Franco Tardani

Wilfrid Neri

Cecile Zakri

Hamid Kellay

Annie Colin

Philippe Poulin

Shear Rheology Control of Wrinkles and Patterns in Graphene Oxide Films

Drying graphene oxide (GO) films are subject to extensive wrinkling, which largely affects their final properties. Wrinkles were shown to be suitable in biotechno logical applications; however, they negatively affect the electronic properties of the films. Here, we report on wrinkle tuning and patterning of GO films under stress controlled conditions during drying. GO flakes assemble at an air-solvent interface; the assembly forms a skin at the surface and may bend due to volume shrinkage while drying. We applied a modification of evaporative lithography to spatially define the evaporative stress field. Wrinkle alignment is achieved over cm 2 areas. The wavelength (i.e., wrinkle spacing) is controlled in the μm range by the film thickness and GO concentration. Furthermore, we propose the use of nanoparticles to control capillary forces to suppress wrinkling. An example of a controlled pattern is given to elucidate the potential of the technique. The results are discussed in terms of classical elasticity theory. Wrinkling is the result of bending of the wet solid skin layer assembled on a highly elastic GO dispersion. Wavelength selection is the result of energy minimization between the bending of the skin and the elastic deformation of the GO supporting dispersion. The results strongly suggest the possibility to tune wrinkles and patterns by simple physicochemical routes.

1 undesirable for optical applications in which the film roughness and inhomogeneity yield light scattering.

INTRODUCTION

Graphene oxide (GO) is broadly used for the manufacture of graphene based membranes and films. These are currently investigated for a vast number of technological (electronics, [START_REF] Eda | Chemically Derived Graphene Oxide: Towards Large Area Thin Film Electronics and Optoelectronics[END_REF][START_REF] Yan | Graphene based flexible and stretchable thin film transistors[END_REF][START_REF] Chee | Flexible Graphene Based Supercapacitors: A Review[END_REF] optics, optoelectronics, [START_REF] Eda | Chemically Derived Graphene Oxide: Towards Large Area Thin Film Electronics and Optoelectronics[END_REF][START_REF] Chang | Graphene Based Nanomaterials: Synthesis, Properties, and Optical and Optoelectronic Applications[END_REF] filtration systems [START_REF] Liu | Graphene based membranes[END_REF] ) and biotechno logical (biomedical devices and tissue engineering [START_REF] Kumar | Comprehensive Review on the Use of Graphene Based Substrates for Regenerative Medicine and Biomedical Devices[END_REF] ) applica tions.

The oxygen containing groups make GO soluble in water and thus easy to handle under environmentally friendly conditions. Graphene like properties are quickly restored after chemical [START_REF] Chua | Chemical reduction of graphene oxide: a synthetic chemistry viewpoint[END_REF] or thermal [START_REF] Pei | The reduction of graphene oxide[END_REF] reduction. Moreover, the large shape anisotropy of the sheets assures liquid crystalline (LC) behavior even under dilute conditions. [START_REF] Aboutalebi | Spontaneous Formation of Liquid Crystals in Ultralarge Graphene Oxide Dispersions[END_REF][START_REF] Xu | Aqueous Liquid Crystals of Graphene Oxide[END_REF][START_REF] Kim | Graphene Oxide Liquid Crystals[END_REF] Under shear, GO sheets tend to align and flatten due to the suppression of thermal undulations. [START_REF] Poulin | Superflexibility of graphene oxide[END_REF] Exotic assemblies can be expected if instabilities set up during deposition, by analogy to lamellar phases. [START_REF] Diat | Effect of shear on a lyotropic lamellar phase[END_REF] Some effort was recently devoted to exploiting the LC behavior to tune the film structure. [START_REF] Luo | Nematic Order Drives Macroscopic Patterns of Graphene Oxide in Drying Drops[END_REF][START_REF] Hong | Controlling wrinkles and assembly patterns in dried graphene oxide films using lyotropic graphene oxide liquid crystals[END_REF] Shear or liquid crystalline behavior directly affects the structure of wet films and membranes. However, solid films are used in actual devices and applications, requiring a drying step after wet deposition. Drying mechanisms are expected to play a significant role in the film structure beyond the initial possible ordering in the wet state. This is a critical issue especially for micrometer thick coatings, where drying induced inhomogeneity is likely to occur.

The drying of a colloidal dispersion is a quite complex process. Changes in concentration and viscosity and the setting up of instabilities are only some of the processes the film undergoes during drying. [START_REF] Routh | Drying of thin colloidal films[END_REF] The coffee ring [START_REF] Deegan | Capillary flow as the cause of ring stains from dried liquid drops[END_REF] effect and cracking [START_REF] Singh | Cracking in Drying Colloidal Films[END_REF] may affect the macroscopic homogeneity of the manufactured film. Dry graphene oxide deposits are commonly characterized by extensive wrinkling. [START_REF] Hong | Water front recession and the formation of various types of wrinkles in dried graphene oxide droplets[END_REF][START_REF] Ahmad | Water assisted stable dispersal of graphene oxide in non dispersible solvents and skin formation on the GO dispersion[END_REF] The phenomenon is associated with the application of tensile/compressive stresses in 2D systems. In the case of drying GO dispersions, the stress can arise from water evaporation. During evaporation, GO flakes progressively assemble in a stack fashion. The presence of hydrogen bonds increases interflake friction, and sliding is impeded. This no slide condition results in buckling and folding of the microscopic flake assembly [START_REF] Hong | Water front recession and the formation of various types of wrinkles in dried graphene oxide droplets[END_REF][START_REF] Ahmad | Water assisted stable dispersal of graphene oxide in non dispersible solvents and skin formation on the GO dispersion[END_REF][START_REF] Guo | Hydration Responsive Folding and Unfolding in Graphene Oxide Liquid Crystal Phases[END_REF] as a consequence of volume shrinkage.

The presence of ripples and wrinkles may affect the final properties of the film. [START_REF] Cote | Tunable assembly of graphene oxide surfactant sheets: wrinkles, overlaps and impacts on thin film properties[END_REF] In principle, this effect can be desirable or not depending on the final application. Wrinkling can be sought after in biological applications. Indeed, wrinkled GO films are suitable for anisotropic cell growth [START_REF] Wang | wavelength tunable graphene based surface topographies for directing cell alignment and morphology[END_REF] and antibacterial activity. [START_REF] Zou | Wrinkled Surface Mediated Antibacterial Activity of Graphene Oxide Nanosheets[END_REF] Wrinkling can also improve film flexibility. A wrinkled layer can be stretched or bent with a reduced propensity for cracking. In addition, the excess surface area of wrinkles can be useful to enhance the energy storage capabilities of supercapacitors. However, wrinkling should be suppressed in the preparation of other electronic devices. For instance, wrinkled films showed a very large scattering of measured sheet resistance values. [START_REF] Cote | Tunable assembly of graphene oxide surfactant sheets: wrinkles, overlaps and impacts on thin film properties[END_REF] Wrinkles are also It is therefore critical to tune the formation of wrinkles and to develop a better understanding to control their structure or even to suppress them.

Nowadays, it is possible to induce controlled wrinkle formation through the application of known mechanical stresses to dried films. For instance, mono or few layer graphene films were transferred onto prestrained stretchable substrates. [START_REF] Zang | Multifunctionality and control of the crumpling and unfolding of large area graphene[END_REF][START_REF] Thomas | Controlled Crumpling of Graphene Oxide Films for Tunable Optical Transmittance[END_REF] As the strain was released, films started wrinkling. In another example, the strain appeared sponta neously when ultrathin graphite films were suspended on predefined trenches. [START_REF] Bao | Controlled ripple texturing of suspended graphene and ultrathin graphite membranes[END_REF] In this latter case, a mechanical stress was also thermally induced by the mismatch of the thermal expansion coefficients of the graphene layer and that of the substrate. Concerning GO liquid crystalline systems, only few research efforts can be found in the literature. [START_REF] Luo | Nematic Order Drives Macroscopic Patterns of Graphene Oxide in Drying Drops[END_REF][START_REF] Hong | Controlling wrinkles and assembly patterns in dried graphene oxide films using lyotropic graphene oxide liquid crystals[END_REF] In the first case, [START_REF] Luo | Nematic Order Drives Macroscopic Patterns of Graphene Oxide in Drying Drops[END_REF][START_REF] Hong | Water front recession and the formation of various types of wrinkles in dried graphene oxide droplets[END_REF] the authors related the shear banding of the LC flow alignment to the wrinkles that appear after drying. In a more recent study on wrinkling, thick films were produced at high temperature (>50 °C) and low blade coating velocities. [START_REF] Hong | Controlling wrinkles and assembly patterns in dried graphene oxide films using lyotropic graphene oxide liquid crystals[END_REF] Different types of wrinkles were observed as a function of the drying conditions.

Here, we show by using suspensions with different compositions that the rheological properties of the suspensions are key features in controlling the wrinkles and patterns formed upon drying.

We show in particular how the wavelength of the wrinkles varies with the composition of the GO solution, without any obvious link with pre existing ordering in the GO liquid crystal films and without using prestretched substrates. A correlation with mechanical and shear rheological properties of the GO dispersion is discussed. We demonstrate the possibility to pattern wrinkles through the development of an evaporative lithography method. [START_REF] Harris | Patterning Colloidal Films via Evaporative Lithography[END_REF] The latter is shown to be negative, by contrast to other common colloid films, with the accumulation of particles in regions with lower evaporation rates. The distinctive negative evaporative lithography is also driven by the rheological properties of the GO dispersions. Finally, we show how the inclusion of spherical nanoparticles can be used to reduce, and even completely suppress, the formation of wrinkles, therefore providing a method to make perfectly flat GO films from drying solutions. When used alone, the spherical nanoparticles typically form films which crack upon drying. This mechanism which is opposite to buckling and wrinkling results from the tension of capillary bridges between the particles. Here this mechanism is used to balance the spontaneous tendency of GO films to wrinkle.

The present results therefore provide a comprehensive basis to better control and tune the formation and structure of wrinkles in GO based films.

EXPERIMENTAL SECTION

Materials. Commercial graphene oxide solutions in water from Graphenea were used. The graphene oxide concentration is 4.0 mg/ mL. The flake lateral size is a few micrometers. Ludox HS 40 silica nanoparticles were used as additives. Dispersions are provided by Aldrich with a batch concentration of 40 wt % in water. The average nanoparticle diameter is 12 nm.

Sample Preparation. GO concentrated dispersions were obtained with a two step centrifugation process. [START_REF] Poulin | Superflexibility of graphene oxide[END_REF] The first step was a mild centrifugation (1400g, 20 min) used to remove possible aggregates. The collected dispersion was centrifuged for 45 min at 50 000g. This second step allowed the separation of a concentrated LC pellet of graphene oxide in water. The concentration, determined through the dry extract, was (4.3 ± 0.3) wt %. All the other concentrations were obtained by dilution in water. Care was taken to ensure homogeneous mixing after water addition. Usually, diluted samples were vortex mixed for at least 30 s.

Nanoparticles and GO dispersions were obtained by mixing the two dispersions in a volume ratio of 1:10. After being mixed, the hybrid systems were bath sonicated for 15 min.

Rheology. Rheological measurements were performed with an AR2000 stress controlled rheometer from TA Instruments. All of the samples were analyzed with 40 mm cone and plate geometry at 25.0 °C. Evaporation was avoided by the use of a trap to keep the humidity rate constant during the measurements.

Film Preparation. GO films were prepared with a doctor blade combined with rod coating technology. A drop of dispersion was put on a glass substrate, and coating proceeded at a velocity, v, of 1 in./s (2.5 cm/s). Different velocities did not show any particular effect, so the slowest available was chosen. The average film size was 2.5 × 1.5 cm 2 . To avoid geometrical issues, a constant amount of material was used to obtain films with the same surface and shape. [START_REF] Kassuga | The effect of shear and confinement on the buckling of particle laden interfaces[END_REF] Care was taken to avoid substrate inclination by fixing it with a tape and checking for planarity with a bubble level. The films were produced under controlled conditions of temperature and relative humidity of 26.0 °C and 35%, respectively. The films were not removed until they were completely dry.

Dynamic Evaporative Lithography. Film drying was performed in a confined environment. Soon after coating, a mask was placed above the film, at a distance of ∼300 μm. The mask size was 2.0 × 7.5 cm 2 . The mask ensured evaporation in a well controlled manner. Then the mask was moved at a velocity of 5-10 μm/s, according to the contact line recession speed. Faster or slower velocities caused only a small perturbation in wrinkle order, maintaining the wavelength and alignment the same.

Film Characterization. The films were characterized with optical and electronic microscopy. The thickness was determined through the use of an OM Contour GTK optical profilometer. Data were analyzed with the Vision 64 software under white light illumination vertical scanning interferometry mode (VSI). Films were cut at different points to assess the different regions and to get an average film thickness.

Wavelength Determination. The wrinkle spacing (i.e., wave length, λ) was determined through a Gaussian analysis of distributions of 100 measures of distances extracted from 5 pictures per sample. One dimensional fast Fourier transformation (FFT) of different linear profiles extracted from optical micrographs was also used to confirm the obtained results.

RESULTS

Film Casting. GO films were prepared from water based dispersions in the GO concentration range of 0.6-4.3 wt %. Coating was performed in a controlled shear rate range, γ(= v/ H w ) ≈ 100-500 s -1 , where v (2.5 cm/s) is the velocity of the moving blade and H w is the thickness of the wet film. The investigated systems showed shear thinning behavior over the whole range of tested shear rates in rheology characterizations. In confined geometries, the systems form shear bands, as already observed elsewhere. [START_REF] Luo | Nematic Order Drives Macroscopic Patterns of Graphene Oxide in Drying Drops[END_REF] However, these flow induced structures rapidly disappeared during drying.

The progressive rearrangement of flakes and the setting up of concentration gradients produced wrinkled films with no controlled structures. Control over temperature and humidity conditions was not sufficient to avoid this problem. For this reason, we developed a technique to induce progressive directional evaporation. The choice was motivated by the need to control solvent flow to promote flake assembly and solid film growth along a particular direction. The technique is schematized in Figure 1. Soon after casting, the wet films were covered with masks of different heights H m . The mask-wet film distance (H m -H w ) was fixed at ∼300 μm to stop evaporation under the covered area. The end of the film was left uncovered to start evaporation. The films progressively dried as the mask was retracted at a constant velocity, v m . During the whole process, wrinkles (Figure 1c) on the dry film developed from the wavy surface of the wet part. The controlled drying allowed control over wrinkle patterns. An average alignment was obtained along the drying direction. The volume shrinkage of the film induced a compressive stress that caused elastic instabilities. As observed elsewhere, [START_REF] Guo | Hydration Responsive Folding and Unfolding in Graphene Oxide Liquid Crystal Phases[END_REF] rehydration relaxed and suppressed the formed wrinkles. The wrinkles appeared birefringent under crossed polarizers.

Surface Wrinkling Characterization. As soon as the wet film started drying, wrinkles appeared on the surface at the free air-water interface (i.e., outside the mask). These surface wrinkles were compressed and folded after complete drying. This behavior is reminiscent of the behavior of polymeric systems [START_REF] Pauchard | Mechanical instability induced by complex liquid desiccation[END_REF] for which a skin layer forms and buckles. The pinning of the contact line and surface tension on the (covered) wet side clamped the skin at these two ends (i.e., the wet and the dry film boundaries) of the receding front. Then, evaporation induced a reduction in volume associated with a compressive strain producing wrinkling. This process finally allowed the alignment of the wrinkles perpendicular to the receding liquid front. The spacing of parallel wrinkles was analyzed as indicated in the Experimental Section. In Figure 2, 1D FFTs taken from grayscale profiles of the microscopic pictures are shown. For all of the tested concentrations and film thicknesses, nearly the same spacing was found for wet and dry films from a given system, even though the spacing distribution appears broader for dry films.

Concentration and Thickness Effects. The wrinkle spacing, λ, was determined as a function of GO concentration and dry film thickness, H, as shown in Figure 3. First, the solid content of the dry film was kept constant. The wavelength was measured for films obtained from dispersions of different GO concentrations. To achieve this, the film area was kept constant, and H w was changed proportionally to the GO concentration. As shown in Figure 3a, λ decreases with a power law like decay at increasing dispersion concentration. Second, films of different H (i.e., final solid content) were produced. A certain GO dispersion was deposited at different H w values (i.e., different final deposit, H). The process was repeated for three different GO concentrations. In this second case, λ was found to increase linearly with H (Figure 3b). The slope of this linear increase changes with concentration.

Thin Films. Very thin films were prepared by a modification of the blade coating approach. A flexible scraper was attached to the blade to reproduce the technique used in ref 31 (inset of Figure 3c). At very low thickness (inset of Figure 3c), films showed colors from thin film interference, and no more wrinkles were detected. The thickness boundary h between the wrinkling of thick films and no wrinkling of thin films was determined for different GO concentrations (Figure 3c). The absence of wrinkling in thin films reflects the absence of skin formation during drying. [START_REF] De Gennes | Solvent evaporation of spin cast films: ″crust″ effects[END_REF][START_REF] Bornside | Spin coating: One dimensional model[END_REF] Actually because of its very thin structure, the film dries as a whole gel. Similar behavior was reported for polymer films. Following de Gennes, [START_REF] De Gennes | Solvent evaporation of spin cast films: ″crust″ effects[END_REF] a skin forms when the polymer concentration in the top layer of an evaporating solution increases above a given value. A steady state solvent evaporative current is then established in the skin. By comparing solvent diffusion in the vapor phase with that through the skin, he determined a limiting skin thickness (τ 0 ) of around 70 nm. Therefore, for films of comparable thickness a steady state solvent evaporation is set throughout the whole film height, and no skin formation is expected. The measured h was actually in the same range for polymer systems, representing a good approximation of the skin layer (τ) of a (hypothetical) thick film (i.e., h ≈ τ).

Nanoparticle Effect. The effect of added spherical nanoparticles on film structure was investigated. First, it was checked that the nanoparticles had no effect on the GO dispersion phase behavior in the concentration range of the present study. No phase separation or destabilization was observed for weeks. Then, films were prepared at a fixed H w of 40 μm. A map of the different film structures is reported in Controlled Patterns. By using evaporative lithography, it is possible to control not only wrinkle alignment but also the formation of particular patterns. Two examples are reported in Figure 5. Films were dried under a fixed holed mask. The 1 mm diameter holes were hexagonally arranged with an average pitch of d h ≤ 1 mm. The mask pattern was accurately reproduced on the films. Circular menisci were first formed under the open holes, where evaporation was higher. The meniscus diameter grew with time. Finally, different menisci joined under the covered part of the mask. This differential evaporation was responsible for the formation of hexagonal features with higher edges. Surprisingly, this behavior differs from conventional colloidal evaporative lithography. The present case is peculiar to a negative evaporative lithography, as the higher features were cast under the covered part, in contrast to the case of conventional nanoparticle aqueous dispersions. [START_REF] Harris | Patterning Colloidal Films via Evaporative Lithography[END_REF] The reverse situation was reported only in solvent mixtures (i.e., water-ethanol), when Marangoni effects play a role. [START_REF] Harris | Marangoni Effects on Evaporative Lithographic Patterning of Colloidal Films[END_REF] The same peculiar patterns were reproduced in pure GO as well as in GO NP hybrid systems. However, the presence of NPs completely removed wrinkles. This was particularly evident through cross polarizing microscopy. Hybrid films did not show any birefringence because of the absence of wrinkles. The wrinkle arrangement was the consequence of the stress field arising from uneven evaporation rates. Inside the open holes, small wrinkles were aligned perpendicularly with respect to the meniscus contact line, while higher crumpled regions appeared under the covered parts. In the hybrid systems, no small wrinkles were detected, while higher crumpled deposits were still present. The overall pattern appeared more blurred.

DISCUSSION

Wrinkling is a universal phenomenon when applying a compression/tension to elastic films. The general theory of Cerda and Mahadevan [START_REF] Cerda | Geometry and Physics of Wrinkling[END_REF] defines the wrinkle wavelength, λ, as follows

λ ≈ ⎜ ⎟ ⎛ ⎝ ⎞ ⎠ B K 1/4 ( 1
)
where B is the film bending stiffness (= E f τ 3 , E f is Young's modulus, and τ is the film thickness) and K is the stiffness of an effective elastic foundation. It is possible to address a particular case by knowing the physics of the system. The present system is composed of a rigid film lying on an elastic support made of the GO suspension in a gel like liquid crystal state. In that case, K takes the form of an elastic bulk modulus, giving

λ τ ≈ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ E E f s 1/3 (2)
The wavelength, λ, is proportional to the top layer thickness, τ, with a slope defined by the mismatch of the film, E f , and substrate, E s , Young's moduli. GO exhibits surfactant like behavior, exposing the hydrophobic moiety at the air-water interface. [START_REF] Kim | Graphene Oxide Sheets at Interfaces[END_REF] In particular, the used GO possess the proper C/O ratio to be easily entrapped in the interface. [START_REF] Imperiali | Interfacial Rheology and Structure of Tiled Graphene Oxide Sheets[END_REF] In the used concentration range, due to their large lateral size, flake diffusion is limited. If the evaporation is quite fast, GO flakes tend to accumulate at the interface. Routh and Zimmerman 38 defined a specific Peclet number, Pe ≈ H w E /D, to quantify the ratio between diffusion (D) and the evaporation rate (E ). At Pe ≫1, evaporation overtakes diffusion and skinning is expected. Approximating E ̇ as the water front velocity (∼μm/s) and the diffusion coefficient as that of a sphere of 1 μm size, one obtains Pe ≈ 8, fulfilling the above conditions. Thus, a skin layer is expected to form on the top of the film. An analogous process was already observed in polymeric systems. [START_REF] Ciampi | Skin Formation and Water Distribution in Semicrystalline Polymer Layers Cast from Solution: A Magnetic Resonance Imaging Study[END_REF] The high frictional forces among the flakes and the pinning of the contact line suppress slippage, and the skin is finally folded under compression. [START_REF] Guo | Hydration Responsive Folding and Unfolding in Graphene Oxide Liquid Crystal Phases[END_REF] The resultant in plane compressive stress is directed by the receding water front. In this hypothesis, eq 2 could explain the different dependencies shown in Figure 3.A skin layer of thickness τ is suspended on foundations with different elasticities (i.e., GO bulk dispersions). The shear elastic modulus (G′) of bulk GO dispersions increases with concentration, as observed by rheology experiments. In principle, knowledge of the Poisson ratio for the GO foundation layer allows the determination of the bulk Young modulus, E s . However, the qualitative trend in E s with concentration can still be defined by a rough approximation, Es ≈ G'.

Using eq 2, an apparent bending stiffness B and its dependence on concentration can be inferred from the data in Figure 3, as shown in Figure 6. This bending stiffness increases as a power law (B ≈ conc b ) with an exponent b close to 3 (3.3). Considering the relation B = E f τ 3 , the behavior can be explained by assuming a direct proportionality of skin thickness, τ, with GO concentration. If one considers a Young modulus on the order of ∼10 2 GPa, [START_REF] Jimenez Rioboo | Elastic constants of graphene oxide few layer films: correlations with interlayer stacking and bonding[END_REF] then a skin layer in the range of 10-100 nm is expected. These considerations are actually in agreement with data obtained for very thin films (Figure 3c). The approximations of the Young and bending moduli are quite different from that obtained for monolayer graphene oxide. [START_REF] Imperiali | Interfacial Rheology and Structure of Tiled Graphene Oxide Sheets[END_REF] However, the present situation concerns multilayer assembly and the whole mechanical properties derived in a complex way from the monolayer. Compressive rheology characterization would be more appropriate in this particular situation. [START_REF] De Kretster | Compressive rheology: an overview[END_REF] However, shear and compression properties have been shown to behave in qualitatively the same manner. [START_REF] Zhou | Chemical and physical control of the rheology of concentrated metal oxide suspensions[END_REF] Therefore, the considerations above can give an idea of our wrinkling mechanics. Unfortunately, characterization of the skin layer is challenging, and a complete quantitative characterization of the film is not possible.

Wrinkle alignment is a consequence of a resultant tensile stress applied perpendicularly to the drying front as the volume decreased with the contact lines of the skin pinned. The mask drives the formation of a horizontal drying front. This front separates the dry film from the liquid film under the mask. A skin on the drying front is pinned at the contact line and stretched backward by the liquid surface tension. These constraints drive a unidirectional tensile stress in response to the normal compression due to volume shrinkage. The shape of the mask assures negligible skin formation at the lateral sides controlling the alignment.

The addition of nanoparticles modifies the physics of the system. In particular, we are interested here in particles that form cracks when used alone, a behavior exactly opposite to the case of GO systems which have extra surface area. The relatively low nanoparticles concentration presently used does not affect the phase behavior and the bulk mechanical properties of the dispersions. But other effects are coming into play. As known, above a critical thickness, films made solely of nanoparticles tend to crack. During evaporation, a close packed configuration of NPs forms at the contact line. As the solvent recedes (vertically) from this NP front, a large negative capillary pressure is produced in the menisci between particles. [START_REF] Lee | Why Do Drying Films Crack?[END_REF] This generates a compression stress normal to the surface and a large in plane tensile stress, as the rigid substrate prevents lateral deformation. Crack formation will release the tensile stress, at the expense of surface energy. The present situation is more complicated due to the presence of two different particles with two limiting distinct behaviors, one with the formation of cracks and the other with the formation of wrinkles. Actually, it was observed that the in plane stress induced by NPs at the surface of the films could even balance the stress that normally generates wrinkles. This can result in the total disappearance of wrinkles. The large size difference between flakes and nanoparticles is expected to produce stratification. With the same Pe argument for skin formation, one can expect an uneven distribution of flakes and nanoparticles during drying. [START_REF] Routh | Drying of thin colloidal films[END_REF] In a recent simulation, [START_REF] Cheng | Dispersing Nanoparticles in a Polymer Film via Solvent Evaporation[END_REF] the distribution of NPs in a skinning polymer solution was attributed mostly to NP-polymer and NPsolvent interactions. The nanoparticles used here are hydro philic and have a better affinity for the solvent. In this case, NPs are expected to accumulate under the superficial skin. Actually, no NPs are visible at the top of the dry films in experiments. During evaporation, volume shrinkage will set NP layers under compression. As the NPs are not deformable and their movement is hindered under confinement (i.e., GO stack assembly), in plane tensile stress is produced. This stress will first flatten the GO skin layer until there is enough disposable surface from the wrinkles, then again producing cracks. We can infer that the shape of the boundary lines comes from a complex interplay of wrinkle amplitude, wavelength, and dry thickness effects.

The possibility of patterning a peculiar structure with negative lithography can be explained by considering the yield stress nature of GO liquid crystal dispersions. [START_REF] Poulin | Superflexibility of graphene oxide[END_REF] Menisci are created under the open areas of the mask due to solvent loss (Figure 5a). The stress generated by a Laplace pressure gradient (σ = 2γ/R ≈ 14 Pa, for water surface tension and hole size R = 1 mm) is not high enough to overtake the yield stress (i.e., σ y > 14 Pa for 2.0 wt % GO, SI) and to induce viscous flow. Therefore, the films will retain the resulting deformation, generating the pattern. As already mentioned, a compressive yield stress would be more appropriate, but the consideration still holds.

CONCLUSIONS

By taking advantage of dynamic evaporative lithography and tuning the rheological properties of GO dispersions, we were able to control the wrinkling of GO films. The obtained wrinkles were aligned over a macroscopic area of different GO deposits. The wavelength was tuned by changes in the concentration and thickness of the films. The phenomenon was attributed to the formation of a skin layer, subjected to compressive strain during drying. It was shown that this compressive stress was balanced with a tensile stress to get rid of the wrinkles. The latter was simply obtained with the addition of nanoparticles, making the concept easily imple mentable in applications.

Controlling the phenomenon of wrinkling is critical in the fabrication of particular patterns, as illustrated in the present work. Notably, wrinkling can be altered without affecting the final macroscopic texture. This is actually the case of evaporative lithography in hybrid systems. This is important since wrinkles can affect the properties of the films.

As already mentioned, [START_REF] Zou | Wrinkled Surface Mediated Antibacterial Activity of Graphene Oxide Nanosheets[END_REF] the relation between GO film roughness and its antimicrobial activity has been demonstrated. We expect that tuning the wrinkle spacing may add selectivity. Patterned films can be also used as templates for controlled nanoparticle deposition. In principle, anisotropic wettability may also be obtained.

There are still some open questions related to the physical mechanisms involved in the investigated phenomena. We used a modification of the elasticity theory to qualitatively describe the mechanics of the system. The foundation considered in the theory is a purely elastic material, whereas in our case the material is viscoelastic. Moreover, the whole process is dynamic, as the top layer is forming and growing continuously. Therefore, more accurate theories are needed to consider all of these phenomena. From an experimental point of view, the characterization of the skin layer would also be interesting. The determination of the skin thickness will allow the confirmation of our hypothesis. It would also be interesting to look at the NPs' spatial distribution during film drying. The character ization of NPs and their packing would be helpful in quantifying the stress built up during drying.

Figure 1 .

 1 Figure 1. Film casting and drying is shown in (a) along with the overall wrinkle alignment (2D FFT inset) (b) and the microscopic appearance of a wrinkle (c). v and v m indicate the directions of blade and mask movement, and H m , H w , and H are the mask, wet film, and dry film heights, respectively. Scale bars are (b) 500 μm and (c) 500 nm.

Figure 2 .

 2 Figure 2. Wavelength determination examples, optical micrograph, grayscale profile, and 1D FFT for the wet (a) and dry (b) films. The scale bar is 100 μm.

Figure 4 .

 4 Three different situations were observed: wrinkling (filled squares), flattening (open squares), and cracking (crosses). Under certain conditions of GO and NP concentrations, wrinkling was completely suppressed. The addition of NPs favored the casting of smoother films. The threshold concentration of NPs to remove wrinkling increased with GO concentration. Above a certain threshold of NP content, films underwent cracking. At the highest GO concentration used, it was not possible to reduce wrinkling without cracking the dry film.

Figure 3 .

 3 Figure 3. λ (μm) is reported as a function of GO wt % at a fixed (1.3 ± 0.5 μm) dry thickness (a) and (b) at different dry thicknesses (H, μm) for 4.3 wt % (square, red line), 1.7 wt % (circle, green line), 0.68 wt % (triangle, blue line) GO dispersions. (c) Thin film thickness, h (nm), as a function of GO wt %. (Insets) The blade with a flexible scraper and the appearance of a film (scale bar 1.0 cm).

Figure 4 .

 4 Figure 4. Effect of NP addition. (a) Wrinkling phase diagram for hybrid films at 40 μm fixed H w . Wrinkled (black squares), smooth (empty square), and cracked (cross) films. (b, c) SEM and (d) optical images of smooth, wrinkled, and cracked films, respectively. Scale bars are 20 (b), 30 (c), and 250 μm (d).

Figure 5 .

 5 Figure 5. Patterning of GO (2.0 wt %) and GO + NPs (2.0 and 0.1 wt %, respectively) films: (a) schematic representation of the holed mask with the drying and patterning scheme; cross polarizing micrographs showing the obtained patterns for (b) pure GO and (c) GO NPs film. (Inset) Optical micrograph for the hybrid films, obtained in reflection. The scale bar is 1000 μm.

Figure 6 .

 6 Figure 6. Bending stiffness (B = E s λ 3 , N m) reported as a function of GO wt %.

ACKNOWLEDGMENTS

We acknowledge C. Blanc, collaborator at the GAELIC project, for fruitful discussions and E. Laurichesse for some technical support of the experimental part.

Funding

The research in the manuscript was conducted in the framework of the A.N.R. funded GAELIC project.

AUTHOR INFORMATION

Corresponding Author *E mail: franco.tardani@gmail.com.

Notes

The authors declare no competing financial interest.


Emanuele Caglioti

François Golse

The Boltzmann-Grad limit of the periodic Lorentz gas in two space dimensions

The periodic Lorentz gas is the dynamical system corresponding to the free motion of a point particle in a periodic system of fixed spherical obstacles of radius r centered at the integer points, assuming all collisions of the particle with the obstacles to be elastic. In this Note, we study this motion on time intervals of order 1/r as r → 0 + .

Résumé

La limite de Boltzmann-Grad du gaz de Lorentz périodique en dimension deux d'espace. Le gaz de Lorentz périodique est le système dynamique correspondant au mouvement libre dans le plan d'une particule ponctuelle rebondissant de manière élastique sur un système de disques de rayon r centrés aux points de coordonnées entières. On étudie ce mouvement pour r → 0 + sur des temps de l'ordre de 1/r.

Version française abrégée

On appelle gaz de Lorentz le système dynamique correspondant au mouvement libre d'une particule ponctuelle dans un système d'obstacles circulaires de rayon r centrés aux sommets d'un réseau de R 2 , supposant que les collisions entre la particule et les obstacles sont parfaitement élastiques. Les trajectoires de la particule sont alors données par les formules [START_REF] Boca | The distribution of the free path lengths in the periodic two-dimensional Lorentz gas in the small-scatterer limit[END_REF]. La limite de Boltzmann-Grad pour le gaz de Lorentz consiste à supposer que le rayon des obstacles r → 0 + , et à observer la dynamique de la particule sur des plages de temps longues, de l'ordre de 1/r -voir (3) pour la loi d'échelle de Boltzmann-Grad en dimension 2.

Or les trajectoires de la particule s'expriment en fonction de l'application de transfert d'obstacle à obstacle T r définie par [START_REF] Tartar | Compensated compactness and applications to partial differential equations[END_REF] -où la notation Y désigne la transformation inverse de [START_REF] Golse | On the distribution of free path lengths for the periodic Lorentz gas II. M2AN Modél[END_REF] -application qui associe, à tout paramètre d'impact h ′ ∈ [-1, 1] correspondant à une particule quittant la surface d'un obstacle dans la direction ω ∈ S 1 , le paramètre d'impact h à la collision suivante, ainsi que le temps s s'écoulant jusqu'à cette collision. (Pour une définition de la notion de paramètre d'impact, voir [START_REF] Golse | On the Periodic Lorentz Gas and the Lorentz Kinetic Equation[END_REF].)

On se ramène donc à étudier le comportement limite de l'application de transfert T r pour r → 0 + . Proposition 0.1 Lorsque 0 < ω 2 < ω 1 et α = ω2 ω1 / ∈ Q, l'application de transfert T r est approchée à O(r 2 ) près par l'application T A,B,Q,N définie à la formule (14). Pour ω ∈ S 1 quelconque, on se ramène au cas ci-dessus par la symétrie (15).

Les paramètres A, B, Q, N mod. 2 intervenant dans l'application de transfert asymptotique sont définis à partir du développement en fraction continue (9) de α par les formules (11) et (12).

On voit sur ces formules que les paramètres A, B, Q, N mod. 2 sont des fonctions très fortement oscillantes des variables ω et r. Il est donc naturel de chercher le comportement limite de l'application de transfert T r dans une topologie faible vis à vis de la dépendance en la direction ω. On montre ainsi que, pour tout h ′ ∈ [-1, 1], la famille d'applications ω → T r (h ′ , ω) converge au sens des mesures de Young (voir par exemple [START_REF] Tartar | Compensated compactness and applications to partial differential equations[END_REF] p. 146-154 pour une définition de cette notion) lorsque r → 0 + vers une mesure de probabilité P (s, h|h ′ )dsdh indépendante de ω :

Théorème 0.2 Pour tout Φ ∈ C c (R * + ×] -1, 1[) et tout h ′ ∈] -1, 1[, la limite (16) a lieu dans L ∞ (S 1
) faible-* lorsque r → 0 + , où la mesure de probabilité P (s, h|h ′ )dsdh est l'image de la probabilité µ définie dans (17) par l'application (A, B, Q, N ) → T A,B,Q,N (h ′ ) de la formule (14). De plus, cette densité de probabilité de transition P (s, h|h ′ ) vérifie les propriétés (18).

Le théorème ci-dessus est le résultat principal de cette Note : il montre que, dans la limite de Boltzmann-Grad, le transfert d'obstacle à obstacle est décrit de manière naturelle par une densité de probabilité de transition P (s, h|h ′ ), où s est le laps de temps entre deux collisions successives avec les obstacles (dans l'échelle de temps de la limite de Boltzmann-Grad), h le paramètre d'impact lors de la collision future et h ′ celui correspondant à la collision passée.

Le fait que la probabilité de transition P (s, h|h ′ ) soit indépendante de la direction suggère l'hypothèse d'indépendance (H) des quantités A, B, Q, N mod. 2 correspondant à des collisions successives. Théorème 0.3 Sous l'hypothèse (H), pour toute densité de probabilité f in ∈ C c (R 2 × S 1 ), la fonction de distribution f r ≡ f r (t, x, ω) de la théorie cinétique, définie par (3) converge dans L ∞ (R + × R 2 × S 1 ) vers la limite (22) lorsque r → 0 + , où F est la solution du problème de Cauchy (21) posé dans l'espace des phases étendu (x, ω, s, h) ∈ R 2 × S 1 × R * + ×] -1, 1[. Dans le cas d'obstacles aléatoires indépendants et poissonniens, Gallavotti a montré que la limite de Boltzmann-Grad du gaz de Lorentz obéit à l'équation cinétique de Lorentz (4). Le cas périodique est absolument différent : en se basant sur des estimations (cf. [START_REF] Bourgain | On the distribution of free path lengths for the periodic Lorentz gas[END_REF] et [START_REF] Golse | On the distribution of free path lengths for the periodic Lorentz gas II. M2AN Modél[END_REF]) du temps de sortie du domaine Z r défini dans (1), on démontre que la limite de Boltzmann-Grad du gaz de Lorentz périodique ne peut pas être décrite par l'équation de Lorentz (4) sur l'espace des phases R 2 × S 1 classique de la théorie cinétique : voir [START_REF] Golse | On the Periodic Lorentz Gas and the Lorentz Kinetic Equation[END_REF]. Si l'hypothèse (H) ci-dessous était vérifiée, le modèle cinétique (22) dans l'espace des phases étendu fournirait donc l'équation devant remplacer l'équation cinétique classique de Lorentz (4) dans le cas périodique.

The Lorentz gas

The Lorentz gas is the dynamical system corresponding to the free motion of a single point particle in a periodic system of fixed spherical obstacles, assuming that collisions between the particle and any of the obstacles are elastic. Henceforth, we assume that the space dimension is 2 and that the obstacles are disks of radius r centered at each point of Z 2 . Hence the domain left free for particle motion is

Z r = {x ∈ R 2 | dist(x, Z 2 ) > r} ,
where it is assumed that 0 < r < 1 2 .

(1)

Assuming that the particle moves at speed 1, its trajectory starting from x ∈ Z r with velocity ω ∈ S 1 at time t = 0 is t → (X r , Ω r )(t; x, ω) ∈ R 2 × S 1 given by Ẋr (t) = Ω r (t) and Ωr (t) = 0 whenever X r (t) ∈ Z r ,

X r (t + 0) = X r (t -0) and Ω r (t + 0) = R[X r (t)]Ω r (t -0) whenever X r (t -0) ∈ ∂Z r , (2)
denoting ˙= d dt and R[X r (t)] the specular reflection on ∂Z r at the point X r (t) = X r (t ± 0). Assume that the initial position x and direction ω of the particle are distributed in Z r × S 1 with some probability density f in ≡ f in (x, ω), and define

f r (t, x, ω) := f in (rX r (-t/r; x, ω), Ω r (-t/r; x, ω)) whenever x ∈ Z r . ( 3
)
We are concerned with the limit of f r as r → 0 + in some appropriate sense to be explained below. In the 2-dimensional setting considered here, this is precisely the Boltzmann-Grad limit.

In the case of a random (Poisson), instead of periodic, configuration of obstacles, Gallavotti [START_REF] Gallavotti | Rigorous theory of the Boltzmann equation in the Lorentz gas[END_REF] proved that the expectation of f r converges to the solution of the Lorentz kinetic equation for (x, ω) ∈ R 2 × S 1 :

(∂ t + ω • ∇ x )f (t, x, ω) = S 1 (f (t, x, ω -2(ω • n)n) -f (t, x, ω))(ω • n) + dn , f t=0 = f in . (4)
In the case of a periodic distribution of obstacles, the Boltzmann-Grad limit of the Lorentz gas cannot be described by a transport equation as above: see [START_REF] Golse | On the Periodic Lorentz Gas and the Lorentz Kinetic Equation[END_REF] for a complete proof, based on estimates on the free path length to be found in [START_REF] Bourgain | On the distribution of free path lengths for the periodic Lorentz gas[END_REF] and [START_REF] Golse | On the distribution of free path lengths for the periodic Lorentz gas II. M2AN Modél[END_REF]. This limit involves instead a linear Boltzmann equation on an extended phase space with two new variables taking into account correlations between consecutive collisions with the obstacles that are an effect of periodicity: see Theorem 4.1.

The transfer map

Denote by n x the inward unit normal to Z r at the point x ∈ ∂Z r , consider

Γ ± r = {(x, ω) ∈ ∂Z r × S 1 | ± ω • n x > 0} , (5)
and let Γ ± r /Z 2 be the quotient of Γ ± r under the action of Z 2 by translation on the x variable. For (x, ω) ∈ Γ + r , let τ r (x, ω) be the exit time from x in the direction ω and h r (x, ω) be the impact parameter:

τ r (x, ω) = inf{t > 0 | x + tω ∈ ∂Z r } , and h r (x, ω) = sin(ω, n x ) . (6)
Obviously, the map

Γ + r /Z 2 ∋ (x, ω) → (h r (x, ω), ω) ∈] -1, 1[×S 1 (7)
coordinatizes Γ + r /Z 2 , and we henceforth denote Y r its inverse. For each r ∈]0, 1 2 [, consider now the transfer map

T r : ] -1, 1[×S 1 → R * + ×] -1, 1[ defined by T r (h ′ , ω) = (rτ r (Y r (h ′ , ω)), h r (X r (τ r (Y r (h ′ , ω)); Y r (h ′ , ω)), Ω r (τ r (Y r (h ′ , ω)); Y r (h ′ , ω)))) . (8)
For a particle leaving the surface of an obstacle in the direction ω with impact parameter h ′ , the transition map T r (h ′ , ω) = (s, h) gives the (rescaled) distance s to the next collision, and the corresponding impact parameter h. Obviously, each trajectory (2) of the particle can be expressed in terms of the transfer map T r and iterates thereof. The Boltzmann-Grad limit of the periodic Lorentz gas is therefore reduced to computing the limiting behavior of T r as r → 0 + , and this is our main purpose in this Note.

We first need some pieces of notation. Assume ω = (ω 1 , ω 2 ) with 0 < ω 2 < ω 1 , and α = ω 2 /ω 1 ∈]0, 1[\Q. Consider the continued fraction expansion of α:

α = [0; a 0 , a 1 , a 2 , . . .] = 1 a 0 + 1 a 1 + . . . . ( 9
)
Define the sequences of convergents (p n , q n ) n≥0 and errors (d n ) n≥0 by the recursion formulas

p n+1 = a n p n + p n-1 , p 0 = 1 , p 1 = 0 , d n = (-1) n-1 (q n α -p n ) , q n+1 = a n q n + q n-1 q 0 = 0 , q 1 = 1 , (10)
and let

N (α, r) = inf{n ≥ 0 | d n ≤ 2r 1 + α 2 } , and k(α, r) = - 2r √ 1 + α 2 -d N (α,r)-1 d N (α,r) . ( 11
)
Proposition 2.1 For each ω = (cos θ, sin θ) with 0 < θ < π 4 , set α = tan θ and ǫ = 2r √ 1 + α 2 , and

A(α, r) = 1 - d N (α,r) ǫ , B(α, r) = 1 - d N (α,r)-1 -k(α, r)d N (α,r) ǫ , Q(α, r) = ǫq N (α,r) . ( 12
)
In the limit r → 0 + , the transition map T r defined in ( 8) is explicit in terms of A, B, Q, N up to O(r 2 ):

T r (h ′ , ω) = T A(α,r),B(α,r),Q(α,r),N (α,r) (h ′ ) + (O(r 2 ), 0) for each h ′ ∈] -1, 1[. ( 13
)
In the formula above

T A,B,Q,N (h ′ ) = (Q, h ′ -2(-1) N (1 -A)) if (-1) N h ′ ∈]1 -2A, 1] , T A,B,Q,N (h ′ ) = Q ′ , h ′ + 2(-1) N (1 -B) if (-1) N h ′ ∈ [-1, -1 + 2B[ , T A,B,Q,N (h ′ ) = Q ′ + Q, h ′ + 2(-1) N (A -B) if (-1) N h ′ ∈ [-1 + 2B, 1 -2A] , ( 14
)
for each (A, B, Q, N ) ∈ K :=]0, 1[ 3 ×Z/2Z, with the notation Q ′ = 1-Q(1-B) 1-A .
The proof uses the 3-term partition of the 2-torus defined in section 2 of [START_REF] Caglioti | On the distribution of free path lengths for the periodic Lorentz gas III[END_REF], following the work of [START_REF] Blank | Thom's problem on irrational flows[END_REF].

For ω = (cos θ, sin θ) with arbitrary θ ∈ R, the map h ′ → T r (h ′ , ω) is computed using Proposition 2.1 in the following manner. Set θ = θ -m π 2 with m = [ 2 π (θ + π 4 )] and let ω = (cos θ, sin θ). Then

T r (h ′ , ω) = (s, h) , where (s, sign(tan θ)h) = T r (sign(tan θ)h ′ , ω) . (15)
3. The Boltzmann-Grad limit of the transfer map T r

The formulas (11) and (12) defining A, B, Q, N mod. 2 show that these quantities are strongly oscillating functions of the variables ω and r. In view of Proposition 2.1, one therefore expects the transfer map T r to have a limit as r → 0 + only in the weakest imaginable sense, i.e. in the sense of Young measuressee [START_REF] Tartar | Compensated compactness and applications to partial differential equations[END_REF], pp. 146-154 for a definition of this notion of convergence.

The main result in the present Note is the theorem below. It says that, for each h ′ ∈ [-1, 1], the family of maps ω → T r (h ′ , ω) converges as r → 0 + and in the sense of Young measures to some probability measure P (s, h|h ′ )dsdh that is moreover independent of ω.

Theorem 3.1 For each Φ ∈ C c (R * + × [-1, 1]) and each h ′ ∈ [-1, 1] Φ(T r (h ′ , •)) → ∞ 0 1 -1 Φ(s, h)P (s, h|h ′ )dsdh in L ∞ (S 1 ω ) weak-* as r → 0 + , ( 16
)
where the transition probability P (s, h|h ′ )dsdh is the image of the probability measure on K given by

dµ(A, B, Q, N ) = 6 π 2 1 0<A<1 1 0<B<1-A 1 0<Q< 1 2-A-B dAdBdQ 1 -A (δ N =0 + δ N =1 ) (17) under the map K ∋ (A, B, Q, N ) → T A,B,Q,N (h ′ ) ∈ R + × [-1, 1]
. Moreover, P satisfies:

(s, h, h ′ ) → (1 + s)P (s, h|h ′ ) is piecewise continuous and bounded on

R + × [-1, 1] × [-1, 1],
and P (s, h|h ′ ) = P (s, -h| -h ′ ) for each h, h ′ ∈ [-1, 1] and s ≥ 0.

(

) 18
The proof of (16-17) is based on the explicit representation of the transition map in Proposition 2.1 together with Kloosterman sums techniques as in [START_REF] Boca | The distribution of the free path lengths in the periodic two-dimensional Lorentz gas in the small-scatterer limit[END_REF]. The explicit formula for the transition probability P is very complicated and we do not report it here, however it clearly entails the properties (18).

The Boltzmann-Grad limit of the Lorentz gas dynamics

For each r ∈]0, 1 2 [, denote dγ + r (x, ω) the probability measure on Γ + r that is proportional to ω • n x dxdω. This probability measure is invariant under the billiard map

B r : Γ + r ∋ (x, ω) → B r (x, ω) = (x + τ r (x, ω)ω, R[x + τ r (x, ω)ω]ω) ∈ Γ + r . (19)
For (x 0 , ω 0 ) ∈ Γ + r , set (x n , ω n ) = B n r (x 0 , ω 0 ) and

α n = min(|ω n 1 /ω n 2 |, |ω n 2 /ω n 1 |) for each n ≥ 0, and define b n r = (A(α n , r), B(α n , r), Q(α n , r), N (α n , r) mod. 2) ∈ K for each n ≥ 0. ( 20
)
We make the following asymptotic independence hypothesis: for each n ≥ 1 and each Ψ ∈

C([-1, 1]× K n ) (H) lim r→0 + Γ + r Ψ(h r , ω 0 , b 1 r , . . . , b n r )dγ + r (x 0 , ω 0 ) = 1 -1 dh ′ 2 S 1 dω0 2π K n Ψ(h ′ , ω 0 , β 1 , . . . , β n )dµ(β 1 ) . . . dµ(β n )
Under this assumption, the Boltzmann-Grad limit of the Lorentz gas is described by a kinetic model on the extended phase space R 2 × S 1 × R + × [-1, 1] -unlike the Lorentz kinetic equation ( 4), that is set on the usual phase space R 2 × S 1 . Theorem 4.1 Assume (H), and let f in be any continuous, compactly supported probability density on R 2 × S 1 . Denoting by R[θ] the rotation of an angle θ, let F ≡ F (t, x, ω, s, h) be the solution of

(∂ t + ω • ∇ x -∂ s )F (t, x, ω, s, h) = 1 -1 P (s, h|h ′ )F (t, x, R[π -2 arcsin(h ′ )]ω, 0, h ′ )dh ′ F (0, x, ω, s, h) = f in (x, ω) ∞ s 1 -1 P (τ, h|h ′ )dh ′ dτ (21) where (x, ω, s, h) runs through R 2 × S 1 × R * + ×] -1, 1[. Then the family (f r ) 0<r< 1 2 defined in (3) satisfies f r → ∞ 0 1 -1 F (•, •, •, s, h)dsdh in L ∞ (R + × R 2 × S 1 ) weak- * as r → 0 + . ( 22
)
For each (s 0 , h 0 ) ∈ R + × [-1, 1], let (s n , h n ) n≥1 be the Markov chain defined by the induction formula (s n , h n ) = T βn (h n-1 ) for each n ≥ 1 , together with

ω n = R[2 arcsin(h n-1 ) -π]ω n-1 , (23)
where β n ∈ K are independent random variables distributed under µ. The proof of Theorem 4.1 relies upon approximating the particle trajectory (X r , Ω r )(t) starting from (x 0 , ω 0 ) in terms of the following jump process with values in R 2 × S 1 × R + × [-1, 1] with the help of Proposition 2.1 (X t , Ω t , S t , H t )(x 0 , ω 0 , s 0 , h 0 ) = (x 0 + tω 0 , ω 0 , s 0 -t, h 0 ) for 0 ≤ t < s 0 , (X t , Ω t , S t , H t )(x 0 , ω 0 , s 0 , h 0 ) = (X τn + (t -s n )ω n , ω n , s n+1 -t, h n ) for s n ≤ t < s n+1 .

(24)

Unlike in the case of a random (Poisson) distribution of obstacles, the successive impact parameters on each particle path are not independent and uniformly distributed in the periodic case -likewise, the successive free path lengths on each particle path are not independent with exponential distribution. The Markov chain (23) is introduced to handle precisely this difficulty.

Figure 1 .

 1 Figure 1. Left: the transfer map (s, h) = Tr(h ′ , v), with h ′ = sin(n x ′ , v) and h = sin(nx, v). Right: Particles leaving the surface of one obstacle will next collide with one of generically three obstacles. The figure explains the geometrical meaning of A, B, Q.


Miroslava Černochová
email: miroslava.cernochova@pedf.cuni.cz

Tomáš Jeřábek
email: tomas.jerabek@pedf.cuni.cz

DIYLab as a way for Student Teachers to Understand a Learning Process

Keywords: Digital Literacy, DIY, DIYLab Activity, Visualisation of Learning Process, Student Teacher

The authors introduce their experiences gained in the EU project Do It Yourself in Education: Expanding digital literacy to foster student agency and collaborative learning (DIYLAB). The project was aimed to design an educational procedure based on DIY philosophy with a student-centred and heuristic approach to learning focused on digital literacy development and later to verify it in teaching practice in primary and secondary schools and HEIs in Finland, Spain and the Czech Republic. In the Czech Republic the project DIYLAB was realized as a teaching approach in initial teacher education with Bachelor and MA degrees for ICT, Biology, Primary Education and Art Education student teachers. DIYLab activities represented occasions for student teachers to bring interesting problems related to their study programmes and also their after-school interests. An integral part of DIYLab activities was problem visualisation using digital technology; visual, film, animation, etc. served as a basis for assessing both pupils' digital competence and their problem-solving capability. The DIYLab have influenced student teachers' pedagogical thinking of how to develop pupils' digital literacy and to assess digital literacy development as a process and not as a digital artefact. Following the project, the DIYLab approach is being included in future Bachelor and MA level initial teacher education with the aim to teach student teachers (1) to design DIY activities for digital literacy development supported inter-disciplinary relations in school education, and (2) to use digital technology to oversee and assess learning as a process.

Introduction

Since 2006, ICT has been included in the curriculum as compulsory for primary, lower and upper secondary education in the Czech Republic with the aim of developing digital literacy and the use of ICT skills so that pupils are enabled to use standard ICT technology. In schools there is, however, a tendency for ICT lessons to focus on outcomes which can be produced by basic formats using a typical office suite (for example, using a word processor, spreadsheet and presentation program) and on the ability to search and to process information (primarily via the Internet).

Some pupils and HEI students manifest their discontentment against how digital technology is used in schools. They show teachers they are more experienced in using digital technology than their teachers. The 2006 ICT curriculum is a thing of the past. It does not sufficiently reflect new and advanced technology and the need to implement innovative teaching approaches in digital literacy development which emphasises the educational potential for new ways and strategies of learning, not just user skills.

Thus the Czech government approved in 2014 the Strategy of Digital Education ( [START_REF] Moeys | Strategie digitálního vzdělávání[END_REF]) that would contribute, among other things [START_REF] Bergold | Participatory Research Methods: A Methodological Approach in Motion [110 paragraphs[END_REF] to create conditions for the development of the digital literacy and computational thinking of pupils, and (2) to create conditions for the development of the digital literacy and teachers' computational thinking.

The DIYLab, being grounded in DIY (do-it-yourself) philosophy, is an example of an innovative approach to education which worked in schools and enabled the development of digital literacy. Pupils at all levels were at first cautious but attracted to the idea and motivated to learn. Teachers were empowered by coming to understand another strategy to enable students' learning.

The DIY Concept

The concept of DIY is not totally new. It can be found when speaking of the development, for example, of amateur radio as a hobby. The DIY movement has developed and spread step by step into different branches (technical education, Art, science, etc.). It has common features: it brings together enthusiastic people (who have the same aim and interest) to solve in a creative way interesting problems in their field and mutually to share "manuals" on how to proceed or how "you can do it yourself".

Globally, there is a generation of DIY enthusiasts and supporters who join in various communities or networks. There is nothing that could limit activities of this generation of creative and thoughtful people; if they need to know something to be able to realise their DIY ideas they learn from one another. The DIY generation very often uses ICT for their creative initiatives. The DIY generation visualises stories to document the process explaining how problems were solved to be shared as tutorials by others. Freedom to make and to create using ICT is perceived as freedom of access, in the choice of tools and technology, and a release from reliance on specific software and hardware tools; it is using a variety of resources, making copies and sharing outcomes and methods.

Implementation of DIY into Education

To apply DIY in schools means enabling pupils to bring into school interesting ideas from the extra-curricular environment and to create conditions for their exploration; to put them within a school curriculum and to (re)arrange circumstances for collaboration and sharing experiences, in a similar way to that in which scientists and experts might work. Through such processes pupils can use their knowledge and skills from different subjects and interests so they can keep discovering new things and interdisciplinary contexts and connections ( [START_REF] Sancho-Gil | Envisioning DIY learning in primary and secondary schools[END_REF]). In such activities pupils organise themselves, their procedures and processes; principles of autonomous learning are thus being put into practice.

According to Y. Kafai and K. Peppler ( [START_REF] Kafai | Youth, Technology, and DIY: Developing Participatory Competencies in Creative Media Production[END_REF]), it is possible to incorporate DIY activities into programming, designing models, constructing robots, creating manuals (tutorials) on how to do or how to learn something (for example, how to count using an abacus). Thus, DIY can potentially contribute to further mastery in the use of digital technology and consequently improve digital literacy. DIY in school contexts corresponds with the concept of learning as "the natural, unstoppable process of acquiring knowledge and mastery" and being aware that "the vast majority of the learning in your life doesn't happen when you're a kid in school" ( [6, p. 22]).

This article focuses on DIYLab activities implemented through the project DIYLab in teacher education at the Faculty of Education but also includes an example from a school since one of the participant teachers was also a student teacher at the Faculty.

First, it was necessary to map ideas of student teachers and teacher educators if, why and how to bring topics from outside students' activities into their study programme. The student teachers were expected to integrate their own interesting problem related to their life or hobbies, but unfortunately the majority of them did not come with any own initiative waiting for a task formulated by their teachers. It emerged that student teachers who participated in DIYLab had not been used to bringing their extra-curricular interests, hobbies, or expertise into university study. After that, it was necessary to specify a framework and key features of the DIY activities which were consequently realised by student teachers.

A Model of DIYLab Activity

Imperative in the practice of DIYLab is that all who apply the DIY idea in their activities endeavour to share with the outside world how they proceeded and how they solved a problem. They develop tutorials which visually (using movie, animation, etc.) document the process, explaining how problems were solved and what was learned. This means of transmitting to others how to proceed may be perceived as an author's self-reflection of his/her learning. Story-telling -a narrative assemblage -is a very important attribute of the DIY creation process ([4, p. 300]). The publishing of a procedure or a manual on how to create or produce something or how something was made can help others to produce something similar; it can help others to learn new methods or to create something completely new and original. The concept of DIY aligns with young people's experiences who point out that in schools "we miss so much of the richness of real learning, which relies on failure, trial and error, getting to know people, and reaching for things you didn't think were possible" ([6, p. 75]).

Key features of DIYLab activities

A model of DIYLab activities was based on DIY philosophy which is a studentcentred, heuristic approach to learning and problem-solving and which implies six pedagogical principles for approaches to learning (Table 1). A key aim of DIY activities beyond solving a problem is to provide a manual on how to solve the problem. This "handbook" is then published in a form which can be shared with others -the best and easy to understand way is in a visible form (e.g., video, animation). (2) to have the characteristics of inquiry-based teaching and learning methods DIY communities dedicate their time to original problems which have not been solved and which are different to traditional school tasks.

(

3) to support transdisciplinary knowledge

To enable pupils to bring into school interesting ideas from the extra-curricular environment and to create conditions for their exploration. If pupils have an interesting problem to be solved, they do not worry about which school subject it relates to.

J. Sancho et al. ([10])

(4) to contribute to autonomous / selfregulated learning Documenting how to proceed for others may be perceived as an author's self-reflection of his/her learning. DIY communities enjoy to find a solution "Building new tools and paths to help all of us learn". The requirement to visualise a learning process about "how the problem can be solved" as a message for others follows several reasons. Firstly, visual tools are normally understood as comprehensible regardless of which languages we speak. Secondly, we are all, student teachers included, increasingly surrounded by visual stories (e.g. YouTube, animated instructions for passengers how to behave during a flight). The skills required to use digital technology for visualisation fully correspond to a concept of digital literacy defined by Y. Eshet-Alkalai ([3, p. 93]): "digital literacy involves more than the mere ability to use software or operate a digital device; it includes a large variety of complex cognitive, motor, sociological, and emotional skills, which users need in order to function effectively in digital environments." Eshet-Alkalai's conceptual model of digital literacy consists of five digital literacy thinking skills, including the "photo-visual digital thinking skill: Modern graphic based digital environments require scholars to employ cognitive skills of "using vision to think" in order to create photo-visual communication with the environment. This unique form of digital thinking skill helps users to intuitively "read" and understand instructions and messages that are presented in a visualgraphical form, as in user interfaces and in children's computer games." ([3, p. 93])

Specification of a Research Field

The implementation of the DIYLab project in teacher education was an opportunity to focus on the development of pedagogical thinking in student teachers; primarily to enrich by using an innovative didactical approach to digital literacy development, and to understand better the learning process.

The project DIYLab was expected to answer questions such as how much student teachers are capable of visualising their learning, which type of visual description (narration) student teachers would produce or how difficult it is for them to visualise their learning process. The student teachers had not been used to considering why and how to visualise the learning process. They had studied learning theory in aspects of pedagogy and psychology. Therefore, it was expected that they would find visualising their learning processes challenging because they had never undertaken such a pedagogical task. During their HEI study, they mainly do self-reflections from didactic situations, teaching processes or learning only in oral or written forms, but not in a visual manner.

Research Methodology

Participatory Action Research -PAR ( [START_REF] Bergold | Participatory Research Methods: A Methodological Approach in Motion [110 paragraphs[END_REF]; ( [START_REF] Kemmis | Exploring the relevance of critical theory for action research: Emancipatory action research in the footsteps of Jürgen Habermas[END_REF]; [START_REF] Kemmis | Participatory action research. Communicative action and the public sphere[END_REF]) was the research methodology adopted since it allowed active engagement, intervention and the opportunity for participant observation. The approach was also consonant with the democratic values implicit in the above-stated DIY philosophy. The impact of DIY approach on teacher education was studied using qualitative research methods (focus groups, questionnaire surveys, interviews, observations and analyses of student teachers' DIY outcomes). Teacher educators evaluated not only the originality of student teachers' DIYLab activity procedures, but also how much these DIY activities corresponded to the six pedagogical principles (see Fig. 1) and to what extent student teachers managed to visualise a process and their ways of thinking and learning.

Characteristics of student teachers participated in DIYLab activities

From January 2015 to January 2016, at the Faculty of Education, 192 part-time and full-time student teachers (aged at least 20 years) and eight teacher educators from four departments (IT and Technical Education, Art Education, Biology and Environmental Studies, and Primary Education) were introduced to the DIY philosophy within compulsory courses focused on pedagogy, ICT education, computing education, biology, educational technology, multimedia etc.

Analysis of Some DIYLab Activities Performed by Student Teachers

The ). Some of them were published on the HUB (hub.diylab.eu).

Ways in which the DIYLab activities met the defined requirements (1) Collaborative learning

The collaborative approach to DIYLab activities was the most irregular one, and was dependent on each particular process and students. For the part-time student teachers who live and work in different places of the Czech Republic and only meet during classes at the Faculty of Education collaboration and co-operation with their coscholars are more fitting and appropriate than for full-time students. Some DIYLab activities were extremely specialised and tasks had to be solved individually.

(2) Inquiry-based teaching and learning DIYLab activities were not for student teachers routine tasks usually assigned in seminars. In some cases, the student teachers faced technological problems (see Building android apps or the specific solution for Installing a camera in a birdhouse for the subject Multimedia Systems), in another cases they faced more theoretical didactic problems (see Collection of examples of problems which human cannot solve without using computer).

(

3) Trans-disciplinary knowledge

Almost all of DIYLab activities had trans-disciplinary overlap. In some cases, the trans-disciplinary co-operation became obvious only thanks to the DIYLab project and had an impact in forming a student teachers' professional competence of selfreflection (e.g. How I'm becoming a teacher). Nearly thirty per cent (28.5%) of ICT student teachers stated that in their DIYLab activity they did not use knowledge from other subjects; if there was any required knowledge from other subjects, it was mostly from physics, mathematics, English, geography, medicine or cinematography or computer science, and rarely from biology, chemistry or art. Nevertheless, they appreciated the opportunity to collaborate with students of other study specialisations very much, and due to such collaboration from their point of view they learned a great deal.

(4) Autonomous / Self-regulated learning

The dimensions of independent learning and self-regulation underpin the whole process and were actively promoted, taking due account of the diversity of the students and their willingness to learn by new means. The student teachers appreciated the DIYLab approach to learning from two perspectives: they learned (a) another approach to solve an issue, and (b) how to properly lay out their work, to visualise and organise tasks in order to find solutions.

(

5) Digital literacy improvement / Digital competence

In carrying out the DIYLab activities the student teachers worked with quite a narrow range of hardware and software which was largely determined by the technical equipment available at the Faculty of Education or in the resources available through their respective Bachelor's and Master's degree studies. Most of the student teachers involved in DIYLab activities were ICT students. In general, in the case of ICT student teachers it was virtually impossible to determine any improvement or progress in their digital literacy. Based on the outputs, the students were mostly using video, presentation and text editors. For ICT student teachers, the majority of DIY activities were only an opportunity (sometimes a routine) to apply their digital literacy skills to solving problems, while for Art or Biology Education student teachers DIY activities distinctively contributed to improvement of their skills in using digital technology. As a result of involvement in DIYLab they learned to create animations etc. DIY activities with ICT student teachers increased their didactic thinking about the role and possibilities of digital technology in education; besides that, they assisted their Art Education peers to be able to do animations or Biology student teachers to design a technological solution and to install a camera in a birdhouse.

(6) Connection to study programmes / curriculum

The student teachers did their DIYLab activities during one semester as a part of their final work with the aim of gaining credits and grades. Each DIYLab outcome consisted of two main parts: (i) a product as a solution of the problem (e.g., SW application, a set of 3D tools, models, database, mechanical drawing, electric circuit, robot), and (ii) a digital object (e.g. video, movie, animation) which visualises a process demonstrating how student teachers were progressing, how they learned a problem and how they managed to resolve a DIYLab activity. Fig. 1 shows the results of a questionnaire survey focused on how teacher educators evaluated their DIYLab activities connecting the six pedagogical principles. From this evaluation the following average values for each item are derived: contribution to autonomous / self-regulated learning (4.8), digital competence improvement (4.4), connection with the curriculum (4.3), support to trans-disciplinary knowledge (3.9), inquiry-based teaching and learning (3.6), and support for collaborative learning (3.1). The majority of problems solved in DIYLab were not characteristic of inquirybased problems. Teacher educators invested a lot of time facilitating student teachers to develop a DIYLab idea. For the teacher educators it was not always easy to motivate their students to bring their own projects. Students seemed to be afraid to step into new territory. The main motivation to carry out their DIY activities for some student teachers lay in getting credits, not in solving problems. In part-time study, there was not much time for defining and understanding a problem for inquiry-based teaching and inter-disciplinary links. Potentially, this is an advantage since it may have contributed to increased online collaboration between students and an increase in collaborative learning. There were several factors (teacher educator, student, solved problem, study specialisation, motivation, experiences, etc.) that influenced a way how particular pedagogical principles were accomplished in each individual DIYLab activity (Fig. 1).

Examples of DIYLab activities carried out by Bc degree student teachers

The student teachers on ICT Bachelor Studies' courses counted on their teachers to assign them a topic. Despite some of them work in computer companies or specialise in some aspect of computing, they rarely came up with their own proposals. When they had some ideas for topics for DIYLab activities these were related to their hobbies (e.g. diving, gardening, theatre). Some of them were surprised that they had to do something linking knowledge and experiences from different branches or disciplines.

For example, there was a student who was interested in scuba diving and who proposed a project, Diver's LogBook (see http://hub.diylab.eu/2016/01/27/diverslogbook/). Another student is part of a theatre group, Kašpárek and Jitřenka, and she decided to initiate a project entitled, Database Development -database of theatre ensembles (http://hub.diylab.eu/2016/01/27/database-development-database-oftheatre-ensembles/). Bc. student teachers of Biology who studied the life of birds in a nesting box directed their activities to a project, Bird House (http://hub.diylab.eu/2016/01/27/bird-house/). In courses focused on digital technology, one ICT student looked for a solution as to How to create an animated popup message in Adobe After Effects.

Bachelor student teachers weren't used to thinking about what and how they had learned, much less how to visualise their own learning process. They didn't consider thinking about learning and reflection on DIYLab activity to be "professional". Unlike Art Education students, the ICT student teachers are advanced in digital technology, but they lack knowledge and skills to observe and to visually display and present processes. Bachelor ICT student teachers very often reduced a visualisation of their DIYLab procedure to a set of screenshots. They recognised DIYLab only from the technological point of view and the extent to which software and hardware were applied. Generally, for Bc. student teachers it was very difficult to visualise their learning in DIY activities. They were not particularly interested in the pedagogical concept of learning and how to visualise its process because in Bc. study programmes the main focus is on acquiring knowledge from particular branches (Biology, ICT, Art, etc.), rather than understanding the learning process involved.

Examples of DIY activities carried out by MA degree student teachers

The MA degree student teachers did their DIYLab activities predominantly within didactic subjects or courses making limited use of technology. The majority of them were part-time students who work in schools as unqualified teachers of ICT or Informatics subjects, and so most of them tried to apply the DIYLab idea in their teaching with their pupils.

MA student teachers thought about and mediated the topics and the purpose of DIYLab activities more deeply than Bachelor-level students mainly due to the fact that they realised their DIYLab activities primarily in courses focused on didactics' aspects and contexts. MA student teachers elaborated some general themes proposed by their teacher educators.

The requirement to record and visualise a learning process did not surprise MA student teachers. They understand how important it is to visualise a learning process from a pedagogical point of view. Data taken from such visualisation can help teachers to understand better the learning outcomes of their pupils. However, they had no experience in the process of visualisation. Similarly to the Bachelor-level students, they very often reduced a visualisation of DIYLab procedure to a set of screenshots. A few of them did an animation of their way of thinking about the DIYLab activity. (e.g., Problems which a human cannot solve without using a computertomography). Some of them did a tutorial (An animated story about a small wizard, https://www.youtube.com/watch?v=QA1skX4GiBI). Some of them developed a methodical guide how to work with pupils (DIY_Little Dances in Scratch -Start to move_CZ, http://hub.diylab.eu/2016/01/11/little-dances-in-scratch-start-tomove/diy_little-dances-in-scratch-start-to-move_cz/), some did a comic strip.

Examples of DIY activities carried out by pupils and completed in lessons managed by ICT student teachers on school practice

Some part-time ICT student teachers decided to apply DIYLab to their class teaching in schools where their pupils did similar DIY activities. All these experiences from schools demonstrate a great enthusiasm and motivation to learn and to solve problems related to after-school activities and through which they develop their digital literacy. For example a girl (aged 15) enjoys recording and editing digital sounds in her free time. She designed a DIYLab activity as a sound story-telling of a boy who would like to meet his girlfriend (https://www.youtube.com/watch?v=a8TzZCAzxKo). She describes how she produced the story-telling as a movie in which she explains what she did, how she collected sounds and which software applications were used in her work (https://www.youtube.com/watch?v=jbSID9_B72k).

Conclusions

Although the EU project has now ended, the Faculty of Education will continue DIYLab activities as a compulsory assignment and an integrated component on courses for Bachelor degree ICT study and for full-time and part-time student teacher of MA degree ICT study. Great attention will be given to (i) ways how to motivate student teachers to choose appropriate topics from their after-school interests and hobbies and to design DIY activities for inquiry-based learning, (ii) methodological approaches as to how to visualise a process of learning in Informatics, Computing or ICT subjects; diaries, scenarios, process-folios or log books used in Art Education or in technical or technological oriented branches will be used as an inspiration for such an approach in ICT teacher education, and (iii) how to support a close interdisciplinary collaboration among student teachers and teacher educators.

The challenge for the Czech context is to change the culture from teacherdependency to students as independent, autonomous learners in the classroom. Creativity in content, methods and pedagogy are absolute requirements to achieve this goal. The DIYlab project showed differences and limits in the culture of approaches to creativity from a pedagogical point of view: If we compare the DIY learning in educational practice in Prague with approaches to creative learning in Barcelona, which is a popular place for international creative artists and DIY communities, in the Czech context the DIYLab will need much longer to break free from the bureaucratic concept of teaching and the assessment of learning outcomes.

The value of the evaluative criteria in framing the DIY process and the parameters used enabled analysis and could support the design and thinking by teachers considering using the DIY method.

K

  voluntarily spend a lot of time in intense learning, they tackle highly technical practices, including film editing, robotics, and writing novels among a host of other activities across various DIY networks." To develop photo-visual digital thinking skill as a component of digital literacy Yto be connected with the curriculum School curriculum / study program for HEI students

Fig. 1 .

 1 Fig. 1. Teacher educator's evaluation of their DIYLab activities using a scale 0-5 with 0-being no accomplishment and 5-being maximal accomplishment. (Source: [2])

Table 1 .

 1 Six pedagogical principles for a design of DIYLab activities

	Feature of DIYlab	Idea	Authors,
	activity		resources
	(1) to support	Members of DIY communities collaborate mutually.
	collaborative learning

  student teachers worked on 16 themes for DIYLab activities and produced within one semestre 81 digital outcomes of different quality and content: Multimedia project (6 DIY digital objects/ 11 students), Design of Android applications (4/11),

	Little
	dances with Scratch (4/12), Collection of examples of problems which human cannot
	solve without using computer (6/28), Contemporary trends in WWW pages
	development (5/9), Teaching learning object development (4/12), Wiki of teaching
	activities (1/8), Educational robotics project (6/16), Anatomy and morphology of
	plants (5/20), Biological and geological technology -field trips (2/3), How I´m
	becoming a teacher (17/23), Animated stories (11/13), and Teaching with tablets
	(10/26