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Computational Materials Science

journal homepage: www.elsevier.com/locate/commatsci

Interactions between displacement cascade and dislocation and theirinfluences on Peierls stress in Fe-20Cr-25Ni alloys

Zizhe Lu (芦子哲)a, Liubin Xu (徐刘彬)a, Tianyi Chenb, Lizhen Tanb, Haixuan Xua,⁎

a Department of Materials Science and Engineering, University of Tennessee, Knoxville, TN 37996, USAbMaterials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA

A R T I C L E I N F O

Keywords:Fe-20Cr-25NiCascadePeierls stressDislocationRadiation-induced hardening

A B S T R A C T

The interactions between dislocations and displacement cascades are investigated using molecular dynamicssimulations. The primary knock-on atoms (PKAs) are placed at different distances to edge and screw dislocationlines. Specifically, an fcc Fe-20Cr-25Ni system is studied; the alloy is the based alloy for Alloy 709. The numberof defects is calculated using Wigner-Seitz analysis in order to illustrate the effect of dislocations on cascades.The displacement cascades in systems containing dislocations tend to generate more total defects compared withbulk systems, but eventually lead to less surviving defects in the materials matrix due to interactions withdislocations. The changes in atomic structures of the dislocations after interacting with cascades are analyzed tounderstand how the primary damage affects dislocations. The displacement cascade potentially causes dis-location climb in edge dislocations and cross-slip in screw dislocations, which could serve as additional me-chanisms contributing to the radiation-induced hardening (RIH) compared with conventional RIH models. Toreveal this, Peierls stresses are calculated before and after cascades using molecular static simulations. This studyprovides critical information of how defect production and dislocations are correlated, especially when totaldose is high, which needs to be taken into account in upper scale models, such as mean-field rate theory.

1. Introduction

In nuclear power plants, many structural components are under theinfluence of intense radiation, which can result in permanent damage inmaterials, such as radiation-induced hardening (RIH), embrittlement,and creep. The initial stage of radiation damage, particularly the defectproduction from displacement cascades, has been extensively studiedusing molecular dynamics (MD) simulations in bulk bcc, fcc and hcpmetals [1–9]. The number of surviving defects after displacement cas-cade depends on the energy of the PKA atom [10–12], and sometimessub-cascade events may occur [13]. In addition, the size distribution ofthe surviving defects is also influenced by the PKA energies, such as inbcc iron [10,14].

As the microstructure evolves as a function of irradiation dose, thedislocations and dislocation loops could become the dominant defectsin materials [15,16]. Because of the presence of these defects in ma-terials, the displacement cascade processes may interact with thesedefects, leading to a different defect production rate, which is needed inthe mean field rate theory model to accurately describe the defectevolution for predictions of long-term material performance. However,to our best knowledge, the influence of dislocations or dislocation loops

on displacement cascades is not extensively studied, and the currentunderstanding of this is rather limited, especially compared with theextensive displacement cascade simulations in bulk materials. Becauseof the presence of the dislocation, the defects may go into the dis-location and eventually may lead to different defect production effi-ciency and distributions. An MD simulation of interactions betweencascades and an edge dislocation in Zr suggests that the presence of theedge dislocation can promote the nucleation of vacancy clusters andcan even help to generate an experimental-scale vacancy loop [17].

In addition, the displacement cascade processes will influence thedislocation structure [18,19], which may subsequently lead to pinningof dislocations. Conventional RIH mainly focuses on the interactionsbetween a dislocation and obstacles created by the radiation [20,21].For instance, immobile dislocation loops or voids could serve as ob-stacles to dislocation motion, leading to RIH [22–24]. This is usuallyaccompanied by the loss of ductility and fracture toughness, which isidentified as one of the major challenges that limits the lifetime ofstructural materials. Early theories of RIH are based on an elastic theorydescription of the interaction between a single dislocation and an ob-stacle and are empirical or semi-empirical. The dislocation barriermodel [25] was developed based on geometrical consideration and has

https://doi.org/10.1016/j.commatsci.2018.12.018Received 6 November 2018; Received in revised form 6 December 2018; Accepted 8 December 2018

⁎ Corresponding author.E-mail address: xhx@utk.edu (H. Xu).

Computational Materials Science 160 (2019) 279–286

Available online 23 January 20190927-0256/ © 2018 Elsevier B.V. All rights reserved.

T

been applied to the description of strong obstacles. Friedel–Kroupa–-Hirsch (FKH) developed an alternative model for weak obstacles[26,27]. However, these models do not address the potential hardeningeffect caused by the structural change of dislocation cores directlycaused by irradiation.

In this study, we investigate the interactions between dislocationsand displacement cascades using MD simulations in Fe-20Cr-25Ni alloy.The Fe-20Cr-25Ni is the base system of Alloy 709 [28,29], which iscurrently considered as a candidate of structural materials for next-generation nuclear reactors [30]. Particularly in this study, the effect ofdislocations on the surviving defect number is determined. The struc-tural changes of dislocations caused by displacement cascades areanalyzed at the atomic scale. Finally, the Peierls stress simulations arecarried out to study the hardening effect of displacement cascades ondislocations. The rest of the paper is organized as follows. Section 2contains the edge and screw dislocation models, simulation setups, vi-sualization, analysis tools, and the defect counting method. Section 3contains the results of the cascade simulations and Peierls stress cal-culations. A summary is given in Section 4.

2. Methodology and simulation setup

2.1. Generation of edge and screw dislocations

In order to allow accurate Peierls Stress simulations and to avoidany potential surface effects, a dislocation model that has periodicboundary conditions (PBC) in not only the dislocation line direction butmore importantly in the dislocation gliding direction is used.Specifically, the periodic array of dislocations (PAD) model proposedinitially by Daw and Baskes [31] is followed. The PAD model has beenwidely used in previous studies to create edge and screw dislocations[32–37]. The following sections detail how to construct an edge andscrew dislocation using the PAD model.

2.1.1. Edge dislocation modelThe schematic of creating an edge dislocation is shown in Fig. 1. We

first divide the perfect crystal into two half crystals, the upper (blue1)and the lower (red) crystals. The spacing b shown in Fig. 1 representsthe length of the Burger’s vector. The number of atomic planes withinspacing b (a Burger’s vector) is determined by the crystalline structures(bcc, fcc, hcp). Initially, we have an equal number of planes in theupper and lower crystals. Then, the planes within one spacing b in thelower part of the crystal are removed. The length of the lower crystal inthe y-direction becomes (N−1) * b. In order to reduce the strain effectson the simulation box, a scheme developed by Osetsky et al. [38] isemployed. Following this approach, the length of the upper crystal inthe y-direction is changed by −b/2 and the length of the lower crystalin the y-direction is changed by +b/2. By doing so, the PBCs in bothdislocation line direction (x-direction) and dislocation gliding direction(y-direction) are achieved. Although the dislocation may interact withits own periodic image, the interaction can be easily controlled by in-creasing the simulation cell size in the x/y direction.

2.1.2. Screw dislocation modelCompared with achieving PBC in the gliding direction for an edge

dislocation, the implementation of PBC in the gliding direction for ascrew dislocation is more complicated. The schematic of constructing ascrew dislocation is shown in Fig. 2(a), also following the PAD model[36]. The initial displacements of a screw dislocation determined byisotropic elasticity theory [39] are given in Eq. (1),

⎜ ⎟= = ⎛⎝

⎞⎠

−u b θπ

b tan zy

·2

· /2x1

(1)

The dislocation line and Burger’s vector are along the x-direction,and the dislocation gliding direction is along the y-direction as shownin Fig. 2(a). PBC in the x-direction is achieved inherently because of theindependence of ux on x. However, PBC in the y-direction is brokenbecause of the± b/2 shift in the x-direction across the± y boundaries.To achieve PBC in the y-direction, a± b/2 displacement can be addedto the x coordinates when atoms cross the± y boundaries as shown inFig. 2(b). In Fig. 2(b), the blue lines represent the upper half y-z planes,and the red dotted lines represent the lower half y-z planes. Specifically,we can see that the x coordinates in the upper half crystal at the yboundary is shifted by 1/2 b, while the x coordinates in the lower halfcrystal at the y-boundary is shifted by −1/2 b. In addition, this is es-sentially equivalent to applying a shear strain to the simulation box,especially when the strain is small, which is employed in this study. So,a small shear strain is applied in the x-direction (y-axis is tilted) toachieve this shifted boundary condition. When Lx/Lz ratio is smaller,the effect of the tilted simulation cell on Peierls stress is minimized.After testing, Lx/Lz should be less than 0.5 in order to obtain the correctPeierls stress behavior.

2.2. Cascade simulations

The schematic of a cascade simulation cell is shown in Fig. 3. Thesimulation cell contains one dislocation in the center. The energy of thePKA is 10 KeV, and the simulation temperature is 300 K. When the PKAis placed close to the dislocation within 30 Å, the cascade event willdirectly interact with the dislocation. While when the PKA is placed faraway from the dislocation, the cascade will not interact with the dis-location so that it is the same as the scenario of cascade events in theperfect crystal. For each scenario, 12 independent cases are simulated.Within the 12 simulated cases, the position of the PKA changes whilethe direction of the PKA remains the same. The orientation of the si-mulation cell is based on the dislocation type. The details of the si-mulation cell orientations and the sizes of the simulation cells used inour study are shown in Table. 1. Based on the above orientation setup,the x-axis is along the dislocation line, and the y-axis is along the dis-location gliding direction for all the simulated systems. As mentionedpreviously, all the simulated systems have PBCs in the x and y direc-tions and a non-periodic condition in the z-direction.

LAMMPS is used as the molecular dynamics engine [40]. The systemcontains around 3.2 million atoms. The interatomic potential developedby Bonny et al. [41] is employed for fcc Fe-Ni-Cr alloy systems. Thealloy elements are randomly distributed. The Ziegler-Biersack-Littmark(ZBL) potential [42] is used for describing short-range interactions for ahigh-energy collision between atoms. During cascade simulations, anadaptive time step method [40], is used to ensure the maximum dis-placement of atoms per time step less than 0.014 Å.

2.3. Peierls stress simulation

Peierls stress simulations are carried out to investigate any potentialhardening effect caused by the interaction between dislocations andcascade events. The schematic of the Peierls stress simulation for edgeand screw dislocations are shown in Fig. 1(a) and Fig. 2(a) respectively.The Peierls stress simulations are carried out using molecular staticcalculations. Conjugate gradient is used to find the relaxed atomicconfigurations. For both of edge and screw dislocations, the top andbottom layers are fixed to be a rigid block. The thickness of the layerneeds to exceed the range of the interatomic potentials. We apply strainto the top rigid block and keep the bottom rigid block immobile. Theapplied shear strain at each step is 4 * 10−5. For edge dislocation asshown in Fig. 1(a), the applied strain is along the y-direction (Burger’svector direction), and the gliding direction is also along the y-direction.

1 For interpretation of color in Figs. 2, 4, and 6, the reader is referred to theweb version of this article.

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While for screw dislocation shown in Fig. 2(a), the applied strain isalong the x-direction (Burger’s vector direction) and the gliding direc-tion is along the y-direction.

2.4. Analysis and visualization tools

Common Neighbor Analysis (CNA) is used to distinguish the atomsin fcc, bcc, hcp and other environments [43,44]. The Wigner-Seitzanalysis is used to count the number of interstitials and vacanciescaused by the cascade [45]. OVITO is used as the visualization software[46], and the Dislocation Analysis Algorithm (DXA) implemented inOVITO is used to extract dislocation lines and identify their corre-sponding Burger’s vectors [47].

2.5. Defect counting method

The defect analysis in these systems which contain complicateddefects is delicate. We will demonstrate how the defects are analyzedafter the interaction between cascades and dislocations. Fig. 4 showsthe atomic structure of dislocations after cascade simulations. First, thetotal number of defects of interstitial (It) and vacancy (Vt) are calcu-lated using Wigner-Seitz analysis. As shown in Fig. 4 highlighted by thered circle, some portion of the defects move into dislocations or formdislocation loops well connected to the original dislocation. We definethis portion of the defects as Id and Vd. The other portion of the defects

Fig. 1. (a) Schematic of an edge dislocation in a simulation box. (b) The method used to create an edge dislocation.

Fig. 2. (a) Schematic of a screw dislocation in a simulation box. (b) Achieving periodic boundary condition by shifting x coordinates by +1/2 b or −1/2 b whenatoms cross the boundary in y-direction [36].

Fig. 3. Schematic of the cascade simulations.

Table 1Type of dislocations, orientations and size of the simulation cells.

x y Z

Fcc edge 1/2 ⟨1 1 0⟩{1 1 1}

⟨1 1−2⟩,27.2 nm

⟨1−1 0⟩,26.2 nm

⟨1 1 1⟩,53.1 nm

Fcc screw 1/2 ⟨1 1 0⟩{1 1 1}

⟨−1 1 0⟩,26.4 nm

⟨1−1 2⟩,27.2 nm

⟨1 1 1⟩, 53 nm

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created by cascades resides in the material’s matrix, which are definedas Im and Vm. So, It equals (Im+ Id) and Vt equals (Vm+Vd).

3. Results

3.1. Cascade simulation of Fe-20Cr-25Ni alloy

In this section, the cascade simulation results of Fe-20Cr-25Ni alloysystem are analyzed from two perspectives. One aspect is the defectproduction in terms of number of defects, interstitials, and vacancies.The other aspect is the cascade induced atomic structural changes of thedislocations.

3.1.1. The influence of dislocations on defect productionFig. 5 shows the number of defects generated by the cascade for Fe-

20Cr-25Ni alloy. The data point is the average number of defects oftwelve independent simulations. And the error bars represent thestandard deviation. In Fig. 5(a), “Close” means that the cascade inter-acts with the dislocation and “Far” means the cascade does not interactwith the dislocation. As described in Section 2.5, the defects that go intodislocations or form dislocation loops well connected to the originaldislocations are counted separately from the defects that formed in thematerials. From Fig. 5(a), it can be seen that cascades tend to generatemore total defects (It + Vt) when they interact with both edge andscrew dislocations in Fe-20Cr-25Ni. Due to the statistical variation, thedifference is insignificant. The interaction between cascades and

dislocations results in the structural changes of the dislocations, anddefects can go into dislocations and form dislocation loops intertwinedwith the original dislocations. Comparatively, Fig. 5 (b), there are lesssurviving defects which are still left in the materials after the cascades,excluding the defects that go into the dislocations or form dislocationloops well connected to the original dislocations (“1/2Close” vs Im). Ifwe only focus on the surviving defects generated in the matrix, thenumber of defects (Im plus Vm in Fig. 5(b)) is actually less than that inthe “Far” (non-interacting) cases for both edge and screw dislocations.So the defect production efficiency in the materials with high disloca-tion density is lower than the bulk case. For edge dislocations, an al-most equal number of interstitials (Id) and vacancies (Vd) move into thedislocation or form dislocation loops well connected to the originaldislocation as shown in Fig. 5(b). For screw dislocations, more inter-stitials move into dislocation or form dislocation loops well connectedto the original dislocation. Further discussions of defect populations aregiven in Section 3.1.2.

3.1.2. Atomic structural changes of dislocationsFig. 6 shows the atomic structures of edge and screw dislocations

before and after a cascade simulation for Fe-20Cr-25Ni alloy, using theCNA analysis to extract the dislocation, defect, and stacking faultstructures. DXA analysis is used to construct dislocations and extract theinformation on dislocation type and Burger’s vector. The Fe-20Cr-25Nialloy has a fcc lattice. As a result, the full dislocation slips into twoShockley partials connected by the stacking fault after relaxation. The

Fig. 4. Illustration of atomic structures of edge and screw dislocations after cascade simulations for Fe-20Cr-25Ni alloy. (a) The atomic structure of an edgedislocation after a cascade simulation. (b) The atomic structure of a screw dislocation after a cascade simulation. The atoms in the fcc environment have beenremoved. The black atoms are either interstitials or vacancies calculated by Wigner-Seitz analysis.

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dislocation dissociation can be described as → +[101̄] [21̄1̄] [112̄]12

16

16 .

As a result, the initial dislocation structures after relaxation before in-teracting with cascade for both edge and screw dislocations are twopartial dislocations with stacking faults in between as shown in Fig. 6(a)and (e).

For an edge dislocation, first, the cascade eliminates some of thestacking faults as shown in Fig. 6(b). The PKA introduces extra highenergy into the system. As a result, part of the stacking fault structurebecome amorphous as a result of the cascade processes and changes toperfect fcc lattice when it cools down. So the cascade has the potentialability to reduce the stacking fault structures. Second, the interactionbetween a cascade and an edge dislocation causes stair-rod dislocations(Lomer-Cottrell lock) to form, see the magenta dislocation line inFig. 6(b). The dislocation reaction to form Lomer-Cottrell lock can bedescribed as + →[1̄21] [21̄1̄] [110]1

616

16 . The stair-rod dislocation is

sessile. The formation of stair-rod dislocations can make the dislocationstructure more difficult to glide. In addition, a small amount of Hirthdislocations also forms. Third, the cascade facilitates dislocation climbas shown in Fig. 6(b) and (d). The dislocation climb shown in Fig. 6induced by the cascade is a negative climb. For the negative climb, an

additional row of atoms is added to the extra half plane. While for thepositive climb, a row of atoms is removed from the extra half plane. So,for a negative climb to happen, the edge dislocation needs to absorbinterstitials, while the edge dislocation needs to absorb vacancies tohave a positive climb. The preference of positive or negative climb isrelated to the type of defects that go into dislocations.

For screw dislocations, the interaction shares many similarities withedge dislocations. Part of the stacking fault structure is eliminated bythe cascade as shown in Fig. 6(f). The stair-rod dislocation also formsdue to the interaction between a screw dislocation and a cascade, asshown in Fig. 6(f). Unlike the dislocation climb of edge dislocations, thescrew dislocation may cross slip when interacting with a cascade asshown in Fig. 6(f) and (h). The screw dislocations cross slip from (1 1 1)plane to (1−1−1) plane. Out of the total 12 cases, five cases of thecross-slip behaviors have been observed. The formation of stair-roddislocations and the cross slip of screw dislocations may further hinderthe motion of the screw dislocations.

The structural changes of the dislocation cores are shown in Fig. 7.Three out of twelve cases of edge and screw dislocations are exhibited.For edge dislocations, from Fig. 7(a)–(c), both vacancy and interstitial

Fig. 5. The number of defects generated by the cascade for Fe-20Cr-25Ni, when the PKA is placed close to and far away from the dislocations. (a) For “Close” and“Far” cases, the number of defects equals to the number of interstitials (I) plus vacancies (V). (b) “1/2Close” and “1/2Far” represent the number of interstitials (I) orVacancies (V). The “Im” and “Vm” represent the number of interstitials and vacancies that do not move into dislocations nor form dislocation loops that wellconnected to the original dislocations, respectively.

Fig. 6. (a, b) The atomic structure of an edge dislocation. (c, d) The atomic structure of an edge dislocation from [1–10] perspective (parallel to the gliding plane). (e,f) The atomic structure of a screw dislocation. (g, h) The atomic structure of a screw dislocation from [1 1−2] perspective (parallel to the gliding plane).

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loops well connected to dislocations form as a result of dislocationclimb. The availability of vacancies and interstitials determines whe-ther positive climb (vacancy loop) or negative climb (interstitial loop)can form. In our 12 simulated cases of edge dislocations, the possibilityof a positive climb and a negative climb is almost equal. It explains thenearly equal number of interstitials (It− Im) and vacancies (Vt− Vm)being absorbed by edge dislocations in our 12 simulated cases. Forscrew dislocations, as shown from Fig. 7(d) to (f), more interstitial loopswell connected to the dislocations form in our 12 simulated cases,which explains fewer surviving interstitials (Im) in materials matrix.However, the surviving defect population and type of defects for in-dividual interaction may vary depending on the distance/position anddirection of the PKA with respect to the dislocation line.

3.2. Peierls stress simulation

After interacting with a cascade, the Peierls stress simulations arecarried out to evaluate the effects of displacement cascades on dis-location motion. For each scenario, three cases are tested. The testeddislocation structures contain the structures shown in Fig. 6 and twoadditional cases. Fig. 8 shows the comparison of Peierls stress beforeand after cascade. The stress required to move edge and screw dis-locations both increases. In Section 3.1, the assumption that the for-mation of stair-rod dislocations (sessile), radiation-induced dislocationclimb and cross-slip can contribute to dislocation hardening is

confirmed by the Peierls stress simulations. The structural changes ofthe dislocations caused by a cascade make it more difficult for dis-locations to glide or cross-slip. From the Peierls stress point of view,radiation-induced cross slip in screw dislocations causes more hard-ening than the radiation-induced dislocation climb in edge dislocations.From Fig. 9(a)–(b), the edge dislocation shows almost no bowing outbefore it starts to glide. While for screw dislocations when cross sliphappens, due to the pinning defect, the dislocation bowing out can beclearly observed as shown from Fig. 9(c)–(d). So the atomic structuralchanges of dislocations caused by cascades lead to more hardening forscrew dislocations than edge dislocations.

The Peierls stress simulation confirms the hardening effects as aresult of the atomic structural changes of dislocations caused by dis-placement cascades. And the number of surviving defects in materialsmatrix when interacting with dislocations is slightly less than that ofperfect bulk materials. So, when dislocations are present, the hardeningeffects estimated based on the traditional RIH models that consider theinteractions between dislocations and surviving defects in materialsmay decrease slightly. While, additional hardening effects arising fromthe atomic structural changes of dislocations cannot be ignored, espe-cially for screw dislocation in this material. It should be noted that thechanges in Peierls caused by the interactions with displacement cas-cades depend on many factors, such as the PKA energies, directions, anddistance between PKA and dislocations. To establish a quantitative re-lationship between these factors and radiation induced changes in

Fig. 7. CNA and DXA analysis of the dislocations’ structure after interacting with cascades. (a–c) Edge dislocation. (e–f) Screw dislocation. Atoms in fcc environmenthave been removed.

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Peierls stress requires further systematic studies.

4. Summary

Cascade simulations and Peierls stress simulations are carried out tostudy the interaction between displacement cascades and edge andscrew dislocations in Fe-20Cr-25Ni alloy. Due to the presence of dis-locations, the defect production efficiency changes. The cascade tendsto generate more total defects when interacting with dislocations.However, a portion of these defects is in the form of defect clusters ordislocation loops that are entangled with the original dislocations, re-sulting in less surviving defects in the materials matrix compared withthat of the non-interacting (pristine case). Since the surviving defects inthe materials matrix may migrate and interact with other defects in thesystem, leading to microstructural evolution and properties changes,this change in defect production efficiency needs to be properly con-sidered in upper scale models, such as mean-field rate theory. In ad-dition, dislocation climb and cross-slip have been observed for edgedislocations and screw dislocations, respectively, because of displace-ment cascades. The sessile stair-rod dislocations also form because ofinteractions between cascades and dislocations.

The dislocation core structures also undergo significant changesbecause of the cascade, which serve as additional contributions to RIH

compared to the conventional dislocation-obstacle interaction models.The Peierls stress simulations confirm this additional hardening effect.The interaction between cascades and other defects in materials, suchas grain boundaries, voids, and precipitates, may also influence thedefect production efficiency and microstructural changes of irradiatedmaterials. Therefore, this study provides insight into how defect pro-duction and extended defects are correlated, especially when total in-fluence is high, which needs to be taken into account for modeling long-term performance of irradiated structural alloys.

CRediT authorship contribution statement

Zizhe Lu: Data curation, Formal analysis, Investigation,Methodology, Validation, Visualization, Writing - original draft,Writing - review & editing. Liubin Xu: Formal analysis, Investigation,Methodology, Writing - review & editing. Tianyi Chen: Investigation,Writing - review & editing. Lizhen Tan: Funding acquisition,Investigation, Writing - review & editing. Haixuan Xu:Conceptualization, Formal analysis, Funding acquisition,Investigation, Methodology, Project administration, Resources,Software, Supervision, Validation, Visualization, Writing - review &editing.

Acknowledgement

This material is based upon work supported by the U.S. Departmentof Energy, Office of Nuclear Energy, the Nuclear Energy UniversityProgram (project no. 14-6346) under contract number DE-NE0008271.This research used resources of the National Energy Research ScientificComputing Center, a DOE Office of Science User Facility supported bythe Office of Science of the U.S. Department of Energy under ContractNo. DE-AC02-05CH11231.

Data availability

The raw/processed data required to reproduce these findings cannotbe shared at this time as the data also forms part of an ongoing study.

Appendix A. Supplementary material

Supplementary data to this article can be found online at https://doi.org/10.1016/j.commatsci.2018.12.018.

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Fig. 8. Comparison of Peierls stress before and after cascade for Fe-20Cr-25Ni alloy. (a) Edge dislocation. (b) Screw dislocation.

Fig. 9. Atomic structure of edge and screw dislocations in Peierls stress simu-lation. (a, b) Edge dislocation before and after applying shear strain. (c, d)Screw dislocation before and after applying shear strain.

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