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The effects of interaction geometry on pinning strength induced by interstitial dislocation loop in BCC-Fe
Nuclear Inst. and Methods in Physics Research B 456 (2019) 103–107
Contents lists available at ScienceDirect
Nuclear Inst. and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
The effects of interaction geometry on pinning strength induced by interstitial dislocation loop in BCC-Fe Lixia Jia, Xinfu He , Yankun Dou, Dongjie Wang, Shi Wu, Han Cao, Wen Yang ⁎
T
⁎
China Institute of Atomic Energy, P.O.Box 275-51, 102413 Beijing, People’s Republic of China
ARTICLE INFO
ABSTRACT
Keywords: Irradiation hardening Pinning strength Dislocation loop Molecular dynamics Edge dislocation
Irradiation induced dislocation loops in Fe-based alloys can cause material hardening by impeding the mobility of dislocation. Using molecular dynamics (MD), the interaction between 1/2 < 1 1 1 > {1 1¯ 0} edge dislocation (ED) and 1/2 < 1 1 1 > interstitial dislocation loops with Burgers vector (BV) along different directions in BCC (body centered cubic)-Fe were investigated, and the influence of interaction geometry (means the relative position between dislocation glide plane (GP) and loop equator plane) on the CRSS (critical resolved shear stress) for dislocation to move again induced by loops was analyzed. It was found that the pinning strength of 1/ 2 < 1 1 1 > loop was estimated to be 0.2–0.8, varying with the BVs direction of loops and the interaction geometries. The present study clearly showed that when parameterizing the dislocation dynamics, more interaction geometries should be considered in order to get a more quantitative pinning strength.
1. Introduction
penetrating obstacle, the relative position between dislocation GP and loop equator plane may be arbitrary. Hatano et al. [4] and Haghighat et al. [5] used MD method to investigate the dependence of the cutting stress (CRSS) on the distance from the center of voids, and they found that the cutting stress decreased with increasing distance. Using MD, Haghighat [6] found that the interaction geometry between dislocation GP and center of He bubble had influence on the release stress for dislocation. Grammatikopoulos [7] had investigated the influence of interaction geometry on the pinning strength of voids and copper precipitates in iron. By comparing the strength factor of voids in BCC-Fe estimated by Orowan equation and by direct TEM approach, Ryosuke Nakai deem that the difference was due to the positional relationship between dislocations and voids [8]. These results lead to the conclusion that different positional relationship between microstructure and dislocation GP possess different pinning strength. For purpose of getting the accurate pinning strength for dislocation dynamics simulation, the pinning strength induced by loops should consider the effects of interaction geometry. By now, it is not clear, or other, it has not been studied systematically. In this study, using MD method, the interaction between ED and interstitial dislocation loops were studied at the same temperature and strain rate, and five different geometries were considered.
Iron-chromium ferritic/martensitic (F/M) steels are considered as candidate structural materials for the next generation of nuclear reactors due to their swelling, corrosion and creep resistance. However, during service, low temperature irradiation-induced embrittlement, caused by vast increase of DBTT (ductile-to-brittle transition temperature), will influence the safety of nuclear plants. To predict the performance of F/M steels during service, it is important to know the embrittlement mechanism. The motion of dislocation will be impeded by defects, such as voids, loops, precipitate, on or near their slip plane, thereby causing materials hardening and loss of ductility, leading to embrittlement. Upon irradiation, a mixed population of interstitial dislocation loops with Burgers vector (BV) = 1/2 < 1 1 1 > and < 1 0 0 > can be found in ferritic steels. Herein we firstly focused on the interaction between an edge dislocation (ED) and 1/2 < 1 1 1 > loops. Terentyev [1] found that the solute Cr in matrix mainly affect the friction stress, not the actual effect of the obstacle. So, for simplicity, we consider their interaction in pure BCC-Fe. Molecular dynamics (MD) can be used to study dislocation-loop interactions. A lot of work had been done on the study of interaction between dislocation and dislocation loop [2,3]. In general, the critical resolved shear stress (CRSS) increases with increasing loop size and decreasing temperature. For most cases, the center of the loop were below the slip plane. However, when
⁎
Corresponding authors. E-mail addresses: [email protected] (X. He), [email protected] (W. Yang).
https://doi.org/10.1016/j.nimb.2019.06.024 Received 14 August 2018; Received in revised form 5 May 2019; Accepted 17 June 2019 0168-583X/ © 2019 Elsevier B.V. All rights reserved.
Nuclear Inst. and Methods in Physics Research B 456 (2019) 103–107
L. Jia, et al.
Fig. 1. Schematic picture of the simulation MD box (a). Schematic representation of the five different geometries studied for the interaction between ED and loops (b). The gray circle represents the loop. The black symbol is the ED.
2. Simulation methods
Fig. 2. Stress-Strain plots for 1/2[1 1 1] loops.
The simulation model developed by Osetsky and Bacon [9] was employed. The principal axes x, y and z of the simulated box were oriented along the [1 1 1], [1¯ 1¯ 2] and [1 1¯ 0] directions, respectively. Periodic boundary conditions were applied along the x and y directions. Along z direction the box was divided into three parts: the upper and the lower parts were consisted of several fixed atomic planes, whereas atoms in the inner region were free to move in the MD cycles. The Burgers vector (BV) of the ED (1/2 < 1 1 1 > {1 1¯ 0} ) was along x direction and the dislocation line was along y direction. Four kinds of 1/2 < 1 1 1 > interstitial dislocation loops (with diameter = 4 nm) were considered, as shown in Fig. 1(a). These loops had the shape of hexagonal. The size of inner region of the box was chosen to be 160 × 3, 20 × 6, 24 × 2 atomic planes along the x, y and z, respectively. The volume of the box was 40 × 14 × 10 nm3 and contained about 0.5 × 106 freely mobile atoms. The size of the box is large enough for simulating the interaction between the ED and loop with 4 nm diameter. The length of the box ensures that there is no interaction between dislocations along the x, no interaction between loops along the y. The loops were placed about 12 nm away from the ED to avoid a strong interaction with ED before it glides. The five different geometries (represents different relative position between loop equator plane and ED GP) were showed in Fig. 1(b). Here, R is the radius of loop. Model R(-R) means the equator plane of loop is higher (lower) of R than the ED GP. For model R/2(-R/2), the equator plane of loop is located at R/2 above (below) the GP. The equator plane of loop lies on the GP is model 0. The simulation temperature is 600 K and the applied strain rate is 5 × 107 s−1. Through the relative displacement of the upper rigid blocks in the x direction, the glide force on the ED was generated, which represent simple shear strain. The corresponding resolved shear stress induced by the applied deformation was calculated as = Fx / Axy , Fx is the total force along the x direction on the lower outer block from all atoms in the inner region, it can be got by adding the forces of all atoms in the mobile region along x direction; and Axy is the area of the x-y plane. Based on the analysis of atomic disregistry, the location of the ED core atoms can be determined, thus helping visualize the interaction between ED and loops. The many-body interatomic potential for Fe developed by Ackland [10] was used for all simulations here.
temperature of 600 K, the interaction geometry is model -R. Owing to a long distance (12 nm) between the ED and loop, when applied glide force, the ED started gliding along its GP in an otherwise perfect crystal with less shear stress. Once the ED was near the loop (without contacting the loop physically), their stress fields interacted with each other. As the ED started to pass through the loop, the stress decreased. It is because that the plastic strain of crystal induced by ED movement is larger than the imposed strain [11]. The ED was pinned by the loop, so it needs more shear stress to overcome obstruction interaction applied by the loop. The shear stress for the ED to start slip again is called the critical resolved shear stress (CRSS). After interacting with the loop and breaking away from it, the stress decreased rapidly. The forms of stressstrain curves are similar for these loops, and the difference is the value of CRSS. The BV of 1/2[1 1 1] loop is parallel with the ED GP, and the direction is same as the BV of ED. The migration energy of this loop is very low, about 0.05 eV [3]. It is very easy to be pushed or dragged by the ED to move together, thus showing weak obstacle. For the other loops with different BV, the CRSS is higher. After analyzed the CRSS for different type of loop, we found that the dependence of the reaction mechanism on the relative position were different with BV of the loop. For the 1/2[1 1 1] loop, as said before, it showed weak obstacle to dislocation movement. The difference between CRSS for different position is very small, and it is equal to the friction stress for the dislocation to move in the matrix. Known from the reference [3], when the center of loop is below (or above) the glide plane, the glide force induced by the dislocation on loops is dependent on the distance between center of loops and glide plane: the distance is bigger, the glide force is smaller. 3.1. The interaction between ED and 1/2[1 1¯ 1¯] loop BVs of both 1/2[1 1 1] loop and 1/2[1 1¯ 1¯] loop are paralleled to the GP of ED. So they can move when interacting with ED. Different from the 1/2[1 1 1] loop, whose BV is in the same direction with BV of ED, the glide direction of 1/2[1 1¯ 1¯] loop is at 70.5° to that of ED. Fig. 3(a) represents the spatial relationship between BVs of ED and 1/2[1 1¯ 1¯] loop in the [1 1¯ 0] projection. When within the stress field of the ED, the interaction force between the ED and 1/2[1 1¯ 1¯] loop has component along the y axis, which leading to the glide of loop along the ED in the y direction. It needs more stress for 1/2[1 1¯ 1¯] loop to move along the x axis then the 1/2[1 1 1] loop, thus higher CRSS. Z. rong [3], found that when the 1/2[1 1¯ 1¯] loop center was placed below the GP of ED, it can be dragged together with the moving ED, in an interaction process similar
3. Results and discussions By analyzing stress-strain curves, the interaction process can be clearly seen. Fig. 2 presents the applied stress as function of the applied strain, for 1/2[1¯ 1 1] dislocation loop with 4 nm diameter at the 104
Nuclear Inst. and Methods in Physics Research B 456 (2019) 103–107
L. Jia, et al.
with the 1/2[1 1 1] loop. In this work, for model -R and -R/2, where the loop center is below the GP, the loop can move together with the ED. For Model 0, R/2 and R, where the loop center is on or above the GP, a different interaction process is shown: ED just break away from the loop and move again lonely leaving the loop behind. Fig. 3(b) shows the dependence of CRSS for ED to breaking away from 1/2[1 1 1] loop on interaction geometry. Knowing from it, for 1/2[1 1¯ 1¯] loop, the strongest obstacle was -R, followed by -R/2, 0, R/2 and R. The interaction between Cu precipitates and ED showed the similar result: the lower down the precipitate was located, the more it resisted ED motion, due to the different stress state (compressive stress or tensile stress) above or below the ED GP [7]. It can be speculated here that the strength order for different position is related with the ED stress field. For model 0, R/2 and R, more parts of the loop is above the GP, means on the compressive stress field; while for model -R/2 and -R, more parts of the loop is on the tensile stress field. Tensile stress induced the absorption of loop by ED in order to balance the hydrostatic pressure caused by the extra atomic plane, while the movement for the ensemble (ED plus loop) needs much stress, thus higher CRSS. 3.2. The interaction between ED and 1/2[1 1¯ 1] loop, 1/2[1¯ 1 1] loop The BVs of these two loops are inclined to the GP of ED, and it means that they are sessile. When interacting with the ED, a segment with BV of [0 1 0] [1,12] or [1 0 0] was formed respectively for 1/2[1 1¯ 1] loop and 1/2[1¯ 1 1] loop. The formed < 0 0 1 > segment was sessile in the GP of ED. The ED was pinned by the sessile segment, bow between the loop, and move forward difficultly. Under external strain, a dipole of two short screw segments was formed at the bowed ED. With the increasing stress, the < 0 0 1 > segment started gliding and screw segments began crossing-slip [12]. Due to high temperature used in this work (600 K), the mobility of < 0 1 0 > segment and screw segments were enhanced by thermal vibrations, Finally, the remaining part of the loop transformed into a pair of superjogs in the ED, and moved together with it, leaving some vacancies behind. Fig. 4 shows the atom-core visualization of the interaction of 1/2[1 1¯ 1] loop with the ED. The angle between BVs of 1/2[1¯11] loop and ED is obtuse, while the angle between BVs of 1/2[1¯ 1 1] loop and ED is acute. The angle
Fig. 3. Schematic representation of the spatial relationship between ED and
1/2[111] loop in [110]projection (a). Dependence of CRSS on interaction geo-
metry for 1/2[111] loop (b).
Fig. 4. Atom-core visualization of the interaction of 1/2[111] loop with the ED.
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Nuclear Inst. and Methods in Physics Research B 456 (2019) 103–107
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Table 1 The critical angles (°) for 1/2[1 1 1] loop and 1/2[1 1 1] loop at different models. Model
R
R/2
0
-R/2
-R
1/2[1 1 1]
79.1
78.3
129.1
48.1
86.8
53.1
74.3
0
66.2
0
1/2[1 1 1]
relationship determined that their interaction with ED are symmetrical. Fig. 5 shows the dependence of CRSS on interaction geometry for these two loops. It is clearly that the curve shape is inverse for these two loops. The CRSS is corresponding with the critical line shape for ED passing through the loop. Fig. 6 shows the line shape of ED just before breaking away from these two loops with different interaction geometry. By analyzing the critical line shape, the angel between bowing ED can be got. The critical angles for different models are shown in Table 1. For a periodic array of the obstacle, the critical resolved shear stress (CRSS) for ED to break away from the obstacles can be represented by Gb [13]: c = L cos 2c , where G is the shear modulus; b represents the BV of ED; L is the spacing between obstacles, here means loops, c is the critical angle between two tangential vectors of the bowing ED. The pinning strength, characterized by the critical angle, is an important parameter to analyze extent of hardening [4,8]. Knowing from this formula, smaller angle represents higher pinning strength, thus higher CRSS. For 1/2[1 1¯ 1] loop, due to largest angle, model 0 shows weakest pinning strength; while for 1/2[1¯11] loop, CRSS is highest at model 0 because of smallest angle.
( )
4. Discussion It is the interaction force between dislocation and loop, who leads to the above results. For the 1/2[1 1 1], 1/2[1 1¯ 1] and 1/2[1¯ 1 1] loop, the interaction mechanism is independent on the relative position between loop equator and GP of ED. To assist the discussion, it is helpful to know the force on the loop as a function of distance from the glide plane. Using the infinitesimal loop approximation, the interaction energy between an ED and a loop of area A is E= bA, is the normal component of the ED stress field in the direction of b of the loop. The force on the
Fig. 5. Dependence of CRSS on interaction geometry for 1/2[1 1 1] loop (a) and
1/2[1 1 1] loop (b).
Fig. 6. Critical line shape for ED passing through the 1/2[1 1 1] loop (a) and 1/2[1 1 1] loop (b) with different interaction geometry. 106
Nuclear Inst. and Methods in Physics Research B 456 (2019) 103–107
L. Jia, et al.
work mainly focuses on the interaction between ED and 1/ 2 < 1 1 1 > loops, while screw dislocation and < 1 0 0 > loops co-exist in real materials, their interactions also contribute to the pinning strength of loops in experimental results; finally, the alloy composition is very complex, alloy elements can also affect the results; although alloy elements in solution state mainly affect the friction stress, while alloy elements segregated at loop may affect the interaction between dislocation and loop, for instance it is difficult for dislocation to break away from loop with Cr segregation. This work is the preliminary study of the effects of interaction geometry on pinning strength induced by interstitial dislocation loop. And the results showed that when parameterizing the dislocation dynamics simulation, the randomness relationship between obstacle and dislocation should be considered. Here we only considered five models. The dislocation is pinned even when it is not in contact with the obstacle. For many kinds of obstacle (loops, void, He bubble, precipitate), systematic researches should be done in order to study the dislocation motion and the resulting material hardening accurately. Fig. 7. Force on the 1/2[1 1 1] loop at model R and R/2.
5. Conclusion
Table 2 Pinning strength for different BV of 1/2 < 1 1 1 > loops at different models.
1/2[1 1 1]
1/2[1 1 1] 1/2[1 1 1] 1/2[1 1 1]
R
R/2
0
-R/2
-R
0.10 0.18
0.09 0.33
0.07 0.48
0.10 0.50
0.10 0.82
0.63
0.61
0.32
0.73
0.51
0.78
0.58
0.82
0.58
0.83
Based on the MD, the influence of interaction geometry on pinning strength induced by 1/2 < 1 1 1 > interstitial dislocation loop was studied. By analyzing the CRSS for ED to overcome loop with different type at different model, it is concluded that the CRSS is related with the relative geometry, and the relation is different for different BVs of loops. For 1/2[1 1 1] loop, it showed weak obstacle due to its easy mobility together with the ED. For 1/2[1 1 1] loop, the interaction mechanism is different when the equator plane of loop is above or below the GP, CRSS is lower when more parts of loop is on the compressive field of the ED. The trends for the variation of CRSS with interaction geometries is inverse for 1/2[1 1 1] and 1/2[1 1 1] loop owing to the angles between their BVs and BV of ED is different. Based on the calculated CRSS and the hardening model, the estimated pinning strength due to dislocation loop is 0.2–0.8.
loop in the ED slip direction is then given byF= dE / dx . The stress filed of ED is symmetrical about the GP, so is the force. It is the reason why the curve shape of relationship between CRSS and relative position is symmetrical for these three loops (see Fig. 5). Fig. 7 shows force on the 1/2[1 1 1] loop at model R and R/2, it is clearly that when at model R, the force is lower than at R/2, thus lower CRSS. Due to the different angle between the loop and ED, dependence of CRSS on relative position for 1/2[1 1 1] loop showed the reversed trend to the 1/2[1 1 1] loop. While for the 1/2[1 1 1] loop, the interaction mechanism is dependent on the relative position between loop equator and GP of ED, the curve shape for the dependence of CRSS on position is not symmetrical. Knowing from above results, interaction geometry has influence on the CRSS for ED to break away from interstitial dislocation loop. The hardening induced by the interstitial loop can be expressed by equation [8] c = Gb (Nd )1/2 ,where c is the critical shear stress, G is the shear modulus (82 GPa), b is the magnitude of Burgers vector (2.49 Å), N and d represent the number density and diameter of the loop, respectively. is the strength factor of loops. Using the calculated CRSS, size and density of the loops in the simulation box, the pinning strength of the 1/ 2 < 1 1 1 > loops with 4 nm diameter at different geometries was calculated, as shown in Table 2. 1/2[1 1 1] loop is the weakest obstacle, nearly no hinder impact on the mobility of dislocation. For the other loops, from this table, it is clearly that when interacting with different geometry, the pinning strength induced by interstitial loop showed large variance, within the range of 0.2–0.8. Pinning strength factors of 0.15–0.6 for dislocation loops have been reported by experimental studies [14,15]. Difference between this work and reference results exists. There are some reasons for this, firstly, owing to the limitation of computing technology and algorithm development, the strain rate in this simulation is 5 × 107 s−1, which is typically orders of magnitude higher than the experimental ones (10−2 s−1), higher strain rate results in higher CRSS; secondly, this
Acknowledgements This research was supported by the National Key Research and Development Program of China (Grant Number: 2017YFB0202304), and partially supported by National Natural Science Foundation of China (Grant Number: U1867217, 11575295 and 11375270) and the CNNC Centralized Research and Development Project (FA16100820). References [1] D. Terentyev, L. Malerba, D.J. Bacon, Y.N. Osetsky, J. Phys.: Condens. Matter 19 (2007) 456211. [2] D.J. Bacon, Y.N. Osetsky, Z. Rong, Phil. Mag. 86 (2006) 3921–3936. [3] Z. Rong, Y.N. Osetsky, D.J. Bacon, Phil. Mag. 85 (2005) 1473–1493. [4] T. Hatano, H. Matsui, Phys. Rev. B. 72 (2005) 094105. [5] S.M. Hafez Haghighat, M.C. Fivel, J. Fikar, R. Schaeublin, J. Nucl. Mater. 386–388 (2009) 102–105. [6] S.M. Hafez Haghighat, R. Schaeublin, Phil. Mag. 90 (2010) 1075–1100. [7] P. Grammatikopoulos, D.J. Bacon, Y.N. Osetsky, Modelling simul. Mater. Sci. Eng. 19 (2011) 015004. [8] R. Nakaia, K. Yabuuchi, S. Nogamia, A. Hasegawa, J. Nucl. Mater. 471 (2016) 233–238. [9] D.J. Bacon, Y.N. Osetsky, Modelling Simul. Mater. Sci. Eng. 11 (2003) 427–446. [10] G.J. Ackland, M.I. Mendelev, D.J. Srolovitz, D. Han, A.V. Barashev, J. Phys.: Condens. Matter 16 (2004) S2629–S2642. [11] D.J. Bacon, Y.N. Osetsky, Phil. Mag. 89 (2009) 3333–3349. [12] D. Terentyev, Y.N. Osetsky, D.J. Bacon, Acta Mater. 58 (2010) 2477–2482. [13] D. Terentyev, S.M. Hafez Haghighat, R. Schaeublin, J. Appl. Phys. 107 (2010) 061806. [14] E. Wakai, A. Hishinuma, K. Usami, Y. Kato, T. Seiichi, K. Abiko, Mater. Trans. 41 (2000) 1180–1183. [15] F. Bergner, C. Pareige, M. Hernández-Mayoral, L. Malerba, C. Heintze, J. Nucl. Mater. 448 (2014) 96–102.
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Contents lists available at ScienceDirect
Nuclear Inst. and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
The effects of interaction geometry on pinning strength induced by interstitial dislocation loop in BCC-Fe Lixia Jia, Xinfu He , Yankun Dou, Dongjie Wang, Shi Wu, Han Cao, Wen Yang ⁎
T
⁎
China Institute of Atomic Energy, P.O.Box 275-51, 102413 Beijing, People’s Republic of China
ARTICLE INFO
ABSTRACT
Keywords: Irradiation hardening Pinning strength Dislocation loop Molecular dynamics Edge dislocation
Irradiation induced dislocation loops in Fe-based alloys can cause material hardening by impeding the mobility of dislocation. Using molecular dynamics (MD), the interaction between 1/2 < 1 1 1 > {1 1¯ 0} edge dislocation (ED) and 1/2 < 1 1 1 > interstitial dislocation loops with Burgers vector (BV) along different directions in BCC (body centered cubic)-Fe were investigated, and the influence of interaction geometry (means the relative position between dislocation glide plane (GP) and loop equator plane) on the CRSS (critical resolved shear stress) for dislocation to move again induced by loops was analyzed. It was found that the pinning strength of 1/ 2 < 1 1 1 > loop was estimated to be 0.2–0.8, varying with the BVs direction of loops and the interaction geometries. The present study clearly showed that when parameterizing the dislocation dynamics, more interaction geometries should be considered in order to get a more quantitative pinning strength.
1. Introduction
penetrating obstacle, the relative position between dislocation GP and loop equator plane may be arbitrary. Hatano et al. [4] and Haghighat et al. [5] used MD method to investigate the dependence of the cutting stress (CRSS) on the distance from the center of voids, and they found that the cutting stress decreased with increasing distance. Using MD, Haghighat [6] found that the interaction geometry between dislocation GP and center of He bubble had influence on the release stress for dislocation. Grammatikopoulos [7] had investigated the influence of interaction geometry on the pinning strength of voids and copper precipitates in iron. By comparing the strength factor of voids in BCC-Fe estimated by Orowan equation and by direct TEM approach, Ryosuke Nakai deem that the difference was due to the positional relationship between dislocations and voids [8]. These results lead to the conclusion that different positional relationship between microstructure and dislocation GP possess different pinning strength. For purpose of getting the accurate pinning strength for dislocation dynamics simulation, the pinning strength induced by loops should consider the effects of interaction geometry. By now, it is not clear, or other, it has not been studied systematically. In this study, using MD method, the interaction between ED and interstitial dislocation loops were studied at the same temperature and strain rate, and five different geometries were considered.
Iron-chromium ferritic/martensitic (F/M) steels are considered as candidate structural materials for the next generation of nuclear reactors due to their swelling, corrosion and creep resistance. However, during service, low temperature irradiation-induced embrittlement, caused by vast increase of DBTT (ductile-to-brittle transition temperature), will influence the safety of nuclear plants. To predict the performance of F/M steels during service, it is important to know the embrittlement mechanism. The motion of dislocation will be impeded by defects, such as voids, loops, precipitate, on or near their slip plane, thereby causing materials hardening and loss of ductility, leading to embrittlement. Upon irradiation, a mixed population of interstitial dislocation loops with Burgers vector (BV) = 1/2 < 1 1 1 > and < 1 0 0 > can be found in ferritic steels. Herein we firstly focused on the interaction between an edge dislocation (ED) and 1/2 < 1 1 1 > loops. Terentyev [1] found that the solute Cr in matrix mainly affect the friction stress, not the actual effect of the obstacle. So, for simplicity, we consider their interaction in pure BCC-Fe. Molecular dynamics (MD) can be used to study dislocation-loop interactions. A lot of work had been done on the study of interaction between dislocation and dislocation loop [2,3]. In general, the critical resolved shear stress (CRSS) increases with increasing loop size and decreasing temperature. For most cases, the center of the loop were below the slip plane. However, when
⁎
Corresponding authors. E-mail addresses: [email protected] (X. He), [email protected] (W. Yang).
https://doi.org/10.1016/j.nimb.2019.06.024 Received 14 August 2018; Received in revised form 5 May 2019; Accepted 17 June 2019 0168-583X/ © 2019 Elsevier B.V. All rights reserved.
Nuclear Inst. and Methods in Physics Research B 456 (2019) 103–107
L. Jia, et al.
Fig. 1. Schematic picture of the simulation MD box (a). Schematic representation of the five different geometries studied for the interaction between ED and loops (b). The gray circle represents the loop. The black symbol is the ED.
2. Simulation methods
Fig. 2. Stress-Strain plots for 1/2[1 1 1] loops.
The simulation model developed by Osetsky and Bacon [9] was employed. The principal axes x, y and z of the simulated box were oriented along the [1 1 1], [1¯ 1¯ 2] and [1 1¯ 0] directions, respectively. Periodic boundary conditions were applied along the x and y directions. Along z direction the box was divided into three parts: the upper and the lower parts were consisted of several fixed atomic planes, whereas atoms in the inner region were free to move in the MD cycles. The Burgers vector (BV) of the ED (1/2 < 1 1 1 > {1 1¯ 0} ) was along x direction and the dislocation line was along y direction. Four kinds of 1/2 < 1 1 1 > interstitial dislocation loops (with diameter = 4 nm) were considered, as shown in Fig. 1(a). These loops had the shape of hexagonal. The size of inner region of the box was chosen to be 160 × 3, 20 × 6, 24 × 2 atomic planes along the x, y and z, respectively. The volume of the box was 40 × 14 × 10 nm3 and contained about 0.5 × 106 freely mobile atoms. The size of the box is large enough for simulating the interaction between the ED and loop with 4 nm diameter. The length of the box ensures that there is no interaction between dislocations along the x, no interaction between loops along the y. The loops were placed about 12 nm away from the ED to avoid a strong interaction with ED before it glides. The five different geometries (represents different relative position between loop equator plane and ED GP) were showed in Fig. 1(b). Here, R is the radius of loop. Model R(-R) means the equator plane of loop is higher (lower) of R than the ED GP. For model R/2(-R/2), the equator plane of loop is located at R/2 above (below) the GP. The equator plane of loop lies on the GP is model 0. The simulation temperature is 600 K and the applied strain rate is 5 × 107 s−1. Through the relative displacement of the upper rigid blocks in the x direction, the glide force on the ED was generated, which represent simple shear strain. The corresponding resolved shear stress induced by the applied deformation was calculated as = Fx / Axy , Fx is the total force along the x direction on the lower outer block from all atoms in the inner region, it can be got by adding the forces of all atoms in the mobile region along x direction; and Axy is the area of the x-y plane. Based on the analysis of atomic disregistry, the location of the ED core atoms can be determined, thus helping visualize the interaction between ED and loops. The many-body interatomic potential for Fe developed by Ackland [10] was used for all simulations here.
temperature of 600 K, the interaction geometry is model -R. Owing to a long distance (12 nm) between the ED and loop, when applied glide force, the ED started gliding along its GP in an otherwise perfect crystal with less shear stress. Once the ED was near the loop (without contacting the loop physically), their stress fields interacted with each other. As the ED started to pass through the loop, the stress decreased. It is because that the plastic strain of crystal induced by ED movement is larger than the imposed strain [11]. The ED was pinned by the loop, so it needs more shear stress to overcome obstruction interaction applied by the loop. The shear stress for the ED to start slip again is called the critical resolved shear stress (CRSS). After interacting with the loop and breaking away from it, the stress decreased rapidly. The forms of stressstrain curves are similar for these loops, and the difference is the value of CRSS. The BV of 1/2[1 1 1] loop is parallel with the ED GP, and the direction is same as the BV of ED. The migration energy of this loop is very low, about 0.05 eV [3]. It is very easy to be pushed or dragged by the ED to move together, thus showing weak obstacle. For the other loops with different BV, the CRSS is higher. After analyzed the CRSS for different type of loop, we found that the dependence of the reaction mechanism on the relative position were different with BV of the loop. For the 1/2[1 1 1] loop, as said before, it showed weak obstacle to dislocation movement. The difference between CRSS for different position is very small, and it is equal to the friction stress for the dislocation to move in the matrix. Known from the reference [3], when the center of loop is below (or above) the glide plane, the glide force induced by the dislocation on loops is dependent on the distance between center of loops and glide plane: the distance is bigger, the glide force is smaller. 3.1. The interaction between ED and 1/2[1 1¯ 1¯] loop BVs of both 1/2[1 1 1] loop and 1/2[1 1¯ 1¯] loop are paralleled to the GP of ED. So they can move when interacting with ED. Different from the 1/2[1 1 1] loop, whose BV is in the same direction with BV of ED, the glide direction of 1/2[1 1¯ 1¯] loop is at 70.5° to that of ED. Fig. 3(a) represents the spatial relationship between BVs of ED and 1/2[1 1¯ 1¯] loop in the [1 1¯ 0] projection. When within the stress field of the ED, the interaction force between the ED and 1/2[1 1¯ 1¯] loop has component along the y axis, which leading to the glide of loop along the ED in the y direction. It needs more stress for 1/2[1 1¯ 1¯] loop to move along the x axis then the 1/2[1 1 1] loop, thus higher CRSS. Z. rong [3], found that when the 1/2[1 1¯ 1¯] loop center was placed below the GP of ED, it can be dragged together with the moving ED, in an interaction process similar
3. Results and discussions By analyzing stress-strain curves, the interaction process can be clearly seen. Fig. 2 presents the applied stress as function of the applied strain, for 1/2[1¯ 1 1] dislocation loop with 4 nm diameter at the 104
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with the 1/2[1 1 1] loop. In this work, for model -R and -R/2, where the loop center is below the GP, the loop can move together with the ED. For Model 0, R/2 and R, where the loop center is on or above the GP, a different interaction process is shown: ED just break away from the loop and move again lonely leaving the loop behind. Fig. 3(b) shows the dependence of CRSS for ED to breaking away from 1/2[1 1 1] loop on interaction geometry. Knowing from it, for 1/2[1 1¯ 1¯] loop, the strongest obstacle was -R, followed by -R/2, 0, R/2 and R. The interaction between Cu precipitates and ED showed the similar result: the lower down the precipitate was located, the more it resisted ED motion, due to the different stress state (compressive stress or tensile stress) above or below the ED GP [7]. It can be speculated here that the strength order for different position is related with the ED stress field. For model 0, R/2 and R, more parts of the loop is above the GP, means on the compressive stress field; while for model -R/2 and -R, more parts of the loop is on the tensile stress field. Tensile stress induced the absorption of loop by ED in order to balance the hydrostatic pressure caused by the extra atomic plane, while the movement for the ensemble (ED plus loop) needs much stress, thus higher CRSS. 3.2. The interaction between ED and 1/2[1 1¯ 1] loop, 1/2[1¯ 1 1] loop The BVs of these two loops are inclined to the GP of ED, and it means that they are sessile. When interacting with the ED, a segment with BV of [0 1 0] [1,12] or [1 0 0] was formed respectively for 1/2[1 1¯ 1] loop and 1/2[1¯ 1 1] loop. The formed < 0 0 1 > segment was sessile in the GP of ED. The ED was pinned by the sessile segment, bow between the loop, and move forward difficultly. Under external strain, a dipole of two short screw segments was formed at the bowed ED. With the increasing stress, the < 0 0 1 > segment started gliding and screw segments began crossing-slip [12]. Due to high temperature used in this work (600 K), the mobility of < 0 1 0 > segment and screw segments were enhanced by thermal vibrations, Finally, the remaining part of the loop transformed into a pair of superjogs in the ED, and moved together with it, leaving some vacancies behind. Fig. 4 shows the atom-core visualization of the interaction of 1/2[1 1¯ 1] loop with the ED. The angle between BVs of 1/2[1¯11] loop and ED is obtuse, while the angle between BVs of 1/2[1¯ 1 1] loop and ED is acute. The angle
Fig. 3. Schematic representation of the spatial relationship between ED and
1/2[111] loop in [110]projection (a). Dependence of CRSS on interaction geo-
metry for 1/2[111] loop (b).
Fig. 4. Atom-core visualization of the interaction of 1/2[111] loop with the ED.
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Table 1 The critical angles (°) for 1/2[1 1 1] loop and 1/2[1 1 1] loop at different models. Model
R
R/2
0
-R/2
-R
1/2[1 1 1]
79.1
78.3
129.1
48.1
86.8
53.1
74.3
0
66.2
0
1/2[1 1 1]
relationship determined that their interaction with ED are symmetrical. Fig. 5 shows the dependence of CRSS on interaction geometry for these two loops. It is clearly that the curve shape is inverse for these two loops. The CRSS is corresponding with the critical line shape for ED passing through the loop. Fig. 6 shows the line shape of ED just before breaking away from these two loops with different interaction geometry. By analyzing the critical line shape, the angel between bowing ED can be got. The critical angles for different models are shown in Table 1. For a periodic array of the obstacle, the critical resolved shear stress (CRSS) for ED to break away from the obstacles can be represented by Gb [13]: c = L cos 2c , where G is the shear modulus; b represents the BV of ED; L is the spacing between obstacles, here means loops, c is the critical angle between two tangential vectors of the bowing ED. The pinning strength, characterized by the critical angle, is an important parameter to analyze extent of hardening [4,8]. Knowing from this formula, smaller angle represents higher pinning strength, thus higher CRSS. For 1/2[1 1¯ 1] loop, due to largest angle, model 0 shows weakest pinning strength; while for 1/2[1¯11] loop, CRSS is highest at model 0 because of smallest angle.
( )
4. Discussion It is the interaction force between dislocation and loop, who leads to the above results. For the 1/2[1 1 1], 1/2[1 1¯ 1] and 1/2[1¯ 1 1] loop, the interaction mechanism is independent on the relative position between loop equator and GP of ED. To assist the discussion, it is helpful to know the force on the loop as a function of distance from the glide plane. Using the infinitesimal loop approximation, the interaction energy between an ED and a loop of area A is E= bA, is the normal component of the ED stress field in the direction of b of the loop. The force on the
Fig. 5. Dependence of CRSS on interaction geometry for 1/2[1 1 1] loop (a) and
1/2[1 1 1] loop (b).
Fig. 6. Critical line shape for ED passing through the 1/2[1 1 1] loop (a) and 1/2[1 1 1] loop (b) with different interaction geometry. 106
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work mainly focuses on the interaction between ED and 1/ 2 < 1 1 1 > loops, while screw dislocation and < 1 0 0 > loops co-exist in real materials, their interactions also contribute to the pinning strength of loops in experimental results; finally, the alloy composition is very complex, alloy elements can also affect the results; although alloy elements in solution state mainly affect the friction stress, while alloy elements segregated at loop may affect the interaction between dislocation and loop, for instance it is difficult for dislocation to break away from loop with Cr segregation. This work is the preliminary study of the effects of interaction geometry on pinning strength induced by interstitial dislocation loop. And the results showed that when parameterizing the dislocation dynamics simulation, the randomness relationship between obstacle and dislocation should be considered. Here we only considered five models. The dislocation is pinned even when it is not in contact with the obstacle. For many kinds of obstacle (loops, void, He bubble, precipitate), systematic researches should be done in order to study the dislocation motion and the resulting material hardening accurately. Fig. 7. Force on the 1/2[1 1 1] loop at model R and R/2.
5. Conclusion
Table 2 Pinning strength for different BV of 1/2 < 1 1 1 > loops at different models.
1/2[1 1 1]
1/2[1 1 1] 1/2[1 1 1] 1/2[1 1 1]
R
R/2
0
-R/2
-R
0.10 0.18
0.09 0.33
0.07 0.48
0.10 0.50
0.10 0.82
0.63
0.61
0.32
0.73
0.51
0.78
0.58
0.82
0.58
0.83
Based on the MD, the influence of interaction geometry on pinning strength induced by 1/2 < 1 1 1 > interstitial dislocation loop was studied. By analyzing the CRSS for ED to overcome loop with different type at different model, it is concluded that the CRSS is related with the relative geometry, and the relation is different for different BVs of loops. For 1/2[1 1 1] loop, it showed weak obstacle due to its easy mobility together with the ED. For 1/2[1 1 1] loop, the interaction mechanism is different when the equator plane of loop is above or below the GP, CRSS is lower when more parts of loop is on the compressive field of the ED. The trends for the variation of CRSS with interaction geometries is inverse for 1/2[1 1 1] and 1/2[1 1 1] loop owing to the angles between their BVs and BV of ED is different. Based on the calculated CRSS and the hardening model, the estimated pinning strength due to dislocation loop is 0.2–0.8.
loop in the ED slip direction is then given byF= dE / dx . The stress filed of ED is symmetrical about the GP, so is the force. It is the reason why the curve shape of relationship between CRSS and relative position is symmetrical for these three loops (see Fig. 5). Fig. 7 shows force on the 1/2[1 1 1] loop at model R and R/2, it is clearly that when at model R, the force is lower than at R/2, thus lower CRSS. Due to the different angle between the loop and ED, dependence of CRSS on relative position for 1/2[1 1 1] loop showed the reversed trend to the 1/2[1 1 1] loop. While for the 1/2[1 1 1] loop, the interaction mechanism is dependent on the relative position between loop equator and GP of ED, the curve shape for the dependence of CRSS on position is not symmetrical. Knowing from above results, interaction geometry has influence on the CRSS for ED to break away from interstitial dislocation loop. The hardening induced by the interstitial loop can be expressed by equation [8] c = Gb (Nd )1/2 ,where c is the critical shear stress, G is the shear modulus (82 GPa), b is the magnitude of Burgers vector (2.49 Å), N and d represent the number density and diameter of the loop, respectively. is the strength factor of loops. Using the calculated CRSS, size and density of the loops in the simulation box, the pinning strength of the 1/ 2 < 1 1 1 > loops with 4 nm diameter at different geometries was calculated, as shown in Table 2. 1/2[1 1 1] loop is the weakest obstacle, nearly no hinder impact on the mobility of dislocation. For the other loops, from this table, it is clearly that when interacting with different geometry, the pinning strength induced by interstitial loop showed large variance, within the range of 0.2–0.8. Pinning strength factors of 0.15–0.6 for dislocation loops have been reported by experimental studies [14,15]. Difference between this work and reference results exists. There are some reasons for this, firstly, owing to the limitation of computing technology and algorithm development, the strain rate in this simulation is 5 × 107 s−1, which is typically orders of magnitude higher than the experimental ones (10−2 s−1), higher strain rate results in higher CRSS; secondly, this
Acknowledgements This research was supported by the National Key Research and Development Program of China (Grant Number: 2017YFB0202304), and partially supported by National Natural Science Foundation of China (Grant Number: U1867217, 11575295 and 11375270) and the CNNC Centralized Research and Development Project (FA16100820). References [1] D. Terentyev, L. Malerba, D.J. Bacon, Y.N. Osetsky, J. Phys.: Condens. Matter 19 (2007) 456211. [2] D.J. Bacon, Y.N. Osetsky, Z. Rong, Phil. Mag. 86 (2006) 3921–3936. [3] Z. Rong, Y.N. Osetsky, D.J. Bacon, Phil. Mag. 85 (2005) 1473–1493. [4] T. Hatano, H. Matsui, Phys. Rev. B. 72 (2005) 094105. [5] S.M. Hafez Haghighat, M.C. Fivel, J. Fikar, R. Schaeublin, J. Nucl. Mater. 386–388 (2009) 102–105. [6] S.M. Hafez Haghighat, R. Schaeublin, Phil. Mag. 90 (2010) 1075–1100. [7] P. Grammatikopoulos, D.J. Bacon, Y.N. Osetsky, Modelling simul. Mater. Sci. Eng. 19 (2011) 015004. [8] R. Nakaia, K. Yabuuchi, S. Nogamia, A. Hasegawa, J. Nucl. Mater. 471 (2016) 233–238. [9] D.J. Bacon, Y.N. Osetsky, Modelling Simul. Mater. Sci. Eng. 11 (2003) 427–446. [10] G.J. Ackland, M.I. Mendelev, D.J. Srolovitz, D. Han, A.V. Barashev, J. Phys.: Condens. Matter 16 (2004) S2629–S2642. [11] D.J. Bacon, Y.N. Osetsky, Phil. Mag. 89 (2009) 3333–3349. [12] D. Terentyev, Y.N. Osetsky, D.J. Bacon, Acta Mater. 58 (2010) 2477–2482. [13] D. Terentyev, S.M. Hafez Haghighat, R. Schaeublin, J. Appl. Phys. 107 (2010) 061806. [14] E. Wakai, A. Hishinuma, K. Usami, Y. Kato, T. Seiichi, K. Abiko, Mater. Trans. 41 (2000) 1180–1183. [15] F. Bergner, C. Pareige, M. Hernández-Mayoral, L. Malerba, C. Heintze, J. Nucl. Mater. 448 (2014) 96–102.
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