晶体相场模型及其在材料微结构演化中的应用
|
摘要
随着计算机技术的快速发展,计算机模拟实验在材料科学中的作用越来越突出。计算机数值模拟技术已经和实验观测、理论模型分析并称为20世纪以来的三大科学研究方法。本文首先简要地从空间特征分辨尺度和时间特征尺度比较了几种重要的计算模拟方法——分子动力学(MD)、传统相场方法(TPF)和晶体相场(PFC)方法的各自适应的特征尺度范围和特点。在模拟纳观尺度的材料微结构演化,PFC在特征时间尺度上更具优势。其次,介绍了PFC模型,及其建立的物理基础和数学基础,以及该方法的特色优势。同时,介绍该PFC模型的拓展与推广,包括二元和多元体系、气-液-固三相体系、双模和多模体系的PFC模型,以及求解PFC模型的动力学方程数值计算的关键技术与主要步骤。再次,结合作者在材料微结构演化方面的研究,着重介绍PFC模型的几个重要方面的应用例子,包括材料纳观缺陷结构演化、凝固的枝晶生长和晶体外延生长、高温预熔化变形和动态回复、纳观尺度的裂纹扩展与分叉、无序-有序金属玻璃转变、石墨烯的缺陷结构、金属互联线电迁移空洞、多铁复合材料的畴结构、金属泡沫结构的生成等。最后,总结并指出PFC模型的拓展方向与今后应用的重点方面和新领域。
With the rapid development of computer technology, the roles of computer numerical simulation technology in materials are more and more prominent. Computer numerical simulation technology,real experimental observation and theoretical model analysis are the same important and are known as three great scientific research methods since the 20 thcentury. In this paper, several important computational numerical simulation methods are briefly compared, firstly, in the spatial characteristic resolution scale and the characteristic time scale, for example, for molecular dynamics(MD), traditional phase field(TPF), and phase field crystal(PFC) method. For simulation of microstructure evolution in nano-scale,the PFC method is of the advantage on the characteristic time scale. Then, the PFC model, and its physical and mathematical basises for establishment, as well as the special feature of the method, are introduced. Next, the development of the PFC models are presented, including the PFC model of binary and multi-element alloys, of gas-liquid-solid three systems, of two-mode and multimode systems, as well as the key technology and the main procedure of the numerical calculation of the dynamic equation solution.After that, combining with the research works of the authors' group in the microstructure evolution of materials, several examples of important aspects of application of the PFC model are presented, including the nanostructure of defects of materials, dendritic growth and heterogenous epitxial growth, premelting under deformation at high temperature and dynamic recovery, extension and bifurcation of cracks on nanoscale, matalllic glass transition, defect structures of graphene, voids formation of electromigration in metal interconnects, microstructure in multiferroic composite matrials, and the formation of the structure of the metal foams. Finally, a summary is given and the development direction and future emphasis application and new fields of the PFC model are pointed out.
With the rapid development of computer technology, the roles of computer numerical simulation technology in materials are more and more prominent. Computer numerical simulation technology,real experimental observation and theoretical model analysis are the same important and are known as three great scientific research methods since the 20 thcentury. In this paper, several important computational numerical simulation methods are briefly compared, firstly, in the spatial characteristic resolution scale and the characteristic time scale, for example, for molecular dynamics(MD), traditional phase field(TPF), and phase field crystal(PFC) method. For simulation of microstructure evolution in nano-scale,the PFC method is of the advantage on the characteristic time scale. Then, the PFC model, and its physical and mathematical basises for establishment, as well as the special feature of the method, are introduced. Next, the development of the PFC models are presented, including the PFC model of binary and multi-element alloys, of gas-liquid-solid three systems, of two-mode and multimode systems, as well as the key technology and the main procedure of the numerical calculation of the dynamic equation solution.After that, combining with the research works of the authors' group in the microstructure evolution of materials, several examples of important aspects of application of the PFC model are presented, including the nanostructure of defects of materials, dendritic growth and heterogenous epitxial growth, premelting under deformation at high temperature and dynamic recovery, extension and bifurcation of cracks on nanoscale, matalllic glass transition, defect structures of graphene, voids formation of electromigration in metal interconnects, microstructure in multiferroic composite matrials, and the formation of the structure of the metal foams. Finally, a summary is given and the development direction and future emphasis application and new fields of the PFC model are pointed out.
引文
[1]Uchic M D,Dimiduk D M,Florando J N,et al.Sample dimensions influence strength and crystal plasticity[J].Science,2004,305:986
[2]Provatas N,Dantzig J A,Athrega B,et al.Using the phase-field crystal method in the multi-scale modeling of microstructure evolution[J].JOM,2007,59(7):83
[3]Chernatynskiy A,Phillpot S R,Lesar R.Uncertainty quantification in multi-scale simulation of materials:A prospective[J].Annu.Rev.Mater.Res.,2013,43:157
[4]Rajan K.Materials informatics:The materials"Gene"and big data[J].Annu.Rev.Mater.Res.,2015,45:153
[5]Gleiter H.Nanostructured materials:Basic concepts and microstructure[J].Acta Mater.,2000,48:1
[6]Meyers M A,Mishra A,Benson D J.Mechanical properties of nanocrystalline materials[J].Prog.Mater.Sci.,2006,51:427
[7]Pande C S,Cooper K P.Nanomechanics of Hall-Petch relationship in nanocryatalline materials[J].Prog.Mater.Sci.,2009,54:689
[8]Steinbach I.Phase-field model for microstructure evolution at the mesoscopic scale[J].Annu.Rev.Mater.Res.,2013,43:89
[9]Chen L Q.Phase-field model for microstructure evolution[J].Annu.Rev.Mater.Res.,2002,32:113
[10]Fan D,Chen L Q.Computer simulation of grain growth using a continuum field model[J].Acta Mater.,1997,45:611
[11]Lobkovsky A E,Warren J A.Phase field model of premelting of grain boundaries[J].Physica,2002,164D:202
[12]Williams P L,Mishin Y.Thermodynamics of grain boundary premelting in alloys.II.Atomistic simulation[J].Acta Mater.,2009,57:3786
[13]Wu X H,Xiang J Z.Modern Material Computation and Design[M].Beijing:Electronic Indusial Press,2002:1(吴兴惠,项金钟.现代材料计算与设计教程[M].北京:电子工业出版社,2002:1)
[14]Qi Y,Krajewski P E.Molecular dynamics simulations of grain boundary sliding:The effect of stress and boundary misorientation[J].Acta Mater.,2007,55:1555
[15]Stefanovic P,Haataja M,Provatas N.Phase field crystal study of deformation and plasticity in nanocrystalline materials[J].Phys.Rev.,2009,80E:046107
[16]Stefanovic P,Haataja M,Provatas N.Phase-field crystals with elastic interactions[J].Phys.Rev.Lett.,2006,96:22504
[17]Elder K R,Grant M.Modeling elastic and plastic deformation in nonequilibrium processing using phase field crystals[J].Phys.Rev.,2004,70E:051605
[18]Emmerch H,Gránásy L,Lowen H.Selected issues of phase-field crystal simulations[J].Eur.Phys.Plus.J.,2011,126:102
[19]Elder K R,Katakowsk M,Haataja M,et al.Modeling elasticity in crystal growth[J].Phys.Rev.Lett.,2002,88:245701
[20]Chan P Y,Goldenfeld G,Dantzig J.Molecular dynamics on diffusive time scales from the phase-field-crystal equation[J].Phys.Rev.,2009,79E:035701.
[21]Wang J,Li X K,Liu C,et al.Phase field simulations of microstructure evolution[J].Chin.Solid J.Mech.,2016,37:1(王杰,李欣凯,刘畅等.材料微结构演化的相场模拟[J].固体力学学报,2016,37:1)
[22]Achim C V,Ramos J A,Karttunen M,et al.Nonlinear driven response of a phase-field crystal in a periodic pinning potential[J].Phys.Rev.,2009,79E:011606.
[23]Ramos J A P,Granato E,Ying S C,et al.Dynamical transitions and sliding friction of the phase-field-crystal model with pinning[J].Phys.Rev.,2010,81E:011121
[24]Kim J.Phase-field models for multi-component fluid flows[J].Commun.Comput.Phys.,2012,12:613
[25]Cross M,Greenside H.Pattern Formation and Dynamic in Nonequilibrium Systems[M].Cambridge:Cambridge University Press,2009:1
[26]Provatas N,Elder K.Phase-Field Methods in Materials Science and Engineering[M].Weinheim:Wiley-VCH,2010:1
[27]Wu K A,Adland A,Karma A.Phase-field-crystal model for fcc ordering[J].Phys.Rev.,2010,81E:061601
[28]Mkhonta S K,Elder K R,Huang Z F.Exploring the complex world of two-dimensional ordering with three modes[J].Phys.Rev.Lett.,2013,111:035501
[29]Schwalbach E J,Warren J A,Wu K A.Phase-field crystal model with a vapor phase[J].Phys.Rev.,2013,88E:023306
[30]Kocher G,Provatas N.New density functional approach for solidliquid-vapor transitions in pure materials[J].Phys.Rev.Lett.,2015,114:155501
[31]Elder K R,Provatas N,Berry J,et al.Phase field crystal modeling and classical density functional theory of freezing[J].Phys.Rev.,2007,75B:064107
[32]Toth G I,Pusztai T,Tegzy G,et al.Amorphous nucleation precursor in highly nonequilibrium fluids[J].Phys.Rev.Lett.,2011,107:175702
[33]Wu K-A,Voorhees P W.Stress-induced morphological instabilities at the nanoscale examined using phase field crystal approach[J].Phys.Rev.,2009,80B:125408
[34]Elder K R,Huang Z F,Provatas N.Amplitude expansion of the binary phase field crystal modeling[J].Phys.Rev.,2010,81E:0011602
[35]Huang Z F,Elder K R,Provatas N.Phase-field-crystal dynamics for binary systems:Derivation from dynamical density functional theory,amplitude equation formalism,and applications to alloy heterostructures[J].Phys.Rev.,2010,82E:021605
[36]Toth G I,Tegzy G,Pusztai T,et al.Hetergeneous crystal nucleation:The effect of lattice mismatch[J].Phys.Rev.Lett.,2012,108:025502
[37]Fallah V,Korinek A,Ofori-Opoku N,et al.Atomic-scale pathway of early-stage precipitation in Al-Mg-Si alloy[J].Acta Mater.,2015,82:457
[38]Gao Y J,Quan S L,Deng Q Q,et al.Phase-field-crystal simulation of edge dislocation climbing and gliding under shear strain[J].Acta Phys.Sin.,2015,64:106104(高英俊,全四龙,邓芊芊等.剪切应变下刃型位错的滑移机理的晶体相场模拟[J].物理学报,2015,64:106104)
[39]Gao Y J,Lu C J,Huang L L,et al.Phase field crystal simulation of disloca-tion movement and reaction[J].Acta Metall.Sin.,2014,50:110(高英俊,卢成健,黄礼琳等.晶界位错运动与位错反应过程的晶体相场模拟[J].金属学报,2014,50:110)
[40]Gao Y J,Deng Q Q,Quan S L,et al.Phase field crystal simulation of grain boundary movement and dislocation reaction[J].Front.Mater.Sci.,2014,8:176
[41]Gao Y J,Huang L L,Deng Q Q,et al.Simulation of epitaxial growth on convex substrate using phase field crystal method[J].Front.Mater.Sci.,2014,8:185
[42]Gao Y J,Luo Z R,Huang C G,et al.Phase-field-crystal modeling for two-dimensional transformation from hexagonal to square structure[J].Acta Phys.Sin.,2013,62:050507(高英俊,罗志荣,黄创高等.晶体相场方法研究二维六角相向正方相结构转变[J].物理学报,2013,62:050507)
[43]Tao Y,Zheng C,Jing Z,et al.Phase field crystal study on the temporal evolution and coarsening mechanism of precipitates during spinodal decomposition[J].Rare Met.Mater.Eng.,2013,42:1773
[44]Galenko P K,Gomez H,Kropotin N V,et al.Unconditionally stable method and numerical solution of the hyperbolic phase-field crystal equation[J].Phys.Rev.,2013,88E:013310
[45]Gomez H,Nogueira X.An unconditionally energy-stable method for the phase field crystal equation[J].Comput.Methods Appl.Mech.Eng.,2012,249-252:52
[46]Cheng M W,Warren J A.An efficient algorithm for solving the phase field crystal model[J].J.Comput.Phys.,2008,227:6241
[47]Shin J,Lee H G,Lee J Y.First and second order numerical methods based on a new convex splitting for phase-field crystal equation[J].J.Comput.Phys.,2016,327:519
[48]Baskaran A,Hu Z Z,Lowengrub J S,et al.Energy stable and efficient finite-difference nonlinear multigrid schemes for the modified phase field crystal equation[J].J.Comput.Phys.,2013,250:270
[49]Tegze G,Bansel G,Toth G,et al.Advanced operator splittingbased semi-implicit spedtral method to solve the binary phasefield crystal equations with variable coefficients[J].J.Comput.Phys.,2009,228:1612
[50]Lee H G,Shin J,Lee J Y.First-and second-order energy stable methods for the modified phase field crystal equation[J].Comput.Methods Appl.Mech.Eng.,2017,321:1
[51]Dehghan M,Mohammadi V.The numerical simulation of the phase field crystal(PFC)and modified phase field crystal(MPFC)models via global and local meshless methods[J].Comput.Methods Appl.Mech.Eng.,2016,298:453
[52]Guan Z,Heinonen V,Lowengrub J,et al.An energy stable,hexagonal finite difference scheme for the 2D phase field crystal amplitude equations[J].J.Comput.Phys.,2016,321:1026
[53]Greenwood M,Sinclair C,Militzer M.Phase field crystal model of solute drag[J].Acta Mater.,2012,60:5752
[54]Stolle J,Provatas N.Characterizing solute segregation and grain boundary energy in binary alloy phase field crystal models[J].Comput.Mater.Sci.,2014,81:493
[55]Greenwood M,Ofori-Opoku N,Rottler J,et al.Modeling structural transformations in binary alloys with phase field crystals[J].Phys.Rev.,2011,84B:064104
[56]Fallah V,Ofori-Opoku N,Stolle J,et al.Simulation of early-stage clustering in ternary metal alloys using the phase-field crystal method[J].Acta Mater.,2013,61:3653
[57]Fallah V,Stolle J,Ofori-Opoku N,et al.Phase-field crystal modeling of early stage clustering and precipitation in metal alloys[J].Phys.Rev.,2012,86B:134112
[58]Fallah V,Korinek A,Ofori-Opoku N,et al.Atomistic investigation of clustering phenomenon in the Al-Cu system:Three-dimensional phase-field crystal simulation and HRTEM/HRSTEM characterization[J].Acta Mater.,2013,61:6372
[59]Kocher G,Ofori-Opoku N,Provatas N.Incorporating noise quantitatively in the phase field crystal model via capillary fluctuation theory[J].Phys.Rev.Lett.,2016,117:220601
[60]Gorkaya T,Molodov K D,Molodov D A,et al.Concurrent grain boundary motion and grain rotation under an applied stress[J].Acta Mater.,2011,59:5674
[61]Gutkin M Y,Ovidko I A.Grain boundary migration as rotational deformation mode in nanocrystalline materials[J].Appl.Phys.Lett.,2005,87:251916
[62]Liu P,Mao S C,Wang L H,et al.Direct dynamic atomic mechanisms of strain-induced grain rotation in nanocrystalline[J].Scr.Mater.,2011,64:343
[63]Cahn J W,Taylor J E.A unified approach to motion of grain boundaries,relative tangential translation along grain boundaries and grain rotation[J].Acta Mater.,2004,52:4887
[64]Trautt Z T,Adland A,Karma A,et al.Coupled motion of asymmetrical tilt grain boundaries:Molecular dynamics and phase field crystal simulations[J].Acta Mater.,2012,60:6528
[65]Wu K A,Vorrhees P W.Phase field crystal simulation of nanocrystalline grain growth in two dimensions[J].Acta Mater.,2012,60:407
[66]Yamanka A,Mcreynolds K,Voorheers P W.PFC simulation of grain boundary motion,grain rotation and dislocation in BCC bicrysatl[J].Acta Mater.,2017,133:160
[67]Bjerre M,Tarp J M,Angheiuta L,et al.Rotation-induced grain growth and stagnation in phase-field crystal models[J].Phys.Rev.,2013,88E:020401
[68]Tallon J L.Premelting near crystal defects[J].Nature,1978,276:849
[69]Alsayed A M,Islam M F,Zhang J,et al.Premelting at defects within bulk colloidal crystals[J].Science,2005,309:1207
[70]Inoko F,Hama T,Tagami M et al.Grain boundary premelting in thin foils of deformed copper bicrystals[J].Ultramicroscopy,1991,39:118
[71]Inoko F,Okada T,Maraga T,et al.Strain induced grain boundary premelting in bulk copper bicrystals[J].Interface Sci.,1997,4:263
[72]Olmsted D L,Buta D,Adland A,et al.Dislocation-pairing transitions in hot grain boundaries[J].Phys.Rev.Lett.,2011,106:046101
[73]Divinski S,Lohmann M,Herzig C,et al.Grain boundary melting phase transition in the Cu-Bi system[J].Phys.Rev.,2005,71B:104104
[74]Shi X M,Luo J.Grain boundaries wetting and premelting in Nidoped Mo[J].Appl.Phys.Lett.,2009,94:251908
[75]Gao Y J,Huang L L,Zhou W Q,et al.Phase field crystal subgrain boundary annihilation and dislocation rotation mechanism under strain at high temperature[J].Sci.Sin.Technol.,2015,45:306(高英俊,黄礼琳,周文权等.高温应变下的亚晶界湮没与位错旋转机制制的晶体相场模拟[J].中国科学:技术科学,2015,45:306)
[76]Gao Y J,Zhou W Q,Deng Q Q,et al.Phase field crystal model of strain effect on dislocation movement of premelting grain boundary at high temoearture[J].Acta Metall.Sin.,2014,50:886(高英俊,周文权,邓芊芊等.晶体相场方法模拟高温应变作用的预熔化晶界的位错运动[J].金属学报,2014,50:886)
[77]Gao Y J,Huang L L,Deng Q Q,et al.Phase field crystal simulation of dislocation configuration evolution in dynamic recovery in two dimensions[J].Acta Mater.,2016,117:238
[78]Meca E,Shenoy V B,Lowengrub J.Phase-field modeling of twodimensional crystal growth with anisotropic diffusion[J].Phys.Rev.,2013,88E:052409
[79]Chan V W L,Pisutha-Arnond N,Thornton K.Thermodynamic relationships for homogeneous crystalline and liquid phases in the phase-field crystal model[J].Comput.Mater.Sci.,2017,135:205
[80]Asadi E,Zaeem M A.A review of quantitative phase-field crystal modeling of solid-liquid structures[J].JOM,2015,67:186
[81]Berghoff M,Nestler B.Phase field crystal modeling of ternary solidification microstructures[J].Comput.Condens.Matter,2015,4:46
[82]Tang S,Wang Z J,Guo Y L,et al.Orientation selection process during the early stage of cubic dendrite growth:A phase-field-crystal study[J].Acta Mater.,2012,60:5501
[83]Tang S,Yu Y M,Wang Y M,et al.Phase-field-crystal simulation of nonequalibrium crystal growth[J].Phys.Rev.,2014,89E:012405
[84]Podmaniczky F,Tóth G I,Tegze G,et al.Phase-field crystal modeling of heteroepitaxy and exotic modes of crystal nucleation[J].J.Cryst.Growth,2017,457:24
[85]Lu Y L,Peng Y Y,Chen Z.A binary phase field crystal study for liquid phase heteroepitaxial growth[J].Superlattices Microstruct.,2016,97:132
[86]Peng Y Y,Lu Y L,Chen Z,et al.A binary phase field crystal study for phase segregation of liquid phase heteroepitaxial growth[J].Comput.Mater.Sci.,2016,123:65
[87]Elder K R,Huang Z F.A phase field crystal study of epitaxial island formation on nanomembranes[J].J.Phys.:Condens.Matter,2010,22:364103
[88]Granato E,Ramos J A P,Achim C V,et al.Glassy phases and driven response of the phase-field-crystal model with random pinning[J].Phys.Rev.,2011,84E:031102
[89]Robbins M J,Archer A J,Thiele U,et al.Modeling the structure of liquids and crystals using one-and two-component modified phasefield crystal models[J].Phys.Rev.,2012,85E:061408
[90]Berry J,Grant M.Phase-field-crystal modeling of glass-forming liquids:Spanning time scales during vitrification,aging,and deformation[J].Phys.Rev.,2014,89E:062303
[91]Zhang W,Mi J.Phase field crystal modelling of the order-to-disordered atomistic structure transition of metallic glasses[J].IOP Conf.Ser.:Mater.Sci.Eng.,2016,117:012056
[92]Pineau A,Benzerga A A,Pardoen T.Failure of metal:Brittle and ductile fracture[J].Acta Mater.,2016,107:424
[93]Kim J H,Park W S,Chun M S,et al.Effect of pre-straining on low temperature mechanical behavior of AISI 304L[J].Mater.Sci.Eng.,2012,A543:50
[94]Hamada A S,J?rvebp??A,Honkaueu M,et al.Effects of cyclic pre-straining on mechanical properties of an austenitic microalloyed high-Mn twinning-induced plasticity steel[J].Procedia Eng.,2014,74:47
[95]Yin D Y,Xiao Q,Chen Y Q,et al.Effect of natural ageing and prestraining on the hardening behavior and microstructural response during artificial ageing of an Al-Mg-Si-Cu alloy[J].Mater.Des.,2016,95:329
[96]Gao Y J,Deng Q Q,Huang L L,et al.Atomistic modeling for mechanism of crack cleavage extension on nano-scale[J].Comput.Mater.Sci.,2017,130:64
[97]Gao Y J,Luo Z R,Huang L L,et al.Phase field crystal study of nano-crack growth and branch in materials[J].Modell.Simul.Mater.Sci.Eng.,2016,24:055010
[98]Hu S,Chen Z,Xi W,et al.Phase-field-crystal study on the evolution behavior of microcracks initiated on grain boundaries under constant strain[J].J.Mater.Sci.,2017,52:5641
[99]Hu S,Xi W,Chen Z,et al.Coupled motion of grain boundaries and the influence of microcracks:A phase-field-crystal study[J].Comput.Mater.Sci.,2017,132:125
[100]Tu K N.Recent advances on electromigration in very-large-scaleintergration of interconnects[J].Appl.J.Phys.,2003,94:5451
[101]De Orio R L,Ceric H,Selberherr S.Physically based model of electromigration:From Black's equation to modern TCAD models[J].Microelectron.Reliab.,2010,50:775
[102]Ogurtania T O,Akgjldiz O.Morphological evolution of voids of surface drift diffusion driven by capillary,electromagration,and thermal-stress-gradients induced by steady-state heat flow in passivated metallic thin films and flip chip solder joints.I.Theory[J].Appl.J.Phys.,2008,104:023521
[103]Mahadeven M,Bradleg R M.Simulations and theory of electromigration-induced slit formation in unpassivated single-crystal metal line[J].Phys.Rev.,1999,59B:11037
[104]Wang N,Bevan K H,Provatas N.Phase-field-crystal model for electromigration in metal interconnects[J].Phys.Rev.Lett.,2016,117:155901
[105]Gusak A M,Tu K N.Interaction between the Kirkendall effect and the inverse Kirkendall effect in nanoscale particles[J].Acta Mater.,2009,57:3367
[106]Elder K R,Thornton K,Hoyt J J.The kirkendall effect in the phase field crystal model[J].Philos.Mag.,2011,91:151
[107]Lu G M,Lu Y L,Hu T T,et al.Phase field crystal study on the grain boundary porosity induced by the Kirkendall effect[J].Modell.Simul.Mater.Sci.Eng.,2016,24:035001
[108]Ma W J,Ke C B,Liang S B,et al.Morphological evolution and growth kinetics of Kirkendall voids in binary alloy system during deformation process—Phase field crystal simulation study[J].Trans.Nonferrous Met.Soc.China,2017,27:599
[109]Ma W J,Ke C B,Zhou M B,et al.Phase-field crystal simulation on evolution and growth kinetics of Kirkendall voids in interface and intermetallic compound layer in Sn/Cu soldering system[J].Acta Metall.Sin.,2015,57:873.(马文婧,柯常波,周敏波等.Sn/Cu互连体系界面和金属间化合物层Kirkendall空洞演化和生长动力学的晶体相场法模拟[J].金属学报,2015,57:873)
[110]Lu Y L,Lu G M,Hu T T,et al.Phase field crystal study on the formation and evolution of phase boundary void induced by the Kirkendall effect[J].Acta Metall.Sin.,2015,57:866(卢艳丽,卢广明,胡婷婷等.晶体相场法研究Kirkendall效应诱发的相界空洞形成和演变[J].金属学报,2015,57:886)
[111]Hubert A,Sch?fer R.Magnetic Domains the Analysis of Magnetic Microstructures[M].Berlin:Springer-Verlag,1998:1
[112]Lawas G,Srinivasan G.Introduction to magnetelectric coupling and multiferrous films[J].J.Phys.,2011,44D:243001
[113]Van Der Boomgaard J,Terrell D R,Born R A,et al.An in situ grown eutectic magnetoelectric composite material[J].J.Mater.Sci.,1974,9:1705
[114]Van Run A M J G,Terrell D R,Scholing J H.An in situ grown eutectic magnetoelectric composite material[J].J.Mater.Sci.,1974,9:1710
[115]Seymour M,Sanches F,Elder K,et al.Phase-field crystal approach for modeling the role of microstructure in multiferroic composite materials[J].Phys.Rev.,2015,92B:184109
[116]Faghihi N,Provatas N,Elder K R,et al.Phase-field-crystal model for magnetocrystalline interactions in isotropic ferromagnetic solids[J].Phys.Rev.,2013,88E:032407
[117]Geim A K,K Novoselov S.The rise of grapheme[J].Nat.Mater.,2007,6:183
[118]Lee C,Wei X D,Kysar J W,et al.Measurement of the elastic properties and intrinsic strength of monolayer grapheme[J].Science,2008,321:385
[119]Lee G H,Cooper R C,An S J,et al.High-strength chemicalvapor-deposited graphene and grain boundaries[J].Science,2013,340:1073
[120]Wei Y J,Wu J T,Yin H Q,et al.The nature of strength enhancement and weakening by pentagon-heptagon defects in grapheme[J].Nat.Mater.,2012,11:759
[121]Seymour M,Provatas N.Structural phase field crystal approach for modeling graphene and other two-dimensional structures[J].Phys.Rev.,2016,93B:035447
[122]Hirvonen P,Ervasti M M,Fan Z Y,et al.Multiscale modeling of polycrystalline graphene:A comparison of structure and defect energies of realistic samples from phase field crystal models[J].Phys.Rev.,2016,94B:035414
[123]Hirvonen P,Fen Z Y,Ervasti M M,et al.Energetics and structure of grain boundary triple junctions in grapheme[J].Sci.Rep.,2016,7:4754
[124]L?wen H.A phase-field-crystal model for liquid crystals[J].J.Condens.Matter,2010,22:364105
[125]Guttenberg N,Goldenfeld N,Dantzig J.Emergence of foams from the breakdown of the phase field crystal model[J].Phys.Rev.,2010,81E:065301
[126]Gao Y J,Yang R L,Wang Y L,et al.Phase field model simulation of bumps and holes pattern of two dimension crystals[J].Guangxi Sci.,2015,22:485.(高英俊,杨瑞琳,王玉玲等.空位晶体相场模型模拟二维晶体相形貌图[J].广西科学,2015,22:485)
[127]Asadi E,Zaeem M A,Baskes M I.Phase-field crystal method for Fe connected to MEAM molecular dynamics simulations[J].JOM,2014,66:429
[2]Provatas N,Dantzig J A,Athrega B,et al.Using the phase-field crystal method in the multi-scale modeling of microstructure evolution[J].JOM,2007,59(7):83
[3]Chernatynskiy A,Phillpot S R,Lesar R.Uncertainty quantification in multi-scale simulation of materials:A prospective[J].Annu.Rev.Mater.Res.,2013,43:157
[4]Rajan K.Materials informatics:The materials"Gene"and big data[J].Annu.Rev.Mater.Res.,2015,45:153
[5]Gleiter H.Nanostructured materials:Basic concepts and microstructure[J].Acta Mater.,2000,48:1
[6]Meyers M A,Mishra A,Benson D J.Mechanical properties of nanocrystalline materials[J].Prog.Mater.Sci.,2006,51:427
[7]Pande C S,Cooper K P.Nanomechanics of Hall-Petch relationship in nanocryatalline materials[J].Prog.Mater.Sci.,2009,54:689
[8]Steinbach I.Phase-field model for microstructure evolution at the mesoscopic scale[J].Annu.Rev.Mater.Res.,2013,43:89
[9]Chen L Q.Phase-field model for microstructure evolution[J].Annu.Rev.Mater.Res.,2002,32:113
[10]Fan D,Chen L Q.Computer simulation of grain growth using a continuum field model[J].Acta Mater.,1997,45:611
[11]Lobkovsky A E,Warren J A.Phase field model of premelting of grain boundaries[J].Physica,2002,164D:202
[12]Williams P L,Mishin Y.Thermodynamics of grain boundary premelting in alloys.II.Atomistic simulation[J].Acta Mater.,2009,57:3786
[13]Wu X H,Xiang J Z.Modern Material Computation and Design[M].Beijing:Electronic Indusial Press,2002:1(吴兴惠,项金钟.现代材料计算与设计教程[M].北京:电子工业出版社,2002:1)
[14]Qi Y,Krajewski P E.Molecular dynamics simulations of grain boundary sliding:The effect of stress and boundary misorientation[J].Acta Mater.,2007,55:1555
[15]Stefanovic P,Haataja M,Provatas N.Phase field crystal study of deformation and plasticity in nanocrystalline materials[J].Phys.Rev.,2009,80E:046107
[16]Stefanovic P,Haataja M,Provatas N.Phase-field crystals with elastic interactions[J].Phys.Rev.Lett.,2006,96:22504
[17]Elder K R,Grant M.Modeling elastic and plastic deformation in nonequilibrium processing using phase field crystals[J].Phys.Rev.,2004,70E:051605
[18]Emmerch H,Gránásy L,Lowen H.Selected issues of phase-field crystal simulations[J].Eur.Phys.Plus.J.,2011,126:102
[19]Elder K R,Katakowsk M,Haataja M,et al.Modeling elasticity in crystal growth[J].Phys.Rev.Lett.,2002,88:245701
[20]Chan P Y,Goldenfeld G,Dantzig J.Molecular dynamics on diffusive time scales from the phase-field-crystal equation[J].Phys.Rev.,2009,79E:035701.
[21]Wang J,Li X K,Liu C,et al.Phase field simulations of microstructure evolution[J].Chin.Solid J.Mech.,2016,37:1(王杰,李欣凯,刘畅等.材料微结构演化的相场模拟[J].固体力学学报,2016,37:1)
[22]Achim C V,Ramos J A,Karttunen M,et al.Nonlinear driven response of a phase-field crystal in a periodic pinning potential[J].Phys.Rev.,2009,79E:011606.
[23]Ramos J A P,Granato E,Ying S C,et al.Dynamical transitions and sliding friction of the phase-field-crystal model with pinning[J].Phys.Rev.,2010,81E:011121
[24]Kim J.Phase-field models for multi-component fluid flows[J].Commun.Comput.Phys.,2012,12:613
[25]Cross M,Greenside H.Pattern Formation and Dynamic in Nonequilibrium Systems[M].Cambridge:Cambridge University Press,2009:1
[26]Provatas N,Elder K.Phase-Field Methods in Materials Science and Engineering[M].Weinheim:Wiley-VCH,2010:1
[27]Wu K A,Adland A,Karma A.Phase-field-crystal model for fcc ordering[J].Phys.Rev.,2010,81E:061601
[28]Mkhonta S K,Elder K R,Huang Z F.Exploring the complex world of two-dimensional ordering with three modes[J].Phys.Rev.Lett.,2013,111:035501
[29]Schwalbach E J,Warren J A,Wu K A.Phase-field crystal model with a vapor phase[J].Phys.Rev.,2013,88E:023306
[30]Kocher G,Provatas N.New density functional approach for solidliquid-vapor transitions in pure materials[J].Phys.Rev.Lett.,2015,114:155501
[31]Elder K R,Provatas N,Berry J,et al.Phase field crystal modeling and classical density functional theory of freezing[J].Phys.Rev.,2007,75B:064107
[32]Toth G I,Pusztai T,Tegzy G,et al.Amorphous nucleation precursor in highly nonequilibrium fluids[J].Phys.Rev.Lett.,2011,107:175702
[33]Wu K-A,Voorhees P W.Stress-induced morphological instabilities at the nanoscale examined using phase field crystal approach[J].Phys.Rev.,2009,80B:125408
[34]Elder K R,Huang Z F,Provatas N.Amplitude expansion of the binary phase field crystal modeling[J].Phys.Rev.,2010,81E:0011602
[35]Huang Z F,Elder K R,Provatas N.Phase-field-crystal dynamics for binary systems:Derivation from dynamical density functional theory,amplitude equation formalism,and applications to alloy heterostructures[J].Phys.Rev.,2010,82E:021605
[36]Toth G I,Tegzy G,Pusztai T,et al.Hetergeneous crystal nucleation:The effect of lattice mismatch[J].Phys.Rev.Lett.,2012,108:025502
[37]Fallah V,Korinek A,Ofori-Opoku N,et al.Atomic-scale pathway of early-stage precipitation in Al-Mg-Si alloy[J].Acta Mater.,2015,82:457
[38]Gao Y J,Quan S L,Deng Q Q,et al.Phase-field-crystal simulation of edge dislocation climbing and gliding under shear strain[J].Acta Phys.Sin.,2015,64:106104(高英俊,全四龙,邓芊芊等.剪切应变下刃型位错的滑移机理的晶体相场模拟[J].物理学报,2015,64:106104)
[39]Gao Y J,Lu C J,Huang L L,et al.Phase field crystal simulation of disloca-tion movement and reaction[J].Acta Metall.Sin.,2014,50:110(高英俊,卢成健,黄礼琳等.晶界位错运动与位错反应过程的晶体相场模拟[J].金属学报,2014,50:110)
[40]Gao Y J,Deng Q Q,Quan S L,et al.Phase field crystal simulation of grain boundary movement and dislocation reaction[J].Front.Mater.Sci.,2014,8:176
[41]Gao Y J,Huang L L,Deng Q Q,et al.Simulation of epitaxial growth on convex substrate using phase field crystal method[J].Front.Mater.Sci.,2014,8:185
[42]Gao Y J,Luo Z R,Huang C G,et al.Phase-field-crystal modeling for two-dimensional transformation from hexagonal to square structure[J].Acta Phys.Sin.,2013,62:050507(高英俊,罗志荣,黄创高等.晶体相场方法研究二维六角相向正方相结构转变[J].物理学报,2013,62:050507)
[43]Tao Y,Zheng C,Jing Z,et al.Phase field crystal study on the temporal evolution and coarsening mechanism of precipitates during spinodal decomposition[J].Rare Met.Mater.Eng.,2013,42:1773
[44]Galenko P K,Gomez H,Kropotin N V,et al.Unconditionally stable method and numerical solution of the hyperbolic phase-field crystal equation[J].Phys.Rev.,2013,88E:013310
[45]Gomez H,Nogueira X.An unconditionally energy-stable method for the phase field crystal equation[J].Comput.Methods Appl.Mech.Eng.,2012,249-252:52
[46]Cheng M W,Warren J A.An efficient algorithm for solving the phase field crystal model[J].J.Comput.Phys.,2008,227:6241
[47]Shin J,Lee H G,Lee J Y.First and second order numerical methods based on a new convex splitting for phase-field crystal equation[J].J.Comput.Phys.,2016,327:519
[48]Baskaran A,Hu Z Z,Lowengrub J S,et al.Energy stable and efficient finite-difference nonlinear multigrid schemes for the modified phase field crystal equation[J].J.Comput.Phys.,2013,250:270
[49]Tegze G,Bansel G,Toth G,et al.Advanced operator splittingbased semi-implicit spedtral method to solve the binary phasefield crystal equations with variable coefficients[J].J.Comput.Phys.,2009,228:1612
[50]Lee H G,Shin J,Lee J Y.First-and second-order energy stable methods for the modified phase field crystal equation[J].Comput.Methods Appl.Mech.Eng.,2017,321:1
[51]Dehghan M,Mohammadi V.The numerical simulation of the phase field crystal(PFC)and modified phase field crystal(MPFC)models via global and local meshless methods[J].Comput.Methods Appl.Mech.Eng.,2016,298:453
[52]Guan Z,Heinonen V,Lowengrub J,et al.An energy stable,hexagonal finite difference scheme for the 2D phase field crystal amplitude equations[J].J.Comput.Phys.,2016,321:1026
[53]Greenwood M,Sinclair C,Militzer M.Phase field crystal model of solute drag[J].Acta Mater.,2012,60:5752
[54]Stolle J,Provatas N.Characterizing solute segregation and grain boundary energy in binary alloy phase field crystal models[J].Comput.Mater.Sci.,2014,81:493
[55]Greenwood M,Ofori-Opoku N,Rottler J,et al.Modeling structural transformations in binary alloys with phase field crystals[J].Phys.Rev.,2011,84B:064104
[56]Fallah V,Ofori-Opoku N,Stolle J,et al.Simulation of early-stage clustering in ternary metal alloys using the phase-field crystal method[J].Acta Mater.,2013,61:3653
[57]Fallah V,Stolle J,Ofori-Opoku N,et al.Phase-field crystal modeling of early stage clustering and precipitation in metal alloys[J].Phys.Rev.,2012,86B:134112
[58]Fallah V,Korinek A,Ofori-Opoku N,et al.Atomistic investigation of clustering phenomenon in the Al-Cu system:Three-dimensional phase-field crystal simulation and HRTEM/HRSTEM characterization[J].Acta Mater.,2013,61:6372
[59]Kocher G,Ofori-Opoku N,Provatas N.Incorporating noise quantitatively in the phase field crystal model via capillary fluctuation theory[J].Phys.Rev.Lett.,2016,117:220601
[60]Gorkaya T,Molodov K D,Molodov D A,et al.Concurrent grain boundary motion and grain rotation under an applied stress[J].Acta Mater.,2011,59:5674
[61]Gutkin M Y,Ovidko I A.Grain boundary migration as rotational deformation mode in nanocrystalline materials[J].Appl.Phys.Lett.,2005,87:251916
[62]Liu P,Mao S C,Wang L H,et al.Direct dynamic atomic mechanisms of strain-induced grain rotation in nanocrystalline[J].Scr.Mater.,2011,64:343
[63]Cahn J W,Taylor J E.A unified approach to motion of grain boundaries,relative tangential translation along grain boundaries and grain rotation[J].Acta Mater.,2004,52:4887
[64]Trautt Z T,Adland A,Karma A,et al.Coupled motion of asymmetrical tilt grain boundaries:Molecular dynamics and phase field crystal simulations[J].Acta Mater.,2012,60:6528
[65]Wu K A,Vorrhees P W.Phase field crystal simulation of nanocrystalline grain growth in two dimensions[J].Acta Mater.,2012,60:407
[66]Yamanka A,Mcreynolds K,Voorheers P W.PFC simulation of grain boundary motion,grain rotation and dislocation in BCC bicrysatl[J].Acta Mater.,2017,133:160
[67]Bjerre M,Tarp J M,Angheiuta L,et al.Rotation-induced grain growth and stagnation in phase-field crystal models[J].Phys.Rev.,2013,88E:020401
[68]Tallon J L.Premelting near crystal defects[J].Nature,1978,276:849
[69]Alsayed A M,Islam M F,Zhang J,et al.Premelting at defects within bulk colloidal crystals[J].Science,2005,309:1207
[70]Inoko F,Hama T,Tagami M et al.Grain boundary premelting in thin foils of deformed copper bicrystals[J].Ultramicroscopy,1991,39:118
[71]Inoko F,Okada T,Maraga T,et al.Strain induced grain boundary premelting in bulk copper bicrystals[J].Interface Sci.,1997,4:263
[72]Olmsted D L,Buta D,Adland A,et al.Dislocation-pairing transitions in hot grain boundaries[J].Phys.Rev.Lett.,2011,106:046101
[73]Divinski S,Lohmann M,Herzig C,et al.Grain boundary melting phase transition in the Cu-Bi system[J].Phys.Rev.,2005,71B:104104
[74]Shi X M,Luo J.Grain boundaries wetting and premelting in Nidoped Mo[J].Appl.Phys.Lett.,2009,94:251908
[75]Gao Y J,Huang L L,Zhou W Q,et al.Phase field crystal subgrain boundary annihilation and dislocation rotation mechanism under strain at high temperature[J].Sci.Sin.Technol.,2015,45:306(高英俊,黄礼琳,周文权等.高温应变下的亚晶界湮没与位错旋转机制制的晶体相场模拟[J].中国科学:技术科学,2015,45:306)
[76]Gao Y J,Zhou W Q,Deng Q Q,et al.Phase field crystal model of strain effect on dislocation movement of premelting grain boundary at high temoearture[J].Acta Metall.Sin.,2014,50:886(高英俊,周文权,邓芊芊等.晶体相场方法模拟高温应变作用的预熔化晶界的位错运动[J].金属学报,2014,50:886)
[77]Gao Y J,Huang L L,Deng Q Q,et al.Phase field crystal simulation of dislocation configuration evolution in dynamic recovery in two dimensions[J].Acta Mater.,2016,117:238
[78]Meca E,Shenoy V B,Lowengrub J.Phase-field modeling of twodimensional crystal growth with anisotropic diffusion[J].Phys.Rev.,2013,88E:052409
[79]Chan V W L,Pisutha-Arnond N,Thornton K.Thermodynamic relationships for homogeneous crystalline and liquid phases in the phase-field crystal model[J].Comput.Mater.Sci.,2017,135:205
[80]Asadi E,Zaeem M A.A review of quantitative phase-field crystal modeling of solid-liquid structures[J].JOM,2015,67:186
[81]Berghoff M,Nestler B.Phase field crystal modeling of ternary solidification microstructures[J].Comput.Condens.Matter,2015,4:46
[82]Tang S,Wang Z J,Guo Y L,et al.Orientation selection process during the early stage of cubic dendrite growth:A phase-field-crystal study[J].Acta Mater.,2012,60:5501
[83]Tang S,Yu Y M,Wang Y M,et al.Phase-field-crystal simulation of nonequalibrium crystal growth[J].Phys.Rev.,2014,89E:012405
[84]Podmaniczky F,Tóth G I,Tegze G,et al.Phase-field crystal modeling of heteroepitaxy and exotic modes of crystal nucleation[J].J.Cryst.Growth,2017,457:24
[85]Lu Y L,Peng Y Y,Chen Z.A binary phase field crystal study for liquid phase heteroepitaxial growth[J].Superlattices Microstruct.,2016,97:132
[86]Peng Y Y,Lu Y L,Chen Z,et al.A binary phase field crystal study for phase segregation of liquid phase heteroepitaxial growth[J].Comput.Mater.Sci.,2016,123:65
[87]Elder K R,Huang Z F.A phase field crystal study of epitaxial island formation on nanomembranes[J].J.Phys.:Condens.Matter,2010,22:364103
[88]Granato E,Ramos J A P,Achim C V,et al.Glassy phases and driven response of the phase-field-crystal model with random pinning[J].Phys.Rev.,2011,84E:031102
[89]Robbins M J,Archer A J,Thiele U,et al.Modeling the structure of liquids and crystals using one-and two-component modified phasefield crystal models[J].Phys.Rev.,2012,85E:061408
[90]Berry J,Grant M.Phase-field-crystal modeling of glass-forming liquids:Spanning time scales during vitrification,aging,and deformation[J].Phys.Rev.,2014,89E:062303
[91]Zhang W,Mi J.Phase field crystal modelling of the order-to-disordered atomistic structure transition of metallic glasses[J].IOP Conf.Ser.:Mater.Sci.Eng.,2016,117:012056
[92]Pineau A,Benzerga A A,Pardoen T.Failure of metal:Brittle and ductile fracture[J].Acta Mater.,2016,107:424
[93]Kim J H,Park W S,Chun M S,et al.Effect of pre-straining on low temperature mechanical behavior of AISI 304L[J].Mater.Sci.Eng.,2012,A543:50
[94]Hamada A S,J?rvebp??A,Honkaueu M,et al.Effects of cyclic pre-straining on mechanical properties of an austenitic microalloyed high-Mn twinning-induced plasticity steel[J].Procedia Eng.,2014,74:47
[95]Yin D Y,Xiao Q,Chen Y Q,et al.Effect of natural ageing and prestraining on the hardening behavior and microstructural response during artificial ageing of an Al-Mg-Si-Cu alloy[J].Mater.Des.,2016,95:329
[96]Gao Y J,Deng Q Q,Huang L L,et al.Atomistic modeling for mechanism of crack cleavage extension on nano-scale[J].Comput.Mater.Sci.,2017,130:64
[97]Gao Y J,Luo Z R,Huang L L,et al.Phase field crystal study of nano-crack growth and branch in materials[J].Modell.Simul.Mater.Sci.Eng.,2016,24:055010
[98]Hu S,Chen Z,Xi W,et al.Phase-field-crystal study on the evolution behavior of microcracks initiated on grain boundaries under constant strain[J].J.Mater.Sci.,2017,52:5641
[99]Hu S,Xi W,Chen Z,et al.Coupled motion of grain boundaries and the influence of microcracks:A phase-field-crystal study[J].Comput.Mater.Sci.,2017,132:125
[100]Tu K N.Recent advances on electromigration in very-large-scaleintergration of interconnects[J].Appl.J.Phys.,2003,94:5451
[101]De Orio R L,Ceric H,Selberherr S.Physically based model of electromigration:From Black's equation to modern TCAD models[J].Microelectron.Reliab.,2010,50:775
[102]Ogurtania T O,Akgjldiz O.Morphological evolution of voids of surface drift diffusion driven by capillary,electromagration,and thermal-stress-gradients induced by steady-state heat flow in passivated metallic thin films and flip chip solder joints.I.Theory[J].Appl.J.Phys.,2008,104:023521
[103]Mahadeven M,Bradleg R M.Simulations and theory of electromigration-induced slit formation in unpassivated single-crystal metal line[J].Phys.Rev.,1999,59B:11037
[104]Wang N,Bevan K H,Provatas N.Phase-field-crystal model for electromigration in metal interconnects[J].Phys.Rev.Lett.,2016,117:155901
[105]Gusak A M,Tu K N.Interaction between the Kirkendall effect and the inverse Kirkendall effect in nanoscale particles[J].Acta Mater.,2009,57:3367
[106]Elder K R,Thornton K,Hoyt J J.The kirkendall effect in the phase field crystal model[J].Philos.Mag.,2011,91:151
[107]Lu G M,Lu Y L,Hu T T,et al.Phase field crystal study on the grain boundary porosity induced by the Kirkendall effect[J].Modell.Simul.Mater.Sci.Eng.,2016,24:035001
[108]Ma W J,Ke C B,Liang S B,et al.Morphological evolution and growth kinetics of Kirkendall voids in binary alloy system during deformation process—Phase field crystal simulation study[J].Trans.Nonferrous Met.Soc.China,2017,27:599
[109]Ma W J,Ke C B,Zhou M B,et al.Phase-field crystal simulation on evolution and growth kinetics of Kirkendall voids in interface and intermetallic compound layer in Sn/Cu soldering system[J].Acta Metall.Sin.,2015,57:873.(马文婧,柯常波,周敏波等.Sn/Cu互连体系界面和金属间化合物层Kirkendall空洞演化和生长动力学的晶体相场法模拟[J].金属学报,2015,57:873)
[110]Lu Y L,Lu G M,Hu T T,et al.Phase field crystal study on the formation and evolution of phase boundary void induced by the Kirkendall effect[J].Acta Metall.Sin.,2015,57:866(卢艳丽,卢广明,胡婷婷等.晶体相场法研究Kirkendall效应诱发的相界空洞形成和演变[J].金属学报,2015,57:886)
[111]Hubert A,Sch?fer R.Magnetic Domains the Analysis of Magnetic Microstructures[M].Berlin:Springer-Verlag,1998:1
[112]Lawas G,Srinivasan G.Introduction to magnetelectric coupling and multiferrous films[J].J.Phys.,2011,44D:243001
[113]Van Der Boomgaard J,Terrell D R,Born R A,et al.An in situ grown eutectic magnetoelectric composite material[J].J.Mater.Sci.,1974,9:1705
[114]Van Run A M J G,Terrell D R,Scholing J H.An in situ grown eutectic magnetoelectric composite material[J].J.Mater.Sci.,1974,9:1710
[115]Seymour M,Sanches F,Elder K,et al.Phase-field crystal approach for modeling the role of microstructure in multiferroic composite materials[J].Phys.Rev.,2015,92B:184109
[116]Faghihi N,Provatas N,Elder K R,et al.Phase-field-crystal model for magnetocrystalline interactions in isotropic ferromagnetic solids[J].Phys.Rev.,2013,88E:032407
[117]Geim A K,K Novoselov S.The rise of grapheme[J].Nat.Mater.,2007,6:183
[118]Lee C,Wei X D,Kysar J W,et al.Measurement of the elastic properties and intrinsic strength of monolayer grapheme[J].Science,2008,321:385
[119]Lee G H,Cooper R C,An S J,et al.High-strength chemicalvapor-deposited graphene and grain boundaries[J].Science,2013,340:1073
[120]Wei Y J,Wu J T,Yin H Q,et al.The nature of strength enhancement and weakening by pentagon-heptagon defects in grapheme[J].Nat.Mater.,2012,11:759
[121]Seymour M,Provatas N.Structural phase field crystal approach for modeling graphene and other two-dimensional structures[J].Phys.Rev.,2016,93B:035447
[122]Hirvonen P,Ervasti M M,Fan Z Y,et al.Multiscale modeling of polycrystalline graphene:A comparison of structure and defect energies of realistic samples from phase field crystal models[J].Phys.Rev.,2016,94B:035414
[123]Hirvonen P,Fen Z Y,Ervasti M M,et al.Energetics and structure of grain boundary triple junctions in grapheme[J].Sci.Rep.,2016,7:4754
[124]L?wen H.A phase-field-crystal model for liquid crystals[J].J.Condens.Matter,2010,22:364105
[125]Guttenberg N,Goldenfeld N,Dantzig J.Emergence of foams from the breakdown of the phase field crystal model[J].Phys.Rev.,2010,81E:065301
[126]Gao Y J,Yang R L,Wang Y L,et al.Phase field model simulation of bumps and holes pattern of two dimension crystals[J].Guangxi Sci.,2015,22:485.(高英俊,杨瑞琳,王玉玲等.空位晶体相场模型模拟二维晶体相形貌图[J].广西科学,2015,22:485)
[127]Asadi E,Zaeem M A,Baskes M I.Phase-field crystal method for Fe connected to MEAM molecular dynamics simulations[J].JOM,2014,66:429