheology of Complex Fluids[END_REF] 𝜂(𝜙) = 𝜂 & (1 -𝜙/𝜙 * ) +, where 𝜂 & is the solvent viscosity and 𝜙 * = 0.65 the maximum-packing fraction. [START_REF] Thai | On the Rheology of Pulmonary Surfactant: Effects of Concentration and Consequences for the Surfactant Replacement Therapy[END_REF] The Krieger-Dougherty behavior is characterized by an exponential increase of the viscosity and a divergence at 𝜙 * . An important conclusion of this study was that in physiological conditions, the biomimetic lung fluid had a viscosity around 10 mPa s and a highvolume fraction (𝜙 ~ 0.25), indicating a crowded environment. With a physiological value close to the maximum-packing fraction, viscosity changes resulting from the addition of particles are expected. During breathing, pulmonary surfactant is subjected to local motion and any structure or viscosity changes could impair the fluid reorganization and recirculation.

In this work, we study the effects of silica and alumina nanoparticles on Curosurf® rheological properties at physiological concentrations. As NP models, silica and alumina were selected because they are manufactured in high volume by the chemical industry, which increases the risk of occupational and environmental exposure. [START_REF] Van Donkelaar | Global Estimates of Ambient Fine Particulate Matter Concentrations from Satellite-Based Aerosol Optical Depth: Development and Application[END_REF] It was also found that silica and alumina NPs inhalation may lead to a series of disease including lung inflammation and cancer [START_REF] Lanone | Comparative Toxicity of 24 Manufactured Nanoparticles in Human Alveolar Epithelial and Macrophage Cell Lines[END_REF][START_REF] Napierska | The Nanosilica Hazard: Another Variable Entity[END_REF][START_REF] Oberdorster | Nanotoxicology: An emerging discipline evolving from studies of ultrafine particles[END_REF] . Recently, Puisney et al. have shown that silicon and aluminum based NPs, including oxides, are produced by brake wear and degradation processes of passenger vehicles and are expected to be present in urban air pollution in significant amounts. [START_REF] Puisney | Brake Wear (Nano)Particle Characterization and Toxicity on Airway Epithelial Cells In Vitro[END_REF] Concerning the choices of nanoscale particles in our study, it has been shown that the particle deposition along the respiratory tract depends primarily on their aerodynamic diameter. It was found that for particles around 20 nm, more than 90% of the inhaled mass fraction deposit in the entire lungs and 50% in the alveolar region. [START_REF] Oberdorster | Nanotoxicology: An emerging discipline evolving from studies of ultrafine particles[END_REF][START_REF] Geiser | Deposition and biokinetics of inhaled nanoparticles[END_REF] For the 40 nm particles put under scrutiny here, these percentages are slightly lower, at respectively 53% and 37%. For microrheology experiments, we exploit the technique of magnetic rotational spectroscopy (MRS) [START_REF] Berret | Local Viscoelasticity of Living Cells Measured by Rotational Magnetic Spectroscopy[END_REF][START_REF] Gu | Ferromagnetic Nanorods in Applications to Control of the In-Plane Anisotropy of Composite Films and for In Situ Characterization of the Film Rheology[END_REF][START_REF] Loosli | Wire-Active Microrheology to Differentiate Viscoelastic Liquids from Soft Solids[END_REF][START_REF] Tokarev | Nanostructured Soft Matter with Magnetic Nanoparticles[END_REF] in which micron-sized wires are submitted to a rotational magnetic field as a function of time and of the angular frequency (𝜔 = 10 -2 -10 rad s -1 ), while their motion is monitored by time-lapse optical microscopy. The MRS technique enables to measure the shear viscosity and elastic modulus of complex fluids, it is suitable for fluids in minute amounts and with heterogeneities at the micron scale. It is found that particles interacting with Curosurf® vesicles induce profound modifications in the surfactant flow properties. At low concentrations, fluidification is observed with silica whilst with alumina, a transition towards an arrested vesicular state occurs at particle concentration above 0.1 g L -1 .

II -Results and discussion

II.1 -Nanoparticles, Curosurf® vesicles and interaction

Recently we studied the phase behavior of a series of nanoparticles and biomimetic surfactants under controlled physico-chemical conditions. [START_REF] Mousseau | Supported Pulmonary Surfactant Bilayers on Silica Nanoparticles: Formulation, Stability and Impact on Lung Epithelial Cells[END_REF][START_REF] Mousseau | The Role of Surface Charge in the Interaction of Nanoparticles with Model Pulmonary Surfactants[END_REF][START_REF] Mousseau | Biophysicochemical Interaction of a Clinical Pulmonary Surfactant with Nanoalumina[END_REF] Interaction diagrams were determined from dilute solutions using various organic and inorganic NPs. From their aggregation behaviors with Curosurf®, the NPs were characterized with regard to their interaction strength parameter. Large strength parameters were obtained for highly charged cationic particles, whereas small or close to zero strength parameters were found for neutral and for anionic particles. In this section, we summarize the main results obtained on nanoparticle-vesicle interactions. Figs. 1a-b display transmission electron microscopy images of positively charged silica and alumina NPs, respectively. Silica are spherical and characterized by an average diameter of 42 nm, whereas alumina appear as irregular platelets of diameter 40 nm and thickness 10 nm. In solution, NP stability is ensured by electrostatic repulsions mediated by cationic surface charges. The NP charge density has been estimated at +0.62e nm -2 and +7.3e nm -2 respectively using a polymer-based titration method. [START_REF] Mousseau | Polyelectrolyte Assisted Charge Titration Spectrometry: Applications to Latex and Oxide Nanoparticles[END_REF] Combining different techniques such as the cryo-transmission electron microscopy (cryo-TEM), nanoparticle tracking analysis, dynamic light scattering and phase-contrast light microscopy, we have shown that Curosurf® lipids self-assemble into vesicles and that the vesicle sizes are broadly distributed, typically between 100 nm and 10 µm. [START_REF] Thai | On the Rheology of Pulmonary Surfactant: Effects of Concentration and Consequences for the Surfactant Replacement Therapy[END_REF] Fig. 1c illustrates a representative cryo-TEM image of Curosurf® at 5 g L -1 showing uni-, multilamellar and multivesicular vesicles. [START_REF] Mousseau | Supported Pulmonary Surfactant Bilayers on Silica Nanoparticles: Formulation, Stability and Impact on Lung Epithelial Cells[END_REF][START_REF] Schleh | The Effect of Titanium Dioxide Nanoparticles on Pulmonary Surfactant Function and Ultrastructure[END_REF][START_REF] Braun | A Freeze-Fracture Transmission Electron Microscopy and Small Angle X-Ray Diffraction Study of the Effects of Albumin, Serum, and Polymers on Clinical Lung Surfactant Microstructure[END_REF][START_REF] Kumar | Enrichment of Immunoregulatory Proteins in the Biomolecular Corona of Nanoparticles within Human Respiratory Tract Lining Fluid[END_REF][START_REF] De Backer | Bio-inspired pulmonary surfactant-modified nanogels: A promising siRNA delivery system[END_REF][START_REF] Waisman | Nanostructure of the Aqueous Form of Lung Surfactant of Different Species Visualized by Cryo-Transmission Electron Microscopy[END_REF] From the lipids in Curosurf® (Supplementary Information S1), we determined the surface charge density on the vesicles at -0.67 e nm -2 and anticipate strong (resp. weak) interactions with cationic (resp. anionic) NPs. [START_REF] Mousseau | Polyelectrolyte Assisted Charge Titration Spectrometry: Applications to Latex and Oxide Nanoparticles[END_REF][START_REF] Belloni | Ionic Condensation and Charge Renormalization in Colloid Suspensions[END_REF] To evaluate these interactions, we use the method of continuous variation developed by P. Job that we modify for light scattering. [START_REF] Mousseau | The Role of Surface Charge in the Interaction of Nanoparticles with Model Pulmonary Surfactants[END_REF][START_REF] Job | Studies on the Formation of Complex Minerals in Solution and on Their Stability[END_REF][START_REF] Renny | Method of Continuous Variations: Applications of Job Plots to the Study of Molecular Associations in Organometallic Chemistry[END_REF] Figs. 1d and 1e display the hydrodynamic diameter obtained from mixed dispersions as a function of the concentration ratio 𝑋 = 𝑐 12 /𝑐 31 , where 𝑐 12 and 𝑐 31 denote the lipid and nanoparticle weight concentrations at a given 𝑋. Note that in the Job plots the total concentration 𝑐 = 𝑐 12 + 𝑐 31 is held constant and maintained in the dilute regime. In Fig. 1d and 1e, the hydrodynamic diameters exhibit a maximum in the intermediate range 𝑋 = 1 -10, which we ascribe to the formation of aggregates. Examples of aggregates seen in optical microscopy are shown in the insets. Results on aggregates were obtained at room and body temperatures i.e. below and above the gel-to-fluid transition of Curosurf® membrane at 29.5 °C (Supplementary Information S2). [START_REF] Mousseau | Supported Pulmonary Surfactant Bilayers on Silica Nanoparticles: Formulation, Stability and Impact on Lung Epithelial Cells[END_REF][START_REF] Mousseau | Biophysicochemical Interaction of a Clinical Pulmonary Surfactant with Nanoalumina[END_REF] The outcomes of Fig. 1 suggest a strong electrostatic attraction between the oppositely charged NPs and vesicles. According to this scenario, the NPs adsorb at the lipid membranes and play the role of stickers for vesicles. Instances of particle sticking at the vesicular membrane were found with silica NPs using cryo-TEM experiments. [START_REF] Mousseau | The Role of Surface Charge in the Interaction of Nanoparticles with Model Pulmonary Surfactants[END_REF] As a negative control, we also investigated the effects of an anionic silica NPs (20 nm) with Curosurf®. The Job scattering plot did not show evidence of aggregation and the 𝐷 7 's varied continuously from that of the NPs to that of the vesicles. Results on anionic silica NPs and interaction diagrams are provided in Supplementary Information S3.

II.2 -Nanoparticle loaded surfactant showing a viscous fluid behavior

We now turn to the study of Curosurf® and to the effects of NPs on the rheology. The Curosurf® concentration is fixed at 𝑐 12 = 44 g L -1 , while the NP concentrations are varied from 𝑐 31 = 10 -3 to 0.5 g L -1 . It is shown in Materials and Methods that the lowest range of the above 𝑐 31 -interval is associated with an amount of nanoparticles in the alveolar region (extrapolated to human lungs) of 20 to 200 µg. [START_REF] Schleh | The Effect of Titanium Dioxide Nanoparticles on Pulmonary Surfactant Function and Ultrastructure[END_REF][START_REF] Geiser | Deposition and biokinetics of inhaled nanoparticles[END_REF] Depending on the NP type and concentration, two generic behaviors were observed with the wire-based microrheology. In this section, we describe the features corresponding to the case where the wires rotate steadily in the fluid with a non-zero average velocity, defined as 𝛺(𝜔) = < 𝑑𝜃(𝜔, 𝑡)/𝑑𝑡 > @ , 𝜃(𝜔, 𝑡) being the wire orientation angle at actuating frequency 𝜔. The case where 𝛺(𝜔) ≃ 0 is discussed in the next section. Fig. 2a illustrates the rotation of a 43 µm wire in a Curosurf® dispersion mixed with alumina NPs. The different images of the chronophotograph are taken at fixed time intervals (3.5 s) during a 𝜋-rotation of the object, showing that the wire rotates synchronously with the field. The applied frequency is 0.06 rad s -1 and the magnetic field 𝜇 E 𝐻 = 10.3 mT. In the image background, micron-sized vesicles may be seen and in Movie#1&2, it is shown that they are being sheared by the wire rotation (Supplementary Information, Movies#1&2). With increasing frequencies, the wire exhibits a transition between a synchronous and an asynchronous (i.e. not synced with the magnetic excitation) regime. In Figs. 2b, 2c and 2d, transient 𝜃(𝑡)-traces are represented at angular frequencies 0.06, 0.3 and 3 rad s -1 . The straight lines in red are least squared fitting to the 𝜃(𝑡) in both regimes, and their slopes provide the average rotation velocity 𝛺, here equal to 0.06, 0.12 and 0.03 rad s -1 . In these studies, non-zero 𝛺(𝜔)'s were found for pristine Curosurf® at physiological concentrations [START_REF] Thai | On the Rheology of Pulmonary Surfactant: Effects of Concentration and Consequences for the Surfactant Replacement Therapy[END_REF] and for Curosurf® mixed with silica at all concentrations tested, and with alumina at concentrations below 𝑐 31 = 0.1 g L -1 . Such behaviors are characteristic of viscous or viscoelastic liquids. 57 The rheological nature of the vesicular fluids put under scrutiny is here identified by monitoring the asymptotic behaviors of two measurable quantities: the angular frequency 𝜔 G at which the synchronous-asynchronous transition takes place, and the angle 𝜃 H (𝜔) which is the amplitude of the oscillations in the asynchronous regime. As shown previously, [START_REF] Tokarev | Nanostructured Soft Matter with Magnetic Nanoparticles[END_REF][START_REF] Chevry | Magnetic Wire-Based Sensors for the Microrheology of Complex Fluids[END_REF][START_REF] Helgesen | Nonlinear Phenomena in Systems of Magnetic Holes[END_REF] 𝜔 G is used to determine the fluid viscosity 𝜂 through the expression [START_REF] Chevry | Magnetic Wire-Based Sensors for the Microrheology of Complex Fluids[END_REF] 𝜔 G = 3𝜇 E Δ𝜒𝐻 , /8𝜂𝐿 * , where 𝐿 * = 𝐿 𝐷O𝑔(𝐿 𝐷 ⁄ ) ⁄ and 𝑔(𝐿/𝐷) = 𝑙𝑛(𝐿/𝐷) -0.662 + 0.917𝐷/𝐿 -0.050(𝐷/𝐿) , . In the previous equation, 𝜇 E is the permeability in vacuum, 𝐿 and 𝐷 the length and diameter of the wire, 𝐻 the magnetic excitation amplitude, Δ𝜒 = 𝜒 , (2 + 𝜒) ⁄ denotes the anisotropy of susceptibility between parallel and perpendicular directions and 𝜒 represents the material magnetic susceptibility. Fig. 3a displays the ratio 8𝜂𝜔 G /3𝜇 E Δ𝜒𝐻 , as a function of 𝐿 * for different Curosurf® samples with and without NPs. The definition for 𝜔 G is illustrated in the inset. The concentrations shown in the figure are 𝑐 31 = 0.037 and 0.086 g L -1 for alumina and 𝑐 31 = 0.088, 0.21 and 0.50 g L -1 for silica. The data points are found to collapse on a single master curve displaying the 1/𝐿 * , -dependence (continuous line). The agreement between data and theory is excellent. The measured viscosity is between 3 and 12 mPa s and its concentration dependence will be discussed in Section. III.4. Fig. 3b shows the oscillation amplitude angle 𝜃 H as a function of the reduced frequency 𝜔/𝜔 G for the same 6 samples (see inset for the 𝜃 H -definition). A good superimposition of the data points is observed and is in agreement with the Newtonian constitutive equation prediction which goes as 𝜃 H (𝜔/𝜔 G ) ~ (𝜔/ 𝜔 G ) +[ for 𝜔 ≫ 𝜔 G (continuous line). [START_REF] Chevry | Magnetic Wire-Based Sensors for the Microrheology of Complex Fluids[END_REF][START_REF] Helgesen | Nonlinear Phenomena in Systems of Magnetic Holes[END_REF] Note that in the 𝜃 H (𝜔/𝜔 G )-representation, the continuous line in the figure is obtained with no adjustable parameter. [START_REF] Loosli | Wire-Active Microrheology to Differentiate Viscoelastic Liquids from Soft Solids[END_REF][START_REF] Chevry | Magnetic Wire-Based Sensors for the Microrheology of Complex Fluids[END_REF] In conclusion to this part, it is found that Curosurf® loaded with silica behaves as a Newtonian fluid at all concentrations studied and that the dispersions are characterized by a single rheological parameter, the static shear viscosity 𝜂. A similar type of behavior can be observed for Curosurf® mixed with alumina at concentrations lower than 𝑐 31 = 0.1 g L -1 . In the previous samples, evidence of viscoelasticity was not found. [START_REF] Thai | On the Rheology of Pulmonary Surfactant: Effects of Concentration and Consequences for the Surfactant Replacement Therapy[END_REF][START_REF] Chevry | Magnetic Wire-Based Sensors for the Microrheology of Complex Fluids[END_REF]

II.3 -Alumina loaded surfactant showing a soft solid behavior

Here we describe experiments corresponding to the second generic behavior in which wires oscillate apart from a fixed orientation and have a zero-average velocity. Fig. 4a displays a chronophotograph of a 64 µm wire incorporated to a 44 g L -1 Curosurf® dispersion mixed with alumina particles at 𝑐 31 = 0.40 g L -1 (𝜔 = 0.3 rad s -1 , 𝜇 E 𝐻 = 10.3 mT). The corresponding movie can be watched in Supplementary Information, Movies#3&4. The images in Fig. 4a are recorded every 3.5 s and show that the wire now oscillates between two orientations, here 𝜃 = 85° (as shown in images 1, 4 and 7) and 𝜃 = 55° (images 2, 5 and 8). These back-and-forth oscillations remain steady over long period of time (> 1 h). The top red arrows indicate the counterclockwise rotation occurring at each oscillation. Fig. 4b, 4c and 4d display the orientation angle 𝜃(𝑡) at angular frequencies 0.06, 0.3 and 3 rad s -1 respectively. As shown by the straight lines in red, the average rotation frequency 𝛺(𝜔) is small (of the order of ±10 +_ rad s -1 ) and frequency independent. This inequality 𝛺(𝜔)/𝜔 ≪ 1 observed in these experiments is an indication that the material around the wires is a soft solid and characterized by a yield stress behavior. The results of Fig. 4 are similar to those found in calcium ion crosslinked polysaccharide gels and for which cone-and-plate rheometry evidenced a marked yield stress signature. 45 We now proceed to a quantitative analysis of the data obtained on the soft solid dispersions. Fig. 5a displays the oscillation amplitude 𝜃 H (𝜔) as a function of the frequency at 𝑐 31 = 0.19 g L -1 and for wires comprised between 28 and 92 µm. It is found that 𝜃 H (𝜔) exhibits a plateau at low frequency, noted 𝜃 H,abc and decreases above 1 rad s -1 . At 𝑐 31 = 0.49 g L -1 , the 𝜃 H (𝜔)-response consists essentially in a plateau and no high frequency decrease is observed (Supplementary Information S5). According to the constitutive equations developed for soft solids, [START_REF] Loosli | Wire-Active Microrheology to Differentiate Viscoelastic Liquids from Soft Solids[END_REF] the low frequency limit, here approximated to 𝜃 H,abc is linked to the equilibrium storage modulus 𝐺 ef through the relation lim j→E 𝜃 H (𝜔) = 3𝜇 0 Δ𝜒𝐻 2 /4𝐿 * , 𝐺 ef . For a soft solid, 𝐺 ef characterizes the quasistatic elastic response of the material, i.e. as 𝜔 goes to zero. Figs. 5b and5c show the 𝐿 * -dependence of the maximum angle for two alumina concentrations together with the predicted 𝜃 H,abc ~ 1/𝐿 * , scaling. [START_REF] Loosli | Wire-Active Microrheology to Differentiate Viscoelastic Liquids from Soft Solids[END_REF] The shaded area in the figures highlights the range over which the 1/𝐿 * ,prefactor varies in these experiments. These results suggest that the variability in the modulus could come from elasticity fluctuations at the scale of the wires. The value and physical interpretation of the elastic soft solid properties are discussed in the next section.

To further illustrate the soft solid behavior found for Al2O3 loaded dispersions, we investigated the transmitted light intensity with time-lapse microscopy in pristine and in processed samples. From their structures, the samples with and without NPs appear identical under the phase-contrast microscope (Fig. 5d). Strong differences are observed however with regards to the transmitted intensity light fluctuations (Fig. 5e). For Curosurf®, the vesicles are agitated with thermal motions, leading to noticeable fluctuations of the transmitted light. For Al2O3-loaded samples by contrast, the vesicles appear as stuck to each other and form large immobile clusters (Supplementary Information S6 and Movie#5). The observed vesicular frozen state is illustrated in Fig. 5e, which compares the fluctuation distributions for Curosurf® dispersions with and without alumina particles (𝑐 31 = 0.50 g L -1 ). The distributions were collected at five different locations on each sample by recording the transmitted light intensity as a function of the time and were later adjusted with a Gaussian function. Best fit calculations reveal that the fluctuation amplitudes are decreased by a factor of 10 for alumina compared to those of the initial dispersion and that the data for silica are close to those of pristine dispersion. These results illustrate quantitatively the existence of an arrested state induced by alumina particles, in agreement with the microrheology findings. In conclusion, we found that concentrated alumina loaded Curosurf® behaves as a soft solid material characterized by a yield stress and by an equilibrium storage modulus 𝐺 ef , the latter being measured. For these samples, the static viscosity 𝜂 is infinite. [START_REF] Larson | The Structure and Rheology of Complex Fluids[END_REF]

II.4 -Concentration dependences

Fig. 6 shows the concentration dependences of the rheological properties of Curosurf® loaded with silica and alumina NPs. The main frame displays the static viscosity 𝜂 as a function of the nanoparticle concentration 𝑐 31 (data from Section III.2). Below 10 -3 g L -1 , the viscosity remains at the level of the pristine fluid around 𝜂 = 9.0 ± 2.0 mPa s. This value agrees well with the recent measurements obtained on Curosurf® as a function of the lipid content. [START_REF] Thai | On the Rheology of Pulmonary Surfactant: Effects of Concentration and Consequences for the Surfactant Replacement Therapy[END_REF] With increasing 𝑐 31 , a similar behavior is observed with the two particles studied. The static viscosity decreases down to 5 ± 1 mPa s and for silica the tendency persists towards the highest concentrations. At 0.5 g L -1 , 𝜂(𝑐 31 ) has been reduced by a third, eventually reaching 3.5 ± 0.7 mPa s. The straight lines in the figure are power laws of the form 𝜂(𝑐 31 ) ~ 𝑐 31 +E.[m , indicating that the viscosity fall is slow and progressive. Note that for the non-interacting anionic silica, the viscosity remains concentration independent (Supplementary Information S7). The reduction of viscosity noted in Fig. 6 is a phenomenon that can be understood in the context of colloidal physics. Recent studies have found that for dense suspensions stabilized by short range repulsion (as it is the case for Curosurf® vesicles), the addition of weak attractive interaction leads to a decrease in viscosity. [START_REF] Minami | Rheological Properties of Suspensions of Thermo-Responsive Poly(N-Isopropylacrylamide) Microgels Undergoing Volume Phase Transition[END_REF][START_REF] Varga | Linear Viscoelasticity of Attractive Colloidal Dispersions[END_REF][START_REF] Weis | Microstructure, Local Dynamics, and Flow Behavior of Colloidal Suspensions with Weak Attractive Interactions[END_REF][START_REF] Weis | Effect of Weak Attractive Interactions on Flow Behavior of Highly Concentrated Crystalline Suspensions[END_REF] In these reports, the colloids are beads and microgels and the attraction is due to depleting non-adsorbing polymers. [START_REF] Minami | Rheological Properties of Suspensions of Thermo-Responsive Poly(N-Isopropylacrylamide) Microgels Undergoing Volume Phase Transition[END_REF][START_REF] Weis | Microstructure, Local Dynamics, and Flow Behavior of Colloidal Suspensions with Weak Attractive Interactions[END_REF][START_REF] Weis | Effect of Weak Attractive Interactions on Flow Behavior of Highly Concentrated Crystalline Suspensions[END_REF] It was found that a minute amount of depleting agents leads to the formation of large cohesive aggregates and thereby to a decrease of the effective volume fraction. As a result, the suspension bulk viscosity decreases, in proportions similar to those of Fig. 6. [START_REF] Varga | Linear Viscoelasticity of Attractive Colloidal Dispersions[END_REF][START_REF] Weis | Microstructure, Local Dynamics, and Flow Behavior of Colloidal Suspensions with Weak Attractive Interactions[END_REF] With NPs strongly interacting with vesicles (as shown in Fig. 1d and1e), results suggest that the attractive potential is not due here to depletion forces, but mediated by particles acting as physical links between vesicles. [START_REF] Mousseau | Supported Pulmonary Surfactant Bilayers on Silica Nanoparticles: Formulation, Stability and Impact on Lung Epithelial Cells[END_REF][START_REF] Mousseau | Biophysicochemical Interaction of a Clinical Pulmonary Surfactant with Nanoalumina[END_REF] However, in the low concentration regime (𝑐 31 < 0.1 g L -1 ), the particle shape and surface charge density do not play a major role, as the viscosities of silica and alumina loaded dispersion behave similarly. The second important result is the observation of a transition from a viscous liquid to a soft solid for alumina dispersions (inset of Fig. 6). The measured equilibrium modulus 𝐺 ef (𝑐 31 ) is found in the range 10 -2 -10 -1 Pa and increases with the concentration. At such values, the vesicle network is fragile and can be easily broken e.g. by the application of mild flow conditions. To explain the soft solid formation, we again exploit the colloid analogy and assume that with strongly interacting NPs, the vesicle dispersion undergoes a bond percolation transition. Such transitions are common in liquid-to-soft solid phenomena involving polymers or colloids. [START_REF] Larson | The Structure and Rheology of Complex Fluids[END_REF] To substantiate this assertion, we estimate the instantaneous elastic modulus 𝐺 E (𝑐 31 ) by assuming that all alumina particles added in the dispersion are active bonds and participate to the cross-linked network. In this case, 𝐺 E (𝑐 31 ) scales with the product 𝑛 31 𝑘 H 𝑇, where 𝑛 31 is the bond density, 𝑘 H the Boltzmann constant and 𝑇 the absolute temperature. [START_REF] Larson | The Structure and Rheology of Complex Fluids[END_REF] In regular soft solid, the instantaneous and equilibrium moduli are in principle different, but an estimation for 𝐺 E (𝑐 31 ) is sometimes useful to impart orders of magnitude for the elastic response. The straight line in the inset displays the linear behavior predicted for 𝐺 E (𝑐 31 ) and provides moduli in the range 10 -2 -10 -1 Pa, as for the experiments. Concerning the soft solid properties, it is possible that other mechanisms such as crowding or modification of the membrane elasticity contribute to the overall dispersion elasticity. The differences between silica and alumina particles are explained by the charge and shape of the particles. The silicas are spherical and weakly charged, whereas alumina NPs are in the form of platelets and strongly charged. Cryo-TEM images have also shown that the silica particles can be internalized in the vesicles, potentially reducing the number of effective bonds in the network formation. [START_REF] Mousseau | The Role of Surface Charge in the Interaction of Nanoparticles with Model Pulmonary Surfactants[END_REF] In conclusion, it is found that above a threshold concentration, the pulmonary surfactant Curosurf® loaded with alumina particles undergoes a rheological transition from a liquid to a soft solid state. These variations in viscosity are important for alveoli whose function is based on the flow properties of the surfactant layer, as they could cause pulmonary diseases.

III -Conclusion

In this work we study the rheological properties of the biomimetic pulmonary surfactant Curo-surf® in the presence of engineered nanoparticles. We a recent microrheology technique developed in-house and based on the remote actuation of micron-sized wires via the application of a rotating magnetic field. The wire microrheology is quantitative as it can measure the static shear viscosity and storage modulus of complex fluids. It also enables to differentiate between fluid-type and soft solid-type behaviors, [START_REF] Chevry | Magnetic Wire-Based Sensors for the Microrheology of Complex Fluids[END_REF] an outcome that is often hard to achieve with classical cone-and-plate rheometers. In a recent work we investigated the viscosity of Curosurf® dispersions as a function of the lipid concentration, and we found that in physiological conditions, the vesicle volume fraction is high and close to the liquid-to-soft solid transition. This finding prompted us to examine the effect of NPs on the surfactant rheology. Here, we extend the work on this biomimetic lung fluid and study its rheological behavior in presence of sub-100 nm NPs. Three types of particles are put under scrutiny, silica particles bearing either positive or negative charges at their surface and positively charged alumina. For particles strongly interacting with lipid membranes via electrostatic attraction, such as positive silica and alumina, significant changes to the dispersion viscosity are observed. For the positive silica, a reduction of the viscosity by a factor of 3 is found from 𝜂 = 9.0 ± 2.0 mPa s for pristine Curosurf® to 3.5 ± 0.7 mPa s at 𝑐 31 = 0.5 g L -1 . For alumina, we observe a liquid-to-soft solid transition at 𝑐 31 = 0.1 g L -1 that is interpreted in terms of a bond percolation transition, the particles acting as cross-links in regards to the vesicular network. These results show that inhaled particles interacting with lipid membrane can significantly alter the surfactant flow properties and vesicle reorganization taking place during breathing, a phenomenon that has not been evaluated yet.

IV -Materials and Methods

IV.1 -Materials

Nanoparticles Aluminum (Al2O3) and silicon oxide (SiO2) nanoparticle synthesis and characterization have been described recently. [START_REF] Mousseau | Supported Pulmonary Surfactant Bilayers on Silica Nanoparticles: Formulation, Stability and Impact on Lung Epithelial Cells[END_REF][START_REF] Mousseau | The Role of Surface Charge in the Interaction of Nanoparticles with Model Pulmonary Surfactants[END_REF][START_REF] Mousseau | Biophysicochemical Interaction of a Clinical Pulmonary Surfactant with Nanoalumina[END_REF][START_REF] Mousseau | Polyelectrolyte Assisted Charge Titration Spectrometry: Applications to Latex and Oxide Nanoparticles[END_REF] The NPs particles were chosen based on their importance in the nanomaterial production worldwide and on the inherent risks of inhalation exposure. A list of the production volumes and industrial applications for these NPs can be found in Supplementary Information S8. Positive silica were synthetized via the Stöber synthesis using tetraethyl orthosilicate silica precursor (TEOS, Aldrich). [START_REF] Mousseau | Supported Pulmonary Surfactant Bilayers on Silica Nanoparticles: Formulation, Stability and Impact on Lung Epithelial Cells[END_REF] Functionalization by amine groups was carried out, resulting in a positively charged coating. Dispersions were prepared at 40 g L -1 and diluted with DI-water at pH 5 for further use. The geometric diameter was determined from TEM at 𝐷 E = 41.2 nm (dispersity 0.11, Fig. 1a). Negative silica (trade name CLX®, diameter 20 nm, dispersity 0.20) were purchased from Sigma Aldrich at the concentration of 450 g L -1 . The batch was diluted down to 50 g L -1 and dialyzed against DI-water at pH 6.4 for 48 h. Alumina dispersions were prepared from Disperal® (SASOL, Germany) powder at the concentration of 10 g L -1 (pH 5) and thoroughly sonicated. TEM images show irregular platelets of length 40 nm and thickness 10 nm, the dispersity being 𝑠 = 0.30 (Fig. 1b). The dispersion stability was assessed using dynamic light scattering at pH 5 and 6.4 over extended periods of time (> months) [START_REF] Mousseau | The Role of Surface Charge in the Interaction of Nanoparticles with Model Pulmonary Surfactants[END_REF] and the hydrodynamic diameters were found to be time independent at 𝐷 7 = 64 nm, 60 nm and 34 nm, respectively. The surface charge density was determined using the polyelectrolyte assisted charge titration spectrometry at 𝜎 = +7.3e, +0.62e, and -0.31e nm -2 (Table I).

Table I:

List of nanoparticles and their characteristics. 𝐷 E and 𝐷 7 stand for the geometric and hydrodynamic diameters determined by transmission electron microscopy and dynamic light scattering. 𝑠 denotes the size dispersity (ratio between the standard deviation and average size of the distribution). The electrostatic charge densities 𝜎 was obtained using polyelectrolyte assisted charge titration spectrometry. [START_REF] Mousseau | Polyelectrolyte Assisted Charge Titration Spectrometry: Applications to Latex and Oxide Nanoparticles[END_REF] Pulmonary surfactant Curosurf® (Chiesi Pharmaceuticals, Parma, Italy) is a porcine minced pulmonary surfactant extract used in maternity hospitals for the treatment of premature newborns with respiratory distress syndrome. [START_REF] Curstedt | A Unique Story in Neonatal Research: The Development of a Porcine Surfactant[END_REF] It is produced as an 80 g L -1 whitish dispersion at pH 6.4 and contains phosphatidylcholine (PC) lipids, sphingomyelin(SM), phosphatidylethanolamine (PE), phosphatidylinositol (PI), phosphatidylglycerol (PG) and the hydrophobic proteins SP-B and SP-C. [START_REF] Braun | A Freeze-Fracture Transmission Electron Microscopy and Small Angle X-Ray Diffraction Study of the Effects of Albumin, Serum, and Polymers on Clinical Lung Surfactant Microstructure[END_REF][START_REF] Panaiotov | Effect of Hydrophobic Protein Sp-C on Structure and Dilatational Properties of The Model Monolayers of Pulmonary Surfactant[END_REF][START_REF] Camacho | Effect of Ph on the Interfacial Adsorption Activity of Pulmonary Surfactant[END_REF] Curosurf® is considered as a model biomimetic surfactant because its composition and bulk properties are constant from one batch to another and because of its long-term shelf stability. The Curosurf® formulation is compared to pulmonary surfactant obtained from lung lavage in Supplementary Information S1. The main differences between the two formulations are the percentages in sphingomyelin, in cholesterol and in phosphatidylglycerol, as well as the absence of the hydrophilic proteins, SP-A and SP-D. 75 µl aliquots were frozen, stored at -20 °C and thawed before use. Curosurf® was kindly provided by Dr. Mostafa Mokhtari and his team from the neonatal service at Hospital Kremlin-Bicêtre, Val-de-Marne, France and used as received or diluted in phosphate buffer saline (PBS, Aldrich). Recently, we used the microrheology MRS technique described below to measure Curosurf® viscosity as a function of the volume fraction [START_REF] Thai | On the Rheology of Pulmonary Surfactant: Effects of Concentration and Consequences for the Surfactant Replacement Therapy[END_REF] and observed an exponential increase in the range 0 -80 g L -1 . The 𝜂(𝜙)-data are provided in Supplementary Information S9.

Sample preparation

For the evaluation of the interaction strength parameters (data from Fig. 1), dilute stock solutions of surfactant and NPs were prepared in the same conditions of pH and concentration. The dispersions were then mixed at different ratios 𝑋 = 𝑐 12 /𝑐 31 following the continuous variation protocol. [START_REF] Mousseau | The Role of Surface Charge in the Interaction of Nanoparticles with Model Pulmonary Surfactants[END_REF][START_REF] Job | Studies on the Formation of Complex Minerals in Solution and on Their Stability[END_REF][START_REF] Renny | Method of Continuous Variations: Applications of Job Plots to the Study of Molecular Associations in Organometallic Chemistry[END_REF] The total concentration was kept constant at 𝑐 = 𝑐 12 + 𝑐 31 = 0.1 or 1 g L -1 , i.e. below the lipid physiological concentration. For alumina, the pH of the stock solution was adjusted at pH 5 to ensure that particles do not aggregate as a result of pH changes. Positive and negative silica were studied at the surfactant physiological pH. For microrheology experiments, the Chiesi formulation at 𝑐 12 = 80 g L -1 was used as received and diluted by addition of a NP dispersion in PBS, ensuring physiological conditions of pH and ionic strength. In the end, the lipid concentration was at 𝑐 12 = 44 g L -1 and that of the nanoparticles was varied from 𝑐 31 = -3 to 0.50 g L -1 . From these values, it is possible to derive the total mass present in the alveolar region, a quantity that is of interest for the comparison with actual exposure data. Assuming for the human alveolar region a total volume of pulmonary surfactant of 25 mL, [START_REF] Notter | Lung Surfactant: Basic Science and Clinical Applications[END_REF][START_REF] Schleh | The Effect of Titanium Dioxide Nanoparticles on Pulmonary Surfactant Function and Ultrastructure[END_REF] we found that the lowest concentration used, 𝑐 31 = 10 -3 g L -1 corresponds to 24 µg of NPs in the entire alveolar region, and that the first effect on the viscosity, observed around 10 -2 g L -1 corresponds to 240 µg. These two values are associated with a total number of particles per alveolus of 1000 and 10000 respectively. Detailed calculations, including those for alumina and negative silica are given in Supplementary Information S10.

IV.2 -Methods

Optical and cryo-transmission electron microscopy

Phase-contrast and bright field images were acquired on an IX73 inverted microscope (Olympus) equipped with 20× and 100× objectives. An EXi Blue camera (QImaging) and Metaview software (Universal Imaging Inc.) were used as acquisition system. For cryo-TEM, few microliters of a 𝑐 12 = 5 g L -1 Curosurf® dispersion were deposited on a lacey carbon coated 200 mesh (Ted Pella Inc.). The drop was blotted with a filter paper using a FEI Vitrobot TM freeze plunger. The grid was then quenched rapidly in liquid ethane to avoid crystallization and later cooled with liquid nitrogen. [START_REF] Dubochet | On the Development of Electron Cryo-Microscopy (Nobel Lecture)[END_REF] The membrane was transferred into the vacuum column of a JEOL 1400 TEM microscope (120 kV).

Active microrheology (including sample preparation)

The magnetic wire microrheology technique has been described in previous accounts. [START_REF] Berret | Local Viscoelasticity of Living Cells Measured by Rotational Magnetic Spectroscopy[END_REF][START_REF] Loosli | Wire-Active Microrheology to Differentiate Viscoelastic Liquids from Soft Solids[END_REF][START_REF] Chevry | Magnetic Wire-Based Sensors for the Microrheology of Complex Fluids[END_REF] In brief, wires were synthesized by electrostatic co-assembly of 6.7 nm iron oxide NPs and with poly(diallyldimethylammonium chloride) (PDADMAC, Aldrich, 𝑀 x > 100000 g mol -1 ) 67 (Supplementary Information S10). The magnetic wires used in this study have lengths between 5 and 100 µm and diameters between 0.6 and 2 µm. The wire diameters were determined independently by optical microscopy and scanning electron microscopy, leading to the expression, 𝐷(𝐿) = 0.619𝐿 E.,E, (Supplementary Information S11). As already mentioned, the NP-lipid dispersions were prepared at 𝑐 12 = 44 g L -1 and 𝑐 31 = 10 -3 and 0.50 g L -1 . A volume of 0.5 µL containing 10 5 wires in PBS was then added to the previous dispersion and gently stirred. 25 µL of this dispersion were then deposited on a glass plate and sealed into a Gene Frame ® (Abgene/Advanced Biotech, dimensions 10×10×0.25 mm 3 ). The glass plate was introduced into a homemade device generating a rotational magnetic field, thanks to two pairs of coils (23 Ω) working with a 90°-phase shift. An electronic set-up allowed measurements in the frequency range 𝜔 = 10 -3 -10 2 rad s -1 and at magnetic fields 𝜇 E 𝐻 = 0 -15 mTesla. The microrheology protocol used is based on the Magnetic Rotational Spectroscopy technique (Supplementary Information S12). [START_REF] Gu | Ferromagnetic Nanorods in Applications to Control of the In-Plane Anisotropy of Composite Films and for In Situ Characterization of the Film Rheology[END_REF][START_REF] Loosli | Wire-Active Microrheology to Differentiate Viscoelastic Liquids from Soft Solids[END_REF][START_REF] Tokarev | Nanostructured Soft Matter with Magnetic Nanoparticles[END_REF][START_REF] Chevry | Magnetic Wire-Based Sensors for the Microrheology of Complex Fluids[END_REF] For each condition of magnetic field and angular frequency, a movie was recorded for a period of time of 10/𝜔 or longer and then treated using the ImageJ software (https://imagej.nih.gov/ij/). For calibration, MRS was performed on a series of water-glycerol mixtures of increasing viscosities, 4.95, 34.9, 48.9 and 80.0 mPa s, corresponding to glycerol concentrations of 49.8%, 81.0%, 84.5% and 89% (T = 25 °C), leading to a susceptibility anisotropy coefficient Δ𝜒 = 0.054 ± 0.006 (Supplementary Information S13). [START_REF] Thai | On the Rheology of Pulmonary Surfactant: Effects of Concentration and Consequences for the Surfactant Replacement Therapy[END_REF] microscopy -S13: Magnetic field rotating device and spatial distribution of the magnetic field -S14: Calibrating the magnetic wires using water-glycerol solutions of different viscosity, leading to the determination of the magnetic susceptibility anisotropy.

TOC Image

Figure 1 :

 1 Figure 1: a, b) Transmission electron microscopy (TEM) images of positively charged silica and alumina nanoparticles. Their sizes are respectively 42 nm and 40 nm (platelet diameter) and their size dispersities (ratio between the standard deviation and the mean) are 𝑠 = 0.11 and 0.30. c) Cryogenic TEM images of native Curosurf® obtained from a 5 g L -1 dispersion at 25 °C. The arrows are pointing to multi-lamellar vesicles and multi-vesicular vesicles. Additional images are available in Supplementary Information S4. d, e) Hydrodynamic diameter of nanoparticle-vesicle dispersions as a function of the concentration ratio 𝑋 = 𝑐 12 /𝑐 31 obtained from dynamic light scattering at 𝑇 = 25 °C for silica (d) and for alumina (e). In the figures the lipid and nanoparticle weight concentrations as a function of 𝑋 read: 𝑐 12 (𝑋) = 𝑋𝑐/(1 + 𝑋) and 𝑐 31 (𝑋) = 𝑐/(1 + 𝑋), with 𝑐 being the total concentration. To avoid multiple scattering, the total active concentration was adjusted at 𝑐 = 1 g L -1 for silica and 𝑐 = 0.1 g L -1 for alumina. Inset: Optical microscopy images of mixed nanoparticle-vesicle aggregates. The images were obtained from dispersions at 𝑋's corresponding to the maximum seen by light scattering.

Figure 2 :

 2 Figure 2: a) Chronophotograph of a 43 µm wire undergoing a 𝜋-rotation (angular frequency 0.06 rad s -1 , magnetic field 𝜇 E 𝐻 = 10.3 mT) in a 44 g L -1 pulmonary surfactant dispersion loaded with alumina NPs (𝑐 31 = 0.004 g L -1 ). The time interval between two images is 3.5 s. The upper arrows indicate the direction of rotation (here clockwise). b) Time dependence of the rotation angle 𝜃(𝑡) in the synchronous regime (𝜔 = 0.06 rad s -1 ) observed in the sample of Fig. 2a. c and d) Same as in Fig. 2b for the asynchronous regime at 𝜔 = 0.3 and 3 rad s -1 . The average rotation frequency 𝛺(𝜔) is defined from the straight lines indicated in red.

Figure 3 :

 3 Figure 3: a) Normalized critical frequency 8𝜂𝜔 G /3𝜇 E 𝛥𝜒𝐻 , as a function of the reduced wire length 𝐿 * obtained for Curosurf® dispersions with and without particles (𝑇 = 25 °C). The Curo-surf® concentration is set at the physiological value, 44 g L -1 and the nanoparticle concentrations are 𝑐 31 = 0.037 to 0.086 g L -1 for alumina and 0.088 to 0.50 g L -1 for silica. The parameters featuring in the normalized critical frequency are given in the text. The straight line displays the 1/𝐿 * , -dependence predicted from the viscous fluid constitutive equation. Inset: Angular frequency dependence of the average rotation velocity 𝛺(𝜔) illustrating the definition of 𝜔 G . b) Oscillation amplitude 𝜃 H (𝜔/𝜔 G ) observed in the asynchronous regime for the same samples as in a) The straight line is for Newtonian fluids. Inset: Time-dependent rotation angle of a wire in the asynchronous regime illustrating the definition of 𝜃 H .

Figure 4 :

 4 Figure 4: a) Chronophotograph of a 64 µm wire undergoing back-and-forth oscillations in the mimetic pulmonary surfactant Curosurf® loaded with alumina NPs (𝑐 31 = 0.40 g L -1 ). The angular frequency is set at 0.3 rad s -1 and the magnetic field at 10.3 mT. Time interval between two images is 3.5 s. The upper grey (resp. red) arrows indicate a clockwise (resp. anticlockwise) movement of the wire. b, c and d) Time dependences of the rotation angle 𝜃(𝑡) in the sample shown in a) at 𝜔 = 0.06, 0.3 and 3 rad s -1 , respectively. The average rotation frequency 𝛺(𝜔) is defined from the straight lines indicated in red. Movies corresponding to Figs. 4b and 4d are in Supplementary Information, Movies#3&4).

Figure 5 :

 5 Figure 5: a) Oscillation amplitude 𝜃 H (𝜔) measured in Curosurf® loaded with alumina particles (𝑐 31 = 0.19 g L -1 ) using wires of different lengths. The arrows at low frequencies indicate the maximum values 𝜃 H,abc detected in frequency sweeps. b, c) 𝜃 H,abc (𝐿 * ) for Curosurf® dispersions loaded with alumina particles at 𝑐 31 = 0.19 and 0.49 g L -1 respectively, 𝐿 * being the reduced wire length defined in the text. The thick red line indicates the 1/𝐿 * , -behavior expected from the soft solid model. 45 The orange areas denote the range over which the 1/𝐿 * , -prefactor varies in these experiments. These ranges are associated with a standard error of 25%. d) Phase contrast microscopy images of Curosurf® samples without and with alumina particles. The squares indicate spatial domains where the transmitted light intensity was collected as a function of the time. e) Distribution of the transmitted light fluctuation amplitudes for Curosurf® and Curosurf® treated with alumina NPs (𝑐 31 = 0.50 g L -1 ). The data in grey are distributions taken in 5 different locations, and the continuous curves are best fit calculations using a Gaussian function. The standard deviations for the continuous lines in blue and red are 0.38 and 0.05, respectively.

Figure 6 :

 6 Figure 6: Static shear viscosity of Curosurf® loaded with silica (red symbols) and alumina (blue symbols) as a function of the nanoparticle concentration 𝑐 31 . The straight lines in grey are least squared calculations using power laws of the form 𝜂(𝑐 31 ) ~ 𝑐 31 +n with 𝛼 ~ 0.15. The vertical arrow shows the transition between a liquid and a soft solid behavior. Inset: Equilibrium elastic modulus 𝐺 ef for alumina loaded surfactant above 𝑐 31 = 0.1 g L -1 . The straight line is calculated from the concentration dependence of the instantaneous elastic modulus assuming that all alumina particles are active bonds in the percolated vesicular network.

  [START_REF] Mousseau | Polyelectrolyte Assisted Charge Titration Spectrometry: Applications to Latex and Oxide Nanoparticles[END_REF]

	Nanoparticle	𝑫 𝟎 (nm)	𝒔	𝑫 𝑯 (nm)	𝝈 (nm -2 )
	Alumina	40	0.30	64	+7.3e
	Silica (positive)	42	0.11	60	+0.62e
	Silica (negative)	20	0.20	34	-0.31e

Acknowledgments

We thank Armelle Baeza-Squiban, Victor Baldim, Yong Chen, Marcel Filoche, Mélody Merle, Mostafa Mokhtari and Chloé Puisney for fruitful discussions. Imane Boucema is acknowledged for letting us use the Anton Paar rheometer for the cone-and-plate rheology. We also thank Stéphane Mornet from the Institut de Chimie de la Matière Condensée de Bordeaux (Université Bordeaux 1) for the synthesis of the aminated silica nanoparticles. ANR (Agence Nationale de la Recherche) and CGI (Commissariat à l'Investissement d'Avenir) are gratefully acknowledged for their financial support of this work through Labex SEAM (Science and Engineering for Advanced Materials and devices) ANR 11 LABX 086, ANR 11 IDEX 05 02. We acknowledge the Ima-goSeine facility (Jacques Monod Institute, Paris, France), and the France BioImaging infrastructure supported by the French National Research Agency (ANR-10-INSB-04, « Investments for the future »). This research was supported in part by the Agence Nationale de la Recherche under the contract ANR-13-BS08-0015 (PANORAMA), ANR-12-CHEX-0011 (PULMONANO), ANR-15-CE18-0024-01 (ICONS), ANR-17-CE09-0017 (AlveolusMimics) and by Solvay.

Supplementary Information

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsnano/xxxxxx S1: Curosurf® composition and comparison with the native surfactant -S2: Fluid-to-Gel transition from the Curosurf® phospholipid bilayer studied using by differential scanning calorimetry -S3: Dynamic light scattering on Curosurf®/negative silica NPs showing the absence of interaction between the two species -S4: Additional cryo-TEM images of Curosurf® vesicles -S5: Additional oscillation amplitude data 𝜃 H obtained for Curosurf® loaded with alumina particles -S6: Transmitted light intensity for Curosurf® without and with NPs measured by phase contrast optical microscopy -S7: Mircrorheology data for Curosurf® treated with negatively charged silica -S8: List of production volumes and industrial applications for silica and alumina nanoparticles for 2012 -S9: Curosurf® viscosity as a function of the vesicle volume fraction -S10: Correspondence between the NP concentration in Curosurf® and the amount of nanoparticles in the alveolar region -S11: Wire synthesis scheme -S12: Magnetic wire characterization using optical


Jean Boulvert
email: jean.boulvert@univ-lemans.fr

Josué Costa-Baptista

Théo Cavalieri

Maxime Perna

Edith Roland Fotsing

Vicente Romero-García

Gwenael Gabard

Annie Ross
email: annie.ross@polymtl.cac

Jacky Mardjono
email: jacky.mardjono@safrangroup.comd

Jean-Philippe Groby
email: jean-philippe.groby@univ-lemans.fr

Gwénaël Gabard

Acoustic modeling of micro-lattices obtained by additive manufacturing

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

Additive manufacturing has proved to be a versatile and promising tool to produce sound absorbing materials. Helmholtz resonators 1 , straight 2 , inclined 3 and coiled tubes 1 , sonic crystals 4 , micro perforated panels 5 and porous treatments such as micro-lattices 68 are among some interesting applications. For all these acoustic materials, simple analytical models can be used to describe, analyze and optimize the geometry. Numerical or analytical models [START_REF] Auriault | Homogenization of coupled phenomena in heterogenous media[END_REF],10 showed that the acoustic parameters of the porous material strongly depend on several geometric parameters of the micro-structure and can be strongly aected by the manufacturing, such as change in the laments shape of micro-lattice 11 , presence of sinusoidal rugosity in micro slits [START_REF] Song | Modeling of roughness eects on acoustic properties of micro-slits[END_REF] , and surface irregularities in packed micro tubes 13 .

Therefore, the impact of additive manufacturing imprecision on acoustic parameters could be very important and needs to be well described. Usually discrepancies between models and experimental results are attributed to the additive manufacturing inaccuracy or its inability to reproduce an idealized geometry [START_REF] Zielinski | Pore-size eects in sound absorbing foams with periodic microstructure: modelling and experimental verication using 3d printed specimens[END_REF][START_REF] Deshmukh | Design of periodic foam structures for acoustic applications: Concept, parametric study and experimental validation[END_REF] .

In this work a combined numerical and experimental approach is used to describe the dierences between a 3D printed micro-lattice and its idealized representation in terms of the geometry and the acoustic response. The 3D printed micro-lattices can be acoustically described by the Johnson-Champoux-Allard-Lafarge (JCAL) model [START_REF] Lafarge | Dynamic compressibility of air in porous structures at audible frequencies[END_REF] which accounts for the thermal and viscous losses in the eective bulk modulus and density, respectively. They are expressed in terms of air parameters and six parameters depending on the micro-structure of the porous medium. These JCAL parameters can be obtained by applying inverse characterization techniques from the measurements of the scattering coecients of a layer of porous material. In particular, the method proposed by Niskanen et al.

17 can be used. This method is based on a statistical inversion using Bayes' formula and does not require any previous knowledge on the sample except its thickness. Moreover, in the particular case where a

Representative Elementary Volume (REV) having a subwavelength scale dimension can be dened, these JCAL parameters can be also numerically computed using the Multi-Scale Asymptotic Method (MAM) [START_REF] Auriault | Homogenization of coupled phenomena in heterogenous media[END_REF] . In the case of periodic porous medium such as micro-lattices, the REV is the constitutive unit cell (an elementary periodic volume). Through these two techniques, the eects of the manufacturing defects on the acoustic response of the 3D printed material are analyzed. The methodology presented in this work can be extended to other 3D printed porous materials of dierent micro-structure or manufacturing process.

The article is organized as follows. First the micro-lattice fabrication is presented. The additive manufacturing procedure, used to create the micro-structure that governs the material acoustic behavior is detailed. Then, corresponding expected and actual micro-lattices are analyzed. The discrepancies are identied in terms of micro-geometry dierences. In the second part, the acoustic simulation and experimental characterization procedures are explained. Finally, the acoustic impact of the additive manufacturing defects is analyzed.

The procedure that must be applied in order to obtain an accurate acoustic prediction model is presented.

II. IDENTIFICATION OF THE MANUFACTURING DEFECTS

In this section, the manufacturing process and the expected resulting micro-lattices are described. Then, the manufacturing induced defects are analyzed on the basis of geometrical observations.

A.

Additive manufacturing procedure

The Fused Deposition Modeling (FDM) Pro2 printer supplied by RAISE3D is used to produce 30 mm thick cylindrical samples with a diameter of 30 mm. The extruded material is polylactic acid (PLA). The diameter of the nozzle is D n = 400 µm. The revolution axis of the sample is placed vertically. The nozzle temperature is set to 200 • C; its displacement speed is 60 mm.s -1 and the extrusion multiplier is 0.9. The slicing software Simplify3D is used to generate the les providing the instructions to the printer.

The samples are printed in orthogonal direct conguration: any given micro-layer is made of parallel rods; the next micro-layer of parallel rods is orthogonal to the previous layer.

This conguration is also known as "woodpile" or "interwoven perpendicular arrangement".

The printer nozzle extrudes a layer of rods in the horizontal plane, then the build plate shifts vertically and allows the next orthogonal layer to be printed. The build plate displacement is set to 300 µm. This way, two consecutive orthogonal layers are separated by 300 µm and two consecutive parallel layers are separated by 600 µm. Since the lament diameter D is close to D n , the contact between two layers is substantial as it can be seen in the idealized geometry shown in Fig. 1.

The in-plane spacing between the laments, i.e., the lattice constant of the micro-lattice, a, can also vary. A normalized lattice parameter is dened as A = a/D (A ≥ 1). Samples are homogeneous if A is kept constant trough the sample's thickness, or multilayered if

A varies within the thickness. An intuitive representation of the micro-lattice would be a superposition of perfectly cylindrical parallel rods, alternating in orthogonal in-plane directions, with a 300 µm spacing between layers (Fig. 1). In this case, the rods diameter would be equal to the nozzle diameter: D = D n .

The Simplify3D slicing software does not allow to directly control the lattice constant a.

The manufacturing variable that regulates the lattice constant is the "inll factor" (I F ).

It is inversely proportional to the spacing between two adjacent laments and can only take integer percent values. The relation between the inll factor, IF , and normalized lattice parameter, A, is simply obtained from the printer's instruction les and veried experimentally IF (%) = 100/A.

(

) 1
Two pictures of a representative homogeneous sample are presented in Fig. 2. To the naked eye, the printed geometry is consistent with the idealized one. The laments seem to be well aligned and no macroscopic defect can be seen. It is worth noting that a 800 µm thick solid layer surrounds the porous micro-lattice to provide consistency during the acoustic measurements.

B. Idealized Representative Elementary Volume

From the idealized geometry (see Fig. 1), an indivisible periodic pattern, called "unit cell", can be extracted. The idealized unit cell is represented in Fig. 3. In the homogenization theory 9 , the unit cell plays the role of the "Representative Elementary Volume" (REV)

because the porous medium is structured in a regular pattern. More details are given below on how this theory is used to predict a porous material's acoustic behavior. C.

Manufacturing geometry inherent defects

The printed samples were analyzed by means of both optical microscope and scanning electron microscope (SEM) in order to identify the dierent possible manufacturing geometrical defects. The optical microscope has a 64× zoom and is equipped with a digital camera for the acquisition of images. The SEM produces 2D images of the sample's surface (i.e., the depth of surface irregularities cannot be assessed). The accuracy of SEM enables distinguishing details smaller than 1 µm. Samples were not coated since their conductivity was high enough to obtain high-quality images. Three types of defects were identied.

Filament elliptical section

Figure 4 shows an image obtained from the optical microscope of a manually sliced sample.

The image reveals two important observations. First, the lament section is not perfectly circular but rather elliptical with an horizontal major axis. The attening of the micro-rods depends on the printing parameters. In our case, the minor axis represents in average 92% of the major axis; this leads to lament width equal to the nozzle diameter in the horizontal direction (400 µm ) and 92% of the nozzle diameter (368 µm ) in the vertical direction. This defect does not depend on the lattice constant. The second important observation is that the spacing between two consecutive orthogonal layers is respected and is equal to 300 µm.

Filament section shrinkage

Due to the extrusion process, the laments are subjected to section shrinkage. The lament is extruded by the printer then pulled by the nozzle before touching a rod from the previous layer. The shrinkage between two junctions is clearly visible as shown in Fig. 5:

the section of the laments decreases between two consecutive junctions in the direction of the material deposition. Figure 5 shows that the widths D1 and D1

MFWP = D n (0.8A -1), A ∈ [1.5; 11]. (2)
In Fig. 5, D1 is a measurement of the M F W at the M F W P . These relations are used in Section IV-A to account for this manufacturing defect in the simulations. They are derived based on the microscope measurement of 10 samples with A varying between 1.5 and 11.

The measurement of M F W is only satisfactory when A is higher than 2. Below value of lattice constant, the shrinkage is too small to be correctly measured.

Filament micro-grooves

The bright lines appearing on the microscopic images of micro-lattices (see Fig. 5) draw attention to nano-scale defects. The surface of the rods is not perfectly smooth. Figure 6 shows the image of a SEM analysis revealing impurities scattered all over the lament surface. Moreover, many micro-grooves are clearly identiable. These grooves are aligned with the lament direction. Their width ranges between 5 and 20 µm. However, their exact distribution and depth could not be assessed from the SEM images. A cross-sectional view would not reveal more details of these grooves. In fact, slicing the sample always has the detrimental eect of damaging the cross-section surface of the lament. A simple explanation for the grooves origin would be that the printer nozzle's inner surface is not smooth.

As a consequence, this defect might be specic to the printer nozzle and its state of wear. FDM 3D printer nozzles are mainly made of brass, stainless and hardened steel or special materials such as ruby. The resistance to abrasion can dier from one material to another.

Thus, the type and wear speed of a nozzle depend on both its constitutive material and the nature of the extruded material (PLA, Nylon...). The manufacturing technique of a nozzle also depends on the manufacturer. Among others, the shape of the hole of a brand-new nozzle depends on the drilling technique and on the type of post-drilling coating applied.

Acoustic model for porous media

The energy of acoustic waves penetrating a porous medium is dissipated mainly through the interaction of the wave between the frame and the pore saturating uid (in our case air)

resulting in viscous and thermal losses. If the skeleton is suciently dense and rigid, it can be decoupled from the air. Then, the porous medium can be considered as an equivalent uid characterized by its complex and frequency dependent eective bulk modulus K eq and density ρ eq . The properties of the equivalent uid depend on the micro-structure of the porous medium and govern its acoustic behavior 18 . There are three options to predict the properties of a porous medium's equivalent uid. The rst one consists in using empirical laws correlating a few parameters 18 . However, these laws only apply to small ranges of mate- rials, similar to each other. The second option can be applied if the porous medium is made of aligned, non-crossing cylindrical micro-rods 4 . For this precise case, an analytical solution has been derived for porosity larger than 0.7, from a semi-phenomenological approach 18 .

This kind of model is not limited to a few applications, but it requires more parameters than empirical or analytical laws. This is the case for the JCAL model 16 that provides the expression of the density and bulk modulus of the porous material as follows

ρ eq (ω) = ρ 0 φ α(ω), (4)
K eq (ω) = γP 0 φ γ - γ -1 α (ω) -1 , ( 5
)
where ω is the angular frequency, ρ 0 the density of the saturating uid, i.e., the air medium, P 0 the static pressure, γ the specic heat ratio, φ the porous material porosity, α(ω) its dynamic tortuosity, and α (ω) its thermal tortuosity. The two latter are dened as

α(ω) = α ∞ - iν ω φ q 0 1 + iω ν 2α ∞ q 0 φΛ 2 , (6)
α (ω) = 1 - iν ω φ q 0 1 + iω ν 2q 0 φΛ 2 , (7)
where ν = η/ρ 0 is the kinematic viscosity of the saturating uid, η its dynamic viscosity, ν = ν/P r, P r the Prandtl number, α ∞ the porous material geometrical tortuosity, Λ its viscous characteristic length, Λ its thermal characteristic length, q 0 its visco-static permeability and q 0 its thermo-static permeability.

Once the equivalent uid parameters are known, the acoustic behavior of the considered porous medium can be predicted. The equivalent uid wave number k eq and characteristic impedance Z eq take the forms

k eq = ω ρ eq K eq , (8)
Z eq = ρ eq K eq .

(

) 9
The acoustic behavior of a porous medium is then described by its six JCAL parameters:

φ, α ∞ , Λ, Λ , q 0 and q 0 .

Transfer Matrix Method

The Transfer Matrix Method (TMM) is a well-known method used to predict the acoustic behavior of a monolayer or multilayer sound absorber [START_REF] Allard | Propagation of sound in porous media: modelling sound absorbing materials[END_REF] . In section V, absorption coecients of monolayer and multilayer sound absorbers are predicted by the TMM. The transfer matrix of a porous layer is expressed in terms of its thickness, equivalent uid wave number and characteristic impedance. The total transfer matrix of a multilayer is the product of the layers' matrices and leads to the surface impedance Z s of the multilayer. The sound absorption coecient A of a monolayer or a multilayer can be calculated by

A = 1 - Z s -Z 0 Z s + Z 0 2 , (10)
where Z 0 is the acoustic impedance of the air. As stated above, the porous samples are framed by a thin solid circular wall. Its eect is considered by correcting the surface impedance by the ratio S sample /S porous 18 . In Eq. ( 10), Z s must be replaced by Z s * dened as

Z s * = Z s S sample S porous . ( 11
)
The radiation eect of the solid wall is negligible.

Porous media: Multi-scale Asymptotic Method

In this work MAM 9 is used to numerically evaluate the JCAL parameters of a porous material made from the idealized REV. This method solves fundamental equations in the REV's uid domain and then computes the JCAL parameters. The MAM can be implemented in a Finite Element Method (FEM) code from which the JCAL parameters are computed for any open-cell provided that the scales separation 10 is respected. It is a well suited method for the numerical evaluation of micro-geometry defects because the defects can be controlled and considered separately or simultaneously.

The MAM is implemented in the commercial FEM software Comsol Multiphysics. Three problems, Eqs. [START_REF] Song | Modeling of roughness eects on acoustic properties of micro-slits[END_REF][START_REF] Kulpe | Computation of acoustic absorption in media composed of packed microtubes exhibiting surface irregularity[END_REF][START_REF] Zielinski | Pore-size eects in sound absorbing foams with periodic microstructure: modelling and experimental verication using 3d printed specimens[END_REF] are solved in order to retrieve the six JCAL parameters dening the porous medium's equivalent uid when the pressure wave travels in the z direction (Fig. 3). The thermal problem equation, taking the limit in ω → 0, reads

         div(grad(θ 0 )) = -1, θ 0 = 0 on Γ f s , θ 0 Ω-periodic, (12)
where θ 0 plays the role of the temperature eld, Γ f s is the uid solid interface and Ω the REV. The visco-inertial problem, taking the limit in ω → 0, is

                   div(grad(k 0 z )) = ∂ξ 0 j ∂z -1, ∂k 0 z ∂z = 0, k 0 z = 0 on Γ f s , ξ 0 z Ω = 0, k 0 z and ξ 0 Ω-periodic, (13)
where • Ω is the REV averaging. The visco-inertial problem, taking the limit in ω → ∞, becomes

                   iωρ 0 η k ∞ z = ∂ξ ∞ z ∂z -1, ∂k ∞ z ∂z = 0, k ∞ z • n = 0 on Γ f s , ξ ∞ z Ω = 0, k ∞ z and ξ ∞ z Ω-periodic, ( 14
)
where n is the solid surface outer normal vector. In Eqs. [START_REF] Kulpe | Computation of acoustic absorption in media composed of packed microtubes exhibiting surface irregularity[END_REF][START_REF] Zielinski | Pore-size eects in sound absorbing foams with periodic microstructure: modelling and experimental verication using 3d printed specimens[END_REF] k z plays the role of the z component of the velocity eld and ξ z of its associated pressure.

The JCAL parameters are obtained by integrating the solution elds of these equations, over the uid domain Ω f or uid-solid interface Γ f s . They are expressed as

φ = Ω f dΩ f Ω dΩ , α ∞ = φ 1 - ∂ξ ∞ z ∂z -1 , Λ = 2 Ω f k ∞ z • k ∞ z dΩ f Γ f s k ∞ z • k ∞ z dΓ f s , Λ = 2 Ω f dΩ f Γ f s dΓ f s , q 0 = k 0 z Ω , q 0 = Ω f θdΩ f Ω f dΩ f . B.

Acoustic experimental characterization

The acoustic impact of the geometry defects can be assessed numerically by means of the MAM. As a complement to this numerical approach, the samples are also characterized by using experimental acoustic techniques.

Acoustic direct characterization

The acoustic scattering parameters of the samples are measured using a 30 mm diameter impedance tube following the ASTM E1050 and ISO 10534-2 standards. The twomicrophones technique with hard backing conguration is used for the absorption coecient measurement. The data required for the inverse characterizations (see below) are obtained using the four-microphones technique with an anechoic termination. The measurements are made between 500 Hz and 6000 Hz.

Homogeneous samples are tested in both direct and reverse orientations to make sure they are symmetric and thus, most likely homogeneous. Each sample is measured three times in both orientations. For each measurement, the sample is removed and installed again in the tube. The absorption curves are always superimposed, regardless of the measurement orientation.

Acoustic inverse characterization

While the MAM can be used to compute the JCAL parameters from a REV description, the measured reection and transmission coecients of an homogeneous porous layer can also be used to recover its JCAL parameters. Diverse methods have been discussed in the literature; in the present work, the inverse method proposed by Niskanen et al.

17 is used. To perform the procedure, a sample is tested in a four-microphones impedance tube to measure the reection coecient on both sides along with the transmission coecient. During the inversion, the JCAL parameters are bounded to realistic values (eg. φ ∈ [0; 1]) and forced to respect the natural physical conditions Λ ≤ Λ and q 0 ≤ q 0 . In some cases, the reection and transmission coecients are strongly correlated to a subset of JCAL parameters while the other parameters have a small impact. Therefore, precise inverse characterization of the JCAL parameters is complicated. Adjusting parameters bounds to account for their prior knowledge, obtained from the numerical model, limits the eld of inverse characterization research and leads to more accurate values. This way, the retrieved JCAL parameters follow realistic trends, as a function of lament spacing.

IV. EFFECT OF MANUFACTURING DEFECTS ON THE JCAL PARAMETERS: NUMERICAL EVALUATION

Three types of defects inherent to additive manufacturing been identied. It is suspected that their impact on the acoustic response depends on the spacing of the rods. In this section, their impact is numerically studied for the range of A ∈ [1.2; 25].

A. Elliptical section and section shrinkage

The impact of the elliptical section and shrinkage of the laments are analyzed in this section by modifying the idealized REV. The micro-lattice composed of 400 µm perfectly cylindrical rods is taken as a reference. The considered elliptical section has a 368 µm vertical minor axis and a 400 µm horizontal major axis. The geometry of the shrinkage is described by Eqs. ( 2) and (3). Figure 8 of the horizontal major axis.

tions. An example of the visco-static MAM eld is given in Figs. 8(b) and (c) for the case of perfectly cylindrical and elliptical rods, respectively. This eld is used to compute the viscous permeability q 0 . The section shrinkage is negligible when A is lower than 2 and is then not accounted for in the unit cell. The impacts of the defects are summarized in • the elliptical lament section signicantly increases the porosity φ and permeabilities q 0 and q 0 when the laments are very close to each other, and slightly increases the thermal length Λ . Figures 8(b) and (c) show that the volume occupied by the laments is slightly smaller when their section is elliptical. Moreover, looking at the color map, the velocity eld component in incident wave direction surrounding the cylindrical rods seems higher than the one surrounding the elliptical rods. Both observations are coherent with the fact that the porosity and viscous permeability are higher when considering elliptical laments;

• the lament section shrinkage increases the thermal length Λ when the laments are far from each other and therefore the shrinkage is strong (see Eq. 2);

• the combination of both defects results in a signicant increase of the porosity φ and permeabilities q 0 and q 0 when the laments are very close to each other. Moreover, an increase of the thermal length Λ is achieved when the lament spacing is large. any singularity in the resolution of the high-frequency limit of the visco-inertial problem, Eq. ( 14).

Figure 9 shows the dimensions of the geometry, the converged mesh and the visco-inertial MAM velocity eld. The latter is used to compute the viscous length Λ. On the one hand, the porosity φ, tortuosity α ∞ , viscous q 0 and thermal q 0 permeabilities are not aected by the grooves, or very little when A is close to 1.2. Then, their relative variations are 5%, 1%, 4% and 4% respectively. On the other hand, the characteristic lengths Λ and Λ are impacted by the grooves. Λ is reduced by 16% while Λ is reduced by 26%. These variations are constant except when A is close to 1.2. The variation of Λ was expected as it is dened by the ratio of the uid volume over the uid-skeleton surface 18 . Thus, the presence of grooves leads to the variation of the uid-skeleton surface without changing the uid volume, except when the rods are very close to each other and the porosity is aected by the grooves. The JCAL parameters variations with respect to the idealized REV are summarized in Table II.

V. EFFECT OF MANUFACTURING DEFECTS ON THE JCAL PARAMETERS: EXPERIMENTAL EVALUATION

The defects inherent to additive manufacturing have been numerically studied. In this section, their impact is experimentally studied for the range of A ∈ [1.4; 6.7], along with that of potentially non identied defects. Sample 2: IF = 30%, i.e., A = 3.33.

For each sample, an inverse characterization is performed in order to retrieve the JCAL parameters from experiments. The JCAL parameter bounds are set around the FEM values of the unit cell with elliptical rods section and no shrinkage. The grooves should signicantly reduce the characteristic lengths while the laments shrinkage should aect porosity, thermal length and permeabilities. During the inverse characterization process, the characteristic lengths can vary between 1.1 and 1/3 times the unit cell values. The porosity φ and tortuosity α ∞ could vary between ±0.1 of the unit cell value, and permeabilities q 0 and q 0 are ranged between 3 and 1/3 time the values of the unit cell .

Inverse characterization of the homogeneous samples leads to the JCAL parameters corresponding to certain inll factors. To obtain a complete parametric model linking the JCAL parameters to the inll factor, piece-wise cubic interpolations (Matlab, cubicinterp ) of the data was processed for the seven homogeneous samples. Simple analytic expressions of the variation of the JCAL parameters with respect to the inll factor could not be derived without important loss of accuracy. The trends of the experimental JCAL parameters with respect to the inll factor are similar to those considering idealized REV. In details, it can be noted that:

• The values of the porosity obtained experimentally are higher than those obtained numerically using the idealized REV. The dierence is more marked when the inll factor is high, i.e., the lattice constant is low. It is 27% higher for IF = 70%. This eect was already observed during the numerical study.

• The tortuosity is fairly well predicted numerically considering the idealized REV, which is in accordance with the simulations of the defects. The biggest shift is observed for IF = 40% where the experimentally retrieved tortuosity is only 0.06 higher than that of the idealized REV. Such dierence of tortuosity has only a small impact on the acoustic response of a porous material.

• The experimental characteristic lengths are lower than those obtained numerically considering the idealized REV. The viscous length is on average 30% lower experimentally and the shift is almost constant. A near constant down-shift was numerically predicted and is due to the presence of grooves on the laments surface.

The experimental thermal length is close to the one obtained numerical considering idealized REV for medium and high inll factors, i.e., medium and low lattice constants. Moreover, for IF = 15%, the experimental one is 40% lower. The simulations of the defects suggest that the down-shifting of the thermal length caused by the presence of grooves on the laments surface may be partially compensated by elliptical and shrinked lament section for high lattice constant; which is not the case however.

• The permeabilities were predicted to be higher in presence of defects, especially for low lattice constant. This forecast is experimentally conrmed. For IF = 70% the experimental viscous permeability is 15% higher than the one numerically predicted considering an idealized REV and the thermal permeability is 3 times higher. However, For illustration, the JCAL parameters of two homogeneous samples used for the inversion process (I F = 50%, i.e., A = 2 and IF = 30%, i.e., A = 3.33) are summarized in Table III considering the values from idealized micro-structure simulation, and in Table IV considering values from inverse characterization. The absorption coecients of these samples are presented in Fig. 11. As described in Ref. [START_REF] Zielinski | Pore-size eects in sound absorbing foams with periodic microstructure: modelling and experimental verication using 3d printed specimens[END_REF], the measured absorption coecient is up to 0.1 higher than the coecient simulated considering idealized micro-structure. Both curves follow the same trend. Still, the disparity between measured and simulated absorption coefcients is not the unique consequence of acoustic measurement imprecision. It is evidenced that the measured acoustic absorption coecients of homogeneous samples do not match the simulations with idealized micro-structure, and it is clear that the printed geometry diers from the idealized simulated one, which causes important acoustic discrepancies.

Finally, the inverse characterization gives very satisfactory results for three reasons. First, the JCAL parameters retrieved from inverse characterization follow realistic monotonic trends (Fig. 10). Second, the shifting of the JCAL parameters due to the presence of defects is oriented in the same direction numerically and experimentally. Third, the obtained absorption curves are almost perfectly overlaid to the measurement curves, when considering the JCAL parameters either from inverse characterization or from the corrected parametric model (Fig. 11).

Table IV. JCAL parameters of homogeneous samples, from inverse characterization. D n = 400 µm.

IF(%)

φ α ∞ Λ(µm ) Λ (µm ) q 0 (10 -9 m 2 ) q 0 (10 12 presents the absorption coecient measured and predicted on both sides of the multilayer samples. The bi-layer sample is composed of two 15 mm thick layers, the three-layers sample is composed of three 10 mm thick layers and the six-layers sample is composed of six 5 mm thick layers. It can be observed that the model considering idealized REV do not t well the measured absorption coecients while the experimentally corrected model, (Fig. 10), is in very good agreement with the experimental values.

C. Accurate JCAL parametric model routine

The approach to account for the impact of the geometry defects is summarized in Fig. 13.

It can be applied to other porous materials and to other manufacturing techniques. It consists in simulating the JCAL parameters of the unit cell using the MAM. Experimental geometric and acoustic measurements, including inverse characterization of homogeneous samples, provide the parametric model. Defects that can be simply and accurately described, such as elliptical section or lament section shrinkage, can be accounted for in the denition of the unit cell, while complicated or random defects such as grooves are considered by tuning the equivalent uid parametric model. Prior knowledge of the parameters scopes simplies the experimental retrieval of realistic JCAL parameters values.

TMM with the corrected model is applied to multilayered samples and compared to experimental results in order to validate the routine.

Figure 1 .

 1 Figure 1. (Color online) Idealized geometry. a) Top view. b) Cross section.

Figure 2 .Figure 3 .

 23 Figure 2. (Color online) Pictures of homogeneous sample.

Figure 4 .

 4 Figure 4. (Color online) Microscopic image of a micro-lattice cross section. Dimensions in micrometers.

Figure 5 .

 5 Figure 5. (Color online) Microscopic images of two micro-lattices having dierent lattice constant, top view. ED arrows indicate the extrusion direction. Dimensions in micrometers.

Figure 7 presents

 7 microscopic images of two new nozzles made by the same manufacturer. They are made of brass. In addition, the nozzle with a 200 µm diameter hole (Figs.7(c) and 7(d)) is plated with a nickel composite. The hole of the 200 µm nozzle is closer to a circle and has less imperfections than the other nozzle. This comparison does not benchmark manufacturers, materials or nozzle diameter but illustrates that using dierent nozzles certainly changes the nature of the defects of 3D printed micro-lattices. order to assess the impact of each defect separately, a modeling procedure is implemented. It considers the acoustic wave propagation in homogeneous and multilayer porous media as a function of microscopic parameters computed by a numerical method.

Figure 6 .Figure 7 .

 67 Figure 6. (Color online) SEM image of the micro-lattice, plane vue. The circle indicates an impurity and the arrow indicates a micro-groove.

Figure 8 .

 8 Figure 8. (Color online) 3D micro-lattice MAM visco-static simulation with A = 1.2. a) Unit cell. b) Velocity eld component in incident wave direction, with cylindrical rods. c) Velocity eld component in incident wave direction, with elliptical rods. The vertical minor axis is equal to 92%

Figure 9 .

 9 Figure 9. (Color online) Grooved lament 2D MAM simulation. a) Unit cell and visco-inertial, ω → ∞, MAM velocity eld. b) Grooves geometry and mesh. c) Visco-inertial, ω → ∞, MAM velocity eld detail. All dimensions are in microns.

Figure 10 .Figure 11 .

 1011 Figure 10. (Color online) JCAL parametric model with respect to the inll factor, considering idealized REV (blue line), from inverse characterization of homogeneous samples (red line). The green dashed lines indicates the authorized research scope during inverse characterization. The black markers spot the inverse characterization values.

Figure 10 presents

 10 Figure 10 presents the parametric model of the JCAL parameters with respect to the inll factor. The values obtained during inverse characterization and the interpolated curves dening the experimentally corrected JCAL parameters model are shown along with the values for the idealized REV and the authorized research scope.

Figure 12 .

 12 Figure 12. (Color online) Hard backed absorption coecient of 30 mm thick multilayers, D n = 400 µm , samples, measured (black lines) and simulated without considering the defects (blue marks) and simulated by the experimentally corrected model (red lines) on both sides. a) IF = [60, 30]%. b) IF = [44, 55, 33]%. c) IF = [15, 25, 35, 45, 55, 65]%.

Figure 13 .

 13 Figure 13. Routine diagram. Following the black lines leads to experimentally corrected JCAL parameters variation with respect to the considered sample parameter. Green lines indicate how to conrm the accuracy of the corrected model.

Table I .

 I JCAL parameters relative dierence (%) with respect to perfectly cylindrical rods. Data obtained from FEM numerical simulations. A positive value indicates that the corresponding defect increases the considered JCAL parameter, and vice versa.

	Defect	A	φ	α ∞	Λ	Λ	q 0	q 0
		1.2	15	2	0	5	19	22
	Elliptic	10	1	0	0	4	3	1
		25	0	0	0	4	2	0
	Shrinkage	10	1	0	1	9	4	5
		25	1	1	0	12	2	2
	Elliptic	1.2	15	2	0	5	19	21
	and	10	2	0	1	13	5	6
	Shrinkage	25	1	0	0	17	2	3

Table I with the following main ndings:

Table II .

 II JCAL parameters relative dierence (%) of grooved rods with respect to perfectly circular rods. Data obtained from 2D numerical simulations. A positive value indicates that the corresponding defect increases the considered JCAL parameter, and vice versa.

	Defect	A	φ	α ∞	Λ	Λ	q 0	q 0
		1.2	5	1	-14	-24	4	4
	Grooves	10	0	0	-16	-26	0	0
		25	0	0	-16	-26	0	0

Table III .

 III JCAL parameters of homogeneous samples, numerically computed considering the idealized REV. D = 400 µm. high lattice constant, the trend is inverted. For IF = 15% both experimental permeabilities are 45% lower. It is worth noting that any permeability higher than ≈ 2•10 -8 m 2 will give overlaid absorption curves, providing all other JCAL parameters are kept constant. Thus, the permeabilities obtained from inverse characterization of the high lattice constant samples (I F < 30%) should be viewed with caution.

	A	φ	α ∞	Λ(µm )	Λ (µm )	q 0 (10 -9 m 2 )	q 0 (10 -9 m 2 )
	2	0.51	1.28	217	252	2.89	4.05
	3.33	0.71	1.16	447	515	20.0	23.7

for

  -9 m 2 ) one three-layers (I F = [44, 55, 33]%, i.e., A = [2.27, 1.82, 3.03]) and one sixlayers (I F = [15, 25, 35, 45, 55, 65]%, i.e., A = [6.67, 4.00, 2.86, 2.22, 1.82, 1.54]) samples were printed and simulated by the TMM. It should be noted that some of the considered IF are homogeneous samples used for the inverse characterization. Each sample was printed in one step.Figure

	50	0.56	1.29	145	216	2.45	5.02
	30	0.72	1.18	358	393	10.3	15.7
	B.	Validation
	To validate the corrected predictive model, one bi-layers (I F = [60, 30]%, i.e., A =
	[1.67, 3.33]),

ACKNOWLEDGMENTS

The authors acknowledge Safran Aircraft Engines, the Natural Sciences and Engineering Research Council of Canada (NSERC) for supporting and funding this research. They acknowledge nancial support from ANR industrial chair MACIA (ANR-16-CHIN-0002).

They also would like to thank L. Schwan for useful discussion on homogenization theory.