An atomistic study of dislocation-solute interaction in Mg-Al alloys

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1 IOP Conference Series: Materials Science and Engineering An atomistic study of dislocation-solute interaction in Mg-Al alloys To cite this article: Luming Shen et al 21 IOP Conf. Ser.: Mater. Sci. Eng View the article online for updates and enhancements. Related content - Dislocation motion in magnesium: a study by molecular statics and molecular dynamics S Groh, E B Marin, M F Horstemeyer et al. - Solute softening and defect generation during prismatic slip in magnesium alloys Peng Yi, Robert C Cammarata and Michael L Falk - Orbital-free density functional theory simulations of dislocations in magnesium Ilgyou Shin and Emily A Carter Recent citations - Atomistic modeling of grain boundary behavior under shear conditions in magnesium and magnesium-based binary alloys M.K. Nahhas and S. Groh - First principles study of the dislocation core structures on basal plane in magnesium Tou-Wen Fan et al - Effects of Zn atoms on the basal dislocation in magnesium solution from Peierls Nabarro model Tou-Wen Fan et al This content was downloaded from IP address on 7/4/218 at 19:6

2 An atomistic study of dislocation-solute interaction in Mg-Al alloys Luming Shen 1,*, Gwénaëlle Proust 1, 2, 3 and Gianluca Ranzi 1 1 School of Civil Engineering, University of Sydney, NSW 26, Australia 2 Australian Key Centre for Microscopy & Microanalysis, University of Sydney, NSW 26, Australia 3 School of Aerospace, Mechanical and Mechatronic Engineering, University of Sydney, NSW 26, Australia * Corresponding author. Luming.Shen@sydney.edu.au Abstract. In this study, atomistic simulations are performed to study the dislocation-al solute effect on the strength of Mg alloys. At temperature T = K, molecular mechanics simulations are carried out to investigate the effect of Al solute concentration on the Peierls stress of basal plane edge and screw dislocations. It is found that the Peierls stress will increase by at least one order of magnitude when.25 at% Al atoms are randomly introduced in the Mg alloys with an edge dislocation. Generally, the Peierls stress will increase with the increase of the Al concentration up to 8 at%. Furthermore, the interaction between the basal plane edge dislocation and Al solute at T = 3 K is studied using molecular dynamics. It appears from the simulations that the critical shear stress increases with the Al solute concentration. Comparing with the solute effect at T = K, however, the critical shear stress at T = 3 K is lower since the kinetic energy of the atoms can help the dislocation conquer the energy barriers created by the Al atoms. 1. Introduction Magnesium alloys are being actively developed for various structural applications owing to their light weight and high specific strength [1], although they have limited formability and mechanical anisotropy arising from their hexagonal close packed (HCP) structure. It has long been known that the strength and ductility of alloys can be influenced by the effect of solute content on solid-solution hardening. Improving these mechanical properties of magnesium alloys necessitates a fundamental understanding of the microstructure-property relations at different length scales, from the atomic structure to the continuum level. In particular, studies at the atomistic level can help elucidate important aspects of dislocation-solute interaction and dislocation motion on the different slip planes. Atomistic simulations can explicitly describe the dislocation core structure, including its splitting into partials, and solute/core interactions in a physical and realistic manner, as compared with ad hoc continuum models. They also permit interatomic interactions, as relevant, among solutes, and the explicit consideration of solute configurations. Molecular mechanics (MM) simulations can be performed at zero temperature, where system energy is minimized, to analyse the unpinning of c 21 Published under licence by Ltd 1

3 dislocation, or molecular dynamics (MD) can be carried out at finite temperatures to understand the thermal escape of dislocation from the energy barriers due to solutes. There has been some atomistic work on dislocations in various hexagonal close packed (HCP) materials [2-4], but these works do not relate specifically to Mg. Only until recently there are two atomistic studies focused on the dislocation behaviour in pure Mg crystal. Yasi et al. [5] studied the core structures of screw and edge dislocations on the basal and prism planes in Mg, and the associated gamma surfaces, using an ab initio method and the embedded-atom-method interatomic potentials developed by Sun et al [6] and Liu et al. [7]. Their results showed that the dislocation core structures from ab initio were well represented by the Sun potential in all cases while the Liu potential showed some notable differences [5]. Groh et al. [8] investigated the motion of dislocations with 1)3 [1 1-2 ] Burgers vector lying on the basal, prismatic and pyramidal slip planes in pure Mg under static and dynamic loading conditions using molecular mechanics (MM) and molecular dynamics (MD). It appears that no atomistic simulations of the dislocation-solid solute interactions in Mg alloys can be found in the open literature. Dislocations in the alloys are pinned through interactions with the solute atoms. Compared with dislocation motion in pure crystal, higher stresses are therefore required for dislocation movement through a solute field. Strengthening is typically divided into two categories: strong pinning, in which individual solutes are sufficient to pin the dislocation at the solute position; and weak pinning, in which the dislocation is pinned by a favourable collective configuration of solutes in a region of the material [9]. In many cases, solute strengthening is the results of both effects. As the dislocation attempts to move, or bow out in regions where it is not pinned, the remaining pinned dislocation segments experience increased forces, resulting from the dislocation curvature, which drive the unpinning. Thus, the amount of strengthening achieved by solutes is a function of the solute concentration, the fundamental strength and spatial range of the dislocation-solute interaction, the dislocation line tension, and the spatial configuration of solutes [9]. The overall picture, however, is not clear and many unanswered questions remain with regards to the dislocation-solid solute interactions and the basic strengthening mechanisms in the alloys. In HCP crystals, dislocations lying on different slip and twin systems can accommodate the plastic deformation [1]. Basal, prismatic and <c + a>-pyramidal are the most active slip systems in Mg and are commonly used to model the plastic behaviour at the mesoscale [8]. Research has shown that the Peierls/critical shear stress of basal plane dislocation is much lower than those in the other two planes [5, 8, 11-13]. This study is thus focused mainly on the role that the basal plane dislocations play on the plasticity of Mg alloys. We present extensive MM/MD simulation results aimed at elucidating the effect of random Al solute additions on dislocation motion in a binary Mg-Al alloy and the associated solute strengthening as a function of concentration. In particular, edge and screw dislocation structures in the basal plane of Mg/Al alloy will be investigated using molecular mechanics at zero temperature and molecular dynamics at finite temperature. Quantitative comparisons among available experimental and numerical results will be presented. The work here on the solute dislocation interactions could also provide a starting point for future physically-based crystal plasticity model for Mg alloys. The remaining sections of the paper are arranged as follows. The computation procedures for the MM and MD simulations of dislocation-solute interaction are described in Section 2, which is followed by the demonstration of simulation results and discussions in Section 3. The concluding remarks are given in the last section. 2. Simulation procedure In this study, both molecular mechanics (MM) and molecular dynamics (MD) simulations are performed to investigate the interaction between basal plane dislocation and Al solid solute and to study the effect of Al solute on the critical shear stress of Mg alloys at different temperatures. In the atomistic simulations, MD code LAMMPS [14] is adopted and the embedded atom method [15] is used to describe Mg-Mg, Al-Al and Mg-Al interactions with the potential model parameters being obtained from Mendelev et al. [16]. The computational configuration of the proposed MD/MS simulations is developed based on the work of Osetsky and Bacon [17] who set up their simulation model to allow the analysis of edge/screw dislocations in an infinite periodic glide plane. The material 2

4 is an Mg crystal with randomly inserted Al atoms deforming only through dislocation slip. The analysis is focused mainly on (i) analyzing the core structure of basal plane edge and screw dislocations, (ii) studying the effect of Al concentration on the Peierls stresses in the basal plane slip system at zero temperature, and (iii) investigating the effect of Al concentration on the critical shear stress of the basal slip plane at a finite temperature. c z T A H x y B D c W Figure 1. Schematic diagram of the computational super-cell for simulating the behaviour of a dislocation in the HCP crystal. As shown in Figure 1, the computational super-cell is a three-dimensional box where x, y and z are the global coordinates. The simulation cell consists of two parts. One part is referred to as the active zone (labeled as A ) in which the atoms move according to the interactions among the neighboring atoms; the other part, labeled with a letter T or B (see Figure 1), is referred to as the boundary zone, where the atoms are either fixed or assigned with a prescribed displacement to simulate a shear loading. The dimensions of the active zone are indicated by D, W and H, while the thickness of each boundary zone (T or B) in the z-direction is c= 3 a with a being the lattice parameter of Mg. The simulation cell uses a z coordinate normal to the glide plane. In standard HCP crystallographic notation, the crystal orientation has x = [1-1 ], y = [1 1-2 ] and z = [ 1] for the basal edge dislocation, and x =[ ], y = [-1-1 ] and z = [ 1] for the basal screw dislocation. Free surfaces are used along the z direction and periodic boundary conditions are applied along the x- and y-directions. In this case, the system can be represented as an array of dislocations periodically repeating in space Edge dislocation The following procedure is used to form a basal plane edge dislocation: (i) The ideal crystal (supercell) is evenly divided into the upper and lower half regions in the z-direction; (ii) In the lower half region, two adjacent atomic half-planes (1 1-2 ) are removed from the ideal crystal structure; (iii) The y-coordinators of the remaining atoms in the lower half region are readjusted so that the distance between any two adjacent half-planes is the same while their x- and z-coordinates are kept the same; (iv) The atoms in the T and B regions are then frozen while the atoms in region A are relaxed to minimize the potential energy of the entire system. It is necessary to emphasize the importance of the final stage in the above procedure because, at this stage, the pure dislocation, which is energetically less favourable for the HCP lattice, splits into two partial edge dislocations: [1 1-2 ] [1-1 ] + [ 1-1 ]

5 Figure 2 demonstrates the dissociation of an Mg basal edge dislocation into two Shockley partials. In this study, the centro-symmetry deviation (CSD) method [18] is used to differentiate the surface and dislocation atoms from the bulk atoms, and only the atoms near the free surfaces and partial dislocations are shown in Figure 2. The atoms with CSD parameter falling in the range of.5å 2 < P < 4 Å 2 correspond to the cores of the partial dislocations, while the range of P > 2 Å 2 to the free surfaces. To study the behaviour of basal edge dislocation under shear loading, the atoms is region B are fixed while the atoms in region T are subjected to a small displacement in the y-direction until a given shear strain is reached, as shown in Figure 2. z [ 1] y [1 1-2 ] x [1-1 ] Figure 2. Dissociation of an Mg basal edge dislocation into two Shockley partials 2.2. Screw dislocation The following procedure is used to form a basal plane screw dislocation: (i) By putting the origin at the centre of the super-cell, the ideal crystal is evenly divided into the upper-left, upper-right, lowerleft and lower-right regions in the y-z plane; (ii) the atoms in the upper-right region are displaced by a distance in the x-direction. The displacement applied to each atom is linearly increased from to a based on their y-coordinate; (iii) The atoms in the T and B regions are frozen while the atoms in region A are relaxed to minimize the potential energy of the entire system. Again the pure screw dislocation is unstable and will dissociate into tow partial dislocations on the basal plane as shown in Figure 3. Similarly, only atoms near the free surfaces and the partial dislocations are presented in the figure using the CSD method. To study the basal screw dislocation under shear loading, the atoms in region B are fixed while the atoms in region T are subjected to a small displacement in the x- direction until a given shear strain is reached. 4

6 z [ 1] x [ ] y [-1 1 ] Figure 3. Relaxed screw dislocation lying on the basal plane Molecular mechanics simulations In the MM simulations, the atoms in region B are always fixed. In order to capture the Peierls mechanism using the MM, a strain increment is applied on the atoms in region T each step and the system energy is then minimized using the conjugate gradient relaxation algorithm. The minimum potential energy is assumed to be reached when one of the following stopping criteria is satisfied: (i) the change in total energy is lower than 1 14 or (ii) the mean gradient per atom is lower than 1 14 ev/a, where a is the lattice parameter. Research has shown that the calculated Peierls stress is dependent on the super-cell size and strain increment size [8]. To evaluate the effect of these two factors on the Peierls stress, simulations are performed with active zone dimensions of 1b 1b 4b and 1b 5b 4b, where the magnitude of b is a, 3 a and 8 / 3 a in the x-, y- and z-directions, respectively, under strain increments of 1 6, 1 7 and 1 8 for edge dislocation. Many studies [5, 8] used only one unit cell in the y-direction due to the application of PBC. This is not a problem for pure Mg crystal since it is reasonable to assume that the unit cell will repeat in the dislocation line direction, namely, the y-direction. To make sure that the Al atoms are randomly distributed in each direction, however, it is important to have many unit cells in the y-direction as well. Figure 4(a) reports the stress-strain curve of each case obtained using MM. As can be seen from Figure 4(a), a converged result can be obtained using the combination of cell size 1b 5b 4b and strain increment 1 7. Hence this combination will be adopted in the future edge dislocation simulations. The stress-strain behaviour for basal plane screw dislocations using different cell size and strain increment is demonstrated in Figure 4(b). It appears the obtained Peierls stress for the screw dislocation is not significantly affected by the cell size or strain increment. Hence, the cell size of 1b 5b 4b and strain increment of 1 7 will be adopted for the future screw dislocation study Molecular dynamics simulations After the basal edge dislocation is dissociated into two Shockely partials at zero temperature using MM, MD simulation will be performed at a finite temperature. Those atoms in the boundary zone are fixed and the system is equilibrated in NVT ensemble for 5 ps. The Nose/Hoover temperature thermostat [19] is used to maintain the constant temperature. With the atoms in region B being fixed, a constant velocity in the y-direction is then applied on the atoms in region T to simulate a displacement-controlled shear loading. During the loading period, the temperature of the system is kept at constant using Nose/Hoover thermostat after excluding the y-component of the atom velocity. The time step is 1 fs in the MD simulation. Due to the thermal fluctuation of atoms at a finite temperature, the noise in the simulation result is significant. To improve the signal to noise ratio, average result from 1 independent MD simulations will be used. 5

7 Shear Stress (MPa) b 1b 1b 1b 1b 1b 1b 5b 1b 5b 1b 5b Shear Strain (a) 25 2 Shear Stress (MPa) b 1b 5b@1^-6 1b 1b 5b@1^-7 2b 1b 5b@1^-6 2b 1b 5b@1^-7 4b 1b 5b@1^ Shear Strain (b) Figure 4. Stress-strain behaviour obtained by MS for basal plane (a) edge and (b) screw dislocations using different cell size and strain increment. 3. Simulation results 3.1. Dissociation of core dislocation It is found the core basal edge and screw dislocations are unstable and will disassociate into two Shockley partial dislocations. Table 1 reports the comparison of the simulated split distance of a basal plane dislocation using ab initio and MM with different EAM potentials. As can be seen from the table, the split distances of the two partial dislocations obtained using EAM potential from [16] match well with previous ab initio calculations [5]. The simulation results validate the accuracy and effectiveness of the potential model. Table 1. Simulated split distance of a basal plane dislocation (Å) 6

8 Dislocation ab initio [5] This study MM with Sun 26 EAM [5] MM with Liu 1996 EAM [5] edge screw MM simulation results To study the effect of Al concentration on the Peierls stress of basal plane edge dislocation, different number of Al atoms are inserted to randomly replace the Mg atoms. The resulting alloy is a HCP Mg- Al system with an edge dislocation. The active zone size is 1b 5b 4b and strain increment is 1-7. Figure 5 shows the stress-strain behaviour of Mg alloyed with different at% Al atoms with an edge dislocation at temperature T = K. As can be seen from the figure, the Peierls stress increases dramatically from.6 MPa to 12.3 MPa when the Al concentration increases from at% to.25 at %, which suggests that the dislocation could be effectively pinned by small amount of solute atoms. In general, the Peierls stress increases with the increase of Al concentration up to 8 at%, although the strengthening is not as effective as the first.25% concentration increase. 7 Shear Stress (MPa) at% Al.25 at% Al.5 at% Al 1. at% Al 2. at% Al 4. at% Al 6. at% Al 8. at% Al Shear Strain Figure 5. Stress-strain curve of Mg/Al alloy with basal plane edge dislocation obtained by MS. To study the effect of Al concentration on the Peierls stress of basal plane screw dislocation, different number of Al atoms are randomly inserted to replace the Mg atoms, which results in a HCP Mg-Al system with a basal plane screw dislocation. The active zone size is 2b 1b 5b and strain increment is 1-6. Figure 6 shows the stress-strain curve of Mg alloyed with different at% Al with a screw dislocation at temperature T = K. Similarly, the Peierls stress increases with the increase of Al concentration up to 8 at% MD simulation results To study the effect of Al concentration on the critical shear stress of basal plane edge dislocation at a finite temperature, the Mg-Al system is subjected to shear loading in the MD simulation. The active zone size is 1b 5b 4b and the strain rate is 1 8 s -1. Due to the statistical fluctuations, the thermal noise level of the stress-strain curve from one MD simulation could be significant. Hence, 1 independent MD simulations are performed and thermodynamic quantities such as stress are averaged in order to obtain statistically meaningful results. Figure 7 shows the stress-strain curve of Mg crystal with different at% Al and a basal plane edge dislocation at temperature T = 3 K. Since the kinetic energy can help the dislocation unpin from the Al solute barriers at the finite temperature, the critical shear stress at T=3 K is generally lower than the corresponding Peierls stress at T= K. 7

9 7 Shear Stress (MPa) at% Al.25 at% Al.5 at% Al 1. at% Al 2. at% Al 4. at% Al 6. at% Al 8. at% Al Shear Strain Figure 6. Stress-strain curve of Mg/Al alloy with basal plane screw dislocation obtained by MS at% Al.25 at% Al.5 at% Al 1. at% Al 2. at% Al 4. at% Al 6. at% Al 8. at% Al Shear Stress (MPa) Shear Strain Figure 7. Stress-strain curve of Mg/Al alloy with basal plane edge dislocation at T=3K. Figure 8 demonstrates the comparison of the stress required to move an edge/screw dislocation at different temperature. Since dislocation in the alloy is pinned through interactions with the solute atoms, the critical shear stress increases with the increase of Al concentration regardless of the dislocation type and temperature. 8

10 7 Peierls/Critiical Shear Stress (MPa) Edge ( K) Screw ( K) Edge (3 K) at.% Al Figure 8. Effect of Al solute concentration on the critical shear stress of edge/screw basal dislocation at different temperatures. 4. Conclusions In this study, atomistic simulations have been performed to simulate the interaction between basal plane edge/screw dislocation and Al solute. In the atomistic simulations, the Mg-Mg, Al-Al and Mg- Al interatomic interactions are described by the embedded atom method [15] with the potential model parameters being obtained from Mendelev et al. [16]. The effectiveness of the potential models has been verified through the simulations of the disassociation of core edge and screw dislocations into two partial dislocations. The simulated split distances match well with ab initio calculations [5]. At zero temperature, MM simulations are undertaken to investigate the effect of Al solute concentration on the Peierls stress of basal plane edge and screw dislocations. It is found that the Peierls stress will increase by at least one order of magnitude when.25 at% Al atoms are randomly introduced in the Mg alloys with an edge dislocation. Generally, the Peierls stress will increase with the increase of the Al concentration up to 8 at%. Furthermore, the interaction between the basal plane edge dislocation and Al solute at T = 3 K is studied using MD. It appears from the simulations that the critical shear stress increases with the Al solute concentration. Comparing with the solute effect at T = K, however, the critical shear stress at T = 3 K is lower due to the kinetic energy of the atoms at the finite temperature. Further MD simulations of basal plane dislocation in Mg-Al alloy at different temperatures are being carried out. The work here on the solute dislocation interactions could not only help better understand the effect of solute concentration on Mg alloy strength, but also provide a starting point for future physically-based crystal plasticity model for Mg alloys. Acknowledgements This research was supported in part by the Australian Research Council Centre of Excellence for Design in Light Metals (CE561574) and was undertaken on the NCI National Facility in Canberra, Australia, which is supported by the Australian Commonwealth Government. Figures 2 and 3 were produced by utilizing software VMD developed by Humphrey et al. [21]. References [1] Kainer K U 2 Magnesium Alloys and Their Applications (Weinheim: Wiley-VCH) [2] Bacon D and Martin J 1981 Phil. Mag. A [3] Bacon D and Liang M 1986 Phil. Mag. A [4] Ando S, Gotoh T and Tonda H 22 Metall. Mater. Trans. A [5] Yasi J A, Nogaret T, Trinkle D R, Qi Y, Hector L G and Curtin W A 29 Modelling Simul. 9

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