Hailong Shi

Weimin Gan

Claude Esling

Xiaojun Wang

Yudong Zhang

Emad Maawad

Andreas Stark

Xiaohu Li

Lidong Wang

Elastic strain induced abnormal grain growth in graphene nanosheets (GNSs) reinforced copper (Cu) matrix composites

Keywords: Texture, Recrystallization, Abnormal grain growth, Anisotropic elasticity, Metal matrix composite

Orientation related grain growth stimuli during primary and secondary recrystallization of the metal matrix in laminated metal/non-metal composites demonstrate unique effect on orientation evolution but has rarely been investigated. In this work, the recrystallization and grain growth of Cu in a graphene nanosheets (GNSs) reinforced laminated Cu matrix composite during sintering was thoroughly investigated. The microstructure, texture and lattice strain evolution of the Cu/GNS composite was examined referenced to the Cu stack without GNSs by ex-situ and in-situ orientation characterization techniques (SEM-EBSD, neutron diffraction and synchrotron radiation) from mesoscale to macroscopic scale. The results evidenced that a strong Cube orientation was produced in the Cu/GNS composite instead of the individual non-Cube orientations in the pure Cu stack without GNSs. Detailed strain-state analysis of the Cu foils in the Cu/GNS composite revealed that the anisotropic expansion behavior of the GNS that is incompatible with that of the Cu foils imposed multiple elastic constraints to the foils during the sintering process, resulting in a biaxial isostrain state in the surface layers and a uniaxial compressive strain state in the central layer of each Cu foil. The elastic anisotropy of Cu favors the growth of the Cube oriented grains to minimize the total strain energy. This work clarified the thermal strain induced abnormal grain growth of selected orientations.

The mechanism revealed can be useful for analysing abnormal grain growth in elastically strained materials and can also be applied to fabrication process for texturization or even monocrystallization.

Introduction

Recrystallization and grain growth are intrinsic processes of metallic materials occurring inevitably when the material is thermally treated after plastic deformation. The driving force arises from the high stored energy in relation to the point defects, dislocations and grain boundaries created during the deformation process [START_REF] Alvi | Cube texture in hot-rolled aluminum alloy 1050 (AA1050)-nucleation and growth behavior[END_REF][START_REF] Lü | Recrystallization kinetics and microstructure evolution during annealing of a cold-rolled Fe-Mn-C alloy[END_REF][START_REF] Kestens | Texture formation in metal alloys with cubic crystal structures[END_REF][START_REF] Humphreys | Recrystallization and Related Annealing Phenomena[END_REF]. Recrystallization makes drastic microstructure and crystallographic texture transformation of the deformed materials. Certain orientation related stimuli, such as orientation related surface energy, may cause abnormal grain growth and completely modify the recrystallized microstructure and texture [START_REF] Lin | Observation of annealing twin nucleation at triple lines in nickel during grain growth[END_REF][START_REF] Carel | Computer simulation of strain energy effects vs furface and interface energy effects on grain growth in thin films[END_REF][START_REF] Zielinski | The influence of strain energy on abnormal grain growth in copper thin films[END_REF][START_REF] Zhang | Competition between surface and strain energy during grain growth in free-standing and attached Ag and Cu films on Si substrates[END_REF][START_REF] Kim | Effect of microelasticity on grain growth and texture evolution: A phase field grain growth simulation[END_REF]. As the microstructural and texture transformation significantly affect the properties of the materials, the determination and interpretation of recrystallization texture are of both fundamental and practical importance in materials processing. Thus, study on the origin of recrystallization and grain growth has been intensive for decades for the purpose of practical property control and theoretical microstructure process simulation. The related investigations were first conducted in mono-phased metals or alloy systems [START_REF] Humphreys | Recrystallization and Related Annealing Phenomena[END_REF][START_REF] Fabregue | Enhanced recovery and recrystallization of metals due to an applied current[END_REF][START_REF] Sun | Effects of Laves phase particles on recovery and recrystallization behaviors of Nb-containing FeCrAl alloys[END_REF][START_REF] Cram | Modelling discontinuous dynamic recrystallization using a physically based model for nucleation[END_REF]. The external influential factors, such as deformation amount, deformation temperature, material purity and initial grain size [START_REF] Chang | Relationships between rolled grain shape, deformation bands, microstructures and recrystallization textures in Al-5%Mg[END_REF][START_REF] Chekhonin | Confined recrystallization of high-purity aluminium during accumulative roll bonding of aluminium laminates[END_REF][START_REF] Miszczyk | The influence of deformation texture on nucleation and growth of cube grains during primary recrystallization of AA1050 alloy[END_REF] have been always the subjects of investigtions. Two theories have been elaborated, the one being Oriented Nucleation (ON) and the other Oriented Growth (OG) [START_REF] Duggan | Oriented nucleation, oriented growth and twinning in cube texture formation[END_REF][START_REF] Engler | The roles of oriented nucleation and oriented growth on recrystallization in commercial purity aluminium[END_REF][START_REF] Fan | Oriented growth during recrystallization revisited in three dimensions[END_REF].

With the demand on materials capacity for wide applications, composite materials integrating suitably distributed two or several chemically and physically distinct constituent components were conceived to provide properties not obtainable from any of the single components. According to the nature of the components, the composite could be classified into two categories, i.e., metal/metal and metal/non-metal composites. For the former, one of the representative composites are the laminar metal composites (LMCs) usually fabricated by serve plastic deformation (SPD) techniques like Accumulative Roll Bonding (ARB) [START_REF] Carpenter | Bulk texture evolution of nanolamellar Zr-Nb composites processed via accumulative roll bonding[END_REF][START_REF] Mccabe | Recrystallization and Grain Growth in Accumulative Roll-Bonded Metal Composites[END_REF][START_REF] Scharnweber | Microstructure, texture, and mechanical properties of laminar metal composites produced by accumulative roll bonding[END_REF] and Repeated Folding and Rolling [START_REF] Battezzati | Ni-Al intermetallics produced by cold-rolling elemental sheets[END_REF][START_REF] Crone | Bulk shape memory NiTi with refined grain size synthesized by mechanical alloying[END_REF]. For the latter, non-metal reinforcements such as ceramic particles (SiC, Al2O3, TiC, etc. [START_REF] Williams | Damage evolution in SiC particle reinforced Al alloy matrix composites by X-ray synchrotron tomography[END_REF][START_REF] Kawabata | Creep deformation behavior of spherical Al2O3 particlereinforced Al-Mg matrix composites at high temperatures[END_REF]), short whiskers and fibers (TiB, Al2O3-SiO2, SiC, etc. [START_REF] Schuh | Whisker alignment of Ti-6Al-4V/TiB composites during deformation by transformation superplasticity[END_REF][START_REF] Okabe | A new fracture mechanics model for multiple matrix cracks of SiC fiber reinforced brittle-matrix composites[END_REF]) and continuous fibers (carbon fiber, W, B, etc. [START_REF] Liu | High-pressure interfacial impregnation by micro-screw in-situ extrusion for 3D printed continuous carbon fiber reinforced nylon composites[END_REF][START_REF] Sherman | Mesoscale characterization of continuous fiber reinforced composites through machine learning: Fiber chirality[END_REF]) are usually adopted. Recently, carbon materials (graphite, carbon nanotubes and graphene) received great attention as ideal reinforcements for metal matrix composites because of their superior mechanical and physical properties [START_REF] Fan | Enhanced interfacial bonding and mechanical properties in CNT/Al composites fabricated by flake powder metallurgy[END_REF][START_REF] Q.-H. Yuan | Interfacial structure in AZ91 alloy composites reinforced by graphene nanosheets[END_REF]. For the metal/metal LMCs, the recrystallization and grain growth processes have been intensively investigated by the ex-situ and in-situ electron back-scattered diffraction (EBSD) and transmission electron microscopic (TEM) characterization. The creation and redistribution of dislocations, grain boundaries, and phase boundaries during the ARB and the subsequent annealing process have been thoroughly investigated. For the metal/non-metal composites, the recrystallization and grain growth processes happening inevitably during sintering are more complicated. On the one hand, the different thermal expansion coefficients (TEC) of the non-metal reinforcement from those of the matrix metal introduces giant thermal constraints to the recrystallization and grain growth process of the metal matrix. On the other hand, the non-metal reinforcements are much finer. A huge thickness difference exists between the matrix metal and the non-metal reinforcements. Thus, the effect of the non-metal reinforcements subsides with the distance into the metal matrix. The constraints are more in a gradient form, further complicating their influence on the recrystallization and grain growth of the metal matrix. So far, the effect of the reinforcements on the recrystallization and grain growth of the metal matrix has rarely been investigated and stays unclear.

Motivated by such an observation, we conducted the present work. The composite components used are cold-rolled Cu foils with a thickness of 30 µm and multi-layered graphene nanosheets (GNSs) each with a diameter of 1-3 μm and a thickness of 1-5 nm that were deposited on the Cu surfaces. The asdeposited cold-rolled Cu foils were stacked and hot-pressing sintered. The evolutions of the microstructure and orientation were first thoroughly examined through ex-situ orientation characterization techniques in multi-scales (SEM-EBSD and neutron diffraction). In particular, the advanced in-situ synchrotron radiation technique was used to characterize the complicated strain state by measuring the lattice strain of the Cu matrix during the recrystallization and grain growth processes.

To reveal the underlying mechanism on selective grain growth, the experimental output was further analyzed in the frame of the elasticity theory. The results provide useful information for the theory of recrystallization and grain growth of metallic materials and helpful for analysing abnormal grain growth in elastically strained materials, such as structural products produced by additive fabrication.

The stimulus for selective grain growth can be conceived to fabricate strongly textured or even monocrystallized materials.

Experimental procedures

Material and sample preparation

The materials used in this study are cold-rolled Cu foils with a thickness of about 30 μm and the graphene nanosheets (GNSs) with specification thicknesses around 1~5 nm (the thickness of monolayer graphene is 0.34 nm) that was further verified by Atomic Force Microscopy (AFM) analysis as detailed in Appendix A1). The chemical composition of the Cu foils is displayed in Table 1. The fabrication of the bulk composite with the two materials was through two steps: electrophoretic deposition (EPD) of the GNSs on the two surfaces of each Cu foil and hot-pressing sintering of the stacked Cu foils with the GNSs, as detailed below. The electrophoretic deposition (EPD) process was conducted in an electrolytic cell. A pure Cu plate of 50×50×5 mm 3 was employed as the anode and the cold-rolled Cu foils with the same sized surface area as the cathode. An electrolyte suspension solution with GNSs of 0.004 g•mol -1 was utilized and the distance between the two electrodes was 50 mm. The deposition on each Cu foil was conducted at 30 V for 30 s. The coverage of the GNSs on the Cu foil surfaces was estimated, as detailed in Appendix A1. After deposition, the as-deposited Cu foils were dried in air and stored in a vacuum tank for the subsequent stacking and hot-pressing sintering. For comparison, the Cu foils without the deposition of the GNSs were also prepared. 100 Cu foils each with an area of 4×4 mm 2 with and without the GNSs were further stacked to 3 mm thick bulks. The stacks were made along the direction normal to the foil plane, i.e., the normal direction (ND) in the orthonormal rolling coordinate system (the rolling direction (RD), the transverse direction (TD) and the ND) by keeping the respective RD and TD consistent for each foil. The stacks with the GNSs were denoted Cu/GNS and the stacks without the GNSs pure Cu. The hot-pressing sintering process of the Cu/GNS composite and the pure Cu was identically conducted in vacuum (<10 -2 Pa) under a compressive stress of about 50 MPa applied along the ND. The stacks were first heated to 950 °C at a rate of about 5 ℃• min -1 , isothermally held for 2 hours and then cooled inside the furnace to room temperature.

In order to study the influences of the GNSs on the microstructure and texture evolution during the sintering process by SEM-EBSD for the former and by neutron diffraction measurements for the latter, the as-deposited Cu/GNS samples and the pure cold-rolled Cu samples were sintered to 150 ℃, 170 ℃, 230 ℃, 450 ℃ and 700 ℃, respectively, in a dilatometer (DIL 805A/D), and then quenched in compressed argon flux to room temperature to preserve the as-heated microstructure.

Microstructural characterization and texture analysis

Microstructural features of the pure Cu and the Cu/GNS samples at the states of as-cold-rolled, asfully-sintered and as-heated to different temperatures were characterized by scanning electron microscopy -electron backscatter diffraction (SEM-EBSD), using a JEOL 6500F SEM equipped with an EBSD acquisition camera and the Aztech online acquisition software package (Oxford Instruments).

In order to achieve the surface quality required for EBSD measurement, the samples were mechanically polished with grit silicon carbide wet sandpaper up to #4000, and then electro-polished with a solution of 20% nitric acid and 80% methanol at a temperature lower than 15 °C under a voltage of 10 V for 5 s. The EBSD measurements were performed under the beam controlled mode with a step size of 70 nm for the as-cold-rolled and the as-heated samples and of 0.35 μm for the as-sintered samples. The measurement results were analyzed with the Channel 5 (Oxford Instruments) and homemade ATEX [START_REF] Beausir | Analysis Tools for Electron and X-ray diffraction[END_REF] software packages.

The bulk texture of the pure Cu and the Cu/GNS samples at the states of as-cold-rolled, as-fullysintered and as-heated to different temperatures were measured by neutron diffraction at the diffractometer STRESS-SPEC located at FRM II (Heinz Maier-Leibnitz Zentrum (MLZ), Garching, Germany). The PG (200) monochromator was selected to produce neutrons with a dual-wavelength of 1.11 and 1.65 Å. The (111), (220) and (200) diffraction patterns were recorded at 2θ = 51° with a detector window of 15°. The software StressTextureCalculator (SteCa2-2.0) [START_REF] Randau | StressTextureCalculator: a software tool to extract texture, strain and microstructure information from area-detector measurements[END_REF] was used to extract the pole figure intensity data.

In-situ synchrotron radiation measurement

In order to examine the lattice strain evolution during the sintering process, the in-situ high energy X-ray diffraction (HEXRD) experiments were performed at the P07-HEMS beamline operated by the Helmholtz-Zentrum Geesthacht at PETRA III (Deutsches Elektronen-Synchrotron (DESY), Hamburg Germany), using a DIL805A/D dilatometer [START_REF] Schell | The High Energy Materials Science Beamlime (HEMS) at PETRA III[END_REF]. The main experimental parameters of the HEXRD setup are summarized in Table 2. The instrument calibration was performed using a standard LaB6 powder sample. During the measurement, the as-cold-rolled pure Cu sample and Cu/GNS composite samples each was heated from the ambient temperature (< 28 °C) to 950 ℃ at a heating rate of 5 ℃•min -1 , then cooled in an argon atmosphere. The incident beam was set to be parallel to the TD of the cold-rolled Cu foils. The geometrical configuration of the measurement layout is illustrated in Fig. 1. During the whole heating process, the full Debye-Scherrer rings were recorded at every 8 seconds.

The measured 2D Debye-Scherrer patterns were processed using the FIT-2D [START_REF] Hammersley | Two-dimensional detector software: From real detector to idealised image or two-theta scan[END_REF] by which the Debye-Scherrer rings were integrated radially at selected azimuthal angles to yield the 1D diffraction patterns that were further analyzed using a homemade program, A-fit. The information contained in the peak profile, such as the integrated intensity (I), the Bragg angles (2𝜃) of the peaks and the full width at half maximum (FWHM) at each measured temperature were extracted. For the fitting of the diffraction peaks, a Voigt profile (Pseudo-Voigt function) that results from the convolution of the Gaussian and the Lorentzian profile was used to mimic the exact Voigt function and to describe approximately both the instrumental and the intrinsic sample profiles. In order to subtract the instrumental broadening from the observed profile, an internal standard calibration technique was used in which a standard LaB6 powder sample (with a thickness of 5 mm in the beam direction) diffraction data was recorded under the identical experimental condition at room temperature. After that, the diffraction data for the standard LaB6 powder was calibrated by the Maud software package [START_REF] Villalobos-Portillo | Texture, 2D diffraction and piezoelectricity[END_REF][START_REF] Lutterotti | MAUD -Materials Analysis Using Diffraction[END_REF][START_REF] Chateigner | Quantitative texture analysis and combined analysis[END_REF][START_REF] Chateigner | Combined Analysis[END_REF] to deduce the instrumental values so that the instrumental broadening could be excluded at different Bragg angles. It should be mentioned that the thickness of the LaB6 sample is very close to our pure Cu and Cu/GNS composite samples (4 mm), thus, the shapes and the volumes of the LaB6 powder sample and our samples irradiated by the beam are very close and the effect from size difference should be largely eliminated. The integration of the raw Debye-Scherrer rings to the diffraction patterns was performed with the fit2D package, and the data from the LaB6 powder measurement was used to subtract the instrumental effect. To analyze the characteristic lattice strain states of the Cu foils in the Cu/GNS sample with respect to those of the pure Cu, the in-plane (parallel to the foil plane) and out-of-plane (perpendicular to the foil plane) thermal expansion characteristics were further analyzed for the two kinds of samples. Under the geometrical arrangement of the synchrotron diffraction measurements (Fig. 1), the samples were fixed during the whole heating process, and the incident beam was set to be parallel to the TD of each However, when we compare the slopes of the dhkl curves of the Cu/GNS sample with those of the pure Cu sample in the same direction, we can find that with the increase of the temperature, 𝑑 ℎ𝑘𝑙 𝐶𝑢 (𝑇)

Results

Initial microstructure and texture of as-cold-rolled Cu foil

increases more rapidly than 𝑑 ℎ𝑘𝑙 𝐺𝑁𝑆/𝐶𝑢 (𝑇) , especially at higher temperatures. This indicates that the deposited GNSs on the surfaces of the Cu foils indeed imposed constraints to the thermal expansion of the Cu foils. To figure out the constraints from the GNSs, we calculated the relative lattice strain (Belonging to Type I, macro strain [START_REF] Chateigner | Combined Analysis[END_REF], giving rise to a shift of the diffraction peaks) of the Cu in the Cu/GNS sample with respect to that of the Cu in the pure Cu sample in the consistent direction at each temperature as expressed by Eq. ( 1):

𝜀 = 𝑑 ℎ𝑘𝑙 𝐺𝑁𝑆/𝐶𝑢 (𝑇) -𝑑 ℎ𝑘𝑙 𝐶𝑢 (𝑇) 𝑑 ℎ𝑘𝑙 𝐶𝑢 (𝑇) , (1)
where 𝑑 ℎ𝑘𝑙 𝐺𝑁𝑆/𝐶𝑢 (𝑇) and 𝑑 ℎ𝑘𝑙 𝐶𝑢 (𝑇) are the dhkl of Cu in the Cu/GNS and in the pure Cu sample at temperature T. The results are displayed in Fig 7c and7d. It is seen from Fig. 7c that the GNSs induced a compressive strain in the plane parallel to the Cu foil plane (in-plane strain). This in-plane strain possesses bi-axial character, as the Cu layers in the sample are in thin foil shape, especially when the recrystallization was complete and the banded microstructure was replaced by the equiaxed recrystallized grains. Interestingly, the {200}Cu and {111}Cu strains were consistent during the whole heating process, inferring that the Cu foils possess an isostrain feature. Similarly, in the ND (Fig. 7d), the GNSs also introduced a compressive strain in the direction normal to the foil plane (out-of-plane strain). Although the value of the {111}Cu strain is not exactly the same as that of the {200}Cu strain at each temperature, the magnitudes of the two strains are very close. In the two cases, the absolute values of the strains increased linearly with the temperature, demonstrating an elastic condition.

Further examination of the diffraction patterns revealed that different levels of peak broadening occurred to the {200}Cu peaks in the two kinds of samples. Because a wide diffraction peak corresponds to a spread of d-spacing of the reflective lattice plane, the broadening of the peak infers a large variation of the lattice strain (Type II and III [START_REF] Chateigner | Combined Analysis[END_REF], mesoscopic and micro strain, manifested as broadening of the diffraction peaks), from a negative strain to a positive strain, if the peak is symmetric, which is the case in the present study. Thus, the Full Width at Half Maximum (FWHM) [START_REF] Adler | Simplifications in the x-ray line-shape analysis[END_REF][START_REF] Leineweber | Notes on the order-of-reflection dependence of microstrain broadening[END_REF] of the representive {200}Cu peaks of the pure Cu and the Cu/GNS in the respective RD and ND was extracted and displayed in Fig. 8a andb, respectively. As the FWHM are very sensitive to crystal defects, especially dislocations (the Type III strain [START_REF] Chateigner | Combined Analysis[END_REF]) that possess both lattice tension and compression around the dislocation lines, the progressive decrease and then a rapid decrease of the FWHM values below 210°C for the pure Cu sample and below 170°C for the Cu/GNS sample are related to the progressive and then rapid annihilation of dislocations during the recovery and recrystallization processes. After these temperatures, the FWHM values of the two kinds of samples stay relatively constant (Fig. 8a andb). However, the RD-FWHM of the Cu/GNS sample behaves very abnormally, as shown in Fig. 8b. Instead of decrease, a drastic increase appeared although recrystallization had already completed and the dislocations had been largely eliminated (corresponding to the disappearance of the Type III strain [START_REF] Chateigner | Combined Analysis[END_REF]). Such a high level RD-FWHM infers that the compressive biaxial strain in the Cu foil plane is not constant everywhere. With respect to the average strain (corresponding to the diffraction peak), some parts of the foil were compressed and the other parts tensed. This result infers that the GNSs indeed imposed complicated constraints to the Cu foil (the Type II strain [START_REF] Chateigner | Combined Analysis[END_REF]). The multiple strain states of Cu will be analyzed in detail in the next part.

Discussion

Incompatible thermal expansion induced elastic strain states in Cu foil

The above results demonstrated that the GNSs deposited on the Cu surfaces imposed obvious constraints to the Cu foils during the heating process that significantly affected the growth of the recrystallized grains and modified the texture of the Cu foils. As the constraints occurred during the heating process and thermal expansion of the two materials is inevitable, the different thermal expansion behavior of the two should be at the origin. Fig. 9 displays the TECs of Cu, graphite and graphene. The difference between graphite and graphene lies in the number of the two-dimensional hexagonally structured carbon layers. For the multi-layered graphene (<15 layers) used in the present study, the expansion behavior shows different in-plane and out-of-plane features. The in-plane expansion behavior should be between that of graphene and graphite (> 10 layers [START_REF] Ni | Graphene Thickness Determination Using Reflection and Contrast Spectroscopy[END_REF]), i.e., with negative TEC values (Fig. 9), and the out-of-plane behavior should be close to that of graphite, i.e., with large positive TEC values even higher than those of Cu. For Cu, the expansion is isotropic with large positive TEC values. Thus, at elevated temperatures, the Cu of the Cu/GNS composite tended to expand in the planes parallel to the foil plane (in-plane), whereas the GNSs adhered on the Cu tended to shrink. The negative thermal expansion of the GNSs exerted a biaxial in-plain compressive constraint to the expansion of the Cu foil. However, in the out-of-plain direction, both the GNSs and the Cu foils expanded during heating. It should be noted that the samples were fixed by a holding frame in this direction during the heating process. Thus, with higher hardness and larger TEC, the GNSs exerted a uniaxial compressive constraint to the Cu foils in the direction perpendicular to the foil plane. Finally, a biaxial strain parallel to the Cu foil plane and a uniaxial compressive strain perpendicular to the Cu foil plane were produced in the Cu foils by the deposited GNSs on the Cu foil surfaces.

As the GNSs used in the present composite are very thin (1 ~ 5nm in thickness) with respect to the Cu foils (30 μm thickness), the biaxial constraint should be mainly situated in the surface layers of each Cu foil. With the increase of the depth to the central layer, this constraint decreases and the uniaxial compressive constraint in the foil normal direction (ND) becomes dominant. This means that the central layer experienced a tension in the foil plane from the uniaxial compression other than the compression from the in-plane shrinkage of the GNSs. Thus, the strain in RD changed from a compression in the surface layer to a tension in the central layer and resulted in a large variation of dhkl and in turn the increased RD-FWHM (Fig. 8b). Thus, during the heating process, each Cu foil was in a composite strain state, i.e., the two surface layers were in biaxial compressive isostrain state in the foil plane and the central layer was in the uniaxial compression state in the direction normal to the foil plane. Clearly, it should be this complicated strain state that sustained the growth of the Cube grains in the Cu/GNS composite. The reason for this is certainly related to the interplay between the strain state and the crystal orientation related elastic feature of Cu. Thus, the orientation-resolved elastic strain energy as the result of the interplay was further investigated.

Fig. 9. Variations of thermal expansion coefficients (TECs) of Cu [START_REF] Wang | Thermal Expansion of Copper[END_REF] and graphene [START_REF] Mounet | First-principles determination of the structural, vibrational and thermodynamic properties of diamond, graphite, and derivatives[END_REF] with temperature, and variations of in-plane and out-of-plane TECs of graphite [START_REF] Mounet | First-principles determination of the structural, vibrational and thermodynamic properties of diamond, graphite, and derivatives[END_REF] with temperature.

Orientation dependent elastic strain energy density of strained crystals

With the above analyzed strain state of the Cu foils, the stress state can be further derived and the elastic strain energy density can be calculated. In the theory of elasticity, if a crystal is elastically strained, the stress tensor 𝜎 𝑖𝑗 is related to the strain tensor 𝜀 𝑘𝑙 through the stiffness constant tensor 𝐶 𝑖𝑗𝑘𝑙 , as expressed in the following equation in Nye's notation [START_REF] Nye | Physical properties of crystals[END_REF]:

𝜎 𝑖𝑗 = 𝐶 𝑖𝑗𝑘𝑙 • 𝜀 𝑘𝑙 , (2)
where the subscripts i, j, k and l take the integer values 1, 2 and 3. The matrix expression of Eq. ( 2) is given as Eq. (A1) in Appendix. A2. Due to the cubic crystal symmetry of Cu, the independent elastic constants of Cu in the crystal coordinate system are C11, C12 and C44 with the corresponding values of 169, 122 and 75.3 GPa [START_REF] Zielinski | The influence of strain energy on abnormal grain growth in copper thin films[END_REF][START_REF] Knowles | The Biaxial Moduli of Cubic Materials Subjected to an Equi-biaxial Elastic Strain[END_REF], respectively. Eq. ( 2) for cubic crystals is given in Eq. (A2) in Appendix.

A2. As the elastic strains were determined in the sample coordinate system (RD-TD-ND), the stress and strain are further related by

𝜎′ 𝑖𝑗 = 𝐶′ 𝑖𝑗𝑘𝑙 • 𝜀′ 𝑘𝑙 (3)
in the sample coordinate system, where 𝐶′ 𝑖𝑗𝑘𝑙 =𝐴 𝑖𝑚 𝐴 𝑗𝑛 𝐴 𝑘𝑜 𝐴 𝑙𝑝 𝐶 𝑚𝑛𝑜𝑝 (i, j, k, l, m, n, o and p all take integral values from 1 to 3) and 𝐴 𝑟𝑡 (r = i, j, k, l and t = m, n, o, p) are direction cosine of the basis vectors of the crystal coordinate system with respect to the basis vectors of the sample coordinate system. Thus, the elastic strain energy density can be calculated as follows:

𝑊 = 1 2 𝜎′ 𝑖𝑗 •𝜀′ 𝑖𝑗 . (4)

Biaxial strain condition

In the present sample coordinate system, axes 1 and 2 stand for the respective RD and TD that define the plane parallel to the Cu foil plane and axis 3 stands for the ND normal to the foil plane. For the biaxial strain condition, the strain in the planes parallel to the foil plane is

𝜀′ 11 = 𝜀′ 22 = 𝜀′. (5)
The stress components that are not in the plane of the Cu foil can be considered to be zero, i.e.,

𝜎′ 23 = 𝜎′ 13 = 𝜎′ 33 = 0. (6)
As the GNSs were firmly embedded in the Cu foils,

𝜎′ 12 = 0. (7)
Under these boundary conditions and with the cubic symmetry of Cu, the following equations can be obtained: Hence, the linear mean value of the biaxial modulus for a plane (h k l) subjected to biaxial strain, 𝑀 ̅ ℎ𝑘𝑙 , is:

𝜎′
𝑀 ̅ ℎ𝑘𝑙 = 1 2 (𝑀 1 + 𝑀 2 ). ( 10
)
It should be noted that an equivalent way to obtain the coordinate transformation 𝐴 𝑟𝑡 between the orthonormal sample coordinate system and the orthonormal crystal coordinate system can be made by representing the relation between the two coordinate systems with a set of triple rotations, i.e., Euler angles (𝜑 1 , Ф, 𝜑 2 ), in Bunge notation [START_REF] Bunge | The influence of crystal and sample symmetries on the orientation distribution function of the crystallites in polycrystalline materials[END_REF][START_REF] Bunge | The role of the inversion centre in texture analysis[END_REF], and expressing it in the rotation matrix, as follows

𝐴 𝑟𝑡 = R(𝜑 1 ) 𝑅(Ф) 𝑅(𝜑 2 ) = [ 𝑢 1 𝑣 1 𝑤 1 𝑢 2 𝑣 2 𝑤 2 ℎ 𝑘 𝑙 ]. (11)
In such a matrix, all the row elements are related to the foil plane. The (h k l) represents the crystalline plane parallel to the foil plane or normal to axis 3 (the ND), the [𝑢 1 𝑣 1 𝑤 1 ] and [𝑢 2 𝑣 2 𝑤 2 ] are two orthogonal in-plane unit vectors parallel to the respective axis 1 and axis 2, i.e., the RD and the TD.

Once the matrix is constructed, the crystalline plane that is parallel to the foil plane and the directions parallel to the RD and the TD are known. Thus, in the biaxial strain condition, the elastic strain energy density can be calculated with the following equation:

𝑊 = 1 2 𝑀 ̅ ℎ𝑘𝑙 •𝜀 ′ (ℎ𝑘𝑙) 2 . ( 12
) 15
Then, the strain energy density associated with the uniaxial compression is:

W = 1 2 𝐸′ 33 •𝜀 ′ 33 2 (16)

Simulated elastic modulus for the two strain conditions

The above analyses show that both the biaxial elastic strain energy and the uniaxial compressive elastic strain energy are crystal orientation related. The former is proportional to the biaxial modulus 𝑀 ̅ ℎ𝑘𝑙 and the latter to the Young's modulus in the direction normal to the foil plane or the (h k l) plane, 𝐸′ [START_REF] Randau | StressTextureCalculator: a software tool to extract texture, strain and microstructure information from area-detector measurements[END_REF] . For the grains in the surface layers of the Cu foils in the Cu/GNS sample, they demonstrated a biaxial isostrain feature (Fig. 7c), meaning that for differently oriented grain the elastic strains induced by the GNSs are the same. Thus, the strain energy is solely determined by 𝑀 ̅ ℎ𝑘𝑙 . As Cu is elastically anisotropic, if differently oriented grains were subjected to the same biaxial strain, the strain energies of the grains should be very different. Thus, we simulated the 𝑀 ̅ ℎ𝑘𝑙 of all possibly oriented Cu crystals and the orientation-resolved 𝑀 ̅ ℎ𝑘𝑙 represented in the orientation space is displayed in Fig. 10. It is seen that the 𝑀 ̅ ℎ𝑘𝑙 is indeed very anisotropic, varying from 114.18 GPa to 261.26 GPa. The lowest 𝑀 ̅ ℎ𝑘𝑙 (114.86 GPa) is associated with the grains having their {002}Cu parallel to the foil plane and the highest 𝑀 ̅ ℎ𝑘𝑙 (261.26 GPa) with those possessing their {111}Cu parallel to the foil plane. Thus, in the biaxial isostrain state, the grains with the {002}Cu parallel to the foil plane possess the lowest elastic strain energy and are energetically advantageous to grow. Consequently, during the heating process when grain growth occurred, the growth of the Cube oriented grains was stimulated, as their growth can effectively minimize the elastic strain energy and hence the total energy of the system. For the grains in the central layers of the Cu foils, they mainly experienced a uniaxial compressive strain. Although the strains in differently oriented grains are slightly different, they are very close (Fig. 7d). Thus, the strain energy should also be dominated by the Young's modulus in the compression direction, i.e., 𝐸′ 33 . We also calculated the 𝐸′ 33 of all possibly oriented Cu grains and the obtained orientation-resolved 𝐸′ 33 represented in the orientation space is displayed in Fig. 11. It is seen that the 𝐸′ 33 is also very anisotropic, varying from 66.70 GPa to 191.27 GPa. The lowest 𝐸′ 33 (66.70

GPa) is associated with the grains with their <002>Cu parallel to the foil plane normal direction and the highest 𝐸′ 33 (191.27 GPa) is associated to those with their <111>Cu parallel to the foil plane normal direction. Thus, in the uniaxial compression (quasi-isostrain) condition, again the grains with the <002>Cu parallel to the foil plane normal direction or grains with the {002}Cu parallel to the foil plane possess the lowest elastic strain energy and are energetically advantageous to grow. As a result, during the heating process, the growth of the Cube oriented grains was stimulated, as their growth can minimize the elastic strain energy and hence the total energy of the system. These analysis results indeed indicate that the complicated strain state induced by the GNSs provided the unique driving force for the abnormal grain growth of the Cube oriented grains in the Cu/GNS composite.

Summary

In this study, the complicated elastic strain state of the Cu matrix induced by the thermal constraints from the GNS reinforcements in the Cu/GNS metal/non-metal composite was experimentally investigated through ex-situ and in-situ characterization of microstructure, orientation and lattice strain by means of SEM-EBSD in mesoscale, neutron diffraction and synchrotron radiation in macroscopic scale. Its selective simulating effect on the specifically oriented grains was quantitatively analyzed by integrated theories of crystallography and elasticity. Its orientation-related potential on orientation selectivity was mapped in the orientation space.

The results revealed that during the heating process of the sintering, the anisotropic and incompatible expansion of the GNS imposed multiple elastic constraints to the Cu foils. In the surface layers, the Cu foils mainly experienced a biaxial compressive isostrain parallel to the foil plane, and, in the central layer, they were subjected a uniaxial compressive strain perpendicular to the foil plane. The anisotropic elasticity of Cu favored the further growth of the Cube oriented grains. The growth of such grains allowed the smallest biaxial modulus planes ({200}Cu plane) to be situated in the biaxial isostrain planes and the smallest Young's modulus direction (<200>Cu direction) to be in the uniaxial compression direction, hence minimizing the total strain energy.

The mechanism of elastic strain induced grain growth of selected orientations revealed in the present work provides new information on recrystallization and grain growth of metallic materials and useful for analyzing abnormal grain growth in elastically strained materials, such as the cases in additive fabrication. The orientation related grain growth stimulus can be applied to fabrication process to realize texturization or even monocrystallization of metallic materials.

Acknowledgement

This work was supported by "National Natural Science Foundation of China" (Grant Nos. 51871074, 51971078 and 51671066). This work is based upon experiments in the framework of the proposal No. It should be mentioned that the determination of the coverage of GNSs is very approximate, as many GNSs are very thin and transparent, as indicated in Fig. A1-1c. This part of GNSs were neglected by the estimation. When the as-deposited Cu-GNS foils were stacked together, the coverage of GNSs should be increased, as the GNSs from the adjacent Cu foils may fill the uncovered regions of the two foils on the surfaces in contact. The average thickness of the GNS is about 3.85 nm that is consistent with the specification thickness of the purchased GNSs.

A2. Tensor expression used in this work

Matrix expression of Eq. ( 2) in Nye's notation [START_REF] Nye | Physical properties of crystals[END_REF]:

[

Fig. 1 .

 1 Fig. 1. Schema of in-situ synchrotron radiation measurement.

Fig. 2a displays

  Fig. 2a displays an SEM-EBSD RD-axis inverse pole figure (RD-IPF) micrograph of the RD-ND plane of the as-cold-rolled Cu foil. It is seen that the cold-rolled Cu foil possesses banded microstructure composed of grains elongated in the RD. The orientation of the deformed grains were further analyzed and represented with the respective texture component micrograph in Fig. 2b. It is clearly seen that the cold-rolled grains possess the Copper, the Brass, the Goss, the Cube and the S orientations that are the typical texture components of the cold-rolled FCC metals.Fig. 2c shows the φ2 = 0°, 45° and 60° sections of the orientation distribution function (ODF) of the cold-rolled foils measured by through-volume neutron diffraction. For reference, the ideal positions of the cold-rolled texture components are also indicated in the figure. It is seen from the measured ODF sections that macroscopically the texture of the present cold-rolled foils is also characterized by the five typical cold-rolled texture components: Copper, S, Brass, Goss and Cube, among which the Copper component is the strongest, the S the second and the Cube the weakest. Moreover, the Brass component spreads toward the Goss orientation along the α skeleton line (φ2=0° and 45° sections) and

Fig. 2 .

 2 Fig. 2. Microstructure and texture of initial as-cold-rolled Cu foils. (a) SEM-EBSD RD inverse pole figure (RD-IPF) micrograph. (b) SEM-EBSD micrograph colored according to orientation of cold-rolled texture components of FCC metals. (c) Typical φ2= 0°, 45° and 65° sections of orientation distribution function (ODF) [40] measured by neutron diffraction (ideal texture components of cold-rolled FCC materials are detailed and indicated).

3. 2

 2 Microstructure and texture of as-sintered pure Cu and Cu/GNS After sintering, the microstructures and textures in the two kinds of samples are in great contrast, although the initial banded structure in each Cu foil in both the pure Cu and the Cu/GNS sample was replaced by fully recrystallized coarse grains, as shown in Fig.3, the SEM-EBSD RD-IPF micrographs from the RD-ND plane of the as-fully-sintered multilayered pure Cu and Cu/GNS samples together with the corresponding neutron diffraction ODF sections. For the pure Cu (Fig.3a), it is composed of several coarse grains with the sizes in millimetre range and the texture is characterized by strong individual orientations. Although the microstructure of the pure Cu was obtained from the sample containing 100 layers of the cold-rolled foils, no layer interfaces can be identified within the coarse grains. The grain boundary migration during grain growth already overcame the confinement of the layer surfaces. However, for the Cu/GNS composite, the microstructure is clearly layered and mainly composed of rectangular blocks delineated by the layer interfaces (the horizontal ones) where the GNSs were deposited and with the boundaries nearly perpendicular to the layer interfaces (the vertical ones). Moreover, most of the boundaries are low-angle boundaries (< 15°, yellow lines in Fig.3b).Although each block has its own distinct orientation, they are very close to the Cube orientation. Very occasionally, non-Cube oriented grains can be spotted (light to bright blue ones). Most of these grains are twin related with their neighbouring Cube oriented blocks and the boundaries between them are the Σ3 coincident site lattice (CSL) boundaries. Such boundaries are less mobile[START_REF] Humphreys | Recrystallization and Related Annealing Phenomena[END_REF], hence survived the abnormal growth of the Cube grains. The through-volume neutron diffraction ODF sections (φ2 = 0° and 45°) in Fig.3bconfirmed that in the whole measured sample volume (4×10×2 mm 3 ) the Cube orientation is clearly dominant, thus, the microstructure displayed in Fig.3bis throughout the whole treated sample. The layered microstructure in the Cu/GNS sample suggests that recovery, recrystallization and then the abnormal grain growth occurred only within each foil and did not overpass the foil surfaces on which the GNSs were deposited. The orientation feature of each Cu foil infers that the Cube oriented grains had the absolute privilege to survive and grow by consuming the other orientation grains until they joined each other.

Fig. 3 .

 3 Fig. 3. EBSD RD-IPF micrographs and representative ODF φ2 sections from neutron diffraction of as-fully-sintered (a) pure Cu (φ2= 15° and 65°) and (b) Cu/GNS composite (φ2= 0° and 45°).

3. 3

 3 Microstructure and orientation evolution of pure Cu and Cu/GNS during heatingTo investigate the microstructure and texture evolution features of the Cu/GNS composite samples with respect to the pure Cu samples during the sintering process, the as-cold-rolled Cu and Cu/GNS samples were heated to 5 different temperatures (150 ℃, 170 ℃, 230 ℃, 450 ℃ and 700 ℃) and then quenched in compressed argon flow to preserve the as-heated microstructure and texture. Fig.4shows the EBSD Kernel Average Misorientation (KAM) micrographs of single-foil Cu of the pure Cu and Cu/GNS samples heated to 150℃ and 170℃ (Fig.4band Fig.4c), respectively, and the EBSD texture component micrographs of the single-foil Cu of the pure Cu and Cu/GNS samples heated to 230°C, 450°C and 700°C (Fig.4d, 4e and 4f), respectively. For reference, the KAM micrograph of the initial cold-rolled Cu is also displayed (Fig.4a). It is seen from Fig.4bthat with the increase of the heating temperature from 150°C to 170°C the microstructures are still in band form, however, at 150°C in both the pure Cu and the Cu/GNS sample, the density of the disorientation that represents the density of Geometrical Necessary Dislocations (GNDs) decreased in certain regions in the bands, as indicated with the white arrows in the figures, indicating the occurrence of recovery or the early recrystallization nucleation. When the heating temperature was increased to 170°C, small equiaxed and distortion-free grains started to appear in the two samples, as indicated with the yellow arrows in the micrographs, indicating the onset of recrystallization. When the samples were further heated to 230°C, recrystallization continued. In the Cu/GNS sample, the initially banded microstructure was totally replaced by recrystallized grains, whereas in the pure Cu sample, the recrystallization was still in progress with the microstructure composed of recrystallized grains and deformed bands. The orientations of the recrystallized grains in the two cases are close to those of the cold-rolled texture components, as evidenced by the consistent colors of the texture components in Fig.4d. The Cube oriented grains did not show any preference in growth. When the heating temperature was elevated to 450°C, abnormal grain growth occurred in the Cu/GNS sample (Fig.4e2). The Cube grains demonstrate strong growth preference. However, in the pure Cu sample, the recrystallization fully terminated and some recrystallized grains started to grow. Compared with the Cu/GNS sample, the grain growth did not occur preferentially to the Cube oriented grains. When the samples were further heated to 700°C, although abnormal grain growth occurred in the two samples, almost all the Cu/GNS sample is occupied by the Cube grains, whereas the pure Cu sample is mainly occupied by the Goss oriented grains in the observed area. These results infer that the GNSs deposited on the cold-rolled Cu foils indeed promoted the growth of the Cube oriented grains.

Fig. 4 .

 4 Fig. 4. Kernel Average Misorientation (KAM) micrographs of (a) Cu cold-rolled at ambient temperature and of Cu in pure Cu and Cu/GNS samples heated to (b) 150 ℃ and (c) 170 ℃, respectively. EBSD texture component micrographs of Cu in pure Cu and Cu/GNS heated to (d) 230 ℃, (e) 450 ℃ and (f) 700 ℃, respectively. The texture evolution during the recrystallization and grain growth processes was also quantitatively examined in a large sample volume by neutron diffraction. Fig. 5 shows the ODF sections of the pure Cu and the Cu/GNS composite sample heated to different temperatures (150 ℃, 170 ℃, 230 ℃ and 450 ℃) calculated from the neutron diffraction results measured from the whole sample volume. It is seen from Fig. 5a1, 5a2, 5b1 and 5b2 that both the pure Cu and the Cu/GNS samples still possess similar texture components but with different intensities with respect to those of the cold-rolled Cu. The spread of the Brass component along the α skeleton line toward the Goss orientation in the cold-rolled microstructure disappeared and the Cube component became strengthened, especially in the Cu/GNS sample. When the heating temperature was increased to 230 ℃, the texture evolutions in the two samples were quite different. In the Cu/GNS sample, the non-Cube orientations greatly weakened, whereas the Cube orientation strengthened. However, in the pure Cu sample, the Cube component slightly weakened, whereas the Copper component strengthened. When the heating temperature increased to 450°C, in the Cu/GNS sample, the Cube component continued to develop and the other orientations largely disappeared, whereas, in the pure Cu sample, the existing orientations all continued to decrease their intensities. These results indicate that the GNSs indeed provided a unique driving force for the growth of the Cube grains during the grain growth process.

Fig. 5 .

 5 Fig. 5. Neutron diffraction ODF φ2= 0°; 45°; 65° sections of pure Cu samples heated to (a1) 150°C; (b1) 170°C; (c1) 230°C; (d1) 450°C and of Cu/GNS samples heated to (a2) 150°C; (b2) 170°C; (c2) 230°C; (d2) 450°C.

Fig. 6 .

 6 Fig. 6. HEXRD patterns for (a) the initial cold-rolled Cu sample obtained using a 2D detector and as-integrated 1D diffraction patterns of (b) pure Cu and (c) Cu/GNS at temperatures from ambient to 800℃. The intensity profile of each pattern was obtained by integrating the intensity of corresponding 2D Debye-Scherrer rings in the full azimuth angle range.

  Cu foil in the pure Cu and Cu/GNS samples. Thus, the change of the d-spacing of the lattice planes perpendicular to the RD was used to analyze the in-plane strain, and the change of the d-spacing of the lattice planes perpendicular to the ND the out-of-plane strain. Fig.7a and 7bshows the RD and ND dhkl of the {111}Cu and {200}Cu planes of the Cu foils in the pure Cu and in the Cu/GNS samples. It is seen that, in general, the dhkl of each lattice plane increases linearly with the temperature. The difference in dhkl between the RD and ND is due to the 2D feature of the thin Cu foils.

Fig. 7 .

 7 Fig. 7. Comparison of dhkl of (a) {111}Cu, (b) {200}Cu of the two samples and relative lattice strains of Cu in Cu/GNS with respect to that of Cu in pure Cu (c) in RD (in-plane) and (d) in ND (out-of-plane).

Fig. 8 .

 8 Fig.8. FWHM of {200}Cu peaks for (a) pure Cu and (b) Cu/GNS in RD and ND on heating.

Fig. 10 .

 10 Fig. 10. φ2=0° to 90° sections of mean biaxial modulus 𝑀 ̅ ℎ𝑘𝑙 as a function of crystal orientation in orientation space.

Fig. 11 .

 11 Fig. 11. φ2=0° to 90° sections of Young's modulus in the uniaxial compression direction 𝐸′ 33 as a function of crystal orientation in orientation space.

  Fig.A1-1shows the surface morphology micrographs of the raw cold-rolled Cu foil and the asdeposited GNSs on the Cu foil surface. The coverage of GNSs on Cu foil was estimated using the gray levels of the GNSs. The result shows that at least about 40% of the Cu surface is covered by the GNSs.

Fig. A1- 1 .

 1 Fig. A1-1. SEM-secondary electron (SE) micrographs of the surface morphology of (a) raw cold-rolled Cu foil, (b) Cu foil deposited with GNSs (low magnification) and (c) Cu foil deposited with GNSs (high magnification).

Fig. A1- 2 .

 2 Fig. A1-2. (a) AFM image of raw graphene nanosheet deposited on a single-crystal silicon substrate; (b) the corresponding thickness profile along the white line in (a).

Table 1 .

 1 Chemical composition of Cu foil (wt.%).

	Cu	Pb	Fe	Sb	S	As	Bi
	≥ 99.90	≤ 0.005	≤ 0.005	≤ 0.002	≤ 0.005	≤ 0.002	≤ 0.001

Table 2 .

 2 Experimental parameters and devices of in-situ synchrotron radiation measurement.

	Energy [keV]	Wavelength [Å]	Beam size [mm 2 ]	Sample-Detector Distance [mm]	Acquisition time [s]	Detector	Dilatometer
	103.3	0.12	0.8×0.8	1306.5	8	Perkin Elmer XRD 1621	DIL 805A/D

  11 = 𝐶′ 1111 𝜀′ 11 + 𝐶′ 1122 𝜀′ 22 + 𝐶′ 1133 𝜀′ 33 𝜎′ 22 = 𝐶′ 2211 𝜀′ 11 + 𝐶′ 2222 𝜀′ 22 + 𝐶′ 2233 𝜀′ 33 (8) 𝜎′ 12 = 𝐶′ 1211 𝜀′ 11 + 𝐶′ 1222 𝜀′ 22 + 𝐶′ 1233 𝜀′ 33 .

	Thus, the biaxial moduli of plane (h k l) in the two in-plane directions (1 or RD and 2 or TD) is given
	by:
	𝑀 1 =	𝜎′ 11 𝜀	= 𝐶′ 1111 + 𝐶′ 1122 -	1 𝐶′ 3333	(𝐶′ 1133	2 + 𝐶′ 1133 𝐶′ 2233 )
							(9)
	𝑀 2 =	𝜎′ 22 𝜀	= 𝐶′ 2211 + 𝐶′ 2222 -	1 𝐶′ 3333	(𝐶′ 2233	2 + 𝐶′ 1133 𝐶′ 2233 ).

  Under this condition, the crystal is only subjected to a uniaxial stress in axis 3 (the ND), i.e., 𝜎′ 33 and all other normal stress components and all shear stress components are equal to zero, i.e.,𝜎′ 11 = 𝜎′ 22 = 𝜎′ 23 = 𝜎′ 13 = 𝜎′ 12 = 0. (13)Utilizing Eq. (13) as boundary conditions and with cubic crystal symmetry, the following stress and strain relations can be obtained:𝜎′ 11 = 0 =𝐶′ 1111 𝜀′ 11 +𝐶′ 1122 𝜀′ 22 +𝐶′ 1133 𝜀′ 33 𝜎′ 22 = 0 = 𝐶′ 2211 𝜀′ 11 +𝐶′ 2222 𝜀′ 22 +𝐶′ 2233 𝜀′ 33 (14) 𝜎′ 33 = 𝐶′ 3311 𝜀′ 11 +𝐶′ 3322 𝜀′ 22 +𝐶′ 3333 𝜀′ 33 .Thus, the elastic modulus in the compressive stress direction can be deduced and is as follows:

	𝐸′ 33 =	𝜎′ 33 𝜀′ 33	= (𝐶′ 1111 𝐶′ 2233	2 +𝐶′ 2222 𝐶′ 1133	2 -2𝐶′ 1122 𝐶′ 1133 𝐶′ 2233 )/( 𝐶′ 1122	2 -𝐶′ 1111 𝐶′ 2222 )+ 𝐶′ 3333 .
						(
						)
	4.2.2 Uniaxial compression condition

  𝐶 1111 𝐶 1122 𝐶 1133 𝐶 1123 𝐶 1113 𝐶 1112 𝐶 2211 𝐶 2222 𝐶 2233 𝐶 2223 𝐶 2213 𝐶 2212 𝐶 3311 𝐶 3322 𝐶 3333 𝐶 3323 𝐶 3313 𝐶 3312 𝐶 2311 𝐶 2322 𝐶 2333 𝐶 2323 𝐶 2313 𝐶 2312 𝐶 1311 𝐶 1322 𝐶 1333 𝐶 1323 𝐶 1313 𝐶 1312 𝐶 1211 𝐶 1222 𝐶 1233 𝐶 1223 𝐶 1213 𝐶 1212 ]

	[ 𝜎 22 𝜎 11 𝜎 33 𝜎 23 𝜎 13 𝜎 12 ]	=	[ 𝐶 11 𝐶 12 𝐶 12 0 𝐶 12 𝐶 11 𝐶 12 0 𝐶 12 𝐶 12 𝐶 11 0 0 0 0 0 0 0 0 0 0 0 𝐶 44 0 0 0 0 0 0 0 0 𝐶 44 ] 0 𝐶 44 0 0	𝜀 11 𝜀 22 [ 𝜀 12 ] 𝜀 13 • 𝜀 33 𝜀 23		(A2)
	𝜎 11				𝜀 11
	𝜎 22				𝜀 22
	𝜎 33 𝜎 23	=		•	𝜀 33 𝜀 23	(A1)
	𝜎 13				𝜀 13
	𝜎 12 ]		[	[	𝜀 12 ]
	For cubic crystals, Eq. (A1) is in the following form:


Eva Bedna ´r ˇova ´

Martin Drac ˇı ´nsky ´

S ˇtefan Malatinec

Ivana Cı ´sar ˇova ´

Fre ´de ´ric Lamaty
email: frederic.lamaty@umontpellier.fr

Martin Kotora
email: martin.kotora@natur.cuni.cz

Synthesis of a Bolm's 2,2'-Bipyridine Ligand Analogue and Its Applications

Keywords: catalysis, enantioselective synthesis, cyclotrimerization, bipyridines, chiral ligands

A new method of synthesis of an analogue of Bolm's 2,2'-bipyridine ligand based on the catalytic [2 + 2 + 2] cyclotrimerization of 1-halodiynes with nitriles was developed. Crucial step of the whole synthesis turned out to be homodimerization of a substituted 2-bromopyridine to the corresponding bipyridine, that was studied and optimized. The newly prepared bipyridine (S,S)-2 was then tested as a chiral ligand in metalcatalyzed enantioselective reactions. Out of the studied reactions the most promising results were obtained in epoxide ring opening (82% yield, 98% ee) and Mukaiyama aldol reaction (> 96% yield, 99/1 dr, 92% ee). In the case of Mukaiyama-aldol reaction as well as in the Michael addition, novel ligand 2 proved its robustness compared to Bolm's ligand as it was less sensitive to the purity of used reagents.

Introduction

2,2'-Bipyridines and their derivatives form an important class of heteroaromatic compounds with application in various fields of chemistry such as supramolecular and macromolecular chemistry, photochemistry and electrochemistry but more significantly, thanks to their unique coordination abilities, as ligands able to complex almost all imaginable metals. [1] On the contrary to regularly used phosphine or cyclopentadienyl ligands, bipyridine ligands are generally more stable both to moisture and oxidation (air), which is simplifying their synthesis, handling, and storage. In this manner bipyridines have found numerous applications in transition metal-catalyzed racemic and enantioselective reactions. For example, chiral 2,2'-bipyridine ligands have found applications in metal-catalyzed asymmetric reactions such as cyclopropanation of olefins, alkylation and allylation of aldehydes, hydrosilylation of ketones, allylic oxidation of cycloalkenes. [2] It is also worth mentioning that the bipyridine scaffold can be found in several natural compounds such as caerulomycins, [3] orellanine, [4] lavendamycin, [5] (AE)-streptonigrin, [6] and (+)-complanadine [7] (Figure 1).

One of the most comprehensively studied member of this class of compounds is diol 1 possessing the 2,2'bipyridine scaffold (Bolm's ligand) with (R,R) configuration (Figure 2) that was designed and synthesized by Bolm et al. in 1990. [8] The synthetic approach was slightly modified by Kobayashi et al. in 2005, who prepared (S,S)-1. [9] Bolm's ligand 1 proved to be a useful chiral ligand in various types of metal-catalyzed asymmetric reactions as e. g. allylation of aldehydes, [10] opening of epoxides, [11,12,13] Michael addition, [14] Mukaiyama-aldol reaction, [15] hydroxymethylation, [16] CÀH activation of indols, [17] and most recently Diels-Alder reaction. [18] In addition, its complexes with non-toxic metal salts (e. g. Fe, Bi, Sc) also showed interesting behavior in the aqueous media making it an important player in the field of green chemistry. [16,19,20] Despite the abovementioned interesting results, studies of the dependence of the activity of the ligand on changes in its structure have been almost exclusively limited to the variations of the t-Bu part of the alcohol moiety in 1 to some other groups (Me, i-Pr, Ph, Napth etc.). [9,11] Only one example of a derivative with substituents in 3,3' position (Me), was efficiently applied in alcoholysis and aminolysis of meso-epoxides. [11,21] 2 Results and Discussion Ligand design considerations. From the point of view ligand design, further elaboration of the basic skeleton of Bolm's ligand is desirable, because it could lead to development of new ligands capable of producing even higher asymmetric induction. One option is to add additional rings onto the pyridine scaffold that would restrict rotation along the CÀC bond connecting the pyridine rings as in the modified Bolm's ligand 2 (Figure 3). This could potentially induce a conformationally dynamic chirality. [START_REF] Walsh | For a typical example[END_REF] If the pyridine rings cannot lie in one plane, resulting conformational rigidity of the transition state will be higher. This has been already demonstrated in the case of Co [START_REF] Suzuki | [END_REF] or Re [24] complexes, and may serve as typical examples. In addition, it is reasonable to expect that the presence of an alkyl group in the 3 and 3' as well as 4 and 4' positions would increase, through its inductive effect, electron density on the nitrogen atom [START_REF] Grandberg | For a typical example of influence of inductive effect[END_REF][START_REF] Al-Aljouni | For an exemplary demonstration of steric and electronic effects in bipyridine ligands see[END_REF] and thus enhance the coordination to a transition metal. In this respect it has been noticed that the presence of alkyl group in the positions 3 and 3' and as well as 4 and 4' of the bipyridine ligand had a positive effect on the performance of Cr-bipyridine catalytic system. [27] Our intended approach to the modified Bolm's ligand was based on the following retrosynthetic analysis (Scheme 1): a) catalytic homocoupling of a bicyclic 2-halopyridine that in turn could be prepared by b) catalytic cyclotrimerization of 1-halohepta-1,6diyne with electron-poor nitriles. We have recently developed a new type of Ru-complex catalyzed [2 + 2 + 2] cyclotrimerization of halodiynes with electronpoor nitriles, [28] which would support this approach.

Synthesis of ligand 2. Starting hepta-1,6-diyne 3 was deprotonated with LiHMDS followed by reaction with trimethylsilyl chloride to furnish the monosilylated diyne 4. Its subsequent deprotonation with n-BuLi and reaction with NBS provided diyne 5 in 52% isolated yield after two steps (Scheme 2). It should be noted that the standard conditions for the synthesis of bromoalkynes by using AgNO 3 , NBS could not be applied, because of the unwanted removal of the TMS-group during the course of the reaction. Ensuing cyclotrimerization of diyne 5 with pivaloyl cyanide under the standard conditions [28] (Cp*RuCl(cod) (10 mol%), dichloroethane, 20 8C) provided a mixture

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asc.wiley-vch.de of two regioisomers 6 a and 6 b in 3.5:1 ratio furnishing the former in 56% and the latter in 15% isolated yields. As expected, the desired regioisomer 6 a was formed preferentially. Unfortunately, the use of a simpler starting material such as 1-bromo-1,6-heptadiyne was not feasible, because of undesirable homocyclotrimerization. The TMS-group was then removed by treatment with tetrabutylammonium fluoride yielding ketone 7 (94%). It was necessary to remove the TMS-group prior to the reduction of the carbonyl, because its steric hindrance hampered the reaction. Ketone 7 was then reduced by enantioselective transfer hydrogenation catalyzed by RuCl[(S,S)-Tsdpen](pcymene) [9] to the chiral alcohol 8 (96%, 94% ee). Its absolute S configuration was unequivocally confirmed by a single crystal X-ray analysis [29] (Figure 4). The crucial step of the whole pathway to bipyridine 2 was the reductive dimerization of (S)-8. Although, a plethora of various conditions for homocoupling of pyridines has been reported, they mainly deal with simple 2-halopyridines. [START_REF] Kotora | Palladium-Catalyzed Homocoupling of Organic Electrophiles or Organometals in Handbook of Organopalladium Chemistry for Organic Synthesis[END_REF] Moreover, it was observed that yields were highly dependent on the structure of the starting halopyridine. Out of a number of available reaction conditions four best (in terms of yields) were selected to screen the homocoupling of (S)-8 (Table 1). Among these belong: a) Pd(0)-catalyzed homocoupling by using Pd(II)/tetrakis(dimethylamino) ethylene (TDAE) (Entry 1) the original method used by Kobayashi et al. in the synthesis of Bolm's ligand 1; [9] b) Pd-catalyzed cross-coupling of the in situ generated tributylstannyl pyridine (Entry 2); [START_REF] Barolo | [END_REF] c) Ni(0)-catalyzed dimerization by using a mixture of NiBr 2 (PPh 3 ) 2 /Zn/Bu 4 NI in THF (Entry 3); [32] and finally, d) Ni(0)-catalyzed coupling by using a mixture of NiCl 2 /Zn/PPh 3 in DMF (Entry 4), the original method used by Bolm et al. in the first synthesis of bipyridine 1. [8] In all cases, the main side product was dehalogenated pyridine 9 (Table 1), the formation of this type of product was also observed in most of the published procedures. It should be noted that formation of the side product could be reduced by using carefully degassed solvents. The unexpected products were compounds 10 (formed when method b was used), and 11 (observed in method c) arising after coupling with butyl and phenyl group respectively. The highest HPLC yield (50%) of compound 2 was obtained by Ni(0)-catalyzed homodimerization (Entry 4), but because of the complicated separation, the product was isolated in just 35% yield.

Although in most of the publications studying the dimerization of similar compounds the tolerance of free hydroxyl groups is described, in our hands the reactions proceeded only in low to moderate yields (Entries 1-4 Table 1). We decided to study homocoupling of ketone 7 to diketone 12 with intent to reduce it to the diol afterwards. Homocouplings were carried out under the same reaction conditions as in the above mentioned cases (Table 2). The yields of diketone 12 obtained under the used conditions were improved in all cases and the best yield of 12 (70% HPLC yield, 66% isolated yield) was obtained when the catalytic system composed of a mixture of NiCl 2 /Zn/PPh 3 was used (Entry 4). Surprisingly, the next step, proved to be more challenging.

Although the racemic reduction of diketone 12 by using NaBH 4 provided a mixture of rac and meso product in 90% combined isolated yield (See SI), enantioselective reduction turned out to be more challenging (Table 3). Reduction with (À)-Ipc 2 BCl brought about no conversion of the starting material (Entry 1). On the other hand, transfer hydrogenation by using 5 mol% of RuCl[(S,S)-Tsdpen](p-cymene) [9] Scheme 2. Synthesis of (S)-8. and HCOOH/Et 3 N mixture at 20 8C gave rise to product 16, in which only one of the ketone moieties was reduced (Entry 2). Both the higher amount of catalyst (10 mol%, Entry 3) and heating the reaction mixture to 40 8C (Entries 3 and 4) provided compound 16 in higher yields (35% and 50%, respectively), but only with traces of the desired diol 2. Increasing the temperature to 60 8C along with 10 mol% of the Rucatalyst finally furnished the diol 2 in excellent enantiomeric excess (97% ee), although only in a moderate yield of 48% (Entry 5).

The next strategy, for the synthesis of diol 2 in high yields, was based on homocoupling of the acetylprotected alcohol 17. Chiral alcohol 8 was protected to give 17 in 99% isolated yield under the standard conditions (Ac 2 O, pyridine), then was homocoupled under the same four conditions as in the previous cases (Table 4). The best results were obtained again by Ni-catalyzed dimerization (Entry 4) furnishing the diacetate 18 in 68% isolated yield.

Deprotection of the hydroxyl groups gave the desired bipyridine diol 2 as an optically pure compound after isolation (Scheme 3). The S,S stereo-chemistry of the furnished diol 2 was confirmed by a single crystal X-ray analysis [29,33] (Figure 5).

We intended to determine the rotational barrier in compounds 2 and 12 by 1 H-NMR spectroscopy, because of speculation that the barriers might be close to that of 2,2'-dimethylbiphenyl (DG # = 18.1 kcal/mol at 25 8C, see Figure SI-19). [34,35] Although non-equivalency of the hydrogen atoms of the CH 2 groups in 2 due to diastereotopicity was observed in the spectra of compound 2 (the result of the presence of centers of chirality in the molecule), FULL PAPER asc.wiley-vch.de no signal splitting due to the restricted rotation around the axis of chirality in 1 H-NMR spectra of this compound as well as of compound 12 was observed even at À80 8C (Figure SI-19). This observation may be explained either by very low rotational barrier or by very small difference of resonance frequencies of the pairs of diastereotopic hydrogen atoms.

Because the 1 H-NMR determination of rotational barriers in compounds 2 and 12 was not conclusive, we performed density functional theory (DFT) calculations of the barrier in compound 2. For a validation of the computational method, we included calculations of the rotational barrier in 2,2'-dimethylbiphenyl. An energy scan upon the rotation around the biphenyl/ bipyridine main axis revealed a striking difference between compound 2 and 2,2'-dimethylbiphenyl (Fig-ure SI-20). While 2,2'-dimethylbiphenyl has a clear maximum, which has to be passed to achieve racemization, no such maximum is apparent in the scan of compound 2. On the other hand, the conformation with planar (antiperiplanar) arrangement of the two pyridine rings has minimal energy. This can be explained by stabilization due to extended conjugation in the planar conformation and at the same time, a decrease of steric hindrance due to missing hydrogen atoms in positions 1 and 1' in bipyridine. These data are in agreement with previously reported calculations regarding rotational barrier of 2,2'-bipyridine. [36] Optimization of the transition-state structure of the racemization in 2,2'-dimethylbiphenyl provided the rotational barrier of 16.7 kcal/mol, which is reasonably close to the experimental value of 18.1 kcal/mol. [34] In order to acquire some information on chiral transition metal pre-catalyst, we tried to crystallize various transition metal salts (FeBr 3 , Fe(ClO 4 ) 2 , Bi(OTf) 3 , CoCl 2 , [RuCl 2 (p-cymene)] 2 , Cu(OTf) 2 , Cu(BF 4 ) 2 , CuCl 2 , [CuCl(cod)] 2 , Cu(MeCN) 4 PF 6 ) with 2. Out of these attempts only crystallizations with various Cu salts provided partially useful results. However, unlike the previous reports that described Cu complexes with 1 as monomeric species with square pyramidal, [37] or trigonal bipyramidal [38] geometry for Cu(II) or Y-shaped planar [38] geometry for Cu(I), we obtained different types of species. Thus the crystallization of [CuCl(cod)] 2 with 2 provided light green crystals suitable for X-ray analysis. Interestingly, unlike in other cases of Bolm's ligand Cu-complexes, a cluster composed of four Cu atoms and two ligands 2-CuCl was obtained [29] (Figure 6). In addition, green crystals 2-CuX1 were obtained from other crystallization attempts with CuCl 2 . It was also a cluster, but with tetragonal geometry, in which four Cu atoms were coordinated to the chloride anion (Figure SI-25). In a similar manner blue crystals were obtained by Application of 2 in enantioselective catalysis. Application of ligand 2 in several catalytic enantioselective reactions was examined. For this purpose five typical reactions in which Bolm's ligand 1 proved its activity were chosen: a) Fe(II)-catalyzed Mukaiyamaaldol reaction, [15,19] b) Bi(III)-catalyzed hydroxymethylation, [20] c) Sc(III)-catalyzed Michael addition, [14a] d) Pd(II)-catalyzed CÀH activation of indols, [17] and e) Sc(III)-catalyzed opening of the epoxides. [13] The conditions from the literature were used without any optimization. In order to ensure comparability of the results obtained with (S,S)-Bolm's ligand 1 and our new analogue (S,S)-2, both of the reactions were done in parallel.

Ligand 2 was applied to iron(II)-catalyzed Mukaiyama-aldol reaction of propiophenone-derived silicon enolate ((Z)-trimethyl(1-phenylprop-1-enyloxy) silane) 22 and benzaldehyde, for which were selected three conditions reported by Kobayashi et al. [15] and Ollevier et al. [19] (Table 5). Conditions A used Fe(OTf) 2 , pyridine and a ligand, all in catalytic amounts in dimethoxyethane-water mixture (DME/ H 2 O 9/1) at 0 8C. In conditions B the source of iron was Fe(ClO 4 ) 2 and benzoic acid was used instead of FULL PAPER asc.wiley-vch.de pyridine in DME/H 2 O 7/3. In conditions C were used Fe(II) dodecylsulfate (Fe(DS) 2 ), benzoic acid and a ligand in pure water at 20 8C. The reaction under conditions A with ligand 2 proceeded to give the corresponding aldol product 23 in quantitative yield with excellent diastereoselectivity (99/1) and very high enantioselectivity of 92% ee (Entry 3). The results were comparable with the reported results obtained by using Bolm's ligand 1 [15] (Entry 1) or that carried out by us (Entry 2). The reaction was performed under conditions B by using ligand 2 provided aldol 23 in quantitative yield, good diastereoselectivity (98/2) and a good enantioselectivity of 90% ee (Entry 6) that were almost same as by using Bolm's ligand (Entries 4 and 5). It should be emphasized, that during the course of this study we found that the results of aldol reaction carried out with Bolm's ligand 1 were highly sensitive to the purity of all reactants. In order to achieve reproducible results all reactants had to be meticulously purified prior to the reactions. This can be nicely exemplified by comparison of reactions carried out with the reactants purified (distilled) prior to the reactions (see Entries 1-6) with those used as purchased (see Entries 7-10). Interestingly, the decrease in enantioselectivity was more pronounced in the cases when ligand 1 was used, than in cases when ligand 2 was used.

By using conditions C the published level of selectivity (dr 91/9, 92% ee) with Bolm's ligand 1 was not achieved and 23 was obtained in dr 88/12 and 84% ee (Entries 11 and 12). [19] This could be attributed to difficulties encountered during the preparation of Fe(DS) 2 . Gratifyingly, the use of ligand 2 gave aldol 23 with better diastereoselectivity (93/7) and enantioselectivity (88% ee) (Entry 13).

The second application was Bi(III)-catalyzed hydroxymethylation of the same silicon enolate 22 with formaldehyde (Table 6). [20] By using ligand 2, product 24 was furnished in quantitative yield and 86% ee (Entry 3) a result similar to the one obtained with Bolm's ligand (Entry 2). Ligand 2 was also used in Sc(III)-catalyzed Michael addition of b-ketoester 25 to methyl vinyl ketone (MVK) in dichloromethane (Table 7). [14a] The Michael adduct 26 was obtained in quantitative yield and 81% ee (Entry 3), while using Bolm's ligand 1 its yield was 94% and 90% ee (Entry 2). However, without drying of Sc(III) triflate and distillation of MVK prior to use ee dropped to 67% (Entry 4). On the other hand, under the same conditions using ligand 2 ee remained unchanged at 81% (Entry 5).

Ligand 2 was also tested in CÀH activation of indol 27 followed by conjugated addition to 28 catalyzed by Pd(II) in the presence of dodecyl sulfate salt in water (Table 8). [17] In this case were the yield and ee of product 29 noticeably lower (69%, 72% ee, Entry 3) compared to the use of Bolm's ligand (94%, 89% ee, Entry 2).

Table 5. Fe(II)-catalyzed Mukaiyama-aldol reaction of silicon enolate 22 and benzaldehyde in the presence of Bolms ligand 1 and ligand 2.

Entry Cond. [a] Ligand Yield (%) [b] dr (syn/ anti) [c] ee (%) [d,e] (syn) )

1 [f] A ( S,S)-1 90 97/3 91 2 A (S,S)-1 > 96 [c] 97/3 93 3 A (S,S)-2 > 96 99/1 92 4 [f] B ( S,S)-1 85 97/3 88 5 B (S,S)-1 95 [c] 97/3 91 6 B (S,S)-2 > 96 98/2 90 7 [g] A ( S,S)-1 > 96 [c] 95/5 86 8 [g] A ( S,S)-2 96 97/3 92 9 [g] B ( S,S)-1 > 96 [c] 96/4 76 10 [g] B ( S,S)-2 90 97/3 83 11 [h] C ( S,S)-1 80 91/9 92 12 C (S,S)-1 88 [c] 88/12 84 13 C (S,S)-2 87 93/7 88 [a] Conditions A: Fe(OTf) 2 (3 mol%), pyridine (7.2 mol%), ligand (3.6 mol%), DME/H 2 O 9/1, 0 8C. [15] Conditions B: Fe(ClO 4 ) 2 (3 mol%), PhCOOH (3.6 mol%), ligand (3.6 mol%), DME/H 2 O 7/3, 0 8C. [15] Conditions C: Fe(DS) 2 (5 mol%), PhCOOH (5 mol%), ligand (15 mol%), H 2 O, 20 8C. [19] [b] Isolated yields unless otherwise noted. [c] Determined by 1 H-NMR. [d] Determined by HPLC with a chiral stationary phase. [e] S,S configuration in all cases. [f] Data reported in ref. [15]. [g] Reactants were not purified prior to the reaction. [h] Data reported in ref. [19]. Table 6. Bi(III)-catalyzed hydroxymethylation of silicon enolate 22 with formaldehyde in the presence of Bolms ligand 1 and ligand 2.

Entry

Ligand Yield (%) [a] ee (%) [b,c] 1 [d] (S,S)-1 93 91 2 ( S,S)-1 > 96 89 3 ( S,S)-2 > 96 86

[a] Isolated yields. [b] Determined by HPLC with a chiral stationary phase. [c] S configuration in all cases. [d] Data reported in ref. [20].

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Last but not least, the activity of ligand 2 was also tested in Sc(III)-catalyzed epoxide ring opening of cisstilbene oxide 30 by p-methoxybenzyl alcohol in dichloromethane (Table 9). [13] By using Bolm's ligand 1, product 31 was obtained with 90% yield ( 1 H-NMR) and 95% ee while using ligand 2, product 31 was obtained in 82% isolated yield and with excellent enantioselectivity of 98%.

Conclusion

We have demonstrated that Ru-catalyzed cyclotrimerization of a 1-halodiyne with nitriles is a suitable reaction allowing preparation of substituted pyridines. Specifically, cyclotrimerization of (7-bromohepta-1,6diyn-1-yl)trimethylsilane with pivaloyl cyanide provides regioselectively 1-bromo-6,7-dihydro-5H-cyclopenta [c]pyridine that can be easily converted into Bolm's ligand analog 2. One of the crucial steps along the synthetic route was homodimerization of the bromopyridine that was optimized to give a bipyridine intermediate in high yield.

As far as the use of 2 as ligand in enantioselective reactions is concerned, the results indicate its potential applicability. Although the use of 2 in hydroxymethylation, Michael addition, and CÀH activation gave products with lower enantioselectivity than the reported values obtained with Bolm's ligand 1, other reactions were more promising. The use of 2 in aldol reaction gave products within the same range of enantioselectivity as Bolm's ligand 1. However, it should be pointed out that 2 is more robust, because it gave exactly the same results with reactants that were not purified prior to the reaction. In this instance the use of Bolm's ligand 1 gave products with decreased enantioselectivity by 6-7% ee. Its robustness was also noticed in the case of Michael addition with reactants that were not purified where the product was obtained with enantioselectivity of 81% ee, whereas with 1 the obtained ee was 67%. Gratifyingly, in the ring opening of epoxides it gave products with slightly higher enantioselectivity of 98% ee (95% ee with 1 in our hand, 97% ee the reported value).

Experimental Section

Entry

Ligand Yield (%) [a] ee (%) [b,c] 1 [d] (S,S)-1 94 92 2 ( S,S)-1 94 [e] 90 3 ( S,S)-2 > 96 81 4 [f] (S,S)-1 > 96 67 5 [f] (S,S)-2 > 96 81 [a] Isolated yields unless otherwise noted. [b] Determined by HPLC with a chiral stationary phase. [c] R configuration in all cases. [d] Data reported in ref. [14a].

[e] 1 H-NMR yield. [f] Reactants were not purified prior to the reaction. Entry Ligand Yield (%) [a] ee (%) [b,c] 1 [d] (S,S)-1 94 92 2 ( S,S)-1 94 [e] 89 3 ( S,S)-2 69 72

[a] Isolated yields unless otherwise noted. [b] Determined by HPLC with a chiral stationary phase. [c] R configuration in all cases. [d] Data reported in ref. [17].

[e] 1 H-NMR yields.

Table 9. Sc(III)-catalyzed ring opening of cis-stilbene oxide 30 in the presence of Bolms ligand 1 and ligand 2.

Entry Ligand Yield (%) [a] ee (%) [b,c] 1 [d] (R,R)-1 82 97 [e] 2 ( S,S)-1 90 [f] 95 3 ( S,S)-2 82 98 respective spot on TLC), volatiles were evaporated under reduced pressure. Column chromatography of the residue on silica gel (gradient 10/1!2/1 hexanes/DCM) provided 566 mg (6 a, 56%) and 151 mg (6 b, 15%) of the title compounds as colorless solids.

6 a: Mp= 65.7 8C. ((S,S)-2). To a solution of 321 mg of 18 (321 mg, 0.65 mmol) in MeOH (5 mL) in 25 mL flask, K 2 CO 3 (675 mg, 4.9 mmol) was added. After stirring of the mixture for 5 hours at 20 8C, the reaction was quenched with water and extracted with DCM (4 3 30 mL). The combined organic layers were dried over MgSO 4, filtered and concentrated under reduced pressure. Column chromatography of the residue on silica gel (gradient 5/1!3/1 hexanes/EtOAc) provided 254 mg (96%, > 99% ee) of the title compound 2 as colorless crystals. Mp = 214.0 8C. (1S,2S)-2-((4-Methoxybenzyl)oxy)-1,2-diphenylethan-1-ol (31). To a solution of Sc(OTf) 3 (15 mg, 0.03 mmol) and (S,S)-2 (13 mg, 0.03 mmol) in dry DCM (1.5 mL) in 4 mL vial under the argon atmospehe were after 5 min of stirring at 20 8C subsequently added cis-stilbene oxide 30 (59 mg, 0.3 mmol) and p-methoxybenzyl alcohol (86 mg, 0.6 mmol). After 12 hours of stirring at 20 8C, the corresponding spot of stilbene disappeared, so the volatiles were removed under reduced pressure. Column chromatography of the residue on silica gel (5/1 hexanes/Et 2 O) furnished 83 mg (82%, 98% ee) of the title compound 31 as colorless crystals.

The recorded values were in agreement with the published data. [13]

Figure 1 .

 1 Figure 1. Natural compounds possessing the 2,2'-bipyridine scaffold.

Figure 2 .

 2 Figure 2. Bolms 2,2'-bipyridine ligand (R,R).

Figure 3 .

 3 Figure 3. Bolms ligand analog 2.

Figure 4 .

 4 Figure 4. View on the molecule of 8. Displacement ellipsoids are drawn on 30% probability level.

  Scheme 3. Synthesis of the target compound 2 by deprotection of the diacetate 18.

  Isolated yield unless otherwise noted. NR -no reaction; ND -not detected.[b] 1 H-NMR yield.

Figure 5 .

 5 Figure 5. View on the molecule of 2. Displacement ellipsoids are drawn on 30% probability level.

  (isolated yield). ND -not detected.

Figure 6 .

 6 Figure 6. View on the molecule of 2-CuCl. The displacement ellipsoids are drawn on 30% probability level. The solvating methanol molecules as well as second position of disordered C14 atom are omitted for clarity. The complex exhibits the symmetry of two fold axis of the space group C2, therefore position of half of the atoms are generated via operation with symmetry code (i) 1-x, y, 1-z.

1-( 1 -

 1 Bromo-4-(trimethylsilyl)-6,7-dihydro-5H-cyclopenta[c] pyridin-3-yl)-2,2-dimethylpropan-1-one (6 a) and 1-(4-bromo-1-(trimethylsilyl)-6,7-dihydro-5H-cyclopenta[c]pyridin-3-yl)-2,2-dimethylpropan-1-one (6 b). To a dried argon flushed 50 mL flask with Cp*RuCl(COD) (110 mg, 0.29 mmol), DCE (14 ml) followed by pivaloyl cyanide (0.7 ml, 5.8 mmol) were added. Then diyne 5 (700 mg, 2.9 mmol) dissolved in DCE (18 mL) was added slowly by the syringe pump during the course of 1 hour and the reaction mixture was stirred at 20 8C After the consumption of 5 (disappearance of the

Table 7 .

 7 Sc(III)-catalyzed Michael addition of b-ketoester 25 and MVK in the presence of Bolms ligand 1 and ligand 2.

  0580. R f (10/1 hexane/EtOAc) = 0.28. a ½ 20 D = + 1.528 (CHCl 3 , c = 2.62 g/100 mL) (S,S)-(6,6',7,7'-Tetrahydro-5H,5'H-[1,1'-bi(cyclopenta[c] pyridine)]-3,3'-diyl)bis(2,2-dimethylpropane-1,1-diyl) diacetate(18). To a dry argon flushed 100 mL flask with NiCl 2 (305 mg, 2.3 mmol) dry degassed DMF (13.5 mL) was added. The mixture was heated to 70 8C, then PPh 3 (2.4 g, 9.24 mmol) and Zn (166 mg, 2.5 mmol) were added. The reaction mixture was stirred at 70 8C for 1 hour, then 17 (628 mg, 1.9 mmol) in dry degassed DMF (13.5 mL) was added. After stirring for 14 hours at 70 8C, the reaction mixture was quenched with water (25 mL) and extracted with Et 2 O (33 100 mL). The combined organic layers were washed with brine (50 mL), dried over MgSO 4 , filtered and concentrated under reduced pressure. Column chromatography of the residue on silica gel (gradient 20/1!5/1 hexanes/ EtOAc) furnished 321 mg (68%) of the title compound 18 as a colorless solid. Mp = 173.6 8C. 1 H-NMR (400 MHz, CDCl 3 ) d 7.11 (s, 2H, 2 3 ArÀH), 5.56 (s, 2H, 2 3 CH), 3.49-3.38 (m, 2H, 2 3 CH), 3.14-3.03 (m, 2H, 2 3 CH), 2.92 (t, J = 7.6 Hz, 4H, 2 3 CH 2 ), 2.12 (s, 6H, 2 3 CH 3 ), 2.11-1.94 (m, 4H, 2 3 CH 2 ), 1.00 (s, 18H, 6 3 CH 3 ).13 C-NMR (100 MHz, CDCl 3 ) d 170.43, 154.99, 154.63, 152.27, 138.88, 117.76, 84.08, 35.16, 33.01, 32.99, 26.53, 24.99, 21.32. IR (drift KBr) u max 2959, 2905, 2870, 1735, 1585, 1558, 1427, 1396, 1370, 1244, 1023, 972 cm À1 . HRMS (ESI) m/z calculated for C 30 H 41 N 2 O 4 (M + H) 493.30608, found 493.30600. R f (10/1 hexane/EtOAc) = 0.23.(S,S)-1,1'-(6,6',7,7'-Tetrahydro-5H,5'H-[1,1'-bi(cyclopenta[c] pyridine)]-3,3'-diyl)bis(2,2-dimethylpropan-1-ol)

  28521. R f (3/1 hexanes/EtOAc) = 0.33. a ½ 20 D = À20.58 (CHCl 3 , c = 0.367 g/ 100 mL).

Table 1 .

 1 Reductive dimerization of alcohol (S)-8 to diol (S)-2.

	Yield (%) [a]

[a] HPLC-yield (isolated yield). ND -not detected.

Table 2 .

 2 Reductive dimerization of the ketone 7 to diketone 12.

	Yield (%) [a]

4 20 + 10 (n-Bu) 6 Sn 2 (1.15 eq.) -toluene MW 150 + 180 4 + 2 20 30 20 20 3 NiBr 2 (PPh 3 ) 2 100 Zn (5 eq.) Bu 4 NI (5 eq.) THF 60 120 30 60 ND 10 4 NiCl 2 120 Zn (1.3 eq.) PPh 3 (4.8 eq.) DMF 70 16 70 (66) 30 ND ND

Table 3 .

 3 Transfer hydrogenation of the diketone 12 to diol 2.

Table 4 .

 4 Reductive dimerization of acetate

	18.
	17 to diacetate

Table 8 .

 8 Pd(II)-catalyzed CÀH activation of indol 27 in the presence of Bolms ligand 1 and ligand 2.

  1 H-NMR (300 MHz, CDCl 3 ) d 3.09 (t, J = 7.6 Hz, 2H, CH 2 ), 2.92 (t, J = 7.6 Hz, 2H, CH 2 ), 2.06 (p, J = 7.6 Hz, 2H, CH 2 ), 1.41 (s, 9H, 3 3 CH 3 ), 0.28 (s, 9H, 3 3 CH 3 ).13 C-NMR (75 MHz, CDCl 3 ) d 210.59, 162.76, 161.12, 141.90, 137.81, 130.33, 44.17, 36.91, 32.79, 28.31, 23.64, 1.69. IR (drift KBr) u max 2956, 1681, 1520, 1249, 1054, 919, 901, 845 cm À1 . HRMS (ESI) m/z calculated for C 14 H 25 NOBrSi (M + H) 354.0889, found 354.0889. R f (20/1 hexane/EtOAc) = 0.38.

	(S)-1-(1-Bromo-6,7-dihydro-5H-cyclopenta[c]pyridin-3-yl)-
	2,2-dimethylpropan-1-ol (8). To a dried argon flushed 10 mL
	flask with solution of 7 (332 mg, 1.2 mmol) and RuCl[(S,S)-
	Tsdpen](p-cymene) (40 mg, 0.06 mmol) in DCM (1 mL), 2/1
	mixture of Et 3 N/HCOOH (1.8 mL) was added. After stirring
	for 3 days at 20 8C, the reaction was quenched with water
	(10 mL) and extracted with EtOAc (3 3 30 mL). The com-
	bined organic layers were dried over MgSO 4 , filtered and
	concentrated under reduced pressure. Column chromatogra-
	phy of the residue on silica gel (gradient 10/1!7/1 hexanes/
	EtOAc) furnished 319 mg (96%, 94% ee) of the title
	compound 8 as an amorphous solid. Optically pure com-
	pound could be obtained by crystallization from CHCl 3 by
	slow evaporation of solvent.
	7.6 Hz, 2H, CH 2 ), 0.90 (s, 9H, 3 3 CH 3 ). 13 C-NMR (100 MHz,
	CDCl 3 ) d 159.75, 155.84, 140.65, 137.61, 118.37, 80.47, 36.36,
	33.91, 32.84, 26.01, 23.75. IR (drift KBr) u max 2956, 2903,
	1598, 1541, 1453, 1415, 1364, 1238, 1108, 1063, 1016 cm À1 .
	HRMS (ESI) m/z calculated for C 13 H 18 BrNO (M) 283.0572,
	found 283.

M.p. = 65.9 8C. 1 H-NMR (400 MHz, CDCl 3 ) d 7.01 (s, 1H, ArÀH), 4.27 (s, 1H, CH), 3.60 (brs, 1H, OH), 3.00 (t, J = 7.6 Hz, 2H, CH 2 ), 2.92 (t, J = 7.6 Hz, 2H, CH 2 ), 2.13 (p, J =

  1 H-NMR (400 MHz, CDCl 3 ) d 7.09 (s, 2H, 2 3 ArÀH), 4.71 (brs, 2H, 2 3 OH), 4.38 (s, 2H, 2 3 CH), 3.37-3.27 (m, 2H, 2 3 CH), 3.25-3.15 (m, 2H, 2 3 CH), 2.97 (t, J = 7.6 Hz, 4H, 2 3 CH 2 ), 2.16-2.05 (m, 4H, 2 3 CH 2 ), 0.97 (s, 18H, 6 3 CH 3 ). 13 C-NMR (100 MHz, CDCl 3 ) d 156.47, 155.68, 151.06, 138.76, 118.74, 80.50, 36.66, 33.10, 32.89, 26.29, 25.40. IR (drift KBr) u max 2969, 2870, 1588, 1558, 1435, 1363, 1226, 1066, 1051, 1014 cm À1 . HRMS (ESI) m/z calculated for C 26 H 37 N 2 O 2 (M + H) 409.28495, found 409.

Adv. Synth. Catal. 2018, 360, 2869 -2878

Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

Acknowledgements E. B., S ˇ. M., and M. K. would like to acknowledge financial support from the Czech Science Foundation (grant No. 17-07707S), Grant Agency of Charles University (grant No. 243-250362) and French embassy in Prague and M. D. from Ministry of Education of the Czech Republic (Project LO1304, Program "NPU-I"). E. B. and F. L. would like to thank Universite ´de Montpellier, CNRS and French Ministry of Higher Education and Research for funding. Last but not least, M. K. would like to express his gratitude to the Institute for Molecular Science, Okazaki, Japan for funding of his visiting professorship.


D Rontani
email: damien.rontani@supelec.fr

A Locquet

M Sciamanna

D S Citrin

Spectrally Efficient Multiplexing of Chaotic Light

We numerically demonstrate multiplexing and demultiplexing of two distinct chaotic optical signals with strongly overlapped spectra, which are generated by mutually coupled external-cavity semiconductor lasers. Demultiplexing is performed by complete chaos synchronization, while the lasers at the receiving end are optically injected with the same multiplexed signal that couples the emitters together. Such a configuration could lead to spectrally efficient chaos-based optical encryption and decryption of multiple data streams.

Optical chaos-based communication between two users has attracted considerable attention since high-bit-rate physical layer security can be achieved [START_REF] Vanwiggeren | Communication with chaotic lasers[END_REF][START_REF] Argyris | Chaos-based communications at high bit rates using commercial fibre-optic links[END_REF]. When more than two users are involved, and a single optical channel is available, it is necessary to multiplex the chaotic signals of the different users. In conventional optical communications, time-and wavelength-division multiplexing (TDM and WDM) are well known protocols that make use of different time slots and wavelengths, respectively, to convey each user's signal. It has been proposed to apply WDM [START_REF] Uchida | Dual synchronization of chaos in microchip lasers[END_REF][START_REF] Matsuura | Chaotic wavelength division multiplexing for optical communication[END_REF][START_REF] Zhang | Wavelength division multiplexing of chaotic secure and fiber-optic communications[END_REF][START_REF] White | Multichannel communication using an infinite dimensional spatiotemporal chaotic system[END_REF][START_REF] Buldú | Multimode synchronization and communication using unidirectionally coupled semiconductor lasers[END_REF][START_REF] Lee | Two-mode chaos synchronisation using a multi-mode external-cavity laser diode and two single-mode laser diodes[END_REF] to optical chaotic communications. Chaotic WDM can be achieved using either multiple chaotic single-mode lasers operating at detuned wavelengths [START_REF] Uchida | Dual synchronization of chaos in microchip lasers[END_REF][START_REF] Matsuura | Chaotic wavelength division multiplexing for optical communication[END_REF][START_REF] Zhang | Wavelength division multiplexing of chaotic secure and fiber-optic communications[END_REF] or multi-mode lasers [START_REF] White | Multichannel communication using an infinite dimensional spatiotemporal chaotic system[END_REF][START_REF] Buldú | Multimode synchronization and communication using unidirectionally coupled semiconductor lasers[END_REF][START_REF] Lee | Two-mode chaos synchronisation using a multi-mode external-cavity laser diode and two single-mode laser diodes[END_REF]. Users on the receiving end synchronize their chaotic fluctuations with those of the same-frequency emitter. Though interesting, chaotic WDM has the disadvantage of requiring a large frequency separation between channels to avoid interference between each user's wide spectrum [START_REF] Zhang | Wavelength division multiplexing of chaotic secure and fiber-optic communications[END_REF]. The application of the other standard multiplexing approach, TDM, would not lead to any improvement of the spectral efficiency of the communication.

In this letter, we show how to use more efficiently the large bandwidth of optical chaos for multiplexed communications. For this purpose, we propose an optical analogy of active-passive decomposition (APD) [START_REF] Kocarev | General approach for chaotic synchronization with applications to communication[END_REF] to create a multiplexed signal that equally drives the dynamics of the master and slave lasers. The separation of each carrier is performed through complete synchronization of each master/slave laser pair. The synchronization properties are numerically investigated in the special case of two pairs of lasers.

Figure 1 shows a two-user setup composed of two mutually coupled master lasers (M 1 and M 2 ) unidirectionally coupled to two slave lasers (S 1 and S 2 ). Each master is subjected to delayed optical feedback from mirror M f and to delayed optical injection from the other master. A linear combination of the two masters' delayed complex optical fields is thus injected in each master, but with Fig. 1. (Color online) Multiplexing scheme based on optically coupled semiconductor lasers. LD: laser diode (labeled M1, M2 for the masters and S1, S2 for the slaves), CS: current source, M, M f : mirrors, VA 1 ,VA 2 : variable attenuators, BS: 50/50 beam splitter, OI: optical isolator. E T is the multiplexed field sent to the slave lasers. specific strength (different variable attenuators), phase, and delay (different optical paths) for each field. The same linear combinations are then optically injected, after propagation on a free-space optical channel, into the uncoupled slave semiconductor lasers (S 1 and S 2 ). As a result, the dynamics of each slave is driven by an identical, though time-shifted, copy of the signal that drives the corresponding master. This coupling architecture can be considered an application of APD to multi-user optical delayed systems.

We have investigated theoretically the synchronization properties of our setup based on the assumption of single-mode semiconductor lasers modeled by the Lang-Kobayashi equations [START_REF] Lang | External optical feedback effects on a semiconductor injection laser properties[END_REF]. The system of equations for n = 0 GHz. The internal parameters are:

β sp = 1000 s -1 ). The operational parameters are J m 1 = J s 1 = 2.75J th , J m 2 = J s 2 = 2.5J th , and η m 11 = η c 11 = 10 GHz, η m 22 = η c 22 = 15 GHz, η m 12 = η m 21 = η c 12 = η c 21 = η m 1 η m 2 ,
α m,s 1 = 5, α m,s 2 = 4, τ m,s p1 = 2 ps, τ m,s p2 = 1 ps, γ m,s s1 = 0.5 GHz, γ m,s s2 = 1 GHz, ε m,s 1 = 5 × 10 -7 , ε m,s 2 = 2.5 × 10 -7 , g m,s 1 = 1.5 × 10 -4 s -1 , g m,s 2 = 1 × 10 -4 s -1 , N m,s 01 = 1.5 × 10 8 , N m,s 02 = 2 × 10 8 .
emitter-receiver laser pairs reads

dE m k dt = 1 2 (1 + iα m k ) (G m k -1/τ m pk )E m k + F m k + n j=1 η m jk exp(-iω m 0j τ m jk + i∆ω m/m jk t)E m j t -τ m jk (1)
dE s k dt = 1 2 (1 + iα s k ) (G s k -1/τ s pk )E s k + F s k + n j=1 η c jk exp(-iω s 0j τ c jk + i∆ω m/s jk t)E m j t -τ c jk (2) dN m,s k dt = J m,s k -γ m,s sk N m,s k -G m,s k |E m,s k | 2 (3)
where the subscript k denotes the kth lasers pair and superscripts m, s master or slave variables, respectively, independent Gaussian white noises with unitary variance. The geometry of our system leads, with the assumption of identical perfectly reflecting mirrors and identical optical coupling efficiencies in all laser cavities, to the following relations between flight times and coupling strengths:

E m,s k = |E m,s k |e iφ m,s k is the slowly varying complex elec- tric field, N m,s k the carrier number, G m,s k = g m,s k (N m,s k - N m,s 0k )/(1 + ε m,s k |E m,s k | 2 )
τ m jk = τ m kj = τ m jj + ∆τ m kj /2 with ∆τ m jk = -∆τ m kj = τ m jj -τ m kk and η m kj = η m jk = η m kk η m jj .
Additionally, our scheme has been devised in such a way that in each pair M k /S k is subjected to all the master chaotic optical fields E m j with identical relative time shifts. Mathematically, this means that τ m kk -τ m jk = τ c kk -τ c jk . Furthermore, it is necessary to set η m jk = η c jk to ensure that each coupling field has the same weight in the master and slave equations within a pair. This necessary condition naturally extends the coupling requirement for complete chaos synchronization between a single master and a single open-loop slave laser [START_REF] Locquet | Synchronization regimes of optical-feedback-induced chaos in unidirectionally coupled semiconductor lasers[END_REF]. Under the conditions above, a regime of complete synchronization exists in each pair M k /S k , in the absence of noise and of parameter mismatch between the lasers of a pair, and are mathematically described by the following equations:

E s k (t) = E m k (t -∆τ k ), φ s k (t) = φ m k (t -∆τ k ) -ω m 0k ∆τ k (mod 2π), and N s k = N m k (t -∆τ k )
, where ∆τ k is a time lag given by ∆τ k = τ c kk -τ m kk . Figure 2 presents theoretical synchronization diagrams of our system described by Eqs. ( 1)-( 3) with n = 2. They represent the evolution of the optical intensity, defined by

I m,s k = |E m,s k | 2
, of each master as a function of its respective slave. They are plotted without and with spontaneous noise emission in Fig. 2-case (a) and -case (b), respectively. Additionally, the synchronization quality is evaluated with the cross-correlation coefficient

C jk (τ ) = [I m j (t + τ ) -I m j ][I s k (t) -I s k ] I m j (t + τ ) -I m j I s k (t) -I s k 1/2 , ( 4
)
where • is the time-averaging operator. In Figs. We find that the degree of decorrelation between each pair strongly depends on the value of coupling parameters, as will be detailed elsewhere. The introduction of noise destroys the perfect synchronization [C 11 (∆τ 1 ) ≈ C 22 (∆τ 2 ) ≈ 0.97 and C 12 (∆τ 1 -∆τ 2 ) ≈ 0.07], but the trajectories in the plane (I m k , I s k ) remain sufficiently close to the noiseless synchronization manifolds [Fig. 2(b1)-(b4)] for our scheme to be used for communication applications.

We determine that perfect synchronization remains robust to parameter mismatch between lasers in a pair as well if the mismatch level is comparable to that encountered in single emitter/receiver laser configurations [START_REF] Sánchez-Díaz | Encoded Gbit/s digital communications with synchronized chaotic semiconductor lasers[END_REF]. Additionally, it must be noted that our APD-based coupling configuration does not limit the amount of mismatch between two different pairs (so long as they remain in chaotic regimes). By way of illustration, the case of two laser pairs has been presented, but this number can be increased: if each new master laser shares the original external cavity, and each slave laser remains decoupled and properly injected. Simulations suggest that complete synchronization still exists for a large range of coupling parameters and remains stable as n is increased. This architecture has thus the potential to handle chaos-based communications between numerous users.

Finally, we investigate the spectral efficiency of the proposed multiplexing scheme by computing the optical spectrum of each master laser M k and of the optical field E T propagated on the transmission channel. Figure 3 shows the spectra of the two free-running master lasers (inset) and of the coupled external-cavity master lasers. The free-running spectra are centered at ∆f = 0 GHz. The mutual coupling and feedback induce chaotic behaviors of both M 1 and M 2 resulting in spectral broadening. The two chaotic spectra significantly overlap and have identical bandwidth (taken 20 dB below the spectrum's maximum) ∆f E m 1 ≈ ∆f E m 2 ≈ 25 GHz. Since the transmitted signal E T has a bandwidth ∆f E T ≈ 25 GHz, it presents similar spectral properties to each master chaotic optical field but with more optical power [Fig. 3]. Assuming each master signal E m k (k = 1, 2) encodes a different information-bearing message, the message bit rate will be double compared to the single-ECL emitter case, while using a similar bandwidth. Our scheme could thus lead to a dramatic increase of the spectral efficiency of chaotic communications. Additionally, the transmitted signal E T may also mislead an eavesdropper trying to attack the system, who could consider it is as resulting from a single chaotic external-cavity semiconductor laser.

In this letter, we have shown for the first time to our knowledge the possibility to multiplex and Fig. 3. (Color online) Optical spectrum of each master laser and of the transmitted signal, calculated using Welch's method. Free-running (inset) and chaotic spectra are represented. Parameters are the same than in Fig. 2.

demultiplex chaotic optical fields with a strong spectral overlap and produced by two semiconductor lasers with identical free-running wavelength using both a coupling architecture based on APD concepts and complete chaos synchronization. Our results pave the way toward spectrally efficient optical encryption of multiple datastreams using chaotic semiconductor lasers.

Fig. 2 .

 2 Fig. 2. (Color online) Synchronization diagrams in planes (I m k=1,2 (t -∆τ k ), I s j=1,2 (t)) without (in case (a)) and with spontaneous-emission noise (in case (b), β sp = 1000 s -1 ). The operational parameters are J m 1 = J s 1 = 2.75J th , J m 2 = J s 2 = 2.5J th , and η m 11 = η c 11 = 10 GHz, η m 22 = η c 22 = 15 GHz, η m 12 = η m 21 = η c 12 = η c 21 = η m 1 η m 2 , and τ m 11 = 1 ns, τ m 22 = 4 ns, τ c 11 = 10 ns, τ c 22 = 15 ns, and ∆ω

and τ m 11 =

 11 Fig. 2. (Color online) Synchronization diagrams in planes (I m k=1,2 (t -∆τ k ), I s j=1,2 (t)) without (in case (a)) and with spontaneous-emission noise (in case (b), β sp = 1000 s -1 ). The operational parameters are J m 1 = J s 1 = 2.75J th , J m 2 = J s 2 = 2.5J th , and η m 11 = η c 11 = 10 GHz, η m 22 = η c 22 = 15 GHz, η m 12 = η m 21 = η c 12 = η c 21 = η m 1 η m 2 , and τ m 11 = 1 ns, τ m 22 = 4 ns, τ c 11 = 10 ns, τ c 22 = 15 ns, and ∆ω m/s 11 = ∆ω m/s 22 = ∆ω m/m 12

  is the nonlinear gain with g m,s k the differential gain, N m,s 0k the carrier number at transparency, ε m,s k the gain saturation coefficient, τ m,s pk the photon lifetime, α m,s k is the linewidth enhancement factor, γ m,s sk is the carrier decay rate, J m,s k is the pumping current density, ω m,s 0k is the free-running laser frequency, τ m jk (τ c jk ), η m jk (η c jk ), ∆ω m/m jk = ω m 0j -ω m 0k (∆ω m/s jk = ω m 0j -ω s 0k ) are the flight time, injection strength, and detuning between the jth and the kth master laser (the jth master laser and the kth slave laser). Spontaneous emission noise is modeled by Langevin sources F m,s k = 2β m,s k N m,s k ξ m,s k with β sp the spontaneous emission rate and ξ m,s k

2 (

 2 a1)-(a4), straight lines are observed and correspond to perfect chaos synchronization of each laser pair M 1 /S 1 and M 2 /S 2 [C 11 (∆τ 1 ) = C 22 (∆τ 2 ) = 1] with time lags ∆τ 1 = 9 ns and ∆τ 2 = 11 ns. Figures 2(a1)-(a4) display a cloud of points which reveals a low degree of correlation between each laser pair [C 12 (∆τ 1 -∆τ 2 ) ≈ 0.04].

The authors acknowledge the support of Fondation Supélec, of Conseil Régional de Lorraine and of National Science Foundation by ECCS Grants No. 0925713 and No. 0523923.