- Email: [email protected]
Atomistic simulations of dislocation behavior in a model FCC multicomponent concentrated solid solution alloy
Accepted Manuscript Atomistic Simulations of Dislocation Behavior in a Model FCC Multicomponent Concentrated Solid Solution Alloy
S.I. Rao, C. Woodward, T.A. Parthasarathy, O. Senkov PII:
S1359-6454(17)30460-3
DOI:
10.1016/j.actamat.2017.05.071
Reference:
AM 13832
To appear in:
Acta Materialia
Received Date:
30 January 2017
Revised Date:
31 May 2017
Accepted Date:
31 May 2017
Please cite this article as: S.I. Rao, C. Woodward, T.A. Parthasarathy, O. Senkov, Atomistic Simulations of Dislocation Behavior in a Model FCC Multicomponent Concentrated Solid Solution Alloy, Acta Materialia (2017), doi: 10.1016/j.actamat.2017.05.071
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT
(a) 5K
(b) 150K
(c) 300K Structure of a/2[1-10] edge dislocation core under a) applied stresses of 0.00375, 0.004375 and 0.005 at 5K, b) applied stresses of 0.00375 and 0.004375 at 150K, and c) applied stresses of 0.0025, 0.003125 and 0.00375 at 300K in a FCC Co30Fe16.67Ni36.67Ti30 alloy. The core structure is shown for molecular dynamics steps of 0, 25, 50 and 100 ps. Atoms with a centrosymmetry parameter greater than 4 is shown in the plot. For clarity, core structure at timesteps of 0, 25, 50 and 100 ps are displaced by -2, -1, 0 and 1 periodic units, respectively, along the ‘y’ direction.
ACCEPTED MANUSCRIPT
Atomistic Simulations of Dislocation Behavior in a Model FCC Multicomponent Concentrated Solid Solution Alloy S.I. Rao,2,3 C.Woodward1, T.A.Parthasarathy2 and O.Senkov2 1 Air
Force Research Laboratory, Materials and Manufacturing Directorate, WPAFB, OH 45433 2 3
UES Inc, 4401 Dayton-Xenia Road, Dayton, OH 45432
Institute of Mechanical Engineering, EPFL, Lausanne 1015
Abstract In this work, molecular statics and molecular dynamics simulations of a/2<110> dislocation behavior for a model FCC Co30Fe16.67Ni36.67Ti16.67 alloy are discussed. It is shown that the single FCC phase is elastically stable in this alloy. Local stacking fault energies for the FCC alloy are determined as a function of average composition. The core structure of a/2<110> screw and edge dislocations in the FCC Co30Fe16.67Ni36.67Ti16.67 alloy is shown to be planar with significant variations in the Shockley partial splitting along the dislocation line (factor of ~3) due to concentration fluctuations. The correlation lengths for dislocation line fluctuations in this alloy are determined and discussed. The critical stress to move both a/2<110> screw and edge dislocations at 0K in the model FCC Co30Fe16.67Ni36.67Ti16.67 alloy is of the order of 0.0025 – 0.005, where is the (111) shear modulus, and is significantly higher than that of pure FCC Ni. Molecular dynamics simulation results on the critical stress to move a/2<110> screw and edge dislocations in the model FCC concentrated solid solution alloy show that it decreases with increasing temperature, similar to solid-solution strengthened FCC metals. These molecular dynamics simulation results are in reasonable agreement with experimental tensile yield strength data for an analogous FCC concentrated solid solution alloy. It is also shown that local fluctuations in the concentration of solutes has a strong effect on the effective cross-slip activation energy of screw dislocations in the random alloy. 1.0 Introduction Recent works have shown that when 5 or more elements are combined in nearly equiatomic concentrations, a unique class of materials termed high entropy alloys with unusually high strengths results [1-3]. High Entropy Alloys (HEA) derive their name from the idea that entropy of mixing may stabilize novel disordered phases. The number of phases in a HEA system can be significantly smaller than the maximum number of phases predicted by the well-
ACCEPTED MANUSCRIPT
known Gibbs phase rule [4]. While many alloys that meet the definition of an HEA [1,4,5] contain multiple phases, including intermetallics [6,7], there are some well characterized singlephase solid solution systems [6,8]. The other interesting behavior of these alloys is their unusually high yield strength while maintaining room temperature ductility. In FCC multicomponent alloys such behavior is thought to originate from dislocation –solute interactions [9]. We hypothesize that the high strength must have its origin in the core structure of the dislocations that form in these systems as well as their interaction with solutes. This hypothesis can be examined using atomistic simulations. Atomistic modeling of multicomponent alloys is challenging since it requires accurate interatomic potentials for all pairwise interacting elements. The modeling community has focused mainly on binary systems, (e.g. Ni-Al, Ti-Al, Al-Mg) with very detailed tuning of fitting parameters to achieve quantitative predictive capability. Some years ago a suite of potentials for 16 different metallic elements (Cu, Ag, Au, Ni, Pd, Pt, Al, Pb, Fe, Mo, Ta, W, Mg, Co, Ti, Zr) was developed by Johnson, Chou and co-workers [10]. They introduced a unified functional form for the interaction potentials between different species which opens the possibility of modeling complex compositions. Fit mainly to elemental properties, such potentials are not expected to give quantitative descriptions of the properties of multiphase components. However, the ability to generate model systems of arbitrary chemical complexity presents a huge opportunity to explore and discover mechanisms of deformation in systems where concepts developed for elemental and dilute binary alloys may not apply. The development of high entropy alloys remains Edisonian, relying mostly on insight and experimental search. Similarly the concept of building alloys from a periodic table using interatomic potentials has just begun [11], and remains largely untested for structural materials. While the former is of great engineering interest, the latter is of high scientific interest especially with respect to the Materials Genome Initiative. Previously, we have examined the behavior of a/2[111] screw and edge dislocations in a model BCC multicomponent alloy, Co16.67Fe36.67Ni16.67Ti30 [12]. It was shown that the core structure of a/2[111] screw dislocations is non-planar, similar to simple BCC metals. However, there were significant core structure fluctuations along the dislocation line in the multicomponent alloy, leading to much higher effective kink-pair activation energy and much shallower fall-off in critical stress with temperature, as compared to simple BCC metals and a mean-field version of
ACCEPTED MANUSCRIPT
the same alloy composition [12]. Also, the large static distortions around the Ti atoms in the multicomponent alloy resulted in a high critical stress for motion of the edge dislocation as opposed to simple BCC metals and a mean-field version of the same alloy composition, where the Peierl’s barrier of a/2[111] edge dislocations is very low, < 10MPa [12]. The screw/edge anisotropy of strength in the BCC random alloy was found to be much smaller (~1) than in a mean-field version of the same composition (~100). The atomistic approach was able to capture the major geometrical factors, such as atomic size and scale of local bonding that influence dislocation mobility in the BCC alloys. With this as proof of concept we turn our attention to the chemically complex FCC alloys. Here, we examine the behavior of a/2[110] screw and edge dislocations in a FCC multicomponent alloy, Co30Fe16.67Ni36.67Ti16.67. As before, we use the suite of EAM potentials developed by the Johnson and co-workers for 16 different metallic elements such that any combination of these elements can be handled in a computer simulation [10]. Here, we take advantage of these potentials to probe a model FCC HEA material precisely to uncover heretofore unknown mechanisms of plastic deformation. If this approach can be verified it offers a tremendous potential for alloy discovery. We hypothesize that this approach can be used in solid-solution alloys to understand the unusual mechanical behavior of FCC high entropy alloys and possibly lead to alloy discovery. 2.0 Computational model of the FCC Co30Fe16.67Ni36.67Ti16.67 random alloy A four element, Co-Fe-Ni-Ti, interatomic potential developed by Zhou et.al [10,13] was interfaced with the parallel molecular dynamics (MD) code, LAMMPS [14]. An atomistic simulation cell was generated (~ 2 million atoms) consisting of a FCC lattice with periodic boundary conditions along three orthogonal directions, [1-10], [110] and [001], with a cell size of approximately 300A along each side, and with an average lattice parameter of pure Ni, 3.52A. The cell was populated randomly with Co, Fe, Ni and Ti atoms with an average composition (in at. %) of 30Co:16.67Fe:36.67Ni:16.67Ti. The energy of the system was determined as a function of the average lattice parameter of the FCC cell and a minimum-energy lattice parameter of 3.65A was found for the random alloy. By straining the lattice, the elastic constants, C11, C12 and C44 were determined. The produced FCC lattice was elastically stable and the elastic constants were determined to be C11 = 166.9, C12=131.7 and C44 = 83.6 GPa.
ACCEPTED MANUSCRIPT
The shear elastic constants, (C11-C12)/2 and C44 were 17.6 and 83.6 GPa, respectively. The (111) shear modulus, , is [C44(C11-C12)/2]0.5 ~ 38.4 GPa. The EAM potential gives the chosen composition as highly elastically anisotropic, with an anisotropy ratio, 2C44/(C11-C12) of 4.75. The Voigt average shear modulus (3C44+C11-C12)/5 is 57.2 GPa. The Voigt average shear modulus of pure Ni, Fe, Co and Ti are 76, 82, 78 and 44 GPa, respectively [ 15]. Using a rule of mixtures, one obtains for the selected composition of 30Co, 16.67Fe, 36.67Ni and 16.67Ti, a Voigt average shear modulus of 72 GPa, which is noticeably higher than the atomistically measured value of 57.2 GPa. A second FCC simulation cell, with x, y and z axes along [1-10], [11-2], and [111] directions, and with dimensions of approximately 300A in each direction (~2 million atoms), was generated and relaxed using free surface boundary conditions along z and periodic boundary conditions along the x and y directions. The cell was populated randomly with Co, Fe, Ni and Ti atoms with an average composition (in at. %) of 30Co:16.67Fe:36.67Ni:16.67Ti. Next, the upper half of the simulation cell was translated with respect to the bottom half by the unstable extrinsic stacking fault (ESF) vector, stable intrinsic stacking fault vector (ISF) and ¼[1-10] vector on the (111) plane. For the intrinsic stacking fault vector translation, the whole cell was relaxed along all three directions. The intrinsic stacking fault was found to be stable with an average energy of 20 mJ/m2. For the other two gamma surface translations, atoms were freely relaxed along z. However, the x and y relaxations were performed by ensuring that the total force on all the atoms along the x and y directions to be zero. The energy of the unstable ESF and ¼[1-10] gamma surface translations were found to be 940 and 530 mJ/m2. Similar gamma surface calculations were performed using the Mishin potential for Ni [15]. The energy of the unstable ESF and ¼[1-10] gamma surface translations for Ni were found to be 1480 and 1030 mJ/m2. The Ni potential also gives a stable stacking fault energy of 131 mJ/m2. These gamma surface results suggest that the model Co30Fe16.67Ni36.67Ti16.67 FCC alloy has lower gamma surface fault energies as compared to pure Ni. This also shows that the FCC lattice is stable to large shear deformations on the (111) glide plane. The average composition of the random FCC alloy chosen in this manuscript is somewhat arbitrary. For these potentials, a 4-element composition with significant content of each element where the FCC lattice is elastically stable and stable with respect to large shear deformation on the (111) plane was searched for and obtained. The average composition was also similar to the composition of the matrix in several engineering superalloys. Due to the stability of
ACCEPTED MANUSCRIPT
the FCC lattice to Bain deformation, elastic stresses and large shear deformation on the (111) plane, a study of the properties of the <110> glide dislocations in the FCC lattice of the concentrated random alloy becomes feasible. Stacking fault energies were also determined as a function of average composition of the 2 million atom simulation cell. The average concentration of Co was varied from 0.2-0.5, Fe 0. 0 -0.4, Ni 0.2-0.7 and Ti 0.0-0.3. These composition ranges were picked by looking at local concentration fluctuations at the a/2<110> screw dislocation core to be discussed in the next section. The local composition was defined as the composition up to and including the first neighboring shell of an atom at the core. Figure.1 gives a plot of the stacking fault energy as a function of Ni content, for various Co, Fe and Ti concentrations. In general, increasing Co and Ti concentrations tended to decrease the stacking fault energy whereas increasing Fe concentration tended to increase the stacking fault energy. The stacking fault energy was approximately linear with Ni content, increasing with increasing Ni content. For the composition space studied, the stacking fault energy shows significant variations, ranging from ~ 0 – 70mJ/m2. These results suggest that in the model FCC alloy of interest, Co30Fe16.67Ni36.67Ti16.67, local concentration fluctuations could lead to significant local variations in stacking fault energy, which would have an impact on the local core structure of dislocations. The FCC lattice was unstable for average Ti compositions > 0.30. To study dislocation structure and dynamics, a ½[1-10] screw dislocation, lying along the x axis, was introduced into the simulation cell as two Shockley vectors separated by 5b along the [11-2] direction on the (111) plane using the anisotropic elasticity displacement field of the two Shockley partials. The simulation cell was 300A along x, [1-10], dislocation line direction, 1200A along y, [11-2], perpendicular to the dislocation line direction lying in the (111) glide plane and 300A along z, [111], glide plane normal. In total, the simulation had ~ 8 million atoms. Some simulations were run with 1200A along the x or dislocation line direction, with a total simulation of ~ 32 million atoms, to ensure that the periodic length along the dislocation line contains several units of correlation length for glide in the random alloy. All the cells were populated randomly with Co, Fe, Ni and Ti atoms with an average composition (in at. %) of 30Co:16.67Fe:36.67Ni:16.67Ti. Similarly, a ½[1-10] edge dislocation, lying along the y axis, was introduced into the simulation cell using the anisotropic elasticity displacement field. The
ACCEPTED MANUSCRIPT
simulation cell was 1200A along x, [1-10], perpendicular to the dislocation line direction lying in the [111] glide plane, 300A along y, [11-2], dislocation line direction and 300A along z, [111], glide plane normal, in total, ~ 8 million atoms. As before, some simulations were run with 1200A along the dislocation line direction, giving a simulation of ~ 32 million atoms. Initially, conjugate gradient energy minimization was performed on the initial atomic positions using periodic boundary conditions along the dislocation line direction and fixed boundary conditions along the other two directions. Molecular dynamics simulations were performed on the larger cells with 1200A length along the dislocation line direction, to determine the variation of critical stress for overcoming the resistance to motion of the dislocation as a function of temperature (5, 150 and 300K) in the random alloy. The initial conditions for the MD simulations were a molecular statics relaxed core under the applied stress of interest. Additional forces were applied at the boundary of the MD cell to mimic the state of applied stress. The system was permitted to relax under NVT conditions at various temperatures for 100ps. The position of the dislocation was then monitored as a function of time to follow its motion. A schematic of the simulation cell used in the molecular dynamics simulations is given in Fig.2. Note that portions of the a/2<110> dislocations (both Shockley partials) move a significant amount perpendicular to their line direction (~50A) before uninterrupted free movement in the lattice. As a result, the dimension of the cell perpendicular to the dislocation line direction needs to be large to avoid significant image stress effects from the free surfaces. Here, this dimension was kept at a large value of 1200A. In the random alloy, at a stress just below the critical stress for motion, portions of the dislocation (both Shockley partials) glide a distance of the order of 50A in a stick-slip fashion and come to an eventual stop. At a stress just above the critical stress for motion, the dislocation overcomes the lattice resistance and does not come to an eventual stop. The former stress is termed the lower bound of the critical stress for motion of the dislocation whereas the latter stress is termed the upper bound of the critical stress for motion of the dislocation. This behavior is shown later in the manuscript in Fig.5 for the movement of ½[1-10] edge dislocations in the model Co30Fe16.67Ni36.67Ti16.67 FCC alloy at temperatures of 5, 150 and 300K. For depiction of core structures in the FCC multicomponent concentrated solid solution alloy, we use the method developed by Stukowski and Albe [16] which extracts dislocation lines and their associated Burgers vectors from three-dimensional atomistic simulations (DXA, dislocation
ACCEPTED MANUSCRIPT
extraction algorithm). It is based on a fully automated Burgers circuit analysis, which locates dislocation cores and determines their Burgers vector. The transition from the atomistic system to a discrete dislocation representation is achieved through a subsequent vectorization step. Sophisticated information, including dislocation reactions and junctions, can be obtained from this analysis. Recent work has shown that, for various functional forms of interatomic potentials, an “average atom” (A-atom) interatomic potential can be rigorously constructed for any random alloy composed [17]. Most importantly, many properties of the true random alloy, such as lattice constant, elastic constants, and stacking fault energies, are accurately reproduced by the single Aatom potential. Thus, the A-atom interatomic potential is the mean-field representation of the true random alloy, capturing all properties that do not depend on local composition fluctuations in the material. We have created the A-atom potential for the Co30Fe16.67Ni36.67Ti16.67 alloy, and have verified that it has essentially the same reference properties as the true random alloy. The A-atom potential gives a fcc lattice parameter of 3.649A, elastic constants C11 = 175 GPa, C12 = 124 GPa, C44 = 90 GPa and = 48 GPa, and an intrinsic stacking fault energy, sf of 16 mJ/m2. These values compare favorably with the values of these parameters for the true random alloy of the average composition Co30Fe16.67Ni36.67Ti16.67. We have also used direct atomistic simulations to determine the volume change due to Co, Fe, Ni and Ti atoms in the average A-atom FCC lattice. One obtains V values of -1.00, -0.60, -1.47 and 5.16 A3 for the Co, Fe, Ni and Ti atoms, respectively. This suggests that the Ti atom is a very strong misfit strain center in the model fcc multicomponent alloy whereas Ni, Co and Fe atoms have lower misfit strain values. Therefore, strong lattice distortions are expected around the Ti atoms in the model FCC multicomponent alloy. We have also used this A-atom potential in simulations of both edge and screw dislocations using the geometries and methods described below. 3. Results 3.1 Core structure of a/2<110> dislocations in a model FCC Co30Fe16.67Ni36.67Ti16.67 alloy Figure 3 gives a DXA plot of the relaxed ½[1-10] screw dislocation core in a model Co30Fe16.67Ni36.67Ti16.67 concentrated solid solution FCC alloy. The screw dislocation shows a classical Shockley partial splitting along the dislocation core, similar to pure FCC metals. Also, the core is planar with the core atoms are confined to a single (111) glide plane. However, there
ACCEPTED MANUSCRIPT
are significant variations in the Shockley partial splitting along the dislocation line, so that the dislocation core structure changes significantly along the dislocation line. Pure shear stresses on the (111) glide plane were applied on the screw dislocation core by straining the atoms and using fixed boundary conditions along the y and z directions. The screw dislocation starts moving when applied shear stresses reach 0.0025 -0.00375 giving a critical stress for motion at 0K at least one order of magnitude higher compared to pure FCC metals. Figure 4 gives a DXA plot of the relaxed a/2[1-10] edge dislocation core in a model Co30Fe16.67Ni36.67Ti16.67 concentrated solid solution alloy. Similar to the screw core, the edge dislocation core is planar and shows significant fluctuations in Shockley partial splitting along the dislocation line. Pure shear stresses on the (111) glide plane were applied on the edge dislocation core by straining the atoms and using fixed boundary conditions along the x and z directions. The edge dislocation moves when the applied shear stresses reach 0.00375 0.005 giving a critical stress for motion at 0K at least one to two orders of magnitude higher as compared to pure fcc metals. Both a/2[110] screw and edge dislocations were planar in the mean-field alloy, dissociated into Shockley partials on the (111) plane, with constant splitting distances of 27 and 80A, respectively. The width of the Shockley partials belonging to the a/2[110] edge dislocation was estimated as ~ 15A in the mean-field alloy, from Burger’s vector distribution plots obtained from atomistic simulations. The critical stress for motion at 0K for both a/2[110] screw and edge dislocations in the mean-field alloy were very low, < 10 MPa, similar to simple fcc metals. 3.2 Critical resolved shear stress for the screw and edge dislocations and comparison with experiment Figures 5a, b and c give a (111) projection of atoms with a centrosymmetry parameter > 4 in the FCC random alloy with an a/2[1-10] edge dislocation at applied stresses of 0.00375, 0.004375 and 0.005 at 5K, 0.00375 and 0.004375 at 150K and applied stresses of 0.0025, 0.003125 and 0.00375 at 300K, for different molecular dynamics timesteps of 0, 25, 50 and 100ps. Such a plot shows atoms at the stacking fault region between the Shockley partials of the a/2[1-10] edge dislocation. The profiles at 0, 25, 50 and 100 ps have been shifted by -2, -1, 0, and 1 periodic units along the dislocation line along the ‘y’ direction for clarity of viewing. There are significant Shockley partial splitting variations along the dislocation line, as much as
ACCEPTED MANUSCRIPT
factors of 3, similar to variations seen in recent high resolution transmission electron microscopy observations of local Shockley partial splitting widths at an a/2[1-10] 600 dislocation in an analogous 5-element FCC multicomponent alloy, Cantor alloy [19]. The experimentally measured average stacking fault energy in the Cantor alloy is 25mJ/m2 [19], similar to the present multicomponent FCC alloy. The edge dislocation moves by continuous expansion and contraction of these local stacking fault splitting widths. Some portions of the dislocation line move a significant amount, of the order of 50A, below the critical stress. At a stress of 0.004375 at 5K, all portions of the edge dislocation overcome the lattice resistance to dislocation motion. Similarly, the edge dislocation overcomes the lattice resistance to motion at stresses of 0.00375 and 0.003125 at 150K and 300K, respectively. Figure 6 gives a variation of critical stress as a function of temperature obtained from the MD results, for both a/2<110> screw and edge dislocations. The upper and lower bounds of the critical stress are shown. The critical stress shows a fall-off with increasing temperature, similar to solid-solution strengthened fcc metals [9]. Also shown in Fig.4 are experimental results on yield [18] from a similar 5-component concentrated solid solution alloy, Co20Cr20Fe20Mn20Ni20. The experimental yield strength data was for polycrystalline materials with varying grain sizes. The experimental results were extrapolated to an infinite grain size and divided by a Taylor factor of 3 to obtain approximately the critical resolved shear stress for flow in single crystals of such an alloy. Such single crystal CRSS data were compared with the present MD simulation in Fig.6. Overall, the simulation data gives similar values of strength as a function of temperature as compared to experiment. Note that the experimental alloy composition is different from the composition used in these simulations and the comparison should be taken only as qualitative. 3.3 Cross-slip of screw dislocations Figures 7a and b give a (111) [y] ,(11-2) [z] , (11-1) [y] and (112) [z] projection of atoms with a centrosymmetry parameter > 4 in the FCC random alloy with an a/2[1-10] screw dislocation at an applied stress of 0.001875, which is slightly above the critical stress, for molecular dynamics timesteps of 0 and 100ps, at 300K. The profile at 100 ps has been shifted by 1 periodic unit along the dislocation line along the ‘x’ direction for clarity of viewing. At 0 ps, the a/2[1-10] screw dislocation is completely on the [111] glide plane, seen in Fig.7a in the [112] [z] projection as one atomic plane thick of atoms across the dislocation line. At 100ps, small
ACCEPTED MANUSCRIPT
lengths of the dislocation line have cross-slipped, which is clear from the spreading seen in the [11-2][z] projection in Fig.7a at portions of the dislocation line and in Fig.7b in the [112] [z] projection as one atomic plane thick of atoms across the dislocation line. The lengths of dislocation line that have cross-slipped is relatively small presumably due to the low applied shear stress on the cross-slip plane in the simulations ( ~1/3 the stress applied on the glide plane). This suggests that the effective cross-slip activation energy in this concentrated solid solution alloy Co30Fe16.67Ni36.67Ti16.67 is very low, of the order of 0.30 eV (12kT with T = 300K, and k is the Boltzmann constant). The average stacking fault energy, s, of the simulated alloy is 20 mJ/m2 (~ 50% of pure Cu) and the shear modulus, 40 GPa ( ~ equal to that of pure Cu). The atomistically calculated value for the cross-slip activation energy in pure Cu is ~ 1.6-1.8 eV [20,21]. The cross-slip activation energy in FCC metals is proportional to (d/b)[ln(d/b)]0.5 [22] where d is approximately equal to b2/16 [23] . Using the scaling factors given above, the cross-slip activation energy in the simulated FCC alloy based on average stacking fault energy and shear modulus parameters is estimated to be 3.8 – 4.3 eV. Also, using the mean-field potential, the cross-slip activation energy for the mean stacking fault energy and shear modulus parameters was also directly atomistically calculated using the procedure given in reference [21] as 4.6 – 4.9 eV. However, molecular dynamics simulations show that the effective cross-slip activation energy in the concentrated solid solution alloy, Co30Fe16.67Ni36.67Ti16.67 is very low, of the order of 0.3 eV. This suggests that local fluctuations in the concentration of solutes has a strong effect on the effective cross-slip activation energy akin to what has been suggested in the recent model for cross-slip activation energy in alloys given in reference [20]. 3.4 Correlation length for a/2<110> dislocation line fluctuations in a model FCC Ni36.67Co30Fe16.67Ti16.67 alloy Figure 8a is a plot of the shape profile in the (111) glide plane of the leading Shockley of an a/2<110> edge dislocation in a model FCC Co30Fe16.67Ni36.67Ti16.67 random alloy at a stress of 0.00375, slightly below the critical stress for motion of the edge dislocation at 0oK, obtained from DXA analysis. The dislocation line is not straight, and shows significant fluctuations along the ‘y’ direction under an applied stress overcoming varying solute-dislocation interactions along the dislocation line. The y-axis scale has been expanded relative to the x-axis scale to accentuate the fluctuations. The effective ‘x’ period of such fluctuations can be obtained from a one-
ACCEPTED MANUSCRIPT
dimensional correlation analysis of such fluctuations. The correlation function for a separation ‘x0’ can be defined as C(x0) =
[y(x)-][(y(x+x0))-]dx
(1)
where is the average height of the fluctuations. Figure 8b shows a plot of C(x0) obtained from such an analysis as a function of x0. The correlation function starts at a peak value and goes to zero at or around, x0 = 100A. Therefore, the correlation length for the a/2<110> edge dislocation fluctuations is ~ 100A in this model FCC Co30Fe16.67Ni36.67Ti16.67 random alloy, which is approximately a factor of 12 smaller than the dislocation line length, 1200A, considered in the molecular dynamics simulations. This suggests that the periodic boundary conditions along the dislocation line do not have a significant effect on the line fluctuations arising from overcoming solutes in the random alloy in the molecular dynamics simulations. Similar correlation length analysis for the a/2<110> screw dislocation at a stress of 0.0025, gives a value of ~ 125A for the correlation length of dislocation line fluctuations for the a/2<110> screw dislocation. 4. Summary Ordinary dislocation behavior in a model FCC multicomponent alloy Co30Fe16.67Ni36.67Ti16.67 has been modeled using EAM potentials. The dislocation core structure of the multi-component alloy shows similarity to the dislocation structure in monoatomic material; the core structure of a/2<110> dislocations in the fcc multicomponent alloy is planar with classical Shockley partial splitting, similar to pure Ni. However, because of the concentration fluctuations, there are significant core structure variations along the dislocation line for the multicomponent alloy. This manifests intself as Shockley partial splitting variations along the dislocation line ( upto factors of 3). The critical stress to move a/2<110> dislocations in the FCC multicomponent alloy at 0K ( ~ 0.0025 – 0.005) is one to two orders of magnitude higher as compared to pure Ni. Since this strength does not have it’s origin in the core structure (core is planar), it is argued that the high strength results from varying dislocation-solute interactions at the dislocation core [9]. The strength decreases with increasing temperature, similar to conventional FCC solid solution alloys.The correlation length for dislocation line
ACCEPTED MANUSCRIPT
fluctuations in this alloy is ~ 100 - 125A. Therefore, molecular dynamics simulations where the dislocation line length chosen is much greater than 100A (~1200A) should be representative of bulk behavior. Also, local fluctuations in the concentration of solutes has a strong effect on the effective cross-slip activation energy of screw dislocations in the random alloy, decreasing it from a value of 4.6 – 4.9 eV (atomistically calculated from average stacking fault energy and shear modulus parameters) to a value of ~ 0.3 eV. Acknowledgement The authors thank Dr. C. Varvenne, Mr. W. Noehring, and Prof. W. A. Curtin for technical assistance with the development and use of the mean-field interatomic potential presented in this work; their efforts were supported by the European Research Council through the Advanced Grant ``Predictive Computational Metallurgy'', ERC Grant agreement No.339081 - PreCoMet. The authors acknowledge use of the Molecular dynamics code, LAMMPS, which was developed at Sandia National Laboratory by Dr. Steve Plimpton and co-workers. This work was supported by the AFOSR, and by a grant of computer time from the DOD High Performance Computing Modernization Program, at the Aeronautical Systems Center/Major Shared Resource Center. Work by ONS and TAP was supported through the Air Force onsite contract No. FA8650-15-D-5230 managed by UES, Inc. References: 1) M.H. Tsai and J.W. Yeh, ‘High entropy alloys – A critical review’,Mater.Res.Lett., 2 (2014) 107. 2) Y. Zhang, T.T. Zuo, Z. Tang, M.C. Gao, K.A. Dahmen, P.K. Liaw and Z.P. Lu, ‘Microstructures and properties of high-entropy alloys’, Progress. Mater. Sci., 61 (2014) 1. 3) D.B. Miracle, J.D. Miller, O.N. Senkov, C. Woodward, M.D. Uchic and J. Tiley, ‘Exploration and development of high entropy alloys for structural applications’, Entropy, 16 (2014) 494. 4) S. Curtarolo, G.L.W. Hart, M.B. Nardelli, N. Mingo, S. Sanvito and O. Levy, ‘The high-
throughput highway to computational materials design’, Nature Materials, 12 (2013) 191. 5) B.S. Murty, J.W. Yeh and S.Ranganathan, ‘High Entropy Alloys’, ButterworthHeinemann, 2014.
ACCEPTED MANUSCRIPT
6) M. Poletti and L. Battezzati, ‘Electronic and thermodynamic criteria for the occurrence of
high entropy alloys in metallic systems’, Acta Mater., 75 (2014) 297. 7) J.W. Yeh, S.J. Lin, T.S. Chin, J.Y. Gan, S.K. Chen, T.T. Shun, C.H. Tsau and S.Y. Chou,
‘Formation of simple crystal structures in Cu-Co-Ni-Cr-Al-Fe-Ti-V alloys with multiprincipal metallic elements’, Metall.Mater.Trans., 35A (2004) 2533. 8) J.W. Yeh, S.K. Chen, S.J. Lin, J.Y. Gan, T.S. Chin, T.T. Shun, C.H. Tsau and S.Y. Chang,
‘Nanostructured high-entropy alloys with multiple principal elements – Novel alloy design concepts and outcomes’, Adv.Engr.Mater., 6 (2004) 299. 9) C. Varvenne, A. Luque and W.A. Curtin, ’Theory of strengthening in fcc high entropy alloys’, Acta Mater., 118 (2016) 164. 10) X.W. Zhou et.al., ‘Atomic scale structure of sputtered metal multilayers’, Acta Mater., 49
(2001) 4005. 11) F.C. Campbell, ‘Phase Diagrams – Understanding the Basics’, ASM International, 2012. 12) S.Rao, C. Varvenne, C. Woodward, T.A. Parthasarathy, D. Miracle, O.N. Senkov and W.A. Curtin, ‘Atomistic Simulations of Dislocations in a Model BCC Multicomponent Concentrated Solid Solution Alloy’ Acta Mater., 125 (2017) 311. 13) www.ctcms.nist.gov/potentials/ 14) S.J. Plimpton, ‘ Fast parallel algorithms for short-range molecular dynamics’, J. Comput.
Phys., 117 (1995) 1. 15) G. Simmons and H. Wang, ‘Single crystal elastic constants – A Handbook 2nd edition’, MIT Press, 1971. 16) A. Stukowski and K. Albe, ‘Structure identification methods for atomistic simulations of crystalline materials’, Model.Simul.Mater.Sci.Eng., 18 (2010) 085001. 17) C. Varvenne, A. Luque, W.G. Nöhring and W.A. Curtin, ‘Average-atom interatomic potential for random alloys’, Phys. Rev. B 93 (2016) 104201. 18) F.Otto, A. Dlouhy, Ch. Somsen, H. Bei, G. Eggeler and E.P. George, ‘The influence of temperature and microstructure on the tensile properties of a CoCrFeMnNi high-entropy alloy’, Acta. Mater., 61 (2013) 5743. 19) T.M. Smith, M.S. Hooshmand, B.D. Esser, F. Otto, D.W. Mccomb, E.P. George, M. Ghazisaeidi and M.J. Mills, Acta Materialia 110 (2016) 352. 20) W.Goehring and W.A. Curtin, submitted for publication in Acta Materialia.
ACCEPTED MANUSCRIPT
21) S.I. Rao, D.M. Dimiduk, T.A. Parthasarathy, J. El-Awady, C. Woodward and M.D. Uchic, ‘Calculations of intersection cross-slip activation energies in fcc metals using the nudged elastic band method’, Acta Materialia 59 (2011) 7135. 22) S.I. Rao, T.A. Parthasarathy and C. Woodward, ‘Atomistic simulation of cross-slip processes in model fcc structures’, Philosophical Magazine A 79 (1999) 1167. 23) M. Chassagne, M. Legros and D. Rodney, ‘Atomic-scale simulation of screw dislocation / coherent twin boundary interaction in Al, Au, Cu and Ni’, Acta Materialia 59 (2011) 1456.
Figure captions: Fig.1: Stacking fault energy, sf on the (111) plane as a function of Ni concentration, XNi, in Co, Fe, Ni, Ti based multicomponent random FCC solid solution alloys. Fig.2: A schematic illustration of the simulation cell used in molecular dynamics simulations of ½[1-10] screw and edge dislocations in a model FCC Co30Fe16.67Ni36.67Ti16.67 alloy. For the screw dislocation, periodic boundary conditions are applied along the ‘x’ direction and free surface boundary conditions along the ‘y’ and ‘z’ directions. For the edge dislocation, periodic boundary conditions are applied along the ‘y’ direction and free surface boundary conditions along the ‘x’ and ‘z’ directions. Additional forces are applied on top and bottom boundary surface layers ~ 12A thick along the ‘z’ direction to model the applied stress. Fig.3: DXA plot of relaxed ½[1-10] screw dislocation core in model FCC Co30Fe16.67Ni36.67Ti16.67 alloy. The length of the dislocation line is 300A. Fig.4: DXA plot of relaxed ½[1-10] edge dislocation core in model FCC Co30Fe16.67Ni36.67Ti16.67 alloy. The length of the dislocation line is 300A. Fig.5: Structure of a/2[1-10] edge dislocation core under a) applied stresses of 0.00375, 0.004375 and 0.005 at 5K, b) applied stresses of 0.00375 and 0.004375 at 150K, and c) applied stresses of 0.0025, 0.003125 and 0.00375 at 300K in a FCC Co30Fe16.67Ni36.67Ti30 alloy. The core structure is shown for molecular dynamics steps of 0, 25, 50 and 100 ps. Atoms with a centro-symmetry parameter greater than 4 is shown in the plot. For clarity, core structure at time-
ACCEPTED MANUSCRIPT
steps of 0, 25, 50 and 100 ps are displaced by -2, -1, 0 and 1 periodic units, respectively, along the ‘y’ direction.
Fig. 6: Measured critical resolved shear stress scaled by the (111) shear modulus (39 GPa) necessary to achieve on-going glide as a function of temperature, for the a/2[1-10] screw and edge dislocations in the model FCC Co30Fe16.67Ni36.67Ti30 alloy. The upper and lower bounds of the critical resolved shear stress are shown in the plot. Also shown in Figure 4 is the experimentally measured strength for the 5-element Co20Cr20Fe20Mn20Ni20 multicomponent Cantor alloy. Fig.7: Structure of a/2[1-10] screw dislocation core under an applied stress of 0.001875 in a FCC Co30Fe16.67Ni36.67Ti30 alloy at 300K. The core structure is shown for molecular dynamics steps of 0 and 100 ps. Atoms with a centro-symmetry parameter greater than 4 are shown in the plot. For clarity, core structure at a time-step of 100ps is displaced by 1 periodic unit along the ‘x’ direction. The (a) (111) [y] and (11-2) [z] and b) (11-1) [y] and (112) [z] projections are shown. Fig.8: DXA plot of the shape profile in the (111) glide plane of the leading Shockley of an a/2<110> edge dislocation in a model FCC Co30Fe16.67Ni36.67Ti16.67 random alloy at a stress of 0.00375, slightly below the critical stress for motion of the edge dislocation at 0K with both different scaling for the x and y-axes and with identical scaling for the x and y-axes. b) Plot of the one-dimensional correlation function obtained for the a/2[1-10] edge dislocation profile shown in (a) as a function of distance along dislocation line.
ACCEPTED MANUSCRIPT
Fig.1:
ACCEPTED MANUSCRIPT
Fig.2:
ACCEPTED MANUSCRIPT
Fig.3:
ACCEPTED MANUSCRIPT
Fig.4:
ACCEPTED MANUSCRIPT
(a) 5K
(b) 150K
(c) 300K
Fig.5:
ACCEPTED MANUSCRIPT
Fig.6:
ACCEPTED MANUSCRIPT
(a)
(b) Fig.7
ACCEPTED MANUSCRIPT
(a)
(b)
Fig.8:
S.I. Rao, C. Woodward, T.A. Parthasarathy, O. Senkov PII:
S1359-6454(17)30460-3
DOI:
10.1016/j.actamat.2017.05.071
Reference:
AM 13832
To appear in:
Acta Materialia
Received Date:
30 January 2017
Revised Date:
31 May 2017
Accepted Date:
31 May 2017
Please cite this article as: S.I. Rao, C. Woodward, T.A. Parthasarathy, O. Senkov, Atomistic Simulations of Dislocation Behavior in a Model FCC Multicomponent Concentrated Solid Solution Alloy, Acta Materialia (2017), doi: 10.1016/j.actamat.2017.05.071
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT
(a) 5K
(b) 150K
(c) 300K Structure of a/2[1-10] edge dislocation core under a) applied stresses of 0.00375, 0.004375 and 0.005 at 5K, b) applied stresses of 0.00375 and 0.004375 at 150K, and c) applied stresses of 0.0025, 0.003125 and 0.00375 at 300K in a FCC Co30Fe16.67Ni36.67Ti30 alloy. The core structure is shown for molecular dynamics steps of 0, 25, 50 and 100 ps. Atoms with a centrosymmetry parameter greater than 4 is shown in the plot. For clarity, core structure at timesteps of 0, 25, 50 and 100 ps are displaced by -2, -1, 0 and 1 periodic units, respectively, along the ‘y’ direction.
ACCEPTED MANUSCRIPT
Atomistic Simulations of Dislocation Behavior in a Model FCC Multicomponent Concentrated Solid Solution Alloy S.I. Rao,2,3 C.Woodward1, T.A.Parthasarathy2 and O.Senkov2 1 Air
Force Research Laboratory, Materials and Manufacturing Directorate, WPAFB, OH 45433 2 3
UES Inc, 4401 Dayton-Xenia Road, Dayton, OH 45432
Institute of Mechanical Engineering, EPFL, Lausanne 1015
Abstract In this work, molecular statics and molecular dynamics simulations of a/2<110> dislocation behavior for a model FCC Co30Fe16.67Ni36.67Ti16.67 alloy are discussed. It is shown that the single FCC phase is elastically stable in this alloy. Local stacking fault energies for the FCC alloy are determined as a function of average composition. The core structure of a/2<110> screw and edge dislocations in the FCC Co30Fe16.67Ni36.67Ti16.67 alloy is shown to be planar with significant variations in the Shockley partial splitting along the dislocation line (factor of ~3) due to concentration fluctuations. The correlation lengths for dislocation line fluctuations in this alloy are determined and discussed. The critical stress to move both a/2<110> screw and edge dislocations at 0K in the model FCC Co30Fe16.67Ni36.67Ti16.67 alloy is of the order of 0.0025 – 0.005, where is the (111) shear modulus, and is significantly higher than that of pure FCC Ni. Molecular dynamics simulation results on the critical stress to move a/2<110> screw and edge dislocations in the model FCC concentrated solid solution alloy show that it decreases with increasing temperature, similar to solid-solution strengthened FCC metals. These molecular dynamics simulation results are in reasonable agreement with experimental tensile yield strength data for an analogous FCC concentrated solid solution alloy. It is also shown that local fluctuations in the concentration of solutes has a strong effect on the effective cross-slip activation energy of screw dislocations in the random alloy. 1.0 Introduction Recent works have shown that when 5 or more elements are combined in nearly equiatomic concentrations, a unique class of materials termed high entropy alloys with unusually high strengths results [1-3]. High Entropy Alloys (HEA) derive their name from the idea that entropy of mixing may stabilize novel disordered phases. The number of phases in a HEA system can be significantly smaller than the maximum number of phases predicted by the well-
ACCEPTED MANUSCRIPT
known Gibbs phase rule [4]. While many alloys that meet the definition of an HEA [1,4,5] contain multiple phases, including intermetallics [6,7], there are some well characterized singlephase solid solution systems [6,8]. The other interesting behavior of these alloys is their unusually high yield strength while maintaining room temperature ductility. In FCC multicomponent alloys such behavior is thought to originate from dislocation –solute interactions [9]. We hypothesize that the high strength must have its origin in the core structure of the dislocations that form in these systems as well as their interaction with solutes. This hypothesis can be examined using atomistic simulations. Atomistic modeling of multicomponent alloys is challenging since it requires accurate interatomic potentials for all pairwise interacting elements. The modeling community has focused mainly on binary systems, (e.g. Ni-Al, Ti-Al, Al-Mg) with very detailed tuning of fitting parameters to achieve quantitative predictive capability. Some years ago a suite of potentials for 16 different metallic elements (Cu, Ag, Au, Ni, Pd, Pt, Al, Pb, Fe, Mo, Ta, W, Mg, Co, Ti, Zr) was developed by Johnson, Chou and co-workers [10]. They introduced a unified functional form for the interaction potentials between different species which opens the possibility of modeling complex compositions. Fit mainly to elemental properties, such potentials are not expected to give quantitative descriptions of the properties of multiphase components. However, the ability to generate model systems of arbitrary chemical complexity presents a huge opportunity to explore and discover mechanisms of deformation in systems where concepts developed for elemental and dilute binary alloys may not apply. The development of high entropy alloys remains Edisonian, relying mostly on insight and experimental search. Similarly the concept of building alloys from a periodic table using interatomic potentials has just begun [11], and remains largely untested for structural materials. While the former is of great engineering interest, the latter is of high scientific interest especially with respect to the Materials Genome Initiative. Previously, we have examined the behavior of a/2[111] screw and edge dislocations in a model BCC multicomponent alloy, Co16.67Fe36.67Ni16.67Ti30 [12]. It was shown that the core structure of a/2[111] screw dislocations is non-planar, similar to simple BCC metals. However, there were significant core structure fluctuations along the dislocation line in the multicomponent alloy, leading to much higher effective kink-pair activation energy and much shallower fall-off in critical stress with temperature, as compared to simple BCC metals and a mean-field version of
ACCEPTED MANUSCRIPT
the same alloy composition [12]. Also, the large static distortions around the Ti atoms in the multicomponent alloy resulted in a high critical stress for motion of the edge dislocation as opposed to simple BCC metals and a mean-field version of the same alloy composition, where the Peierl’s barrier of a/2[111] edge dislocations is very low, < 10MPa [12]. The screw/edge anisotropy of strength in the BCC random alloy was found to be much smaller (~1) than in a mean-field version of the same composition (~100). The atomistic approach was able to capture the major geometrical factors, such as atomic size and scale of local bonding that influence dislocation mobility in the BCC alloys. With this as proof of concept we turn our attention to the chemically complex FCC alloys. Here, we examine the behavior of a/2[110] screw and edge dislocations in a FCC multicomponent alloy, Co30Fe16.67Ni36.67Ti16.67. As before, we use the suite of EAM potentials developed by the Johnson and co-workers for 16 different metallic elements such that any combination of these elements can be handled in a computer simulation [10]. Here, we take advantage of these potentials to probe a model FCC HEA material precisely to uncover heretofore unknown mechanisms of plastic deformation. If this approach can be verified it offers a tremendous potential for alloy discovery. We hypothesize that this approach can be used in solid-solution alloys to understand the unusual mechanical behavior of FCC high entropy alloys and possibly lead to alloy discovery. 2.0 Computational model of the FCC Co30Fe16.67Ni36.67Ti16.67 random alloy A four element, Co-Fe-Ni-Ti, interatomic potential developed by Zhou et.al [10,13] was interfaced with the parallel molecular dynamics (MD) code, LAMMPS [14]. An atomistic simulation cell was generated (~ 2 million atoms) consisting of a FCC lattice with periodic boundary conditions along three orthogonal directions, [1-10], [110] and [001], with a cell size of approximately 300A along each side, and with an average lattice parameter of pure Ni, 3.52A. The cell was populated randomly with Co, Fe, Ni and Ti atoms with an average composition (in at. %) of 30Co:16.67Fe:36.67Ni:16.67Ti. The energy of the system was determined as a function of the average lattice parameter of the FCC cell and a minimum-energy lattice parameter of 3.65A was found for the random alloy. By straining the lattice, the elastic constants, C11, C12 and C44 were determined. The produced FCC lattice was elastically stable and the elastic constants were determined to be C11 = 166.9, C12=131.7 and C44 = 83.6 GPa.
ACCEPTED MANUSCRIPT
The shear elastic constants, (C11-C12)/2 and C44 were 17.6 and 83.6 GPa, respectively. The (111) shear modulus, , is [C44(C11-C12)/2]0.5 ~ 38.4 GPa. The EAM potential gives the chosen composition as highly elastically anisotropic, with an anisotropy ratio, 2C44/(C11-C12) of 4.75. The Voigt average shear modulus (3C44+C11-C12)/5 is 57.2 GPa. The Voigt average shear modulus of pure Ni, Fe, Co and Ti are 76, 82, 78 and 44 GPa, respectively [ 15]. Using a rule of mixtures, one obtains for the selected composition of 30Co, 16.67Fe, 36.67Ni and 16.67Ti, a Voigt average shear modulus of 72 GPa, which is noticeably higher than the atomistically measured value of 57.2 GPa. A second FCC simulation cell, with x, y and z axes along [1-10], [11-2], and [111] directions, and with dimensions of approximately 300A in each direction (~2 million atoms), was generated and relaxed using free surface boundary conditions along z and periodic boundary conditions along the x and y directions. The cell was populated randomly with Co, Fe, Ni and Ti atoms with an average composition (in at. %) of 30Co:16.67Fe:36.67Ni:16.67Ti. Next, the upper half of the simulation cell was translated with respect to the bottom half by the unstable extrinsic stacking fault (ESF) vector, stable intrinsic stacking fault vector (ISF) and ¼[1-10] vector on the (111) plane. For the intrinsic stacking fault vector translation, the whole cell was relaxed along all three directions. The intrinsic stacking fault was found to be stable with an average energy of 20 mJ/m2. For the other two gamma surface translations, atoms were freely relaxed along z. However, the x and y relaxations were performed by ensuring that the total force on all the atoms along the x and y directions to be zero. The energy of the unstable ESF and ¼[1-10] gamma surface translations were found to be 940 and 530 mJ/m2. Similar gamma surface calculations were performed using the Mishin potential for Ni [15]. The energy of the unstable ESF and ¼[1-10] gamma surface translations for Ni were found to be 1480 and 1030 mJ/m2. The Ni potential also gives a stable stacking fault energy of 131 mJ/m2. These gamma surface results suggest that the model Co30Fe16.67Ni36.67Ti16.67 FCC alloy has lower gamma surface fault energies as compared to pure Ni. This also shows that the FCC lattice is stable to large shear deformations on the (111) glide plane. The average composition of the random FCC alloy chosen in this manuscript is somewhat arbitrary. For these potentials, a 4-element composition with significant content of each element where the FCC lattice is elastically stable and stable with respect to large shear deformation on the (111) plane was searched for and obtained. The average composition was also similar to the composition of the matrix in several engineering superalloys. Due to the stability of
ACCEPTED MANUSCRIPT
the FCC lattice to Bain deformation, elastic stresses and large shear deformation on the (111) plane, a study of the properties of the <110> glide dislocations in the FCC lattice of the concentrated random alloy becomes feasible. Stacking fault energies were also determined as a function of average composition of the 2 million atom simulation cell. The average concentration of Co was varied from 0.2-0.5, Fe 0. 0 -0.4, Ni 0.2-0.7 and Ti 0.0-0.3. These composition ranges were picked by looking at local concentration fluctuations at the a/2<110> screw dislocation core to be discussed in the next section. The local composition was defined as the composition up to and including the first neighboring shell of an atom at the core. Figure.1 gives a plot of the stacking fault energy as a function of Ni content, for various Co, Fe and Ti concentrations. In general, increasing Co and Ti concentrations tended to decrease the stacking fault energy whereas increasing Fe concentration tended to increase the stacking fault energy. The stacking fault energy was approximately linear with Ni content, increasing with increasing Ni content. For the composition space studied, the stacking fault energy shows significant variations, ranging from ~ 0 – 70mJ/m2. These results suggest that in the model FCC alloy of interest, Co30Fe16.67Ni36.67Ti16.67, local concentration fluctuations could lead to significant local variations in stacking fault energy, which would have an impact on the local core structure of dislocations. The FCC lattice was unstable for average Ti compositions > 0.30. To study dislocation structure and dynamics, a ½[1-10] screw dislocation, lying along the x axis, was introduced into the simulation cell as two Shockley vectors separated by 5b along the [11-2] direction on the (111) plane using the anisotropic elasticity displacement field of the two Shockley partials. The simulation cell was 300A along x, [1-10], dislocation line direction, 1200A along y, [11-2], perpendicular to the dislocation line direction lying in the (111) glide plane and 300A along z, [111], glide plane normal. In total, the simulation had ~ 8 million atoms. Some simulations were run with 1200A along the x or dislocation line direction, with a total simulation of ~ 32 million atoms, to ensure that the periodic length along the dislocation line contains several units of correlation length for glide in the random alloy. All the cells were populated randomly with Co, Fe, Ni and Ti atoms with an average composition (in at. %) of 30Co:16.67Fe:36.67Ni:16.67Ti. Similarly, a ½[1-10] edge dislocation, lying along the y axis, was introduced into the simulation cell using the anisotropic elasticity displacement field. The
ACCEPTED MANUSCRIPT
simulation cell was 1200A along x, [1-10], perpendicular to the dislocation line direction lying in the [111] glide plane, 300A along y, [11-2], dislocation line direction and 300A along z, [111], glide plane normal, in total, ~ 8 million atoms. As before, some simulations were run with 1200A along the dislocation line direction, giving a simulation of ~ 32 million atoms. Initially, conjugate gradient energy minimization was performed on the initial atomic positions using periodic boundary conditions along the dislocation line direction and fixed boundary conditions along the other two directions. Molecular dynamics simulations were performed on the larger cells with 1200A length along the dislocation line direction, to determine the variation of critical stress for overcoming the resistance to motion of the dislocation as a function of temperature (5, 150 and 300K) in the random alloy. The initial conditions for the MD simulations were a molecular statics relaxed core under the applied stress of interest. Additional forces were applied at the boundary of the MD cell to mimic the state of applied stress. The system was permitted to relax under NVT conditions at various temperatures for 100ps. The position of the dislocation was then monitored as a function of time to follow its motion. A schematic of the simulation cell used in the molecular dynamics simulations is given in Fig.2. Note that portions of the a/2<110> dislocations (both Shockley partials) move a significant amount perpendicular to their line direction (~50A) before uninterrupted free movement in the lattice. As a result, the dimension of the cell perpendicular to the dislocation line direction needs to be large to avoid significant image stress effects from the free surfaces. Here, this dimension was kept at a large value of 1200A. In the random alloy, at a stress just below the critical stress for motion, portions of the dislocation (both Shockley partials) glide a distance of the order of 50A in a stick-slip fashion and come to an eventual stop. At a stress just above the critical stress for motion, the dislocation overcomes the lattice resistance and does not come to an eventual stop. The former stress is termed the lower bound of the critical stress for motion of the dislocation whereas the latter stress is termed the upper bound of the critical stress for motion of the dislocation. This behavior is shown later in the manuscript in Fig.5 for the movement of ½[1-10] edge dislocations in the model Co30Fe16.67Ni36.67Ti16.67 FCC alloy at temperatures of 5, 150 and 300K. For depiction of core structures in the FCC multicomponent concentrated solid solution alloy, we use the method developed by Stukowski and Albe [16] which extracts dislocation lines and their associated Burgers vectors from three-dimensional atomistic simulations (DXA, dislocation
ACCEPTED MANUSCRIPT
extraction algorithm). It is based on a fully automated Burgers circuit analysis, which locates dislocation cores and determines their Burgers vector. The transition from the atomistic system to a discrete dislocation representation is achieved through a subsequent vectorization step. Sophisticated information, including dislocation reactions and junctions, can be obtained from this analysis. Recent work has shown that, for various functional forms of interatomic potentials, an “average atom” (A-atom) interatomic potential can be rigorously constructed for any random alloy composed [17]. Most importantly, many properties of the true random alloy, such as lattice constant, elastic constants, and stacking fault energies, are accurately reproduced by the single Aatom potential. Thus, the A-atom interatomic potential is the mean-field representation of the true random alloy, capturing all properties that do not depend on local composition fluctuations in the material. We have created the A-atom potential for the Co30Fe16.67Ni36.67Ti16.67 alloy, and have verified that it has essentially the same reference properties as the true random alloy. The A-atom potential gives a fcc lattice parameter of 3.649A, elastic constants C11 = 175 GPa, C12 = 124 GPa, C44 = 90 GPa and = 48 GPa, and an intrinsic stacking fault energy, sf of 16 mJ/m2. These values compare favorably with the values of these parameters for the true random alloy of the average composition Co30Fe16.67Ni36.67Ti16.67. We have also used direct atomistic simulations to determine the volume change due to Co, Fe, Ni and Ti atoms in the average A-atom FCC lattice. One obtains V values of -1.00, -0.60, -1.47 and 5.16 A3 for the Co, Fe, Ni and Ti atoms, respectively. This suggests that the Ti atom is a very strong misfit strain center in the model fcc multicomponent alloy whereas Ni, Co and Fe atoms have lower misfit strain values. Therefore, strong lattice distortions are expected around the Ti atoms in the model FCC multicomponent alloy. We have also used this A-atom potential in simulations of both edge and screw dislocations using the geometries and methods described below. 3. Results 3.1 Core structure of a/2<110> dislocations in a model FCC Co30Fe16.67Ni36.67Ti16.67 alloy Figure 3 gives a DXA plot of the relaxed ½[1-10] screw dislocation core in a model Co30Fe16.67Ni36.67Ti16.67 concentrated solid solution FCC alloy. The screw dislocation shows a classical Shockley partial splitting along the dislocation core, similar to pure FCC metals. Also, the core is planar with the core atoms are confined to a single (111) glide plane. However, there
ACCEPTED MANUSCRIPT
are significant variations in the Shockley partial splitting along the dislocation line, so that the dislocation core structure changes significantly along the dislocation line. Pure shear stresses on the (111) glide plane were applied on the screw dislocation core by straining the atoms and using fixed boundary conditions along the y and z directions. The screw dislocation starts moving when applied shear stresses reach 0.0025 -0.00375 giving a critical stress for motion at 0K at least one order of magnitude higher compared to pure FCC metals. Figure 4 gives a DXA plot of the relaxed a/2[1-10] edge dislocation core in a model Co30Fe16.67Ni36.67Ti16.67 concentrated solid solution alloy. Similar to the screw core, the edge dislocation core is planar and shows significant fluctuations in Shockley partial splitting along the dislocation line. Pure shear stresses on the (111) glide plane were applied on the edge dislocation core by straining the atoms and using fixed boundary conditions along the x and z directions. The edge dislocation moves when the applied shear stresses reach 0.00375 0.005 giving a critical stress for motion at 0K at least one to two orders of magnitude higher as compared to pure fcc metals. Both a/2[110] screw and edge dislocations were planar in the mean-field alloy, dissociated into Shockley partials on the (111) plane, with constant splitting distances of 27 and 80A, respectively. The width of the Shockley partials belonging to the a/2[110] edge dislocation was estimated as ~ 15A in the mean-field alloy, from Burger’s vector distribution plots obtained from atomistic simulations. The critical stress for motion at 0K for both a/2[110] screw and edge dislocations in the mean-field alloy were very low, < 10 MPa, similar to simple fcc metals. 3.2 Critical resolved shear stress for the screw and edge dislocations and comparison with experiment Figures 5a, b and c give a (111) projection of atoms with a centrosymmetry parameter > 4 in the FCC random alloy with an a/2[1-10] edge dislocation at applied stresses of 0.00375, 0.004375 and 0.005 at 5K, 0.00375 and 0.004375 at 150K and applied stresses of 0.0025, 0.003125 and 0.00375 at 300K, for different molecular dynamics timesteps of 0, 25, 50 and 100ps. Such a plot shows atoms at the stacking fault region between the Shockley partials of the a/2[1-10] edge dislocation. The profiles at 0, 25, 50 and 100 ps have been shifted by -2, -1, 0, and 1 periodic units along the dislocation line along the ‘y’ direction for clarity of viewing. There are significant Shockley partial splitting variations along the dislocation line, as much as
ACCEPTED MANUSCRIPT
factors of 3, similar to variations seen in recent high resolution transmission electron microscopy observations of local Shockley partial splitting widths at an a/2[1-10] 600 dislocation in an analogous 5-element FCC multicomponent alloy, Cantor alloy [19]. The experimentally measured average stacking fault energy in the Cantor alloy is 25mJ/m2 [19], similar to the present multicomponent FCC alloy. The edge dislocation moves by continuous expansion and contraction of these local stacking fault splitting widths. Some portions of the dislocation line move a significant amount, of the order of 50A, below the critical stress. At a stress of 0.004375 at 5K, all portions of the edge dislocation overcome the lattice resistance to dislocation motion. Similarly, the edge dislocation overcomes the lattice resistance to motion at stresses of 0.00375 and 0.003125 at 150K and 300K, respectively. Figure 6 gives a variation of critical stress as a function of temperature obtained from the MD results, for both a/2<110> screw and edge dislocations. The upper and lower bounds of the critical stress are shown. The critical stress shows a fall-off with increasing temperature, similar to solid-solution strengthened fcc metals [9]. Also shown in Fig.4 are experimental results on yield [18] from a similar 5-component concentrated solid solution alloy, Co20Cr20Fe20Mn20Ni20. The experimental yield strength data was for polycrystalline materials with varying grain sizes. The experimental results were extrapolated to an infinite grain size and divided by a Taylor factor of 3 to obtain approximately the critical resolved shear stress for flow in single crystals of such an alloy. Such single crystal CRSS data were compared with the present MD simulation in Fig.6. Overall, the simulation data gives similar values of strength as a function of temperature as compared to experiment. Note that the experimental alloy composition is different from the composition used in these simulations and the comparison should be taken only as qualitative. 3.3 Cross-slip of screw dislocations Figures 7a and b give a (111) [y] ,(11-2) [z] , (11-1) [y] and (112) [z] projection of atoms with a centrosymmetry parameter > 4 in the FCC random alloy with an a/2[1-10] screw dislocation at an applied stress of 0.001875, which is slightly above the critical stress, for molecular dynamics timesteps of 0 and 100ps, at 300K. The profile at 100 ps has been shifted by 1 periodic unit along the dislocation line along the ‘x’ direction for clarity of viewing. At 0 ps, the a/2[1-10] screw dislocation is completely on the [111] glide plane, seen in Fig.7a in the [112] [z] projection as one atomic plane thick of atoms across the dislocation line. At 100ps, small
ACCEPTED MANUSCRIPT
lengths of the dislocation line have cross-slipped, which is clear from the spreading seen in the [11-2][z] projection in Fig.7a at portions of the dislocation line and in Fig.7b in the [112] [z] projection as one atomic plane thick of atoms across the dislocation line. The lengths of dislocation line that have cross-slipped is relatively small presumably due to the low applied shear stress on the cross-slip plane in the simulations ( ~1/3 the stress applied on the glide plane). This suggests that the effective cross-slip activation energy in this concentrated solid solution alloy Co30Fe16.67Ni36.67Ti16.67 is very low, of the order of 0.30 eV (12kT with T = 300K, and k is the Boltzmann constant). The average stacking fault energy, s, of the simulated alloy is 20 mJ/m2 (~ 50% of pure Cu) and the shear modulus, 40 GPa ( ~ equal to that of pure Cu). The atomistically calculated value for the cross-slip activation energy in pure Cu is ~ 1.6-1.8 eV [20,21]. The cross-slip activation energy in FCC metals is proportional to (d/b)[ln(d/b)]0.5 [22] where d is approximately equal to b2/16 [23] . Using the scaling factors given above, the cross-slip activation energy in the simulated FCC alloy based on average stacking fault energy and shear modulus parameters is estimated to be 3.8 – 4.3 eV. Also, using the mean-field potential, the cross-slip activation energy for the mean stacking fault energy and shear modulus parameters was also directly atomistically calculated using the procedure given in reference [21] as 4.6 – 4.9 eV. However, molecular dynamics simulations show that the effective cross-slip activation energy in the concentrated solid solution alloy, Co30Fe16.67Ni36.67Ti16.67 is very low, of the order of 0.3 eV. This suggests that local fluctuations in the concentration of solutes has a strong effect on the effective cross-slip activation energy akin to what has been suggested in the recent model for cross-slip activation energy in alloys given in reference [20]. 3.4 Correlation length for a/2<110> dislocation line fluctuations in a model FCC Ni36.67Co30Fe16.67Ti16.67 alloy Figure 8a is a plot of the shape profile in the (111) glide plane of the leading Shockley of an a/2<110> edge dislocation in a model FCC Co30Fe16.67Ni36.67Ti16.67 random alloy at a stress of 0.00375, slightly below the critical stress for motion of the edge dislocation at 0oK, obtained from DXA analysis. The dislocation line is not straight, and shows significant fluctuations along the ‘y’ direction under an applied stress overcoming varying solute-dislocation interactions along the dislocation line. The y-axis scale has been expanded relative to the x-axis scale to accentuate the fluctuations. The effective ‘x’ period of such fluctuations can be obtained from a one-
ACCEPTED MANUSCRIPT
dimensional correlation analysis of such fluctuations. The correlation function for a separation ‘x0’ can be defined as C(x0) =
[y(x)-
(1)
where
ACCEPTED MANUSCRIPT
fluctuations in this alloy is ~ 100 - 125A. Therefore, molecular dynamics simulations where the dislocation line length chosen is much greater than 100A (~1200A) should be representative of bulk behavior. Also, local fluctuations in the concentration of solutes has a strong effect on the effective cross-slip activation energy of screw dislocations in the random alloy, decreasing it from a value of 4.6 – 4.9 eV (atomistically calculated from average stacking fault energy and shear modulus parameters) to a value of ~ 0.3 eV. Acknowledgement The authors thank Dr. C. Varvenne, Mr. W. Noehring, and Prof. W. A. Curtin for technical assistance with the development and use of the mean-field interatomic potential presented in this work; their efforts were supported by the European Research Council through the Advanced Grant ``Predictive Computational Metallurgy'', ERC Grant agreement No.339081 - PreCoMet. The authors acknowledge use of the Molecular dynamics code, LAMMPS, which was developed at Sandia National Laboratory by Dr. Steve Plimpton and co-workers. This work was supported by the AFOSR, and by a grant of computer time from the DOD High Performance Computing Modernization Program, at the Aeronautical Systems Center/Major Shared Resource Center. Work by ONS and TAP was supported through the Air Force onsite contract No. FA8650-15-D-5230 managed by UES, Inc. References: 1) M.H. Tsai and J.W. Yeh, ‘High entropy alloys – A critical review’,Mater.Res.Lett., 2 (2014) 107. 2) Y. Zhang, T.T. Zuo, Z. Tang, M.C. Gao, K.A. Dahmen, P.K. Liaw and Z.P. Lu, ‘Microstructures and properties of high-entropy alloys’, Progress. Mater. Sci., 61 (2014) 1. 3) D.B. Miracle, J.D. Miller, O.N. Senkov, C. Woodward, M.D. Uchic and J. Tiley, ‘Exploration and development of high entropy alloys for structural applications’, Entropy, 16 (2014) 494. 4) S. Curtarolo, G.L.W. Hart, M.B. Nardelli, N. Mingo, S. Sanvito and O. Levy, ‘The high-
throughput highway to computational materials design’, Nature Materials, 12 (2013) 191. 5) B.S. Murty, J.W. Yeh and S.Ranganathan, ‘High Entropy Alloys’, ButterworthHeinemann, 2014.
ACCEPTED MANUSCRIPT
6) M. Poletti and L. Battezzati, ‘Electronic and thermodynamic criteria for the occurrence of
high entropy alloys in metallic systems’, Acta Mater., 75 (2014) 297. 7) J.W. Yeh, S.J. Lin, T.S. Chin, J.Y. Gan, S.K. Chen, T.T. Shun, C.H. Tsau and S.Y. Chou,
‘Formation of simple crystal structures in Cu-Co-Ni-Cr-Al-Fe-Ti-V alloys with multiprincipal metallic elements’, Metall.Mater.Trans., 35A (2004) 2533. 8) J.W. Yeh, S.K. Chen, S.J. Lin, J.Y. Gan, T.S. Chin, T.T. Shun, C.H. Tsau and S.Y. Chang,
‘Nanostructured high-entropy alloys with multiple principal elements – Novel alloy design concepts and outcomes’, Adv.Engr.Mater., 6 (2004) 299. 9) C. Varvenne, A. Luque and W.A. Curtin, ’Theory of strengthening in fcc high entropy alloys’, Acta Mater., 118 (2016) 164. 10) X.W. Zhou et.al., ‘Atomic scale structure of sputtered metal multilayers’, Acta Mater., 49
(2001) 4005. 11) F.C. Campbell, ‘Phase Diagrams – Understanding the Basics’, ASM International, 2012. 12) S.Rao, C. Varvenne, C. Woodward, T.A. Parthasarathy, D. Miracle, O.N. Senkov and W.A. Curtin, ‘Atomistic Simulations of Dislocations in a Model BCC Multicomponent Concentrated Solid Solution Alloy’ Acta Mater., 125 (2017) 311. 13) www.ctcms.nist.gov/potentials/ 14) S.J. Plimpton, ‘ Fast parallel algorithms for short-range molecular dynamics’, J. Comput.
Phys., 117 (1995) 1. 15) G. Simmons and H. Wang, ‘Single crystal elastic constants – A Handbook 2nd edition’, MIT Press, 1971. 16) A. Stukowski and K. Albe, ‘Structure identification methods for atomistic simulations of crystalline materials’, Model.Simul.Mater.Sci.Eng., 18 (2010) 085001. 17) C. Varvenne, A. Luque, W.G. Nöhring and W.A. Curtin, ‘Average-atom interatomic potential for random alloys’, Phys. Rev. B 93 (2016) 104201. 18) F.Otto, A. Dlouhy, Ch. Somsen, H. Bei, G. Eggeler and E.P. George, ‘The influence of temperature and microstructure on the tensile properties of a CoCrFeMnNi high-entropy alloy’, Acta. Mater., 61 (2013) 5743. 19) T.M. Smith, M.S. Hooshmand, B.D. Esser, F. Otto, D.W. Mccomb, E.P. George, M. Ghazisaeidi and M.J. Mills, Acta Materialia 110 (2016) 352. 20) W.Goehring and W.A. Curtin, submitted for publication in Acta Materialia.
ACCEPTED MANUSCRIPT
21) S.I. Rao, D.M. Dimiduk, T.A. Parthasarathy, J. El-Awady, C. Woodward and M.D. Uchic, ‘Calculations of intersection cross-slip activation energies in fcc metals using the nudged elastic band method’, Acta Materialia 59 (2011) 7135. 22) S.I. Rao, T.A. Parthasarathy and C. Woodward, ‘Atomistic simulation of cross-slip processes in model fcc structures’, Philosophical Magazine A 79 (1999) 1167. 23) M. Chassagne, M. Legros and D. Rodney, ‘Atomic-scale simulation of screw dislocation / coherent twin boundary interaction in Al, Au, Cu and Ni’, Acta Materialia 59 (2011) 1456.
Figure captions: Fig.1: Stacking fault energy, sf on the (111) plane as a function of Ni concentration, XNi, in Co, Fe, Ni, Ti based multicomponent random FCC solid solution alloys. Fig.2: A schematic illustration of the simulation cell used in molecular dynamics simulations of ½[1-10] screw and edge dislocations in a model FCC Co30Fe16.67Ni36.67Ti16.67 alloy. For the screw dislocation, periodic boundary conditions are applied along the ‘x’ direction and free surface boundary conditions along the ‘y’ and ‘z’ directions. For the edge dislocation, periodic boundary conditions are applied along the ‘y’ direction and free surface boundary conditions along the ‘x’ and ‘z’ directions. Additional forces are applied on top and bottom boundary surface layers ~ 12A thick along the ‘z’ direction to model the applied stress. Fig.3: DXA plot of relaxed ½[1-10] screw dislocation core in model FCC Co30Fe16.67Ni36.67Ti16.67 alloy. The length of the dislocation line is 300A. Fig.4: DXA plot of relaxed ½[1-10] edge dislocation core in model FCC Co30Fe16.67Ni36.67Ti16.67 alloy. The length of the dislocation line is 300A. Fig.5: Structure of a/2[1-10] edge dislocation core under a) applied stresses of 0.00375, 0.004375 and 0.005 at 5K, b) applied stresses of 0.00375 and 0.004375 at 150K, and c) applied stresses of 0.0025, 0.003125 and 0.00375 at 300K in a FCC Co30Fe16.67Ni36.67Ti30 alloy. The core structure is shown for molecular dynamics steps of 0, 25, 50 and 100 ps. Atoms with a centro-symmetry parameter greater than 4 is shown in the plot. For clarity, core structure at time-
ACCEPTED MANUSCRIPT
steps of 0, 25, 50 and 100 ps are displaced by -2, -1, 0 and 1 periodic units, respectively, along the ‘y’ direction.
Fig. 6: Measured critical resolved shear stress scaled by the (111) shear modulus (39 GPa) necessary to achieve on-going glide as a function of temperature, for the a/2[1-10] screw and edge dislocations in the model FCC Co30Fe16.67Ni36.67Ti30 alloy. The upper and lower bounds of the critical resolved shear stress are shown in the plot. Also shown in Figure 4 is the experimentally measured strength for the 5-element Co20Cr20Fe20Mn20Ni20 multicomponent Cantor alloy. Fig.7: Structure of a/2[1-10] screw dislocation core under an applied stress of 0.001875 in a FCC Co30Fe16.67Ni36.67Ti30 alloy at 300K. The core structure is shown for molecular dynamics steps of 0 and 100 ps. Atoms with a centro-symmetry parameter greater than 4 are shown in the plot. For clarity, core structure at a time-step of 100ps is displaced by 1 periodic unit along the ‘x’ direction. The (a) (111) [y] and (11-2) [z] and b) (11-1) [y] and (112) [z] projections are shown. Fig.8: DXA plot of the shape profile in the (111) glide plane of the leading Shockley of an a/2<110> edge dislocation in a model FCC Co30Fe16.67Ni36.67Ti16.67 random alloy at a stress of 0.00375, slightly below the critical stress for motion of the edge dislocation at 0K with both different scaling for the x and y-axes and with identical scaling for the x and y-axes. b) Plot of the one-dimensional correlation function obtained for the a/2[1-10] edge dislocation profile shown in (a) as a function of distance along dislocation line.
ACCEPTED MANUSCRIPT
Fig.1:
ACCEPTED MANUSCRIPT
Fig.2:
ACCEPTED MANUSCRIPT
Fig.3:
ACCEPTED MANUSCRIPT
Fig.4:
ACCEPTED MANUSCRIPT
(a) 5K
(b) 150K
(c) 300K
Fig.5:
ACCEPTED MANUSCRIPT
Fig.6:
ACCEPTED MANUSCRIPT
(a)
(b) Fig.7
ACCEPTED MANUSCRIPT
(a)
(b)
Fig.8: