Unprecedented grain size effect on stacking fault width

Unprecedented grain size effect on stacking fault width A. Hunter and I. J. Beyerlein Citation: APL Mater. 1, 032109 (2013); doi: 10.1063/1.4820427 View online: http://dx.doi.org/10.1063/1.4820427 View Table of Contents: http://aplmaterials.aip.org/resource/1/AMPADS/v1/i3 Published by the AIP Publishing LLC.

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APL MATERIALS 1, 032109 (2013)

Unprecedented grain size effect on stacking fault width A. Hunter1,a and I. J. Beyerlein2 1

X Computational Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA 2 Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA (Received 23 May 2013; accepted 26 July 2013; published online 9 September 2013)

Using an atomistic-phase field dislocation dynamics model, we isolate and investigate grain size and stress effects on the stacking fault width created by partial dislocation emission from a boundary. We show that the nucleation stress for a Shockley partial is governed by size of the boundary defect and insensitive to grain size. We reveal a grain size regime in which the maximum value the stacking fault width attains increases with grain size. © 2013 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported License. [http://dx.doi.org/10.1063/1.4820427]

Several experimental and all-atom studies have shown that nanostructured materials (nanowires, thin films, 2D nanocomposites, micro-electro-mechanical systems (MEMS) devices) deform by the motion of partial dislocations, which leave behind stacking faults.1–11 The increased number of Shockley partial dislocations and stacking faults in nano-sized face-centered cubic (fcc) grains can lead to noticeable changes in deformation mechanisms, such as twinning, boundary mobility, and nucleation.12–16 The observed transitions from perfect- to partial-dislocation-mediated slip are often associated with a reduction in grain size from bulk to nano-scale dimensions.3, 53 Moreover, within this fine-scale regime, molecular dynamics (MD) simulations have shown that these partial dislocations are emitted from grain boundaries and do not develop within the grain interior.6, 8, 17 As a dynamic process that involves both nucleation and propagation, many variables, apart from grain size, can affect the distance that the partial dislocation travels into the grain before a trailing partial is emitted. These variables include the boundary source for the Shockley partial, the local stress, and the material γ -surface (including the intrinsic stacking fault energy (SFE) and the unstable SFE).1, 6, 14, 20–23 Because it has been challenging in theory, simulation, and experiment to represent and decouple these effects, the dependence of emitted stacking fault widths (SFWs) on nano-grain size is not fully understood. While density functional theory (DFT) can calculate the entire material γ -surface, it is problematic to calculate wide SFWs using this technique within finite-sized crystals due to length-scale limitations. While these issues are overcome in MD simulations, MD involves large applied stresses and strain rates whose effects cannot be separated from grain size effects. Larger length- and time-scale models,24–27 such as discrete dislocation dynamics, utilize rules that only account for the intrinsic SFE to model stacking faults, rather than the entire γ -surface.25, 26 Many analytical models for SFW calculation are derived from continuum dislocation theory also rely solely on the intrinsic SFE.24, 27–29 Using a phase field dislocation dynamics (PFDD) model with the full material γ -surface incorporated, we have successfully decoupled applied stress, grain size, and the material γ -surface in order to study grain size effects on SFWs emitted from boundary ledges in nano-grains. In contrast to current thinking, we report a grain size regime in which an unexpected grain size effect prevails, wherein the SFW increases with increasing grain size up to a saturation value. In addition, we show that the partial dislocation nucleation stress has little to no dependence on grain size.

a [email protected]

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© Author(s) 2013

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FIG. 1. Initial orientation and dimensions.

The PFDD model is particularly advantageous for such a study because the calculations are based on the minimization of the total system energy, and the time- and length-scales are much larger than those in DFT or MD alone, making them more representative of laboratory conditions.30–33 The PFDD model does not rely on rules or indirect passage of atomic-scale information. Rather, our PFDD model can directly incorporate the entire material γ -surface calculated by either DFT or MD into the calculation.33–35 For a wide range of fcc metals (e.g., Pd, Ag, Cu, Ni), single, isolated dislocations attain equilibrium SFWs in agreement with available results from DFT, MD, and experiments.33, 34 In principle, inclusion of the entire γ -surface in the system energy calculations leads to more accurate predictions than those from numerical and analytical methods that only consider the intrinsic SFE.24–27 Within the PFDD model, we construct a finite-sized crystal in the shape of a cube with grain boundaries on all sides and periodic boundary conditions. Grain boundaries are modeled as an array of pinned obstacles of width, w = 1.25 nm. As mentioned, prior experimental and atomistic simulation studies have shown that Shockley partials nucleate from defects, kinks, and ledges at grain boundaries and free surfaces.36–39 To potentially produce a stacking fault in the model grain, a single ledge is added to one of the grain boundaries. The ledge is given a length, L, and a height, w L , thicker than the width of the grain boundary, w, yielding a total ledge thickness of w L + w at the grain boundary defect. The ledge has an effective Burgers vector equal to that of a full edge ¯ where a is the lattice parameter. The system is oriented such that the slip dislocation, a/2[101], ¯ aligns with the x-axis; and plane normal, [111], is parallel with the z-axis; the slip direction, [101], ¯ [121] corresponds to the third orthogonal direction. A schematic of this configuration is shown in Figure 1. To investigate the effects of grain size on SFW, it is desirable to decouple grain size and applied stress. For most experimental or all-atom systems,3, 18 this is not straightforward. Varying the characteristic length-scale, such as grain size or layer thickness, is often accompanied by changes in the local and macroscopic stress state.7, 40–47 In the present calculations, we found a way to achieve decoupling by closely studying the critical conditions needed for nucleating a Shockley partial dislocation from the grain boundary ledge. We apply a uniform and constant stress, σ xz , to a model Ni grain and find that at a critical value, σ xz = 3.53 GPa, the grain boundary ledge forms a leading ¯ that expands into the crystal and on the (111) glide plane. partial loop with Burgers vector a/6[21¯ 1] Repeating this simulation over a wide range of applied stresses indicates that this critical stress is, in fact, a partial dislocation nucleation stress. When the applied stress is lower than this critical value, no dislocation is emitted from the grain boundary. When the applied σ xz exceeds this critical ¯ emerges shortly after the leading one. value, the trailing partial loop (with Burgers vector a/6[112]) Thereafter, the two partials make a perfect dislocation whose core is extended by the equilibrium

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FIG. 2. Dependence of the nucleation stress on (a) grain size, D, and (b) length of boundary ledge, 2r, which produces an initial partial dislocation half loop with radius, r, in a Ni grain.

SFW (w0 = 1.743 ± 0.125 nm).33 Escaig stresses are the non-glide stresses acting on the Shockley partials of the extended dislocation.48 For this orientation, the Escaig stresses correspond to the resolved shear stress on the oppositely signed screw components of the Shockley partials, and for this applied stress state they are zero. Removing the Escaig effect is another way in which our simulations have decoupled stress and grain size effects on the SFW. In addition, an interesting finding is that after the trailing partial dislocation emerges, the nano-grain size had no effect on the equilibrium SFW of the perfect dislocation. The nucleation stress was calculated for ledge half-lengths ranging from 0.5L = r = 1.25–2.81 nm and grain sizes ranging from D = 7.97–63.74 nm. Our key finding is that the nucleation stress is not sensitive to D. Figure 2(a) shows the variation in the nucleation stress with D for different r. Except for the smallest defect size within the smaller grains, which are sensitive to localized stress fields from the grain boundaries, the nucleation stress exhibits little to no dependence on grain size. The effects of these stress fields are less apparent in small grains with large defects because the defect is nearly the size of the grain and the stress concentration at the defect is not localized. The distribution of the stress concentration favors nucleation, which counteracts the stress fields due to presence of grain boundaries. Conversely, Figure 2(b) shows that the ledge r has a strong effect on the nucleation stress, which is to be expected. However, the grain size has no effect on the rate at which the nucleation stress decays with increasing r. In Figure 2(b), we compare the magnitudes of the predicted nucleation stresses with the Orowan expression for the critical stress to expand a half loop unstably, i.e., μb/2r, where b is the magnitude of the Shockley partial dislocation

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Burgers vector, and μ is the shear modulus. The reasonable consistency with the Orowan equation to activate a dislocation source confirms that our calculated nucleation stress is primarily determined by r and explains why it is not noticeably affected by D. The insensitivity of the nucleation stress with grain size may at first seem to contradict conventional thought. It has been previously suggested that the nucleation stress varies inversely with grain size.3, 18 An implicit assumption that lead to this conclusion, however, was that the size of the boundary defect increased proportionally with grain size.18 In light of the present results, when the boundary defect increases with increasing grain size, the nucleation stress would decrease and vice versa. Moreover, the independence of the nucleation stress from grain size predicted here agrees with observations made in MD simulations.11, 49 With this critical insight, we can proceed to examine SFW behavior in grains of different sizes under the same applied stress. 3D grains of varying sizes containing the same grain boundary ledge are subjected to the same applied stress equal to the nucleation stress characteristic of this ledge. For the studies that follow, we chose L = 2r = 2.49 nm and w L = 0.498 nm. In the PFDD model, dislocation nucleation and propagation are completely governed by energy minimization.30, 31 In carrying out the above mentioned suite of constant-stress simulations, we observe the following sequence of events. Initially, for all D, the leading partial dislocation forms at the grain boundary ledge and glides into the grain. The expansion of the Shockley partial loop accommodates the applied stress at the expense of creating a stacking fault behind it. As the leading partial continues to propagate, the stacking fault area will expand until the energetic cost to continue growing the leading partial loop overcomes the energy barrier required to nucleate the trailing partial dislocation. When the grain size is below a threshold Dmin , the leading partial dislocation can propagate to the other side of the grain without reaching the critical point at which nucleating the trailing partial becomes energetically favored. In this case, the largest value the SFW attains equals the grain size, SFW = D < Dmin . On the other hand, when D is greater than a threshold Dmin , the trailing partial dislocation emerges from the ledge before the leading partial propagates across the entire grain. Before the trailing partial is emitted, the SFW has extended to a size greater than Dmin . The largest SFW attained for these grain sizes is SFW > Dmin > D. Hereinafter, for any D we refer to the maximum SFW as the critical SFW. When the leading partial traverses the entire grain length without nucleating the trailing partial, the grain size limits SFW expansion, placing a lower bound on the critical SFW, i.e., critical SFW =D < Dmin . As the grain size increases beyond Dmin , however, the grain size provides plenty of room for the stacking fault to expand and the critical SFW > Dmin . The natural questions to ask next are how far can the SFW extend and what is the effect of D on maximum extension? To determine whether D also places an upper bound on the critical SFW, we must systematically quantify the critical SFW. Here, we use the sum of the vector projections of the three active slip systems in the slip direction32, 33 to locate the leading partial dislocation loop just prior to nucleation of the trailing partial dislocation, (x) =

3 

ζi (x)si · sp ,

(1)

i=1

where ζ i (x) are the active phase field variables on the slip plane, si are the slip directions of the respective phase field variables, and sp is the Burgers vector direction of the perfect dislocation created by summing the two partial dislocations. The gradient of  (x) across the grain in the slip direction, d/dx, exhibits peaks where the partial dislocations are located and hence the boundaries of the stacking fault area. The stacking fault area just prior to nucleation of the trailing partial dislocation defines the critical stacking fault area. To demonstrate, we consider a case in which the grain size is sufficiently large such that the trailing partial nucleates before the leading partial propagates across the grain, i.e., critical SFW > Dmin . Figure 3 uses the gradient  (x) to expose the location of the leading partial dislocation along the centerline of the grain at times (colors) before and after nucleation of the trailing partial dislocation. The outer peak is the leading partial and the peak of the same color closer to the grain boundary ledge is the trailing partial. The inset in Figure 3 shows the corresponding glide evolution

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[121]

Ledge 1.2

1 400 s

500 s

450 s

0.8

dΔ dx

[101]

0.6

Time = 400 s

Leading Partials 0.4

Time = 450 s Time = 500 s

Trailing Partials

0.2

0 0 -0.2

5

10

15

20

25

30

35

Grain Size (nm)

FIG. 3. Evolution of the extended dislocation structure during the emission of the trailing partial dislocation in a (31.87 nm)3 size grain in Ni. A nucleation stress of 3.53 GPa is applied. The inset shows the evolution of slip by the leading (green) and trailing (red) partials. The dotted lines indicate the grain centerline where the gradient of Eq. (1) was taken. The simulation time is indicated in the lower corner. White arrows indicate the direction of the partial Burgers vectors, and green regions indicate stacking faults.

of the leading and trailing partial dislocations on the (111) slip plane. As shown, before the trailing partial nucleates, the stacking fault area has already reached two neighboring boundaries. The partial dislocation loops are not generally symmetric because each dislocation loop moves to reduce the amount of its edge component, which has a higher self energy than the screw component. Most published experimental and numerical studies on stacking fault behavior discuss findings in terms of SFW. To make the connection with reported literature, we can associate the critical SFW with the distance along the centerline of the grain between the leading partial and freshly nucleated trailing partial of the critical stacking fault area. The unprecedented grain size effect that we report in this letter applies to both the stacking fault area and centerline SFW. The critical SFW is calculated for Ni grains of different sizes within the range of D = 7.97 nm–127.49 nm. Figure 4 shows the relationship between grain size and both stacking fault area (Figure 4(a)) and SFW (Figure 4(b)). For the smallest grain size tested (D = 7.97 nm), the leading partial traverses the entire grain and a trailing dislocation does not emerge from the grain boundary. In this case, the critical SFW equaled the grain size. For the remaining grain sizes, the critical SFW was less than the grain size. Intriguingly, for these cases, Figure 4(b) shows that the critical SFW increases with grain size. The leading partial is able to glide a greater distance within the larger grain sizes before the trail emerges. Thus, it is revealed that grain size does have an effect on the critical SFW. Significantly, the calculations also show that as the grain size increases, approaching bulk sizes, the critical stacking fault area saturates to ∼576 nm2 (Figure 4(a) shows the square root of the stacking fault area, which saturates at 24 nm) and the critical SFW to SFWsat = 20–21 nm. The saturated value SFWsat can be related to the maximum grain size Dmax at which stacking faults (and twins) are visible in experiment. It is substantially larger than the theoretical equilibrium value and stable dynamic SFW for the same extended edge dislocation (1.743 ± 0.125 nm).24, 27, 33 Notably, the unusually large 21 nm value agrees with studies on electrodeposited nanocrystalline Ni films that report stacking faults and twins extending through the full width of the grains with an average diameter of 25 nm.50 It is also consistent with predictions by an analytical model that identifies 19–27 nm as the optimal grain size range for twinning.28 The foregoing grain size effect elucidated by our calculations has not been previously reported. At first, this finding may appear to contradict prior results from experiments12 and atomistic simulations,14, 19 which generally report that SFWs are wider in smaller grains or that partialmediated deformation prevails as grain size decreases into the nano-scale regime. We find that as the grain size increases, the critical stacking fault area also increases but the fraction of the grain

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140

100

SF Area

√ nm2

r

pe

p Co

120

80 60 40

Nickel

20

(a)

0 0

20

40

60 80 √ 100 Grain Area nm2

120

140

140 r

pe

p Co

120 SFW (nm)

100 80 60

W SF

40

=D

SFWsat

20

Nickel

0

(b)

0

20

40

60 80 100 Grain Size (nm)

120

140

FIG. 4. Critical SFW for several grain sizes in Ni and Cu. Panel (a) shows the relationship of the square root of the stacking fault area with the square root of the grain area. Similar to the SFW case, we see that a saturation area is reached in Ni. Panel (b) shows how the critical SFW in Ni compares the limiting case where the critical SFW is equal to the grain size in all simulations.

cross-sectional area taken up by the critical stacking fault region decreases. These grain size effects are intriguing and cannot be explained by any model to date. As we have decoupled stress from grain size effects on SFW, we envision that this unusual effect arises due to the loop geometry and close interactions with the boundary. This notion is supported by the fact that the SFW saturates as the grain size increases. The calculations up to this point are focused on one material, Ni. It is well known that the material γ -surface can have a significant impact on the SFW. Our calculations are repeated for Cu, which directly incorporates the Cu γ -surface and elastic moduli from DFT.34 There are three ways in which a change in material is expected to impact the calculations. First, following the Orowan stress for the activation of a dislocation loop, σ = μb/2r, from a defect of length 2r, Cu is expected to have a lower nucleation stress than Ni due to its lower μ. In accordance, for the same grain boundary ledge, we calculated a nucleation stress of 2.2 GPa for Cu, which is much lower than the 3.53 GPa calculated for Ni. Second, the γ -surface for Cu contains values for the unstable SFE, γ US , and intrinsic SFE, γ I , that are lower than those for Ni. Nucleation of a Shockley partial, whether leading or trailing, has also been connected with either γ US or δγ = γ US − γ I .21, 22 The γ US for Ni (212 mJ/m2 ) is larger than that for Cu (163.7 mJ/m2 ), but their δγ values are approximately the same. Therefore, while the former nucleation criterion based on γ US would differentiate between Ni and Cu, the latter one based on δγ would not. Our PFDD calculations support the former, predicting that a lower nucleation stress for Cu than Ni. Based on γ US alone, nucleation would be expected to be easier in Cu, which is consistent with the PFDD calculation of a lower nucleation stress. Last, lower γ US and γ I values mean the energy penalty associated with forming

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FIG. 5. Progression of partial dislocations traversing a 63.74 nm grain in Ni, and a 65.79 nm grain in Cu. Black arrows indicate the direction of the partial Burgers vectors and green regions indicate stacking faults. The red regions are areas that have been traversed by both the leading and trailing dislocations.

FIG. 6. Conceptual map illustrating three regimes of grain size effect on SFW. Region I presents a lower bound in which the critical SFW is equal to the grain size. Region II describes a previously unknown grain size effect where the critical SFW increases with increasing grain size up to a saturation width. Finally, in region III, the grain size has no effect on the size of the stacking fault, which has reached its maximum width.

the stacking fault is lower. Hence, wider SFWs can be tolerated in Cu than in Ni, a well established trend that has been appreciated in many simulation and analytical models.3, 24, 27, 28, 34 However, to date the effects of the γ -surface on the lower and upper bounds of emitted stacking faults are unknown. To elucidate these γ -surface dependencies, we study partial dislocation emission from the same grain boundary defect discussed above in a Cu and Ni grain of approximately the same size, i.e., a 65.79 nm grain in Cu and a 63.74 nm grain in Ni. Figure 5 shows that in the Ni grain, the trailing dislocation emerges before the leading defect propagates across the entire grain. By contrast, the leading dislocation fully traverses the Cu grain without a trailing dislocation emerging from the grain boundary. This same outcome is observed for grain sizes up to 131.78 nm in Cu. Therefore, SFWsat for Cu is over six times greater than that for Ni (20–21 nm). This gigantic effect of the γ -surface would not have been anticipated by current analytical models. In support of our prediction, the large critical SFW for Cu (>132 nm) is consistent with experimental observations of stacking faults and twins within Cu grains approximately 100 nm and larger in size. In situ studies on Cu nanowires have shown that when the diameter is reduced to less than 150 nm, extensive twinning and large stacking faults occur.51 Microstructural analysis on rolled nanolamellar Cu-Ag composites reported fully twinned layers of Cu, which ranged from 60–100 nm

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in thickness.52 Finally, a study on rolled Cu-Nb nanocomposites observed the onset of Cu twins in layers about 135 nm thick.53 In conclusion, using a combined atomistic-phase field dislocation dynamics model, we reveal a previously unknown grain size effect on the stacking faults emitted from grain boundaries. The calculations prove that the critical stress to emit the leading partial is directly dependent on the size of the source defect in the grain boundary and insensitive to grain size. On this basis, we were able to decouple stress and grain size effects to show that there exist three regions of grain size effect, as depicted in Figure 6. Most unusual is the grain size range, region II from Dmin to Dmax in Figure 6, in which the SFW increases as the grain size D increases. Grain sizes below a minimum Dmin (region I) limit stacking fault expansion and the stacking fault propagates across the entire grain. In this region I, the SFW cannot be less than D. For grain sizes above Dmax (region III), the SFW saturates to SFWsat , its largest possible value. Hence, in region III, grain size has no effect on SFW. Notably, the grain sizes at which SFWs are predicted to prevail agree with those reported experimentally in deformed nanostructured Cu and Ni, unlike any model to date. This newfound mapping of the grain size effect on SFW has implications on the design of nanostructured metals in controlling the relative activities of perfect dislocation slip, partial dislocation slip, and deformation twinning. A.H. would like to acknowledge support from the Los Alamos National Laboratory Directed Research and Development (LDRD) Project 20130745ECR. I.J.B. gratefully acknowledges support from the Los Alamos National LDRD Project DR20110029. The authors gratefully acknowledge valuable discussions with Dr. Timothy C. Germann. 1 H.

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