Kamila Łępicka

Piotr Pieta

Gregory Francius

Alain Walcarius

Wlodzimierz Kutner

Grégory Francius

Structure-Reactivity Requirements with Respect to Nickel-Salen Based Polymers for Enhanced Electrochemical Stability

come

Structure-reactivity requirements with respect to

nickel-salen based polymers for enhanced electrochemical stability

INTRODUCTION

Nickel-salen based polymers are Schiff base semiconducting macromolecular compounds composed of multiple monomer units of the same salen metal complex bound in para positions of their aromatic rings. Salen-metal complexes are an important class of tetradentate [O, N, N,

and O] bis-Schiff base ligands capable to coordinate cations of different transition metals, e.g., Cr, Ni, Cu, and Co. The undisputed advantage of salen ligands is their relatively easy synthetic preparation by condensation of two equivalents of salicylaldehyde or its derivatives with one equivalent of ethylenediamine or with its derivatives [START_REF] Cozzi | Metal-Salen Schiff base complexes in catalysis: practical aspects[END_REF]. The metal cation is coordinated through the imine nitrogen atom and the oxygen atom after phenolic proton removal. During anodic electropolymerization of nickel-salen complexes, electroactive films of conductive nickel-salen based polymers are formed. This electropolymerization proceeds only in moderately electron-donating solvents, including acetonitrile or propylene carbonate [START_REF] Łępicka | Spectroelectrochemical Approaches to Mechanistic Aspects of Charge Transport in meso-Nickel(II) Schiff Base Electrochromic Polymer[END_REF][START_REF] Goldsby | Oxidation of Nickel(II) bis(salicylaldimine) Complexes: Solvent Control of the Ultimate Redox Site[END_REF][START_REF] Hoferkamp | Surface-Modified Electrodes Based on Nickel(II) and Copper(II) bis(salicylaldimine) Complexes[END_REF]. However, it does not proceed in electrodonating solvents or in the presence of other electrondonor compounds capable of axial coordination to the nickel metal center of the salen monomer [START_REF] Łępicka | Spectroelectrochemical Approaches to Mechanistic Aspects of Charge Transport in meso-Nickel(II) Schiff Base Electrochromic Polymer[END_REF][START_REF] Goldsby | Oxidation of Nickel(II) bis(salicylaldimine) Complexes: Solvent Control of the Ultimate Redox Site[END_REF][START_REF] Castro | EPR and electrochemical study of nickel(III) complexes of bis(3,5dichlorosalicylaldehyde) o-phenylenediimine. Evidence for adduct formation with pyridines[END_REF][START_REF] Freire | Spectroscopic characterisation of electrogenerated nickel(III) species. Complexes with N2O2 Schiff-base ligands derived from salicylaldehyde[END_REF]. Then, oxidation of the resulting complex leads to formation of Ni(III) with no polymerization occurring. Besides, the effect of ligand polarization caused by polymer-based nickel-salen modified electrodes interaction with aromatic amines has already been studied [START_REF] Vereshchagin | Interaction of amines with electrodes modified by polymeric complexes of Ni with salen-type ligands[END_REF].

It was proposed that axial coordination of amine to the nickel center of the polymer or its bonding to salen ligand oxygen atoms was responsible for changes observed during electrooxidation of different nickel-salen polymers in the presence of aromatic amines [START_REF] Vereshchagin | Interaction of amines with electrodes modified by polymeric complexes of Ni with salen-type ligands[END_REF].

For application reasons, it is important to comprehend the difference between nickelsalen based polymers and hybrid polymers with incorporated metal-salen complexes or salen ligands [START_REF] Sukwattanasinitt | An Electrochemical Sensor from a Soluble Polymeric Ni-salen Complex[END_REF]. For instance, electropolymerization of thiophenes substituted with salen metal complexes results in formation of the so-called polythiophene hybrids of transition-metal bis(salicylidenimine)s. These hybrids are readily formed via polymerization of thiophene moieties, thus leading to thiophene based polymers with incorporated metal-salen centers [START_REF] Kingsborough | Polythiophene Hybrids of Transition-Metal Bis(salicylidenimine)s: Correlation between Structure and Electronic Properties[END_REF]. These hybrids were applied as catalysts for heterogeneous asymmetric catalysis [START_REF] Zulauf | Electropolymerization of chiral chromium-salen complexes: new materials for heterogeneous asymmetric catalysis[END_REF] and as recognition materials for fabrication of electrochemical sensors [START_REF] Eloi | Metallopolymers with emerging applications[END_REF][START_REF] Holliday | Chemoresistive Gas-Phase Nitric Oxide Sensing with Cobalt-Containing Conducting Metallopolymers[END_REF]. Importantly, electrochemical properties of such polythiophene hybrids of transition-metal bis(salicylidenimine)s or any other salen containing hybrid polymers and conductive metalsalen based polymers are markedly different with respect to their both charge transfer and polymerization mechanism.

There is a significant difference between so-called sterically hindered metal-salen complexes [START_REF] Clarke | Electronic structure elucidation in oxidized metalsalen complexes[END_REF] and conducting metal-salen polymers with or without steric hindrance on the imine bridge. That is, the former are unable to polymerize. In the latter, bulky substituents, inserted in ortho and para positions of the salen ligand of a metal-salen complex, stabilize the one-electron electro-oxidation products [START_REF] Thomas | Ligand-centred oxidative chemistry in sterically hindered salen complexes: an interesting case with nickel[END_REF]. Depending on such factors as the type of these substituents, metal coordination sphere, solvent, and temperature, the electronic structure of these products can be tuned between different forms containing Ni(II) or Ni(III). However, these products are not conducting in a whole continuous potential range because of their valence electrons lack of delocalization in the electric field. Instead, nickel-salen based polymers are electrochemically active in the positive potential range because of the presence of Ni(II)bridged conjugated electron-donating phenolic moieties.

Nickel-salen based polymer films reveal a mixed redox and π-conjugated conductivity.

Their faradic-capacitive charge storage mechanism is attributed to the transfer of delocalized valence electrons in polymer semiconductors, such as nickel-salen based polymers [START_REF] Akinwolemiwa | Mechanisms and Designs of Asymmetrical Electrochemical Capacitors[END_REF]. This transfer occurs in a continuous range of potentials involving two forms of oxidized Ni(II) bridged bis-Schiff base ligands, coupled in the polymer chain [START_REF] Vilas-Boas | New Insights into the Structure and Properties of Electroactive Polymer Films Derived from [Ni(salen)][END_REF][START_REF] Sizov | Redox transformations in electroactive polymer films derived from complexes of nickel with Salen-type ligands: computational, EQCM, and spectroelectrochemical study[END_REF]. The structures of oxidized bis-Schiff base ligands bridged with Ni(II) metal centers are responsible for the charge transfer in the nickel-salen based polymers. These oxidized structures were recognized spectroelectrochemically as bisphenolic radical cations and bisphenolic dications [START_REF] Łępicka | Spectroelectrochemical Approaches to Mechanistic Aspects of Charge Transport in meso-Nickel(II) Schiff Base Electrochromic Polymer[END_REF].

Understanding of the charge transfer mechanism allowed us earlier devising stable and durable unsubstituted in phenyl rings meso form of conductive nickel-salen based polymer, vis., poly[meso-N,N'-bis-(salicylidene)-2,3-butanediaminonickel(II)], abbreviated as poly(meso-Ni-SaldMe) [START_REF] Łępicka | Spectroelectrochemical Approaches to Mechanistic Aspects of Charge Transport in meso-Nickel(II) Schiff Base Electrochromic Polymer[END_REF].

The faradaic process engaging a conductive polymer film involves transfer of charges (electrons and ions) across the electrode-electrolyte solution interface. The widely used as charge storage materials derivatives of polypyrrole, polyaniline, or polythiophene, electroactive in the positive potential range, suffer from low charge-discharge cycling stability [START_REF] Liu | Polyaniline and Polypyrrole Pseudocapacitor Electrodes with Excellent Cycling Stability[END_REF][START_REF] Zotti | Irreversible processes in the electrochemical reduction of polythiophenes. Chemical modifications of the polymer and charge-trapping phenomena[END_REF]. This is because of the polymer volume change accompanying the counter ion shuttling in and out of the polymer during its charging and discharging [START_REF] Frackowiak | Supercapacitors based on conducting polymers/nanotubes composites[END_REF][START_REF] Vorotyntsev | Ionic exchange of the polypyrrole film with the PC lithium perchlorate solution during the charging process[END_REF][START_REF] Vorotyntsev | Charging process in polypyrrole films: effect of ion association[END_REF]. Therefore, we have recently applied a carbon nanostructure composite of synthetized poly(meso-Ni-SaldMe) as an anode material for the asymmetric supercapacitor [START_REF] Łępicka | Spectroelectrochemical Approaches to Mechanistic Aspects of Charge Transport in meso-Nickel(II) Schiff Base Electrochromic Polymer[END_REF][START_REF] Łępicka | A redox conducting polymer of a meso-Ni(II)-SaldMe monomer and its application for a multi-composite supercapacitor[END_REF]. Moreover, we emphasized extraordinary stability of this unsubstituted in phenyl rings meso Ni-salen polymer analog under electrochemical conditions, the feature uncommon as for nickel-salen based polymers. Hence, high stability of the poly(meso-Ni-SaldMe) film upon electrochemical charging and discharging directly arose from reversibility of concomitant viscoelastic and nanomechanical changes [START_REF] Łępicka | A redox conducting polymer of a meso-Ni(II)-SaldMe monomer and its application for a multi-composite supercapacitor[END_REF].

The encouraging results of our former studies focused on unsubstituted in phenyl rings poly(meso-Ni-SaldMe) stability and charge transfer mechanism [START_REF] Łępicka | Spectroelectrochemical Approaches to Mechanistic Aspects of Charge Transport in meso-Nickel(II) Schiff Base Electrochromic Polymer[END_REF][START_REF] Łępicka | A redox conducting polymer of a meso-Ni(II)-SaldMe monomer and its application for a multi-composite supercapacitor[END_REF][START_REF] Łępicka | High Electrochemical Stability of meso-Ni-Salen Based Conducting Polymer Manifested by Potential-Driven Reversible Changes in Viscoelastic and Nanomechanical Properties[END_REF] prompted us to find the first structure-reactivity requirement whit respect to nickel-salen based polymers ensuring its electrochemical stability. The first structure-reactivity requirement for preparation of electrochemically stable nickel-salen polymer is its meso form governed by the two methyl substituents placed on the imine bridge ensuring its electrochemical stability [START_REF] Łępicka | Spectroelectrochemical Approaches to Mechanistic Aspects of Charge Transport in meso-Nickel(II) Schiff Base Electrochromic Polymer[END_REF][START_REF] Łępicka | A redox conducting polymer of a meso-Ni(II)-SaldMe monomer and its application for a multi-composite supercapacitor[END_REF][START_REF] Łępicka | High Electrochemical Stability of meso-Ni-Salen Based Conducting Polymer Manifested by Potential-Driven Reversible Changes in Viscoelastic and Nanomechanical Properties[END_REF].

In order to extend the electroactivity range of meso nickel salen polymers (highly desirable for devising and fabricating novel energy storage materials) we synthetized a new analog of meso nickel salen polymer, vis. poly[meso-N,N'-bis-(3-methylosalicylidene)-2,3butanediaminonickel(II)], abbreviated as poly(meso-Ni-SaldMe-3dMe). Then, we compared its electrochemical stability with two known nickel-salen polymers, differing in seemingly negligible structural changes within the substituents placed on the Ni-salen imine bridge. For this comparison we selected earlier synthetized and studied by us unsubstituted in phenyl rings meso form of nickel salen polymer, poly(meso-Ni-SaldMe) and, nickel-salen polymer containing two dimethyl substituents on the imine bridge, poly[2,3-dimethyl-N,N'-bis-(salicylidene)-2,3-butanediaminonickel(II)], abbreviated as poly(NiSaltMe) [START_REF] Vilas-Boas | Spectroelectrochemical Characterisation of poly[Ni(saltMe)]-Modified Electrodes[END_REF][START_REF] Vilas-Boas | Electrochemical Characterization of a Novel Salen-Type Modified Electrode[END_REF].

Preparation of new poly(meso-Ni-SaldMe-3dMe) improved meso form of the nickelsalen based polymer with methyl substituents in the 3-positions of each salicylaldehyde moiety was inspired by results of the research focused on electrochromic behavior of nickel-salen polymers behavior, which was different in the presence and absence of two dimethyl substituents on each carbon atom of the imine bridge [START_REF] Nunes | High-Performance Electrochromic Devices Based on Poly[Ni(salen)]-Type Polymer Films[END_REF]. Surprisingly, the nickel-salen polymer analog without these substituents, used as the mentioned above electrochromic material, exhibited higher electrochemical stability than its tetra substituted analog. This finding is consistent with the results of our previous [2, [START_REF] Łępicka | A redox conducting polymer of a meso-Ni(II)-SaldMe monomer and its application for a multi-composite supercapacitor[END_REF][START_REF] Łępicka | High Electrochemical Stability of meso-Ni-Salen Based Conducting Polymer Manifested by Potential-Driven Reversible Changes in Viscoelastic and Nanomechanical Properties[END_REF] and present experimental studies complemented with the literature studies [START_REF] Nunes | High-Performance Electrochromic Devices Based on Poly[Ni(salen)]-Type Polymer Films[END_REF][START_REF] Dmitrieva | Radical formation in polymeric nickel complexes with N2O2 Schiff base ligands: An in situ ESR and UV-vis-NIR spectroelectrochemical study[END_REF] defining and explaining structure-reactivity requirements with respect to electrochemically stable nickel-salen based polymers revealing enhanced charge storage capability.

EXPERIMENTAL SECTION

Chemicals

Tetra(n-butyl)ammonium hexafluorophosphate, (TBA)PF6, (puriss, electrochemical grade), propylene carbonate (anhydrous, electrochemical grade) were purchased from Sigma-Aldrich and used without further purification. The monomeric nickel-salen based complexes, vis., meso-Ni-SaldMe-3dMe, meso-Ni-SaldMe, and NiSaltMe, were synthetized in our laboratory according to procedures described earlier [START_REF] Łępicka | Spectroelectrochemical Approaches to Mechanistic Aspects of Charge Transport in meso-Nickel(II) Schiff Base Electrochromic Polymer[END_REF][START_REF] Łępicka | A redox conducting polymer of a meso-Ni(II)-SaldMe monomer and its application for a multi-composite supercapacitor[END_REF].

Electrochemical cell configuration and electrodes

The Au coated glass slide, and ITO electrodes with ~0.25 cm 2 and ~0.78 cm 2 electrochemically active area, respectively, were used as the working electrodes. A Pt tab and a silver wire were used as the counter and pseudo-reference electrode, respectively.

Apparatus

Electrochemical experiments including polymer deposition by potentiodynamic electropolymerization and multi-scan cyclic voltammetry (CV) measurements were performed with a VSP electrochemistry system of BioLogic (SP 150) Science Instruments controlled by EC-Lab software of the same manufacturer. Surface of the polymer film coated electrodes was imaged with scanning electron microscopy (SEM) using a FEI Nova NanoSEM 450 microscope.

Atomic force microscopy (AFM) images were carried out in PeakForce tapping ™ mode in air using a Bioscope Resolve (Bruker Nano Surface, Bruker France SAS, Palaiseau, France).

Silicon nitride cantilevers of conical shape were purchased from Bruker (NPG-10, Bruker France SAS, Palaiseau, France) with spring constant of about 0.35 nN/nm. The applied force between the tip and the surface was carefully monitored and minimized at about 0.20 nN and all images were collected with a resolution of 512  512 pixels and a scan rate of 0.5 Hz. These experimental conditions are necessary to prevent mechanical damages of the sample surface due to the lateral scanning and the corresponding shear stress. AFM images with scan size of 2 µm × 2 µm were then analyzed with Nanoscope Analysis for extracting roughness, thickness and textural feature of the polymer films.

Profilometry measurements were carried out using a Detak stylis profiler DetakXT (Bruker Nano Surface, Bruker France SAS, Palaiseau, France).

RESULTS AND DISCUSSION

Electrochemical synthesis of nickel-salen based polymer films

Nickel-salen based polymer films were prepared in a three-electrode electrochemical cell by electropolymerization under potentiodynamic conditions on Au coated glass slide and (or) ITO working electrodes (separately). The polymer films were grown oxidatively from propylene carbonate solutions of 1 mM nickel-salen complexes (meso-Ni-SaldMe-3dMe, meso-Ni-SaldMe, and NiSaltMe) in 0.1 M (TBA)PF6 by potential multi-cycling between 0.0 and 1.20 V vs. Ag/Ag + at a scan rate of 100 mV s - S1). For all nickel-salen based polymer films, deposition was pointed out by the current increase in consecutive potential cycles. Moreover, this increase indicated that the deposited films were conductive. After polymer electrodeposition, the resulted film coated electrode was rinsed with the abundant propylene carbonate solvent before further use. The electrochemical responses of the so-fabricated films were recorded, in a threeelectrode system, in a monomer-free solution of 0.1 M (TBA)PF6 in propylene carbonate, this medium being selected from previous observations of stable doping and dedoping of poly(meso-Ni-SaldMe) film as performed in such medium over 10 000 times without stability loss [START_REF] Łępicka | A redox conducting polymer of a meso-Ni(II)-SaldMe monomer and its application for a multi-composite supercapacitor[END_REF]. However, in case of the use of, e.g., 0.1 M (TBA)ClO4 in acetonitrile, the ClO4 - anions were entrapped even in the neutral poly(meso-Ni-SaldMe) [START_REF] Łępicka | Spectroelectrochemical Approaches to Mechanistic Aspects of Charge Transport in meso-Nickel(II) Schiff Base Electrochromic Polymer[END_REF]. Similar entrapment of ClO4 -anions in electro-chemically generated nickel-salen based polymer films was observed by others [START_REF] Nunes | High-Performance Electrochromic Devices Based on Poly[Ni(salen)]-Type Polymer Films[END_REF].

For efficient electrodeposition of nickel salen based polymer films, it is crucial to exceed the potential value of nickel-salen complex (monomer) oxidation in the first polymerization cycle. Oxidation of the meso-Ni-SaldMe-3dMe on the gold plate electrode occurs at ~0.9 V vs.

Ag/Ag + (Figure 1b), while it happens at a more anodic value on the ITO electrode ~1.2 V (Figure S1 a). The same anodic shift when passing from gold to ITO electrode is observed for the oxidation of meso-Ni-SaldMe and NiSaltMe (occurring at ~1.0 V vs. Ag/Ag + on gold ( Figures 2b and3b), and at ~1.3V on ITO). The oxidation potential of these nickel-salen monomers depends on its chemical structure (reflected in different electron density distribution). Furthermore, oxidation potential of nickel salen complexes can shift in dependence of working electrode substrate used for its electrodeposition.

Electropolymerization of nickel-salen complexes without electrodonating substituent in phenyl rings at too low potential values leads to preparation of unstable nickel-salen polymer films [START_REF] Alekseeva | Polymeric nickel complexes with salen-type ligands for modification of supercapacitor electrodes: impedance studies of charge transfer and storage properties[END_REF].

Electrochemical stability of nickel-salen based polymer films

Chemical structure of nickel-salen based polymers and conditions of their films preparation determine electrochemical stability and the amount of charge that is reversibly stored in these potential-driven π-conjugated polymer redox systems. Hence, studies of their stability under the applied potential conditions are essential for understanding, and then improving their electrochemical properties. Therefore, we herein studied electrochemical stability of carefully selected nickel-salen polymers with different chemical structures, namely, poly(meso-Ni-SaldMe-3dMe) (Scheme 1a), poly(meso-Ni-SaldMe) (Scheme 1b), and poly(Ni-SaltMe) (Scheme 1c), with respect to potential multi-cycling under CV conditions, from 0.0 to 1.2 V vs.

Ag/Ag + for 200 cycles in monomer-free solutions (Figure 4). Furthermore, our previous [2, [START_REF] Łępicka | A redox conducting polymer of a meso-Ni(II)-SaldMe monomer and its application for a multi-composite supercapacitor[END_REF][START_REF] Łępicka | High Electrochemical Stability of meso-Ni-Salen Based Conducting Polymer Manifested by Potential-Driven Reversible Changes in Viscoelastic and Nanomechanical Properties[END_REF] and present experimental studies of nickel-salen based polymers supplemented with literature [START_REF] Nunes | High-Performance Electrochromic Devices Based on Poly[Ni(salen)]-Type Polymer Films[END_REF][START_REF] Dmitrieva | Radical formation in polymeric nickel complexes with N2O2 Schiff base ligands: An in situ ESR and UV-vis-NIR spectroelectrochemical study[END_REF] studies of their chemical structure modifications were required for preparation of electrochemically stable nickel-salen polymer films indicating extended conductivity in the positive potential range.

In general, the charge transfer mechanism is ultimately ligand-based for all nickel-salen polymers [START_REF] Łępicka | Spectroelectrochemical Approaches to Mechanistic Aspects of Charge Transport in meso-Nickel(II) Schiff Base Electrochromic Polymer[END_REF][START_REF] Vilas-Boas | New Insights into the Structure and Properties of Electroactive Polymer Films Derived from [Ni(salen)][END_REF][START_REF] Vilas-Boas | Spectroelectrochemical Characterisation of poly[Ni(saltMe)]-Modified Electrodes[END_REF][START_REF] Tedim | Modulating spectroelectrochemical properties of [Ni(salen)] polymeric films at molecular level[END_REF]. However, the pathway of charge conduction and delocalization of charge carriers are affected by differences in the polymer chemical structure [START_REF] Dmitrieva | Radical formation in polymeric nickel complexes with N2O2 Schiff base ligands: An in situ ESR and UV-vis-NIR spectroelectrochemical study[END_REF]. Nickel-salen polymers with no substituents in the 3-position of each salicylaldehyde moiety, like poly(meso-NiSaldMe) and poly(NiSaltMe), should conduct charge linearly, i.e., along the -O=Ph=Ph=O-(Ni)-O=Ph=Ph-O-conduction path [START_REF] Łępicka | Spectroelectrochemical Approaches to Mechanistic Aspects of Charge Transport in meso-Nickel(II) Schiff Base Electrochromic Polymer[END_REF][START_REF] Vilas-Boas | New Insights into the Structure and Properties of Electroactive Polymer Films Derived from [Ni(salen)][END_REF]. Phenolic oxygen atoms, located next to the nickel atom, are redox centers of salen polymers [START_REF] Łępicka | Spectroelectrochemical Approaches to Mechanistic Aspects of Charge Transport in meso-Nickel(II) Schiff Base Electrochromic Polymer[END_REF] while linearly coupled semi-quinoid and quinoid polymer structures, containing the 𝜋-conjugated system, transfer the charge between the redox centers. In nickel-salen polymers with unsubstituted phenyl rings, the redox and electronic conductivity is expected to be balanced. However, this balance can be disrupted either by introduction of substituents in the 3-position of each salicylaldehyde moiety of the polymer or steric hindrance, differing in electron-donor nature, imposed on the imine bridge. A proper way of modification of chemical structure of nickel-salen based polymers for enhancing their conductivity and increasing their electrochemical stability is not obvious. Even small structural changes of these polymers affect their electrochemical stability and the charge storage mechanism [START_REF] Dmitrieva | Radical formation in polymeric nickel complexes with N2O2 Schiff base ligands: An in situ ESR and UV-vis-NIR spectroelectrochemical study[END_REF]. Seemingly, introduction of strongly electrodonating methoxyl substituents into phenyl rings of nickel-salen based polymers should increase the amount of charge that can be reversibly stored in these polymers [START_REF] Alekseeva | Polymeric nickel complexes with salen-type ligands for modification of supercapacitor electrodes: impedance studies of charge transfer and storage properties[END_REF]. Therefore, their electron density increase is expected in the 𝜋-conjugated system. However, introduction a strongly electrodonating substituent into 3-position of each salicylaldehyde of the nickel-salen polymer largely inhibits linear coupling [START_REF] Dmitrieva | Radical formation in polymeric nickel complexes with N2O2 Schiff base ligands: An in situ ESR and UV-vis-NIR spectroelectrochemical study[END_REF], which is responsible for effective charge transfer between two neighboring phenolate moieties bridged by the nickel metal center. The presence of strong electron donors in both 3-positions of phenolic rings in the nickel-salen polymer causes ligand-radical localization reflected by disruption of the balance between the redox and electronic conductivity. Therefore, introduction of relatively weak electron donors, such as methyl substituents, in both 3-positions of phenolic rings allows increasing the amount of charge that can be reversibly stored by these polymers. This is because the balance between redox and electronic conductivity is not disrupted.

Spectroelectrochemical experiments clearly indicated that sterically hindered nickelsalen polymers containing two dimethyl substituents on each carbon atom of the imine bridge of the poly(NiSaltMe) analogs did not allow for substantial charge inter-delocalization [START_REF] Dmitrieva | Radical formation in polymeric nickel complexes with N2O2 Schiff base ligands: An in situ ESR and UV-vis-NIR spectroelectrochemical study[END_REF].

Moreover, the presence of two dimethyl substituents on each carbon atom of the imine bridge strongly influences pathways and delocalization of charge carriers in the poly(NiSaltMe), thus decreasing its electrochemical stability. The charge decrease in each consecutive cycle (Figure 4c and4d) for the poly(NiSaltMe) under CV conditions confirms that even small chemical structure changes lead to the decrease of electrochemical stability of the nickel-salen based polymer while meso forms of nickel-salen polymers, namely, poly(meso-Ni-SaldMe-3dMe)

(Figure 4a and4d) and poly(meso-Ni-SaldMe) (Figure 4b and4d) containing two moderately electrodonating methyl substituents located on the imine bridge in the one plane of monomer unit, reveal high electrochemical stability under these conditions.

Introduction of two methyl substituent in the 3-positions of each salicylaldehyde moiety in poly(meso-Ni-SaldMe-3dMe) allows extending the conductivity in the positive potential range without decreasing the electrochemical stability (curve 1 in Figure 5). The electroactivity range of ~0.3 to ~1.20 V vs. Ag/Ag + for the poly(meso-Ni-SaldMe-3dMe) is wider than those for the poly(meso-Ni-SaldMe) and poly(NiSaltMe) being ~0.45 to ~1.2 V vs. Ag/Ag + and ~0.6to ~1.2 V vs. Ag/Ag + , respectively, (Figure 5).

Figure 6 shows CV curves at different potential scan rates for the poly(meso-Ni-SaldMe-3dMe) film (Figure 6a) as well as the anodic and cathodic peak currents dependence for this polymer on the potential scan rate (Figure 6b). The linear dependence of the logarithm of both anodic and cathodic peak current on the logarithm of the potential scan rate (Figure 6b) indicates that the rate of the electrode processes is controlled by the rate of finite diffusion [START_REF] Bard | Electrochemical Methods: Fundamentals and Applications[END_REF]. Apparently, peak potentials are independent of the potential scan rate up to ~30 mV s -1 . Moreover, anodic and cathodic peak potentials separation of ~0.05-0.10 mV (caused by ohmic resistance) evidences reversibility of the electrode processes (Figure 6c). Figure 6d shows the dependence of the accumulated charge on the potential scan rate for the poly(meso-Ni-SaldMe-3dMe) film.

The amount of charge decreases with the increase of the potential scan rate because the rate of diffusional permeation of counter ions into the film cannot keep up with high scan rates.

Moreover, the volumetric capacitance of the poly(meso-Ni-SaldMe-3dMe) film is higher than that for the poly(meso-Ni-SaldMe) and poly(NiSaltMe) film (Figure 5b). This advantageous feature favors this methyl-phenyl ring substituted meso nickel-salen based polymer as a new anode material for energy storage applications.

Figure 7 presents SEM images of (a) poly(meso-NiSaldMe-3dMe), (b) poly(meso-NiSaldMe), (c) poly(NiSaltMe) films. Apparently, morphology of poly(meso-NiSaldMe) and poly(Ni-SaltMe) films is similar. These films reveal a non-ordered globular structure (Figure 7b and7c) . However, the full inspection of these films by AFM at high resolution show (Figure S3, S4 and S5) some differences in the graininess size and roughness. Indeed, poly(meso-NiSaldMe) exhibits globules of ~138 nm diameter while for poly(NiSaltMe) the globules size is ~234 nm. Furthermore, the analysis evidenced that the average distance between neighbouring globular structures remains at ~60 nm for both poly(meso-NisaldMe) and poly(NiSaltMe) films. This peculiar surface architecture is governed by continuous counter ions shuttling in and out of the polymer film occurring under potentiodynamic electropolymerization conditions of film deposition (Figures 1 and2). It can emphasized that highest roughness is measured for poly(NiSaltMe) and the lowest for poly(meso-NisaldMe-3dMe) films. The latter is the most homogeneous/continuous and did not exhibit any globular structures as reported in Figure 8a. Then, the film thickness of each sample was measured by AFM upon images performed on scratch zones. As reported in figures S3b, S4b and S5b, thicknesses of 53, 83 and 115 nm were measured for poly(meso-NiSaldMe-3dMe), poly(meso-NiSaldMe) and poly(NiSaltMe) films, respectively. Indeed, the local thicknesses obtained by AFM measurements are in good accordance with macroscopic ones performed by profilometry as reported in tables S1, S2 and S3. The reported values reflect that thinner is the film and the lowest roughness were obtained. Film decrease in both thickness and roughness regarding the polymer because of smaller counter ions entrapment within the matrix and also depending on adapted structure/conformation.

CONCLUSIONS

Electrochemical stability improvement in meso forms of nickel-salen based polymers results from introduction of moderately electrodonating steric hindrance between polymer layers within the imine bridges, thus ensuring efficient counter ions doping. Two methyl substituents in the meso nickel-salen polymer forms repel polymer layers one from the other. That way, space is generated for counter ion movement in and out of the polymer for doping and dedoping, respectively [START_REF] Łępicka | High Electrochemical Stability of meso-Ni-Salen Based Conducting Polymer Manifested by Potential-Driven Reversible Changes in Viscoelastic and Nanomechanical Properties[END_REF]. However, introduction of two dimethyl substituents on each carbon atom of the imine bridge decreases electrochemical stability of the presently and previously studied [START_REF] Nunes | High-Performance Electrochromic Devices Based on Poly[Ni(salen)]-Type Polymer Films[END_REF][START_REF] Dmitrieva | Radical formation in polymeric nickel complexes with N2O2 Schiff base ligands: An in situ ESR and UV-vis-NIR spectroelectrochemical study[END_REF]

Scheme 1 .Figure 1 .

 11 Scheme 1. Structural formulas of (a) poly(meso-NiSaldMe-3dMe), (b) poly(meso-NiSaldMe), (c) poly(NiSaltMe).

Figure 4 .b 1 Figure 5 .c

 415 Figure 4. Multi-cycle CV curves for (a) poly(meso-NiSaldMe-3dMe), (b) poly(meso-NiSaldMe), and (c) poly(NiSaltMe) film coated Au electrodes in 0.1 M (TBA)PF6,

Figure 6 .

 6 Figure 6. (a) CV curves for the poly(meso-NiSaldMe-3dMe) film in 0.1 M (TBA)PF6, in propylene carbonate for different potential scan rates. (b) The logarithm of the (1) anodic and (2) cathodic peak current dependence on the logarithm of the potential scan rate. (c) The anodic and cathodic peak potential dependence on the logarithm of the potential scan rate. (d) The dependence of the accumulated charge on the potential scan rate calculated from the CV measurements depicted in Panel a.

Figure 7 .

 7 Figure 7. SEM images at two different magnifications of (a) poly(meso-NisaldMe-3dMe), (b) poly(meso-NisaldMe), (c) poly(NiSaltMe).

Figure 8 .

 8 Figure 8. Topographical AFM images of 2µm × 2µm scan size. (a) poly(meso-NisaldMe-3dMe), (b) poly(meso-NisaldMe), (c) poly(NiSaltMe).


E Bayart

A Boudaoud

M Adda-Bedia

On the tearing of thin sheets

Keywords: A. Crack path, Thin plate theory, Crack stability, B. Crack mechanics

We perform an experimental study to investigate the propagation of one or two cracks in a thin elastic sheet of a brittle material torn in an out-of-plane shear mode. We observe that a single crack always follows a straight path, whereas two cracks propagating simultaneously follow curved paths and merge, forming a tongue-like shape. The present experimental setup allows the understanding of how the energy introduced at a large scale is focused at the crack tip. We find that the geometry of the sheet is determined by the direction of a large scale force, applied to the crack tip, which is perpendicular to cracked surfaces. While the material is deformed at large scales under mode III loading, the geometry of system in the vicinity of the crack tip adapts such that the material is locally broken under a pure mode I loading.

Introduction

Crack path prediction is one of the main challenges in the field of fracture mechanics. Indeed, a satisfactory equation of motion of a crack tip requires a fundamental understanding of material separation mechanisms (1; 2; 3; 4). Within the framework of Linear Elastic Fracture Mechanics (LEFM), the propagation of a crack is mainly governed by the singular behavior of the stress field in the vicinity of its tip (1; 2). For a two-dimensional quasi-static crack submitted to in-plane loading (mode I+II), this behavior is given by

σ ij (r, φ) = K I √ 2πr Σ I ij (φ) + K II √ 2πr Σ II ij (φ) + O(r 0 ) , (1)
where Σ I ij (φ) and Σ II ij (φ) are universal functions describing the angular variation of the stress field, and K I and K II are the stress intensity factors (SIFs).

The evolution of the crack tip is governed by the Griffith energy criterion (1; 2; 5), which states that the intensity of the loading necessary to induce propagation is given by

G = Γ , (2)
where G is the energy release rate and Γ is the fracture energy of the material, that is the energy needed to create new free surfaces. While the Griffith criterion is very useful in predicting crack initiation, it cannot predict the direction of the crack tip, and therefore it is not sufficient to determine the actual path of the crack. In order to achieve this, several suggestions have been made. Among them, the Principle of Local Symmetry (PLS) states that the crack advances in such a way that in-plane shear stress vanishes in the vicinity of the crack tip, or explicitly

K II = 0 , (3)
This rule was proposed for in-plane quasi-static cracks (6; 7), and then generalized to rapidly moving cracks [START_REF] Adda-Bedia | Generalized Griffith criterion for dynamic fracture and the stability of crack motion at high velocities[END_REF]. A different approach, known as the maximum energy release rate criterion [START_REF] Erdogan | On the crack extension in plates under plane loading and transverse shear[END_REF] states that the crack advances in a direction that maximizes its energy release rate. Interestingly, even though this approach is quite different from the PLS, it produces very similar results to those obtained using the PLS to such an extent that they were even conjectured to coincide [START_REF] Bilby | The crack with a kinked tip[END_REF]. However, as shown in [START_REF] Amestoy | Crack paths in plane situations-II. Detailed form of the expansion of the stress instensity factors[END_REF] for kinked cracks and in [START_REF] Katzav | Theory of dynamic crack branching in brittle materials[END_REF] for branched cracks, the results are not exactly the same and a clear distinction can be made between the two. In fact, it has been shown that the PLS is the only self-consistent criterion (7; 2).

While the stability of crack paths under in-plane deformations has received extensive attention, see e.g. (3; 12), there exists few experimental observations and theoretical predictions on the stability of crack propagation under out-ofplane shear deformations (mode III) in brittle materials (13; 14). Note however that from a historical point of view, the first criterion for crack path selection, based on symmetry arguments similar to the PLS, was first formulated for mode III crack propagation [START_REF] Barenblatt | On brittle cracks under longitudinal shear[END_REF]. In the case of thin films, such a configuration is illustrated by the tearing of a sheet of paper. In order to characterize crack propagation in torn thin films, we built a controlled setup enabling the study of both the stability of a single crack path and the interaction between two propagating cracks. Moreover, this experimental setup allowed the description of the focusing of the bending elastic energy from large scales to the crack tip, leading to the creation of new surfaces. Note that the interaction between two cracks was investigated in the case of thin films adhering to a hard substrate (17; 18): when a stripe of the film made by initiating two notches is pulled from the substrate, two straight, non parallel cracks propagate and merge forming a triangular shape [START_REF] Hamm | Tearing as a test for mechanical characterization of thin adhesive films[END_REF].

Here, we investigate the propagation of one or two cracks in thin sheets, with a large scale mode III loading. In section 2, we describe the experimental setup. We first focus on single cracks, reporting experimental results in sections 3 and a first interpretation in section 4. Finally, in section 5, we consider the paths of two cracks propagating simultaneously.

Experimental setup

Fig. 1. The tearing machine was designed for an out-of-plane shear loading at a constant velocity. A thin film of width 2L, initially cut at its center, is introduced between the two upper cylinders, which are free to rotate. Here, one side of the cut (in front) is coloured in blue while the other (back) is coloured in yellow. Each stripe of the torn film is fixed, throughout its width, to one of the two lower counter-rotating cylinders, which are driven at the same constant velocity. The crack tip is located between the free cylinders, roughly at the same height as the top of cylinders. The distance between the axes of the two upper cylinders is 2d.

Inspired by the classical trousers test [START_REF] Mai | The essential work of fracture for tearing of ductile metals[END_REF], we built an apparatus to control crack propagation at constant velocity in a thin elastic sheet submitted to an out-of-plane shear loading, mimicking the common way one tears a sheet of paper. The experimental setup consists of four parallel Plexiglas cylinders, of radius 8mm and length 20cm, set between two parallel metallic plates (Fig. 1).

The two upper cylinders are free to rotate, while the two lower cylinders are driven at the same constant velocity by a motor, so that the velocity at the surface of the cylinders is in the range 0.05-1.5mm.s -1 . The elastic films are made of polypropylene and have isotropic mechanical properties. In order to study the propagation of a single crack, we initiate a straight centered cut in a long rectangular sample that we insert between the free cylinders. Each stripe of the torn sheet is fixed on each driven cylinder after passing above a free rotating cylinder. The distance between the two upper cylinders imposes the crack free length. The lower cylinders collect the torn parts of the sheet. The crack tip is generally located midway between the upper cylinders (Fig. 1). In order to study the propagation of two cracks, we use the same procedure: we start with two cuts that are symmetric with respect to the central axis of the sheet, determining three parts in the sheet; the two external parts are fixed on one of the driven cylinders while the middle part is fixed on the opposite cylinder.

The experimental setup allows us to control the following parameters: the distance 2d between the two upper cylinders (d being the length of the free part of the crack), the crack propagation velocity, the sample width 2L and the thickness h of the sheet. We use films of bidirectional polypropylene of thickness 15, 30, 50 and 90µm and Young's modulus 2.2±0.4GP a. At ambient temperature and in the range of the imposed tearing velocities, the fracture process is brittle for this material. We varied the width of the film from 2 to 16cm and the crack free length d from 1.8 to 5cm. This experiment requires a careful adjustment of the parallelism of the cylinders' axes in order to keep a symmetric loading.

The critical fracture toughness K c of polypropylene films was measured for cracks submitted to pure tensile loading. A constant displacement was applied on the edges of a rectangular sample of dimensions 22cm × 10cm × 30µm in which we initiated a centered notch of length l. We measure the value of the applied force when the crack starts to propagate. Initial values of crack length l range from 0.4 to 6cm. Using the approximation of the infinite plane for the stress intensity factor, as given by K = σ πl/2, where σ is the magnitude of the far-field stress, and applying the Irwin criterion we obtained K c = 2.6 ± 0.3M P a.m 1 2 for the fracture toughness.

3 Single crack propagation experiments

Stability analysis

In a first experiment, we studied the stability of a single propagating crack. We found that within the present configuration the crack always follows a straight path parallel to the central axis of the sample even when the crack is initiated off the center of the sheet. In order to study the effect on the crack path when external perturbation are applied, we used a film of width 8cm with an initially centered straight notch, in which we made three incisions of 1.4cm at an angle of 45 • with respect to the direction of the initial notch (Fig. 2). The distance between two successive incisions, which have been made to perturb the crack path, was 5cm. In this configuration, the crack propagation starts by follow-ing a straight path. When the crack tip meets an incision, the configuration becomes asymmetric; nevertheless the crack keeps following an off-centered straight path that is parallel to the central axis of the sample (Fig. 2). Therefore, the crack path remains stable with respect to any perturbations. This makes the behaviour of the crack propagation in the tearing configuration fundamentally different from the case of in-plane propagation under opening mode for which the crack path can be either straight, propagating on the axis of the sample, or unstable (3; 12).

Geometry of the tearing film

In a second set of experiments, we considered the three-dimensional geometry of the film during the tearing process: three regions can be distinguished (see Figs. 1 and3). A first region, far from the crack, is bent under the action of gravity and does not seem to be influenced by the presence of the crack (region 1 in Fig. 3). A second region hangs under the apparatus and is tilted by an angle θ with respect to the cylinders axes. As shown in Fig. 4(a), this angle is constant (≃ 18 • ), independently of L/d, and thus, of the geometry of the experiment. The third region which is located in the neighborhood of the crack tip (region 3 in Fig. 3) has a shape close to a triangle with a vertex at the crack tip. The contour of this region delimits the area where the film is flat. The flatness of the film in this region can be interpreted as a sign that it is stretched and then contains most of the elastic energy that is used to open the crack. Notice that the geometry of the region 2 is mainly governed by the actual tearing process. The torn part is purely stretched near the crack surfaces (region 3) and it is purely bended far from it (region 1); the vertical part of the sheet being free to adapt its shape, must be tilted by angle θ in order suppress the excess of curvature at the boundaries of the sheet. In order to investigate the geometry of the stretched part in Fig. 3, we performed strain measurements of the deformed elastic sheet. Unlike the experimental technique used in [START_REF] Yan | Mixed-mode crack growth in ductile thin-sheet materials under combined in-plane and out-ofplane loading[END_REF], we performed direct image analysis of the three-dimensional shape of the film as explained hereafter. First, we note that the curvature in the x-direction is very small in comparison with the curvature in the y-direction, so that we can neglect it. In order to visualize the curvature of the film, we draw on it parallel lines in the y-direction, equidistant by 2mm: during crack propagation these lines will be straight in the stretched region only. We used degree-2 polynomial fits to measure the curvatures of these lines. We find that the flat region is bounded by the crack line and a straight line emanating from the crack tip, yielding a characteristic angle α (see Fig. 3). We measured α for an extensive range of experimental parameters (Fig. 4(b)) and found an approximately constant value α = 58 • ± 4 • that is insensitive neither to the aspect ratio L/d nor to the film thickness h. Moreover, Fig. 4(b) shows that this angle is not correlated to the geometrical angle tan -1 (L/d), thus the geometry of the stretched region is insensitive to the experimental setup and α characterizes the local concentration of stresses in the vicinity of the crack tip. As sketched in Fig. 3, in addition to the geometrical angles α and θ, there exists a third angle that characterizes the tearing process across the thickness of the film. Indeed, post-mortem analysis of the crack shows a bevel in the thickness of the sample, which is absent when the sheet is torn under tensile loading (see Fig. 5(a)). Then this angle is a characteristic of the mode III loading induced by the present tearing process. Using a top view of the crack surface, we can deduce the bevel angle β from the thickness of the film and the apparent width of the crack surface. Fig. 5(b) shows that for bidirectional polypropylene β = 45 • ±3 • independently of the experimental parameters L/d and of the film thickness. Therefore, the latter three angles satisfy the identity α ≃ β + θ.

Interpretation of the experimental results

The curve limiting the flat region marks the transition from a region of zero curvature from a bent region of finite curvature. Therefore, this separation curve induces a line of maximum tension in the sheet which would correspond to a large scale pulling force on the crack tip. The fact that the different angles satisfy the condition α = β + θ, means that this line of maximum tension is perpendicular to the crack surfaces (Fig. 3). Consequently, even for thin sheets the system adapts itself geometrically so that the crack propagates under the action of a local pure tensile loading (mode I), although a large scale out-of- plane shear mode is imposed. Our result can be seen as a generalization of the Principle of Local Symmetry for a crack propagating in a thin plate teared mode III loading at large scales.

Notice that the stress field induced by the crack tip is divergent and cannot be simply described by a force. The force we introduced above should be understood as the resultant of the stress field far from the tip. In order to validate our interpretation, we built a method to measure the intensity of this large scale force. We performed experiments where we increased the mass of the film by hanging weights at the free edge of the sample below the crack tip. The masses ranged from 1.45 to 7g (the film mass is 0.8g). A first observation is that the angles α, β and θ are not affected by the mass of the film. However, the crack tip departs from the plane determined by the top of the free cylinders (see Fig. 6). We measured the height of the crack tip z as a function of the applied mass m (Fig. 6). As the bevel angle does not vary, the area of the new surfaces is not affected by the weights, so that the energy needed to create new surfaces remains constant. As a consequence, we may assume that the intensity of the force is independent of the experimental geometrical parameters. The balance of vertical forces is simply given by

2F 0 cos α sin δ = mg (4)
with tan δ = z/d. In the experiments performed we had z/d. Then

z = gd 2F 0 cos α m (5)
Indeed, Fig. 6 shows we that z varies linearly with d in the experimental range 0.1 ≤ z/d ≤ 0.3. We could not investigate a wider range of masses because when m is increased further a stick-slip-like phenomenon occurs, such that crack propagation is no more quasi-static. The force intensity is obtained with a linear fit as F 0 = 0.25±0.05N . Then we compare this value with the minimal force necessary to propagate a crack in this material, as given by the critical toughness K c . In the experiment, we observed that a force is applied in the horizontal plane far from crack tip. The typical distance l between the effective application point of this force and the crack tip is of the order of the thickness.

The necessary force to propagate the crack is given by F th = K c h 2l π . For the 30µm thickness films, we obtain F th = 0.3N which coincides, within the experimental uncertainty, with the measured value 0.25 N . Thus the present experimental approach can be viewed as a new tool to measure the fracture toughness of thin sheets submitted to out of plane loading (a modified version of the trouser's test experiments).

Cracks interaction experiments

The interaction between two cracks was previously investigated in the case of thin films adhering to a hard substrate (17; 18). Following these previous studies, we performed experiments to investigate the simultaneous propagation of two cracks using our experimental setup. We initiate two cracks symmetrically with respect to the central axis of the sample. The main control parameters of this experiment are the initial distance between the two cracks l and the width of the sample L. We checked that the crack free length d does not influence our results. In this new configuration, the two cracks trajectories are curved toward the central axis. Then after the merging of the cracks, the central piece of the sample detaches itself from the rest. The torn part of the sheet has a symmetric tongue-like shape (Fig. 7); When the ratio of the initial width of the central part with the width of the sample is small enough (l ≤ 0.1 L), experiments are reproducible. If the middle part is larger (l ≥ 0.1 L), experiments are less reproducible and the shape of the tear is not always symmetric. This could be due to the high sensitivity of the two-cracks propagation experiment to the imperfections in the parallelism of the cylinders' axes. prefactors of the power laws increase with the thickness, while the exponents appear to remain constant. Fig. 8 also shows that there is transition in the power law behavior when the width of the samples becomes small; i.e. close to the tip of the strip. The exponent at small scales is close to 1 and far from the tip, the value of the exponent is approximately 0.6. The transition occurs when the width of the sample is around 5mm. The change in the behavior of the two cracks paths when approaching their merging point might be related to the following observations on the resulting tear:

(1) During the tearing process, the cracks tips positions do not remain stationary and are not always located at the mid-distance d between the two cylinders because the distance between the cracks tips is decreasing. This leads to an asymmetry in the loading configuration which in turn increases the bending deformation of the central part of the sheet. (2) After tearing, far from the tip, the strip recovers its flatness indicating a brittle fracture process, while it is permanently curved in a small region close to the tip which is a signature of plastic deformations. (3) Post-mortem analysis of the fracture surface across the film thickness shows that the bevel angle is far from the tip is constant along the strip and equal to 45 • , in agreement with a single crack propagation. However, at approximately 3.5mm from the cracks merging position, a transition occurs; the bevel angle suddenly increases until reaching the value of 63 • at the tip of the strip. When the width becomes of the order of the thickness, problem becomes intrinsically 3D.

These observations indicate that there are two regimes of fracture process. When the distance between cracks is large, the imposed loading for tearing is low leading to a brittle fracture behavior. In this regime, the strip is described by a power law shape with a well-defined exponent of approximately 0.6. A transition occurs when the distance between the crack tips is of order of few thicknesses of the film. In this region, the fracture process in no more brittle because of the high loading imposed by bending deformations. In this plastic regime, the strip becomes sharper and is well-described by a triangular shape.

During the first regime of propagation when the distance between the two cracks tips is large, the bevel angle of both crack surfaces is the same as for a single crack propagation. This suggests that the local geometry at the crack tip, constrained by the necessity to break under mode I loading, is not modified.

Conclusion

The main result of the present experimental study is that crack propagation in a thin film submitted to out-of-plane tearing mode at large scale occurs such that the crack tip is opened locally under the action of an in-plane tensile loading. In other words, the fracture process is always ruled by the Griffith criterion and the Principal of Local Symmetry, whatever is the loading mode at large scales and the dimensionality of the material. Moreover, we have shown that, within our experimental setup, a single crack always propagate such that its path is always straight and parallel to the lateral boundaries of the sample and that it remains stable in the presence of imposed perturbations on its path. The present work can be put in the framework of the more general problem of stability of crack paths under mixed mode loadings.

Concerning the interaction of two cracks, we have shown that in the tearing mode two cracks propagating simultaneously attract each other and merge to form a tear of reproducible shape. Image analysis show that the obtained pattern is well described by a power law function with an exponent of order 2/3 in the brittle fracture regime. The interpretation of this behavior will be the subject of a future study.

Fig. 2 .

 2 Fig. 2. Stability of the crack path. (a) Schematic of the film before tearing: we initiate a centered cut in the film and prepare inclined incisions in order to perturb the crack path. (b) Scan of the film after tearing: the crack path remains straight in the presence of perturbations along is path, even when it is off-centered or close to the boundary of the sample.

Fig. 3 .

 3 Fig. 3. (a) Zoom of Fig. 1. Region 1, which is located far from the crack surface, is bent due to gravity. Region 2 is located hangs vertically under the apparatus and is tilted by an angle θ with respect to the cylinders axes. Region 3 is a flat region whose a shape is close to a triangle ending at the crack tip. (b) Schematic of a top view of the experimental setup: the red rectangle shows a horizontal cut across the vertical part of the film, with a magnified thickness. The opening angle of region 3 at the crack tip is α, the bevel angle across the thickness of the torn film is β while θ is the angle between the hanging part and the axes of the cylinders.

Fig. 4 .

 4 Fig. 4. (a) The angle θ as a function L/d for a film of thickness 30µm. (b) The angle α of the "triangular" flat region as a function L/d for three value of the film thickness. The opening angle is given by tan -1 (L/d) and is defined by the crack line and the line joining the crack tip to the contact point of the film edge with the top cylinder.

Fig. 5 .

 5 Fig. 5. (a) Magnified view with a microscope of the crack surface. A: a crack propagated in out-of-plane shear mode using the the present tearing apparatus. The crack surface is darker because it is rough. The width of the dark region shows that the crack surface is tilted, defining a bevel across the thickness of the film. B: a crack propagated under tensile loading. The crack surface looks sharper, showing that it is normal to the surface of the film. (b) Bevel angle β of the crack surface, as a function of the parameter L/d for films of thickness 30µm.

Fig. 6 .

 6 Fig. 6. The height of the crack tip z as a function of the added mass m at the bottom of the film, for a film of thickness h = 30 µm, d = 2.6 cm and L = 4 cm.

Fig. 7 .

 7 Fig. 7. Schematic of the experiment (left) and a scan of the resulting pattern from the simultaneous propagation of two cracks (right).

3 Fig. 8 .

 38 Fig. 8. Log-Log plots of the width of the strip along the direction of tearing. The merging point of the two cracks defines the origin of x-axis. The points corresponding to films of thickness 90µm are multiplied by 10 2 for clarity. Experiments with the same film thickness and different distances l between the two initial cuts clearly superimpose.

We digitized the tongue-like shapes and measured their width as function of the distance to their tips. Fig. 8 is the main result of this section. It shows that for a given thickness of the film, profiles can be superimposed for different values of the initial width l and are well-described by power laws. Moreover, the

Acknowledgements

We are grateful to Laurent Quartier for his help in building the experimental apparatus and to Benoit Roman for insightful discussions.