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Cerium (Ce), a rare earth metal, undergoes a significant (14%−17%) and discontinuous volume shrinkage when subjected to ~0.7 GPa compression at ambient temperature: there happens a first-order isostructural phase transition from γ-Ce phase to α-Ce phase (these two phases are both face-centered-cubic (fcc) phase). Because of the α→ γ transition in Ce under shock compression, the shock front in cerium exhibits a 3-wave configuration: elastic precursor, plastic shock wave in γ-Ce, and phase transition wave corresponding to the γ → α transition according to the experimental observation. In this paper, a recently developed embedded-atom-method (EAM) potential for fcc Ce is employed in the large-scale molecular dynamics simulations of shock loading onto single crystal Ce to study its dynamic behavior, especially the shock-induced α→ γ phase transition, and the orientation dependence with [001], [011] and [111] shock loading. The simulation results show single-wave or multi-wave configuration for shock wave profiles. Under the shock loading along the [001] or [011] crystallographic orientation, the shock wave possesses a 2-wave structure: an elastic precursor and a phase transition wave, while under shock loading along the [111] crystallographic orientation, the obtained shock wave shows a 3-wave profile as observed experimentally. Thus the shock wave structure is obviously dependent on loading orientation. The Hugoniot data obtained in MD simulation show good agreement with the experimental results. The shock loading MD simulation shows lower phase transition pressure than hydrostatic loading, indicating an accelerant role of the deviatoric stress played in the shock induced γ → α phase transition in Ce. The local lattice structure before and after shocked are recognized with polyhedral template matching and confirmed with radial distribution functions. Under the [011] and [111] loading, the lattice structure maintains the fcc before and after the shocks, and experiences a collapse during the last shock (the second shock for the [011] loading and the third shock for the [111] loading). The lattice structure also maintains fcc before and after the first shock for the [001] loading, while after the second shock the structure type is considered to be body-centered-tetragonal (bct) which is a meta-stable structure resulting from the used EAM potential for Ce. The fcc lattice rotation after shock is observed in the [011] and [111] loading after the phase transition, while no re-orientation occurs in the [001] loading.
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Keywords:
- shock induced phase transition /
- isostructural phase transition /
- molecular dynamics /
- cerium
[1] Koskenmaki D C, Gschneidner K A 1978 Handbook on the Physics and Chemistry of Rare Earths (Vol. 1) (Amsterdam: Elsevier North-Holland) pp337−377
[2] 潘昊, 胡晓棉, 吴子辉, 戴诚达, 吴强 2012 61 206401
Google Scholar
Pan H, Hu X M, Wu Z H, Dai C D, Wu Q 2012 Acta Phys. Sin. 61 206401
Google Scholar
[3] Wang Y, Hector Jr L G, Zhang H, Shang S L, Chen L Q, Liu Z K 2008 Phys. Rev. B 78 104113
Google Scholar
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Google Scholar
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Google Scholar
[6] Borisenok V A, Simakov V G, Volgin V A, Bel'skii V M, Zhernokletov M V 2007 Combust. Expl. Shock Wavea 43 476
Google Scholar
[7] Simakov V G, Borisenok V A, Bragunets V A, Volgin V A, Zhernokletov M V, Zocher M A, Cherne F J 2007 Shock Compression of Condensed Matter-2007, Pts 1 and 2 Kohala Coast, Hawaii, June 24−29, 2007 pp105−108
[8] Yelkin V M, Kozlov E A, Kakshina E V, Moreva Y S 2006 Shock Compression of Condensed Matter-2005 Baltimore, Maryland July 31−August 5, 2005 pp77−80
[9] El'kin V M, Kozlov E A, Kakshina E V, Moreva Y S 2006 Phys. Met. Metall. 101 232
[10] El'kin V M, Mikhaylov V N, Petrovtsev A V, Cherne F J 2011 Phys. Rev. B 84 094120
Google Scholar
[11] Hu X, Pan H, Dai C, Wu Q 2012 Shock Compression of Condensed Matter - 2011, Pts 1 and 2 Chicago, Illinois, June 26-July 1, 2011 pp1567−1570
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Google Scholar
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Google Scholar
Diwu M J, Hu X M 2019 Acta Phys. Sin. 68 203401
Google Scholar
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Google Scholar
[16] Faken D, Jónsson H 1994 Comput. Mater. Sci. 2 279
Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
[21] 李俊, 吴强, 于继东, 谭叶, 姚松林, 薛桃, 金柯 2017 66 146201
Google Scholar
Li J, Wu Q, Yu J D, Tan Y, Yao S L, Xue T, Jin K 2017 Acta Phys. Sin. 66 146201
Google Scholar
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Google Scholar
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图 1 不同加载晶向和强度(up)的密度剖面图(t = 80 ps)
Figure 1. Density profiles of different loading orientation and strength (up) for t = 80 ps.
图 2 [111]晶向冲击加载后晶体中的微结构 (a) 全部原子; (b) DXA分析显示位错并隐去了fcc结构原子. 绿色原子为局部fcc, 红色为hcp, 蓝色为bcc. (b) 中用管表示位错: 绿色为Shockley偏位错, 深蓝色为全位错, 浅蓝色为梯杆位错. up = 150 m·s–1, 时间t = 80 ps
Figure 2. Microstructure of the sample shocked along [111]: (a) All atoms are shown; (b) only non-fcc atoms are shown. Color coding: Green for local fcc atoms; red for hcp; blue for bcc. Dislocations are illustrated with tubes in (b): Green for Shockley partials; deep blue for perfect fcc dislocations; light blue for stair-rod dislocations. up = 150 m·s–1, t = 80 ps.
图 3 冲击加载后晶体中的微结构. 冲击晶向为 (a) [001]; (b), (c) [011]; (d), (e) [111]; (c), (e)中隐去了fcc结构原子. up = 200 m·s–1, 时间t = 80 ps
Figure 3. Microstructure of the shocked samples. The shock orientation is along (a) [001], (b) and (c) [011], (d) and (e) [111], respectively. Atoms in fcc structure are hidden in (c) and (e). up = 200 m·s–1, t = 80 ps.
图 5 不同晶向加载下的压力剖面图, up = 200 m·s–1, 时间t = 80 ps
Figure 5. Pressure profile for each loading orientation at up = 200 m·s–1 and t = 80 ps.
图 6 冲击相变的相变温度-压力状态与静水压相变条件
Figure 6. Temperature-pressure condition of shock-induced and hydrostatic phase transition.
图 7 冲击前后的径向分布函数
Figure 7. Radial distribution function of the sample before and after the shocks.
图 8 Ce冲击相变分界面, 隐去了原子体积较大的fcc. up = 200 m·s-1, 时间t = 80 ps (a) [001]; (b) [011]; (c) [111]
Figure 8. Phase boundary of shock induced transition. Shock orientation: (a) [001]; (b) [011]; (c) [111]. The atoms of fcc structure with larger atomic volume are hidden. up = 200 m·s-1, t = 80 ps.
图 9 [001]晶向不同加载强度下相变波后的微结构 (a) up[001] = 150 m·s–1; (b) up[001] = 200 m·s–1; (c) up[001] = 250 m·s–1; (d) up[001] = 300 m·s–1; (e) up[001] = 400 m·s–1; (f) up[001] = 500 m·s–1
Figure 9. Microstructure of the sample after phase transition shock along [001] with listed piston velocity: (a) up[001] = 150 m·s–1; (b) up[001] = 200 m·s–1; (c) up[001] = 250 m·s–1; (d) up[001] = 300 m·s–1; (e) up[001] = 400 m·s–1; (f) up[001] = 500 m·s–1.
图 10 沿四方变形路径(原子体积不变)以及单轴压缩(a不变仅改变c/a)路径变形的能量
Figure 10. Comparison of the energy along tetragonal deforma-tion path (atomic volume preserved) and the path of constant a.
表 1 单晶Ce计算模型详细参数
Table 1. Parameters of single crystal Ce sample for MD simulation.
加载
晶向x 轴晶向及
尺寸/nmy 轴晶向及
尺寸/nmz 轴晶向及
尺寸/nm模型
原子数[001] [100] [010] [001] 5.00×106 25.8 25.8 255.5 [011] [100] $ [0 1 \bar1] $ [011] 4.83×106 25.8 25.5 251.9 [111] $ [\bar1 \bar1 2] $ $ [1 \bar1 0] $ [111] 4.88×106 25.9 25.6 253.7 
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表 2 [001]晶向加载相变波后区域微结构组分(依据PTM分析)(%)
Table 2. Fraction for each type of microstructure (analyzed with PTM algorithm) in the part after phase transition shock along [001] (%).
up[001] /m·s–1 150 200 250 300 400 500 fcc 66.1 44.1 24.6 11.8 2.1 0.4 bcc 30.1 51.6 71.4 84.9 96.1 97.4 其他 3.8 4.8 4.0 3.3 1.8 2.2
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[1] Koskenmaki D C, Gschneidner K A 1978 Handbook on the Physics and Chemistry of Rare Earths (Vol. 1) (Amsterdam: Elsevier North-Holland) pp337−377
[2] 潘昊, 胡晓棉, 吴子辉, 戴诚达, 吴强 2012 61 206401
Google Scholar
Pan H, Hu X M, Wu Z H, Dai C D, Wu Q 2012 Acta Phys. Sin. 61 206401
Google Scholar
[3] Wang Y, Hector Jr L G, Zhang H, Shang S L, Chen L Q, Liu Z K 2008 Phys. Rev. B 78 104113
Google Scholar
[4] Decremps F, Belhadi L, Farber D L, Moore K T, Occelli F, Gauthier M, Polian A, Antonangeli D, Aracne-Ruddle C M, Amadon B 2011 Phys. Rev. Lett. 106 065701
Google Scholar
[5] Pavlovskii M N, Komissarov V V, Kutsar A R 1999 Combust. Expl. Shock Waves 35 88
Google Scholar
[6] Borisenok V A, Simakov V G, Volgin V A, Bel'skii V M, Zhernokletov M V 2007 Combust. Expl. Shock Wavea 43 476
Google Scholar
[7] Simakov V G, Borisenok V A, Bragunets V A, Volgin V A, Zhernokletov M V, Zocher M A, Cherne F J 2007 Shock Compression of Condensed Matter-2007, Pts 1 and 2 Kohala Coast, Hawaii, June 24−29, 2007 pp105−108
[8] Yelkin V M, Kozlov E A, Kakshina E V, Moreva Y S 2006 Shock Compression of Condensed Matter-2005 Baltimore, Maryland July 31−August 5, 2005 pp77−80
[9] El'kin V M, Kozlov E A, Kakshina E V, Moreva Y S 2006 Phys. Met. Metall. 101 232
[10] El'kin V M, Mikhaylov V N, Petrovtsev A V, Cherne F J 2011 Phys. Rev. B 84 094120
Google Scholar
[11] Hu X, Pan H, Dai C, Wu Q 2012 Shock Compression of Condensed Matter - 2011, Pts 1 and 2 Chicago, Illinois, June 26-July 1, 2011 pp1567−1570
[12] Kadau K, Germann T C, Lomdahl P S, Holian B L 2005 Phys. Rev. B 72 064120
Google Scholar
[13] Dupont V, Chen S P, Germann T C 2010 EPJ Web of Conferences Paris, France, May 24−28, 2010 p00009
[14] 第伍旻杰, 胡晓棉 2019 68 203401
Google Scholar
Diwu M J, Hu X M 2019 Acta Phys. Sin. 68 203401
Google Scholar
[15] Plimpton S 1995 J. Comput. Phys. 117 1
Google Scholar
[16] Faken D, Jónsson H 1994 Comput. Mater. Sci. 2 279
Google Scholar
[17] Tsuzuki H, Branicio P S, Rino J P 2007 Comput. Phys. Commun. 177 518
Google Scholar
[18] Larsen P M, Schmidt S, Schiøtz J 2016 Modell. Simul. Mater. Sc. Eng. 24 055007
Google Scholar
[19] Stukowski A, Albe K 2010 Modell. Simul. Mater. Sc. Eng. 18 085001
Google Scholar
[20] Jensen B J, Cherne F J, Cooley J C, Zhernokletov M V, Kovalev A E 2010 Phys. Rev. B 81 214109
Google Scholar
[21] 李俊, 吴强, 于继东, 谭叶, 姚松林, 薛桃, 金柯 2017 66 146201
Google Scholar
Li J, Wu Q, Yu J D, Tan Y, Yao S L, Xue T, Jin K 2017 Acta Phys. Sin. 66 146201
Google Scholar
[22] 郭扬波, 唐志平, 徐松林 2004 固体力学学报 25 417
Google Scholar
Guo Y B, Tang Z P, Xu S L 2004 Acta Mech. Solida Sin. 25 417
Google Scholar
[23] Blank V D, Estrin E I 2013 Phase Transitions in Solids under High Pressure (Boca Raton: CRC Press) pp193−198
[24] Casadei M, Ren X, Rinke P, Rubio A, Scheffler M 2016 Phys. Rev. B 93 075153
Google Scholar
[25] Sheng H W, Kramer M J, Cadien A, Fujita T, Chen M W 2011 Phys. Rev. B 83 134118
Google Scholar
[26] Germann T C, Kadau K 2009 AIP Conf. Proc. 1195 1209
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