Spiro configured compounds constitute one of the most important class of organic semiconductors used in all the devices of the emerging technologies of organic electronics. The origin of the success story of spiro compounds is linked to their particular geometry and the consequences on the structural, physical and electronic properties. Spiro compounds are constituted by the association of two molecular fragments linked by a shared tetrahedral carbon. Thus, spiro compounds possess a particular 3D geometry with the two fragments set along two orthogonal planes. This geometry is particularly appealing as the two branches are only electronically slightly connecting (spiroconjugation), allowing a fine control of the electronic properties.

Since the display of the 'spiro concept' for organic electronics in the nineties by the group of Salbeck, [1] the 9,9'-spirobifluorene (SBF), made by the association of two fluorene units via a shared spiro carbon has become the flagship molecular structure of spiro compounds and is nowadays a central molecular scaffold in organic electronics. [2,3] One of the particularity of the SBF fragment (and of spiro compounds in general) is its capacity to improve the thermal and morphological properties of the organic semi-conductors in which it has been introduced. This is a key property for organic technologies.

Thus, for the last ten years, the spiro architecture has been particularly used to design host materials for Phosphorescent OLEDs [4] and for TADF OLEDs. [5,6] Indeed, in these two technologies, the host material should gather an electron-rich and electron-poor fragment and these two fragments should be spatially separated. The spiro design appears herein as ideal.

In the present collection, the great diversity of spiro compounds in the emerging technologies of organic electronics is presented. I have selected outstanding papers published during recent years in Journal of Materials Chemistry C and Materials Advances that highlight advances in spiro compounds for electronic applications. These articles provide exciting results that span a wide range of applications including host materials for phosphorescent OLEDs, fluorescent and thermally activated emitters for OLEDs, visual sensory materials, non-fullerene acceptors for solar cells, hole transporting materials etc


A Comandini

D Nativel

Nabiha Chaumeix
email: chaumeix@cnrs-orleans.fr

Laminar flame speeds and ignition delay times of gasoline/air and gasoline/alcohol/air mixtures: the effects of heavy compared to light alcohol

Keywords: Bio-fuels, Renewable fuels, iso-pentanol, gasoline, ethanol, laminar flame speeds, ignition delay times

The oxidation of gasoline and gasoline/alcohol blends is studied in a shock tube and a spherical reactor. A commercial oxy-free gasoline and two alcohols (ethanol and iso-pentanol) were used in this study. The spherical reactor experiments were conducted at an initial temperature of 483 K, initial pressure of 0.1 MPa, and equivalence ratios from 0.65 to 1.36. Ignition delay times were measured behind reflected shock waves. The shock tube experiments were conducted at 2 MPa over a temperature range from 955 to 1284 K and for two equivalence ratios (0.5 and 1). The experimental measurements indicate that replacing ethanol by iso-pentanol on a gasoline/alcohol blend results in flame speeds which are closer to the ones of a commercial gasoline at nearly stoichiometric conditions (an in-engine applications). On the other hand, the ignition delay times are more affected by the presence of iso-pentanol than the ethanol case. Two different surrogate fuels composed of n-heptane, iso-octane, and toluene were also tested against the newly obtained experimental results (oxy-free gasoline and mixtures with ethanol) using kinetic modeling with a reduced model [L. Cai, H. Pitsch, Combust. Flame 162 (2015) 1623-1637]. While a good agreement between real fuel and surrogate properties was observed for the laminar flame speeds, discrepancies were obtained between the measured and calculated ignition delay times especially in the lower temperature range of our study. Additional simulation analyses on the ignition delay times were performed with a different chemical kinetic model (detailed LLNL model for gasoline surrogates) and a more-complex four-component surrogate. The results show a considerable improvement in the prediction capabilities of the ignition properties of the oxy-free gasoline and the oxy-free gasoline + ethanol mixtures. The ignition delay time data are also in agreement with the correlations provided in the literature for gasoline fuels and their surrogates.

Introduction

In view of the necessity to reduce greenhouse effects and the possibility to develop renewable sources of oxygenated fuels, their blending with gasoline is becoming a common as well as necessary practice worldwide. Besides engine modifications to allow an efficient use of the new blended fuels, the study of the fuel combustion properties is an important subject of investigation in order to foresee any fuel reformulation (with blending amount as high as 20% v/v) that could be necessary in order to comply with the present and future environmental regulations. Today, ethanol is the most commonly used biofuel. However, its rather low energetic density is a major drawback for the users. To overcome this aspect, the use of heavier alcohols with higher energy densities such as iso-pentanol could be a solution if suitable supply sources are developed [1]. The Research Octane Number (RON) for ethanol, iso-pentanol and a typical commercial gasoline are 109 [2,3], 94 [3] and ~88-98 [3], respectively, while the Motor Octane Number (MON) are 90, 84, and 80-88, respectively. This suggests the use of iso-pentanol as a better solution compared to ethanol in view of the higher energy density (21.4 MJ/L and 27.8 MJ/L [4] for ethanol and iso-pentanol, respectively, compared to 30-33 MJ/L for gasolines) without degradation of the octane rating [START_REF] Kalghatgi | SAE paper[END_REF] (iso-pentanol and gasolines have similar RON and sensitivity S = RON -MON). On the other hand, the lower RON of iso-pentanol compared to ethanol is a drawback for the use of such biofuel as its addition will not improve the octane rating of the commercial fuels. In the recent years, a large number of studies concerning the combustion of ethanol and gasoline-ethanol blends have been published. The review article on alcohol combustion by Sarathy [3] contains a detailed summary of these investigations. In particular, more and more interest has also been addressed to the combustion properties of gasolines.

The first high-temperature shock tube study reporting ignition delay time measurements of gasoline/air mixtures was performed by Gauthier et al. [START_REF] Gauthier | [END_REF] at pressures between 15 and 60 atm, from fuel lean to fuel rich conditions. This study was subsequently complemented with experimental results from the same research group on different gasolines [7][8]. In particular, the recent work by Davidson et al. [7] reports very high-pressure (up to 250 atm) ignition delay time measurements and species time-history profiles on four gasolines characterized by very different compositions, including three oxy-free fuels. Cai et al. [9] studied the RON95E10 fuel supplied by Shell Global Solutions GmbH in a rapid compression machine and a shock tube, comparing the results with and without addition of exhaust gas recirculation (EGR) gases. A similar combination of experimental techniques was used by Sarathy et al. [10][11] to investigate the ignition properties of FACE (Fuels for Advanced Combustion Engines) gasolines. Concerning the laminar flame speed measurements of gasoline/air mixtures, Jerzembeck et al. [12] used a spherical reactor to monitor the propagation of gasoline/air and surrogate/air mixtures at an initial temperature of 373 K and initial pressures from 10 to 25 bar. A similar technique was implemented by Manaa et al. [13] to study the properties of three FACE fuels at initial temperature and pressure of 358 K and 0.6 MPa. Finally, Sileghem [14] and Dirrenberger et al. [2] studied commercial gasoline fuels at 298 K and 358 K with a flat flame adiabatic burner. So far, only pure iso-pentanol experimental studies are reported in the literature [15][16][17][18][19][20]. In the present study, laminar flame speeds and ignition delay times of gasoline, isopentanol/gasoline and ethanol/gasoline mixtures in air were measured using spherical reactor and shock tube techniques, respectively. The experimental datasets obtained herein provide essential information on the fundamental properties of gasoline/alcohol blended fuels at engine relevant conditions which can be also used as a benchmark for future fuel reformulations. In particular, the aim of the study is to understand the effects of blending different oxygenated fuels with a commercial gasoline in terms of the global properties of the fuel itself.

Experimental Facilities

Spherical reactor

Laminar flame experiments were performed in a 56 L heated stainless steel spherical vessel made of two concentric spheres, the space between the 2 spheres allows the flowing of a thermal fluid which is regulated to the desired initial temperature. This system permits to fix the initial temperature to any value between 302 and 483 K and thanks to the circulation pump to ensure the homogeneity over the entire vessel. Two different thermocouples, one at the wall of the vessel and one protruding inside the combustion chamber are used to regulate and monitor the initial temperature with an uncertainty of ±1 K. the facility is insulated to limit the heat losses of the vessel to the ambient temperature and to ensure that no cold spot exist on the entire vessel. The combustion takes place in the inner sphere characterized by an internal diameter of 476 mm. The maximum operating pressure is 5 MPa and a piezo-electric pressure transducer (Kistler 601A) is used to monitor the pressure evolution during combustion. The sensor is mounted flush with the internal surface of the reactor.

The ignition of the combustible mixture is obtained via a spark generated by a spark created between two tungsten electrodes linked to a high voltage generator. The average energy delivered by the high voltage generator is 1.82 mJ with a standard deviation of 0.48 mJ. This energy has been measured using a high voltage probe and a current probe connected to the electrodes. Both U and I signals are recorded on an oscilloscope during the spark formation and hence the energy delivered by the spark (E = ∫U•I) is estimated. The spherical reactor is equipped with two opposite quartz windows (100 mm diameter, 50 mm thickness). The visualization of the flame was obtained using a Z-shape Schlieren apparatus. A white continuous lamp is used to illuminate the flame via two lenses and two concave spherical mirrors. A high-speed camera (PHANTOM V1610) with an acquisition rate of 25000 images per second records the Schlieren images of the growing flame. The frame size was fixed to 768 × 768 pixels 2 . More details can be found in [21][22].

The synchronization of the pressure measurement and the flame recording is obtained by using the voltage breakdown during the spark formation to trigger a TTL (transistor-transistor logic) generator which in turns triggers the camera and the oscilloscopes.

Shock tube

The shock tube has been described in detail in a previous publication [23]. Briefly, experiments were carried out behind reflected shock waves in a stainless-steel shock tube with a 2.00 m long driver section and a 5.15 m driven section with a 52.4 mm internal diameter. This tube is designed for initial pressures up to 4 MPa and initial temperatures up to 403 K. The maximum error in the temperature is 1%, thus around 10-15 K. Both shock tube portions were evacuated using two primary vacuum pumps. Mixtures of helium and nitrogen were used as driver gas, and the shock wave was initiated by the bursting of a double diaphragm. Four piezoelectric pressure transducers (Chimie Metal, Model A25L05B) with a sensitive surface area of 0.75 mm 2 and a rise time of 0.4 µs were used to measure the shock velocity over the last section (0.7 m long) and were mounted flush with the inner surface of the tube, the last one being at 10 mm from the end-wall. The precision on the distance is ±0.1 mm. The very small surface area coupled with a short rise time allows precise determination of the shock wave velocity and, consequently, more accurate evaluation of the temperature and pressure conditions behind the reflected shock wave (reflected shock pressure, P5, and temperature, T5, were calculated using the classical shock tube equations [START_REF] Gaydon | The shock-tube in high-temperature chemical physics[END_REF]). Additionally, a Kistler pressure transducer (model 603B1) is mounted at the end-wall for monitoring of the pressure time-history behind the reflected shock waves. The outputs from the pressure transducers were recorded using two numerical oscilloscopes (Tektronix TDS5034B).

Reagents

For the laminar flames study, the mixtures were prepared inside the spherical reactor using the partial pressure method. The introduction of air creates turbulences that ensured a good mixing of the various components. The waiting time between the introduction of air and the flame ignition was around 5 minutes. Several experiments were repeated for the same initial conditions but increasing the waiting time with no substantial differences in the results. This suggests that the initial mixture is homogeneous while the turbulence levels are too low to play a relevant role in the experimental measurements. Partial pressures of fuel as well as dry air were measured using capacitive manometers (MKS) of two different scales (13.33 and 133.3 kPa). For the ignition delay time study, the reactive gas mixtures were prepared manometrically in a 30 L stainless steel reservoir equipped with magnetic fans to ensure homogeneous composition before injection into the shock tube. Four manometers were used to accurately measure the pressure during the mixture preparation, three MKS Baratron type 631 with different scales (1.33, 13.33, 133.3 kPa) and one Kistler type 7505, with a 1 MPa scale. The mixture preparation rig (tubing and reservoir) is heated to a temperature of 423 K in order to avoid any condensation or adsorption on the walls. For both studies, according to the precision of the manometers, the mixtures were obtained with an accuracy of 99.5%.

As mentioned above, two alcohols were used in the present investigation for blending with gasoline, ethanol (LiChrosolv ® , Merck ≥ 99.9%) and iso-pentanol (Sigma Aldrich anhydrous ≥ 99%). The synthetic air used in the experiments is composed of 20.9 % O2 + 79.1 % N2 (Air Liquide, grade alphagaz 2, purity > 99.9999 %). Finally, the gasoline was provided by TOTAL group. The main chemical families of the gasoline constituents and its composition expressed in molar percent are reported in Table 1. In particular, no oxygenated compounds are present in the reference gasoline fuel, which in the rest of the manuscript will be called OFG (for OxyFreeGasoline). Its global formula is C6.384H11.842 and it is characterized by a RON number equal to 92.5. A more detailed analysis of the single species present in the gasoline was provided by TOTAL so that the corresponding thermodynamic data and the thermodynamic data of the OFG fuel could be accurately estimated. In particular, 26 species constitute 97.72% of the total mass percentage. For these species, the thermodynamic properties were taken from the literature data or estimated with the software THERM [START_REF] Ritter | [END_REF] using the group additivity method. Two different E10 mixtures were prepared by mixing either ethanol or iso-pentanol with the oxy-free gasoline (OFG): the E10 resulting from the ethanol/OFG blend will be labeled E10-Ethanol, the E10 resulting from the blending with iso-pentanol will be labeled E10-Iso-Pentanol. The percentages of ethanol and iso-pentanol in the mixtures are specified in Table 2 and have been chosen in order to match the current commercial E10 gasolines in terms of oxygen content. According to the linear relation for the RON calculation, the RON for E10-Ethanol and E10-Iso-Pentanol mixtures are 95.2 and 92.8, respectively.

Results and discussions

Laminar flame speeds

The methodology for the data acquisition and processing was well detailed in recent papers [START_REF] Barone | Unsupervised analysis of experiments of laminar flame propagation in a spherical enclosure[END_REF][START_REF] Goulier | Numerical assessment of accurate measurements of laminar flame speed[END_REF].

For this study, the same methodology was adopted. Briefly, the Schlieren images of the growing flames are processed using a home-made code based on Matlab [28] to determine the evolution of the spatial speed VS = dRf/dt. Usually between 110 and 350 images are used to derive the flame speed. Great care has been taken when performing these experiments in order to obtain flames that are spherical and without significant distortions. An example of recorded images is given in Fig. 1 together with the corresponding pressure trace. The expansion of the burned gases leads to a substantial increment in the pressure which reaches a maximum value at around 0.8 seconds in the specific case. On the other hand, the flame measurements are obtained over much shorter times, so that the pressure rise during data acquisition is negligible as presented in Fig. 1 (bottom-right graph). For expanding flames, the flame is stretched due to curvature and strain and the total stretch for this configuration is given by κ = 2(Vs/Rf). The flame propagation depends on the stretch rate. Knowing this dependence allows to derive the flame speed at zero stretch, Vs 0 , and Lb, the Markstein length in the burnt gases, derived here with the non-linear relation proposed by Kelley and Law [START_REF] Kelley | [END_REF]:

0 b b 2 0 b b 2 0 b b S L 2 S S ln S S κ ⋅ - =         ⋅         (1)
The accuracy of this mathematical approach has been discussed in detail in a previous publication where various extrapolation methods were implemented and compared [22]. In a similar way, the errors associated to the extrapolation technique applied to the current experimental datasets are below the uncertainties due to the measurement of the radii as discussed in the following paragraphs.

Thus, the extrapolation model does not seem to have a significant influence on the derivation of the laminar flame speeds. In support of this result, a different criterion which was originally proposed by Wu et al. [30] based on the analyses of the product between Markstein and Karlovitz numbers was also used in order to estimate the uncertainty of the extrapolation process. Fig. 2 contains the results of such analysis which was performed considering the linear Markstein number and the Karlovitz number at the middle radius for all the experiments performed. Over the entire range of equivalence ratios considered here, the product (Ma x Ka) is between -0.05 and 0.15, condition for which negligible extrapolation uncertainty is expected. Fig. 3 illustrates examples of the processed flame speed versus the stretch rate for different OFG/air mixtures. Fitting curves corresponding to the solutions of eqn. 1 have been presented in Fig. 3, while the experimental points have been obtained from the gradients of the radii. The slopes of such fitting curves are directly related to the Markstein lengths, which clearly decrease with increasing equivalence ratios (Lb becomes negative for φ = 1.36). The minimum flame radius considered is between 10 and 15 mm in order to ensure that the ignition phase does not affect the results and that the stretch rate is not too high. The maximum radius is limited by the appearance of non-idealities in the experimental results, and it is normally included in the range between 30 and 35 mm.

Subsequently, the observation time is limited to the initial stage of the flame expansion where the pressure remains nearly constant as shown in Fig. 1 (for a maximum radius of 48 mm only about 0.8% of the total fuel is burned). Therefore, the relation SL 0 = VS 0 /σ can be used to determine the laminar flame speed at zero stretch SL 0 , where σ = ρu/ρb and ρu, ρb are the unburned and burned densities of the mixture, respectively. The renormalized burned Markstein length L′, is determined with the relation L′ = Lb/σ. L′ is presented for consistency to the fact that the unstretched laminar flame speed of the unburned gases is discussed in the manuscript (in fact we could re-write Eq. The laminar flame speeds were measured at an initial pressure and temperature equal to 0.1 MPa and 483±2 K, respectively, and for φ from 0.65 to 1.36. In particular, numerous experiments were performed nearly stoichiometric conditions which are typically encountered in gasoline engines. The experimental error was estimated to be around 2% maximum based on the error assessment procedure established in previous papers [20][21][22] (the absolute values of the errors on the flame speeds vary between 0.2 and 1.3 cm/s). No flame wrinkling due to flame instabilities was observed for the flame studied even for the highest equivalence ratios considered herein. Finally, the error due to heat loss by radiation was estimated based on the equation proposed by Yu et al. [START_REF] Yu | [END_REF]. For all the experimental conditions, the radiation-correction terms is around 0.7 cm/s, thus of the same order of magnitude of the experimental uncertainties. If the corrections were applied to all the experimental points, a simple shift of the profiles by 0.7 cm/s would be obtained. Such shift would not modify the considerations on the comparisons between the different fuel mixtures. The evolution of SL 0 with the equivalence ratio of the three mixtures tested is presented in the Fig. 4.

0,8 1 1,2 1,4
For the clarity of the discussion, the experimental results are compared in panels A, B, and C, while in the central panel containing all the experimental results only the polynomial fits of the experimental measurements are reported, together with the flame speed measurements for ethanol and iso-pentanol performed by our group and presented in previous publications [20,22].

The polynomial fits well represent the experimental data. The discrepancy between the curves and the experimental data points is always below the experimental uncertainties (for 90% of the points, such discrepancy is below 0.5 cm/s).

In the lean region (panel A in Fig. 4), the experimental fits for the OFG/air mixtures and the E10-Iso-Pentanol/air mixtures overlay, thus the addition of iso-pentanol does not have any effect on the flame propagation of the oxy-free gasoline. The two curves start diverging at an equivalence ratio of around 1.1, thus at slightly fuel-rich conditions (panel B), while at even higher equivalence ratios (panel C) the E10-Iso-Pentanol/air mixtures propagate much faster than the corresponding OFG/air mixtures. At φ ∼ 1.3, the difference between the experimental curves reaches ∼ 3 cm/s, thus the presence of iso-pentanol significantly enhances the fuel propagation speed.

Different considerations apply to the E10-Ethanol/air mixtures. In this case, the addition of ethanol leads to an increment of the flame speed over the entire range of equivalence ratios studied herein.

Such increment is relatively small and uniform over the experimental data, with a corresponding average value of 0.6 cm/s based on the fitting curves. Although the error on the single measurement is larger than this value, the use of fitting curves which reproduce a large number of experimental data leads to a compensation of the opposite random errors, making the observed increment in the flame speed significant. As we will discuss later in the text, such increment is consistent with the modeling results. As for the case of iso-pentanol, the addition of ethanol does not change the position at which the maximum experimental flame speed is observed (φ ∼ 1.07).

From an engineering point of view, for the best efficiency in a spark ignition engine the global 4, at φ = 1, SL 0 is almost maximal and no equivalence ratio is around 1. According to Fig.

difference between the OFG/air mixtures and the E10-Iso-Pentanol/air mixtures is observed. This indicates that blending iso-pentanol with the gasoline does not deteriorate the fuel properties in relation to the flame propagation compared the base case where pure gasoline is used. On the other hand, the addition of ethanol has beneficial effects, although the increment on the flame speed is minimal (∼0.5% at φ = 1). Thus, the effects on the laminar flame speed of blending the two alcohols studied here with the OFG fuel are nearly equivalent.

Figure 4 also reports the flame speeds of ethanol/air and iso-pentanol/air mixtures at 473 K [20,22].

The comparison between the experimental sets shows that the ethanol/air flames propagate much faster compared to the case of iso-pentanol and gasoline over the entire range of equivalence ratios.

On the other hand, the experimental profiles for iso-pentanol and the gasoline fuel are very close at fuel lean conditions, which may explain the fact that the two profiles for OFG and E10-Iso-Pentanol in Fig. 4 are almost identical at these conditions, while they diverge only at fuel rich conditions. It is interesting to notice how the effects of the addition of iso-pentanol to the OFG at high equivalence ratios are much more pronounced compared to the addition of ethanol, although the ethanol/air flames propagate faster compared to the iso-pentanol/air mixtures. On the other hand, the blending of ethanol and gasoline does not follow a linear mixing rule. The difference between the flame speeds of ethanol/air and OFG/air mixtures increases monotonically with the equivalence ratio, from ∼4

cm/s at φ ∼ 0.7 to ∼16 cm/s at φ ∼ 1.3, while the effects of the blending between OFG and ethanol are nearly independent of the equivalence ratio (as discussed above).

In addition to the laminar flame speeds, the experiments were used to derive the renormalized Markstein lengths L′ with an estimated maximum uncertainty of 10%. The evolution of L′ is represented in Fig. 5. It seems that the three mixtures exhibit similar L′. A slight deviation is observed at the highest equivalence ratios, but more experimental points would be necessary in order to verify such hypothesis. The similarity between the experimental results presented in Fig. 5 suggests that the three fuels will have a similar behavior with respect to the high stretch encountered in gasoline engines especially for equivalence ratios close to unity. Moreover, the results on the Markstein lengths support the observation that blending iso-pentanol with gasoline does not deteriorate the fuel properties compared to the ethanol/gasoline blends or the pure gasoline cases.

Ignition delay times

The ignition delay time is defined as the interval between the pressure rise caused by the arrival of the shock wave at the end wall and the extrapolation to the baseline of the maximum rate of rise in the pressure signal due to the exothermic ignition event (Fig. 6, left graph). The maximum uncertainty on the ignition delay using this method is ±2 %. In the lowest temperature range of our study, thus for long ignition delay times (normally above 1.1 ms), evidences of pre-ignition pressure rise could be observed in the case of stoichiometric mixtures. An example is reported in the right graph of Fig. 6. The results associated with strong pre-ignition pressure rises are presented in Fig. S1.

As shown, all the data with ignition delay times higher than 1.1 ms are affected by such phenomenon. In order to exclude the possibility that such pressure rise is due to non-idealities in the experimental technique, experiments were also performed removing the oxygen from the initial mixture, which is now composed only of the fuel and N2. In this case, no pressure rise was observed behind the reflected shock wave as shown in Fig. 6, suggesting that the pre-ignition is not due to non-ideal effects. However, in order to identify if this initial rise of the pressure is not due to an inhomogeneous ignition [33,34], more experiments are needed with the implementation of a visualization of the ignition process by implementing a window at the end-flange which would allow the recording of the ignition and the identification of any inhomogeneous ignition with a flame propagation before the main ignition event. This will be addressed in the very near future. For the shock tube study, the initial temperature was fixed at 403±2 K to avoid any condensation and the temperature behind the reflected shock wave was varied between 955 and 1284 K. Considering the precision on the velocity and the initial temperature, the global uncertainty on T5 is ±15 K. The reflected shock pressures were equal to 2.13±0.33 MPa. Two different nominal equivalence ratios (0.5 and 1.0) were studied. The mixtures composition used are listed in Table 3 and the experimental data are presented in the supplemental file. The results of the ignition delay time experiments for OFG, E10-Ethanol and E10-Iso-Pentanol are summarized in Fig. 7a. The data of the three mixtures at φ = 0.5 are characterized by a small scatter.

On the other hand, the difference between the experimental data obtained at stoichiometric conditions is more marked. In order to better appreciate these results, the fits of the experimental results based on the power law have been plotted in Fig. 7b. Prior to fitting the data, the pressure scaling of P -1.05 was adopted from the study of Gauthier et al. [START_REF] Gauthier | [END_REF] and all the data of this study were normalized to 2 MPa in Fig. 7b. At φ = 0.5, almost no difference on the delay is observed for the three mixtures even if the data of E10-Iso-Pentanol/air are slightly lower (~6% on average in the range 1050-1260 K based on the experimental fits) compared to the E10-Ethanol and OFG fuels. The effects of ethanol addition are in this case almost negligible (around 1% average decrement in the ignition delay times). At stoichiometric conditions, a larger difference between the three curves was obtained. Indeed, the data for the oxy-free gasoline are higher than those for both blended fuels. In the range between 990 K and 1260 K the ignition delay times of ethanol/gasoline and isopentanol/gasoline blends in air are lower by ~12% and ~14% on average, respectively, compared to the OFG/air mixtures. It is interesting to notice how at φ = 1.0 and in the highest temperature range of our study the effects of ethanol and iso-pentanol additions seem to be similar, while the isopentanol/gasoline blends are characterized by lower ignition delay times at lower temperatures. The results presented in Fig. 7 indicate that replacing ethanol with iso-pentanol leads to reduced ignition delay times of the corresponding blend fuel. This should be considered in engine applications since it could be directly related to knock problems.

The OFG/air data are also compared with the data of Gauthier et al. [START_REF] Gauthier | [END_REF] in Fig. 4b. In their paper, Gauthier et al. [10] studied the auto-ignition characteristics of n-heptane/air, gasoline/air, and ternary surrogate/air mixtures in a shock tube. We can observe an overall good agreement at φ = 0.5, with a difference of less than 10% for the two measurements performed by Gauthier et al. [START_REF] Gauthier | [END_REF]. At stoichiometric conditions, the agreement between the experimental sets is excellent at low temperatures, although for T > 1100 K the difference increases to around 16%. Such discrepancy in the high temperature range of the studies may be mainly due to the different gasoline compositions.

Although both gasolines do not contain oxygenated compounds, the fuel investigated by Gauthier et al. [START_REF] Gauthier | [END_REF] contains more C5 napthenes and C5-C7 paraffins than the fuel used herein as well as less aromatics and isoparaffins. conditions encountered in the current experiments using COSILAB  [START_REF] Cosilab | The Combustion Simulation Laboratory, Version 3.3.2[END_REF]. In particular, for the flame speed calculations the one-dimensional freely-propagating flame system was solved, with GRAD and CURV values equal to 10 -5 and a spatial domain from -1 to 10 cm. For these GRAD and CURV values, the final solution is independent from the adaptative gridding procedure. All the flame calculations were performed using multi-component transport properties and including the Soret effects. The ignition delay times were calculated based on the pressure profiles assuming adiabatic and constant volume conditions behind the reflected shock wave. The experimental laminar flame speed results and corresponding surrogate simulations are reported in Fig. 8. The experimental results on the real fuels are well reproduced by the simulations performed with both fuel surrogates. In particular, the shape of the experimental and modeling profiles are in excellent agreement for the case of the surrogate based on Morgan et al. [35], with a slight overprediction of the maximum laminar flame speeds (∼2 cm/s, which corresponds to ∼2% of the experimental values). On the other hand, the profiles obtained with the surrogate based on Cai and Pitsch [36] presented in Fig. 8b are slightly shifted towards higher equivalence ratios compared to the experimental profiles although the maximum laminar flame speed values are better reproduced (discrepancy ∼1%). This is somehow expected since the flame propagation is affected by the rate of heat released during combustion, thus function also of the carbon and hydrogen content in the fuel. A better match of the fuel molecular formula as in the method proposed by Cai and Pitsch [36] should lead to a better prediction of the heat release and consequently of the absolute value of the maximum flame speed. More importantly, for both surrogates the effects of the ethanol addition are similar to the ones observed experimentally, with an increment of around 1 cm/s on average in the flame speed (against 0.6 cm/s from experiments). Thus, both surrogate formulations can be used to predict the flame speeds of real fuels also in the presence of ethanol addition. A detailed kinetic analysis of the reasons for the increase of the laminar flame speed with addition of ethanol is behind the scope of the present study. On the other hand, preliminary analyses were performed. In the equivalence ratio range considered, the percentage increment in the calculated laminar flame speed with ethanol addition is 1.7%, which might be due effects of thermal diffusivity, flame temperature, and/or chemistry. The adiabatic flame temperatures with and without ethanol differ by no more than 5 K over the entire equivalence ratio range, while the thermal diffusivity by no more than 0.06%. Thus, the only factor which may explain the increment in the laminar flame speed is the fuel chemistry. Sensitivity analyses at an equivalence ratio of 1.1 showed that the reactions which influence the speed calculations are the same in the two cases with and without ethanol. On the other hand, the reactions involving HCO, CH3, HO2, and C2H3 show increased sensitivity coefficients when ethanol is added to the fuel mixture (above 8% increase).

Indeed, most of these species are produced in larger amounts in the flame with ethanol added to the OFG. The detailed analyses of the reaction pathways responsible for the formation of these species in both flames is quite complicated and not considered in the present study.

The ignition delay time simulations obtained with the two surrogate fuels are presented in Fig. 9 together with the experimental data obtained in the present work. The surrogate based on the method by Morgan et al. [35] is characterized by ignition delay times which are longer than the experimental ones for both equivalence ratios (Fig. 9a). Better agreement can be observed for the results in Fig. 9b which are related to the surrogate formulation by Cai and Pitsch [36]. Once again, the fact that in this case the molecular formula of the fuel is considered should explain the better accuracy of the surrogate formulation. In particular, the experimental results for temperatures above ∼ 1100 K are very similar to the ignition delay times of the corresponding surrogates with the only exception of the OFG stoichiometric data, while at lower temperatures a divergence can be observed. It is interesting to notice how for the fuel lean case the addition of ethanol to the surrogate mixture leads to a shift in the ignition curve towards lower values with an average decrement of ∼16%. This value is much higher compared to the experimental results (the ignition delay times of OFG/air and E10-Ethanol/air mixtures are very similar). In addition, the two stoichiometric curves in Fig. 9b converge in the low temperature range of the current study, while at high temperatures the surrogate ignition delay times are larger than the OFG/air ones. Despite these discrepancies, the results in Fig. 9b are quite promising considering the fact that a complex fuel as the one studied here is compared to a mixture of only three components. In order to further investigate the reasons behind the discrepancies observed between the simulations and the ignition delay time measurements, additional simulations were performed with a different chemical kinetic model [37] and i) the surrogate based on the work by Cai and Pitsch [36] or ii) the (iso-octane/toluene/n-heptane/1-hexene) surrogate as proposed by McCormick et al. [38]. The latter was also experimentally investigated by Cooper et al. [39] using shock tube techniques for ignition delay time measurements in their work on gasoline surrogates and their blends with ethanol and methyl acetate. It is worth mentioning that the surrogate by McCormick et al. [38] has estimated RON and MON values of 90.3 and 84.7, thus slightly different than the current OFG fuel. On the other hand, the mole fractions of the different components are quite representative of the corresponding classes in the OFG fuel. In fact, the surrogate is composed of 14.4% n-heptane, 33.2% toluene, 46.7% iso-octane, and 5.6% 1-hexene (in mole fraction), while the OFG fuel of 10.8% paraffins + naphthenes, 34.1% aromatics, 50.9% iso-paraffins, and 4.1% olefins (analysis based on the known components). The tested kinetic model is the well-established LLNL model for gasoline surrogates [37], validated against experimental data on n-heptane, iso-octane, toluene, and C5-C6 olefins. The results are presented in Fig. 10a for the pure OFG fuel and Fig. 10b for the OFG/ethanol mixtures. The use of the LLNL model improve sensibly the prediction capabilities even with the three-component surrogate based on the work by Cai and Pitsch [36] (blue lines). The addition of the 1-hexene component as in the formulation by McCormick et al. [38] further improves the correspondence between the simulations and the experiments (red lines) although the difference between the two surrogates is not negligible only in the low-temperature range of our experiments. If we don't consider the points corresponding to simulations above 2 ms ignition delay times where the experimental measurements may be affected by the observed pressure rise for stoichiometric conditions, the discrepancies between all the experimental data but one and the simulations is below 25%. The chemical kinetic model also reproduces very well the influence of the ethanol addition on the ignition delay time properties of the fuel. As mentioned in the experimental section, the average decrements in the ignition delay times for fuel-lean and stoichiometric mixtures are around 1% and 12%, while the simulated ones are 3% and 9%, respectively. It is now possible to conclude that the two surrogates well represent the current studied OFG fuel, although the utilization of a more complex mixture with addition of the olefin component slightly improves the prediction capabilities. In addition to perform experiments on the four-components surrogate, Cooper et al.

[39] used their data together with other shock tube data available in the literature on different fuel surrogates and gasoline fuels to provide a correlation for the ignition delay times as function of the equivalence ratio, the pressure, and the temperature conditions. The comparison between the correlation results applied to our experimental conditions and the current experimental results are presented in Fig. 11.

Considering the scattering present in the data analyzed by Cooper et al. [39], the match between the correlation and the data is quite good, confirming once again the validity of the present experimental data sets compared to the previous studies focused on gasoline fuels and surrogates.

Conclusions

The combustion properties of gasoline/air, gasoline/iso-pentanol/air, and gasoline-ethanol/air mixtures were investigated. New measurements of laminar flame speeds at an initial temperature of 483 K, an initial pressure of 0.1 MPa and equivalence ratios between 0.65 and 1.36 were obtained in a spherical reactor. These data were compared with pure ethanol and iso-pentanol flames from the literature. It was observed that the iso-pentanol flame speeds are closer to the three mixtures tested in this study, showing a similar behavior of iso-pentanol with gasoline and gasoline/alcohol blends. The ignition delay times were measured behind reflected shock waves in a high pressure heated shock tube. The experiments were conducted at 2 MPa over a temperature range from 955 to 1284 K. Two equivalence ratios were tested, 0.5 and 1. The ignition delay times of the commercial gasoline were compared with the literature and an overall good agreement was found for both equivalence ratios. From these experiments in shock tube and spherical reactor, it was shown that at nearly stoichiometric conditions (typical of optimized engines) the addition of iso-pentanol does not significantly affect the flame speed of the fuel/air mixture compared to the ethanol case, while the contrary is true concerning the ignition delay time measurements. In fact, the addition of iso-pentanol reduces the time necessary for ignition in a more significant proportion than ethanol. Two surrogate mixtures were also tested against the real fuel measurements (OFG and E10-Ethanol) using kinetic modeling with a reduced model from the literature. The comparison between surrogate and real fuel properties are quite promising although some discrepancies were observed especially for the ignition delay time measurements. Additional analyses showed how the use of a different detailed chemical kinetic model and a more complex surrogate fuel leads to accurate predictions of the ignition properties.

Fig. 1 .

 1 Fig.1. Example of images recorded using the high-speed camera in case of stoichiometric OFG/air mixture and corresponding pressure profile. Pini = 0.1 MPa, Tini = 483 K.

Fig. 2 .

 2 Fig. 2. (MaLinear x Kamid) as function of the equivalence ratio. Blue triangles: OFG, red circles: E10-Ethanol, green squares: E10-Iso-Pentanol.

Fig. 3 .

 3 Fig. 3. Evolution of burning speed versus stretch rate for OFG / air mixtures. Pin = 0.1 MPa, Tin = 483 K.

Fig. 4 .

 4 Fig. 4. Laminar flame speeds of OFG/air, E10-Ethanol/air, and E10-Iso-Pentanol/air mixtures at 0.1 MPa and 483 K. The lines represent the polynomial fits of the experimental results: --OFG, --E10-Ethanol, -• -E10-Iso-Pentanol. Results from the literature at 473 K and 0.1 MPa for:  ethanol [9];  iso-pentanol [11]. A, B and C: zoom of the experimental data of the present study represented by symbols: blue triangles: OFG, red circles: E10-Ethanol, green squares: E10-Iso-Pentanol.

Fig. 5 .

 5 Fig. 5. Evolution of L′ versus equivalence ratio at 0.1 MPa and 483 K. Blue triangles: OFG, red circles: E10-Ethanol, green squares: E10-Iso-Pentanol.

Fig. 6 .

 6 Fig. 6. Examples of end-wall pressure profiles for E10-Iso-Pentanol/air experiments at φ = 1 (black curves). Left: definition of the ignition delay time in this study. Right: comparison with a shock E10-Iso-Pentanol/N2 (green curve).

Fig. 7 .

 7 Fig. 7. Experimental ignition delay time measurements. (a) Data from this study: φ = 1.0 -blue empty triangles: OFG, red empty circles: E10-Ethanol, green empty squares: E10-Iso-Pentanol; φ = 0.5 -blue full triangles: OFG, red full circles: E10-Ethanol, green full squares: E10-Iso-Pentanol. (b) Fits of the data normalized to 2 MPa from this study compared with the literature [6]: φ = 1.0 ---OFG, --E10-Ethanol, --E10-Iso-Pentanol; φ = 0.5 ---OFG, --E10-Ethanol, --E10-Iso-Pentanol.

Fig. 8 .

 8 Fig. 8. Laminar flame speeds of OFG/air, E10-Ethanol/air, and corresponding surrogate/air mixtures at 0.1 MPa and 483 K. Experimental results for real fuels: blue triagles: OFG, red circles: E10-Ethanol. Simulations for surrogates based on a) Morgan et al. [35] and b) Cai and Pitsch [36]: --surrogate, --surrogate/ethanol.

Fig. 9 .

 9 Fig. 9. Ignition delay times of OFG/air, E10-Ethanol/air, and corresponding surrogate/air mixtures. Experimental results for real fuels: φ = 1.0 -blue empty triangles: OFG, red empty circles: E10-Ethanol; φ = 0.5 -blue full triangles: OFG, red full circles: E10-Ethanol. Simulations for surrogates based on a) Morgan et al. [35] and b) Cai and Pitsch [36]: φ = 1.0 ---surrogate, --surrogate/ethanol; φ = 0.5 ---surrogate, --surrogate/ethanol. Data normalized to 2 MPa.

Fig. 10 .

 10 Fig. 10. Ignition delay times of OFG/air, E10-Ethanol/air, and corresponding surrogate/air mixtures. Experimental results for real fuels: φ = 1.0 -blue empty triangles: OFG, red empty circles: E10-Ethanol;

Fig. 11 .

 11 Fig. 11. Ignition delay times of OFG/air mixtures and correlation functions by Cooper et al. [39].

Table 1

 1 Composition of the gasoline OxyFree (OFG)

	Family component name	mass%
	Iso-paraffins	46.17
	Aromatics	37.96
	Naphtenes	6.38
	Paraffins	4.59
	Olefins	4.32
	Cyclo-Olefins	0.54
	Di-Olefins	0.01
	Oxygenates	0.00

Table 2

 2 Alcohol percent (by weight and by volume) in order to obtain a commercial E10 gasoline

		Ethanol	Iso-Pentanol
	% w	10.64 %	20.81 %
	% v	9.90 %	18.91 %

stretch rate (1/s) 200 300 400 burning speed (cm/s) φ =1.07 0 200 400 600 stretch rate (1/s) φ =1.14 0 200 400 600 stretch rate (1/s) φ =1.28 0 200 400 600 stretch rate (1/s) φ =1.36

  The densities of unburned and burned gases, ρu and ρb, were calculated using COSILAB [START_REF] Cosilab | The Combustion Simulation Laboratory, Version 3.3.2[END_REF] with the Equilibrium code.

					1 in
	terms of Su , 0 u 300 400 S , and L′). 200 φ =0.70	φ =0.80	φ =0.92	φ =1.00
	0	200	400	600

Table 3

 3 Mixture compositions for ignition delay experiments

	Φ	% fuel	% O2	% N2
		Fuel = OFG
	0.5	1.11	20.67	78.22
	1	2.18	20.44	77.38
		Fuel = E10-Ethanol
	0.5	1.31	20.63	78.06
	1	2.58	20.36	77.06
		Fuel = E10-Iso-Pentanol
	0.5	1.15	20.66	78.19
	1	2.26	20.43	77.31

Table 4

 4 Surrogate mixture compositions (mole fractions in %)

	Fuel	C7H16 i-C8H18 C6H5CH3 C2H5O
		Morgan et al.
	OFG	16.40	29.66	53.94	0.00
	E10-Ethanol	13.17	23.81	43.30	19.72
		Cai and Pitsch
	OFG	15.95	47.25	36.80	0.00
	E10-Ethanol	12.62	37.36	29.10	20.92

Acknowledgements

The authors acknowledge the financial support of le Label CAPRYSSES and the help of Claire Grégoire for some of the shock tube experiments.

Surrogate fuel simulations

The formulation of surrogate fuels able to mimic the physical and chemical properties of real fuels is an essential area of research for the transition between the experimental results on real fuels and the kinetic modeling applied to engine simulations. In the present work, two surrogate fuels have been tested based on the works by Morgan et al. [35] and Cai and Pitsch [36]. The surrogates have been determined implementing the methods proposed in the above-mentioned studies based on the specific properties of the current OFG fuel. Both surrogates, whose compositions are reported in Table 4, are composed of n-heptane, iso-octane, and toluene (PRF/toluene). Indeed, the method proposed by Cai and Pitsch [36] is an extension of the work by Morgan et al. [35] in the sense that the surrogate formulation is based not only on the MON/RON values as in [35] but also on the fuel CxHyOz molecular formula. The percentages of n-heptane in the two formulations are very similar (around 16%). On the other hand, the main component of the surrogate calculated based on Morgan et al. [35] is toluene (mole percentage of ∼54%) compared to ∼30% of iso-octane, while the opposite is true for the formulation based on Cai and Pitsch [36] (the percentages are around 37% and 47% for toluene and iso-octane, respectively). Since the kinetic model by Cai and Pitsch [36], selected here to perform the modeling simulations, was validated against experimental ignition delay time and laminar flame speed measurements of PRF/toluene/ethanol single-components and mixtures, the fuel surrogates with addition of ethanol were also considered (Table 4). The compositions of the surrogate/ethanol surrogate/iso-pentanol mixtures were calculated by considering the mass percentage of ethanol as in Table 2 and re-adjusting the mole fraction of the surrogate fuel components (n-heptane, iso-octane, toluene) in the same proportions. The development of a novel sub-mechanism for iso-pentanol is beyond the scope of the current investigation, thus the modeling section will be focused on the OFG and the E10-Ethanol results. Simulations of laminar flame speeds and ignition delay times for the different surrogate mixtures were performed at the D.J. Gaspar, J. Gladden, A. George, SAE Int. J. Fuels Lubr. 10 (2017) 442-460.

39. S. P. Cooper, O. Mathieu, I. Schoegl, E. L. Petersen, Fuel 275 (2020) 118016.


N Combe
email: combe@cemes.fr

F Mompiou
email: mompiou@cemes.fr

M Legros
email: legros@cemes.fr

Multiple coupling modes to relax shear strain during grain boundary migration

Keywords: Disconnection and dislocation, Grain boundaries, Vacancies, Plastic deformation, Molecular Dynamics

Shear-coupled grain boundary (GB) migration is an effective plastic mechanism in absence of dislocation activity, ie. more favorably in nanocrystalline metals. For a given GB, several stress induced migration mechanisms, referred as coupling modes participate to the decrease of the elastic energy produced by the shear. They operate through the nucleation and motion of interfacial defects known as disconnections, carrying elementary shear strain characterized by their Burgers vector. However, so far, the coupling modes have been studied only under a simple shear, a situation much less complex than expected in a strained polycrystal, where multiple components of the stress tensor are present. Here we propose a more systematic investigation of the coupling modes when a composite shear is applied. This promotes the activation of new coupling modes. Using Molecular Dynamics simulations, we evidence these multiple coupling modes and the operation of their associate disconnections. Moreover, we also show that, even at low temperature, GB migration may occur by the successive occurrence of two modes: the relaxed shear appears then as an effective parameter, resulting from the combination of two operating elementary mechanisms.

In absence of dislocations activity, grain boundary (GB) migration was found to be an efficient plastic deformation mechanism [START_REF] Gianola | Stress-assisted discontinuous grain growth and its effect on the deformation behavior of nanocrystalline aluminum thin films[END_REF]. Among different GB migration mechanisms, shear coupled grain boundary migration (SCGBM) is considered particularly efficient, especially at low temperature, because it is a conservative mechanism, analogous to dislocation slip. For this reason, SCGBM has drawn a significant attention. During SCGBM, the GB migrates over a distance h coincidentally with the relative displacement b of the two grains. The resulting shear, also known as the coupling factor β = b h is commonly used to characterize the mechanism.

In the early seminal work of Cahn et al., a theoretical description of SCGBM was first proposed, based on the motion of intrinsic GB dislocation networks [START_REF] Sutton | Interfaces in crystalline materials[END_REF][START_REF] Cahn | Coupling grain boundary motion to shear deformation[END_REF][START_REF] Molodov | Grain boundary mediated plasticity: On the evaluation of grain boundary migrationshear coupling[END_REF]. The nature and density of these dislocations can be formally determined through the Frank-Bilby equation (FBE) [START_REF] Read | Dislocation models of crystal grain boundaries[END_REF]. The multiplicity of solutions of the FBE, potentially leads to several coupling modes associated to different coupling factors. Cahn et al. [START_REF] Cahn | Coupling grain boundary motion to shear deformation[END_REF] have predicted for instance, in FCC crystals, four coupling modes in symmetric [001]-tilt GBs. While three of them have been evidenced in Molecular dynamics simulations [START_REF] Cahn | Coupling grain boundary motion to shear deformation[END_REF][START_REF] Cahn | Duality of dislocation content of grain boundaries[END_REF][START_REF] Frolov | Effect of interfacial structural phase transitions on the coupled motion of grain boundaries: A molecular dynamics study[END_REF][START_REF] Combe | Disconnections kinks and competing modes in shear-coupled grain boundary migration[END_REF][START_REF] Combe | Shear-coupled grain-boundary migration dependence on normal strain/stress[END_REF], only two modes have been experimentally observed in macroscopic sheared Al bicrystals [START_REF] Gorkaya | Stress-driven migration of symmetrical 100 tilt grain boundaries in al bicrystals[END_REF][START_REF] Molodov | Migration of the σ7 tilt grain boundary in al under an applied external stress[END_REF][START_REF] Molodov | Dynamics of grain boundaries under applied mechanical stress[END_REF].

More recently, it has been found that the elementary processes of SCGBM are not induced by the motion of the intrinsic dislocations but by the nucleation and motion of disconnections [START_REF] Rajabzadeh | Elementary mechanisms of shear-coupled grain boundary migration[END_REF][START_REF] Han | Grain-boundary kinetics: A unified approach[END_REF]. A disconnection is a GB defect with both a step and a dislocation character. Both experiments and simulations have evidenced the role of these disconnections during the SCGBM [START_REF] Wan | Shear response of the σ9 110 221 symmetric tilt grain boundary in fcc metals studied by atomistic simulation methods[END_REF][START_REF] Khater | The disconnection mechanism of coupled migration and shear at grain boundaries[END_REF][START_REF] Rajabzadeh | Evidence of grain boundary dislocation step motion associated to shear-coupled grain boundary migration[END_REF][START_REF] Rajabzadeh | The role of disconnections in deformation-coupled grain boundary migration[END_REF][START_REF] Rajabzadeh | Elementary mechanisms of shear-coupled grain boundary migration[END_REF][START_REF] Race | Role of the mesoscale in migration kinetics of flat grain boundaries[END_REF][START_REF] Combe | Disconnections kinks and competing modes in shear-coupled grain boundary migration[END_REF][START_REF] Combe | Shear-coupled grain-boundary migration dependence on normal strain/stress[END_REF][START_REF] Deng | Size and rate dependent grain boundary motion mediated by disconnection nucleation[END_REF][START_REF] Zhu | In situ atomistic observation of disconnection-mediated grain boundary migration[END_REF][START_REF] Chen | Grain boundary shear coupling is not a grain boundary property[END_REF]. The coupling factor of the SCGBM is directly related to the ratio of the (norm of the) Burgers vector to the step height of

Shear strain directions

The SCGBM of perfect GBs occurs through the (homogeneous) nucleation and further motion of disconnections [START_REF] Hirth | Steps, dislocations and disconnections as interface defects relating to structure and phase transformations[END_REF][START_REF] Khater | The disconnection mechanism of coupled migration and shear at grain boundaries[END_REF][START_REF] Rajabzadeh | Elementary mechanisms of shear-coupled grain boundary migration[END_REF][START_REF] Mackain | Atomic-scale modeling of twinning disconnections in zirconium[END_REF][START_REF] Han | Grain-boundary kinetics: A unified approach[END_REF]. Both processes are thermally activated. The SCGBM activation energy is dominated by the disconnections nucleation [START_REF] Rajabzadeh | Elementary mechanisms of shear-coupled grain boundary migration[END_REF][START_REF] Mackain | Atomic-scale modeling of twinning disconnections in zirconium[END_REF]. This nucleation activation energy barrier typically involves the disconnection core energies, the elastic disconnection-disconnection interaction energy and finally the Peach-Koehler energy due to the work of the shear stress. In the following, we consider a GB plane normal to the x direction (see a sketch in Fig. 1), and apply a shear stress σ (σ ii = 0 with i ∈ {x, y, z}). Glissile disconnections have thus, Burgers vectors perpendicular to x. Following [START_REF] Rajabzadeh | Elementary mechanisms of shear-coupled grain boundary migration[END_REF][START_REF] Mackain | Atomic-scale modeling of twinning disconnections in zirconium[END_REF][START_REF] Han | Grain-boundary kinetics: A unified approach[END_REF], assuming a linear isotropic elastic crystal (Poisson ratio ν, and shear modulus µ), the energy interaction of a migrating GB displaying two (straight) disconnections with opposite Burgers vectors ( b = b y e y + b z e z ) separated by a distance δ writes:

E(δ) = 2E core (δ 0 ) + µ 2π(1 -ν) ln δ δ 0 -(σ xy b y + σ xz b z )δ (1)
δ 0 and E core (δ 0 ) are the core radius and core energy of the disconnections. The activation energy barrier E act of the migration can be deduced from the maximum E max of this energy E(δ) as a function of δ.

E max = 2E core (δ 0 ) + µ 2π(1 -ν) ln µ 2πδ 0 (1 -ν)(σ xy b y + σ xz b z ) -1 (2)
The maximum energy E max is a decreasing function of the applied shear σ xy and σ xz . Obviously, the application of a shear stress reduces the maximum energy and thus the activation energy barrier, and favors the occurrence of the migration [START_REF] Rajabazdeh | Experimental and theoretical study of the shear-coupled grain boundary migration[END_REF][START_REF] Combe | Disconnections kinks and competing modes in shear-coupled grain boundary migration[END_REF][START_REF] Combe | Shear-coupled grain-boundary migration dependence on normal strain/stress[END_REF]. Eq. ( 1) and ( 2) can be written for every potential coupling modes (or disconnections). Single or multiple [START_REF] Cahn | Coupling grain boundary motion to shear deformation[END_REF][START_REF] Chen | Grain boundary shear coupling is not a grain boundary property[END_REF] coupling modes can operate in order to accommodate the shear.

In most studies [START_REF] Cahn | Coupling grain boundary motion to shear deformation[END_REF][START_REF] Homer | Grain boundary plane orientation fundamental zones and structure-property relationships[END_REF], the applied stress is a simple shear stress σ xz perpendicular to tilt axis in symmetric tilt GB, while in a real poly-crystal, the local shear stress can have any direction. This variety of potential stresses can presumably activate alternative disconnections: indeed, from Eq. ( 2), the nucleation of a given disconnection (and thus a given SCGBM coupling mode) will be favored if E max is minimized, i.e. if the shear stress is applied in the direction of the Burgers vector of the disconnection. Note that for simplicity, we have considered a SCGBM induced by the motion of two infinite straight disconnections in Eq. ( 2), but considering the expansion of a disconnection loop would yield the same conclusion [START_REF] Race | Role of the mesoscale in migration kinetics of flat grain boundaries[END_REF].

In the following, we identify the shear directions corresponding to Burgers vectors of mobile disconnections using the dichromatic pattern and shear a set of GBs in these directions.

Molecular dynamics simulations

We perform MD simulations in a FCC material using an embedded-atom potential for copper [START_REF] Mishin | Structural stability and lattice defects in copper: Ab initio, tight-binding, and embedded-atom calculations[END_REF]. We have investigated a series of [001] tilt GB with tilt angles ranging between 16.26 ), Σ13(320)(67.38 • ), Σ25(430)(73.74 • ). In addition, we have also studied an asymmetric tilt GB Σ11(557)(771)[001] and a [001] pure twist GB Σ13(320). MD simulations have been performed at 0K and in several cases at finite temperature.

Each of these GB has been studied using the same procedure. Figure 1 shows a sketch of the simulation cell in the case of a [001] tilt boundary (the cells for the asymmetric and twist GB are similar). The simulation cell contains two grains of a perfect FCC crystal disorientated relatively to each other around the [001] direction and separated by a perfect GB. Periodic boundary conditions (PBCs) are applied in the y-and z-directions. Typical simulation cell x-, y-, and z-sizes are, respectively, 10.8, 5.9, and 4.3 nm. The cell typically contains from about 30000 to 100 000 atoms depending on the investigated GB. Equilibrium structures of the GB are obtained by minimization of the interface energy between the two grains and we have checked that they agree with GB structures reported in the literature. Two 1.7-nm-thick slabs at the top and bottom of the cell contain atoms with relative positions frozen to the perfect lattice ones. The slabs are translated relatively to each other parallel to GB plane by small increments in order to impose a shear stress on the GB. After each translation, a conjugate gradient algorithm minimizes the potential energy during which slabs can move freely in the x-directions. Elementary mechanisms of GB migrations are determined following the procedure reported in Ref. [START_REF] Rajabzadeh | Elementary mechanisms of shear-coupled grain boundary migration[END_REF]: from the configurations before and after the GB migration at a given shear strain, the disconnections involved during the GB migration are evidenced from Nudge Elastic Band (NEB) calculations [START_REF] Henkelman | A climbing image nudged elastic band method for finding saddle points and minimum energy paths[END_REF]. In addition, on few GBs, some simulations of GB migration at finite temperature were performed using a Nose Hoover thermostat to check the robustness of our findings. All simulations have been performed using the LAMMPS simulations sofware [START_REF] Plimpton | Fast parallel algorithms for short-range molecular dynamics[END_REF][START_REF] Plimpton | Particle-mesh ewald and rrespa for parallel molecular dynamics simulations[END_REF]. Below, in order to avoid a fastidious catalog of the observed GB migration for all investigated GB and all shear directions, we have chosen to detail our results for two symmetric tilt GBs Σ17(410)[001] and Σ13(320)[001]), an asymmetric tilt GB Σ11(557)(771)[001] and a pure twist GB Σ13[001]. Results for other tilt GBs will be synthesized in a figure.

x y z Grain 2 Grain 1 G1-[100] G1-[010] G1-[001] G2-[100] G2-[010] G2-[001]

Disconnection notation

The notation commonly used to designate a disconnection is the so-called p/q notation introduced in Ref. [START_REF] Serra | A new model for 1012 twin growth in hcp metals[END_REF]: p and q are related to the step height in each grain (perfect lattice). Unfortunately, this notation, though very simple is ambiguous because several different disconnections have the same p/q notation. In the present study, we need to unambiguously designate a disconnection. We thus propose a notation based on the crystal lattice coordinate systems (Fig. 1). In the following we propose to identify a disconnection by the triplets (n 1 m 1 p 1 )/(n 2 m 2 p 2 ): the Burgers vector b (n1m1p1)/(n2m2p2) and step height h (n1m1p1)/(n2m2p2) of this disconnection are respectively:

b (n1m1p1)/(n2m2p2) = t 2 (n 2 m 2 p 2 ) -t 1 (n 1 m 1 p 1 ) (3)
h (n1m1p1)/(n2m2p2) = 1 2 t 1 (n 1 m 1 p 1 ) + t 2 (n 2 m 2 p 2 ) . u x
where t i (nmp) refers to the vectors expressed in grains i:

t i (nmp) = 1 2 [n i , m i , p i ] (4)
and u x is a unitary vector normal to the GB. The factor 1

Results

Σ17(410) -[001]

We first explicit below our finding for the Σ17(410) -[001] symmetric tilt GB. Fig. 1 shows a sketch of the proposed system with x,y and z axis respectively along [410], [ 140] and [00 1] crystal (grain 1) directions. Fig. 2 shows the dichromatic pattern of the Σ17(410) -[001] symmetric tilt GB and the grain orientation.

Disconnection and shear direction

Figure 2 shows numerous potential disconnections labeled using the notation introduced above. All these disconnections have Burgers vectors parallel to the GB plane and are thus glissile. Each of these disconnections can be related to a disconnection with opposite Burgers vectors and step height (associated to the same coupling mode): for clarity, we have chosen to mention only disconnections with positive Burgers vector y-components.

Let's first focus on the (101)/( 101) [4/4] and ( 11 0)/(110)[-5/ -5] disconnections. The glide of such disconnections respectively induces a GB migration under the so-called 100 and 110 modes [START_REF] Cahn | Coupling grain boundary motion to shear deformation[END_REF][START_REF] Combe | Disconnections kinks and competing modes in shear-coupled grain boundary migration[END_REF]. The Burgers vector norm (step height) of the disconnections (101)/( 101) [4/4] and ( 11 0)/(110)[-5/ -5] are respectively 0.088 nm (0.175 nm) and 0.26 nm (0.22nm). Since the Burgers vectors b (101)/( 101) [4/4] and b ( 11 0)/(110)[-5/-5] are both parallel to the y-direction, the most efficient shear stress to activate these modes is given by σ xy = 0 (σ xz = σ yz = 0), which is indeed verified [START_REF] Cahn | Coupling grain boundary motion to shear deformation[END_REF]. The activation barriers of these two modes decrease as a function of an applied pure shear stress: the occurrence of a given mode depends on the relative values of the activation barriers of these modes [START_REF] Combe | Disconnections kinks and competing modes in shear-coupled grain boundary migration[END_REF].

Let's now consider the glissile disconnection b ( 211)/(130)[-7/-7] . Its Burgers vector norm, 0.28 nm, and step height, 0.31nm, are comparable to those of the ( 11 0)/(110)[-5/ -5] disconnection, ie. 0.26 nm and 0.22 nm, respectively. Hence, the activation energies for the ( 211)/(130)[-7/-7] and ( 11 0)/(110)[-5/-5] modes are expected to be of the same order of magnitude [START_REF] Han | Grain-boundary kinetics: A unified approach[END_REF]. However, to our knowledge, the ( 211)/(130)[-7/-7] mode has never been evidenced experimentally nor numerically.

The shear stress that is the most efficient to activate the nucleation of the disconnection ( 211)/(130)[-7/-7] at 0 K, has the direction of its Burgers vector. During the shear, we observe a stick slip displacement of the GB evidencing a succession of elastic and plastic regimes. Fig. 3a shows the shear stresses σ xy and σ xz as a function of the y-slab displacement. Each stress drop corresponds to a plastic event, here a GB migration. This behavior is consistent with previous numerical results [START_REF] Cahn | Coupling grain boundary motion to shear deformation[END_REF][START_REF] Rajabzadeh | Elementary mechanisms of shear-coupled grain boundary migration[END_REF]. These curves suggest two kind of GB migration: type i) occurs at 0.17 nm y-slab displacement, and type ii) occurs at 0.31nm, 0.41nm and 0.52 nm... For a reason that will appear below, we first focus on type ii) GB migration. Fig. 3b shows the atomic configurations along the minimum energy path (MEP) evidencing the disconnections that homogeneously nucleate during this type ii) GB migration. This disconnection actually corresponds to the ( 211)/(130)[-7/-7] as can be revealed by a Burgers circuit analysis. The nucleation of this disconnection is also corroborated by the stress-strain curve: the successive yield stresses σ xy and σ xz for all type ii) GB migrations are equal (except small variations due to finite sizes of the simulations cells): this coupling mode exactly dissipates the elastic energy accumulated while displacing the slabs in the direction b ( 211)/(130)[-7/-7] . Analyzing the type i) GB migration using the same method, we evidence the nucleation of a b ( 110)/(031)[-3/-3] disconnection that was not expected. The Burgers vector norms (and step height) of the ( 110)/(031)[-3/ -3] and ( 211)/(130)[-7/ -7] being respectively 0.35 nm ( 0.22 nm) and 0.28nm (0.31 nm), the core energy of the ( 110)/(031)[-3/ -3] and ( 211)/(130)[-7/ -7] disconnections are expected to be of the same order of magnitude [START_REF] Han | Grain-boundary kinetics: A unified approach[END_REF]. But more important, the activation energy of the disconnections nucleation depends on both shear stress components σ xy and σ xz : the nucleation of disconnections ( 110)/(031)[-3/ -3] occurs for yield stresses σ xy = 1.72GP a and σ xz = 1.58GP a and allows to accommodate 90% of the shear stress along xz

G1-[100] G1-[010] G1-[001] G2-[100] G2-[010] G2-[001]

Other directions

We also tried to activate other disconnections by shearing in the direction of the (011)/( 130)[1/1], (020)/( 121)[2/2], ( 110)/(031)[-3/ -3], ( 101)/(040)[-4/ -4], (110)/( 231)[5/5)], (121)/( 220)[6/6] disconnections Burgers vectors at 0 K: all these shear directions are reported in the dichromatic pattern in Fig. 2.

For numerous cases, we have observed a shear coupled GB migration under the expected or an alternative coupling mode or a periodic succession of different coupling modes. The GB structure is conserved during this GB migration and hence, the nucleated disconnections can be characterized. But we have also observed shear coupled GB migration mechanisms during which the GB structure is not conserved and beyond, the GB sometimes does not remain planar. In such cases, the characterization of the disconnections and especially the elaboration of a relevant Burgers circuit is tricky: to our knowledge, a recent method of disconnections analysis available in this case has been reported in the literature [START_REF] Frolov | Dislocation content of grain boundary phase junctions and its relation to grain boundary excess properties[END_REF]. GB defect between two different GB structure can be characterized by a Burgers vectors b DSC + b def ect where b DSC and b def ect respectively do and do not belong to the DSC lattice. Because GB migration conserving the GB structure are numerous in our study, we have focused on them and did not try to address cases with GB structures changes. In the following, we report only results concerning cases where the GB structure is conserved and when disconnections can be analyzed.

Tab. 1 summarizes all the disconnections observed in the NEB calculations while investigating the Σ17(410) -[001] GB.

By varying the shear directions, the disconnections ( 110)/(031)[-3/-3], (101)/( 10 1)[4/4], ( 11 0)/(110)[-5/-5], ( 211)/(130)[-7/ -7] have been activated. These disconnections are reported with blue arrows in Fig. 2. Note that though their p/q notations are identical, the (101)/( 10 Burgers vectors parallel to the applied shear stress. In these cases, the shear stress is relaxed by a periodic succession of several coupling modes: an illustrative example of such a case will be detailed below in the case of the Σ13(320) GB. Finally, when shearing along the (121)/( 220)[6/6] directions, several elementary GB migrations reported in the Tab. 1 are first observed before the GB changes its structure.

Finite temperature simulations

Several MD simulations at 100 K and 200 K have been performed. They have evidenced the nucleation of the ( 11 0)/(031)[-3/-3] and ( 211)/(130)[-7/-7] disconnections in the Σ17(410)-[001] GB. However, apart from few simulations that have allowed an observation and characterization of the nucleated disconnections, the GB structure is most often not conserved during the SCGBM yielding to the impossibility to directly analyze the elementary mechanism in terms of disconnections. However, we emphasize that we have always observed a shear coupled GB migration, though we did not try to characterize this coupling in term of elementary mechanisms. Below we detail an interesting behavior previously mentioned: the successive operation of different disconnections.

Σ13(320) -[001]

Fig. 4b shows the stress-strain curve measured when shearing the Σ13(320) -[001] GB at 0 K along the ( 110)/( 121)[-1/ -1] direction. A periodic succession of two types of plastic events operates. A first one (in Fig. 4b at 0.24nm, 0.39nm, 0.54 nm...) induces a drop in the σ xy stress but no σ xz drop : the examination of this event using the NEB method shows the operation of ( 11 0)/(110)[-5/-5] disconnections (corresponding to the operation of 110 coupling mode). Accordingly, the absence of component along the z-direction of the Burgers vector of the disconnection ( 11 0)/(110)[-5/ -5] agrees with the absence of σ xz stress drop. The corresponding coupling factor is negative. The second plastic event (at 0.27nm, 0.42nm, 0.57nm..) induces stress drops for both σ xy and σ xz : the examination of this event using the NEB method reports the operation of the (101)/( 10 Hence, even at 0 K, the SCGBM can successively involve several different elementary events, each of them associated to a disconnection and thus a coupling factor. Surprisingly, the imposed shear is not relaxed by the single ( 110)/( 121)[-1/ -1] mode (and its corresponding disconnection). This result has already been established at finite temperature: SCGBM is a thermally activated process so that different coupling modes can compete with each other [START_REF] Cahn | Coupling grain boundary motion to shear deformation[END_REF][START_REF] Combe | Disconnections kinks and competing modes in shear-coupled grain boundary migration[END_REF][START_REF] Chen | Grain boundary shear coupling is not a grain boundary property[END_REF]. Here, we extend it to the low temperature case. While in the literature, most authors, including us, used to believe that at low temperature, the disconnection mode with the lowest energy barrier was the only activated one [START_REF] Rajabzadeh | Elementary mechanisms of shear-coupled grain boundary migration[END_REF][START_REF] Combe | Disconnections kinks and competing modes in shear-coupled grain boundary migration[END_REF][START_REF] Chen | Grain boundary shear coupling is not a grain boundary property[END_REF], our present results precise this assertion: several disconnections can compete with each other even at low temperature since the activation energy is a multivariate function of the shear stresses σ xy and σ xz .

Since the coupling factors (along the y-direction) of the ( 11 0)/(110)[-5/-5] and (101)/( 10 If we relate this value to experimental measurements, the successive operation of multiple coupling modes seems not to be a relevant argument to explain the low coupling factors reported by experiments on poly-crystalline metals.

Symmetric [001]-tilt GB: a synthesis

Fig. 5 presents a synthesis of our results on all investigated [001] tilt GBs. Fig. 5 shows the misorientation angle of the GB as a function of the coupling factors β y = by h and β z = bz h of the disconnections that we have evidenced. Both solid lines mention the 100 and 110 coupling modes derived from the FBE [START_REF] Cahn | Coupling grain boundary motion to shear deformation[END_REF]. Fig. 5 clearly evidences that the GB disposes of numerous coupling modes to accommodate the shear stress. The 100 and 110 coupling modes only represent a small fraction of available coupling modes. This conclusion is general and applies for all investigated GBs. of the evidenced disconnections. Blue and Red solid lines draw the 100 and 110 coupling modes as described in [START_REF] Cahn | Coupling grain boundary motion to shear deformation[END_REF]. In order to show the generality of our procedure to generate alternative coupling modes, we have considered an asymmetric GB Σ11(557)(771) -[110] which dichromatic pattern is showed in Fig. 6. The response of Σ11(557)(771) -[110] GB to a pure shear stress σ xy > 0 has been previously studied by Zhang et al. [START_REF] Zhang | The shear response of copper bicrystals with σ11 symmetric and asymmetric tilt grain boundaries by molecular dynamics simulation[END_REF]: Zhang et al. showed that the deformation mechanism consisted in the emission of the dissociated partial dislocations and the nucleation and propagation of the partial dislocations from the GB plane. They did not report that the GB migrates and couples with the shear. We found the same results. However, applying a negative component σ xy < 0 of the shear stress, we observed a GB migration coupled to the shear. We have thus investigated numerous shear stress directions (all of them involve a negative value of the xy shear stress component σ xy < 0) corresponding to the glissile disconnections reported in Fig. 6. Two coupling modes corresponding to the disconnections with Burgers vectors indicated by blue arrows in the dichromatic patterns were observed. In some cases, a periodic succession of these two coupling modes similarly to our observations for the Σ13(320) -[001] tilt GB were evidenced.

1/4(112) / (100) [1/1] G1-[110] G1-[110] G1-[001] G2-[110] G2-[110] G2-[001] (111) / (300) [3/3] 1/2(110) / 1/4(116) [5/5] 1/4(112) / 1/4(1110) [-1/-1] 1/2(112) / 1/4(710) [-2/-2] (112) / 1/2(710) [-4/-4]

Twist GB: Σ13(001)

Finally, we would like to address a question hardly mentioned in the literature: does a twist GB shear couple ? Fig. 7a) and b) show the dichromatic patterns of the Σ13(001) twist GB projected along [001] (x direction) and [320] (z direction), respectively. Lattice planes above and below the GB plane are identical in the dichromatic pattern, because (001) is a mirror plane of the DSC. Hence, applying a shear stress (parallel to the GB plane) in any direction, is not expected to lead to GB migration since its migration in the opposite direction is totally equivalent: for any potential glissile disconnections of Burgers vector b and step height h, there exists a disconnection of Burgers vectors b and step height -h. However, GB sliding an operate: there exists disconnections with Burgers vectors b = 0 and no step heigh h = 03 . Applying a shear stress on a twist GB, we thus expect that the average position4 of the GB will not evolve with time, and that the shear elastic energy will be relaxed by a sliding of the two grains. We have performed MD simulations at 0 K to shear the twist Σ13(320) -(001) GB in directions parallel to various disconnection Burgers vectors parallel to the GB plane. We have actually observed some sliding without migration. Seldom, we have observed some fluctuations of the GB position around its average position. This behavior is reminiscent of the random nucleation of disconnections with identical Burgers vector but with opposite heights h = ±0.18nm as expected.

Discussion and conclusion

Investigating numerous GB types (symmetric, asymmetric, twist) and numerous shear directions, this work clearly highlights that SCGBM generally operates through the motion of disconnections. The operation of a given disconnection can be tentatively predicted and favored by imposing a shear stress in the Burger vector direction. This approach combining geometry, local stress field distribution and disconnections nucleation energetic considerations is thus expected to provide a description of the coupling modes at the grain scale.

At the beginning of this manuscript, we mentioned that the coupling factor for SCGBM has been previously explained by: 1) modeling the GB by an intrinsic dislocation network using the FBE and 2) predicting the coupling factor from the glide of these dislocations [START_REF] Cahn | Coupling grain boundary motion to shear deformation[END_REF]. As mentioned previously, for a symmetric tilt GB in a FCC crystal, this model predicts four potential coupling modes [START_REF] Cahn | Coupling grain boundary motion to shear deformation[END_REF] in disagreement with our present results (evidencing the operation of numerous disconnections/coupling modes). However, as previously noticed [START_REF] Molodov | Grain boundary mediated plasticity: On the evaluation of grain boundary migrationshear coupling[END_REF], and as a consequence of the FBE multiplicity, several intrinsic dislocation networks can describe a given GB. Any of the coupling modes we have evidenced above can be explained by one of these descriptions [START_REF] Molodov | Grain boundary mediated plasticity: On the evaluation of grain boundary migrationshear coupling[END_REF]. In that sense, both dislocation and disconnection models are completely equivalent from a geometrical point of view: both deals with transport of a shear either using an intrinsic dislocation or a disconnection formalism. Note that pure geometric models [START_REF] Mompiou | Smig model: A new geometrical model to quantify grain boundary-based plasticity[END_REF][START_REF] Caillard | Grain-boundary shear migration coupling. ii. geometrical model for general boundaries[END_REF] equivalent to these two models have also been elaborated. We think that the disconnections model is more physical to describe the SCGBM . Indeed, the SCGBM effectively operates through the nucleation and motion of disconnections. Moreover, the disconnections model relies on the dichromatic pattern, a characteristic of the GB that includes all the potential disconnections and equally describes them. Hence, we agree with the conclusion of Ref. [START_REF] Chen | Grain boundary shear coupling is not a grain boundary property[END_REF] that the SCGBM is not an intrinsic property of the GB. In that sense, GB based plasticity is not different from dislocation based crystal plasticity, except the variety of disconnections has to be taken into account.

Another important conclusion of the present work is that the coupling factor measured at the meso-scale is not an intrinsic property of the underlying elementary mechanisms (disconnection nucleation and motion), even at very low temperature. Hence, an investigation of the GB migration at the atomic scale in order to characterize disconnections is necessary to raise a reliable conclusion on the elementary mechanisms operating during a GB migration. The analysis of the elementary processes of the GB migration is unfortunately difficult at both high and low temperatures both experimentally and numerically. Experimentally, only in situ HRTEM have the required atomic scale resolutions [START_REF] Zhu | In situ atomistic observation of disconnection-mediated grain boundary migration[END_REF]. But these experiments are technically highly challenging and require ultra fine samples where diffusion effects at free surfaces need to be taken into account. Numerically, two main issues restrains the analysis of the elementary mechanisms: first, GB based plasticity can involve several elementary mechanisms that may operate simultaneously and potentially cooperatively [START_REF] Spearot | Shear-driven motion of mg 1012 twin boundaries via disconnection terrace nucleation, growth, and coalescence[END_REF]. Second, the atomic structures of GBs can evolve during their migration and to our knowledge, there is no method to analyze disconnections in these cases. The understanding and knowledge of GB based plasticity would highly benefit from significant advances on these two issues.

In this manuscript, we have focused on shear directions that could be accommodated by a disconnection present in the dichromatic pattern. But we have also tentatively investigated several general shear direction corresponding to none of the disconnection displayed in the dichromatic pattern. We have most often observed the operation of the disconnections that accommodate most but not totally the shear and then, a destabilization of the GB yielding a rough GB caused mainly by large fluctuation of the GB plane. The rough GB then migrates in order to accommodate the stress. We think that in rough GB, the presence of multiple step creates a lot of potential nucleation sources of mobiles disconnections allowing presumably the GB to relax the elastic energy through the operation of different types of disconnections. Unfortunately, though this interpretation seems reasonable, identification of GB steps in rough GB is far from obvious so that a definitive conclusion can not be raised from our present simulations. It seems however rather clear that a perfectly flat GB has less channels to relax the elastic energy than a rough GB. The understanding of the migration of rough GBs is clearly an important research perspective for the GB based plasticity.

Finally, we have limited the present study to homogeneous disconnections nucleation and have observed the operation of numerous disconnections (or coupling mode). But numerous other potential disconnections (the one represented by black arrows) present in the dichromatic patterns did not activate. These latter ones may be activated by heterogeneous disconnection nucleation: vacancies [START_REF] Chen | Atomistic mechanism for vacancy-enhanced grain bouandary migration[END_REF][START_REF] Larranaga | Role of sessile disconnection dipoles in shear-coupled grain boundary migration[END_REF], triple junctions [START_REF] Thomas | Disconnection description of triple-junction motion[END_REF], sessile disconnections [START_REF] Combe | Heterogeneous disconnection nucleation mechanisms during grain boundary migration[END_REF][START_REF] Kvashin | Atomic processes of shear-coupled migration in 112 twins and vicinal grain boundaries in bcc-fe[END_REF] or other GB imperfections can be a source of glissile disconnections and could potentially produce disconnections that have not been observed.

This work was performed using HPC resources from CALMIP (Grant No. 2012-12172) and was funded by ANR-17-CE08-0007.

Figure 1 :

 1 Figure 1: (color online) Sketch of the simulated system. A GB perpendicular to the x direction separates two grains G 1 and G 2 .

4. 1 . 2 .

 12 The example of the ( 211)/(130)[-7/ -7] shear direction The Σ17(410) -[001] symmetric tilt GB is sheared at 0 K in a direction parallel to b ( 211)/(130)[-7/-7] . Note that the Burgers vector b ( 211)/(130)[-7/-7] has a screw-component, i.e. along the [001] tilt axis (z-axis).

(Figure 2 :Figure 3 :

 23 Figure 2: (color online) Projection of the dichromatic pattern of the Σ17(410) -[001] GB along the [001] z-direction reporting the shearing direction (black and blue arrows) and the observed disconnections (blue arrows). The GB plane is reported by a dashed black line. Black (Grey) and Red (Pink) atoms belong to different grains. Black(Red) and Grey(Pink) atoms have not the same z-coordinate. Green solid lines correspond the coincident lattice site cell.

  1)[4/4] disconnection differs from the (101)/( 101)[4/4] one (mentioned previously as operating under the 100 coupling mode). Indeed, the (101)/( 10 1)[4/4] Burgers vector has a screw component along the [001] direction, while the (101)/( 101)[4/4] has not. Note that since Fig. 2 shows the projection of the dichromatic pattern on the [001] direction, both disconnections (101)/( 10 1)[4/4] and (101)/( 101)[4/4] superimpose on Fig. 2. When shearing along the ( 110)/(031)[-3/ -3] and ( 211)/(130)[-7/ -7] directions, the coupling modes and disconnections ( 110)/(031)[-3/ -3] and ( 211)/(130)[-7/ -7] are activated and accordingly allow to relax the applied shears. Surprisingly, when shearing at 0K along the ( 110)/(231)[5/5], (011)/( 130)[1/1] or ( 101)/(040)[-4/ -4] directions, none of the observed disconnections ( 11 0)/(110)[-5/ -5] and ( 110)/(031)[-3/ -3] have their

Fig. 4a shows

  Fig. 4a shows the dichromatic pattern of the Σ13(320) -[001] GB and several potential glissile disconnections (black and blue arrows): we have investigated the shear of this GB in all the Burgers vectors directions of these disconnections at 0K. The nucleation of the (020)/( 211)[4/4], ( 11 0)/(110)[-5/ -5] and (101)/( 10 1)[3/3] disconnections reported by blue arrows in Fig. 4a have been observed. Note again that the (101)/( 10 1)[3/3] coupling mode differs from the (100)/( 100)[3/3] (operating during the 100 coupling mode) since it has a screw component (along the [001] axis).

Figure 4 :

 4 Figure 4: (color online) a) Projection of the dichromatic pattern of the Σ13(320) -[001] along the [001] z-direction reporting the shearing direction (black and blue arrows) and the observed disconnections (blue arrows). The GB plane is in the (yz) plane. Same atoms color code as in Fig. 2. b) Shear stress-strain curve Σ13(320) -[001] as a function of the y-slab position when shearing along the b ( 110)/( 121)[-1/-1] .

  1)[3/3] disconnection. The Burgers vector of such disconnections have both a component on the y-and the z-directions allowing the relaxation of the stress in both directions. The corresponding coupling factor is positive. The combination of these two plastic events ( 11 0)/(110)[-5/ -5] and (101)/( 10 1)[3/3] fully accommodates the shear stresses when shearing along ( 110)/( 121)[-1/ -1]. Indeed, we note: b ( 11 0)/(110)[-5/-5] + b (101)/( 10 1)[3/3] = 2 b ( 110)/( 121)[-1/-1] h ( 110)/(110)[-5/-5] + h (101)/(10 1)[3/3] = 2h ( 110)/( 121)[-1/-1] This relation can be derived from the sum of the coefficients ( 11 0)/(110)[-5/ -5] + (101)/( 10 1)[3/3] = (0 11)/(01 1)[-2/ -2] moreover, invoking ( 23 1)/( 233)[-13/0] a vector of the coincident site lattice, we get: (0 11)/(01 1)[-2/ -2] + ( 23 1)/( 233)[-13/0] = ( 220)/( 242)[-2/ -2] = 2( 110)/( 121)[-1/ -1]

  1)[3/3] disconnections are opposite and their values are respectively β [-5/-5] = by[-5/-5] h[-5/-5] = -0.399 and β [3/3] = by[3/3] h[3/3] = 1.33, the effective coupling factor is β e = by[-5/-5]+by[3/3] h[-5/-5]+h[3/3] = 3.00 2 .

Figure 5 :

 5 Figure 5: Misorientation angle of the investigated tilt-[001] GBs as a function of the coupling factors βy = by h and βz = bz h

Figure 6 :

 6 Figure 6: Projection of the dichromatic pattern of the Σ11(557)(771) -[001] GB along the [001] z-direction reporting the shear direction (black and blue arrows) and the observed disconnections (blue arrows). The GB plane is in the (yz) plane. Same atoms color code as in Fig. 2. For clarity, two CSL cell have been shown.

4. 4 .

 4 Asymmetric GB : Σ11(557)(771)[110]

Figure 7 :

 7 Figure 7: a) and b) Projections of the dichromatic pattern of the twist Σ13(320) -(001) GB along the [001] direction (x direction) and the [320] y-direction. Same atoms color code as in Fig. 2.

Table 1 :

 1 Summery of the Σ17(410) -[001] GB behavior depending on the shear direction. All observed and characterized disconnections are reported.

The motion of a disconnection that would have a Burgers vector non parallel to the GB plane would imply climb and then long range atomic diffusion, which is not considered here.

in Eq. (4) could have been eluded, but it avoids numerous fractional numbers when designating disconnections in the following. We adopt this notation (n 1 m 1 p 1 )/(n 2 m 2 p 2 ) to designate a disconnection in this manuscript. But in order to avoid a confusing notation for readers familiar with the common p/q notation, we also mention this latter one in bracket when designating a disconnection. Note that the introduced notation (as the p/q one) does not provide any information on the core structure of the disconnection.

Note that the effective factor is not the average nor a linear combination of coupling factors

Note that some authors reserve the term disconnection for GB defects with h = 0 and b = 0.

We refer here to a statistical average.