Research on atomistic simulation of the coexistence of multiple interfacial states at heterogeneous solid-liquid interface

doi:10.3969/j.issn.1000-5641.2024.03.006

Abstract

Abstract:

Engineering interfacial complexion (or phase) transitions has been a growing trend in grain boundary and solid surface systems. In addition, little attention has been paid to chemically heterogeneous solid-liquid interfaces. In this study, atomistic simulations are conducted to reveal the coexistence of novel in-plane multi-interfacial states in a Cu(111)/Pb(L) interface at a temperature just above the Pb freezing point. Four monolayer interfacial states, that is, two CuPb alloy liquids and two pre freezing Pb solids, are observed to coexist within two interfacial layers sandwiched between the bulk solid Cu and bulk liquid Pb. Computation of the spatial variations of various properties along the direction normal to the in-plane solid-liquid boundary lines for both interfacial layers presents a rich and varied picture of inhomogeneity and anisotropy in the mechanical, thermodynamical, and dynamical properties. The “bulk” values extracted from the in-plane profiles suggest that each interfacial state examined has distinct equilibrium values and significantly deviates from those of the bulk solid and liquid phases. It also indicates that the “complexion (or phase) diagrams” for the Cu(111)/Pb(L) interface bear a resemblance to those of the eutectic binary alloy systems as opposed to the monotectic phase diagram for the bulk CuPb alloy. The reported data supports the development of interfacial complexion (or phase) diagrams and interfacial phase rules and provides new guidelines for regulating heterogeneous nucleation and wetting processes.

Key words: heterogeneous alloy, solid-liquid interface, prefreezing transition, interfacial phase diagram, molecular dynamics

CLC Number:

O469

Jiaojiao LIU, Hongtao LIANG, Yang YANG. Research on atomistic simulation of the coexistence of multiple interfacial states at heterogeneous solid-liquid interface[J]. Journal of East China Normal University(Natural Science), 2024, 2024(3): 54-63.

Figures/Tables 5

Fig.1

Fig.2

Fig.3

Fig.4

Table 1

Thermodynamic parameters extracted from an equilibrium 620 K Cu(111)/Pb(L) interface with in-plane coexistence structure of interfacial solid and interfacial liquid states"

第一界面层 Pb(S)-alloy(L)		第二界面层 Pb(S)-alloy(L)		块体 Cu(111)/Pb(L)
${\widetilde{\rho } }^{ {\rm{S} },\mathrm{P}\mathrm{b} }/{\mathrm{n}\mathrm{m} }^{-3}$	40.90(1)	${\widetilde{\widetilde{\rho } } }^{{\rm{S}},\mathrm{P}\mathrm{b} }/{\mathrm{n}\mathrm{m} }^{-3}$	33.36(2)	${\rho }^{\mathrm{S,C}\mathrm{u} }/{\mathrm{n}\mathrm{m} }^{-3}$	83.063(1)
${\widetilde{\rho } }^{ {\rm{L,alloy} } }/{\mathrm{n}\mathrm{m} }^{-3}$	45.03(8)	${\widetilde{\widetilde{\rho } } }^{\rm{L,alloy} }/{\mathrm{n}\mathrm{m} }^{-3}$	31.51(5)	${\rho }^{\mathrm{L,P}\mathrm{b} }/{\mathrm{n}\mathrm{m} }^{-3}$	31.406(7)
${\widetilde \delta _{10 \sim 90} }/{\rm{nm} }$	2.7(2)	${\widetilde {\widetilde \delta }_{10 \sim 90} }/{\rm{nm} }$	2.4(3)	${\delta _{10 \sim 90} }/{\rm{nm} }$	0.8
${\widetilde{X} }_{\mathrm{C}\mathrm{u} }^{ {\rm{S} },\mathrm{P}\mathrm{b} }/\mathrm{\%}$	0.2(1)	${\widetilde{\widetilde{X} } }_{\mathrm{C}\mathrm{u} }^{\mathrm{S,P}\mathrm{b} }/\mathrm{\%}$	0.2(1)	${X}_{\mathrm{C}\mathrm{u} }^{\mathrm{S,C}\mathrm{u} }/\mathrm{\%}$	100
${\widetilde{X} }_{\mathrm{C}\mathrm{u} }^{\text{L},\text{alloy} }/\mathrm{\%}$	22.0(6)	${\widetilde{\widetilde{X} } }_{\mathrm{C}\mathrm{u} }^{\text{L,alloy} }/\mathrm{\%}$	1.8(1)	${X}_{\mathrm{C}\mathrm{u} }^{\mathrm{L,P}\mathrm{b} }/\mathrm{\%}$	0.64(1)
$\widetilde{E}_{\mathrm{P},\mathrm{Cu} }^{ {\rm{S} },\mathrm{P}\mathrm{b} }/\mathrm{e}\mathrm{V}$	–3.951(2)	$\widetilde{\widetilde{E} }{}_{\mathrm{P},\mathrm{Cu} }^{{\rm{S}},\mathrm{P}\mathrm{b} }/\mathrm{e}\mathrm{V}$	–4.156(2)	$E_{\mathrm{P,C}\mathrm{u} }^{\mathrm{S,C}\mathrm{u} }/\mathrm{e}\mathrm{V}$	–3.4547(1)
$\widetilde{E}_{\mathrm{P},\mathrm{Cu} }^{\text{L},\text{alloy} }/\mathrm{e}\mathrm{V}$	–3.905(2)	$\widetilde{\widetilde{E} }{}_{\mathrm{P},\mathrm{Cu} }^{\rm{L},\rm{alloy} }/\mathrm{e}\mathrm{V}$	–4.094(4)	$E{}_{\mathrm{P,C}\mathrm{u} }^{\mathrm{L,P}\mathrm{b} }/\mathrm{e}\mathrm{V}$	–4.1033(2)
$\widetilde{E}{}_{\mathrm{P},\mathrm{Pb} }^{ {\rm{S} },\mathrm{P}\mathrm{b} }/\mathrm{e}\mathrm{V}$	–1.076(1)	$\widetilde{\widetilde{E} }{}_{\mathrm{P},\mathrm{Pb} }^{{\rm{S}},\mathrm{P}\mathrm{b} }/{\mathrm{e}\mathrm{V}}$	–1.914(1)	$E{}_{\mathrm{P,P}\mathrm{b} }^{\mathrm{S,C}\mathrm{u} }/\mathrm{e}\mathrm{V}$	N/A
$\widetilde{E}{}_{\mathrm{P},\mathrm{Pb} }^{\text{L},\text{alloy} }/\mathrm{e}\mathrm{V}$	–0.933(6)	$\widetilde{\widetilde{E} }{}_{\mathrm{P},\mathrm{Pb} }^{\rm{L},\text{alloy} }/\mathrm{e}\mathrm{V}$	–1.788(8)	$E{}_{\mathrm{P,P}\mathrm{b} }^{\mathrm{L,P}\mathrm{b} }/\mathrm{e}\mathrm{V}$	–1.8873(3)
${\widetilde{p} }_{xy}^{ {\rm{S} },\mathrm{P}\mathrm{b} }/\mathrm{k}\mathrm{b}\mathrm{a}\mathrm{r}$	–6.30(4)	${\widetilde{\widetilde{p} } }_{xy}^{\mathrm{S,P}\mathrm{b} }/\mathrm{k}\mathrm{b}\mathrm{a}\mathrm{r}$	9.42(5)	${p}_{xy}^{\mathrm{S,C}\mathrm{u} }/\mathrm{k}\mathrm{b}\mathrm{a}\mathrm{r}$	0.002(4)
${\widetilde{p} }_{zz}^{ {\rm{S} },\mathrm{P}\mathrm{b} }/\mathrm{k}\mathrm{b}\mathrm{a}\mathrm{r}$	–2.61(1)	${\widetilde{\widetilde{p} } }_{zz}^{\mathrm{S,P}\mathrm{b} }/\mathrm{k}\mathrm{b}\mathrm{a}\mathrm{r}$	–0.66(1)	${p}_{zz}^{\mathrm{S,C}\mathrm{u} }/\mathrm{k}\mathrm{b}\mathrm{a}\mathrm{r}$	0.001(1)
${\widetilde{p} }_{xy}^{ {\rm{L} },\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{o}\mathrm{y} }/\mathrm{k}\mathrm{b}\mathrm{a}\mathrm{r}$	1.00(1)	${\widetilde{\widetilde{p} } }_{xy}^{\mathrm{L,a}\mathrm{l}\mathrm{l}\mathrm{o}\mathrm{y} }/\mathrm{k}\mathrm{b}\mathrm{a}\mathrm{r}$	3.55(2)	${p}_{xy}^{\mathrm{L,P}\mathrm{b} }/\mathrm{k}\mathrm{b}\mathrm{a}\mathrm{r}$	0.003(6)
${\widetilde{p} }_{zz}^{ {\rm{L} },\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{o}\mathrm{y} }/\mathrm{k}\mathrm{b}\mathrm{a}\mathrm{r}$	–2.16(2)	${\widetilde{\widetilde{p} } }_{zz}^{\mathrm{L,a}\mathrm{l}\mathrm{l}\mathrm{o}\mathrm{y} }/\mathrm{k}\mathrm{b}\mathrm{a}\mathrm{r}$	0.56(2)	${p}_{zz}^{\mathrm{L,P}\mathrm{b} }/\mathrm{k}\mathrm{b}\mathrm{a}\mathrm{r}$	–0.004(4)
${\widetilde{D} }_{xy}^{\text{S},\text{Pb} }/( {10}^{5}\;{\mathrm{c}\mathrm{m} }^{2}\cdot {\mathrm{s} }^{-1})$	0.047(2)	${\widetilde{\widetilde{D} } }_{xy}^{\rm{S,Pb} }/({{10}^{5}\;\mathrm{c}\mathrm{m} }^{2}\cdot {\mathrm{s} }^{-1})$	0.064(2)	${D}^{\mathrm{S,C}\mathrm{u} } /\left({10}^{5}\;{\mathrm{c}\mathrm{m} }^{2}\cdot {\mathrm{s} }^{-1}\right)$	0.002(1)
${\widetilde{D} }_{xy}^{\text{L},\text{alloy} }/\left( {10}^{5}\;{\mathrm{c}\mathrm{m} }^{2}\cdot {\mathrm{s} }^{-1}\right)$	0.509(5)	${\widetilde{\widetilde{D} } }_{xy}^{\text{L,alloy} }/\left({10}^{5}\;{\mathrm{c}\mathrm{m} }^{2}\cdot {\mathrm{s} }^{-1}\right)$	0.947(7)	${D}^{\mathrm{L,P}\mathrm{b} } /\left({10}^{5}\;{\mathrm{c}\mathrm{m} }^{2}\cdot {\mathrm{s} }^{-1}\right)$	1.540(2)

Table 1

References 56

1	TANG M, CARTER W C, CANNON R M.. Diffuse interface model for structural transitions of grain boundaries. Physical Review B, 2006, 73 (2): 024102.
2	FROLOV T, OLMSTED D L, ASTA M, et al.. Structural phase transformations in metallic grain boundaries. Nature Communications, 2013, 4 (1): 1899.
3	BARAM M, CHATAIN D, KAPLAN W D.. Nanometer-thick equilibrium films: The interface between thermodynamics and atomistics. Science, 2011, 332 (6026): 206- 209.
4	BISHOP C M, TANG M, CANNON R M, et al.. Continuum modelling and representations of interfaces and their transitions in materials. Materials Science and Engineering a-Structural Materials Properties Microstructure and Processing, 2006, 422 (1-2): 102- 114.
5	CANTWELL P R, TANG M, DILLON S J, et al.. Grain boundary complexions. Acta Materialia, 2014, 62, 1- 48.
6	KUNDU A, ASL K M, LUO J, et al.. Identification of a bilayer grain boundary complexion in Bi-doped Cu. Scripta Materialia, 2013, 68 (2): 146- 149.
7	FROLOV T, OLMSTED D L, ASTA M, et al.. Structural phase transformations in metallic grain boundaries. Nature Communications, 2013, 4, 1899.
8	PEREIRO-LÓPEZ E, LUDWIG W, BELLET D, et al.. Direct evidence of nanometric invasionlike grain boundary penetration in the Al/Ga system. Physical Review Letters, 2005, 95 (21): 215501.
9	LUO J.. Let thermodynamics do the interfacial engineering of batteries and solid electrolytes. Energy Storage Materials, 2019, 21, 50- 60.
10	DILLON S J, TANG M, CARTER W C, et al.. Complexion: A new concept for kinetic engineering in materials science. Acta Materialia, 2007, 55 (18): 6208- 6218.
11	FROLOV T.. Effect of interfacial structural phase transitions on the coupled motion of grain boundaries: A molecular dynamics study. Applied Physics Letters, 2014, 104 (21): 211905.
12	FROLOV T, MISHIN Y.. Phases, phase equilibria, and phase rules in low-dimensional systems. Journal of Chemical Physics, 2015, 143 (4): 044706.
13	FROLOV T, ASTA M, MISHION Y.. Phase transformations at interfaces: Observations from atomistic modeling. Current Opinion in Solid State & Materials Science, 2016, 20 (5): 308- 315.
14	LI X, LU K.. Improving sustainability with simpler alloys. Science, 2019, 364 (6442): 733- 734.
15	HECKMAN N M, FOILES S M, O'BRIEN C J.. New nanoscale toughening mechanisms mitigate embrittlement in binary nanocrystalline alloys. Nanoscale, 2018, 10 (45): 21231- 21243.
16	O'BRIEN C J, BARR C M, PRICE P M, et al.. Grain boundary phase transformations in PtAu and relevance to thermal stabilization of bulk nanocrystalline metals. Journal of Materials Science, 2018, 53 (4): 2911- 2927.
17	PETER N J, FROLOV T, DUARTE M J, et al.. Segregation-Induced nanofaceting transition at an asymmetric tilt grain boundary in copper. Physical Review Letters, 2018, 121 (25): 255502.
18	MEIBERS T, FROLOV T, RUDD R E, et al.. Observations of grain-boundary phase transformations in an elemental metal. Nature, 2020, 579 (7799): 375.
19	BARR C M, FOILES S M, ALKAYYALI M, et al.. The role of grain boundary character in solute segregation and thermal stability of nanocrystalline Pt-Au. Nanoscale, 2021, 13 (6): 3552- 3563.
20	TUCHINDA N, SCHUH C A.. Grain size dependencies of intergranular solute segregation in nanocrystalline materials. Acta Materialia, 2022, 226, 117614.
21	WYNBLATT P, CHATAIN D, RANGUIS A.. STM study of Bi-on-Cu(100). Surface Science, 2007, 601 (6): 1623- 1629.
22	PICKERING I, PALEICO M, SIRKIN Y A P, et al. Grand canonical investigation of the quasi liquid layer of ice: Is it liquid? [J]. The Journal of Physical Chemistry B, 2018, 122(18): 4880-4890.
23	TANG J, LAMBIE S, MEFTAHI N, et al.. Unique surface patterns emerging during solidification of liquid metal alloys. Nature Nanotechnology, 2021, 16 (4): 431- 439.
24	WANG M, PENG Y, WANG H, et al.. Coarsening of polycrystalline patterns in atomically thin surface crystals. Applied Physics Letters, 2021, 119 (12): 123102.
25	GABRISCH H, KJELDGAARD L, JOHNSON E, et al.. Equilibrium shape and interface roughening of small liquid Pb inclusions in solid Al. Acta Materialia, 2001, 49 (20): 4259- 4269.
26	OH S H, KAUFFMANN Y, SCHEU C, et al.. Ordered liquid aluminum at the interface with sapphire. Science, 2005, 310 (5748): 661- 663.
27	MOORTHY S K E, MENDELEV M I, HOWE J M.. The influence of spatial and temporal averaging on interpretation of HRTEM images of solid-liquid interfaces. Ultramicroscopy, 2013, 124, 40- 45.
28	SCHNEIDER M M, HOWE J M.. Observation of interface dynamics and Cu island formation at a crystalline Si/liquid Al-alloy interface. Acta Materialia, 2017, 133, 224- 229.
29	YANG G Q, LI J F, SHI Q W, et al.. Structural and dynamical properties of heterogeneous solid-liquid Ta-Cu interfaces: A molecular dynamics study. Computational Materials Science, 2014, 86, 64- 72.
30	KERN J L, BARRY P R, LAIRD B B.. Characterization of the Al-Ga solid-liquid interface using classical and ab initio molecular dynamics simulation. Physical Review Materials, 2020, 4 (4): 043604.
31	PALAFOX-HERNANDEZ J P, LAIRD B B.. Orientation dependence of heterogeneous nucleation at the Cu-Pb solid-liquid interface. Journal of Chemical Physics, 2016, 145 (21): 211914.
32	YANG Y, ASTA M, LAIRD B B.. Solid-Liquid interfacial premelting. Physical Review Letters, 2013, 110 (9): 096102.
33	LIANG H T, LAIRD B B, ASTA M, et al.. In-plane characterization of structural and thermodynamic properties for steps at faceted chemically heterogeneous solid/liquid interfaces. Acta Materialia, 2018, 143, 329- 337.
34	WYNBLATT P, SHI Z.. Relation between grain boundary segregation and grain boundary character in FCC alloys. Journal of Materials Science, 2005, 40 (11): 2765- 2773.
35	LUO J.. Grain boundary complexions: The interplay of premelting, prewetting, and multilayer adsorption. Applied Physics Letters, 2009, 95 (7): 071911.
36	RICKMAN J M, LUO J.. Layering transitions at grain boundaries. Current Opinion in Solid State & Materials Science, 2016, 20 (5): 225- 230.
37	WYNBLATT P, CHATAIN D.. Solid-state wetting transitions at grain boundaries. Materials Science and Engineering a-Structural Materials Properties Microstructure and Processing, 2008, 495 (1-2): 119- 125.
38	TANG M, CARTER W C, CANNON R M.. Grain boundary transitions in binary alloys. Physical Review Letters, 2006, 97 (7): 075502.
39	CLARKE D R, SHAW T M, PHILIPSE A P, et al.. Possible electrical double-layer contribution to the equilibrium thickness of intergranular glass films in polycrystalline ceramics. Journal of the American Ceramic Society, 1993, 76 (5): 1201- 1204.
40	MISHIN Y, BOETTINGER W J, WARREN J A, et al.. Thermodynamics of grain boundary premelting in alloys. I. Phase-field modeling. Acta Materialia, 2009, 57 (13): 3771- 3785.
41	LUO J.. Developing interfacial phase diagrams for applications in activated sintering and beyond: current status and future directions. Journal of the American Ceramic Society, 2012, 95 (8): 2358- 2371.
42	GAO B, GAO P Y, LU S H, et al.. Interface structure prediction via CALYPSO method. Science Bulletin, 2019, 64 (5): 301- 309.
43	HOYT J J, GARVIN J W, WEBB E B, et al.. An embedded atom method interatomic potential for the Cu-Pb system. Modelling and Simulation in Materials Science and Engineering, 2003, 11 (3): 287- 299.
44	WEBB E B, GREST G S, HEINE D R.. Precursor film controlled wetting of Pb on Cu. Physical Review Letters, 2003, 91 (23): 236102.
45	HEINE D R, GREST G S, WEBB E B.. Surface wetting of liquid nanodroplets: Droplet-size effects. Physical Review Letters, 2005, 95 (10): 107801.
46	PALAFOX-HERNANDEZ J P, LAIRD B B, ASTA M.. Atomistic characterization of the Cu-Pb solid-liquid interface. Acta Materialia, 2011, 59 (8): 3137- 3144.
47	PLIMPTON S.. Fast parallel algorithms for short-range molecular dynamics. Journal of Computational Physics, 1995, 117 (1): 1- 19.
48	YANG Y, OLMSTED D L, ASTA M, et al.. Atomistic characterization of the chemically heterogeneous Al-Pb solid-liquid interface. Acta Materialia, 2012, 60 (12): 4960- 4971.
49	CANTWELL P R, FROLOV T, RUPERT T J, et al.. Grain boundary complexion transitions. Annual Review of Materials Research, 2020, 50, 465- 492.
50	DU H, LIANG H T, YANG Y.. Molecular dynamics simulation of monolayer confined ice-water phase equilibrium. Acta Chimica Sinica, 2018, 76 (6): 483- 490.
51	ZHANG X, LAIRD B B, LIANG H T, et al.. Atomistic characterization of the SiO₂ high-density liquid/low-density liquid interface. Journal of Chemical Physics, 2022, 157 (13): 134703.
52	LU W L, LIANG H T, MA X M, et al.. Atomistic simulation study of the FCC and BCC crystal-melt interface stresses. Surfaces and Interfaces, 2022, 28, 101639.
53	EMUNA M, GREENBERG Y, HEVRONI R, et al.. Phase diagrams of binary alloys under pressure. Journal of Alloys and Compounds, 2016, 687, 360- 369.
54	HUDON P, JUNG I H, BAKER D R.. Effect of pressure on liquid-liquid miscibility gaps: A case study of the systems CaO-SiO₂, MgO-SiO₂, and CaMgSi₂O₆-SiO₂. Journal of Geophysical Research-Solid Earth, 2004, 109 (B3): B03207.
55	MA X M, LIANG H T, LU W L, et al.. Atomistic characterization of the dispersed liquid droplet in immiscible Al-Pb alloy. Journal of Materials Research and Technology, 2021, 15, 2993- 3004.
56	YANG Y, LAIRD B B.. Droplet spreading on a surface exhibiting solid-liquid interfacial premelting. Acta Materialia, 2018, 143, 319- 328.

[1]	Wenjie DING, Wenhui XIE. Research on calculation of two-dimensional transition metal chalcogenides compounds MX₂-MX-MX₂ (M = V, Cr, Mn, and Fe; X = S, Se, and Te) [J]. Journal of East China Normal University(Natural Science), 2024, 2024(3): 45-53.
[2]	Runrun DU, Shan WANG, Xuezhi KE. Structural phase transitions of Th₂N₂S under high pressure: A first-principles calculation study [J]. Journal of East China Normal University(Natural Science), 2024, 2024(3): 36-44.
[3]	Wei ZHAO, Qinghong YUAN. Bandgap tuning of C₃N: A first-principles study [J]. Journal of East China Normal University(Natural Science), 2022, 2022(4): 114-119.
[4]	Yaqiong ZHANG, Wenhui XIE. First-principles calculations investigations of two-dimensional transition metal phosphide M_nT_n₊₁(M = V, Cr; T = P, As, and Sb) slices [J]. Journal of East China Normal University(Natural Science), 2022, 2022(2): 84-92.
[5]	Qinghua XI, Yiqiang HUANG, Jiaxiang CHEN, Er NIE, Zhuo SUN. Study on Fe₂O₃/g-C₃N₄ photocatalytic degradation of Rhodamine B [J]. Journal of East China Normal University(Natural Science), 2021, 2021(3): 151-160.
[6]	LI Peng-fei, WANG Mei-ting, MEI Ye. Comparison of the efficiency of equilibrium and nonequilibrium molecular dynamic simulations of molecular solvation free energies [J]. Journal of East China Normal University(Natural Sc, 2019, 2019(1): 83-92.
[7]	HUANG Xian-zhi, PIAO Xian-qing, CAI Ya-guo. Preparation of photocatalytic materials MIL-125(Ti)/BiOI and photocatalytic performance study [J]. Journal of East China Normal University(Natural Sc, 2019, 2019(1): 93-104,114.
[8]	SHEN Yu-hao, TANG Zheng, PENG Wei. A negatively charged V_SiO_N center for implementation as qubit [J]. Journal of East China Normal University(Natural Sc, 2017, 2017(2): 97-106.
[9]	ZHANG Jian-Feng, DAN Shu-Ping. Properties of the bound magnetopolaron in a triangular quantum well [J]. Journal of East China Normal University(Natural Sc, 2015, 2015(6): 101-107.
[10]	GAO Li-peng, LIU Kai, LUO Chun-hua, LI Jian-qi,WANG Yi-ting, PENG Hui. Preparation and properties study of a new type of T1-T2 dual mode MRI contrast agent [J]. Journal of East China Normal University(Natural Sc, 2013, 2013(5): 102-109.
[11]	SUN Fang, TANG Zheng. Spin interactions in direct-gap semiconductors [J]. Journal of East China Normal University(Natural Sc, 2012, 2012(5): 31-36.
[12]	YUAN Li;YANG Xie-long;ZHAO Zhen-jie. Influence of Joule annealing on GMI effect for Co-based amorphous microwires(Chinese) [J]. Journal of East China Normal University(Natural Sc, 2008, 2008(3): 120-124.
[13]	WANG Jianqiao, LIU dong, ZHOU Jun, XI Qinghua, NIE Er, SUN Zhuo. Application of TiO₂ nanotube arrays for bipolar photocatalytic fuel cells [J]. Journal of East China Normal University(Natural Science), 2020, 2020(1): 93-102.