本研究利用球状双晶分子动力学模型计算了纯铝对称倾斜晶界和扭转晶界的能量，基于计算所得晶界能探讨分析了晶界能与晶界两侧相邻点阵重合度的相关性。结果表明，重位点阵(CSL)和非CSL晶界能量的相对大小不会呈现一定规律性的差别，当两类晶界对应的取向参数相近时，晶界能也随之相近，即晶界能的高低与晶界两侧相邻点阵是否存在重合无关。对于CSL晶界，其能量与晶界对应重位因子间无可见相关性，不能基于重位因子预测晶界能的相对高低。 In this study, the energies ofsymmetrical tilt grain boundaries (GBs) and twist GBs were calculated using the sphere-bicrystal molecular dynamics model, and the correlation between GB energy and coincidence of neighboring lattices on both sides of GB was investigated based on the calculated energy results. The energy difference between coincidence site lattice (CSL) GBs and non-CSL GBs does not show any visible tendency, these two kinds of GB will have quite close ener-gies when their crystallographic parameters are nearly the same. This indicates that the relative magnitude of GB energy is irrelevant to whether neighboring lattices are coincident or not. For CSL GBs, no correlation is observed between their energies and the coincidence index (Ʃ) corresponding to them, i.e., being unable to predict the relative magnitude of GB energy based on Ʃ.
晶界，晶界能，重位因子，分子动力学, Grain Boundary
Grain Boundary Energy
Coincidence Index
Molecular Dynamics
晶界能与晶界两侧相邻点阵重合度的相关性研究<sup> </sup>
现主要有块状双晶 [
6
] 和球状双晶 [
5
] 两种分子动力学模型计算晶界能，这两种模型计算晶界能的原理相同，都是通过获取单位晶界面积上包含晶界的双晶体系总自由能与同等大小单晶体系总自由能差。前者由于采用周期性边界条件，晶界面内两个尺寸方向必须采用周期性边界条件，且相邻两晶粒沿该两方向的点阵周期长度必须存在最小公倍数，因而只适用于重位因子较小的CSL晶界 [
6
] 。后者采用自由表面，在应用时不受晶界取向参数的限制，从而可计算任意晶界的能量 [
5
] 。因此，本研究将采用球状模型计算各组晶界的能量。图1给出了球状双晶模型计算任意晶界能量的主要步骤。如图所示，具有取向 g 0 的球状单晶按一定要求转动后，分别获得取向为 g A 和 g B 的两个晶粒A和B。沿给定方向n将两个晶粒分别分割成两个同等大小的半球，随后将晶粒A和B的两个半球刚性对接成晶粒取向满足取向差 Δ g = g A − 1 g B 、晶界面法向平行于n的双晶。所构建球状单晶和双晶体系还需基于共轭梯度法进行能量最小化驰豫。驰豫后的单双晶体系被认为具有相同的自由表面能，因而单晶总自由能( E 1 )与双晶总自由能( E 2 )间的差异仅源自于双晶中存在的晶界，也即晶界能量 γ 0 可基于下式获取：
γ 0 = E 2 − E 1 S (1)
其中S为球状双晶体系中晶界的面积， S = π ⋅ R 2 ，R为双晶半径。
图1. 基于球状双晶模型构建取向差为 Δ g = g A − 1 g B 和晶界面取向为n的晶界示意图
国内研究者张明亮 [
15
] 发现，球状模型必须引入块状模型中的原子删除操作 [
6
] ，否则所得晶界能会显著高于晶界能的理论值。引入该操作的主要目的是：删除球状模型中晶界附近可能存在的相距过近和能量过高的原子，构建给定晶界取向参数下的不同初始晶界结构，以获取状态更为稳定的晶界结构。该操作具体为：包含任意晶界的双晶体系在结构驰豫前，晶界处间距小于临界间距 d t 的两相邻原子中任意一个原子将被删除；同时，为构建不同初始晶界结构， d t 以一定增量 d s 在最小间距 d min 和最大间距 d max 间取值，最终选定所有初始晶界结构中对应能量最低者为晶界的结构。本研究在利用球状模型计算晶界能时，也引入原子删除操作以获取更为准确的晶界能。
在利用球状模型计算表1中各晶界的能量前，还需利用文献 [
6
] 中的晶体学理论基于各晶界对应的取向差轴和取向差角推导构建双晶模型所需两个晶粒的取向。本研究所采用球状模型的半径为5 nm，原子删除参数为 d min = d n n / 3 , d max = 0.85 d n n 和ds= 0.05 Å。其中， d n n = a 0 / 2 表示纯铝内的最小原子间距，a0为点阵常数4.05 Å。选用文献 [
16
] 提出的2NN-MEAM纯铝原子势函数来描述原子的能量和相互作用力。本研究所有分子动力学模拟都是利用Lammps [
17
] 软件。
黄双凤,杨 亮. 晶界能与晶界两侧相邻点阵重合度的相关性研究Study of the Correlation between Grain Boundary Energy and Coincidence of Neighboring Lattices on Both Sides of Boundary[J]. 材料科学, 2019, 09(08): 821-827. https://doi.org/10.12677/MS.2019.98102
参考文献References
Sutton, A.P. and Balluffi, R.W. (1995) Interfaces in Crystalline Materials. Oxford University Press, Oxford.
Humphreys, J., Rohrer, G.S. and Rollett, A.D. (2017) Recrystallization and Related Annealing Phenomena. Elsevier, Amsterdam. https://doi.org/10.1016/B978-0-08-098235-9.00015-X
Read, W.T. and Shockley, W. (1950) Dislocation Models of Crystal Grain Boundaries. Physical Review, 78, 275-289. https://doi.org/10.1103/PhysRev.78.275
Olmsted, D.L., Foiles, S.M. and Holm, E.A. (2009) Survey of Computed Grain Boundary Properties in Face-Centered Cubic Metals: I. Grain Boundary Energy. Acta Materialia, 57, 3694-3703. https://doi.org/10.1016/j.actamat.2009.04.007
Lee, B.J. and Choi, S.H. (2004) Computation of Grain Boundary Energies. Modelling and Simulation in Materials Science and Engineering, 12, 621-632. https://doi.org/10.1088/0965-0393/12/4/005
Tschopp, M.A. and McDowell, D.L. (2007) Structures and Energies of Σ3 Asymmetric Tilt Grain Boundaries in Copper and Aluminium. Philosophical Magazine, 87, 3147-3173. https://doi.org/10.1080/14786430701255895
Randle, V. (2017) The Measurement of Grain Boundary Geometry. Routledge, London. https://doi.org/10.1201/9780203736401
Merkle, K.L. (1994) Atomic Structure of Grain Boundaries. Journal of Physics and Chemistry of Solids, 55, 991-1005. https://doi.org/10.1016/0022-3697(94)90119-8
Brokman, A. and Balluffi, R.W. (1981) Coincidence Lattice Model for the Structure and Energy of Grain Boundaries. Acta Metallurgica, 29, 1703-1719. https://doi.org/10.2172/6560150
Randle, V. (1998) The Role of the Grain Boundary Plane in Cubic Polycrystals. Acta Materialia, 46, 1459-1480. https://doi.org/10.1016/S1359-6454(97)00338-8
Bishop, G.H. and Chalmers, B. (1968) A Coincidence-Edge-Dislocation Description of Grain Boundaries. Scripta Metallurgica, 2, 133-139. https://doi.org/10.1016/0036-9748(68)90085-9
Hasson, G.C. and Goux, C. (1971) Interfacial Energies of Tilt Boundaries in Aluminium. Experimental and Theoretical Determination. Scripta Metallurgica, 5, 889-894. https://doi.org/10.1016/0036-9748(71)90064-0
Ratanaphan, S., Olmsted, D.L., Bulatov, V.V., et al. (2015) Grain Boundary Energies in Body-Centered Cubic Metals. Acta Materialia, 88, 346-354. https://doi.org/10.1016/j.actamat.2015.01.069
Otsuki, A. and Mizuno, M. (1986) Grain Boundary Energy and Liquid Metal Embrittlement of Aluminium. Transactions of the Japan Institute of Metals (Supplement), 27, 789-796.
张明亮. 晶界能分子动力学计算球形模型的改进与应用[D]: [硕士学位论文]. 长沙: 中南大学, 2015.
Lee, B.-J., Shim, J.-H. and Baskes, M.I. (2003) Semiempirical Atomic Potentials for the FCC Metals Cu, Ag, Au, Ni, Pd, Pt, Al, and Pb Based on First and Second Nearest-Neighbor Modified Embedded Atom Method. Physical Review B, 68, Article ID: 144112. https://doi.org/10.1103/PhysRevB.68.144112
Plimpton, S. (1995) Fast Parallel Algorithms for Short-Range Molecular Dynamics. Journal of Computational Physics, 117, 1-19. https://doi.org/10.1006/jcph.1995.1039
Sutton, A. and Balluffi, R. (1987) Overview No. 61 on Geometric Criteria for Low Interfacial Energy. Acta Metallurgica, 35, 2177-2201. https://doi.org/10.1016/0001-6160(87)90067-8
