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Phase field crystal simulation of the effect of temperature on low-angle symmetric tilt grain boundary dislocation motion

Qi Ke-Wu Zhao Yu-Hong Guo Hui-Jun Tian Xiao-Lin Hou Hua

Phase field crystal simulation of the effect of temperature on low-angle symmetric tilt grain boundary dislocation motion

Qi Ke-Wu, Zhao Yu-Hong, Guo Hui-Jun, Tian Xiao-Lin, Hou Hua
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  • 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.
      Corresponding author: Zhao Yu-Hong, zhaoyuhong@nuc.edu.cn
    [1]

    Paul S K 2018 Comput. Mater. Sci. 150 24

    [2]

    Bobylev S V, Gutkin M Y, Ovid'Ko I A 2006 Phys. Solid State 48 1495

    [3]

    Caturla M, Nieh T G, Stolken J S 2004 Appl. Phys. Lett. 84 598

    [4]

    Peter S, Mikko H, Nikolas P 2009 Phys. Rev. E 80 046107

    [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

    [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

    [7]

    Elder K R, Katakowski M, Haataja M, Grant M 2002 Phys. Rev. Lett. 88 245701

    [8]

    Elder K R, Grant M 2004 Phys. Rev. E 70 051605

    [9]

    Elder K R, Provatas N, Berry J, Stefanovic P 2007 Phys. Rev. B 75 064107

    [10]

    Wang J, Yu L M, Huang Y, Li H J , Liu Y C 2019 Comput. Mater. Sci. 160 105

    [11]

    Qi Y, Krajewski P E 2007 Acta Mater. 55 1555

    [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

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    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

    Kang Y S, Zhao Y H, Hou H, Jin Y C, Chen L W 2016 Acta Phys. Sin. 65 188102

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    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

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    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

    Tian X L, Zhao Y H, Tian J Z, Hou H 2018 Acta Phys. Sin. 67 230201

    [20]

    Zhao Y H, Tian X L, Zhao B J, Sun Y Y, Guo H J, Dong M Y, Liu H, Wang X J, Guo Z H, Umar A, Hou H 2018 Sci. Adv. Mater. 10 1793

    [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

    [23]

    Wu K A, Voorhees P W 2012 Acta Mater. 60 407

    [24]

    Yamanaka A, McReynolds K, Voorhees P W 2017 Acta Mater. 133 160

    [25]

    Olmsted D L, Buta D, Adland A, Foiles S M, Asta M, Karma A 2011 Phys. Rev. Lett. 106 046101

    [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

    Gao Y J, Qin H L, Zhou W Q, Deng Q Q, Luo Z R, Huang C G 2015 Acta Phys. Sin. 64 106105

    [27]

    Berry J, Elder K R, Grant M 2008 Phys. Rev. B 77 224114

    [28]

    Chan V W L, Pisutha-Arnond N, Thornton K 2017 Comput. Mater. Sci. 135 205

    [29]

    Asadi E, Zaeem M A 2015 JOM 67 186

    [30]

    Tang S, Wang Z J, Guo Y L, Wang J C, Yu Y M, Zhou Y H 2012 Acta Mater. 60 5501

    [31]

    Berry J, Grant M 2014 Phys. Rev. E: Stat. Nonlinear Soft Matter Phys. 89 062303

    [32]

    Zhang W, Mi J 2016 IOP Conf. Ser.: Mater. Sci. Eng. 117 012056

    [33]

    Hu S, Chen Z, Xi W, Peng Y Y 2017 J. Mater. Sci. 52 5641

    [34]

    Hu S, Xi W, Chen Z, Wang S, Zhang T H 2017 Comput. Mater. Sci. 132 125

    [35]

    Hu S, Chen Z, Peng Y Y, Liu Y J, Guo L Y 2016 Comput. Mater. Sci. 121 143

    [36]

    谷季唯, 王锦程, 王志军, 李俊杰, 郭灿, 唐赛 2017 物理学报 66 216101

    Gu J W, Wang J C, Wang Z J, Li J J, Guo C, Tang S 2017 Acta Phys. Sin. 66 216101

    [37]

    Hirouchi T, Takaki T, Tomita Y 2010 Int. J. Mech. Sci. 52 309

    [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

    [39]

    Zhao Y H, Deng S J, Liu H, Zhang J X, Guo Z H, Hou H 2018 Comput. Mater. Sci. 154 365

    [40]

    Wen Z Q, Hou H, Tian J Z, Zhao Y H, Li H J, Han P D 2018 Intermetallics 92 15

    [41]

    Wen Z Q, Zhao Y H, Hou H, Wang B, Han P D 2017 Mater. Des. 114 398

    [42]

    Zhao Y H, Qi L, Jin Y C, Wang K, Tian J Z, Han P D 2015 J. Alloys Compd. 647 1104

    [43]

    Tian J Z, Zhao Y H, Hou H, Han P D 2017 Solid State Commun. 268 44

    [44]

    Hirouchi T, Takaki T, Tomita Y 2009 Comput. Mater. Sci. 44 1192

  • 图 1  单模近似下的二维相图(图中阴影部分表示两相区)

    Figure 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

    Figure 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

    Figure 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

    Figure 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

    Figure 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

    Figure 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

    Figure 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

    Figure 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

    Figure 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

    Figure 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

    Figure 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
    A0.2852.8°–0.23
    B0.2852.8°–0.25
    C0.2852.8°–0.28
    D0.2852.8°–0.30
    DownLoad: CSV
  • [1]

    Paul S K 2018 Comput. Mater. Sci. 150 24

    [2]

    Bobylev S V, Gutkin M Y, Ovid'Ko I A 2006 Phys. Solid State 48 1495

    [3]

    Caturla M, Nieh T G, Stolken J S 2004 Appl. Phys. Lett. 84 598

    [4]

    Peter S, Mikko H, Nikolas P 2009 Phys. Rev. E 80 046107

    [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

    [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

    [7]

    Elder K R, Katakowski M, Haataja M, Grant M 2002 Phys. Rev. Lett. 88 245701

    [8]

    Elder K R, Grant M 2004 Phys. Rev. E 70 051605

    [9]

    Elder K R, Provatas N, Berry J, Stefanovic P 2007 Phys. Rev. B 75 064107

    [10]

    Wang J, Yu L M, Huang Y, Li H J , Liu Y C 2019 Comput. Mater. Sci. 160 105

    [11]

    Qi Y, Krajewski P E 2007 Acta Mater. 55 1555

    [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

    [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

    Kang Y S, Zhao Y H, Hou H, Jin Y C, Chen L W 2016 Acta Phys. Sin. 65 188102

    [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

    Tian X L, Zhao Y H, Tian J Z, Hou H 2018 Acta Phys. Sin. 67 230201

    [20]

    Zhao Y H, Tian X L, Zhao B J, Sun Y Y, Guo H J, Dong M Y, Liu H, Wang X J, Guo Z H, Umar A, Hou H 2018 Sci. Adv. Mater. 10 1793

    [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

    [23]

    Wu K A, Voorhees P W 2012 Acta Mater. 60 407

    [24]

    Yamanaka A, McReynolds K, Voorhees P W 2017 Acta Mater. 133 160

    [25]

    Olmsted D L, Buta D, Adland A, Foiles S M, Asta M, Karma A 2011 Phys. Rev. Lett. 106 046101

    [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

    Gao Y J, Qin H L, Zhou W Q, Deng Q Q, Luo Z R, Huang C G 2015 Acta Phys. Sin. 64 106105

    [27]

    Berry J, Elder K R, Grant M 2008 Phys. Rev. B 77 224114

    [28]

    Chan V W L, Pisutha-Arnond N, Thornton K 2017 Comput. Mater. Sci. 135 205

    [29]

    Asadi E, Zaeem M A 2015 JOM 67 186

    [30]

    Tang S, Wang Z J, Guo Y L, Wang J C, Yu Y M, Zhou Y H 2012 Acta Mater. 60 5501

    [31]

    Berry J, Grant M 2014 Phys. Rev. E: Stat. Nonlinear Soft Matter Phys. 89 062303

    [32]

    Zhang W, Mi J 2016 IOP Conf. Ser.: Mater. Sci. Eng. 117 012056

    [33]

    Hu S, Chen Z, Xi W, Peng Y Y 2017 J. Mater. Sci. 52 5641

    [34]

    Hu S, Xi W, Chen Z, Wang S, Zhang T H 2017 Comput. Mater. Sci. 132 125

    [35]

    Hu S, Chen Z, Peng Y Y, Liu Y J, Guo L Y 2016 Comput. Mater. Sci. 121 143

    [36]

    谷季唯, 王锦程, 王志军, 李俊杰, 郭灿, 唐赛 2017 物理学报 66 216101

    Gu J W, Wang J C, Wang Z J, Li J J, Guo C, Tang S 2017 Acta Phys. Sin. 66 216101

    [37]

    Hirouchi T, Takaki T, Tomita Y 2010 Int. J. Mech. Sci. 52 309

    [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

    [39]

    Zhao Y H, Deng S J, Liu H, Zhang J X, Guo Z H, Hou H 2018 Comput. Mater. Sci. 154 365

    [40]

    Wen Z Q, Hou H, Tian J Z, Zhao Y H, Li H J, Han P D 2018 Intermetallics 92 15

    [41]

    Wen Z Q, Zhao Y H, Hou H, Wang B, Han P D 2017 Mater. Des. 114 398

    [42]

    Zhao Y H, Qi L, Jin Y C, Wang K, Tian J Z, Han P D 2015 J. Alloys Compd. 647 1104

    [43]

    Tian J Z, Zhao Y H, Hou H, Han P D 2017 Solid State Commun. 268 44

    [44]

    Hirouchi T, Takaki T, Tomita Y 2009 Comput. Mater. Sci. 44 1192

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  • Received Date:  09 January 2019
  • Accepted Date:  28 May 2019
  • Available Online:  26 November 2019
  • Published Online:  01 September 2019

Phase field crystal simulation of the effect of temperature on low-angle symmetric tilt grain boundary dislocation motion

    Corresponding author: Zhao Yu-Hong, zhaoyuhong@nuc.edu.cn
  • Material Science and Engineering Institute, North University of China, Taiyuan 030051, China

Abstract: 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.

    • 对于多晶材料, 其晶界处结构复杂, 通常是应力集中和杂质聚集的地方, 因此研究晶界处微观结构特性有重要意义. 而小角度对称倾斜晶界结构可利用位错模型进行描述, 此时晶界可看作由一系列相距一定距离的刃型位错构成. 在不同外界条件作用下, 形成的晶界结构不同, 当外界条件稳定时, 晶界将处于一种稳定的平衡状态. 通常, 晶界被视为位错源, 在外加应力作用下, 位错发生迁移运动进而导致晶界偏移甚至湮没[1-4].

      晶界结构及运动[5,6]对固体材料的宏观性能有较大的影响. 在非平衡条件下观察界面结构和迁移十分困难, 实验过程中细致观察界面位错运动和反应也难以实现. 然而, 通过计算机模拟技术可以实现纳米尺度下观察晶粒界面结构变化, 更加清晰直观地研究其变化特征. 近些年来, 由Elder等[7-9]基于密度泛函理论模型提出的晶体相场方法(phase field crystal, PFC)受到广泛关注. 晶体相场法与分子动力学[10]相比, 在扩散时间尺度上可描述纳观原子运动情况, 分子动力学适用的特征时间尺度主要在原子振动时间尺度(10–12—10–15 s), 因而PFC克服了分子动力学在扩散特征时间尺度(10–6 s)的变形行为存在固有的局限性[11]. 对于传统相场方法(traditional phase field, TPF)[12-20]而言, 因其采用平衡状态下空间均匀量的场变量, 失去了晶体相所固有的周期对称性特征, 从而无法反映界面和其他位错缺陷等原子尺度细节以及弹塑性变形、各向异性等物理现象问题. 而PFC模型采用局部时间平均的原子密度场, 可以反映晶体点阵的周期性变化特征, 从而能够从根本上阐明微观组织在原子尺度的演化机理. 目前, PFC模型可用于研究溶质析出偏聚[21,22]、晶界迁移与晶粒旋转[23,24]、高温晶界预熔化与动态回复[25-27]、凝固过程的枝晶生长[28-30]、有序-无序(金属)玻璃转变[31,32]、纳米裂纹扩展[33-35]、石墨烯结构形核[36]等各个方面. 借助PFC模型, 可以研究在不同温度和粒径下纳米多晶结构中的晶粒旋转、位错运动和晶界迁移[37], 以及在双晶体系中, 探讨在温度和取向角的共同作用下小角对称晶界的湮没机理[38]. 虽然PFC模型研究广泛, 但目前从弛豫过程和附加外应力过程研究温度对小角度对称倾斜晶界位错运动影响尚未见到相关报道.

      本文采用晶体相场模型, 针对弛豫过程和附加外应力过程, 在纳米尺度上观察了晶界上位错运动导致的位置变化和体系自由能的变化, 模拟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.

    2.   晶体相场法模型
    • 与TPF不同的是, PFC借鉴了经典密度泛函理论[39-43], 使用具有周期对称性原子密度场函数来代替TPF中的平均场变量, 改变了TPF在介观尺度所使用的序参量. 其自由能是在Swift-Hohenberg方程[7,8]的基础上改写而来, 是局域密度场的一个泛函, 构造形式为原子概率密度函数. 晶体相场模型局部守恒序参数ϕ的演化为

      式中ϕ为原子密度, τ是时间变量, Γ是迁移率, ▽2为拉普拉斯算子, δ为Dirac函数, F是系统的自由能. 在PFC模型中, 自由能函数的最低要求是它应该在一定的参数范围内能够在基态中产生一个周期性的晶格结构. 能够满足这一要求的最简单的能量泛函由Swift-Hohenberg提出[7,8], 通常称为SH型晶体相场模型, 其形式为

      式中的α, λ, u是与拟合材料性质相关的唯象参数; r为空间向量, ΔT为反映体系温度的参数, q0为与平衡晶格间距成反比的常数, ▽2为拉普拉斯算子. 对方程(2)进行无量纲处理, 引进一系列新的变量:

      由此可以得到无量纲化的自由能方程(3)和无量纲化的动力学方程(4):

      式中ρ为原子密度序参量, r为无量纲化后反映体系温度的参数, r值越小, 体系温度越低.

      无量纲化的动力学方程:

      式中$\omega \left( {{\nabla ^2}} \right) = r + {\left( {1 + {\nabla ^2}} \right)^2}$, t为尺度标准化后的时间变量.

      二维情况下, Fρ构造的自由能泛函具有三种平衡相, 分别为液相(常数值)、三角相以及条纹相, 皆为周期函数. 由(4)式可得到二维体系中稳定的晶态三角相无量纲局域密度的单模近似解为

      式中ρ0为均匀原子密度分布,

      $A = \dfrac{4}{5}\left( {{\rho _0} + \dfrac{1}{3}\sqrt { - 15r - 36{\rho _0}^2} } \right)$, $q = \dfrac{{\sqrt 3 }}{2}$.

      利用三种平衡相的极小自由能函数, 按照吉布斯自由能公切线法确定二维相图[8], 如图1所示.

      Figure 1.  Two-dimensional phase diagram as calculated in a one-mode approximation (hatched areas in the figure correspond to coexistence regions).

    3.   计算方法
    • 对(3)式进行半隐式Fourier谱方法求解, 可以得到其离散化形式为

      式中${\rho _{{{k}} ,t}}$为Fourier空间t时刻原子密度, ${\rho _{{{k}} ,t + \Delta t}}$为Fourier空间tt时刻原子密度, ${{k}} $为Fourier空间波矢, 且满足${k^2} = {\left| {\left. {{{k}} } \right|} \right.^2}$.

      在模拟实验过程中, 用二维三角相表征面心立方{111}面的原子点阵结构, 本次实验所选取的模拟参数为ρ0 = 0.285, 晶粒初始位向差θ = 2.8°, 取空间步长为Δx = Δy = ${\text{π}}/4 $ , 时间步长为Δt = 0.5. 其边界条件为周期性边界条件. 对于初始模拟区域设置如下: 设置模拟计算区域面积为Lx × Ly = 512Δx × 512Δy, 模拟区域分为两个初始液相区和两个晶粒区, 液相区设置宽度为d0的带状区域. 通过(5)式设置中间晶粒区域1/4(Ly + 2d0) < y < 3/4(Ly–2d0)范围的晶粒取向为–θ/2, 其上下两侧晶粒区域0 < y < 1/4(Ly–2d0)和1/4(3Ly + 2d0) < y < Ly范围晶粒取向为θ/2, 而0 < x < Lx. 为了保证形成的晶界具有良好原子初始排列, 设置两晶粒间带状液相区宽度为10. 再经过30000步的时间弛豫, 使得液相区完全凝固结晶, 体系由固-液体系向固相体系发生相变, 自由能持续降低, 最终固相晶体体系达到稳定状态.

      外应力作用下会促使位错运动及晶界偏移, 此次模拟过程中采用等面积不变模型[44], 使得模拟区域具有统一性, 其假设条件为

      式中, Δx和Δy为变形前空间步长, Δx和Δy为变形后空间步长. 应变量ε满足$\varepsilon = \dot \varepsilon n\Delta t$(其中, $\dot \varepsilon $为无量纲化应变速率, 本文取$\dot \varepsilon = 6 \times {10^{ - 6}}/\Delta t$, n为时间步数). 现假设在x方向上对体系施加一个拉应力, 则满足:

    4.   模拟结果与分析
    • 所选用的模拟参数如表1所列, 其他初始设置如第3节所述.

      方案初始原子密度ρ0位向差θ温度相关参量r
      A0.2852.8°–0.23
      B0.2852.8°–0.25
      C0.2852.8°–0.28
      D0.2852.8°–0.30

      Table 1.  Parameters used in the simulation.

    • 以方案B为例, 如图2所示, 在n从0到30000的过程中, 原子无序排列的液相区(图2(a)中长条状黑色区域)两侧原子不断吞噬液相, 最后两晶粒接触形成如图3所示的位向角为2.8°的小角度晶界, 模拟体系由固-液体系完全转化为固相体系. 从图2中可以观察到小角度对称倾斜晶界是由6个畸变区组成, 因倾斜角的存在, 降低了晶体体系的界面能使晶界更加稳定, 而每个畸变区由两个呈一定角度的刃型位错组成, 进而形成一个位错对, 因此得到的晶界则由6个具有一定距离的位错对组成. 随着演化时间的推移, 晶格会自发地向自由能较低的方向发生变化, 晶界处位错会产生小距离滑移运动, 以达到能量最低状态. 在15000步之后, 晶界处位错基本不再进行滑移运动, 晶体体系达到稳定状态.

      Figure 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.

      Figure 3.  Snapshot of two grains with an orientation angle of 2.8°.

      图4所示为29500步时不同温度条件下的晶界位错排列. 从中可知, 随着体系温度的降低, 晶界处位错越加趋向于规则排列, 多个位错对呈现出直线排列. 其原因在于初始温度越低, 晶体体系温度降低速率越大, 在相同的时间内体系的能量也就越低, 晶体内原子规则排列就越早进行. 通常在最后凝固部位原子出现不规则排列, 此时就会产生畸变能并且形成位错对. 在温度较低的情况下, 晶体内部趋向于同时凝固, 多数液相区原子在较短时间内就整齐排列, 剩余原子无法规则排列而形成多组位错构成晶界. 晶界处出现位错对后, 由于体系自由能过低而难以越过能量势垒进行原子迁移, 位错对进行滑移运动的可能性也就越低, 晶界越发趋于稳定. 而在温度较高的情况下, 因为有足够的能量使得原子脱离原来的位置进行原子迁移, 所以在弛豫过程中发生位错对小幅度运动的概率较大, 出现位错对随着时间的推移而发生滑移运动.

      Figure 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中可以看出, 开始阶段晶体体系随着温度的降低, 自由能下降速率愈来愈大. 自由能曲线出现拐点的时间步长分别为650步、450步、350步和300步, 表明在高温条件下, 晶体体系趋于稳定需要较长的时间; 而低温条件下, 晶体体系更容易达到稳定状态. 由图4图5可以观察在不同温度下晶界位错的排列和体系自由能变化情况. 以下层晶界左侧第一个位错对为例, 讨论体系温度对晶界位错对运动和自由能的影响. 在温度为r = –0.23时, 下层晶界左侧第一个位错对距离模拟图底层有21个原子层; 在温度为r = –0.25, –0.28, –0.30时, 下层晶界左侧第一个位错对距离模拟图底层分别有19, 18, 18个原子层. 可以看出温度影响着晶体体系自由能的变化, 进而影响晶界位错对排列的最终形态. 同理, 对比其他位错对发现, 在温度为r = –0.28和r = –0.30时, 位错对排列具有一致性, 都在晶界处呈现直线排列. 对比温度为r = –0.23和r = –0.25情形, 研究发现温度高时晶界位错对会出现部分位错对向晶粒内运动, 表明在高温条件下晶体体系自由能较高, 为了使得体系更加稳定, 能量更早达到最低, 位错对向晶内运动可以消耗内能和释放晶界处储存的畸变能. 而在低温条件下, 体系自由能下降快, 且无外应力作用, 晶体体系没有能量补充, 体系能量在短时间内便达到稳定状态, 没有额外的能量使位错对向晶内运动.

      Figure 5.  Effect of temperature on the change of free energy of relaxation process system.

    • 对A, B, C, D四种方案弛豫后得到的试样施加外应力, 在外加应力的作用下, 由于位错滑移所需要的能量较攀移所需要的能量低, 一般情况下位错优先进行滑移运动. 然而在晶界处位错滑移需要额外克服晶界弹性作用, 所需能量大于位错攀移所需能量, 因而晶界处位错会先进行攀移运动. 以表1中的方案A为例, 分析在应力作用下, 温度为r = –0.23条件下位错的运动情况. 弛豫过程结束后, 体系自由能已经达到一个稳定状态, 再对模拟区域x方向施加拉应力, y方向施加压应力. 由图6演化过程可知, 在刚开始应力加持时, 晶界上位错由于受到热力耦合作用均开始进行攀移运动, 上层晶界向右进行正攀移运动, 下层晶界向左进行正攀移运动, 如图6(a)箭头方向所示. 随着应力的不断增加, 体系中能量不断增加, 晶界处位错积累更多的应变能使得运动加剧. 当作用在位错上的力足以克服晶界弹性作用、越过能量势垒时, 晶界处位错开始向晶粒内进行滑移运动. 如图6(b)中所示, 晶界处部分位错向上滑移运动, 其余位错向下滑移运动(图中箭头所指方向为位错对运动方向). 并且在运动过程中, 位错还是以位错对的形式向晶粒内滑移, 并未出现位错对分解. 在后续过程中, 如图6(c)所示位错对首次相遇, Burgers矢量相反的两个位错对相遇湮没(如图6(c)圆圈所标示位置). 对比图6(b)图6(c)可以发现两个位错对相遇湮没后晶界位向差也发生了变化, 与未发生位错对湮没相比, 湮没后的晶界位向差有明显的减小. 此时位错对湮没区域再无畸变, 原子呈现规则排列, 导致体系自由能降低.

      Figure 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.

      随着演化时间的推移, 剩余位错对在外应力的作用下继续进行滑移运动. 在13350步时如图6(d)中可以观察到, 又有四个位错对两两相互靠近, 并且相互靠近的位错对矢量方向相反, 其结果是位错对相遇湮没. 如图6(e)所示, 四个位错对两两发生位错反应最终湮没消失. 此时位错湮没后晶体体系畸变能进一步降低, 体系自由能下降. 从图中可以进一步观察到四个位错对完全湮没后, 晶体的位向差发生了巨大变化, 此时晶体的位向差相比第一次发生位错对湮没时进一步减小. 在外应力下, 未发生湮没消失的位错对持续运动下去直至最后所有位错对相遇抵消, 晶界完全消失, 晶体的位向差为零, 形成如图6(f)所示的单个晶粒. 同理, 温度为–0.25, –0.28和–0.30条件下位错对运动过程分别如图7图9所示.

      Figure 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.

      Figure 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.

      Figure 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是11200步时, 不同温度下的位错对运动演化图. 从图10(b)可以观察到, 当温度r = –0.25时, 此时出现两个位错对将进行第一次相遇抵消(圆圈所标示的两个位错对). 而温度r = –0.23如图10(a)所示, 此时已经完成了位错对的相遇抵消, 并且还有四个位错对两两相互靠近进行下一次的相遇. 图10(c)图10(d)分别为温度r = –0.28和温度r = –0.30下的模拟图, 虽然都未出现位错对的相遇抵消, 但图10(c)中有互相靠近的位错对, 随着演化时间的进行将进行位错对的相遇抵消, 而图10(d)中此时没有位错对相互靠近抵消的运动趋势且位错对运动情况比较复杂. 由图9图10可以发现, 温度影响着小角度对称倾斜晶界位错首次相遇抵消的时间. 在外应力的作用下, 晶体体系吸收应变能使得体系自由能升高, 位错对运动阻力增大, 晶界处位错对的畸变能也随之增大. 在应力不断加持的过程中, 体系能量不断积累, 当晶界畸变区的能量积累到超过位错对运动所需的畸变能时, 位错对便开始运动以减缓体系能量的增长. 外应力作用下, 相同时间内高温晶体体系所积累的能量要比低温晶体体系所积累的能量多, 使得触发位错对开始运动所需时间比低温环境下短, 位错对更早地发生滑移. 在位错对滑移时, 外应力的持续作用使得体系自由能仍处于上升状态, 在高温下位错对运动阻力小, Burgers矢量相反的两个位错对更容易在较短时间内相遇抵消. 反之, 温度越低则会推迟位错对首次相遇抵消的时间. 从不同温度下位错运动过程中还可以看出, 温度越低时位错对运动相对缓慢, 位错对相遇湮没的时间间隔相对而言较长, 没有出现在温度较高时四个位错对同时相遇湮没的情形, 并且位错对运动情形更加复杂.

      Figure 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.

    • 从体系自由能变化的角度分析温度对位错运动的影响. 对比图11不同温度下体系自由能变化曲线可以发现, 在外加应力作用下不同温度的体系自由能都是由若干上升、下降过程组成, 整个体系自由能变化情况大体一致, 晶体体系自由能都呈现上升趋势. 就一般情况而言, 能量曲线上升对应位错沿晶界攀移阶段, 下降对应位错分解、晶粒内攀移、相遇湮没阶段. 在自由能变化过程中, 第一次出现极小值点对应着位错对第一次相遇抵消, 此时因位错对的相互抵消而使晶体体系畸变能降低, 体系自由能随之减低. 以图11(a)温度r = –0.23为例具体分析体系自由能变化. 从图11(a)中可以看出, 晶界位错从第一次相遇抵消到最后晶体内所有位错都完全相互抵消湮没, 形成单个晶粒体系自由能的变化对应着图中A点到E点自由能变化过程(图11(a)中小图为绿色区域放大图, 图11(b)(d)同理). 在初期外应力的作用下, 晶体内位错对进行着简单的攀移运动, 体系自由能缓慢地上升, 当位错对运动一定时间后, 将出现位错对第一次相遇, 此时体系自由能如图中A点所示, 应变为0.0654, 其对应的时间步长为10900步, 对应着晶界位错运动模拟图6(b)所示的运动状态. 同理, 图11(a)中B点应变为0.0681, 时间步长为11350步对应着晶界位错运动模拟图6(c)所示的运动状态. 从图6(b)图6(c)图11(a)可以观察到, 当位错对运动到第一次即将相遇时, 晶体体系自由能也即将出现下降趋势, 在第一次即将相遇的两个位错对互相靠近直至位错对互相反应湮没, 晶体体系自由能在此过程中不断降低, 而且当两个位错对湮没时晶体体系自由能出现极小值, 此后剩余位错对在外应力的作用下继续运动, 体系自由能又呈现上升趋势. 此后位错对又会出现互相靠近直至相遇湮没的过程, 自由能变化如图11(a)中C点到D点的过程, 随着时间推移, 晶体体系中最后两个位错对相遇抵消, 对应图6(f)图11(a)中E点, 此时晶体形成单个晶粒, 并且随后晶体在外应力作用下体系自由能不断上升.

      Figure 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

      对比图11中不同温度下晶体体系自由能曲线图可以观察到, 温度r = –0.23在应变值为0.065时位错对第一次相遇, 而温度为r = –0.25, –0.28, –0.30分别在应变值为0.0666, 0.0708, 0.0720时出现位错对第一次相遇. 并且在温度r = –0.23, –0.25, –0.28, –0.30下应变值分别为0.1467, 0.1509, 0.2022, 0.4587时, 晶体中位错对都已完全相遇抵消形成单个晶粒. 可以发现, 温度影响着晶体内出现位错对第一次相遇抵消和位错对完全消失湮没形成单个晶粒的时间, 而且对位错对首次相遇和最后完全消失过程所需时间有较大影响. 温度越低, 位错对在外应力作用下运动出现第一次相遇所需的时间越长, 晶界完全消失形成单个晶粒所需的时间也越长. 在位错对首次相遇抵消时, 相遇的位错对释放其畸变能, 且位错对运动到晶粒内部, 晶体体系界面能也相应降低. 由于此时外应力作用提供的能量无法满足位错对发生相遇抵消晶体体系所损失的能量, 体系自由能曲线降低. 在随后的过程中, 体系继续积累能量, 以满足第二组位错对发生相遇抵消所需的能量. 但低温下, 相同时间内体系积累能量较少, 使得位错对无法像高温环境那样出现多组位错对同时相遇抵消, 只能随着能量的积累一组接着一组发生抵消直至晶界完全湮没. 由此可得随着温度降低, 晶体体系初始自由能越低, 位错对运动变得愈发缓慢, 出现首次相遇和晶体内位错对完全消失所需时间越长. 进一步得到, 在低温下位错对从首次相遇到所有位错对都消失过程的时间越长, 并且位错对反应变得愈发复杂, 晶体自由能上升段与下降段增多, 晶体内位错对相遇抵消时间趋向于逐对抵消, 不同于较高温度下多个位错对同时相遇抵消.

    5.   结 论
    • 本文采用晶体相场模型研究了温度对小角度对称倾斜晶界位错运动的影响, 研究结果表明:

      1)在不同温度下的弛豫过程中, 随着体系温度降低, 体系自由能下降速率也增大, 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也越短, 并且温度较低时, 位错对进行滑移运动的概率低, 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;

      2)温度越低, 晶体位错对首次相遇时间越长, 位错对完全消失晶体形成单个晶粒所需的时间越长, 位错对首次相遇到晶体内位错对完全消失过程的时间也越长;

      3)不同温度下体系自由能变化趋势大体一致, 体系自由能都由若干上升下降过程组成并均呈现上升趋势, 且在外应力作用下, 体系自由能升高, 能量曲线上升对应位错沿晶界攀移阶段, 下降对应位错分解和相遇湮没阶段; 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, 晶体位错对相遇到晶界湮没的时间间隔也越长, 趋向于逐对抵消.

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