Atomistic modeling of idealized equal channel angular pressing process

Severe plastic deformation (SPD) processes have established themselves as prominent forming techniques for producing materials with very high strengths [1, 2]. Typically in any SPD process, large plastic strains are applied on the feedstock in a single pass of the process. The feedstock is subjected to repeated multiple passes, thus allowing for further accumulation of the large plastic strains. As a result, pronounced grain refinement occurs, with grain sizes in the ultra-fine grained (\(<\,1\,\upmu\)m) and nanocrystalline (\(<\,100\) nm) regime. The decreased grain size results in a significant increase in strength—by up to a factor of 8—and is attributed to the Hall–Petch effect [3]. Generally, any increase in strength is offset by a loss in ductility, often presented as the strength-ductility paradox [4]. Nonetheless, a few studies have indeed reported enhanced ductility in SPD processed materials [5‐8]. This combination of high strength and increased ductility in the processed material makes SPD processes very interesting and attractive for many applications.

A proper description of the enhanced mechanical properties of SPD processed materials requires a fundamental understanding of the grain refinement mechanisms leading to ultra-fine grained (UFG) or nanocrystalline (NC) microstructures, and the complex interactions at play in such microstructures. However, as will be evident by the discussion below, details of such microstructure level processes are still missing. Furthermore, atomistic simulation studies that can provide a fundamental understanding of such critical mechanisms have been rare.

Experimental studies have tried to illuminate the deformation and grain refinement mechanisms during SPD. Investigations on single crystals of Al [9‐11], Cu [12‐15] and Ni [16] have focused on the influence of orientation on the deformation mechanisms and texture evolution. Further studies on polycrystals have tried to elucidate the influence of stacking fault energy (SFE) on the mechanisms [17]. In high and medium SFE materials like Al and Ni, dislocations interact, intersect and tangle with each other to form sub-structures such as cells and cell blocks [18, 19]. With increasing deformation, it is expected that the misorientation between neighboring cells increases. Two types of boundaries are discussed in this regard: geometrically necessary boundaries (GNBs) and incidental dislocation boundaries (IDBs). While GNBs manifest in transmission electron microscopy (TEM) images as parallel extended lamellar structures and are known to form between regions of different strain patterns, IDBs are randomly oriented structures that are thought to form by random trapping of dislocations. Such lamellar boundaries are observed not only in Al [20, 21] and Ni [22] but also in low SFE materials like Cu [23, 24] and Cu-Al alloys [25, 26].

In materials with low SFE, it is expected that deformation twinning plays an important role in grain refinement [17]. Twin boundaries (TBs) act as effective barriers to dislocation motion, forcing pile-up of dislocations and facilitate dislocation TB interactions. The latter ostensibly transform coherent TBs to incoherent TBs and subsequently to high angle grain boundaries. Further emission of partial dislocations result in the formation of secondary twins whose interaction with themselves and grain boundaries (GB), together with nanoscale grain rotation results in the formation of the refined microstructure [27]. Cao et al. [28] suggest a different mechanism in fcc austenite phase of a duplex steel, where grain refinement is still driven by dislocation twin interactions, but via detwinning of the primary twin.

However, finer details on the mechanisms are missing. For example, it is unclear what the characteristics of the cell wall boundaries are, the mechanism that results in cell boundaries transforming into large angle grain boundaries, and if there is a limiting mechanism for the formation of such cells. It is also an open question if grain refinement indeed extends to the nanoscale, i.e., would grain refinement be observable if the initial characteristic length of the sample is already in the nanometer regime.

It is hence evident that the deformation mechanisms during SPD are far from being well-understood and a clear picture is still lacking. Atomistic simulations are ideally placed in this regard. The atomic resolution provided by these techniques allows for tracking the trajectory of individual atoms and has led to unprecedented insights into mechanical behavior of materials [29, 30], leading to the term computational microscopy [31]. Previous studies have been successful in elucidating mechanisms of deformation in NC materials. For instance, it is now well accepted that GBs act as sources and sinks for dislocations in the absence of dislocation generating sources inside grains due to the small grain sizes [32, 33]. Non-equilibrium GBs that can be found in SPD processed materials enhance this effect due to lower activation energy barriers, and furthermore, exhibit reduced resistance to GB sliding and migration [34]. However, rarely have atomistic simulations been used to study SPD processes, or even, deformation behavior at large strains. A recent atomistic study of SPD has raised the possibility GNBs being associated with disclinations [35].

In this work, we perform a large scale atomistic simulations study of the equal channel angular pressing (ECAP) process, which is perhaps the most widely studied SPD process [36, 37]. In ECAP, a billet is passed through a die consisting of two channels which are equal in cross section and meet at an angle at the center of the die (see Fig. 1a). When the billet is pressed through the die, it is subjected to simple shear when it passes through the plane where the two channels meet [38, 39]. As a result, the ECAP process is often idealized via simple shear boundary conditions in modeling studies, and is also the choice of boundary conditions in the current study.

Of particular focus in the current study is the SFE of the material, which is described by the generalized SFE curve of the interatomic potential used. Three different embedded atom method (EAM) interatomic potentials, which have been parameterized for basic material properties of Al, Ni and Cu, are chosen to model high, medium and low SFE materials. The potentials provide a relatively accurate description of both the unstable \(\gamma _\textrm{usf}\) and stable \(\gamma _\textrm{ssf}\) stacking fault energies compared to density functional theory calculations. The study incorporates large computation cells allowing for dislocation loops of significant sizes to form during the simulations. A single pass of ECAP is modeled by subjecting the simulation cell to 200% shear.

Single crystalline samples used in the current work are generated with a native cube orientation of \(\left\langle {100} \right\rangle\) and \(\left\langle {010} \right\rangle\) parallel to the x and y directions of the simulation box [40]. The lattice constants used correspond to that predicted by the three different interatomic potentials used for the simulations. The samples are then relaxed using the FIRE algorithm [41] in standard molecular statics simulations to a force norm of \(10^{-8}\) eV/Å. Periodic boundary conditions are imposed along all three directions of the simulation box. Two different sample sizes—with 50 and 100 nm side lengths—are used in the study.

The relaxed structures are then equilibrated at 300 K for 20 ps with the microcanonical ensemble in molecular dynamics (MD), and subsequently thermalized at 300 K for 80 ps with the npt ensemble using a Nosé–Hoover thermo-barostat [42] to ensure zero stresses in all box directions. These stress-free structures are then subjected to simple shear at a constant strain rate \(\dot{\epsilon }_{xy}\) with the npt ensemble (see Fig. 1b). Simple shear boundary conditions are used to mimic ideal ECAP deformation. Samples are subjected to two different strain rates of \(\dot{\epsilon }_{xy}=2 \times 10^8\) s\(^{-1}\) and \(\dot{\epsilon }_{xy}=2 \times 10^9\) s\(^{-1}\) in the current study. The samples are deformed to a total shear of 200% to reflect the strain after a single ECAP pass. A stable time step of 2 fs is used in all simulations. The interatomic interactions are modeled via the embedded atom method (EAM) potentials of Mishin et al. [43] for Al and Ni, and Mishin et al. [44] for Cu. All simulations are performed using the atomistic simulation code LAMMPS [45].

The simulations are analysed by writing out snapshots 0.05% strain increment. The open source visualization tool Ovito [46] is used for visualization. Defect structures are identified using the common neighbor analysis (CNA) [47] and polyhedral template matching (PTM) [48] algorithms. Dislocations are characterized by the dislocation extraction algorithm [49] in Ovito, and the interactions are analyzed using the Thompson tetrahedron notation [50].

In the current work, large scale atomistic simulations have been performed to study the deformation behavior and grain refinement of single crystalline samples subjected to idealized ECAP conditions. Specifically we seek answers to the following questions: (a) Is grain refinement observed during ECAP if the initial size of samples is in the nanometer regime?, and (b) What are the deformation and grain refinement mechanisms in such samples when deformed to large plastic strains?.

Of particular focus, in this study, has been the influence of SFE of the material on the mechanisms. Three different materials—Al, Ni and Cu—have been used in the current study. All three materials display the formation of twins, albeit to different amounts, during the initial stages of deformation. At the atomic scale, the SFE alone is insufficient to assess the formation of twins. The relative ease of twinning can be better explained by the generalized planar fault energy curves as defined by the interatomic potential of the material [53]. Specifically, the competition between the two ratios \(\kappa _1 = \gamma _{\textsc {ssf}}/\gamma _{\textsc {usf}}\) and \(\kappa _2 = \gamma _{\textsc {ssf}}/\gamma _{\textsc {utf}}\), where \(\gamma _{\textsc {ssf}}\) and \(\gamma _{\textsc {usf}}\) are the stable and unstable stacking fault energies, and \(\gamma _{\textsc {utf}}\) is the unstable twin fault energy, determines the ability to twin. If the two ratios are similar, the probability of twinning is high. A higher \(\kappa _1\), on the other hand, suggests that the barrier for nucleation of the trailing partial dislocation is low; thus the motion of full dislocations is preferred over twinning [54].

For the interatomic potentials used in the current work [43, 44], the values for the two ratios are \(\kappa _1^{Al}=0.87\), \(\kappa _1^{Ni}=0.34\), \(\kappa _1^{Cu}=0.28\) and \(\kappa _2^{Al}=0.65\), \(\kappa _2^{Ni}=0.66\), \(\kappa _2^{Cu}=0.66\). It is hence expected that the likelihood of twinning is greater in Cu when compared to Ni and Al. Further factors, like e.g., \(\gamma _{\textsc {ssf}}/\mu b\), where \(\mu\) is the shear modulus and b is the magnitude of the Burgers vector, have previously been used to rationalize dislocation absorption and slip transmission at GBs [55‐57]. The numerical values for the aforementioned factor are \(3\times 10^{-2}, 1\times 10^{-2}\) and \(7\times 10^{-3}\) for Al, Ni and Cu, respectively, resulting in the same conclusion as with the ratios \(\kappa _1\) and \(\kappa _2\): Twinning is more likely in Cu as compared to Ni and Al.

In the current work, however, we see twins not only in Cu and Ni, but also in Al with the highest amount of twinning in the latter. Furthermore, these twins are nucleated after considerable elastic straining (approx. 13% applied shear) of the initial cube shaped specimen. This suggests that the fault energies computed at 0 K in a strain-free configuration may by insufficient to completely explain the occurrence of twins in a significantly strained sample. Computation of strain dependent fault energy curves for both twins and stacking faults may be required to provide a clearer picture of twin nucleation. Additional factors like local stresses and activation volume are also likely to influence twinning.

Despite the differences in the amount of twinning, the mechanism itself is the same in all cases. Twinning occurs when a series of partial dislocations nucleate on adjacent atomic planes. A mono-layer twin is created when a partial dislocation propagates on the plane adjacent to an existing stacking fault. With the passage of every additional partial dislocation on an adjacent plane, the twin widens. On the basis of the Burgers vector of the individual partial dislocations, two mechanisms have been identified for the formation of twins: monotonic activation of partials (MAP) where the partial dislocations have the same Burgers vector, and random activation of partials (RAP) where the partial dislocations have different Burgers vectors [58]. In polycrystalline materials, MAP is the accepted mechanism in coarse-grained materials, while RAP has been implied as the more plausible mechanism in nanocrystalline materials since a smaller shear needs to be accommodated [59].

Twins in all samples used in the current work form initially via the MAP mechanism. Once a thickness of at least 3 atomic layers has been achieved, the RAP mechanism seemingly takes over. The large shear created by the MAP mechanism can only be accommodated to a certain extent in the sample. Further compatibility of the twin with the applied boundary conditions necessitates the activation of the RAP mechanism, which requires a lesser amount of shear to be accommodated in comparison to the MAP mechanism.

An additional plausible reason for the lack of identification of any clusters with the Orisodata algorithm lies in the methodology used to calculate the local orientation, which forms the basis for delimiting regions as clusters or grains. In Orisodata, the orientation is obtained via the PTM algorithm, which computes the orientation of an atom by identifying the local neighborhood of each atom and comparing it with templates of local structural types [48]. Only the local structural environment is considered in this calculation. Previous history, particularly plastic events, do not play a role in such a calculation. As a result, the orientation of a region after the traversal of a dislocation is restored to that before the motion of the dislocation. Consequently, the calculated misorientation between regions decreases, resulting in fewer clusters being identified by Orisodata. This plausibility was also verified with the grain segmentation algorithms in Ovito. Both the graph clustering and minimum spanning tree algorithms, which also use the PTM algorithm for obtaining the orientation of atoms, also failed to recognize any clusters in snapshots at increased strain.

One method to overcome this limitation is to use continuum metrics like deformation gradient to obtain the orientation [60‐62]. Such metrics require the local neighborhood in the current configuration to be mapped to a reference configuration. As a result, the deformation history is automatically accounted for. The change in orientation with respect to the reference configuration can then be obtained by the polar decomposition of the deformation gradient \(\varvec{ F }= \varvec{ R }\varvec{ U }\), where \(\varvec{ R }\) and \(\varvec{ U }\) are the rotation tensors and the stretch tensors, respectively. Figure 7 shows the difference in orientations computed via PTM and the rotation tensor \(\varvec{ R }\). With the PTM algorithm, two distinct orientations are identified corresponding to the original cube orientation and the orientation of the twinned region. The usage of the rotation tensor clearly show a more heterogeneous distribution of orientations in the untwinned regions. Nonetheless, identifying such regions will require careful calibration of the available parameters, particularly the minimum grain size, within Orisodata or other grain segmentation algorithms, and is reserved for another study.

A primary motivation for the current work has been to investigate the atomic level details of deformation and grain refinement mechanisms during ECAP. Perhaps the most interesting of mechanisms found in the current study, in addition to the mechanism of twinning, is the formation of SFT. These are three dimensional defects in the shape of a tetrahedron consisting of six stair-rod dislocations along the edges of the tetrahedron bounded by triangular stacking faults on four equivalent 111 planes. Such defects were first observed in quenched gold by Silcox & Hirsch [63] who suggested that the tetrahedra were formed by the clustering of quenched-in vacancies to produce a Frank dislocation loop with a Burgers vector \(\varvec{ b }= 1/3 \left\langle {111} \right\rangle\), which then dissociates along the three \(\left\langle {110} \right\rangle\) directions in the 111 plane of the loop. This mechanism, also known as the Silcox-Hirsch mechanism, is one of the three mechanisms that have been proposed for the formation of SFT [64]. The second mechanism, also involving vacancies, has been proposed to explain the occurrence of SFT in high stacking fault energy materials like Al [65]; in situ experiments at very high strain rates showed a significant number of SFT forming, but no dislocations were found to propagate. As a result, it was concluded that isolated vacancies and complex vacancy arrays are involved in SFT production. Atomistic simulations have also provided evidence that SFT are among many major defects that may form under such conditions [66, 67].

The third mechanism proposes the formation of SFT without the involvement of vacancies, but with dislocation glide and cross slip, and has been used to explain the formation of SFT in a Ni-based superalloy after low cycle fatigue [68]. In this mechanism, dislocations interact to form a Frank loop which then follows the Silcox-Hirsch mechanism to produce an SFT. The mechanism of the formation of a triangular Frank loop is only deemed feasible if dislocation motion is able to reverse its glide direction, as is possible under fatigue loading conditions. Wang et al. [69] propose an alternative method that does not involve the Slicox–Hirsch mechanism. They suggest a sequence of dislocation interactions and dissociation that may finally lead to the formation of the SFT. These dislocation reactions were critically reviewed and questioned by Loretto et al. [64] who show that such dislocation reactions are energetically unfavorable compared to the Frank loop.

The novel mechanism observed in the current work is different from the previously observed mechanisms in literature. Firstly, the mechanism does not involve the collapse of vacancies. Secondly, and more importantly, the mechanism does involve the formation of a Frank loop. The observed mechanism is driven by partial dislocations forming stair-rods on three edges of the SFT, similar to the mechanism found by Wang et al. [69]. Note that if the trailing partial dislocations on the three faces were to move to the apex of the tetrahedron, the stacking fault and the stair-rod dislocations would be removed. The stacking faults, however, remain stable resulting in the trailing partial dislocations forming the base of the tetrahedron. The fourth face is now closed by a full dislocation whose partials form stair-rods with the trailing partial dislocations of three faces of the SFT. It must be noted that such reactions seem only possible due to the homogeneous dislocation motion on all four 111 planes. Further details on the dislocation reactions, energetics of dislocation dissociation and SFT formation will be reported elsewhere.

The SFT are highly stable defect structures that can considerably influence mechanical properties [70, 71]. In general, the stair-rod dislocations and Lomer–Cottrell locks that form such SFT are sessile structures, and act as obstacles to the motion of other dislocations. In the current work, we observe that beyond 150% applied shear a significant number of SFT have accumulated resulting in a slight hardening in the stress response (cf. Fig. 2). Furthermore, at low angle boundaries between clusters identified by the Orisodata algorithm, a considerable number of stair-rod dislocations and Lomer–Cotrell locks can be found (cf. Fig. 5c). At higher strains, although clusters could not be identified with Orisodata, dislocation structures identified by Ovito clearly show a network of dislocations of which SFT and stair-rod dislocations form a considerable portion (see Supplementary Fig. 10).

A few comments must be made at this juncture on the idealizations used in the current work, and in general, on the limitations of atomistic simulations. ECAP is modeled here by imposing simple shear boundary conditions. Friction between the billet and die walls, and the complex process parameters that form a part of the real process are simply neglected here. The idealized boundary conditions, however, allow for the usage of simulation cells of 100 nm side lengths, which are considerably large for typical atomistic simulations. The large strains of 200 % required to simulate the process result in computing times of approximately 2.5 Mio cpu hours per sample and simulation. The sample sizes used in the current work essentially denote a compromise between the maximum size usable under the limitations of computing time and the minimum size required for grain refinement to be observed. The single crystalline samples provide the best conditions—in terms of initial size—for grain refinement, and allow for delineating the influence of grain boundaries on the mechanisms if polycrystalline samples were to be used. Both the sample sizes used in the current work are well above the size where a transition of deformation mechanisms with decreasing characteristic size at the nanoscale is observed [72]. Furthermore, the observed mechanisms can be explained solely based on arguments of crystallography, stress state and dislocation theory. It may hence be concluded that the mechanisms hold importance for the experimental situation too, although the observations in the current work have been in nanoscale specimens.

Atomistic simulations of the molecular dynamics kind typically require the usage of very high strain rates that are necessitated by the notorious time limit of MD simulations. In the current work, strain rates of \(10^8\) s\(^{-1}\) and \(10^9\) s\(^{-1}\) have been employed, which are orders of magnitude higher than those used in the real ECAP process. Such high strain rates are a significant constraint in modeling thermally activated processes, and can lead to considerably different quantitative measures like peak and flow stresses. But they do not seem to have much influence on the mechanisms; the reported mechanisms are fairly robust at least in the regime of the two strain rates used in the current study. All mechanisms reported here are associated with dislocation mediated plasticity. Even though the strain rates are high, the overall simulated time is much longer than characteristic time scales of dislocation motion. For instance, it takes only a few picoseconds for a dislocation to move from one obstacle to another or to adjust its velocity in response to a change in stress. By contrast, the total time of the simulated trajectory is in the range of 10 ns, which provides ample time for the underlying dynamics of dislocation behavior. It has been previously suggested that even if flow stress, dislocation density and other rate-dependent measures are likely to be different, the mechanisms themselves can be deemed representative even in the low strain rate regime [31, 73, 74]. In this regard, computational experiments as reported here can be regarded just as quasi-static with respect to the underlying dynamics of dislocation behavior as the comparatively low-rate of straining in laboratory-scale ECAP processes.

Notwithstanding the limitations of the simulation technique, the results of the current work offer additional mechanistic insights into deformation and grain refinement during ECAP, albeit in an idealized version of the process. The formation of large number of stair-rod dislocations and stacking fault tetrahedra is seemingly inherent to severe plastic deformation of materials. Low angle grain boundaries formed during the process contain a significant number of such stair-rod dislocations and SFT.

It is hence evident that atomistic simulations are extremely useful in providing valuable insights into deformation and grain refinement mechanisms during SPD. These computations are particularly attractive since they offer spatial and temporal resolution that is generally not available to experiments [30, 75], and are rightly referred to as computational microscopy [31]. Nevertheless, there are clear reasons as to why such computations have rarely been used to study SPD processes. A single simulation with the 100 nm sample size results in approximately 100 gigabytes of overall processed data, and as mentioned before, requires approximately 2.5 million cpu hours of computing time. Incorporation of larger samples is not merely a question of computing time since modern MD codes scale very well on high performance computing machines. But even with larger simulation cells, the sample dimensions are still likely to be much smaller than the millimeter size specimens used in typical laboratory-scale ECAP processes. The large amount of data generated from such simulations, however, precludes the usage of manual techniques that are currently used to analyze dislocation reactions or to obtain critical stresses. Moving further, significant progress is required on the data curation, analysis and handling front to make such computations amenable for simulating large strains and multiple passes of an SPD process [76]. Automated analysis methods to supplant manual techniques and better algorithms to identify orientations at the atomic scale are necessary to obtain a better picture of microstructure evolution during SPD.

However, we must admit that the limitations of MD simulations pose strong restrictions on expanding our understanding of SPD processes. Simulations with simplified setups as used here can inhibit important mechanisms that might occur in the real world processes [29, 77]. High purity samples without defects or initial dislocation densities are used, a condition rarely satisfied in experimental situations. True rate limiting processes or thermally activated processes are extremely difficult to determine with these computations. The extrapolation of the knowledge to experiments requires a fundamental understanding of the temporal and spatial time scales accessible to such simulations. Since a direct comparison with experiments is seldom possible, the results from simulations must generally be viewed with caution. A complete picture of the deformation mechanism map cannot be obtained by simulations alone. In this regard, such computations must be seen as complementary to experiments; the tractability of the former can be improved with increased synergy with the latter.

The current study has implications not only for understanding the deformation and grain refinement mechanisms during SPD, but for modeling and simulation in general, both for atomic and continuum scale simulations. The usefulness of atomic scale modeling has been discussed above. Continuum scale simulations have been previously used to study various aspects of SPD [78], e.g., process modeling [79, 80], texture and microstructure evolution [81‐83], grain refinement [84], heterogeneity of stress and strain [85, 86], etc. Such modeling approaches, however, rely on microstructure related information from lower length scales [62]. For instance, crystal plasticity formulations at the mesoscale require details on the deformation mechanisms governing plasticity, as well as quantitative information like resolved shear stresses driving individual mechanisms. Such information—both qualitative and quantitative—can be provided by atomistic simulations to improve the tractability of simulations at the continuum scale. On the other hand, continuum scale simulations can be used to provide feedback in terms of realistic boundary and initial conditions that may not be directly realizable in atomistic simulations [77, 87], increasing the synergy between the modeling approaches.

In this work, large scale atomistic simulations of idealized ECAP process have been performed to study the deformation and grain refinement mechanisms during the process. To this end, simulation cells of two different side lengths—50 and 100 nm—with a native cube orientation and periodic boundary conditions are deformed to 200% under simple shear boundary conditions to mimic ideal ECAP conditions.

The findings of the study can be summarized as follows:These results mark the first steps in illuminating atomic scale details of deformation and grain refinement mechanisms during SPD processes. They also provide valuable pointers for future work, which can include realistic initial and boundary conditions to induce strain gradients as well as hardening, providing a possibility of direct comparison with experiments.