Understanding the interaction between heterogeneous precipitates and grain boundaries (GBs) is of great significance for tailoring the stability and mechanical properties of nanograined materials. In this work, the nanoscale interaction between the cylindrical precipitate and the migrating GB is investigated by atomic simulation. The results show that the resistance for GB migration can be increased by decreasing the direction angle \(\alpha\) (the angle between the axis of the precipitate and the tilt axis of GB). For the larger precipitate, the influence of direction angle is more pronounced. With the increase in shear strain, the interaction between the specific precipitate and GB changes the material deformation mechanism from “GB migration” to “GB migration accompanied with activated dislocations or GB deformation,” which contributes to the softening of the material. By simultaneously tuning the direction angle and size of heterogeneous precipitates, the GB deformation can be strongly inhibited and the stability of GBs can be significantly improved.
Notes
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Introduction
Nanocrystalline materials (NCMs) exhibit ultra-high strength owing to the high-density grain boundaries (GBs) [1‐3]. However, the wide application of NCMs is hampered by their weak thermal/mechanical stability [4‐7] due to the activation of GB-mediated deformations [8], such as nanograin rotation [5] and GB migration [7]. The motion of GBs always results in nanograin coarsening and strain softening, which weakens the mechanical properties of NCMs. Among all the GB-mediated deformation mechanisms, GB migration has been identified as the domination [7]. Therefore, controlling the GB migration is the core of improving NCM stability.
Alloying has been suggested as the most effective method to inhibit GB migration [6, 9]. On the one hand, GB segregation reduces the GB free energy, which decreases the capillary driving force for the migration of curved GBs [10, 11]. On the other hand, the solute drag [12] and precipitate pinning [13] effects produce significant resistance to GB migration. Recent studies show that nanoscale precipitates can strongly inhibit GB migration and prevent further nanograin growth at elevated temperatures [14‐16]. The inhibition from precipitates can be understood in part by several ideal models of spherical precipitates [12, 13, 17]. However, there are kinds of precipitates with different shapes in actual materials [18‐20] and their effects on microstructure evolution and material properties are different. A typical example is Mg alloy, for which the Orowan strength [21] and twinning extension [22] are sensitive to the shape of precipitates. Hence, understanding the precipitate-shape-dependent GB stability is also significant for tailoring the nanograin stability and mechanical properties of NCMs.
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Atomic simulation has been widely used in the field of material science and has demonstrated the capability to reconstruct the nanoscale microstructure evolution [7, 22‐25]. Extensive studies have successfully simulated the interactions between second-phase particles and other microstructures, including dislocations and grain boundaries. These works provide abundant information and are of great help in understanding the microstructure and properties of materials. The interaction between dislocations and precipitates in various alloy systems has been investigated by atomic simulation, which is helpful to understand the precipitate strengthening effect [26‐28]. From the simulations on GB motions [7, 23], such as GB sliding and migration, the underlying mechanisms behind the GB-related phenomenon, including the nanograin growth and the softening of NCMs, are revealed. Recent simulations have successfully reconstructed the interactions between moving GBs and other microstructures, which revealed the effect of irradiation defects on GB migration [29, 30] and the influence of precipitates on GB sliding [31].
In this paper, we aim to study the atomic-scale interaction between GBs and cylindrical precipitates using atomic simulation. The GB stability and deformation mechanism are considered by tuning the direction and size of the cylindrical precipitate. The results can provide helpful information for understanding the precipitate-GB interaction, optimizing the existing theoretical models and realizing nanograin-stability prediction with higher accuracy.
Methodology
Nanograined Cu is a sample material that has been widely studied recently [1, 5, 16]. Experiments have obtained that the Cu-Fe alloy has Cu matrix with face-centered-cubic (FCC) structure and Fe precipitates with body-centered-cubic (BCC) structure [20]. Motivated by this, the nanostructured Cu–Fe sample is used in this work. Previous studies suggested that the pinning effect of precipitates mainly relies on geometric characteristics, but is not sensitive to the element type. Hence, the influence of heterogeneous precipitate on GB migration revealed by the present work should be applicable to the systems with face-centered-cubic-structured (FCC) matrix and body-centered-cubic-structured (BCC) precipitates. The simulated model here is a Cu bicrystal with a cylindrical Fe-precipitate embedded, as shown in Fig. 1. The atomic interaction for the Cu-Fe alloy is described by an embedded atom method potential [32], which has been widely used to describe the microstructure interactions in Cu–Fe alloys [26, 33]. The Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) [34] is employed for all simulations, and the simulated results are visualized using OVITO [35].
Figure 1
a Schematic image of the simulation model. b An example to show the simulation block with a 0° precipitate used in this work. The atoms are colored by their local crystal structure which is analyzed by the common neighbor analysis (CNA) analysis in OVITO: green-FCC, blue-BCC and white-unknown. GB atoms are identified as atoms with unknown local crystal structure and belong to the interface between two Cu matrixes with different orientations
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The GB studied in this work is the Σ17 (530)[001] symmetric tilt GB with the misorientation angle of θ = 61.93° (Σ is the reciprocal density of coincident sites, [001] is the tilt axis, and (530) is the GB plane). This GB has been used to simulate the mechanical-driven migration behavior in nanostructured Cu [36]. Geometrical details are presented in Fig. 1b, and the total number of atoms is about 5.6 × 105. The simulated grain size is about 14 ~ 19 nm, which is comparable with the grain sizes of nanostructured Cu or Cu-based alloys obtained in experiments [37, 38] and used in previous simulations [29, 30]. The precipitate is placed horizontally, i.e., the long axis of the precipitate is parallel to the GB plane. The angle between the axis of precipitate and the tilt axis of GB is identified as the direction angle \(\alpha\). Four direction angles (0°, 30°, 60° and 90°) and two precipitate sizes (diameters are 2 and 4 nm) are considered to investigate the direction and size effect of cylindrical precipitates. The length of precipitate is set as infinite to make the effect of the direction angle more obvious. The resulting spacing distance between precipitates is around 15 ~ 17 nm (the dimension in X/Z-direction minus precipitate diameter), which is slightly smaller than the experimental spacing distance 20 ~ 40 nm [20]. Such simulation spacing distance is needed for computational efficiency and is large enough to ensure the numerical convergence of the simulation results, as shown in the later contents.
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At the beginning of all simulations, an energy minimization with the standard conjugate-gradient algorithm is performed to obtain the stable GB configuration. The minimization step is followed by an equilibrium procedure, in which the boundary conditions are fully periodic in all directions. The equilibrium step is performed at a target temperature and constant pressure of 0 bar in all directions for 100 ps, using the NPT ensemble. After that, the boundary condition in the Y direction is switched to non-periodic, and the ensemble is changed to the NVT ensemble to simulate GB migration. Due to the periodicity along the X and Z-orientation, simulations can be considered as an infinite thick plane interacting with an array of precipitates. In this case, the self-interaction between precipitates may influence the GB-precipitate interaction by generating long-scale stress field, as presented in a previous simulation [31]. To avoid this effect, we examine a larger-size system and compare it with a smaller-size system considered in our work, as shown in Fig. 2a. It is demonstrated that there is negligible difference between the shear stress–strain curves for the samples with different model sizes. That means the self-interaction between precipitates can be ignored in our simulations.
Figure 2
The shear stress–strain curves for a the simulations with different model sizes (19 nm and 22 nm) in the Z direction, b the simulations with different initial GB positions (3.5 nm and 10 nm), and c the simulations performed with different strain rates (1 × 107 s−1 and 1 × 108 s−1)
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To induce GB migration, two thin slabs with a thickness of 1 nm at the top and bottom of the box are used to impose a shear strain into the system, as shown in Fig. 1a. For all simulations, the top slab is fixed, while the bottom slab is translated as a rigid body parallel to the GB plane and normal to the tilt axis with a constant velocity of \(v_{\tau }\). This makes GB migrate in a constant velocity of \(v_{n}\) . The shear velocity \(v_{\tau }\) is set appropriately to result in the shear strain rate of 5 × 107 s−1. The resulting shear deformation is much greater than that in practical experiments, but it is necessary to perform the nanosecond-scale simulation under the current model dimensions. The deformation process continues until the total strain reaches about 0.25, depending on whether GBs bypass precipitates. Considering the possible interaction between the bottom slab and the GB, a simulation with larger GB position (≈10 nm) is performed, and the resulting stress–strain curve is comparable with the stress–strain curve obtained by the simulation with 3.5-nm GB position, as shown in Fig. 2b. Different strain rates are also examined, and the similar shear stress–strain curves are obtained, as shown in Fig. 2c. This means the GB position and strain rate used in our simulations are conserved. For all simulations, the time increment is fixed at 0.001 ps and the temperature is kept at 10 K to eliminate the impact of thermal noise. For the samples containing 1-nm precipitates with 0° and 90° direction angles, two simulations with the same simulation setups are performed to confirm the robustness of the simulation results. Other samples are only simulated one time due to the low computational efficiency. Almost identical shear stress–strain curves and microstructure evolutions are obtained in the simulations with identical simulation setups (not shown in this paper), which proves that the simulation setups converge to similar results despite its stochastic nature.
Results
Effect of the direction angle α
Figure 3a shows the shear stress–strain curves for the samples containing small precipitates (diameters are 2 nm) with different direction angles α. Affected by the stochastic process, these curves inevitably exhibit random disturbances, but the average trends of these curves are still clear. During the whole simulations, there is a typical “saw-tooth” behavior of shear stress, which is consistent with previous simulations [23]. The interaction between migrating GBs and precipitates leads to the increase in the average flow stress, owing to the blocking effect on the GB migration. The blocking effect is quantified by the blocking stress τmax, which is estimated by comparing the averaged peak stresses during and before the GB-precipitate interaction, as shown in Fig. 3a. It can be seen from Fig. 3b that the blocking stress τmax almost decreases linearly with the increase in direction angle α, indicating that the stability of a GB is sensitive to the orientation of the cylindrical precipitate.
Figure 3
a The shear stress–strain curves for the samples containing 2-nm-diameter precipitates with different direction angles α. The averaged peak stresses before and during GB-precipitate interactions are figured out, using solid line and dash lines. The example blocking stress τmax for α = 90° is presented. b The blocking stress τmax as a function of the direction angle α. The stress τmax is obtained via averaging 3 ~ 5 sawtooth peak stresses within the shear strain 0.11 ~ 0.15. The error bars indicate the standard deviation of these peak stresses
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Figure 4 shows the atomic details of the interaction between migrating GBs and small precipitates. Under sufficient strain, all GBs finally bypass precipitates, along with some differences in the migration process due to the different direction angle. For \(\alpha\) = 90°, the interaction between GBs and cylindrical precipitates leads to the obvious GB deformation (Figs. 4d), which is similar to the interaction between GBs and spherical precipitates that is described by the Zener model [13, 17]. A contrary phenomenon can be seen from Figs. 4a–c, in which the migrating GBs keep their planar shape during their interaction with the precipitates with α = 0°, 30° and 60°. That emphasizes the special effect of orientation of cylindrical precipitate on GB shape, compared with the spherical-precipitate-GB interaction, during which the GB always deforms. Moreover, in combination with the blocking stresses shown in Fig. 3, these results indicate that there is an intrinsic correlation between the direction angle, GB deformation and blocking stress.
Figure 4
a–d The interactions between GBs and 2-nm-diamater precipitates with different direction angles α. The SFs generated after GBs detach precipitates are figured out by red lines in (a3)–(d3), and the enlarged images of example SFs are presented in (a4)–(d4). In order to assist the view of precipitates, the Fe atoms are colored by “green circle” and the atoms of precipitate boundaries are colored by “violet circle.” Other atoms are colored by their local crystal structure which is analyzed by the CNA analysis: “blue circle”-BCC, “orange circle”-HCP and “grey circle”-unknown. All FCC atoms are removed to assist the view of GBs and precipitates. The GBs are partially outlined by the dashed lines, to assist the view of GB shapes
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Different microstructure evolutions are observed in different samples, after the shear stresses reach their peaks, as shown in Figs. 3 and 4. The decrease in shear stress is caused by different deformation mechanisms for different samples. 1) For α = 90°, the reduction of stress is accompanied by the occurrence of GB deformation, which effectively releases the local stress around precipitates. In this case, both the GB migration and GB deformation are responses to the plastic deformation of systems. After GBs detached precipitates, the partial dislocation nucleation and propagation occur between the migrating GBs and precipitates (Fig. 4d3), which is verified by the formation of stacking faults (SFs). This phenomenon is also observed in other simulations in the present work, and detected in previous experiments [39], which means such partial dislocations generation is common in the GB-precipitate interaction. The emitted dislocations may derive from the decomposition of misfit interfacial dislocations around precipitates and serve as geometrically necessary dislocation to accommodate the geometrical misfit between the matrix and boundaries [31]. Furthermore, the plastic deformation mechanism also changes to “GB migration accompanied with activated dislocation slip.” After GBs moved away from the precipitates, these SFs still connect the GBs and precipitates, as shown in Fig. 4d3. The observations of SFs within boundaries are also detected in previous experiments [40‐42]. After a long simulation time, these SFs may shrink back into the precipitate boundaries or GBs, as shown in a previous simulation [22]. However, limited by the low simulation efficiency, the SFs-shrink-back event does not occur in our simulations. 2) For α = 0°, 30° and 60°, there is no obvious GB deformation during the whole simulations. The reduction in stress is mainly due to the GBs bypassing the precipitates and partially caused by the nucleation of partial dislocations near the precipitates. The nucleation of SFs or partial dislocations occurs when the GBs are about to detach the precipitates, accommodating the serious geometrical misfit and releasing the highly local stress near precipitates.
Effect of the precipitate size
To clarify the effect of precipitate size, the interactions between GBs and 4-nm-diameter precipitates are simulated, and the results are utilized to compare with the cases with 2-nm-diameter precipitates. The shear stress–strain curves for the 4-nm-diameter cases are shown in Fig. 5, in which the decreasing blocking stress τmax with the increase in direction angle α is also observed. However, compared with the cases with 2-nm-diameter precipitates (Fig. 3b), the effect of the precipitate direction is more pronounced for the cases with 4-nm-diameter precipitates. When the direction angle reduces from 90° to 0°, the blocking stress increases three times for the cases with 4-nm-diameter precipitates (Fig. 5b), which greatly exceeds the one-time increment for the cases with 2-nm-diameter precipitates (Fig. 3b). Thus, it is logical to deduce that the GB stability can be greatly improved via simultaneously tailoring the size and direction of heterogeneous precipitates.
Figure 5
a The shear stress–strain curves for the samples containing 4-nm-diameter precipitates with different direction angles α. The blocking stress τmax is also presented. b The blocking stress as a function of the direction angle α. The stress τmax is obtained via averaging 3 ~ 5 sawtooth peak stresses within the shear strain 0.16 ~ 0.20. The error bars indicate the standard deviation of these peak stresses
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The atomic details of the interaction between the GBs and 4-nm-diameter precipitates are presented in Fig. 6. 1). For the cases with α = 0°, 30° and 60°, the GBs are finally pinned by precipitates. This leads to the accumulation of local strain around precipitates, resulting in the nucleation and emission of partial dislocations and SFs. Looking closely at the partial dislocation formation process (Fig. 7), the first dislocation nucleates at the GB-precipitate intersection, which is the region with the highest local stress. This further confirms that the partial dislocations generation is activated by the high local stress. The emission of partial dislocations not only changes the deformation mechanism of the system, but also causes the decrease in shear stress shown in Fig. 5a. This finding agrees well with the results in recent experiments, in which the motion of partial dislocations is activated when the GB migration is inhibited [2, 5, 6, 43]. 2) For the case with α = 90°, the interaction causes GB deformation, which partially releases the highly local strain and thus avoids the nucleation of partial dislocations. However, after the GB detached the 90° precipitate, SFs still generate and connect the migrating GB and precipitate, which is in consistent with that shown in Fig. 4d. As mentioned above, these SFs may be attributed to the serious geometrical misfit between GBs and precipitate boundaries.
Figure 6
a–d Interaction between GBs and 4-nm-diameter precipitates with different direction angles
Figure 7
a–c Activation of the first partial dislocation formation at the intersection of GB and precipitate (diameter = 4 nm, α = 30°). The slip of partial dislocation is accompanied by the growth of SF. (d) As the strain increases further, more SFs nucleate from the precipitate
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Discussions
The GB-precipitate interaction can be understood by modeling GB as an array of GB dislocations [44, 45], as presented in Fig. 8b–c. 1) For α = 0°, the cylindrical precipitate is parallel to the GB dislocations, as shown in Fig. 8b. In this case, some GB dislocations (colored by red in Fig. 8b) are totally blocked by the precipitate with infinite length, leading to a strong suppression of the GB deformation and GB migration. 2) For α = 90°, all GB dislocations are perpendicular to the precipitate, allowing them to bypass the precipitate as the Orowan mechanism does [46, 47]. Such Orowan-based interaction causes dislocations bending, which agrees well with the GB deformation observed in Figs. 4d and 6d. Besides, it provides an easier way for GB dislocations to bypass the precipitate, resulting in the relatively low blocking stress for the case of α = 90°. In terms of the cases α = 30° and α = 60°, the GB-precipitate interaction can be viewed as the combination of that in the cases α = 0° and α = 90°. This provides an explanation for the decrease in the blocking stress τmax with increasing the direction angle α (Figs. 3b and 5b). In the present simulations, these discrete GB dislocations are perfect dislocations with burgers vector b = 1/2 < 110 > , which is analyzed using the dislocation extraction algorithm (DXA). For other symmetrical tilt GBs, both the burgers vector and spacing distance of GB dislocations vary with GB orientations [44, 45, 48], which influence the GB-precipitate interactions.
Figure 8
Schematic of the GB-precipitate interaction process, in which the GB is modeled as an array of discrete GB dislocations (the slender green stripes). a The 0°-precipitate strongly pins several GB dislocations (the red stripes), while other GB dislocations keep moving forward. That makes the GB keep planar. b The 90°-precipitate blocks the migrating GB, and then all GB dislocations bend during the interaction, resulting in GB deformation
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The above discussion indicates that the direction of cylindrical precipitate is significant only when the considered GB is composed of an array of dislocations. It is longstanding knowledge that the low-angle GB can be regarded as an array of discrete dislocations [44], which ensures the applicability of the above model on the interaction between precipitates and low-angle GBs. This is further confirmed by recent high-resolution experiments [45, 48], in which the low-angle GBs exhibit structures that consist of well-aligned 1/2 < 110 > dislocations. Besides, these GB dislocations keep their burgers vectors during GB migration, which is in good agreement with our simulations [48]. In terms of the high-angle GBs that are common in polycrystalline materials, a recent experiment revealed that they are made up by an array of dislocations in NCMs or exist in the form of two-dimensional structural units in coarse-grained materials [45]. Previous studies have confirmed the generality of such shear coupling mechanism over numerous boundaries and boundary types [24, 49]. That indicates the simulation results obtained from our simulations with the symmetric tilt GB are valid for most GBs in actual materials. This conclusion is further verified by simulating the interaction between cylindrical precipitates and other GBs, e.g., the Σ5(210)[001] GB with misorientation θ = 53.1° shown in Fig. 9. Both the obvious GB deformation and the decreasing blocking stress are observed when α = 90°, compared with the case of α = 0°. For the Σ5(210)[001] GB, the blocking stresses τmax(α = 0°) = 1023(± 25) MPa, and τmax(α = 90°) = 731(± 78) MPa. The difference between the two blocking stress is around 292 MPa, which is very close to the 298 MPa for the cases with Σ17(530)[001] GBs. Note that the misorientation angles for both GBs are close, while the GB energies EGB (254.47 J/m2 for Σ17(530)[001], and 945.3 J/m2 for Σ5(210)[001]) deviate a lot. Thus, it is logical to deduce that the GB-precipitate interaction is sensitive to the GB structural characteristics but not the GB energy. More work is needed to verify this conclusion.
Figure 9
Interaction between the cylindrical precipitates (α = 0° and α = 90°) and the Σ5(210)[001] GBs. When α = 90°, obvious GB deformation occurs, and the blocking stress is lower than that for the case with α = 0°
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In combination with the microstructure evolution observation and flow stress change in the period around peak stresses, it is logically to deduce that there is a competition between the GB deformation and dislocation activation, in carrying plastic deformation and softening materials. For the case with low-angle precipitates, the GB deformation is inhibited. In this case, the nucleation and emission of partial dislocations are activated by the highly local stress and enhanced by the low SF energy of Cu matrix. For the case with high-angle precipitates, the GB deformation is easy to activate, serving as an effective way to release the highly local stress around precipitates. The nucleation of partial dislocations is activated until the GB is about to detach the precipitates. After that, the nucleation and emission of partial dislocations, as well as the generation of SFs, begin to carry plastic deformation and further soften materials. These results demonstrate that the deformation mechanism is sensitive to the direction of heterogeneous precipitates. Furthermore, they also indicate that there is a critical direction angle, above which the materials softening is dominated by GB deformation, which below this angle the materials softening is activated by dislocations nucleation. This conclusion is proven by the decreasing blocking stress with the increase in direction angle shown in Figs. 3b and 5b.
Finally, it is worth emphasizing that, limited by the time and size scales and the computational efficiency, atomic simulations are inevitably affected by the atomic-scale noises. The obtained stress–strain data and microstructure evolution for different precipitate directions and size only provide rough conclusions, which may be invalid for specific cases. For instance, all examined GBs in the present work are symmetrical tilt GBs, which limits the general application of the above dislocation-GB model on describing the GB-precipitate interaction. The influences of temperature, composition, and crystal structures are also not examined. Hence, more studies are still needed to verify the validation and generality of the conclusions and above dislocation-GB model, especially for the non-special GBs with complex structures and the cases under complicated conditions.
Conclusion
The interactions between the GBs and the cylindrical precipitates with various directions and sizes are studied by atomic simulations, and their effects on GB stability are evaluated. Via decreasing the direction angle of precipitate, both the GB deformation and migration can be effectively inhibited, leading to the high GB stability. This effect is more pronounced when it comes to precipitates with larger size. This means the GB stability can be greatly enhanced via simultaneously tuning the direction and size of heterogeneous precipitates. For the case with low-direction-angle precipitates, the material softening is dominated by the nucleation of partial dislocations around the precipitates; for the case with high-direction-angle precipitates, the material softening is activated by the GB deformation and further enhanced by the emission of partial dislocations. Both the GB deformation and dislocation nucleation result in the change of deformation mechanism, from the “GB migration” to “GB migration accompanied with GB deformation” and “GB migration accompanied with activated dislocations,” respectively. This work deepens the understanding of the interaction between GBs and heterogeneous precipitates and provides foundation for optimizing the theoretical models that describe the GB-precipitate interactions and the design of materials with high GB stability.
Acknowledgements
The authors would like to deeply appreciate the supports from the National Key Research and Development Program of China (2016YFB0700300) and the National Natural Science Foundation of China (51871092, 11772122, 12072109 and 52020105013).
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Conflict of interest
The authors declare that they have no conflict of interest.
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