Andrew R Puente

Adrien Bessaguet

Noël Pairault

Grégory Pieters

Jeanne Crassous

Prasad L Polavarapu

Isabelle Opalinski

Sébastien Papot

Absolute configuration of a [1]rotaxane determined from vibrational and electronic circular dichroism spectra

Keywords: Rotaxane, absolute configuration, density functional theory, vibrational circular dichroism, electronic circular dichroism, mechanical planar chirality

Introduction

While many interlocked molecules, such as rotaxanes and catenanes, have been described in the literature over the last 30 years, [START_REF] Bruns | The Nature of the Mechanical Bond: From Molecules to Machines[END_REF][START_REF] Waelès | Challenges and Opportunities in the Post-Synthetic Modification of Interlocked Molecules[END_REF][START_REF] Denis | The active template approach to interlocked molecules[END_REF][START_REF] Fielden | Molecular Knots[END_REF][START_REF] Lewis | Metal ions in the synthesis of interlocked molecules and materials[END_REF][START_REF] Gil-Ramírez | Catenanes: Fifty Years of Molecular Links[END_REF][START_REF] Beves | Strategies and Tactics for the Metal-Directed Synthesis of Rotaxanes, Knots, Catenanes, and Higher Order Links[END_REF][START_REF] Crowley | Active metal template synthesis of rotaxanes, catenanes and molecular shuttles[END_REF] only little attention has been paid to the stereoselective synthesis of mechanically chiral rotaxanes to date [START_REF] Maynard | Strategies for the Synthesis of Enantiopure Mechanically Chiral Molecules[END_REF] . In 2014, Goldup and co-workers reported the first method enabling the access to mechanically planar chiral (MPC) [START_REF] Jamieson | Chirality in rotaxanes and catenanes[END_REF] [2]rotaxanes without using chiral separation techniques. [START_REF] Bordoli | An Efficient Approach to Mechanically Planar Chiral Rotaxanes[END_REF] Soon after, they accomplished the diastereoselective synthesis of MPC [2]rotaxanes through the active template Cu-mediated alkyneazide cycloaddition (AT-CuAAC) 12,13 reaction. [START_REF] Jinks | Stereoselective Synthesis of Mechanically Planar Chiral Rotaxanes[END_REF] Recently, Leigh and co-workers developed a single-step enantioselective strategy leading to MPC [2]rotaxane enantiomers in 50% ee. [START_REF] Tian | Single-Step Enantioselective Synthesis of Mechanically Planar Chiral [2]Rotaxanes Using a Chiral Leaving Group Strategy[END_REF] The emergence of novel stereoselective approaches has also highlighted the lack of techniques allowing the assignment of the absolute configuration of such rotaxanes, in most cases determined using single crystal X-ray diffraction. However, obtaining suitable crystals remains highly challenging, pointing out the need to design alternative methods for assigning the absolute mechanical configuration of these interlocked chiral molecules. Within this framework, Goldup and Buma demonstrated that vibrational circular dichroism (VCD) spectroscopy has the potential to determine the absolute stereochemistry of such architectures. [START_REF] Koenis | Vibrational circular dichroism spectroscopy for probing the expression of chirality in mechanically planar chiral rotaxanes[END_REF] Indeed, they showed that computed VCD spectra were in accordance with the corresponding experimental ones, thereby providing a very useful means to resolve the absolute configuration of MPC [2]rotaxanes.

Very recently, we reported on the first diastereoselective synthesis of [1]rotaxane using the AT-CuAAC reaction. [START_REF] Pairault | Diastereoselective synthesis of [1]rotaxanes via an active metal template strategy[END_REF] This method enabled the preparation of mirror-image MPC [1]rotaxanes with a high stereoselectivity. In this approach, the mechanically planar chirality of the obtained [1]rotaxanes 1 was guided by the absolute configuration of the covalent stereogenic center attached at the base of the macrocycle. However, as we failed to produce a single-crystal, we were unable to attribute the absolute stereochemistry, either (S, Rmp) or (S, Smp), [START_REF]When the absolute configuration of the stereogenic covalent center was S, the configurations of the two potential diastereoisomers were either (S, Rmp) or (S, Smp)[END_REF] of the major isomer (Figure 1).

In this article, we report the absolute configuration (AC) of 1 determined using vibrational circular dichroism (VCD) and electronic circular dichroism (ECD) spectra. Circular dichroism is the differential absorption of left and right circularly polarized incident light. [START_REF] Berova | Comprehensive Chiroptical Spectroscopy[END_REF] ECD probes the CD in electronic transitions, [START_REF] Berova | Comprehensive Chiroptical Spectroscopy[END_REF] while VCD probes that in vibrational transitions. [START_REF] Polavarapu | Chiroptical Spectroscopy: Fundamentals and Applications[END_REF][START_REF] Nafie | Vibrational Optical Activity: Principles and Applications[END_REF][START_REF] Stephens | VCD Spectroscopy for Organic Chemists[END_REF][START_REF] Barron | Molecular light scattering and optical activity[END_REF] We present the experimental VCD and ECD spectra for (S)-1 and (R)-1 and their comparison to corresponding predicted spectra for (S, Rmp) or (S, Smp) diastereomers. The predicted spectra were obtained using density functional theory (DFT). This comparison analysis allowed us to establish the hitherto unknown AC of 1.

Figure 1 Structures of the two possible diastereomers of [1]rotaxane (S)-1.

Computational Methods

Initial conformational searches were performed with Conflex program 24 using interlocked starting geometries for the diastereomers, (S, Rmp)-and (R, Rmp)-1, in a 10 kcal.mol -1 energy window. This search produced many structures without a mechanical stereogenic unit, so a second search was performed using the lowest energy interlocked conformation from the first search. This second search provided many low-energy conformations for both diastereomers, with interlocked conformations being identified as the lowest energy structures There were 2650 conformers for (S, Rmp)-1 in in 10 kcal.mol -1 window. For (R, Rmp)-1, since there were only 321 conformers in 10 kcal.mol -1 window, we examined 870 conformers in 20 kcal.mol -1 window. Subsequent hand-filtering of the (R, Rmp)-1 structures, for eliminating the non-interlocked structures, resulted in 430 interlocked structures and these structures were used for quantum mechanical (QM) calculations. All of 2650 (S, Rmp)-1 structures were used for QM calculations, without any handfiltering. This process was repeated using the (R, Smp)-and (S, Smp)-1 structures, but no additional conformations were found.

The numerous conformations and the large size of the [1]rotaxane diastereomers rendered QM calculations time consuming and computer intensive. Following the work of Koenis et al., [START_REF] Koenis | Vibrational circular dichroism spectroscopy for probing the expression of chirality in mechanically planar chiral rotaxanes[END_REF] we performed an initial geometry optimization using a density functional tight-binding (DFTB) method with the 3ob-3-1 parameter set [START_REF] Gaus | Parametrization and Benchmark of DFTB3 for Organic Molecules[END_REF] to identify lower energy structures prior to moving onto more time-consuming DFT optimization and frequency calculations. The DFTB-optimized structures in the lowest 2 kcal.mol -1 energy window were selected for further processing. Again hand-filtering, for eliminating non-interlocked structures, resulted in 456 interlocked conformations for (S, Rmp)and 92 for (R, Rmp)-1, which were carried over for DFT optimization using the B3LYP functional [START_REF] Lee | Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density[END_REF][START_REF] Becke | Density-functional thermochemistry. III. The role of exact exchange[END_REF] and 6-31G* basis set. [START_REF] Hehre | Self-Consistent Molecular Orbital Methods. XII. Further Extensions of Gaussian-Type Basis Sets for Use in Molecular Orbital Studies of Organic Molecules[END_REF] VCD and ECD calculations were then performed on the 76 (S, Rmp)-and 33 (R, Rmp)-1 optimized conformations in the lowest 2 kcalmol -1 energy window. The lowest energy conformer (#269 in the list of conformers generated by Conflex) of (S, Rmp) diastereomer is displayed in Figure 2.

All QM calculations were performed using Gaussian 16. [START_REF] Frisch | Gaussian 16 Rev. C.01[END_REF] Vibrational animations and electron density maps were generated using GaussView program [START_REF] Dennington | GaussView, Version 6.1 Shawnee Mission[END_REF] . The VCD and ECD spectra were simulated using CDSpecTech [START_REF] Covington | CDSpecTech: Computer Programs for Calculating Similarity Measures for Experimental and Calculated Dissymmetry Factors and Circular Intensity Differentials[END_REF][START_REF] Covington | CDSpecTech: A single software suite for multiple chiroptical spectroscopic analyses[END_REF] and contributing conformers were weighted by their Gibb's energies for all spectra. The VCD and ECD spectra for (R, Rmp)-1 were each multiplied by -1 to generate those for (S, Smp)-1.

Results and Discussion

VCD SPECTRAL ANALYSIS

Two separate experimental VCD measurements were performed on (S)-and (R)-1, first with 3000 scans and then another with 9000 scans. Since experimental VCD signals were weak, a weighted average of these two measurements was taken, with each measurement weighted by , where N is the number of scans used for the measurement. The resulting VCD spectra (Figure 3a) have a sharp mirror-image band at ~1540 cm -1 and small bands at ~1246, 1207, 1200 and 1041 cm -1 . The VCD spectrum for (S)-1, obtained as one-half the difference between enantiomer spectra is compared to those predicted for (S, Rmp) and (S, Smp) diastereomers in Figure 3b. The experimental VCD spectrum of (S)-1 has positive VCD band at 1539 cm -1 and there is a hint of adjacent negative VCD at ~1520 cm -1 , but it is barely above the noise level. The predicted VCD spectrum for (S, Rmp) shows positive band at 1541 cm -1 and negative band at 1526 cm -1 , both of roughly equal magnitudes.

The predicted VCD spectrum for (S, Smp) shows a triplet of neighboring bands, with large positive at 1513 cm -1 and two weak negative bands one on each side. It is somewhat difficult to draw a definitive conclusion on the AC of 1 based solely on this region of the experimental VCD spectrum. However, there are two experimental VCD bands that have favorable comparisons with those in the predicted spectrum for (S, Rmp), but not in that for (S, Smp). (a). No significant VCD is apparent in the experimental spectrum in the amide I region (~1700 cm -1 ). The asymmetric stretch of both amides near the stereocenter has a scaled frequency of 1702 cm -1 for (S, Rmp). The associated rotational strength is weak, so much so that it is barely visible in the simulated VCD spectrum for (S, Rmp). This prediction is in line with the experimental observation. In contrast to this experimental observation, the VCD spectrum of (S, Smp)-1 has a sharp negative VCD band at 1714 cm -1 . Visualization of vibrations of (S, Smp)-1 indicates that 1714 cm -1 band arises from the amide group that connects the stereocenter to the macrocycle. (b). The negative experimental VCD band at 1041 cm -1 is reproduced in the predicted spectrum for (S, Rmp)-1, while the corresponding band for (S, Smp)-1 has opposite sign. Visualization of vibrations reveals that 1041 cm -1 band is a result of CH rocking along the lasso portion of the rotaxane, specifically between the alkyl chain connecting the stereocenter and the triazole heterocycle. Based on these two observations, the experimental VCD spectrum of (S)-1 can be preferentially associated with the (S, Rmp) diastereomer. To ensure that this conclusion will not be altered at higher level calculations, we have undertaken VCD calculations using BP86 functional [START_REF] Becke | Density-functional exchange-energy approximation with correct asymptotic behavior[END_REF] and TZVP basis set [START_REF] Schäfer | Fully optimized contracted Gaussian basis sets of triple zeta valence quality for atoms Li to Kr[END_REF] for the lowest energy conformers of (S, Rmp)-and (S, Smp)-1. These spectra, found to be similar to the ones presented in Figure 3, are delegated to Supporting Information. The nature of vibrations associated with scaled frequencies of 1546, 1521 and 1041 cm -1 are depicted in Figure 4.

ECD SPECTRAL ANALYSIS

We have previously reported the experimental ECD spectra of (S)-and (R)-1 along with their synthesis. [START_REF] Pairault | Diastereoselective synthesis of [1]rotaxanes via an active metal template strategy[END_REF] The experimental spectra contain three clear mirror-image bands for enantiomers, in the 220 to 300 nm region, with (S)-1 showing negativepositive-negative sequence of ECD bands. The simulated ECD spectra of (S, Rmp)-and (S, Smp)-1 appear as near mirror-image spectra, despite their diastereomeric relationship. This may indicate that the ECD contribution of the mechanically planar stereogenic unit is much greater than that of the stereocenter, a phenomenon that has been reported for axial chirality. [START_REF] Polavarapu | Determination of the absolute configurations at stereogenic centers in the presence of axial chirality[END_REF] The simulated ECD spectrum of (S, Rmp)-1 better matches the experimental ECD spectrum, than (S, Smp)-1, with negativepositive-negative sequence of bands observed for (S)-1 being reproduced in the simulated spectrum (Figure 5). The simulated spectrum of (S, Rmp)-1 is a result of two distinct classes of conformers, one set with two ECD bands and another set with three (Figure 6). To discuss MO contributions to the simulated ECD spectrum of (S, Rmp), we focus our analysis on two lowest energy structures of (S, Rmp)-1, conformers C269 and C463, which serve as representatives of these two classes and are also the two lowest energy structures of (S, Rmp), with roughly equal populations (~5% by Gibb's energy).

FIGURE 5

Smoothed experimental ECD and EA spectra of (S)-1 (black) and the simulated spectra of (S, Rmp)-1 (red) and (S, Smp)-1 (blue) at the B3LYP/6-31G* level. Conformers are weighted by their Gibb's energies. The intensity of the experimental ECD spectrum has been multiplied by 3 for visibility and the simulated ECD and EA spectra have their wavelengths scaled by 1.01.

FIGURE 6

Simulated ECD spectra for the lowest energy structures of (S, Rmp)-1. These nine structures account for ~38% of the population by Gibb's energy. The horizontal ordering of conformer numbers in the legend matches ordering by Gibb's energy, with C269 and C463 being the lowest energy conformers.

The ECD bands of interest for conformers C269 and C463 are a result of excited states that involve many molecular orbitals (MOs), which obscured the nature of their origin. To further investigate these transitions, we generated the Natural Transition Orbitals (NTOs) [START_REF] Martin | Natural transition orbitals[END_REF] of the excited states that contributed the most overall to the ECD spectrum for C269 and C463. This analysis reveals that conformer C269 has an intense π-π* ECD band centered at ~240 nm. This band is a result of a transition from a π MO along the macrocycle to a π* MO on the stopping group, but this also includes some contribution from the lower portion of the macrocycle (Figure 7). There is another π-π* ECD band centered at ~260 nm that is a result of MO contributions from the macrocycle near where the mechanical stereogenic unit originates (Figure 8). Conformer C469 has three ECD bands of roughly equal intensity at 240, 250, and 265 nm. There is a π-π* band centered at 240 nm that, much like the 240 nm C269 band, involves a transition from a π MO from the macrocycle to a π* MO on the stopper. NTO analysis reveals an additional contributing π-π* transition localized on the left-hand side of the macrocycle (Figure 9b). The second ECD band is another π-π* centered at 250 nm. NTO analysis decomposes the transition into three pairs of MOs, each with roughly equal occupancy ~30%) (see Figure 10). Electron density pictures for 265 nm ECD band are displayed in Figure 11.

The simulated ECD bands for (S, Rmp)-1 show great agreement at the B3LYP/6-31G* level, a level of theory that has been shown to have large errors for charge-transfer interactions in the calculation of Rydberg states. [START_REF] Peach | Excitation energies in density functional theory: An evaluation and a diagnostic test[END_REF][START_REF] Raeber | The Importance of Short-and Long-Range Exchange on Various Excited State Properties of DNA Monomers, Stacked Complexes, and Watson-Crick Pairs[END_REF][START_REF] Mahato | Assessing the Performance of DFT Functionals for Excited-State Properties of Pyridine-Thiophene Oligomers[END_REF] To test the validity of our results at the B3LYP/6-31G* level, we carried out additional single-point TD-DFT calculations on the lowest energy structures of (S, Rmp)-and (S, Smp)-1 at the CAM-B3LYP/6-31G* level using the previously calculated B3LY/6-31G* optimized geometries. CAM-B3LYP has been shown to predict excited-states with better accuracy than B3LYP thanks to the addition of a longrange correction. [START_REF] Yanai | A new hybrid exchangecorrelation functional using the Coulomb-attenuating method (CAM-B3LYP)[END_REF] The corresponding simulated spectra are presented in Figure 12. The computed transition wavelengths require a much larger scale factor (1.075) to be consistent with experiment than the B3LYP transition wavelengths (1.01). However, the calculations reproduce the trend from the B3LYP/6-31G* level, giving near mirror-image signs for ECD bands between (S, Rmp) and (S, Smp). In addition, the negativepositive-negative ECD triplet at 240, 255, and 265 nm for (S, Rmp) is simulated with matching signs observed in the experimental spectrum of (S)-1.

Conclusions

Based on the comparison of experimental VCD spectrum of (S)-1 with those predicted for (S, Rmp) and (S, Smp) diastereomers there is some evidence for associating (S)-1 with (S, Rmp). This AC was confirmed by the comparison of experimental ECD spectrum of (S)-1 with those predicted for (S, Rmp) and (S, Smp) diastereomers. Thus, the current VCD and ECD spectroscopic investigation leads to definitive assignment of AC associated with mechanically planar stereogenic unit in [1]rotaxanes.

FIGURE 2 .

 2 FIGURE 2. Lowest Gibb's Energy conformer of (S, Rmp)-1. Hydrogen atoms are not shown for clarity.

FIGURE 3

 3 FIGURE 3 (a) Experimental VCD spectra of solvent subtracted (S)and (R)-1 and associated noise level. (b) VCD spectrum of (S)-1, obtained as one-half the difference between enantiomer spectra, and the predicted spectra of (S, Rmp)-and (S, Smp)-1 at the B3LYP/6-31G* level. (c) Experimental vibrational absorption (VA) spectrum of (S)-1, and corresponding predicted spectra for (S, Rmp)-and (S, Smp)-1. All simulated frequencies were scaled by 0.97.

Figure 4 .

 4 Figure 4. Depiction of atomic displacements associated with scaled frequencies of 1541, 1526 and 1041 cm -1 for (S, Rmp) diastereomer, as obtained from GaussView program.[START_REF] Dennington | GaussView, Version 6.1 Shawnee Mission[END_REF]

Figure 7 .

 7 Figure 7. Natural Transition Orbitals for the 240 nm ECD band of conformer C269. The electron density in ground state MO is on the left, and that in the corresponding excited state MO is on the right. The contribution from this pair is ~78%.

Figure 8 .

 8 Figure 8. Natural Transition Orbitals for the 260 nm ECD absorption band for conformer C269. The electron density in ground state MO is on the left, and that in the corresponding excited state MO is on the right. The contribution from the top pair (a) is ~40% and that from the bottom pair (b) is 32%.

Figure 9 .

 9 Figure 9. Natural Transition Orbital pairs for the 240 nm band for conformer C463. Ground state MO density is on the left and the corresponding excited state density is on the right. (a). The contribution from the top pair (a) is 43% and that from the bottom pair (b) is 33%.

Figure 10 .

 10 Figure 10. Natural Transition Orbitals for the 250 nm band of C463. On the left are the ground state densities and on the right are the excited-state densities. The contribution from each pair of MOs is roughly the same (~30%).

Figure 11 .

 11 Figure 11. Natural Transition Orbitals for the 265 nm transition of conformer C463. On the left are the ground state densities and on the right are the excited-state densities. Occupancy for the top pair is ~65% and is ~32% for the bottom pair.

Figure 12 .

 12 Figure12. Smoothed experimental ECD and EA spectra of (S)-1 (black) and the simulated spectra of (S, Rmp)-1 (red) and (S, Smp)-1 (blue) at the CAM-B3LYP/6-31G* level. Conformers are weighted by their Gibb's energies. The intensity of the experimental ECD spectrum has been multiplied by 3 for visibility and the simulated ECD and EA spectra have their wavelengths scaled by 1.075.

Acknowledgments

We thank the Ministère de l'Education Nationale, de la Recherche et de la Technologie and the Centre National de la Recherche Scientifique (CNRS). This work was conducted in part using the resources of the Advanced Computing Center for Research and Education (ACCRE) at Vanderbilt University.

Supporting information

Calculated VCD spectra at BP86/TZVP level. This information may be found in the online version of this article at the publisher's website.

Graphical Abstract

This article reports the absolute configuration of an enantiopure [1]rotaxane determination by comparing experimental vs. theoretical ECD and VCD spectra.


Isabelle Charpentier

Hossam Elouati

Hugo Kelhetter

Daniil Kudriashov

Hélène Blanchoud

Mathieu Bonnefond

Sylvie Dousset

Véronique Gouy

C Piscart

Marie-Noëlle Pons

Joint exploration of agriculture-water quality links through the hybrid serious game Exp'Eau

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

LTSER-FR Réseau des Zones Atelier

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  At local level (1 of the 3 boards) -Sharing actors' constraints, understanding lock-in situations -Setting higher value market -Support for training -… -At basin level (the 3 boards) -Location of collective equipment and subsidies -Commitment for improving practices -…

Joint exploration of agriculture-water quality links through the hybrid serious game Exp'Eau

Isabelle Charpentier, Hossam Elouati, Hugo Kelhetter, Daniil Kudriashov, Hélène Blanchoud, Mathieu Bonnefond, Sylvie Dousset, Véronique Gouy, Christophe Piscart, Marie-Noëlle Pons, Olivier Ragueneau, Sébastien Salvador-Blanes, Olivier Barreteau


S Patinet

L Proville

Depinning transition for a screw dislocation in a model solid solution

Keywords: numbers: 62, 20, F-,83, 60, La

On the basis of the classical dislocation theory, the solid solution hardening (SSH) is commonly ascribed to the pinning of the edge dislocations. At the atomic level, the theoretical study of the dislocation cores contrasts with such a prediction. Using the static molecular simulations with some interatomic effective potentials, we demonstrate numerically that the critical resolved shear stress associated with a screw dislocation in a random Ni(Al) single crystal has same order as the edge one. Such a result is imposed by the details of the dislocation stacking fault and the core dissociation into Shockley partials. The SSH statistical theory is employed to tentatively predict analytically the data acquired through our atomistic simulations at different Al concentration.

I. INTRODUCTION

In the alloy manufacturing the solid solution hardening (SSH) is a standard process which allows to increase the yield stress of a material by dispersion of some atomicsized obstacles across the dislocation glide. The choice of the impurity and which proportion is required is an important issue in the commercial alloy design. 1,2 On the condition that these impurities remain in solution, favored either by a thermal treatment or the alloy thermodynamics, the microstructure is unchanged but the dislocations are pinned by the randomly distributed obstacles. The dislocation pinning yields an increase in the material strength without involving large inhomogeneities such as inclusions or grain boundaries, by contrast to the other methods as the precipitation strengthening or the strain hardening.

The dislocation impinging on a random distribution of obstacles is a standard problem of the theoretical metallurgy [START_REF] Mott | Report on the Strength of Solids[END_REF][START_REF] Mott | Imperfections in Nearly Perfect Crystals[END_REF][START_REF] Nabarro | Dislocations and Properties of Real Materials[END_REF][START_REF] Friedel | Dislocations[END_REF][START_REF] Haasen | Dislocations in Solids[END_REF][START_REF] Suzuki | Dislocation Dynamics and Plasticity[END_REF][START_REF] Butt | [END_REF]10 and the statistical physics 11,12,13,14,15 as well. Only recently this problem could have been addressed with some three-dimensional atomistic simulations 16,17,18,19,20,21,22,23,24,25 that shed a new light on points that were still a matter of debate in material science. However the difficulty of developing reliable inter-atomic potentials for modeling dislocations in alloys confines the atomistic simulations to only few systems. The Ni(Al) γ-phase is one of them and corresponds to the prototypical case for the binary substitutional alloys with a high order energy and a high solubility limit, about C Al = 10 at. % which involves a broad concentration range for the stability of the solid solution.

The static atomistic simulations allow to compute the Critical Resolved Shear Stress (CRSS) for an isolated dislocation in a single crystal of a purely random solution. 18,24,25 In the model Ni(Al) solution, the CRSS associated with an edge dislocation was found 25 to increase roughly linearly at a rate of about 30 MPa per atomic percent. This result agrees with the experimental data on the Ni(Al) hardness 1 (H) from which the flow stress σ can be deduced by application of the empirical linear relation [START_REF] Tabor | The Hardness of Metals[END_REF][START_REF] Gutkin | [END_REF] H = 3σ (valid for metallic crystalline materials). The main question raised by such a result bears on the role of the screw dislocation segments. According to a widespread belief drawn on the first order elastic dislocation theory (see text book as Ref. 28), the screw dislocation CRSS would be smaller than the edge one. By contrast, our present atomistic study shows that the screw CRSS has same order as the edge one for different Al densities C Al , between 2 at. % and 12 at. %. We analyze our results in terms of the interaction between the dislocation core and some isolated obstacles either single Al placed at different positions around the glide plane or Al dimers with different positions, orientations and bond lengths. The dominant component of the dislocation pinning is found to be a short range interaction of typically few Burgers vector between the obstacles and the distinct Schokley partials of the dissociated dislocation. This rather short range interaction is to be compared with the partial cores which spread over few lattice spacings. [START_REF] Joos | [END_REF]30 Our systematic study of every dimer configuration allows us to enhance the role of the chemical interaction between nearest solutes which yields some pinning strengths that diverge from the simple superposition of the strain fields due to each solute.

In addition to provide some data about the strength of a single Ni(Al) crystal, our atomistic simulations allow us to challenge the different versions of the statistical theory for the SSH. [START_REF] Mott | Report on the Strength of Solids[END_REF][START_REF] Mott | Imperfections in Nearly Perfect Crystals[END_REF][START_REF] Nabarro | Dislocations and Properties of Real Materials[END_REF][START_REF] Friedel | Dislocations[END_REF][START_REF] Haasen | Dislocations in Solids[END_REF][START_REF] Suzuki | Dislocation Dynamics and Plasticity[END_REF][START_REF] Butt | [END_REF] The corresponding analytical models intend to provide an estimate of the CRSS from the elementary interaction between a single dislocation and an isolated obstacle. Most of the other effects on the dislocation pinning as the presence of the grain boundaries, the dislocation forest and the thermal activation of the solute motion are assumed not to play an important role. This proves to match the conditions of our numerical simulations which permits us to compare the theoretical predictions to the simulation data. A common point to the different versions of the SSH theory is to have been derived in the framework of the continuous line tension model where the dislocation is thought of as an elastic string anchored by a single type of obstacles. The split between the different versions of the theory stems from the various assumptions made on the critical string con-figuration and how the depinning proceeds. For the sake of consistency, the string-obstacle interaction parameters are determined from our atomistic study and the statistical models end results are compared to our direct computation of the CRSS for the Ni(Al) single crystal. Among the theories proposed to compute the CRSS, we show that some of them quantitatively agree with our computations for the concentrated solid solutions, i.e. 2 at. % < C Al < 12 at. %. The concordance of the atomistic simulations and the SSH continuous theory confirms the plausibility of a multi-scale approach to the plastic flow in the inhomogeneous media.

The present paper is organized as follows: In Sec. II, the atomistic method to compute the CRSS for a screw dislocation is described and the results are compared to the edge ones. In Sec. III, the different analytical models for SSH are presented and discussed in regard of our atomistic simulations. In Sec. IV, our study is resumed and our future works announced.

II. MOLECULAR STATIC COMPUTATION OF THE SOLUTION STRENGTH

By contrast to the phenomenological line tension approach applied to the SSH, 31,32 the atomistic simulation allows to capture the main physical aspect of the dislocation core by integrating the nonlinear many-body interactions between the atoms displaced during the dislocation course. In our simulations the atomic interactions are modeled through the embedded atom method (EAM) the detail of which has been published elsewhere. 19,33,34 One must notice that two typo errors must be corrected before implementing the EAM potentials. For the Ni-Ni interaction see Ref. 35 and for the Ni-Al interaction the coefficients g Ni and g Al must be exchanged in Ref. 19. According to private communications with other authors these corrections have been taken into account in other earlier works using the same method. The simulation cell (see Fig. 1) is oriented so as that the horizontal Z planes are the (1 11) of the face centered cubic (fcc) lattice while the Y direction corresponds to the screw Burgers vector b = [110]a 0 /2 and the X direction is orthogonal to Z and Y and points at the dislocation motion. The simulation box size along the directions i = X, Y, Z is denoted by L i . The periodic boundary conditions are applied along X and Y while the external applied stress τ yz is produced by imposing extra forces to the atoms in the upper and lower Z surfaces. 16 In order to form a screw dislocation between the two (1 11) central midplanes, the displacement field of the elastic solution for a dissociated screw with Burgers vector b is applied to the atoms of the simulation box. In order to compensate the Burgers vector shift at the crossing of the boundaries along X, the corresponding periodic boundary conditions are tilted from b alongside Y. The ideal solid solution is then formed by substituting randomly Ni atoms with Al in the proportion fixed by the Al atomic density C Al . The solute distribution depends on the seed of the numerical random generator. For each C Al ranging from 2 to 12 at. %, several distributions have been generated. The Molecular Statics (MS) simulations are performed to minimize the total enthalpy under a fixed applied shear stress. The external applied stress is incremented by 0.3 MPa and for each increment the minimization procedure is repeated until it either converges to a required precision or the dislocation starts to glide. To minimize the size effects in the simulations of the random solution, we chose L y = 300b which allows to neglect the interaction between an obstacle and its periodic image and to achieve satisfactory statistics. The same method with same inter-atomic potentials have been employed in Ref. 25 for the edge dislocation.

In Fig. 1 we reported a snapshot of three typical systems computed for an external shear stress smaller than the CRSS τ c . The atoms involved into the partials are recognized by their default configurations in first neighbor positions. The Al atoms that participate to the (1 11) planes that bound the glide plane have also been reported on these 3 pictures. The screw dislocation oscillates in the crystal in a way similar to the edge one as reported earlier. 25 It is worth noticing that such a wavy profile has been observed experimentally [START_REF] Saka | Dislocations in Solids[END_REF] in some other type of solid solution as Cu(Al) and Cu(Si). The dislocation does not form large bow between well separated pinning points but rather conserves a wavy shape. Such configurations of the dislocation impinged on a random solution do not correspond to the one obtained within a phenomenological elastic string model as the one used for instance in the work of Foreman and Makin. 31 The reason is threefold: (i) the solution is concentrated and thus the isolated point like obstacles are rare, (ii) the interaction dislocation-obstacle is not a point-like force and (iii) the pinning strength of the Al obstacle is much weaker than what can be estimated through the elastic stress-strain field of a Volterra dislocation. As detailed further (see Tab.I, the pinning strength of an isolated Al on the screw dislocation is about one hundredth of the line tension meanwhile the elastic theory in a very first order version would predict one order larger.

In our MS simulations when the dislocation moves freely, we let it glide for ten passages in the simulation box in order to probe possible stronger pinning configurations created by the relative displacement of one Burgers vector b at each passage between the plane above and below the glide plane. Since L x ≈ 32 b, we assume that when the dislocation has glided over 10 × L x we reach the maximum of the applied shear stress for the considered random distribution. Such a glide distance has the same order as the mean free path of a gliding dislocation through the forest dislocations in a metallic polycrystal where the dislocation density may reach 10 9 cm -2 . In most of our computations the dislocation does not encounter a new pinning configuration after 4 passages in the simulation box. For each concentration, the calculations of the CRSS, τ c from various distributions have been reported in Fig. 2 (a) where each open symbol corresponds to a different random distribution. The dispersion on the measure of τ c is related to the finite size effect along Y. The choice of the suitable L y results from a compromise between the computational load and the statistics.

The order of magnitude of the solution strengthening is about 30 MPa per atomic percent of Al which is noteworthily similar to the edge one, obtained from Ref. 25 and reported in Fig. 2 (b) for further comparison with the analytical models. Such a result contrasts seriously with the classical calculations based on the elastic theory of dislocation. 32 However this can be fairly understood from the analysis of the interaction between the dislocation and a single obstacle at the atomistic level. We carried out the same type of MS simulations as for the random solid solution except that only one isolated Al atom is placed in a simulation cell of pure Ni. Then the external shear stress τ yz is incremented from zero to τ m at which the dislocation liberates. Since the simulation cell is periodic along Y, the obstacle and its periodic images form a regular array of obstacles separated by L y . The balance between the Peach-Kohler force and the obstacle pinning leads to f m = (τ m -τ p )bL y where τ p is the screw Peierls stress and b is the Burger vector of the whole dislocation. Because of the nonlinearity of the atomic interactions, we found that the pinning strength of an isolated obstacle depends on its place above or below the glide plane as well as on which partial is concerned. Our results are reported on the first lines of Tab.I where the corresponding strength f m has been normalized by the constant µb 2 to provide the standard pinning coefficient denoted by α. We choose µ = 74600 MPa, the shear modulus for the Ni (111) planes [c 11 -c 12 + c 44 ]/3. Although the normalization has been realized with the value of µb 2 for pure Ni, one must bear in mind that this normalization is a conventional way to present the result. In Tab.I the single obstacle denoted by (a1) corresponds to an isolated Al placed in the (1 11) plane situated just above the glide plane while (b1) is for an Al which participates in the (1 11) plane just below the glide plane. The pinning strength of a single Al atom is found to have the same magnitude as for the edge dislocation 25 which confirms our results about the same solid solution CRSS for both edge and screw dislocations. The strengths of some isolated Al have also been reported in Tab.I, for the farther (1 11) planes. For the second nearest one, the pinning forces are referenced by (a2) and (b2) for the Al above and below the glide plane, respectively. For the third nearest planes, the pinning forces are arranged in same order and referenced by (a3) and (b3).

On the condition that the dislocation core distance to the obstacle remains larger than the core extent, it is possible to analyze the interaction in terms of the continuous elastic theory. We intend applying such an approach for the obstacles situated in the third (1 11) plane from the glide plane. For that case, the potential landscape of the dissociated screw dislocation (dot-dashed line in Fig. 3) has the form βz[1/(z 2 + (x + d) 2 ) -1/(z 2 + x 2 )], i.e. the one for two opposite edge dislocations separated from d, the dissociation distance. [START_REF] Friedel | Dislocations[END_REF][START_REF] Suzuki | Dislocation Dynamics and Plasticity[END_REF][START_REF] Hirth | Theory of dislocations[END_REF] According to the elastic theory, the interaction pre-factor β is given by µb * 3π 1+ν 1-ν (∆V ) where ν is the Ni Poisson coefficient, ∆V is the atomic volume variation due to the Al impurity in Ni. If one neglects the interaction between the obstacle and the farthest partial, the corresponding pinning coefficient for a single partial is thus

α = 1+ν 1-ν (∆V )b * √ 3/(8πb 2 Z 2
). The distance Z between the third (1 11) plane and the glide plane is Z = 5b/ √ 6. This corresponds to two and a half of the inter-plane distance along the [111] direction. To provide an estimate of the pinning force we choose b * = b/ √ 12 which is the edge component of a perfect Shockley partial. The volume variation can be estimated according to a method described in the textbook, [START_REF] Friedel | Dislocations[END_REF] ∆V = 3v N i a -1 0 da 0 /dC Al , where the Vegard's law for Ni(Al) is used to express a -1 0 da 0 /dC Al = 0.0763. Then one found α = 0.0015 which is of same order though still larger than what has been found in our simulations (see lines (a3-b3) in Tab.I). If instead of b * = b/ √ 12, one chooses the effective Burgers vector computed through a Peierls-Nabarro method 30 then α = 0.00083 which agrees better. A more quantitative study would require the account for the modulus misfit. This comparison allows to emphasize that as a consequence of the edge Burgers components of both partials the size effect dominates the modulus misfit effect often invoked in the analysis of the screw-impurity interaction. Since the partials have some Burgers vectors that are not purely screw, a hydrostatic stress field is localized around the partial cores [START_REF] Wen | [END_REF] which renders the partials of a screw similar to the ones of an edge, at the atomic scale. Far away from the center of mass of the dislocation, since the edge components of the leading and the trailing screw partials are opposite, the hydrostatic stress field of both partials annihilate each other and the elastic theory prediction for a purely screw dislocation is recovered. In the first and second (1 11) planes, the shape of the interaction potential (full and dashed lines in Fig. 3) is imposed by the nonlinear atomic interaction involved during the passage of the impurity through the core of the dislocation. Concerning the pinning strength on a single Al atom, we distinguish a common trend for screw and edge dislocations: 25 The anharmonicity enhances the pinning strength in the compressive regions in regard of the tensile ones. In our simulation cell, the compressive region of the edge dislocation is situated above the glide plane whereas for the screw 2 compressive regions can be distinguished, e.g. below the glide plane for the leading partial and above for the trailing partial. The obstacle labeled a1 (b1) in Tab. I visits the compressive region when it crosses the trailing (leading) partial. In agreement, the stronger pinning strength is found for the trailing (leading) partial. Furthermore the comparison of the pinning strengths in the separated compressive regions shows that the trailing partial is anchored more strongly than the leading one. The same trend can be noted for the tensile regions. Worthily the previous remarks are consistent with the results on the edge dislocation. 25 The latter trend finds some substantiations into the fact that the more stable position of the Al solute is inside the stacking fault ribbon, as can be seen in Fig. 3. The potential energy measured with respect to the isolated Al far away from the dislocation core should yield a diffusion current toward the dislocation stacking fault. This is a Suzuki-type effect which operates on both types of dislocation. This could lead to a classical dynamic strain aging of the fcc substitutional alloys. 40 In our computation the diffusion is frozen and such an effect is thus disregarded as it is indeed in the SSH statistical models discussed below. Another consequence of the absence of the solute diffusion is that there is no short range ordering even for the Al concentrated solid solutions. It must be noticed that the interaction potential with the nearest obstacles (see Fig. 3) is far from the simple shape assumed in some more phenomenological approaches [START_REF] Nabarro | Dislocations and Properties of Real Materials[END_REF] and even from an elastic theory 32,41 which neglects the dislocation core geometry. As it is required for the construction of a statistical model, some of the details of the atomic scale must be left off and one must retain only the essential quantities that are identified as important.

According to the SSH theory, an obstacle can be characterized by an interaction range w and a pinning force f m , or equivalently a binding energy U . We thus attempt to estimate such quantities and to that purpose we chose arbitrarily a rounding method. We distinguish only two different cases depending on the obstacle position with respect to the leading and the trailing partial. For the pinning strength α we choose the maximum of the interaction force. The definition of the interaction range w requires a more tactful treatment since in principle the short range obstacle-core interaction is superposed to a long range elastic one and further the interaction potential is not symmetric for an obstacle position ahead and behind the partials, mainly because of the presence of a stacking fault. Around the absolute force maximum, we measure w as the shortest distance to which the force vanishes or falls to a local minimum. We choose to neglect the variations of the force over that limit. We concede that such a point can be discussed and must be kept in mind for further discussion on the analytical models. We emphasize that we have been primarily concerned with finding a manner to estimate one of the key parameters of the theory, w and that there is no well prescribed theoretical way for that. We believe that the SSH theory should be developped to address specifically the case of a dissociated dislocation with an interaction potential similar to the one reported in Fig. 3 rather than the standard Gaussian-like potentials.

In the present study, when the interaction potential shows a force maximum near the position of one of the partials we ascribe the force to this partial. This is a conventional way of arranging the numerous obstacle forces since the two partials are actually interacting through the stacking fault. Further when the obstacle is attractive for the leading partial and repulsive for the trailing one, the force fields overlap and it may then be difficult to identify with accuracy the interaction range. When it has not been possible to separate different force maximum, we reported the corresponding pinning strength as being exerted on the trailing one (see (b2-b3) lines in Tab.I). In the fcc symmetry, for the Al concentration we are concerned with, the density of the Al dimers has same order as the density of single Al, i.e. C Al ≈ nC 2

Al

where n = 12 is the number of nearest fcc neighbors. Above C Al = 1/n the number of isolated Al vanishes in average. It is thus of some interest to study the pinning strength of the Al dimers that might be expected to play a role on the SSH because of the alloy ordering energy. Among the different configurations of pairs, we selected those with a distance between solute atoms corresponding to first, second and some of the third neighbors in the fcc lattice. Either the screw dislocation interacts with preexisting Al dimers referred to as (c-h) in Tab.I or the dislocation passage modifies the Al-Al bond crossing the glide plane (i-p) in Tab.I. For the non-crossing pairs, the (c-e) configurations correspond to the planar dimer situated above the glide plane whereas the (f-h) are below. The directions of the dimer bond before the dislocation passage are [011] (c) and (f), The resulting strength can be even smaller than an isolated Al as seen from the comparison between the single (b1) and the first neighbor dimer (f). It must be also remarked that some of the dimer strengths can be larger than two isolated Al at the same position, e.g. ( b1) and (g) in Tab.I. The second neighbor Al dimers could be expected to be more stable than others with regard to the L1 2 ordering trend of the Ni(Al) alloy. Such an expectation is not reflected by the dimer pinning strengths which are not significantly larger to break the second neighbor pairs although the interatomic potential between Ni and Al particles was adjusted to fit the corresponding order energy. 19 The forces reported on lines (k-m) are not particularly stronger than those concerning the first and third neighbor dimers. Finally if one considers the whole set of the dimer configurations, the pinning strengths for a screw dislocation are not smaller than for the edge ones reported in Ref. 25.

III. ANALYTICAL MODELS FOR COMPUTING THE SOLUTION CRSS

The MS simulations allowed us to perform a direct computation of the CRSS at the atomic scale. In the past, different statistical theories have been proposed to evaluate the CRSS of solid solutions. The common assumption of the various analytical models is that the CRSS can be derived only from a unique interaction of the dislocation with an isolated impurity in the glide plane. Although these statistical models were primary devoted to the computation of the CRSS for an edge dislocation, there is no theoretical argument against the application of these models to a screw dislocation provided that one is able to quantify the model input parameters. In order to compare consistently the model predictions to our data on the SSH, the model parameters are determined from the average of the single Al pinning configurations (lines (a1) and (b1) in Tab.I). We obtained a single valued obstacle strength f m = ᾱµb 2 where ᾱ = 0.007 is the mean pinning coefficient for the isolated Al situated in the nearest (1 11) planes. We also computed the typical interaction range as an average over the different isolated obstacle configurations w = 1.3b. Further, since in principle the models apply to an undissociated dislocation, the leading and the trailing partials are assumed to be tightly bound. In the different models, the end formula is usually presented for a theoretical square lattice which leads to an atomic area of s = b 2 . In our system, this is to be changed for s = b 2 √ 3/2, 53 i.e. the atomic area in the glide plane.

Another quantity required before going into the details of the models is the dislocation stiffness. In our simulation cell, we computed the dislocation line energy as the difference between the whole atomic potential energy of a simulation cell with a dislocation and the one of a perfect single crystal with same geometry and same atom number. It was then possible to obtain the energy per unit length E L of the straight dislocation in pure Ni. One must emphasize that this computation implies the stacking fault energy which is not modeled in the classical estimation of the dislocation energy, i.e., E L = µb 2 /2. For the edge dislocation, we obtained E Le = 0.95E L while for the screw E Ls = 0.61E L . These two quantities depend on the simulation cell geometry (L x ,L z ) in agreement with the logarithmic law derived from elastic theory. Since in the Ni(Al) solid solution (see Fig. 1) the dislocation length depends on the Al distribution, it is not possible to compute the dislocation line energy as done in a pure material where the dislocation is straight. However, one can expect reasonably that E L should vary as the elastic shear modulus µ and the lattice parameter b, according to the Vegard's law which leads to a variation of roughly 1 at. % of C Al . Since such a variation of the dislocation stiffness has not been considered in the SSH theory we shall not account further for such a dependency although for the concentrated solutions it would be worth accounting for it. To obtain the dislocation line tension we apply a well known result from the isotropic elastic theory [START_REF] Hirth | Theory of dislocations[END_REF] which tells us that for the screw Γ s = E Ls (1 + ν)/(1 -ν) while for the edge Γ e = E Le (1 -2ν). We note that the line tension increase with the cell dimensions will impact the dislocation roughness through the obstacles distribution as it is predicted by some of the following models.

A. Fleischer-Friedel theory

One of the classical models to evaluate the strengthening of the solid solution is referred as to the Fleischer-Friedel (FF) model. [START_REF] Fleischer | Proc. Conf. at N.P.L[END_REF] The model's main assumption is that the dislocation line forms some bows 31 between isolated point-like obstacles randomly distributed. The bow shape assumption contrasts with what can be seen from our simulations in Fig. 1. The critical angle associated with the impurity strength is ᾱs = f m /2Γ s . The Friedel length is L F = 2sΓ s /cf m where c is the obstacle concentration which corresponds to c = 4C Al to account for the different configurations, i.e. above and below the glide plane and for both partials (Tab.I). The CRSS is then given by the balance between the Peach-Kohler force and a meanfield regular array of obstacle τ c bL F = f m . The FF end formula reads:

τ c = f 3/2 m √ c b √ 2sΓ s . (1)
The corresponding plot of τ c against the Al density is reported in Fig. 2 (a) and (b) for the screw and the edge dislocation, respectively. The FF theory is found to underestimate the CRSS of our simulations. Foreman and Makin 31,[START_REF] Nogaret | [END_REF] showed that the Eq.1 overestimates by about 10 % the true CRSS for some perfect point-like obstacle random distribution so that the discrepancy between our simulation and the FF model predictions could not be compensated by such a correction. It is compelling that the FF model is not suitable to compute the CRSS in the concentrated random Ni(Al) solution. One must be reminded however that the FF model proved reliable 31 for some stronger obstacles as precipitates and for smaller densities. It would be interesting to study by MS the lower densities but then the finite size sampling of the Al distribution limits the statistics.

B. Mott-Nabarro-Labusch theory

The Mott-Nabarro-Labusch (MNL) theory has been built from different contributions. [START_REF] Mott | Report on the Strength of Solids[END_REF][START_REF] Mott | Imperfections in Nearly Perfect Crystals[END_REF][START_REF] Nabarro | Dislocations and Properties of Real Materials[END_REF]15,44,45 The solute dislocation interaction is assumed to be of a finite range w without presuming of the attractive or the repulsive character of the interaction. We introduce the parameter v = 2 w which agrees with the definition of Nabarro for the interaction range. The dislocation is still idealized by a perfect elastic string and its configuration is assumed to be a quasi-straight line which interacts with several solute atoms. Alongside the dislocation core, in a ribbon of length 2L and width 2v, the number of atomic sites in the planes contiguous to the glide plane is 8vL/s. The ribbon extent allows to account for the attractive and repulsive parts of the interaction. Inside this ribbon a counting of the obstacle gives in average 2n = 8vLc/s. The segment is sustained to a mean restoring force which depends on the stiffness of the string. Mott 4 showed that this force could be written as 2Lf 2 m x/L ′ v 2 Γ s where x is the segment mean position and L ′ is the mean distance between two obstacles situated along the segment 2L. The distance L ′ is fixed by 4L ′ vc/s = 1. Nabarro assumed 5 that the characteristic length L could be identified as the distance above which the string Green's function vanishes:

L = (L ′ vΓ s / √ 2f m ) 2/3
. Following Mott and Nabarro, 3 the counting of the obstacles situated in front and behind the segment 2L leads to a total number of interactions 2n which the average force is ±f m /2. These interactions yield a maximum fluctuation of √ 2nf m /2 which must be equated to the external Peach-Kohler force 2Lbτ c to obtain the CRSS:

τ c = c 2 vf 4 m b 3 s 2 Γ s 1/3 . (2)
Here the obstacles above and below the glide plane have been accounted for. If one assumes that the two partials are bound, one must replace c with 2C Al . The plot of the corresponding CRSS has been reported in Fig. 2 and it is found to underestimate the strengthening for the concentrated solutions. The same results hold for the edge dislocation. It is noteworthy that the MNL model is commonly thought to provide a good description for the high densities. In the case of our Ni(Al) solution, our comparison contrasts with such a belief. However, as for the FF model at smaller densities it is not excluded that the agreement could be recovered. An extended version of the MNL model has been proposed elsewhere 25 to tentatively account for the different types of obstacles, i.e. isolated and clusters with different pinning strength f m and interaction range w. Although this model gave us some satisfactory results for both the edge and the screw dislocation further developments are required to be fully consistent on the theoretical treatment.

C. Butt-Feltham theory

In the Butt-Feltham (BF) theory, [START_REF] Butt | [END_REF]46,47 an undissociated dislocation liberates from a pinning configuration by nucleation of a bulge. This critical bulge can be approximated by a triangle shape of height W which corresponds to the saddle point energy required to unzip the whole dislocation. According to Feltham W is estimated from the dislocation core radius extent and thus for a close-packed metals W ≈ 3b. As other previous models in the BF theory it is proposed to relate the CRSS to the in-plane obstacle density. Along the dislocation line the inter-obstacle distance is roughly b/ √ c.

For a quantitative comparison we distinguish the mean distance in function of the different geometry for the edge and the screw dislocation: λ = √ 2b/ √ c and λ = b/ √ c, respectively. The enthalpy required to form a bulge is : H = U L/λ + 2W 2 Γ/Lτ bW L/2 where U is the binding energy and L is the bulge extent along the dislocation line. In the expression for H, one recognizes the binding energy, the elastic cost for the bulge and the work of the Peach-Kohler force. The bulge curvature is determined by the external stress and thus for a certain W we deduce the corresponding bulge extent L = 8W Γ/τ b. At the critical configuration, the enthalpy cancels which gives the end result:

τ c = 4U bW λ . (3)
We assume that the binding energy can be estimated roughly as U = f m w and we account for the different obstacle configurations by c = 4C Al . The corresponding plot has been reported in Figs. 2 (a) and 2 (b) for both edge and screw. Although the predicted CRSS is quantitatively of same order as the MS simulation data, the BF theory predicts a τ c in √ C Al which underestimates the hardening rate of our simulations. It should be mentioned that Butt and Feltham also derived different power laws according to the variation of the bulge height W . With such an extension the BF theory prediction may agree better with the MS data.

D. Friedel-Mott-Suzuki theory

Another theory proposed by Friedel, Mott and Suzuki (FMS) in their text books [START_REF] Friedel | Dislocations[END_REF][START_REF] Suzuki | Dislocation Dynamics and Plasticity[END_REF] considers the few atoms around an ideal undissociated dislocation line. This theory allows the dislocation to take locally much larger curvatures than in the FF model. The first effect of this pinning of the dislocation is to give the dislocation a "zigzag" shape. The interaction is characterized by a binding energy U which we approach by f m w valid for a linearized force. We briefly recall the model derivation. The amplitude of the zigzag in the glide plane is denoted by W and its wavelength alongside the dislocation line is denoted by L. According to Friedel 6 the number of solute contained in the glide plane area W L/2 is given by the inverse of the surface density, i.e. W L/2 = s/c. However a careful counting leads us to W L = s/c since the construction of a regular twodimensional lattice where the obstacles are separated by L in the Y direction and W in the X direction leads us to W L = s/c which agrees with Suzuki's derivation. [START_REF] Suzuki | Dislocation Dynamics and Plasticity[END_REF] The binding energy per unit of length is E 1 = 2f m w/L. The line tension energy involved in the zigzag is given by E 2 = Γ( (W 2 + L 2 /4) -L/2)/(L/2) ≈ 2ΓW 2 /L 2 at first order in W/L. By minimizing the difference E 2 -E 1 one obtains the optimal value of the zigzag W = (f m ws/4Γc) 1/3 that fixes the line shape and thus E 1 = 2 wW c/s and E 2 = 4Γc 2 W 4 /s 2 . The application of external shear stress deforms the dislocation. It has been assumed that the effect of the external shear stress is to unzip the line from the obstacles at the bottom of the zigzag, where the line tension exerts a maximum force in the glide direction. Considering that the maximum of the potential energy is a straight line (W = 0), bound to the obstacle situated at the top of the former zigzag, the energy difference between this maximum and the zigzag configuration is (-E 1 /2) -(E 2 -E 1 ). Multiplied by the wavelength L this gives the work that must be provided by the external shear stress to overcome the pinning barrier. The area that is comprised between the two configurations of the line, zigzag and straight is W L/2, so the stress work is τ c bW L/2 which must equal (E 1 /2 -E 2 )L. The end result for τ c is:

τ c = f m wc sb . (4)
Noteworthily this expression for the CRSS does not depend on the line tension in contrast to the other models.

To apply the theory to our system, the input parameters are estimated similarly and we assumed that c = 4C Al . In Fig. 2(a) and (b), it is remarkable that the τ c linear dependency in C Al is much closer from the simulations than a fractional power law. The FMS assumption of a zigzag shape is closer from the wavy profile of the dislocation seen in Fig. 1. The agreement between our data and the theory is better for the screw (Fig. 2(a)) than for the edge (Fig. 2(b)) since for the latter the FMS model slightly overestimates the CRSS. With the force model parameters for a screw dislocation, the amplitude of the zigzag is W = 0.43b at C Al = 2 at. % and W = 0.25b at C Al = 10 at. %. Besides the fact that such an amplitude is smaller than what can be depicted in Fig. 1, the model predicts W to decrease with C Al . We studied the dislocation roughness and found the opposite trend. In view of the discrepancy on the edge CRSS, our procedure to compute the model parameters from our atomistic data can be discussed. To improve this transfer from the atomic scale to a continuous theory, we emphasize that some improvements of the model would be required to account for the dissociation of the dislocation core and the various types of obstacles.

IV. SUMMARY AND PERSPECTIVES

The aim of the present paper was to extend the study of CRSS for a solid solution Ni(Al) to a screw dislocation. We carried out two types of computation: (i) the CRSS of a random distribution of solutes at different concentration C Al and (ii) the pinning strength of some different obstacles, isolated Al or dimers. We found that the CRSS and the pinning strengths of the screw dislocation are of the same order as the edge ones studied elsewhere. 25 Such a result could not be expected from a first order elastic theory which conventionally ascribes the screw pinning to a modulus misfit effect in contrast to the dominant size effect in the edge case. For the isolated Al impurities, this result has been clearly identified as a consequence of the dissociation of the dislocation core and the edge components of the Shockley partials cores. The Al clusters pinning forces have been shown not to derive from the linear superposition of the isolated obstacle strain field.

Another issue of the present paper was to tentatively apply some analytical models to compute the CRSS. To that purpose, the elementary input parameters of the models were compiled from our atomistic data. The analytical models that we considered only account for the isolated foreign atoms situated in the planes that bound the glide plane. The Friedel-Mott-Suzuki theory provides us a better agreement for the Ni(Al) model solid solution. The estimation of the interaction range between the dislocation and the obstacle is however to be questioned because of the presence of the stacking fault and other nearest obstacles. Our work emphasizes that the SSH theory has to be developed to integrate more of the atomistic details of the dislocation solute interaction. We believe that the accuracy of the SSH theory in the fcc solid solution requires to integrate the dissociation of the dislocation as well as the possibility for cluster formation in the concentrated solution. The farther obstacles as those situated in the second and third (1 11) planes were found to yield a pinning strength that is not negligible in comparison to the solute atoms that bound the glide plane. We also tested other SSH theories involving mixing laws as the Pythagorean one 48 but they are essentially devoted to model the random distribution of point-like obstacles and finally did not allow us to obtain a better agreement.

To predict quantitatively the CRSS in a variety of materials, we believe that an extended use of the empirical interatomic potentials is not feasible at the present state of our skill in developing such potentials. The recent progress of the first principles calculations for alloys and dislocations 49,50 allows us to expect that the pinning strength could be computed with a better accuracy. However it is difficult to imagine that the statistics could be also studied with such methods because of the associated computational load. The effective interatomic potential can thus be employed to explore the frontier between the atomic details and the SSH statistics as we attempted in the present paper. In the near future, we shall propose to extend our work to the Al(Mg) solid solution. Our motivation rests on the fact that the interatomic potentials were proved physically reliable for the study of the alloy plasticity 20,23 and the physical properties of the Al(Mg) solid solution differ from the Ni(Al) ones, e.g. a larger size effect and a higher stacking fault energy. The temperature effect on the SSH and therefore the dynamics of a dislocation in a disordered media is an eagerly difficult problem at the atomic scale because of the time limit of the molecular dynamics simulations. Moreover some puzzling problems of physics can be anticipated: (i) at the low temperature the inertial effect leads to a loss of strength, 51 (ii) at higher temperature the activation of the diffusion yields a dynamic aging 23 and (iii) for the intermediate temperature an odd athermal plateau on the yield stress has been clearly identified experimentally for various solutions and remains difficult to interpret through a simple theory. [START_REF] Nabarro | Dislocations and Properties of Real Materials[END_REF][START_REF] Suzuki | International Conference Dislocations and Mechanical Properties of Crystals[END_REF] We keep scrutinizing the works about those topics which suffer from a lack of modern investigations.

FIG. 1 :

 1 FIG. 1: (color online) View of the (1 11) glide plane and the dissociated screw dislocation in the Ni(Al) simulated solution at various concentrations: C Al = 2 at. % (lhs), C Al = 6 at. % (center) and C Al = 10 at. % (rhs). The box size is 300 b along Y, 32 b along X and 34 b along Z (orthogonal to this paper sheet). The Shockley partials are colored in violet and the Al atoms in blue. The external shear stress has been fixed to one half of the CRSS.X

FIG. 2 :

 2 FIG. 2: Variation of the critical resolved shear stress τc for a screw dislocation (a) and for an edge dislocation (b) against the Al concentration C Al computed from the MS simulations with different Al random distributions (symbols).The estimations made with the analytical models have also been reported: Fleischer-Friedel 37 (dotted line), Friedel-Mott-Suzuki 6 (full line), Mott-Nabarro-Labusch 5 (dot-dashed line) and Butt-Feltham 9 (dashed line).

FIG. 3 :

 3 FIG.3:The internal potential energy for a dissociated screw dislocation against the dislocation obstacle distance. The obstacle is an isolated Al in the (1 11) planes above the glide plane: the first plane (full line), the second plane (dashed line) and the third one (dot-dashed).

  [10 1] (d) and (g), [110] (e) and (h), [011]. The dimers that cross the glide plane are oriented in the direction [721] (i), [211]

TABLE I :

 I Summary of different pinning obstacles for both the leading (subscript l) and the trailing partials (subscript t) of the screw dislocation, their pinning force α (normalized by µb 2 ) and their force range w.

	Nature	text Ref.	α l and w l	αt and wt
	single	(a1)	0.0020/ 1.14b 0.0111/ 0.88b
	1st planes	(b1)	0.0088/ 1.05b 0.0059/ 2.20b
	single	(a2)	0.0036/ 1.41b 0.0032/ 1.69b
	2nd planes	(b2)	-	0.0053/ 1.52b
	single	(a3)	-	0.0011/ 2.8b
	3rd planes	(b3)	-	0.0009/ 2.85b
	1st neighbor	(c)	0.0022/ 0.77b 0.0264/ 2.19b
	non-crossing pair	(d)	0.0006/ 0.1b 0.0139/ 1.49b
		(e)	0.0033/ 1.03b 0.0168/ 1.14b
		(f)	0.0079/ 0.72b 0.0086/ 2.82b
		(g)	0.0207/ 1.8b 0.0068/ 2.03b
		(h)	0.0153/ 1.05b 0.0106/ 2.33b
	3th neighbor	(i)	0.0094/ 1.24b 0.0147/ 2.45b
	crossing pair	(j)	0.0026/ 0.43b 0.0206/ 2.09b
	2nd neighbor	(k)	0.0137/ 1.24b 0.0142/ 2.45b
	crossing pair	(l)	0.0184/ 2.87b 0.0224/ 2.38b
		(m)	0.0096/ 1.09b 0.0146/ 1.07b
	1st neighbor	(n)	-	0.0157/ 1.79b
	crossing pair	(o)	0.0171/ 1.87b 0.0035/1.14b
		(p)	0.0218/ 2.77b 0.0161/ 1.37b
	(j), [212] (k), [ 12 2] (l), [2 21 ] (m), [4 11] (n), [1 14] (o) and [1 41] (p). Our results on the pinning strength and the
	interaction range are reported on the corresponding lines
	in Tab.I. Besides the variable geometry of the crossing
	pairs, it is worth noting that the pinning strength of a
	non-crossing dimer does not correspond to the simple su-
	perposition of the strength of 2 isolated Al that would
	be placed at the reticular sites occupied by the dimer.