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采用晶体相场法研究了外加应变作用下, 不同取向差的四方相对称倾侧小角度晶界的位错运动与反应及反应过程中的位错组态, 通过采用几何相位法对位错周围应变场进行了表征. 结果表明, 凝固弛豫达到稳态后, 晶界两侧位错平行且方向相反, 随晶界两侧晶粒取向差增大, 位错数目增加, 距离减小, 且体系自由能增加. 在外加应变作用下, 晶界位错经历攀移、发射、反应湮灭等阶段, 体系自由能呈现波动. 当取向差增大时, 位错运动方式由攀移向攀滑移转变, 产生更多类型的位错组构型, 并发生相应的位错与位错组之间的反应. 对于不同构型的位错反应, 正切应变驱动位错靠近, 负切应变驱动位错湮灭.In this paper, the phase field crystal method is used to study the dislocation motion and reaction of the square phase symmetric tilt low-angle grain boundaries, and the dislocation configurations with different misorientation angles are analyzed under the action of applied strain. The geometric phase approach is used to characterize the strain field around the dislocations. The results show that after the solidification relaxation, the interfacial dislocations on both sides of the grain are distributed in parallel but opposite direction. With the increase of misorientation angle between grains, the number of dislocations increases, the spacing between them decreases, and the free energy of the system increases. Imposed by the applied strain, the grain boundary dislocations undergo climbing, launching, and reactive annihilation, with the free energy fluctuating. When the misorientation increases, the dislocation motion mode changes from climbing to climbing-sliping, resulting in more dislocation group configurations, and more reactions between dislocations and dislocation groups. For the dislocation reactions of different configurations, positive shear strain drives dislocations to approach, and negative shear strain drives dislocations to annihilate.
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Keywords:
- phase field crystal /
- square phase /
- dislocation reaction /
- grain boundary
[1] Elder K R, Katakowski M, Haataja M, Grant M 2002 Phys. Rev. Lett. 88 245701Google Scholar
[2] Elder K R, Grant M 2004 Phys. Rev. E 70 051605Google Scholar
[3] Elder K R, Provatas N, Berry J, Stefanovic P, Grant M 2007 Phys. Rev. B 75 064107Google Scholar
[4] 高英俊, 罗志荣, 黄创高, 卢强华, 林葵 2013 物理学报 62 050507Google Scholar
Gao Y J, Luo Z R, Huang C G, Lu Q H, Lin K 2013 Acta Phys. Sin. 62 050507Google Scholar
[5] Yang T, Gao Y, Wang D, Shi R P, Chen Z, Nie J F, Li J, Wang Y 2017 Acta Mater. 127 481Google Scholar
[6] Greenwood M, Rottler J, Provatas N 2011 Phys. Rev. E 83 031601Google Scholar
[7] Gao Y J, Deng Q Q, Liu Z Y, Huang Z J, Li Y X, Luo Z R 2020 J. Mater. Sci. Technol. 49 236Google Scholar
[8] Gao Y J, Huang L L, Deng Q Q, Zhou W Q, Luo Z R, Lin K 2016 Acta Mater. 117 238Google Scholar
[9] 赵宇龙, 陈铮, 龙建, 杨涛 2013 物理学报 62 118102Google Scholar
Zhao Y L, Chen Z, Long J, Yang T 2013 Acta Phys. Sin. 62 118102Google Scholar
[10] 赵宇龙, 陈铮, 龙建, 杨涛 2014 金属学报 27 81Google Scholar
Zhao Y L, Chen Z, Long J, Yang T 2014 Acta Metall. Sin. 27 81Google Scholar
[11] Gao Y J, Luo Z R, Huang L L, Mao H, Huang C G, Lin K 2016 Modell. Simul. Mater. Sci. 24 055010Google Scholar
[12] Hu S, Chen Z, Xi W, Peng Y Y 2017 J. Mater. Sci. 52 5641Google Scholar
[13] Seymour M, Sanches F, Elder K, Provatas N 2015 Phys. Rev. B 92 184109Google Scholar
[14] Faghihi N, Provatas N, Elder K R, Grant M, Karttunen M 2013 Phys. Rev. E 88 032407Google Scholar
[15] Berghoff M, Nestler B 2015 Comput. Condens. Matter 4 46Google Scholar
[16] Tang S, Wang Z J, Guo Y L, et al. 2012 Acta Mater. 60 5501Google Scholar
[17] 高英俊, 卢成健, 黄礼琳, 罗志荣, 黄创高 2014 金属学报 50 110Google Scholar
Gao Y J, Lu C J, Huang L L, Luo Z R, Huang C G 2014 Acta Metall. Sin. 50 110Google Scholar
[18] Hu S, Chen Z, Yu G G, Xi W, Peng Y Y 2016 Comput. Mater. Sci. 124 195Google Scholar
[19] Gutkin M Y, Ovid'ko I A 2001 Phys. Rev. B 63 064515Google Scholar
[20] Stefanovic P, Haataja M, Provatas N 2009 Phys. Rev. E 80 046107Google Scholar
[21] Bobylev S V, Gutkin M Y, Ovid'ko I A 2004 Phys. Solid State 46 2053Google Scholar
[22] Ohno T, Ii S, Shibata N, Matsunaga K, Ikuhara Y, Yamamoto T 2004 Mater. Trans. 45 2117Google Scholar
[23] Humphreys F J 2001 J. Mater. Sci. 36 3833Google Scholar
[24] Hirouchi T, Takaki T, Tomita Y 2009 Comput. Mater. Sci. 44 1192Google Scholar
[25] 龙建, 王诏玉, 赵宇龙, 龙清华, 杨涛, 陈铮 2013 物理学报 62 218101Google Scholar
Long J, Wang Z Y, Zhao Y L, Long Q H, Yang T, Chen Z 2013 Acta Phys. Sin. 62 218101Google Scholar
[26] 鲁娜, 王永欣, 陈铮 2014 物理学报 63 180508Google Scholar
Lu N, Wang Y X, Chen Z 2014 Acta Phys. Sin. 63 180508Google Scholar
[27] 杨涛, 陈铮, 董伟平 2011 金属学报 47 1301Google Scholar
Yang T, Chen Z, Dong W P 2011 Acta Metall. Sin. 47 1301Google Scholar
[28] Berkels B, Ratz A, Rumpf M, Voigt A 2008 J. Sci. Comput. 35 1Google Scholar
[29] Elsey M, Wirth B 2014 Multiscale Model. Sim. 12 1Google Scholar
[30] Singer H M, Singer I 2006 Phys. Rev. E 74 031103Google Scholar
[31] Wang Z J, Li J J, Guo Y L, Tang S, Wang J C 2013 Comput. Phys. Commun. 184 2489Google Scholar
[32] Hytch M J, Snoeck E, Kilaas R 1998 Ultramicroscopy 74 131Google Scholar
[33] Galindo P L, Kret S, Sanchez A M, et al. 2007 Ultramicroscopy 107 1186Google Scholar
[34] Hytch M J, Putaux J L, Penisson J M 2003 Nature 423 270Google Scholar
[35] Guo Y, Wang J, Wang Z, Li J, Tang S, Liu F, Zhou Y 2015 Philos. Mag. 95 973Google Scholar
[36] Wu K A, Adland A, Karma A 2010 Phys. Rev. E 81 061601Google Scholar
[37] Chen L Q, Shen J 1998 Comput. Phys. Commun. 108 147Google Scholar
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图 6 应变作用下
$ \theta ={4} $ °时晶界位错运动模拟图 (a) t = 50; (b) t = 14150; (c) t = 15250; (d) t = 17550; (e) t = 29100; (f) t = 48500. 图6(a)中插图(g)和(h)分别表示刃位错的类型和位错对应的伯氏矢量类型Fig. 6. Simulation of grain boundary dislocation motion under strain with
$ \theta ={4} $ °: (a) t = 50; (b) t = 14150; (c) t = 15250; (d) t = 17550; (e) t = 29100; (f) t = 48500. Inset (g) and (h) represent the type of edge dislocation and the Burgers vector type respectively corresponding to the dislocation in Fig. 6(a).图 7 应变作用下
$ \theta ={6} $ °时晶界位错运动模拟图 (a) t = 50; (b) t = 15000; (c) t = 16350; (d) t = 19550; (e) t = 23500; (f) t = 26950Fig. 7. Simulation of grain boundary dislocation motion under strain with
$ \theta ={6} $ °: (a) t = 50; (b) t = 15000; (c) t = 16350; (d) t = 19550; (e) t = 23500; (f) t = 26950.图 8 应变作用下
$ \theta ={8} $ °时晶界位错运动模拟图 (a) t = 100; (b) t = 16350; (c) t = 17500; (d) t = 20400; (e) t = 25000; (f) t = 34750Fig. 8. Simulation of grain boundary dislocation motion under strain with
$ \theta ={8} $ °: (a) t = 100; (b) t = 16350; (c) t = 17500; (d) t = 20400; (e) t = 25000; (f) t = 34750.图 10 含位错组的晶体位错构型及对应的不同应变取向应变云图 (a) 晶体构型图; (b)
${\varepsilon }_{xx}$ ; (c)${\varepsilon }_{xy}$ ; (d)${\varepsilon }_{yy}$ Fig. 10. Dislocation configuration of crystals containing dislocation groups and corresponding strain contours of different strain orientations: (a) Crystal configuration diagram; (b)
${\varepsilon }_{xx}$ ; (c)${\varepsilon }_{xy}$ ; (d)${\varepsilon }_{yy}$ .图 12 不同位错反应构型演化时应变(
${\varepsilon }_{xy}$ )云图 (a)—(d) R1构型; (e)—(h) R2构型; (i)—(l) R3构型; (m)—(p) R4构型; (q)—(t) R5构型Fig. 12. Strain (
${\varepsilon }_{xy}$ ) contours during the evolution of different dislocation reaction configurations: (a)–(d) R1 configuration; (e)–(h) R2 configuration; (i)–(l) R3 configuration; (m)–(p) R4 configuration; (q)–(t) R5 configuration.表 1 模拟设定的参数
Table 1. Parameters in the simulation.
方案 初始原子密度$ {\overline \psi _0} $ 温度相关参量ε 取向差θ A $-0.40$ $ 0.45 $ $ {4} $° B $-0.40$ $ 0.45 $ $ {6} $° C $-0.40$ $ 0.45 $ $ {8} $° 表 2 不同构型的位错分解发射与位错反应
Table 2. Dislocation decomposition and dislocation reaction of different configuration.
编号 位错分解发射 编号 位错反应 L1 $n1\xrightarrow{\text{分解}}n1+n3+n4$ R1 $ n3+n4\xrightarrow{\text{反应}} 0 $ $ n2 \xrightarrow{\text{分解}} n2+n3+n4 $ R2 $n1+\overline{n2 n4}\xrightarrow{\text{反应} }n4,$
$n2+\overline{n2 n3}\xrightarrow{\text{反应}}n4$L2 $n1\xrightarrow{\text{分解}}\overline{n1 n3}+n4$ R3 $n3+\overline{n2 n4}\xrightarrow{\text{反应} }n2,$
$n4+\overline{n1 n3}\xrightarrow{\text{反应}}n1$$n2\xrightarrow{\text{分解}}\overline{n2 n3}+n4$ R4 $\overline{n1 n3}+\overline{n2 n4}\xrightarrow{\text{反应} }0,$
$\overline{n1 n4}+\overline{n2 n3}\xrightarrow{\text{反应}}0$L3 $n1\xrightarrow{\text{分解}}\overline{n1 n4}+n3$ R5 $ n1+n2\xrightarrow{\text{反应}}0 $ $n2\xrightarrow{\text{分解}}\overline{n2 n4}+n3$ -
[1] Elder K R, Katakowski M, Haataja M, Grant M 2002 Phys. Rev. Lett. 88 245701Google Scholar
[2] Elder K R, Grant M 2004 Phys. Rev. E 70 051605Google Scholar
[3] Elder K R, Provatas N, Berry J, Stefanovic P, Grant M 2007 Phys. Rev. B 75 064107Google Scholar
[4] 高英俊, 罗志荣, 黄创高, 卢强华, 林葵 2013 物理学报 62 050507Google Scholar
Gao Y J, Luo Z R, Huang C G, Lu Q H, Lin K 2013 Acta Phys. Sin. 62 050507Google Scholar
[5] Yang T, Gao Y, Wang D, Shi R P, Chen Z, Nie J F, Li J, Wang Y 2017 Acta Mater. 127 481Google Scholar
[6] Greenwood M, Rottler J, Provatas N 2011 Phys. Rev. E 83 031601Google Scholar
[7] Gao Y J, Deng Q Q, Liu Z Y, Huang Z J, Li Y X, Luo Z R 2020 J. Mater. Sci. Technol. 49 236Google Scholar
[8] Gao Y J, Huang L L, Deng Q Q, Zhou W Q, Luo Z R, Lin K 2016 Acta Mater. 117 238Google Scholar
[9] 赵宇龙, 陈铮, 龙建, 杨涛 2013 物理学报 62 118102Google Scholar
Zhao Y L, Chen Z, Long J, Yang T 2013 Acta Phys. Sin. 62 118102Google Scholar
[10] 赵宇龙, 陈铮, 龙建, 杨涛 2014 金属学报 27 81Google Scholar
Zhao Y L, Chen Z, Long J, Yang T 2014 Acta Metall. Sin. 27 81Google Scholar
[11] Gao Y J, Luo Z R, Huang L L, Mao H, Huang C G, Lin K 2016 Modell. Simul. Mater. Sci. 24 055010Google Scholar
[12] Hu S, Chen Z, Xi W, Peng Y Y 2017 J. Mater. Sci. 52 5641Google Scholar
[13] Seymour M, Sanches F, Elder K, Provatas N 2015 Phys. Rev. B 92 184109Google Scholar
[14] Faghihi N, Provatas N, Elder K R, Grant M, Karttunen M 2013 Phys. Rev. E 88 032407Google Scholar
[15] Berghoff M, Nestler B 2015 Comput. Condens. Matter 4 46Google Scholar
[16] Tang S, Wang Z J, Guo Y L, et al. 2012 Acta Mater. 60 5501Google Scholar
[17] 高英俊, 卢成健, 黄礼琳, 罗志荣, 黄创高 2014 金属学报 50 110Google Scholar
Gao Y J, Lu C J, Huang L L, Luo Z R, Huang C G 2014 Acta Metall. Sin. 50 110Google Scholar
[18] Hu S, Chen Z, Yu G G, Xi W, Peng Y Y 2016 Comput. Mater. Sci. 124 195Google Scholar
[19] Gutkin M Y, Ovid'ko I A 2001 Phys. Rev. B 63 064515Google Scholar
[20] Stefanovic P, Haataja M, Provatas N 2009 Phys. Rev. E 80 046107Google Scholar
[21] Bobylev S V, Gutkin M Y, Ovid'ko I A 2004 Phys. Solid State 46 2053Google Scholar
[22] Ohno T, Ii S, Shibata N, Matsunaga K, Ikuhara Y, Yamamoto T 2004 Mater. Trans. 45 2117Google Scholar
[23] Humphreys F J 2001 J. Mater. Sci. 36 3833Google Scholar
[24] Hirouchi T, Takaki T, Tomita Y 2009 Comput. Mater. Sci. 44 1192Google Scholar
[25] 龙建, 王诏玉, 赵宇龙, 龙清华, 杨涛, 陈铮 2013 物理学报 62 218101Google Scholar
Long J, Wang Z Y, Zhao Y L, Long Q H, Yang T, Chen Z 2013 Acta Phys. Sin. 62 218101Google Scholar
[26] 鲁娜, 王永欣, 陈铮 2014 物理学报 63 180508Google Scholar
Lu N, Wang Y X, Chen Z 2014 Acta Phys. Sin. 63 180508Google Scholar
[27] 杨涛, 陈铮, 董伟平 2011 金属学报 47 1301Google Scholar
Yang T, Chen Z, Dong W P 2011 Acta Metall. Sin. 47 1301Google Scholar
[28] Berkels B, Ratz A, Rumpf M, Voigt A 2008 J. Sci. Comput. 35 1Google Scholar
[29] Elsey M, Wirth B 2014 Multiscale Model. Sim. 12 1Google Scholar
[30] Singer H M, Singer I 2006 Phys. Rev. E 74 031103Google Scholar
[31] Wang Z J, Li J J, Guo Y L, Tang S, Wang J C 2013 Comput. Phys. Commun. 184 2489Google Scholar
[32] Hytch M J, Snoeck E, Kilaas R 1998 Ultramicroscopy 74 131Google Scholar
[33] Galindo P L, Kret S, Sanchez A M, et al. 2007 Ultramicroscopy 107 1186Google Scholar
[34] Hytch M J, Putaux J L, Penisson J M 2003 Nature 423 270Google Scholar
[35] Guo Y, Wang J, Wang Z, Li J, Tang S, Liu F, Zhou Y 2015 Philos. Mag. 95 973Google Scholar
[36] Wu K A, Adland A, Karma A 2010 Phys. Rev. E 81 061601Google Scholar
[37] Chen L Q, Shen J 1998 Comput. Phys. Commun. 108 147Google Scholar
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