Mathieu Scalabrini
email: scalabrini.mathieu@gmail.com

Jonathan Hamon

Isabelle Linossier

Vincent Ferrières
email: vincent.ferrieres@ensc-rennes.fr

Karine Réhel

Pseudomonas aeruginosa resistance of monosaccharide-functionalized glass surfaces

Keywords: surface, functionalization, carbohydrate, bacterial adhesion, Pseudomonas aeruginosa

Introduction

Biofilm is a bacterial organized community strongly held on live or abiotic surfaces in aqueous media. Bacteria are maintained to each other by excreting a matrix of extracellular polymeric substances (EPS) composed mainly of polysaccharides, proteins and DNA. The high viscosity of the matrix protects the bacteria at the centre of biofilm from environment, chemical and/or thermic treatment and immune system, and enhances the bacterial tolerance against anti-bacterial drugs [1,2].

Biofilms occur in many different fields, resulting in major health and economic issues.

Pseudomonas aeruginosa, Escherichia coli, Staphylococcus epidermidis or Staphylococcus aureus are such bacteria adept in forming biofilms on many medical devices from urinary and venous catheters to dental and orthopaedic implants. They consequently cause health complications with nosocomial infections and diseases, that can be fatal mainly with immunodeficiency persons [2][3][4][5]. Furthermore, in addition to accelerate the corrosion and the maintenance frequency of equipment, biofilms on food contact surfaces minimize the effect of cleaning agents, and thus increase the odds of food contaminations by Listeria monocytogenes or Salmonella which, in most cases, trigger potentially fatal foodborne disease [START_REF] Chia | Attachment of different Salmonella serovars to materials commonly used in a poultry processing plant[END_REF][START_REF] Giaouris | Attachment and biofilm formation by foodborne bacteria in meat processing environments: Causes, implications, role of bacterial interactions and control by alternative novel methods[END_REF][START_REF] Carpentier | Review -Persistence of Listeria monocytogenes in food industry equipment and premises[END_REF][START_REF] Cappitelli | Biofilm Formation in Food Processing Environments is Still Poorly Understood and Controlled[END_REF]. Lastly, marine structures such as ships, underwater pipelines, oil rings are continually in contact with water that are profitable to the development of biofouling. This increases the downside risks and the rhythm of cleaning and maintenance processes and then results in substantial economic losses [START_REF] Callow | Marine biofouling: A sticky problem[END_REF][START_REF] Qian | Marine biofilms as mediators of colonization by marine macroorganisms: Implications for antifouling and aquaculture[END_REF].

Biofilm formation is a complex process not clearly elucidated but it is recognized to be subdivided in sequential steps [START_REF] O'toole | Biofilm Formation as Microbial Development[END_REF][START_REF] Donlan | Biofilm Formation: A Clinically Relevant Microbiological Process[END_REF]. Firstly, planktonic bacteria reversibly adhere onto the surface with weak interactions such as van der Waals interactions, electrostatic interactions and hydrogen bonds. This step is immediately followed by an irreversible attachment of the bacteria thanks to a primary excreted film and flagella, pili and some other membrane structures including lectins [START_REF] An | Concise review of mechanisms of bacterial adhesion to biomaterial surfaces[END_REF][START_REF] Hori | Bacterial adhesion: From mechanism to control[END_REF]. Then, bacteria progressively grow in microcolonies leading up to more structured and condensed architectures. After maturation, almost all the area is colonized, and a part of sessile bacteria finally returns in planktonic state to colonize other surfaces. The accumulation of microbial mass on the surface can be limited and disrupted by interfering at one or more stages of the biofilm formation. Antimicrobial or biocidal coatings are the most powerful against biofilm formation [START_REF] Swartjes | Current Developments in Antimicrobial Surface Coatings for Biomedical Applications[END_REF][START_REF] Wagner | Covalent Lectin Inhibition and Application in Bacterial Biofilm Imaging[END_REF]. Nonetheless, using bactericidal strategy may induce dangerous consequences, especially in health domain, with the rising of multiresistant bacteria, resulting from the abuse and misuse of drugs.

A non-biocidal and non-toxic strategy to prevent bioaccumulation lies in the development of surfaces with adhesion resistance properties. One effectiveness approach to prevent microorganism accumulation is the use of amphiphilic copolymers such as silane-PEG which demonstrated a good fouling release properties in complex environment [START_REF] Hawkins | Bacteria and diatom resistance of silicones modified with PEO-silane amphiphiles[END_REF][START_REF] Faÿ | Non-toxic, Anti-fouling Silicones with Variable PEO-silane Amphiphile Content[END_REF]. PEG chains form a hydrated layer that creates a steric barrier preventing close approach of microorganisms as well as unspecific interactions. Indeed, when bacteria enter in contact of PEG entities, the layer is compressed increasing the osmotic pressure that leads to repulsive force against bacteria [START_REF] Hadjesfandiari | Polymer brush-based approaches for the development of infection-resistant surfaces[END_REF][START_REF] Leckband | Grafted poly(ethylene oxide) brushes as nonfouling surface coatings[END_REF]. PEG brush-coated nevertheless have a limited in vivo stability over time increasing research of PEG brush-like substitutes [START_REF] Browning | Determination of the in Vitro Degradation of Pegda[END_REF][START_REF] Inoshima | PEG Hydrogel Degradation and the Role of the Surrounding Tissue Environment[END_REF].

Carbohydrates are biocompatible and non-toxic compounds with numerous functional groups affording an extensively possibility of structural and property modulations. Moreover, chitosan, alginate, hyaluronic acid and dextran are some of polysaccharides described for their ability to minimise protein adsorption, fouling attachment and biofilm formation [START_REF] Jakobi | Amphiphilic Alginates for Marine Antifouling Applications[END_REF][START_REF] Mcarthur | Effect of polysaccharide structure on protein adsorption[END_REF][START_REF] Bauer | Adhesion of marine fouling organisms on hydrophilic and amphiphilic polysaccharides[END_REF][START_REF] Junter | Polysaccharide-based antibiofilm surfaces[END_REF].

Exopolysaccharides isolated from EPS matrix have also been shown to negatively regulate biofilm formation (see Rendueles [START_REF] Rendueles | Antibiofilm polysaccharides[END_REF] for an extensive summary). Hederos and co-workers presented surfaces designed from self-assembled monolayer of partially methylated galactosides which resisted to the adsorption of proteins [START_REF] Hederos | Synthesis and self-assembly of galactoseterminated alkanethiols and their ability to resist proteins[END_REF]. Recent studies on marine bacteria and Ulva linza macroalgae have shown to curtail the adhesion on one side and to aid the cleaning on the other side [START_REF] Ederth | Resistance of galactosideterminated alkanethiol self-assembled monolayers to marine fouling organisms[END_REF]. In vivo studies of immobilized lactose on poly(ether sulfone) (PES) revealed detrimental effects on both protein adsorption and biomass accumulation [START_REF] Myles | Bioinspired Aryldiazonium Carbohydrate Coatings: Reduced Adhesion of Foulants at Polymer and Stainless Steel Surfaces in a Marine Environment[END_REF].

Valotteau has demonstrated that a sophorolipide-functionalized surface (SL) possessed antibacterial properties by destabilizing the cell wall of bacteria [START_REF] Valotteau | Antibacterial properties of sophorolipid-modified gold surfaces against Gram positive and Gram negative pathogens[END_REF]. Further studies by singlecell force spectroscopy showed an inhibition of the bacterial adhesion force with SL surfaces [START_REF] Valotteau | Nanoscale antiadhesion properties of sophorolipid-coated surfaces against pathogenic bacteria[END_REF]. These latter studies strongly suggest that surface-immobilized mono-or disaccharides may impact the bioaccumulation even if, to the best of our knowledge, they are still underexplored on pathogenic bacteria.

Based on mechanism from PEG units and proprieties from carbohydrates, this work explores the anti-bioadhesion activities of amphiphilic monosaccharides layer grafted on glass surface. D-glucose (D-Glc), D-galactose (D-Gal) and D-mannose (D-Man) were simply connected to pre-functionalized alkyl azide glass through a small propargylic linker thanks to copper catalysed alkyne-azide cycloaddition (CuAAC). It is noteworthy that triazole rings were sometimes able to mimic carbohydrate residues that results in virtual increase of the glycosidic chain length [START_REF] François-Heude | Synthesis of highmannose oligosaccharide analogues through click chemistry: True functional mimics of their natural counterparts against lectins?[END_REF][START_REF] Périon | 1,2,3-Triazoles and related glycoconjugates as new glycosidase inhibitors[END_REF]. Two type of linkage were explored, the standard O-glycosidic bonds and S-glycosidic linkages, the latter being more resistant to chemical and enzymatic hydrolysis [START_REF] Pachamuthu | Synthetic Routes to Thiooligosaccharides and Thioglycopeptides[END_REF][START_REF] Witczak | Thiosugars II lY2 . A novel approach to thiodisaccharides The synthesis of 3-deoxy-4-thiocellobiose from levoglucosenone[END_REF]. The glycosidic surfaces were further compared to hydroxylated surface, which bears the triazolic linker terminated by a hydroxyl group which closely akin to glycosidic surface (Figure 1). The resulting surfaces were then characterized by water contact angle and X-ray Photoelectron Spectroscopy (XPS). Adhesion assays were performed with opportunistic pathogen Pseudomonas aeruginosa by means of parallel plate flow chamber system.

Adhesion results are discussed in terms of specific interaction and unspecific interaction by examining the adhesion change with lectin A deficient mutant as well as the surface free energies and the thermodynamic Lifshift-van der Waals -Acid-Base (LW-AB) approach.

Materials and methods

Chemicals and materials

All reagents and solvents were purchased from commercial sources and were used without further purification unless noted. Unless otherwise stated, all reactions were monitored by TLC on Silica Gel 60 F 254. TLC spots were detected by staining with cerium ammonium molybdate solution. Column chromatography was performed on Silica Gel (50 µm). NMR spectra were recorded on Bruker ARX 400 spectrometer at 400 MHz for 1 H and 100 MHz for 13 C. Chemical shifts are given in δ units (ppm) and referenced to TMS. Coupling constants J were calculated in Hertz (Hz). Proton and carbon NMR peaks were unambiguously assigned by J-Mod (J-Modulated spin echo), COSY (double quantum filtered with gradient pulse for selection), HSQC (gradient echo-anti-echo selection and shape pulse) and HMBC (echo-anti-echo gradient selection, magnitude mode) correlation experiments. Alkyl azide-functionalized glass surfaces were purchased from PolyAn -Molecular Surface Engineering (Germany).

Bacterial strain

Pseudomonas aeruginosa wild-type MPAO1 and mutant PAΔlecA (ref: PW5314 lecA-G10::ISphoA/hah) provided from Two-Allele Library Manoil lab (University of Washington). [START_REF] Jacobs | Comprehensive transposon mutant library of Pseudomonas aeruginosa[END_REF] Syto TM 9 Green were purchased on Fisher scientific. Cell were routinely cultivated at 37 °C in LB medium (NaCl 10 g/L, Tryptone 10 g/L, Yeast extract 5 g/L) under agitation (120 RPM).

Synthesis of Propargyl O-D-glycopyranosides 2a-2c

To a solution of peracetylated sugar (1.0 equiv.) in dry CH2Cl2 was subsequently added BF3.OEt2 (1.5 equiv.) and propargyl alcohol (1.5 equiv.) under inert atmosphere (N2). The reactional mixture was stirred overnight at room temperature and the reaction was quenched with K2CO3 (1.5 equiv.). After stirring another 1 hour, the mixture was filtered on Büchner and the filtrate was diluted in CH2Cl2, washed with H2O (2x) and brine solution (1x). The organic layer was dried over MgSO4, filtered and concentrated under reduced pressure to yield the desired product directly used without further purification.

To a solution of propargyl 2,3,4,6-tetra-O-acetyl-β-D-glycopyranoside (1.0 equiv.) in CH3OH/H2O (6:1) was added Et3N (4.5 equiv.). The reaction was allowed to stir at 20 °C until complete deprotection monitored by TLC [CH2Cl2/CH3OH (9:1)]. The mixture was concentrated, and the product was isolated by flash chromatography on silica gel with CH2Cl2/CH3OH (9:1) as eluent. The detailed characterization of compounds is given in Supplementary Information.

Synthesis of Propargyl 1-thio-D-glycopyranosides 4a-4c

To a solution of peracetylated sugar (1.0 equiv.) in dry CH3CN were added thiourea (1.1 equiv.), BF3.OEt2 (3.0 equiv.) and the reactional mixture was stirred under reflux for 1 hour.

After reaching room temperature, propargyle bromide (1.5 equiv., 80%w in toluene) and Et3N (4.5 equiv.) were slowly added and then, the reaction mixture was stirred at 20 °C for 12 hours.

After completion, the mixture was concentrated, the resulting paste was diluted in CH2Cl2, washed with HCl 5% (2x), brine solution (3x) and distilled water (2x). The collected organic layer was dried over Na2SO4, filtered and concentrated under reduced pressure at 40 °C. The residue was finally purified by flash chromatography on silica gel using an eluent gradient of cyclohexane/EtOAc (4:1 to 1:1) to yield the desired product.

To a solution of propargyl 2,3,4,6-tetra-O-acetyl-1-thio-β-D-glycopyranoside (1.0 equiv.) in CH3OH/H2O (6:1) was added Et3N (4.5 equiv.). The reaction was allowed to stir at 20 °C until complete deacetylation monitored by TLC [CH2Cl2/CH3OH (9:1)]. The mixture was further concentrated, and the product was isolated by flash chromatography on silica gel eluting with CH2Cl2/CH3OH (9:1). The detailed characterizations of compounds are given in Supplementary Information.

Grafting protocol

Commercially available alkyl azide-functionalized glass surfaces (75x25x1 mm) were immersed in a solution of H2O/DMSO (1:1). Propargyl glycopyranoside (2.3 mM) and CuSO4.5H2O (1.2 mM) were added and oxygen was removed from the mixture by bubbling with N2 for 20 min. Then, sodium ascorbate (2.3 mM) was added and the mixture was bubbling for another 15 min. The reactor was kept in the dark and incubated at 35 °C under agitation for 48 h. The resulting grafted surfaces were thoroughly washed with H2O and absolute EtOH and dried in hoven at 40 °C. Finally, grafted surfaces were kept at -20 °C prior to further use.

Contact angle and surface free energy

Water drop contact angles were measured with the optical contact angle meter Digidrop (GBX). A drop of 1.5 µL of ultra-pure water at 20 °C was deposited on the surfaces and the profile was recorded. Contact angle was determined from the analysis of the drop shape with the instrument software. The presented contact angles of each surfaces are averaged from ten surfaces from different batches, and six measurements were made on each surface. The determination of surface energy components was achieved using water and formamide as polar solvent and diiodomethane as apolar solvent with known data (values are presented in Table S1).

XPS Analysis

X-ray photoelectron spectroscopy analysis was performed using a Kratos Axis Nova (University of Nantes, France). The spectrometer used a monochromated Al Kα source (1486.6 eV) operating at 300W (20 mA, 15 kV) and an analysis area of 700 µm x 300 µm. The pressure in the analysis chamber was always kept below 10 -9 Torr. High-resolution spectra of different core levels were recorded using a pass energy of 40 eV at θ = 0° take-off angle (angle between the surface normal and the detection direction). All data were acquired using charge compensation to establish a steady state surface potential. All high-resolution spectra were analyzed and fitted using CasaXPS software. All spectra were calibrated in energy by setting adventitious carbon C 1s at 284.8 eV (binding energy). All spectra were fitted using a Tougaard type baseline for the background and a pseudo-Voigt function with Gaussian (70%)-Lorentzian (30%) for each component.

Characterization of MPAO1 cell wall

Culture of MPAO1 were carried out at 37 °C in 100 mL of LB medium under agitation (120 RPM). Cells were harvested by ultra-centrifugation, washed (3x) and suspended in physiological water (NaCl 0.9%) at a concentration of 3 × 10 9 cell/mL. 50 mL of the bacterial suspension were filtered through 0.45 µm pore size filters. Filters were placed on a glass slide and left to air dry for 30 min at 37 °C. [START_REF] Gallardo-Moreno | Insights into bacterial contact angles: Difficulties in defining hydrophobicity and surface Gibbs energy[END_REF][START_REF] Busscher | Measurement of the Surface Freen Energy of Bacterial Cell Surfaces and Its Relevance for Adhesion[END_REF] Finally, a droplet of 1.5 µL of desired solvents (water, formamide and diiodomethane) was deposited on the filter and contact angle was measured after 2 seconds. Two measurements by filters were achieved from three independent experiments.

Bioadhesion assays

Bacterial adhesion was carried with a flowcell system under static condition. Flowcell was composed of 3 independent channels (40x4x1 mm). The surface was stuck onto the flowcell with silicon glue and connected to a peristaltic pump. The system was sterilized with a flow of absolute ethanol for at least 1 hour. Then, the system was washed and equilibrated with physiological water. Each channel was inoculated with a 300 µL of suspension of the desired strain at OD600= 0.1 in physiological water and bacteria were allowed to adhere to the surface at 37 °C for two hours. Non-adhered bacteria were removed with a minimal flow and adhered bacteria were stained with a solution of 5 µM of Syto TM The glycosylated glass surfaces were obtained by grafting the propargyl monosaccharides to the azide group pre-installed on the material, using usual conditions of CuAAC (Figure 3) [START_REF] Liang | The copper(I)-catalyzed alkyne-azide cycloaddition (CuAAC) "click" reaction and its applications. An overview[END_REF][START_REF] Santos | Synthesis and antitubercular evaluation of new 1,2,3-triazole derivatives of carbohydrates[END_REF]. Azide surface was immersed in a solution of the desired carbohydrate 2a-2c or 4a-4c, copper sulphate and sodium ascorbate in dimethyl sulfoxide and water. The reaction was incubated at 35 °C in the dark for 48 hours. To remove the copper outstripped, glycosidic surfaces were thoroughly washed with water and ethanol. A similar procedure was applied to the simpler surface presenting only hydroxyl group, starting from propargyl alcohol (OH surface).

Water Contact Angle

The evolution of the wettability property of the surfaces throughout the different grafting was characterized by the contact angle of a water droplet. The measurement was performed on ten surfaces from different reaction batches and the mean values are reported in Figure 4.

The reference azide surface displayed the most hydrophobic character with an angle values up to 76° ± 2° within the range of value already cited in the literature (77°) [START_REF] Prakash | Click " Modification of Silica Surfaces and Glass Microfluidic Channels method for covalent attachment of various polymers by and within glass microchannels suitable for CE systems . measurements , X-ray photoelectron spectroscopy , and[END_REF]. Modification with a simple hydroxyl group enhanced the water affinity of the OH surface with a weak decrease of the hydrophobicity (contact angle 68° ± 2°). Surprisingly, the angle exhibited by OH was higher than angle previously determined on OH-terminated gold surfaces (average 30°) [START_REF] Alsteens | Direct measurement of hydrophobic forces on cell surfaces using AFM[END_REF]. A difference in the grafting density and the functionalization method might explain the large gap. However, the contribution of the multiple hydroxyl groups of carbohydrates led to an expected decrease of the water contact angle with values lied between 56° ± 2° and 58° ± 2°. Thereover, the similar angle for glycosidic layer indicated that the nature of the sugar and the glycosidic linkage had thus no impact on the hydrophilicity. And the similarity of standard deviations before and after cycloaddition implied a homogeneous functionalization at the organism scale.

X-Ray photoelectron spectroscopy

The cycloaddition reaction was corroborated with the in-depth XPS analysis of atomistic composition change of the surfaces. First of all, information from large spectrum on different surfaces validated the absence of residual copper which could be prejudicial for biological studies (Figure S1). Secondly, the characteristic double signal in the N 1s high resolution scan of azide surface (Figure 5a) was deconvoluted into three peaks: 404.5 eV and 400.0 eV attributed to the azide group (electron-deficient nitrogen -N=N + =N-and the two lateral nitrogen -N=N + =N-, respectively) and one at 401.0 eV assigned to NH2/NH3 + resulting from the known photolytic degradation of the azide group [START_REF] Prakash | Click " Modification of Silica Surfaces and Glass Microfluidic Channels method for covalent attachment of various polymers by and within glass microchannels suitable for CE systems . measurements , X-ray photoelectron spectroscopy , and[END_REF][START_REF] Gouget-Laemmel | Functionalization of azideterminated silicon surfaces with glycans using click chemistry: XPS and FTIR study[END_REF][START_REF] Schulze | A variable temperature x-ray photoelectron spectroscopic study of the surface conversion of diethylaluminum azide to AlN[END_REF]. Figure 5b and Figure S2 show the typical N 1s spectra for functionalized surfaces. The broaden signal at 400 eV was fitted and deconvoluted in two peaks: 402.2 eV and 400.1 eV assigned to the triazole ring (N-N=N and N-N=N, respectively). As expected, the calculated ratio of area under the peaks was 1:2 for grafted surfaces. Finally, the absence of 404.5 eV signal provided evidence that the cycloaddition "click" reaction was nearly quantitative.

Bioadhesion Assays

The Gram-negative Pseudomonas aeruginosa is an opportunistic pathogen involved in nosocomial and chronic lung infections [START_REF] Breidenstein | Pseudomonas aeruginosa: All roads lead to resistance[END_REF][START_REF] Lyczak | Establishment of Pseudomonas aeruginosa infection: Lessons from a versatile opportunist[END_REF][START_REF] Obritsch | Nosocomial infections due to multidrug-resistant Pseudomonas aeruginosa: Epidemiology and treatment options[END_REF]. In most cases, this type of infections is due to a transfer from contaminated devices to the host. Plethora of literature data on its ability to form biofilm on biotic and abiotic surfaces makes P. aeruginosa a reference in the area. The bacterial adhesion is mostly dependent on hydrophobicity and surface charge of both the surface and the bacterium, and can be described with the Derjaguin-Landau-Verwey-Overbeek (DLVO) model for colloidal particles [START_REF] Hermansson | The DLVO theory in microbial adhesion[END_REF][START_REF] Zita | Effects of bacterial cell surface structures and hydrophobicity on attachment to activated sludge flocs[END_REF]. Once initial adhesion, bacteria are strongly anchored to the surface thanks to the excreted EPS and specific interactions through membranes structures such as lectins [START_REF] An | Concise review of mechanisms of bacterial adhesion to biomaterial surfaces[END_REF][START_REF] Bos | Physico-chemistry of initial microbial adhesive interactions -Its mechanisms and methods for study[END_REF].

The adhesion studies were performed with a flow chamber system under static condition at 37 °C. Bacteria were suspended in physiological water, a free-nutriment medium that forced the passage from the planktonic state to the sessile state and limited bacterial growth. The adhered bacteria were subsequently labelled with a solution of Syto 9 Green, snapshots were taken using an inverse scanning laser confocal microscope; and the surface coverage rates were determined from black and white pixel ratio.

From the microscopy images presented in Figure 6a and Figure S3, the adhesion of P. aeruginosa was homogeneous on all surfaces with visually greater coverage for azide surface, followed by OH surface. The summary of surface coverage rates on azide, OH and glycosidic surfaces are shown in Figures 6b (black bars). The colonization on the control azide surface was the most important with a coverage rate at 10.0 %. Addition of a hydroxyl group did not significantly influence the adhesion whereas glycosidic patterns exhibited an anti-bioadhesion activity with the bioaccumulation decrease of almost 40% of adhered bacteria, with nevertheless no difference between carbohydrates. These observations contrast with those observed with sophorolipid-functionalized gold, where no adhesive or anti-adhesive properties were detected with a large number of bacteria [START_REF] Valotteau | Antibacterial properties of sophorolipid-modified gold surfaces against Gram positive and Gram negative pathogens[END_REF]. However, Kesel and his group noticed that D-galactose, D-mannose and D-lactose coating reduced the adhesion of Bacillus subtilis in same way [START_REF] Kesel | Carbohydrate coating reduces adhesion of biofilm-forming Bacillus subtilis to gold surfaces[END_REF]. They suggested that carbohydrates restrained the thiol-Au interaction between proteins in the bacterial cell wall and gold surface.

Even if they are close-packed, presenting carbohydrate pattern may lead to specific interactions with carbohydrate binding proteins present on the bacterial membrane [START_REF] Lindhorst | Small Molecule Ligands for Bacterial Lectins : Letters of an Antiadhesive Glycopolymer Code[END_REF][START_REF] Weber | Switching of Bacterial Adhesion to a Glycosylated Surface by Reversible Reorientation of the Carbohydrate Ligand[END_REF]. Indeed, P. aeruginosa owned a soluble lectin, LecA (or PA-IL), which specifically binds α/β-Dgalactosyl patterns [START_REF] Imberty | Structures of the lectins from Pseudomonas aeruginosa: Insights into the molecular basis for host glycan recognition[END_REF][START_REF] Cioci | Structural basis of calcium and galactose recognition by the lectin PA-IL of Pseudomonas aeruginosa[END_REF]. Moreover, the affinity of LecA to D-galactoside is all the more important with a hydrophobic group on anomeric position [START_REF] Garber | On the specificity of the d-galactose-binding lectin (PA-I) of Pseudomonas aeruginosa and its strong binding to hydrophobic derivatives of d-galactose and thiogalactose[END_REF]. Thus, bioadhesion studies were performed with the LecA deficient PAΔlecA mutant in similar conditions as described for MPAO1 and the resulting coverage rates are presented in Figure 6b (grey bar). The coverage comparison between MPAO1 and PAΔlecA indicated that the bacterial attachment was not affected in absence of LecA, even with galactosidic surfaces. It confirmed that the contribution of LecA was negligible in the adhesion step compared to their key role in the biofilm maturation [START_REF] Diggle | The galactophilic lectin, LecA, contributes to biofilm development in Pseudomonas aeruginosa[END_REF]. Although specific interactions were not excluded, these results suggested that the adhesion of MPAO1 was dependent more on non-specific interactions (or long-range interactions) determined by the physicochemical properties of both the substratum and the bacterium.

3.5.

Surface free energy

The surface free energy (or surface tension) is an important parameter influencing the adhesion. According to the van Oss, Chaudhury and Good (VCC) theory, the surface free energy of a dipolar surface is the sum of a Lifshitz-van der Waals (dispersive) component and Lewis acid-base (polar) component that is depended on acid component and basic component [START_REF] Van Oss | Monopolar Surfaces[END_REF][START_REF] Van Oss | The Role of van der Waals Forces and Hydrogen Bonds in " Hydrophobic Interactions " between Biopolymers and Low Energy Surfaces[END_REF]: Following a combining rule, a general contact angle equation for a liquid l on a solid s was defined as:

2
1 2
Finally, the three unknown terms for the surface , , can be found by using one apolar solvent and two polar solvents with known , , , and properties (see Good [START_REF] Good | Contact angle, wetting, and adhesion: a critical review[END_REF] for a complete review).

Table 1 presents the surface free energies and the components calculated from contact angle values determined with water and formamide as polar solvents and diiodomethane as apolar one (see ESI). Similar surface free energies ( = 45.48-49.99 mJ/m²) were determined for all studied surfaces with a major contribution (>90 %) of the apolar component that was assigned to the alkyl chain initially grafted on glass surface. However, different values of were observed, notably for the Lewis basic components. Indeed, the lowest basic component was calculated for the azide surface ( = 5.19 mJ/m²) resulting from their πelectrons. The grafting of OH functions provided additional hydrogen bonds that mildly increased up to 11.00 mJ/m². Finally, we observed the largest for glycosidic surfaces with a Lewis basic component three times bigger than azide surface ( = 16.83-19.11 mJ/m²).

The multiple hydrogen acceptors from OH as well as with π-electrons of triazole ring gave to the glycosidic surfaces an appreciable electron donor character. Ederth and al. have presented similar value of surface free energy for galactosidic surfaces ( = 46.4 mJ/m 2 ) with, nevertheless, a higher value for basic component ( = 67.0 mJ/m 2 ) [START_REF] Ederth | Resistance of galactosideterminated alkanethiol self-assembled monolayers to marine fouling organisms[END_REF]. It is well-known that VCC theory depends on the solvent used, especially in the determination of the acid-base contribution [START_REF] Drago | Quantitative evaluation and prediction of donor-acceptor interactions[END_REF].

We also determined the surface free energy of the cell wall of MPAO1 with the VCC approach using contact angle goniometry on a bacterial lawn as described by Busscher & al. [START_REF] Busscher | Measurement of the Surface Freen Energy of Bacterial Cell Surfaces and Its Relevance for Adhesion[END_REF]. First of all, MPAO1 exhibited a hydrophilic character with a contact angle value at 26° ± 2° which was in range of contact angle determined for "non-adhesive" P. aeruginosa [START_REF] Roosjen | Bacterial factors influencing adhesion of Pseudomonas aeruginosa strains to a poly(ethylene oxide) brush[END_REF]. The interfacial free energy was further calculated, and a clear predominance of was observed, with a high contribution of Lewis basic component ( = 52.63 mJ/m²). The strong electron donor and the hydrophilic characters of MPAO1 were easily attributed to structures, such as lipopolysaccharides, lipoproteins, phospholipids and proteins, which compose the surface of an outer membrane of a Gram-negative bacterium. From these data, electrostatic repulsions were expected to be more important for glycosidic surfaces (less for OH surface) than for azide surface, since the latter is less negative. The adhesion studies correlated this assumption with the decrease of the coverage rate for glycosidic surfaces. This notwithstanding, the adhesion remained not changed with OH surface, whereas the deviation of surface free energy between OH and azide or glycosidic surfaces were similar.

Adhesion free energy

The thermodynamic Lifshift-van der Waals -acid-base (LW-AB) approach stipulates that the bacterial adhesion is favourable if the adhesion free energy ∆ between surface (s) and bacterium (b) in medium (l) is negative, and is determined from the apolar free energy (∆ ) and polar free energy (∆ ) as: [START_REF] Sharma | Analysis of different approaches for evaluation of surface energy of microbial cells by contact angle goniometry[END_REF] ∆ ∆ ∆ which can be calculated from the interfacial free energy components, according to:

∆ 2 - - and ∆ 2 " #$ $ -$ % $ - - - &
In that respect, the values of ∆ between studied surfaces and P. aeruginosa were calculated with surface free energy components values (Table 2). From this approach, the interaction surface -MPAO1 was thermodynamically unfavourable with positive ∆ values that implied repulsive interactions. The similar and positive ∆ indicated unfavorable hydrophobic interactions. The latter are the predominant interactions during the initial adhesion that occurs at long distance. Differences were noticed at the short length (hydrogen bonds) with higher values of ∆ after surface modifications and corroborated the assumption of greater repulsive interactions for glycosidic and OH surfaces. That being said, the model presents limits to explain the adhesion results. Indeed, adhesion differences were observed with glycosidic surfaces whereas ∆ values were closed to each other. The application of this model was, to our knowledge, the first described in the literature for this type of surfaces.

However, thermodynamic studies were described for the adhesion of many PAO1 strains on glass and PEG surfaces [START_REF] Roosjen | Bacterial factors influencing adhesion of Pseudomonas aeruginosa strains to a poly(ethylene oxide) brush[END_REF]. The authors determined positive ∆ values (which were like our surfaces) with non-adhesive strains whereas negative values were obtained for adhesive strains. In any cases, adhesion occurred even if the adhesion were thermodynamically unfavourable. Bayoudh suggested that the LW-AB approach may explain the influence of the physicochemical properties of surfaces at the first stage of the adhesion [START_REF] Bayoudh | Quantification of the adhesion free energy between bacteria and hydrophobic and hydrophilic substrata[END_REF]. After that, bacteria have undergone biologic changes to establish strong attachment with the surfaces.

In that respect, the LW-AB approach illustrated the adhesion trend of the low adherence of MPAO1 on surfaces. The model was, nevertheless, insufficient to undoubtedly describe the anti-bioadhesive property of glycosidic surfaces. Another phenomenon has to be taken into account, the formation of a hydration-layer with carbohydrate heads that it was suggested for similar surfaces [START_REF] Valotteau | Nanoscale antiadhesion properties of sophorolipid-coated surfaces against pathogenic bacteria[END_REF]. Indeed, a structured layer of water was supposed on polar surface which created, at very short distance, repulsive forces by osmotic pressure [START_REF] Parsegian | Hydration forces: Observations, explanations, expectations, questions[END_REF]. This latter was predicted to be higher on glycosidic surface with multiple hydroxyl, rather than on OH surface.

Conclusions

The ability of bacteria to develop biofilm on any surface triggers health and economic issues.

Even if the process is not yet fully understood, the adhesion is the critical step which involves non-specific interactions (apolar and polar interactions) and specific interaction with adhesins or lectins. The design of materials from carbohydrates is a promising non-toxic and non- The characterisation with water contact angle and XPS spectroscopy showed a quantitative cycloaddition reaction and a dynamic reorganization of the glycosidic layer in polar environment.

Adhesion studies of Pseudomonas aeruginosa MPAO1 were performed in flow chambers and a strong decrease of the adhesion was observed for glycosidic surfaces with respect to the azide surface control. However, the carbohydrate nature and the type of linkage appeared to not affect the attachment of MPAO1. Further adhesion assays with a LecA deficient mutant PAΔlecA suggested that LecA was not essential for the adhesion step, nonetheless, molecular interactions may not be excluded since the complete mechanism of the adhesion remains misunderstood. The exploration of surface free energies interestingly indicated an repulsive effect between the bacterial cell wall and surfaces, expected more pronounced for glycosidic surfaces. Further thermodynamically studies have shown an unfavourable interaction of bacteria with all surfaces. The closed values of ∆ and ∆ suggested that initial adhesion was governed by the physicochemical properties in the same way for all surfaces, and the anti-bioadhesion property of glycosidic surfaces may also result from hydration force that could reject the bacteria.

These encouraging results add more evidences that a few atomic layers of monosaccharides may minimize the bacterial attachment. Interestingly enough, the antibioadhesion activities were observed under static conditions that did not take in consideration the dynamism of the glycosidic layer. In fact, the glycosidic brush-layer may reduce the adhesion force enhancing the unhooking of bacteria under shear stress [START_REF] Nejadnik | Bacterial adhesion and growth on a polymer brush-coating[END_REF]. Finally, envisaging to extend this study by switching the carbohydrate configuration could provide information on the existence of the molecular interactions and their implications.

Figure 1 .

 1 Figure 1. Principle and design of monocarbohydrate-functionalized surfaces.

  [START_REF] Cappitelli | Biofilm Formation in Food Processing Environments is Still Poorly Understood and Controlled[END_REF] Green, a nucleic live/dead cell stain (λexcitation= 485 -486 nm, λemission= 498 -501 nm). After 15 min in the dark, five points of view were taken on each channel with an inverse laser scanning confocal microscopy (Zeiss, LSM 710, Oberkochen, Germany) with an in-air 40x objective. Resulting pictures were converted to binary ones and the surface coverage rate was determined from the black and white pixels ratio with the open access software ImageJ. Finally, the average-to-average statistical comparisons of five independent adhesion assays were determined by Analysis of Variance (ANOVA) with Tukey HSD test.

  , D-galactose and D-mannose are the most abundant carbohydrates that play crucial roles in life such as in structural role, cell-cell interactions or in the regulation of genes. Their multiple hydroxyl functions are very attractive for elaborating material through easy modulation of physicochemical properties. S/O-D-Glucopyranosidic (S/O-Glcp), S/O-Dgalactopyranosidic (S/O-Galp) and S/O-D-mannopyranosidic (S/O-Manp) surfaces were obtained by "clicking" propargyl glycosides on commercially alkyl azide-functionalized glassplates. The facility and the high selectivity in a wide range of solvent and temperature make the CuAAC reaction, the most used "click" reaction in organic chemistry and materials. S/O-Propargyl D-glycosides were already synthesised and described from commercially available peracetylated glycosides in two steps (Figure2). They were obtained by Lewis acid catalysed glycosylation of propargyl alcohol from the desired peracetylated monosaccharide with complete 1,2-trans diasteroselection. Then, acetyl groups were removed under Zemplén basic conditions.1 H NMR spectra showed characteristic signals in accordance with previous published data[START_REF] Morotti | Semi-Synthesis of new glycosidic triazole derivatives of dihydrocucurbitacin B[END_REF]. In the case of propargyl S-glycopyranosides, it was interesting to find a pathway to stave off the risk of the foul odour and toxicity of thiol compounds. In this way, 4a-4c were synthesised in two steps from roots of Ibatullin[START_REF] Ibatullin | A General Procedure for Conversion of S-Glycosyl Isothiourea Derivatives into Thioglycosides, Thiooligosaccharides and Glycosyl Thioesters[END_REF][START_REF] Ibatullin | Reaction of 1,2-trans-glycosyl acetates with thiourea: A new entry to 1-thiosugars[END_REF] giving access to derived Sglycosides in one-pot procedure. Briefly, a glycosyl isothiuronium salt was formed in situ from a peracetylated carbohydrate. The latter intermediate was allowed to react with propargyl bromide in the presence of trimethylamine. The resulting products 3a-3c were subsequently deprotected by transesterification to afford the desired thioglycosides 4a-4c. All compounds and their intermediates were characterized by 1D NMR and obtained data are similar to the literature[START_REF] Lo Conte | Multi-molecule reaction of serum albumin can occur through thiol-yne coupling[END_REF][START_REF] Giguère | Synthesis of stable and selective inhibitors of human galectins-1 and -3[END_REF][START_REF] Lanzetta | Conversion of yeast mannan polysaccharide in mannose oligosaccharides with a thiopropargyl linker at the pseudo-reducing end[END_REF].

Figure 2 .

 2 Figure 2. General procedure for the synthesis of O-and S-propargylated glycosides from peracetylated D-glucoside (a), D-galactoside (b) and D-mannoside (c). i) Propargyl alcohol, BF3.OEt2, DCM, overnight, r.t.; ii) Et3N, H2O/MeOH, 20 °C; iii) Thiourea, BF3.OEt2, MeCN, reflux then propargyl bromide, Et3N, r.t.

Figure 3 .

 3 Figure 3. Grafting pathway of glycosides 2a-c and 4a-c on commercial azide-functionalized glass surface.

Figure 4 .

 4 Figure 4. Contact angle of a water droplet deposited on the surfaces. The values are the means of six measurements on ten surfaces from independent batches. * indicates a significant difference at p < 0.01 with respect to azide surface (ANOVA).

Figure 5 .

 5 Figure 5. XPS narrow scans of the N 1s region of azide-functionalized surface (a) and O-Glucose functionalized surface (b). After the cycloaddition, the initial double signal disappeared to give one broaden signal.

Figure 6 .

 6 Figure 6. Qualitative and quantitative analysis of the bacterial adhesion. (a) Fluorescent confocal images of adhered MPAO1 on Azide, OH, O-Glcp and S-Glcp surfaces (Scale bar = 20 µm). (b) Surface coverage rates of MPAO1 and PAΔlecA. *Significant difference with respect to azide surface (p < 0.01).

  biocidal strategy to limit and to prevent the bacterial attachment. In this respect, O/S-propargyl β-D-glucoside, O/S-propargyl β-D-galactoside or O/Spropargyl α-D-mannoside were grafted by click chemistry giving amphiphilic glass surfaces.

Table 1 .

 1 Determination of the surface free energy of the different surfaces from contact angles of water (W), formamide (FA) and diiodomethane (DI).

			Surface free energy components a		Total
		Azide	42.25 0.50	5.19	3.23	45.48
		OH	38.59 0.76 11.00	5.79	44.37
		O-Glcp	40.89 0.83 19.11	7.94	48.83
		O-Galp	40.50 0.97 18.80	8.56	49.06
		O-Manp 40.35 0.97 18.81	8.56	48.92
		S-Glcp	40.31 1.14 16.83	8.74	49.05
		S-Galp	40.20 1.08 17.93	8.78	48.99
		S-Manp 40.94 1.22 16.84	9.06	49.99
		MPAO1 13.97 8.79 52.63 43.02	57.00
	a Values in mJ/m². The surface free energy components are	: Lifshitz -van der Waals
	component,	: Lewis acid component,	: Lewis basic component,	: acid-base
	component.

Table 2 .

 2 Values (in mJ/m²) of the adhesion free energy between MPAO1 and surfaces.

			MPAO1
		∆	∆	∆
	Azide	3.41	7.59	11.00
	OH	2.87	11.19	14.07
	O-Glcp	3.21	15.43	18.65
	O-Galp	3.16	14.94	18.10
	O-Manp	3.14	15.16	18.29
	S-Glcp	3.13	13.62	16.75
	S-Galp	3.11	14.29	17.41
	S-Manp	3.22	13.46	16.68

Table S1 .

 S1 Values of surface free energy components of solvents using for the determination of surface free energy and adhesion free energy.

	Water (W)	21.8	25.5	25.5	51.0	72.8
	Formamide (FA)	39.0	2.28	39.6	19.0	58.0
	Diiodomethane (DI)	50.8	0	0	0	50.8

Table S2 .

 S2 Contact angle values used for the determination of surface free energy and adhesion free energy.

			Contact angles (°)
			3	45
	Azide	75.6 ± 1.6	47.8 ± 1.1	34.5 ± 0.6
	OH	67.8 ± 1.9	46.7 ± 1.1	42.0 ± 1.8
	O-Glcp	56.5 ± 1.7	36.7 ± 0.8	37.4 ± 1.1
	O-Galp	56.3 ± 1.6	36.1 ± 1.1	38.2 ± 1.8
	O-Manp	56.4 ± 1.9	34.3 ± 0.9	38.5 ± 1.3
	S-Glcp	57.9 ± 1.9	35.7 ± 1.2	38.6 ± 1.4
	S-Galp	57.0 ± 1.8	35.8 ± 1.0	38.8 ± 2.2
	S-Manp	57.1 ± 1.5	33.8 ± 1.1	37.3 ± 2.0
	MPAO1	26.2 ± 2.1	34.8 ± 2.7	87.2 ± 1.4

Acknowledgements

The authors are grateful to the GlycoOuest (http://www.glyco-ouest.fr/) network supported by the Région Bretagne and the Région Pays de la Loire. We thank the Ministère de l'Enseignement Supérieur et de la Recherche, the CNRS for Financial supports

Conflicts of interest

There are no conflicts to declare.

Supplementary Information

NMR data of glycosides, kinetic of water contact angles, XPS large spectra, microscopy images of adhered MPAO1, surface free energy components of solvents and contact angle values with water, formamide and diiodomethane.

Supplementary Information

Pseudomonas aeruginosa resistance of monosaccharide-

Functionalized glass surfaces

Mathieu Scalabrini a , Jonathan Hamon c , Isabelle a , Vincent Ferrières b and Karine Réhel a a Univ Bretagne-Sud, EA 3884, LBCM, IUEM, F-56100 Lorient, France.

b Univ Rennes, école Nationale Supérieure de Chimie de Rennes, CNRS, ISCR -UMR6226, F-35000 Rennes, France. [1] Title compound was synthesized from 1a (2.59 mmol, 1 g) and afforded product (2.11 mmol, 460 mg, 81%) as white powder. Rf = 0.21 in DCM/MeOH (9:1). [1] Title compound was synthesized from 1b (2.59 mmol, 1 g) and afforded product (1.41 mmol, 308 mg, 55%) as white powder. Rf = 0.21 in DCM/MeOH (9:1). 1 H NMR (400 MHz, D2O) δ 4.58 (d, J1,2 = 7.9 Hz, 1H, H1), 4.54 -4.42 (m, 2H, H1'), 3.94 (dd, J4,3 = 3.4 Hz, J4,5 = 1.0 Hz, 1H, H4), 3.83 -3.70 (m, 3H, H6, H5), 3.67 (dd, J3,2 = 9.9, J3,4 = 3.4 Hz, 1H, H3), 3.54 (dd, J2,3 = 9.9 Hz, J2,1 = 7.9 Hz, 1H, H2), 2.92 (t, J3',1' = 2.4 Hz, 1H, H3'). 13 Propargyl 1-thio-β-D-glucopyranoside (4a) [3] Title compound was synthesized from 3a (2.48 mmol, 1 g) and afforded product (1.56 mmol, 364 mg, 63%) as yellow powder. Rf = 0.18 in DCM/MeOH (9:1). 1 H NMR (400 MHz, D2O) δ 4.74 (d, J1,2 = 9.9 Hz, 1H, H1), 3.92 (dd, J6a,6b = 12.5 Hz, J6a,5 = Propargyl 1-thio-β-D-galactopyranoside (4b) [4] Title compound was synthesized from 3b (0.99 mmol, 400 mg) and afforded product (0.64 mmol, 149 mg, 81%) as white powder. Rf = 0.18 in DCM/MeOH (9:1). Propargyl 1-thio-α-D-mannopyranoside (4c) [5] Title compound was synthesized from 3b (1,24 mmol, 500 mg) and afforded product (1.02 mmol, 240 mg, 74%) as colorless oil. Rf = 0.18 in DCM/MeOH (9:1).

Propargyl 1-O-β-D-glucopyranoside (2a)

Propargyl 1-O-β-D-galactopyranoside (2b)

Van Oss, Chaudhury and Good

The surface free energy for a solid (s) is divided into two components, the Lifshift-van der Waals ( ) and the Lewis acid-base ( ). The latter is dependent on the acid component and the base component giving the relation:

2

A combining rule from Young and Fawkes equation provides a relation between the interfacial free energy and the contact angle θ of a liquid (l) with the surface:

Measuring the contact angle of a liquid with known surface free energy components leads to an equation with three unknown variables; , , (Table S1). To solve this, an equation system could be applied with three liquid with known components such as water (W), formamide (FA) and diiodomethane (DI):


Denis Duboule1

organizing potential of the embryoid bodies Stevens and Pierce described in the 1950s is now 47 being resurrected by the increasing interest of the community of developmental and stem 48 cells biologists to engineer in vitro and study various types of pseudo-embryos showing 49 similar self-organizing properties.

Maine, set out to examine the deleterious effect of cigarette paper on mice under the auspices 4 of a grant supported by the tobacco industry. During the course of his research, he noticed 5 that some mice from the 129 inbred strain developed large testicular tumors. Rather than 6 discarding these mice, he bred them and selected a strain (129 Sv ) with a high incidence of 7 tumors that displayed a large realm of differentiated tissue such as teeth, hairs, muscles or 8 endothelial structures, resembling classical human ovarian teratomas (Stevens and Little, 9 1954). After discovering that these tumors were of germ cells origin, several colleagues 10 including Barry Pierce from the University of Colorado in Denver, started to transplant pieces 11 of such tumors back into the animal to see for how long these germ cells tumors could be 12 maintained and the diversity of cell types they could form. By transplanting back into the 13 murine peritoneum, Pierce and Stevens obtained fluid ascitic tumors which contained 14 thousands of free-floating structures that displayed strikingly resembled day 5 or 6 mouse 15 embryos, both in term of the appearance of cellular layers and their organization. 16 17 In this classic paper published in Developmental Biology in 1960 (Stevens, 1960), Stevens 18 tests the developmental capacities of these embryonic structures and compares their 19 morphogenetic potential to that of the embryos themselves. After isolating such embryo-like 20 structures from ascites tumors, he transplanted them singly into the anterior chamber of a 21 mouse eye. A tumor grew, which was isolated, minced and re-transplanted subcutaneously. 22

The grafts were observed daily, 'except Sunday', as specified in the paper. In one graft, 23

Stevens noted a particularly well-developed embryo-like object, which had generated 24 structures ''unmistakably similar to portions of normal embryo of about 9 days'', with 25 neuroepithelial cells, amnion, yolk sac epithelium and mesodermal cells. In the discussion, 26

Stevens explains that Pierce and Dixon had previously observed the presence, in ascites 27 tumors, of a central core of embryonal carcinoma (EC) cells surrounded by a layer of visceral 28 yolk sac, which they had referred to as 'granules ' (Pierce and Dixon, 1959). In 1959, Stevens 29 further analyzed and described these granules and called them teratomatous 'embryoid 30 bodies', a well inspired name that survived 60 years of highly competitive research. In the 31 conclusion of the attached 1960 publication, Stevens writes: '..the embryoid bodies derived 32 from the testicular teratomas of strain 129 mice have similarities in embryonic potency as 33 well as in morphology to normal mouse embryos'. This seminal finding opened the route 34 towards using EC cells, rapidly replaced by ES cells, to engineer modified mice via 35 blastocyst injection (Dewey et al., 1977). 36

the blastocyst stage in the late 1970's, took me to Leroy's house, which was literally built

into the trees in Mount Desert island, Maine. Leroy offered us several drinks and told us

his discovery, yet without any bitterness or resentment. He was a kind, civilized and modest

person. His contributions to the emergence of ES cells and, consequently, to the experimental

and conceptual revolution that followed, were immense. The surprising and beautiful self-

2Department of Genetics and Evolution, University of Geneva, 1211 Geneva 4, Switzerland, 3Collège de

France, Paris, France.

2


G Parry

S Hamade

J Durinck

C Coupeau

J Colin

Influence of interface steps on the buckle delamination of thin films

Keywords: Buckling, Delamination, Layered Material, Finite Elements

The mechanics of thin film buckle delamination has been widely studied, with the assumption of perfect (flat) interfaces. This work is reporting a study of the evolution of an initially straight-sided buckle propagating in a thin film lying on a substrate in the presence of a step at the film/substrate interface, resulting from the plastic deformation of the substrate. Assuming the interface toughness depends on the mode mixity, it has been found using finite elements calculations that depending on the step orientation and height, the initially straight propagation of the buckle can be perturbed so that the buckle eventually propagates along the step. The critical parameters (orientation and height) of the step have been then determined, which characterizes the transition from a straight propagation perpendicular to the compression axis to a propagation along the inclined step. A "behavior" diagram has been finally presented for the buckle, where the propagation, either perpendicularly to the step or along it have been displayed versus the step height and orientation.

Introduction

The mechanical properties of thin films on substrates and more generally of multilayered structures have been the topics of intensive research in the fields of materials science, metallurgy and solid mechanics, because of the numerous applications of such structures in several engineering fields such as microelectronics [START_REF] Freund | Thin Film Materials: Stress, Defect Formation and Surface Evolution[END_REF] or aeronautics for example [START_REF] Chen | On the propagation and coalescence of delamination cracks in compressed coatings: with application to thermal barrier systems[END_REF]. It is now well admitted that during the deposition processes, high residual stress reaching sometimes several gigapascals are generated in the layers [START_REF] Drory | On the decohesion of residually stressed thin films[END_REF][START_REF] Girault | X-ray diffraction analysis of the structure and residual stresses of W/Cu multilayers[END_REF]. When these residual compressive stresses reach critical values, the interfaces can delaminate and the layers buckle [START_REF] Gille | Buckling instability and adhesion of carbon layers[END_REF][START_REF] Spiecker | Self-Assembled Nanofold Network Formation on Layered Crystal Surfaces during Metal Intercalation[END_REF]. Various buckle morphologies such as straight-sided [START_REF] Coupeau | Atomic force microscopy of in situ deformed nickel thin films[END_REF][START_REF] Coupeau | From thin film and coating buckling structures to mechanical properties[END_REF], circular [START_REF] Colin | Buckling patterns of gold thin films on silicon substrates: Formation of superimposed blisters[END_REF][START_REF] Kuznetsov | Hydrogen-induced blistering mechanisms in thin film coatings[END_REF], telephonecord [START_REF] Moon | Buckle delamination on patterned substrates[END_REF][START_REF] Faulhaber | Buckling delamination in compressed multilayers on curved substrates with accompanying ridge cracks[END_REF] buckles or network-like buckling structures [START_REF] Ni | On the growth of buckle-delamination pattern in compressed anisotropic thin films[END_REF] have been thus observed at the nano-and micro-scales with the help of atomic force and optical microscopes. Likewise, the effect of pressure mismatch between the upper and lower surfaces of the delaminated films has been studied and the re-deposition and snap-through phenomena have been analyzed versus the pressure and internal stress [START_REF] Colin | Redeposition of a straight-sided buckle under pressure[END_REF]. Recently, the plasticity of the films has been considered and the buckle morphologies have been characterized when plastic folding is considered at the circumference or at the top of the buckles [START_REF] Colin | Plastic Folding of Buckling Structures[END_REF][START_REF] Durinck | Molecular dynamics simulations of buckling-induced plasticity[END_REF][START_REF] Foucher | How Does Crystalline Substrate Plasticity Modify Thin Film Buckling?[END_REF]. The ageing of such buckles has been also investigated and the formation of telephone-cord structures has been explained by the secondary buckling of straight-sided buckles [START_REF] Thouless | Plane-strain, buckling-driven delamination of thin films: Model experiments and mode-II fracture[END_REF][START_REF] Audoly | Stability of straight delamination blisters[END_REF][START_REF] Colin | Worm-like delamination patterns of thin stainless steel films on polycarbonate substrates[END_REF][START_REF] Audoly | Secondary buckling patterns of a thin plate under in-plane compression[END_REF][START_REF] Chai | One dimensional modelling of failure in laminated plates by delamination buckling[END_REF][START_REF] Parry | Stability diagram of unilateral buckling patterns of strip-delaminated films[END_REF] as well as the formation of donut-and croissant-like structures from initially circular buckles, considering the coupled effects of plasticity and pressure [START_REF] Hamade | Effect of plasticity and atmospheric pressure on the formation of donut-and croissantlike buckles[END_REF]. The propagation of the buckles and the interface delamination have been also widely investigated and the concept of mode mixity (i.e. relative proportion of shear traction component as compared to normal traction component) dependence of the interface toughness has been introduced as a key ingredient for the description of interface cracks at the decohesion fronts of the buckle [START_REF] Hutchinson | Mixed mode cracking in layered materials[END_REF]. In particular, the fact that straight sided buckles indefinitely propagate along their longitudinal axis without lateral expansion has been explained. Recently, finite element simulations have been carried out to explain the propagation of telephone-cord buckles including an adhesion work dependence on the mode mixity [START_REF] Faou | How does adhesion induce the formation of telephone cord buckles?[END_REF]. The pinning of the buckle front in mode II has been thus found to be responsible of the corresponding configurational telephone-cord instability. At the microscopic scale, it has been shown that a variety of interface imperfections such as fracture zone [START_REF] Cordill | The effects of plasticity on adhesion of hard films on ductile interlayers[END_REF][START_REF] Abdallah | Buckle initiation and delamination of patterned ITO layers on a polymer substrate[END_REF], roughness [START_REF] Evans | Effect of interface undulations on the thermal fatigue of thin films and scales on metal substrates[END_REF][START_REF] Mumm | On the role of imperfections in the failure of a thermal barrier coating made by electron beam deposition[END_REF] and chemical impurity [START_REF] Wang | Measurement and analysis of buckling and buckle propagation in compressed oxide layers on superalloy substrates[END_REF] can be preferential areas where the interface delamination may start. Likewise, atomic steps at the interfaces [START_REF] Storåkers | Imperfection sensitivity at delamination buckling and growth[END_REF][START_REF] Foucher | How Does Crystalline Substrate Plasticity Modify Thin Film Buckling?[END_REF] have been identified as topological defects resulting from the plasticity of the multilayered structures where the delamination process can be initiated. Indeed, it has been recently found by means of atomistic simulations that interface steps can play a key role on the interface delamination and lead to interfacial gliding at the edges of buckles providing a physical explanation of the limited growth of the buckle (Ruffini et al., 2013b). Likewise, for Cu-Zr systems, the buckling and post-buckling behaviors have been analyzed versus the film deformation and film thickness [START_REF] Wu | Unique buckling and postbuckling behavior of Cu-Zr amorphous films on compliant substrates[END_REF]. For titanium nitride films deposited on stainless substrates, the cracking and buckling phenomena have been also investigated for low and strong adhesion energy [START_REF] Guo | Substrate slip steps promote cracking and buckling of thin brittle film[END_REF].

In the context of the finite element investigations of interface delamination and thin film buckling [START_REF] Faou | How does adhesion induce the formation of telephone cord buckles?[END_REF][START_REF] Nilsson | A finite element analysis of configurational stability and finite growth of buckling driven delamination[END_REF][START_REF] Jensen | Numerical analysis of buckling-driven delamination[END_REF][START_REF] Gruttmann | A finite element model for the analysis of buckling driven delaminations of thin films on rigid substrates[END_REF], the study of the complex interactions between crack propagation, buckling and interface imperfections such as atomic steps thus appears as a challenging area of research that could provide a more complete description of the thin film ageing. It is the topic of the present work to investigate, from a numerical point of view through finite element simulations, the influence of an interface step on the propagation of a straight-sided buckle. The Paper is organized as follows. The experimental evidences of buckle propagation in the vicinity of interface steps are first presented for a Ni thin film deposited on a LiF substrate. The modeling technique is then described and the coupled effects of the mixed mode in-terface toughness and interface steps are discussed on the buckle propagation. Finally, the influence of the step height and step orientation with respect to the compression axis has been analyzed.

Buckle observations above a stepped interface

Ni thin films of thickness h = 150 nm have been deposited by sputtering physical vapor methods on the (001) surface of LiF single crystals. The coated samples were then plastically deformed up to p = 2% by uniaxial compression of the substrate along the [110] direction (Fig. 1a). As expected from Schmid factors, the slip system (011) [011] was activated, leading to slip traces lying along the [100] direction at α = 45 • from the compression axis. The slip traces are located in this case at the interface between the film and the substrate [START_REF] Foucher | How Does Crystalline Substrate Plasticity Modify Thin Film Buckling?[END_REF] and make an emerging angle of θ = 45 • with respect to the (001) plane. The out-of-plane displacements of these steps can be observed on the lower right part of the optical observation presented in Fig. 1b. Straight-sided buckles are also evidenced in Fig. 1b (see the upper part), propagating at 45 • from the compression axis. A characteristic profile shown in Fig. 1c demonstrates that the buckles are located just above the steps. In this case, the height H of the three involved slip traces (normalized over the thickness of the film) are equal to H = 1.3, 1.3 and 2.0 respectively. This result clearly evidences that the buckles preferentially propagate along the step rather than perpendicularly to the compression axis, as however expected from usual mechanical considerations in the case of a flat surface. One question naturally emerges concerning the possibility for buckles to select their own directions of propagation when the step and the compression axis are not perpendicular. It is the topic of this work to numerically determine the conditions and relevant parameters for such a step-induced deviation to occur.

Modeling

The following simulations of the interface delamination, thin film buckling and buckle propagation have been performed with the help of the finite element software ABAQUS using an explicit formulation [START_REF] Abaqus | Manuals Collection[END_REF]. A geometrically non-linear plate lying in the (Oxy) plane is used to model the thin film of thickness h (see Fig. 2a for axes). The plate described in the (Oxy) plane by a square flat domain of length L and width l has been meshed with the help of triangular shell elements. Half of a delaminated disc (hd) of diameter b hd has been introduced as an adhesion-free domain at x = 0, in the left part of the film, to initiate the buckling once an internal compressive stress has been applied to the film. The out-of-plane displacement along the (Oz) axis is labeled w, the in-plane ones along (Ox) and (Oy), u and v, respectively. The substrate surface is modeled by a rigid plane such that the plate displacements in the lower half-space are not allowed within this unilateral contact description. The film-substrate interface is described by a cohesive zone model. The film-substrate structure has been submitted to two consecutive loading operations:

• Introduction of the interface step

As shown by [START_REF] Foucher | How Does Crystalline Substrate Plasticity Modify Thin Film Buckling?[END_REF] and Ruffini et al. (2013b), an uniaxial compressive deformation of a coated substrate leads to the emergence of dislocations at the film/substrate interface, which results in the formation of an interface step whose height is of the order of the nanometer. In order to mimic the formation of the interface step, an out-of-plane displacement w = H where H is the step height, has been imposed at the interface region that is located under the right part of the film. This step is thus formed along a slip plane inclined at an angle θ = 45 • with respect to the interface plane which corresponds to the shearing of the substrate along the slip plane (110) (see Fig. 2b). The step direction is inclined at an angle α with respect to the (Oy) axis which is the natural direction for the straight-sided buckle to propagate in case of a flat interface. A displacement field is thus imposed on the right part of the cohesive elements, the left part being fixed to zero (see Fig. 2b) such that:

u = -H sin α, (1) v = -H cos α, (2)
w = H. (3)
At the end of this loading operation, a delaminated part of the film is formed above the step as a strip of width b.

• Application of the residual stress

During the second step, in order to generate a compressive residual stress in the simulation, a thermal expansion is applied to the plate with components yy = 0 and xx = -ν f 0 , with 0 > 0. The expansion is actually restrained by the rigid substrate in the initial unbroken flat state, so that the actual total strain components in the film are zero. The eigenstrain created by the thermal expansion is therefore compensated by exactly opposite elastic strain components, generating a uniaxial compressive stress state:

σ xx = 0, (4) σ yy = -E f 0 = -σ 0 .
(5)

The Young's modulus and Poisson ratio of the film are labeled E f and ν f , respectively.

The following boundary conditions have been considered on both plate edges lying along the (Ox) axis:

v = 0, (6) ∂w ∂y = 0. ( 7
)
In the following, the non-dimensional parameters H = H/h, x = x/h, ỹ = y/h, w = w/h, b = b/h, bhd = b hd /h and bc = b c /h have been introduced, where b c is the critical width for the buckle to appear on the stepped film/substrate interface whose implicit expression writes (Ruffini et al., 2013a):

tr 0 = π 2 3 1 b2 c + 3 4 H2 b2 c - H bc tan θ , (8)
with tr 0 > 0, the critical transversal strain related to 0 through the relation:

y y = -tr 0 = -0 (cos 2 α -ν f sin 2 α), (9)
where y y is the strain applied along the (Oy ) direction perpendicular to the step. In order to study the interface delamination, the dependence of the interface toughness with mode mixity has been considered [START_REF] Hutchinson | Mixed mode cracking in layered materials[END_REF]. The dimensionless interface toughness per unit area is then given by [START_REF] Freund | Thin Film Materials: Stress, Defect Formation and Surface Evolution[END_REF]Ruffini et al., 2013b,a):

Γc (ψ) = Γ c (ψ) G 0 , (10)
with G 0 , the elastic energy per unit area stored in the film under the uniaxial compression described in Eqs. 4 and 5:

G 0 = hσ 0 2 2E f . (11)
Quadrilateral shell elements are used to mesh the film whose element size is chosen such that the delaminated strip of the film for the minimal value of b obtained after the first loading operation is divided into 20 of those elements. The plate length is L = L/h = 500 and its width l = l/h = 250. A mixed-mode cohesive zone model is used to describe the interface rupture (as described in [START_REF] Faou | How does adhesion induce the formation of telephone cord buckles?[END_REF]). Cohesive elements are inserted between the plate elements and the substrate. The traction versus separation law is linear/softening. The traction component normal to the (Oxy) interface plane is denoted T n , and the shear traction component, T s (lying in the (Oxy) plane). An effective traction stress T e can be thus defined for the mixed mode as T e = T 2 n + T 2 s . The interface behavior is linear and reversible (

- → T = K - → δ )
until a peak traction is reach. The peak values for the traction components is defined by a quadratic criterion:

T n T 0 n 2 + T s T 0 s 2 = 1, (12)
At this threshold marking damage initiation, the values T n and T s are taken by T n and T s respectively, leading to T e = (T n ) 2 + (T s ) 2 . Once this criterion is achieved, the mode mixity is evaluated depending on the mixity of the loading described by the angle ψ:

tan(ψ) = T s T n , (13)
The normal T n , shear T s and effective T e tractions have been plotted in Fig. 3 as a function of the normal δ n , shear δ s and effective δ e = δ 2 n + δ 2 s separations respectively, for pure normal (mode I), pure shear (mode II) and mixed mode loadings. For the I and II pure loading modes, the tractions T n and T s linearly increase when the relative normal δ n and shear δ s displacements increase, with K the loading stiffness defined as K = T n /δ n = T s /δ s , where T n and T s are the maximum values of T n and T s obtained for δ n and δ s , respectively. It is also underlined that T n and T s linearly decrease for δ n > δ n and δ s > δ s and finally vanish at δ f n and δ f s , respectively. This second decreasing regime describes the damage of the cohesive elements that have been modeled introducing the interface toughness per unit area:

Γ c (0) = G c I = 1 2 T n δ f n , (14)
for pure mode I and:

Γ c (π/2) = G c II = 1 2 T s δ f s , (15)
for pure mode II. For the mixed mode loading, the first reversible regime is characterized by an effective traction T e linearly increasing with its relative displacement δ e , with δ e < δ e . using the expression of Γ c (ψ) given by [START_REF] Hutchinson | Mixed mode cracking in layered materials[END_REF]:

Γ c (ψ) = G c I [1 + tan 2 (ηψ)], (16)
where η is a parameter determined by the condition Γ c (π/2) = G c II , corresponding to the effective adhesion energy in pure mode II. The second regime is then characterized by the irreversible decrease of T e as δ e increases until it vanishes when δ e reaches its final value δ f e = 2Γ c (ψ)/T e . In the framework of this model, the simulations have been performed using the following parameters for the film. The elastic modulus E f = 205 GPa, the Poisson ratio ν f = 0.30 are considered for an elastic plate of thickness h = 20 nm. A value 0 = 0.59% is taken for the eigenstrain parameter in the plate leading to the resulting uniaxial compressive stress σ 0 = 1200 MPa. The non-dimensional adhesion-free half-disk width is bhd = 50. The ratios between the adhesion energy and the elastic energy G 0 have been taken to be G c I /G 0 = 0.546 and G c II /G 0 = 88.7 (corresponding to a value η = 0.95) for the modes I and II, respectively. The maximum cohesive tractions are then T 0 n = 0.02σ 0 and T 0 s = 0.24σ 0 and the stiffness is

K = 6.6 × 10 2 (T 0 n ) 2 /G 0 .

Results and Discussion

Buckle propagation on a stepped interface

The formation (at 0 = 0) of the step of height H has generated a delaminated strip of the film of width b. The values of b obtained for different values of H ranging from 0.05 to 0.30 have been first extracted and plotted in Fig. 4 versus H, together with the critical buckling width bc defined in Eq. ( 8). It is observed in this Fig. 4 that although b linearly increases with H, it is always below the critical buckling width bc calculated for 0 = 0. This explains why the delaminated strip of the film does not spontaneously buckle once the step is formed at 0 = 0. When the system is under strain, i.e. for 0 = 0.59%, the critical value bc depends on α angle. For the different considered values of α (10 • , 15 • , 20 • , 25 • ), b is still below bc despite its α-dependence. The film is thus not assumed to buckle above the step when the strain 0 is applied to it. On the other hand, the initially delaminated half-disk has been sized such that it is allowed to buckle, its diameter bhd = 50 being chosen larger than the critical value.

In order to illustrate the effect of the interface steps on the propagation of a buckle from the half-disk of the film already delaminated and buckled, two snapshots at two different times t 1 and t 2 have been first presented in Figs. 5a andb, for a planar ( H = 0) and stepped ( H = 0.30) interface, with t 2 > t 1 . Applying the loading procedure described in the previous section, it is observed in Fig. 5a that, in case of a planar interface, the straight-sided buckle propagates along the (Ox) axis perpendicular to the compression axis (Oy), leading thus to the formation of the classical Euler column [START_REF] Hutchinson | Mixed mode cracking in layered materials[END_REF]. When an interface step with an tilt angle α = 10 • with respect to (Ox) axis is considered, it is found in Fig. 5b that the buckle is progressively deviated by the step and at time t 2 , the buckle is observed to be aligned with the step. Beyond these two extreme cases, the problem of the determination of the relevant parameters controlling the buckle deviation such as the deviation angle or step height for example can be raised. This point is addressed in the next subsection.

Influence of the interface step height and tilt angle on the buckle propagation

In order to characterize the deviation of the buckle in the vicinity of the step, the following procedure has been used. For each buckle, the maximum deflection of the film has been plotted in Figs. 6 in the (Oxy) plane, the tilt angle α being fixed to 10 • in a first time. In Fig. 6a, it is observed that starting from the straight-sided buckle propagating along the (Ox) axis in case of a plane interface (black line) when H = 0, the buckle propagation is perturbed by the step as its height increases until 0.3. At this point, it has to be underlined that the ultimate buckle deviations observed at the end of the simulations (in the region x = 500) have been checked (but not shown) to be a numerical artifact due to the size of the simulation box. To determine for this particular angle α = 10 • , the critical height beyond which the buckle is definitely attracted by the step, the size of the simulation box has been increased (×3) and adjusted to avoid the boundary effect in Fig. 6b, where it can be deduced that the changing in the buckle direction occurs for 0.15 < H < 0.2. Indeed, for H = 0.15, the buckle overcomes the step attraction and recovers its (Ox) direction of propagation, while for H = 0.2 the buckles finally follows the step direction. In the following, this procedure consisting in enlarging the simulation box has been applied each time it is required for the determination of Hc to avoid the boundary artifact.

The next step of the present work has been to determine the variation of this critical height Hc as a function of the tilt angle α of the step with respect to the compression axis. To do so, the maximum of the buckle deflection has been plotted versus x and ỹ for H ∈ [0, 0.30] and α = 15 • , 20 • and 25 • in Figs. 7. It is observed that as the tilt angle of the step with respect to the compression axis increases, the propensity of the step to capture the buckle decreases. For α = 25 • in Fig. 7c, although locally modified, the buckle trajectory always expands beyond the step, whatever H.

In order to give a physical explanation of this phenomenon identified as buckle deviation, the dimensionless elastic energy density U el /U 0 stored in the stepped film has been first displayed in Figs 8a and8b, for two values of the step height, i.e. H1 = 0.15 and H2 = 0.25, with U 0 the elastic energy stored in the planar film under strain. It is observed that, in both cases, this elastic energy reaches largest values on the step, with maximum values equal to U max1 el /U 0 = 6.3 and U max2 el /U 0 = 9.1 for the step heights H1 and H2 , respectively. Hence, as soon as the buckle delamination front reaches the step, it tends to deviate towards the step direction in order to increase the strain energy released per unit of delaminated area (i.e. to increase the energy release rate G). In the meantime, propagation towards the vertical direction is optimal in the context of a pure horizontal uniaxial compression. The rotation angle of the buckle therefore sets as a compromise between the additional energy release provided by the step and the high driving force provided by the uniaxial compression. Since a higher value of H offers a higher density of strain energy to be released along the step, it is to be expected that (for a given step tilt angle), the rotation rate of the step per unit length of front propagation will increase with H. This is what is observed, for example in Fig. 7.

It is sometimes observed that the buckle tends to start aligning with the step in the first stages of the interaction, then leaves the step to recover a vertical trajectory. In order to explain this, the dependence of the interface toughness with respect to the mode mixity of the loading has to be considered. Everywhere at the crack front, the maximum work of separation that can be spent per unit area corresponds to the whole elastic energy stored in this unit area. This gives a maximum value of the mode mixity angle ψ max above which the buckle can not propagate (since Γ c is a monotonically increasing function of ψ). This hence provides a local delamination arrest criterion in terms of ψ:

G max el = Γ c (ψ max ), (17)
where Γ c is defined in Eq. 16 and G max el = U max el /S, with S the surface of the elements. The evolution of Γ c /G Ic versus ψ has been then plotted in Fig. 9a for η = 0.95 and also for η = 0.7, this last value being discussed later. From this variation in the case where η = 0.95, three maximum values of the mode mixity angle,

ψ 0 = 40 • , ψ 1 max = 76 • and ψ 2 max = 80 • , have been determined for G 0 /G Ic = 1.83, G max1 el /G Ic = 11.53 and G max2 el /G Ic = 16
.65, in the case of the planar film and the stepped one with step height H1 and H2 , respectively. In Figs 9b and9c, ψ has been represented on the surface of the film for η = 0.95 in cases of H1 and H2 . In both cases, it is found that the propagation of the right-hand side of the buckle stops once the mode mixity angle has reached the maximum value of 40 • already defined. On the other side, at the step location, it appears that for H1 , the maximum value of ψ 1 max has been reached, resulting in a propagation front pining somewhat similar to what is observed in [START_REF] Faou | How does adhesion induce the formation of telephone cord buckles?[END_REF], such that the buckle can not propagate along the step anymore and thus deviates (Fig. 9b). Conversely, for H2 , the maximum value ψ 2 max of the angle is never reached and the buckle propagates along the step (Fig. 9c). In addition to this analysis, it is underlined that when η decreases, the toughness of the interface is less sensitive to ψ variations and the interface is considered as more brittle. This point is illustrated for η = 0.7. Indeed, it is deduced from Fig. 9a (green curve) that no maximum value of ψ can be obtained in the range [0 • , 90 • ] for H1 and H2 when η = 0.7. As a consequence, the delamination arrest criterion defined in Eq. 17 is never satisfied and the buckle can propagate along the step as shown in Fig. 9d for H1 = 0.15.

In order to characterize the effect of the step on the behavior of the buckle, the critical step height Hc has been determined from Figs. 6 and 7 and has been used to build a propagation diagram for the step in Fig. 10, where the two different behaviors, i.e. the propagation beyond or the propagation in the neighborhood of the step have been displayed in the (α, H) plane. It is confirmed on this diagram that the "buckle capture" by the step is less and less favorable as the tilt angle of the step increases with respect to the compression axis and the step height decreases.

Finally, it has to be emphasized that, once captured by a step, particular configurations may allow for the buckles to escape from it. It is the case when two buckles propagate towards each other. As an example, the buckle propagation along a step of height H = 0.3 and tilt angle α = 10 • has been studied in Fig. 11 where two buckles are observed to propagate from the two opposite sides of the simulation box towards its center. It is found in this case that when the two propagating buckles are sufficiently close to each other, they can deviate from their initially straight trajectory due to the repulsive interaction of the two stress fields generated ahead of the propagating fronts. Once the buckles are deviated, they are suspected to propagate perpendicularly to the compression axis, each one lying in a different planar regions of the film.

Conclusions

The coupled effects of interface plasticity and mixed mode dependence of the interface toughness has been investigated by means of finite element simulations of the propagation of a buckle appearing in a thin film deposited on a substrate submitted to compressive strain. It has been found from a numerical point of view that the introduction of an interface step can strongly modify the direction of the buckle propagation. Indeed, provided the tilt angle of the step with respect to the compression axis is sufficiently small, the buckle has been found to be captured by the step when its height exceeds a critical value. Based on an analysis of the elastic energy density stored in the vicinty of the step and on the determination of the mode mixity angle, an explanation to the buckle deviation has been proposed through the possibility or not for the mode mixity angle to reach its maximum value when the buckle propagates onto the step. A propagation diagram has been finally displayed versus the height of the step and its tilt angle with respect to the compression axis. The regions characterizing the buckle propagation along the step and far away from it have been thus identified. The possibility that the buckle may escape from the step and propagate towards the planar regions of the film has been finally discussed.

This study shows, for the first time at our knowledge, how buckles can strongly interact with surface heterogeneities during their propagation, and indicate routes for thin films patterning assisted by controlled heterogeneities through substrate patterning.

For Ni thin films deposited on LiF substrates, the possibility of local deviation of the buckles should be now experimentally investigated. Likewise, numerical simulations coupled with a theoretical analysis of the buckle propagation should be also performed to characterize the mechanical behavior of this particular Ni/LiF system.

Acknowledgments

Computations have been performed on the supercomputer facilities of the Mésocentre de calcul Poitou-Charentes.

i δ i δ f i T i δ i K(1 -d) K

  doi:10.1016/j.matlet.2018.11.095.

Figure 1 :

 1 Figure 1: Optical observations of the surface of a Ni thin film deposited on a plastically deformed LiF substrate. (a) General view at the micrometric scale of the buckled surface. (b) AFM observations of the buckles formed above the interface steps. (c) Profile of the thin film surface for the particular area presented in (b).

Figure 2 :

 2 Figure 2: Schematic representation of the thin film on its substrate. (a) The film is initially delaminated on a half-disc of diameter b hd . (b) After the formation of the interface step, a film strip of width b is delaminated along the step.

T

Figure 3 :

 3 Figure 3: Schematic representation of the material law used for the interface delamination where the traction T i has been plotted versus the separation δ i . The I, II and mixed modes are referenced by the indices i = n, s and e respectively.

Figure 4 :

 4 Figure 4: Variation of the width of the delaminated film strip b versus H after the formation of the step. The variations of the critical width bc versus H have been added for 0 = 0 and for 0 = 0.59% (see Eq. 8). For 0 = 0.59%, different values of the step tilt angle have been considered: α = 10 • , 15 • , 20 • , 25 • .

Figure 5 :Figure 6 :

 56 Figure 5: Snapshots of the surface of the film at two different dimensionless time t 1 and t 2 with t 2 > t 1 . (a) Straight propagation of the buckle in a planar film, along the (Ox) axis, when a uniaxial compressive strain is applied in the (Oy) direction. (b) Inclined propagation of the buckle in the vicinity of a step for α = 10 • .

Figure 7 :

 7 Figure 7: Position in the (x, ỹ) plane of the maximum deflection of the buckle for different heights of the step. (a) α = 15 • . (b) α = 20 • . (c) α = 25 • .

Figure 8 :

 8 Figure 8: Strain energy distribution inside the film during the buckle propagation, normalized with respect to the remote unperturbed value, for two values of the step height and α = 10 • . Dark blue indicates the lowest energy areas, red the highest. (a) H = 0.15. (b) H = 0.25.

Figure 9 :

 9 Figure 9: (a) Evolution of the dimensionless toughness Γ c /G Ic of the interface with respect to ψ for η = 0.95 and η = 0.70. Mapping of the mode mixity angle ψ (in degrees) at which the interface has been broken during the buckle propagation for two values of the step height, with η = 0.95. (b) H = 0.15. (c) H = 0.25. (d) H = 0.15 with η = 0.70.

Figure 10 :Figure 11 :

 1011 Figure 10: Propagation diagram for the buckle in the (α, H) plane.