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采用晶体相场法模拟了纳米尺度下小角度对称倾斜晶界的结构和位错运动, 针对弛豫过程和附加外应力过程, 观察了晶界上位错运动的位置变化和体系自由能变化, 分析了温度对小角度对称倾斜晶界的结构和晶界上位错运动的影响规律. 研究表明, 弛豫过程中体系温度越低, 体系自由能下降速率越大, 原子规则排列速率增加, 体系自由能达到稳定状态所需的时间越短, 晶界达到稳定状态时位错对排列愈发整齐, 呈现直线规则排列. 外应力作用下, 温度越低, 晶体位错对首次相遇时间越长, 晶体形成单个晶粒时间越长, 位错对首次相遇到晶体内位错对完全消失过程时间越长; 随着温度的降低, 体系自由能出现多段上升下降, 位错对反应也愈加复杂, 趋向于逐对抵消.For crystal materials, the grain boundary structure is complex, which is usually the place in which stress concentration and impurity accumulate. Grain boundary structure and movement have a great influence on the macroscopic properties of crystal materials, therefore, it is of great significance to study the microstructure of grain boundary. With the phase filed crystal approach, the structure of low-angle symmetric tilt grain boundary and dislocation motion at nanoscale are studied. The low-angle symmetric tilt grain boundary structure can be described by a dislocation model, in which the grain boundary can be regarded as consisting of a series of edge dislocations at a certain distance. For a relaxation process and applied stress process, the position change of dislocation motion at grain boundaries and the change of free energy density of the system are observed. Furthermore, we also analyze the influence of temperature on the grain boundary structure and the dislocation motion. In the relaxation process, the free energy of the crystal system is higher under high temperature conditions. The results show that the motion of dislocation pairs in the grains can consume the internal energy and release the distortion energy stored at the grain boundary, and thus making the system more stable and the energy reach the lowest value earlier. Simulation results show that the lower the temperature of the system, the faster the free energy density decreases, the faster the regular arrangement rate of atoms increases, the shorter the time required for the free energy density to reach a stable state becomes. And when the grain boundary reaches a steady state, the arrangement of the dislocations becomes more and more regular and arranges in a straight line. For an applied stress process, with the decrease of temperature, the time required for the first encounter of dislocation pairs and the time required for the formation of single crystal become longer, and it takes more time for the first encounter of dislocation pairs in crystals to disappear completely. Further studies also show that with the decrease of temperature, the free energy density exhibits a multi-stage ascending and descending process. The rising process of energy curve corresponds to the stage of dislocation climbing along the grain boundary, and the decline process corresponds to the stage of dislocation decomposition and encounter annihilation. At the same time, the dislocation pairs’ reaction becomes more complex. Finally, the dislocations annihilate with each other.
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Keywords:
- phase field crystal /
- temperature /
- grain boundary /
- dislocation
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图 1 单模近似下的二维相图(图中阴影部分表示两相区)
Fig. 1. Two-dimensional phase diagram as calculated in a one-mode approximation (hatched areas in the figure correspond to coexistence regions).
图 2 r = –0.25条件下弛豫过程模拟 (a) n = 300; (b) n = 800; (c) n = 15000; (d) n = 29450
Fig. 2. Simulation of relaxation process under the conditions of temperature r = –0.25 at (a) n = 300, (b) n = 800, (c) n = 15000, (d) n = 29450.
图 3 两晶粒形成夹角为2.8°的位向角
Fig. 3. Snapshot of two grains with an orientation angle of 2.8°.
图 4 弛豫过程29500步时不同温度条件下晶界位错模拟图 (a) r = –0.23; (b) r = –0.25; (c) r = –0.28; (d) r = –0.30
Fig. 4. Simulation of grain boundary dislocation under different temperature conditions at 29500 steps of relaxation process: (a) r = –0.23; (b) r = –0.25; (c) r = –0.28; (d) r = –0.30
图 5 温度对弛豫过程体系自由能变化的影响
Fig. 5. Effect of temperature on the change of free energy of relaxation process system.
图 6 应力作用下r = –0.23时晶界位错运动模拟图 (a) n = 5500; (b) n = 10900; (c) n = 11350; (d) n = 13350; (e) n = 13850; (f) n = 24450
Fig. 6. Simulation diagram of grain boundary dislocation motion under stress with r = –0.23: (a) n = 5500; (b) n = 10900; (c) n = 11350; (d) n = 13350; (e) n = 13850; (f) n = 24450.
图 7 应力作用下r = –0.25时晶界位错运动模拟图 (a) n = 11100; (b) n = 11550; (c) n = 13600; (d) n = 13850; (e) n = 14100; (f) n = 25150
Fig. 7. Simulation diagram of grain boundary dislocation motion under stress with r = –0.25: (a) n = 11100; (b) n = 11550; (c) n = 13600; (d) n = 13850; (e) n = 14100; (f) n = 25150.
图 9 应力作用下r = –0.30时晶界位错运动模拟图 (a) n = 11300; (b) n = 12100; (c) n = 12500; (d) n = 39550; (e) n = 40100; (f) n = 76500
Fig. 9. Simulation diagram of grain boundary dislocation motion under stress with r = –0.30: (a) n = 11300; (b) n = 12100; (c) n = 12500; (d) n = 39550; (e) n = 40100; (f) n = 76500.
图 8 应力作用下r = –0.28时晶界位错运动模拟图 (a) n = 11800; (b) n = 12050; (c) n = 13800; (d) n = 29050; (e) n = 33450; (f) n = 33700
Fig. 8. Simulation diagram of grain boundary dislocation motion under stress with r = –0.28: (a) n = 11800; (b) n = 12050; (c) n = 13800; (d) n = 29050; (e) n = 33450; (f) n = 33700.
图 10 11200步时不同温度条件下的模拟图 (a) r = –0.23; (b) r = –0.25; (c) r = –0.28; (d) r = –0.30
Fig. 10. Simulation diagram under different temperature conditions at n = 11200: (a) r = –0.23; (b) r = –0.25; (c) r = –0.28; (d) r = –0.30.
图 11 不同温度下体系自由能曲线图 (a) r = –0.23; (b) r = –0.25; (c) r = –0.28; (d) r = –0.30
Fig. 11. Free energy curve of system under different degrees of temperature: (a) r = –0.23; (b) r = –0.25; (c) r = –0.28; (d) r = –0.30
表 1 模拟所采用的参数
Table 1. Parameters used in the simulation.
方案 初始原子密度ρ0 位向差θ 温度相关参量r A 0.285 2.8° –0.23 B 0.285 2.8° –0.25 C 0.285 2.8° –0.28 D 0.285 2.8° –0.30 
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[1] Paul S K 2018 Comput. Mater. Sci. 150 24
Google Scholar
[2] Bobylev S V, Gutkin M Y, Ovid'Ko I A 2006 Phys. Solid State 48 1495
Google Scholar
[3] Caturla M, Nieh T G, Stolken J S 2004 Appl. Phys. Lett. 84 598
Google Scholar
[4] Peter S, Mikko H, Nikolas P 2009 Phys. Rev. E 80 046107
Google Scholar
[5] Li X H, Wen X, Zhao H H, Ma Z Q, Yu L M, Li C, Liu C X, Guo Q Y, Liu Y C 2019 J. Alloys Compd. 779 175
Google Scholar
[6] Chen Y Y, Hu Z P, Xu Y F, Wang J Y, Schützendübe P, Huang Y, Liu Y C, Wang Z M 2019 J. Mater. Sci. Technol. 35 512
Google Scholar
[7] Elder K R, Katakowski M, Haataja M, Grant M 2002 Phys. Rev. Lett. 88 245701
Google Scholar
[8] Elder K R, Grant M 2004 Phys. Rev. E 70 051605
Google Scholar
[9] Elder K R, Provatas N, Berry J, Stefanovic P 2007 Phys. Rev. B 75 064107
Google Scholar
[10] Wang J, Yu L M, Huang Y, Li H J , Liu Y C 2019 Comput. Mater. Sci. 160 105
Google Scholar
[11] Qi Y, Krajewski P E 2007 Acta Mater. 55 1555
Google Scholar
[12] 郑晓娟, 赵宇宏, 侯华, 靳玉春, 马庆爽, 田晋忠 2017 中国有色金属学报 27 2098
Zheng X J, Zhao Y H, Hou H, Jin Y C, Ma Q S, Tian J Z 2017 Chin. J. Nonferrous Met. 27 2098
[13] Fan D, Chen L Q 1997 Acta Mater. 45 611
Google Scholar
[14] 孙远洋, 赵宇宏, 侯华, 郑晓娟, 郭慧俊 2018 稀有金属材料与工程 47 3000
Sun Y Y, Zhao Y H, Hou H, Zheng X J, Guo H J 2018 Rare Metal. Mater. Eng. 47 3000
[15] 康永生, 赵宇宏, 侯华, 靳玉春, 陈利文 2016 物理学报 65 188102
Google Scholar
Kang Y S, Zhao Y H, Hou H, Jin Y C, Chen L W 2016 Acta Phys. Sin. 65 188102
Google Scholar
[16] 王锟, 赵宇宏, 杨东然, 侯华, 王欣然, 田园 2015 稀有金属材料与工程 44 939
Wang K, Zhao Y H, Yang D R, Hou H, Wang X R, Tian Y 2015 Rare Metal. Mater. Eng. 44 939
[17] 赵宇宏 2010 材料相变过程微观组织模拟 (北京: 国防工业出版社) 第139页
Zhao Y H 2010 Simulation for the Materials Microstructure Evolution in Phase Transformation Process (Beijing: National Defense Industry Press) p139 (in Chinese)
[18] 孙远洋, 赵宇宏, 侯华, 靳玉春, 郑晓娟 2018 中国有色金属学报 28 71
Sun Y Y, Zhao Y H, Hou H, Jin Y C, Zheng X J 2018 Chin. J. Nonferrous Metal. 28 71
[19] 田晓林, 赵宇宏, 田晋忠, 侯华 2018 物理学报 67 230201
Google Scholar
Tian X L, Zhao Y H, Tian J Z, Hou H 2018 Acta Phys. Sin. 67 230201
Google Scholar
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Google Scholar
[21] Fallah V, Stolle J, Ofori-Opoku N, Esmaeili S, Provatas N 2012 Phys. Rev. B 86 3209
[22] Fallah V, Ofori-Opoku N, Stolle J, Provatas N, Esmaeili S 2013 Acta Mater. 61 3653
Google Scholar
[23] Wu K A, Voorhees P W 2012 Acta Mater. 60 407
Google Scholar
[24] Yamanaka A, McReynolds K, Voorhees P W 2017 Acta Mater. 133 160
Google Scholar
[25] Olmsted D L, Buta D, Adland A, Foiles S M, Asta M, Karma A 2011 Phys. Rev. Lett. 106 046101
Google Scholar
[26] 高英俊, 秦河林, 周文权, 邓芊芊, 罗志荣, 黄创高 2015 物理学报 64 106105
Google Scholar
Gao Y J, Qin H L, Zhou W Q, Deng Q Q, Luo Z R, Huang C G 2015 Acta Phys. Sin. 64 106105
Google Scholar
[27] Berry J, Elder K R, Grant M 2008 Phys. Rev. B 77 224114
Google Scholar
[28] Chan V W L, Pisutha-Arnond N, Thornton K 2017 Comput. Mater. Sci. 135 205
Google Scholar
[29] Asadi E, Zaeem M A 2015 JOM 67 186
Google Scholar
[30] Tang S, Wang Z J, Guo Y L, Wang J C, Yu Y M, Zhou Y H 2012 Acta Mater. 60 5501
Google Scholar
[31] Berry J, Grant M 2014 Phys. Rev. E: Stat. Nonlinear Soft Matter Phys. 89 062303
Google Scholar
[32] Zhang W, Mi J 2016 IOP Conf. Ser.: Mater. Sci. Eng. 117 012056
Google Scholar
[33] Hu S, Chen Z, Xi W, Peng Y Y 2017 J. Mater. Sci. 52 5641
Google Scholar
[34] Hu S, Xi W, Chen Z, Wang S, Zhang T H 2017 Comput. Mater. Sci. 132 125
Google Scholar
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Google Scholar
[36] 谷季唯, 王锦程, 王志军, 李俊杰, 郭灿, 唐赛 2017 物理学报 66 216101
Google Scholar
Gu J W, Wang J C, Wang Z J, Li J J, Guo C, Tang S 2017 Acta Phys. Sin. 66 216101
Google Scholar
[37] Hirouchi T, Takaki T, Tomita Y 2010 Int. J. Mech. Sci. 52 309
Google Scholar
[38] Guo H J, Zhao Y H, Sun Y Y, Tian J Z, Hou H, Qi K W, Tian X L 2019 Superlattices Microstruct. 129 163
Google Scholar
[39] Zhao Y H, Deng S J, Liu H, Zhang J X, Guo Z H, Hou H 2018 Comput. Mater. Sci. 154 365
Google Scholar
[40] Wen Z Q, Hou H, Tian J Z, Zhao Y H, Li H J, Han P D 2018 Intermetallics 92 15
Google Scholar
[41] Wen Z Q, Zhao Y H, Hou H, Wang B, Han P D 2017 Mater. Des. 114 398
Google Scholar
[42] Zhao Y H, Qi L, Jin Y C, Wang K, Tian J Z, Han P D 2015 J. Alloys Compd. 647 1104
Google Scholar
[43] Tian J Z, Zhao Y H, Hou H, Han P D 2017 Solid State Commun. 268 44
Google Scholar
[44] Hirouchi T, Takaki T, Tomita Y 2009 Comput. Mater. Sci. 44 1192
Google Scholar
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