二维Frenkel-Kontorova模型在纳米摩擦学中的应用
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摘要
二十世纪90年代兴起的纳米科学技术为人们能制造出用于在原子尺度上研究材料特性的实验设备提供技术指导。同时,计算方法的巨大变革衍生出了便于理论工作者在该尺度上探索真实滑动体系的新技术——分子动力学技术。这种实验和计算上的进步进一步激发了人们用简单的数学模型来探索复杂体系的浓厚兴趣。在众多模型中,Frenkel-Kontorova(FK)模型便是成功地描述和解释有关复杂摩擦体系的动力学特性的模型之一。
本论文研究了作为上层的正方形晶格和作为下层的正方形基低间存在任意错合角θ,外界驱动力沿任意角度α驱动时,二维FK模型的动力学行为及摩擦特性。本文第一章概述了纳米摩擦学的发展历史及应用前景。第二章我们主要介绍纳米摩擦学的理论研究技术——分子动力学技术。实践证明该技术能有效地用来研究纳米结构及探索微观现象的新规律。因而受到人们越来越广泛的关注,并成功地应用于包括纳米摩擦学在内的许多微观研究领域中。第三章基于分子动力学(molecular dynamics)模拟技术,主要介绍了二维FK模型在纳米摩擦学领域中的应用。我们的数值模拟结果表明:系统的静摩擦力与系统的重要参数有关,如改变系统的硬度(stiffness)、系统的公度和不公度、系统的错合角θ,外界驱动力的方向α以及原子间的强度系数K等都可改变系统的静摩擦力。本文第四章展望了我们在纳米摩擦学领域中的后续工作。
Over the last decade, remarkable developments in nanotechnology have provided the possibility to build experimental devices which are able to perform analysis on materials at the atomic-scale. Simultaneously, tremendous advances in computing methodology (molecular dynamics techniques, MDT) have supplied very detailed information on the atomic scale for realistic sliding system. The accelerations in experiments and computations have deeply stimulated the search and implementation of simpler mathematical model such as the Frenkel-Kontorova (FK) model. The model of friction is capable of describing and interpreting the main features of the complicated dynamics involved.
In this dissertation, we study the dynamical behavior of a two-dimensional FK model with a square symmetry potential for a square lattice layer driven by an external driving force with an arbitrary directionαand an arbitrary misfit angle 9 between upper and lower layer. There are four chapters in the dissertation. The paper is organized as following: in chapter I, the development and prosperity of the nano-tribology is briefly introduced. In the theoretical simulations of the nano-tribology, one of the most important techniques called the MDT will be reviewed in chapter II. It is verified that the technique is effective and coherent in the study of the nanostructure. So, many physicists pay more and more attention on it. In chapter III, by using the MDT, the application of the two dimensional FK model in the nano-tribolody will be presented. With regards to the static friction force, the results show that the parameters of the system, such as the effective elastic constant, the commensurate and incommensurate structures of the system, the misfit angleα, the direction of external driving forceθand the interatomic stiffness strength K can affect the static friction force of the system. In the future, we will focus on the study of quantum behaviors of the FK model. These works are shown in chapter IV.
本论文研究了作为上层的正方形晶格和作为下层的正方形基低间存在任意错合角θ,外界驱动力沿任意角度α驱动时,二维FK模型的动力学行为及摩擦特性。本文第一章概述了纳米摩擦学的发展历史及应用前景。第二章我们主要介绍纳米摩擦学的理论研究技术——分子动力学技术。实践证明该技术能有效地用来研究纳米结构及探索微观现象的新规律。因而受到人们越来越广泛的关注,并成功地应用于包括纳米摩擦学在内的许多微观研究领域中。第三章基于分子动力学(molecular dynamics)模拟技术,主要介绍了二维FK模型在纳米摩擦学领域中的应用。我们的数值模拟结果表明:系统的静摩擦力与系统的重要参数有关,如改变系统的硬度(stiffness)、系统的公度和不公度、系统的错合角θ,外界驱动力的方向α以及原子间的强度系数K等都可改变系统的静摩擦力。本文第四章展望了我们在纳米摩擦学领域中的后续工作。
Over the last decade, remarkable developments in nanotechnology have provided the possibility to build experimental devices which are able to perform analysis on materials at the atomic-scale. Simultaneously, tremendous advances in computing methodology (molecular dynamics techniques, MDT) have supplied very detailed information on the atomic scale for realistic sliding system. The accelerations in experiments and computations have deeply stimulated the search and implementation of simpler mathematical model such as the Frenkel-Kontorova (FK) model. The model of friction is capable of describing and interpreting the main features of the complicated dynamics involved.
In this dissertation, we study the dynamical behavior of a two-dimensional FK model with a square symmetry potential for a square lattice layer driven by an external driving force with an arbitrary directionαand an arbitrary misfit angle 9 between upper and lower layer. There are four chapters in the dissertation. The paper is organized as following: in chapter I, the development and prosperity of the nano-tribology is briefly introduced. In the theoretical simulations of the nano-tribology, one of the most important techniques called the MDT will be reviewed in chapter II. It is verified that the technique is effective and coherent in the study of the nanostructure. So, many physicists pay more and more attention on it. In chapter III, by using the MDT, the application of the two dimensional FK model in the nano-tribolody will be presented. With regards to the static friction force, the results show that the parameters of the system, such as the effective elastic constant, the commensurate and incommensurate structures of the system, the misfit angleα, the direction of external driving forceθand the interatomic stiffness strength K can affect the static friction force of the system. In the future, we will focus on the study of quantum behaviors of the FK model. These works are shown in chapter IV.
引文
[1]Binning G,Rohrer H.Appl.Letter.1982,40:178
[2]白春礼.纳米科学与技术.昆明:云南科技出版社,1995
[3]温诗铸,李娜.中国机械过程,1996,7(2):云南科技出版社,1995
[4]温诗铸.纳米摩擦学,清华大学出版社,1998
[5]温诗铸,卜炎,干东英等.纳米机械学研究形状与展望.北京:机械工业出版社,1996
[6]Amontaons G.Memoires de 1' Academie Royal.Amsterdam:Chez Gerald Kuyer,1778:161
[7]Hardy W B,Hardy J K.Phil.Mag.,1919,38:33
[8]Tomlinson G A.Phil Mag.,1929,7:905
[9]温诗铸.科技与出版,1995,4(3):38
[10]Winer W O.Wear,1990,136:19
[11]Arsenault R J,Beeler J B,Esterling D M.Computer Simulation in Material Science.ASM International,1988
[12]Heermann D.W.Computer Simulation Methods in Theoretiacal Physics.Springer-Verlag Press,1990
[13]Allen M D,Tildesley D J,Computer Simualtion of Liquid.Oxfbrd:Clarendon Press,1987
[14]Ciccotti G,Hoover W G,Molecular Dynamics Simulation of Statistical Mechanical System.Elesevier Science Publisher B.V.,1986
[15]Gunstom V,Berendsen W F.Mol.Phy.,1977,34:1311
[16]Allen M D,Tildesley D J.Computer Simulation in Chemical Physics.Netherlands:Kluwer Academic Publishers,1993
[17]Meeyer M,Pontikis V.Computer Simulation in Materials Science,NATO ASI Seris.Netherlands:Kluwer Academic Publishers,1991
[18]Ciccotti G,Ryckaert J P.Comput.Phys.Rep.,1986 4:345
[19]Edberg R,Evans D J,Morriss G P.J.Chem.Phys.,1986,84:6933
[20]Liem S Y,Chen K Y.Surf.Science,1995,328:119
[21]Cummings P T,Evans D J,Ind.Eng.Chem.Res.,1992,31:1237
[22]胡元中,王慧,郭炎等.纳米摩擦学进展.北京:清华大学出版社,1996
[23]胡元中,王慧,郭炎.仪器仪表学报,1995,1:301
[24]张人佶,沈耀.纳米摩擦学进展.北京:清华大学出版社,1996
[25]J.Robertson.Materials Science and Engineering R,37:129-281,(2002)
[26]B.Bhushan.Modern Tribology Handbook,Ⅱ,CRC Press,:787-825,(2001)
[27]H.X.Li,T.Xu,C.B.Wang,J.M.Chen,H.D.Zhou and H.W.Liu,Tribology International 40 132-138(2007)
[28]D.Dowson,History of Tribology,Longman,New York,(1979)
[29]M.Hirano,Atomistic of Friction,Surface Scince Report,60159-201(2006)
[30]M.Dienwieel,G.S.Verhoeven,N.Pradeep,and J.W.M.Frenken Phys.Rev.Lett.92 126101(2004)
[31]Z.G.Zheng,B.Hu and G.Hu,Phys.Rev.B 58 5453-5461(1998)
[32]M.Hirano,Wear 932-940(2003)
[33]H.Wang,Y.Z.Hu,T.Zhang,Tribology International 40 680-686(2007)
[34]P.R.Yang,J.L.Cui,et al.,J.Tribology-transactions of the ASME,130,041502(2008):
[35]T.Strunz and F.J.Elmer,Phys.Rev.E 58,1601,58,1612(1998).
[36]J.B.Sokoloff,Phys.Rev.Lett.71,3450(1993);
J.Phys.Condens.Matter 10,9991(1998).
[37]Y.Braiman,F.Family,and H.G.E.Hentschel,Phys.Rev.B 55,5491(1997).
[38]n.G.E.Hentschel,F.Family,and Y.Braiman,Phys.Rev.Lett.83,104(1999).
[39]J.Tekic,O,M,Braun,and B.Hu,Phys.Rev.E,1 026104(2005)
[40]A.Vanossi,A.R.Bishop,A.Franchini and V.Bortolani,Surface Science,566-568816-820(2004)
[41]C.Reichardt,C.J.Olson,N.Gr?nbech-Jensen,and F.Nori,Phys.Rev.Lett.86,4354(2001)
[42]G.Blatter et al.,Rev.Mod.Phys.66,1125(1994)
[43]M.J.Higgins and S.Bhattacharya,Phys.Rev.Lett.74,3029(1995)
[44]G.Gr(u|¨)ner,Rev.Mod.Phys.60,1129(1988)
[45]C.Reichhardt and C.J.Olson Reichhardt,Phys.Rev.Lett.92,108301(2004)
[46]Y.Guo,Z.H.Qu,Z.Y.Zhang,Phys.Rev.B 73 094118(2006)
[47]B.Hu,L.Yang,ang Y.Zhang,Phys.Rev.Lett.97 124302(2006)
[48]Y.Zolotaryuli,and M.Salerno,Phys.Rev.E 73 066621(2006)
[49]H.C.Hu,S.Y.Lou,and K.W.Chows,Chaos,Solitons Fractals 31 1213-1222(2007)
[50]S.Cuenda,A,Sanchez,and N.R.Quintero,Physica D 223 214-221(2006)
[51]C.L.Wang,W.S.Duan,X.R.Hong,and J.M.Chen,App1.Phys.Lett.93(15)153116(2008)
[52]X.Y.Gao,C.L.Wang,X.R.Hong,and W.S.Duan,Chin.Phys.17 9 3378-09(2008)
[53]C.L.Wang,J.N.Han,X.Y.Gao,W.S.Duan,Commu.Theor.Phys.,51(1) 90-96(2009)
[54]X.Y.Wang,W.S.Duan,M.M.Lin,G.X.Wan,Chaos,Solitons Fractals,30 909-919(2006)
[55]W.S.Duan,J.Parkes,M.M.Lin,Phys.Plasmas,12,022106(2005)
[56]W.S.Duan,G.X.Wan,X.Y.Wang,M.M.Lin,Phys.Plasmas,11 4408-4413(2004)
[57]Homola A M,Israelachvili J N,McGuiggan P M,Gee M L.Proc.Sth International Congress on Tribology,Finland,1989,1
[58]Bowdon F P,Tabor D.Friction and Lubrication,London:Methuen,1967
[59]Johnson K L,Kendall K,Roberts A D.Proc.Roy.Soc.London,1971,A324:301
[60]Ruan J,Bhushan B.J.APPL.Phys.,1993,75:5741
[61]Mate C M,McClelland G M,Erlandsson R,Chiang S.Phys.Rev.Lett.,1987,59:1942
[62]Ruan J,Bhushan B,ASME,J.Tribology.1994,116:378
[63]Erlandsson R,Hadziioannou G,Mate C M et al.J.Chem.Phys.,1988,89:5190
[64]Ruan J,Bhushan B J.Appl.Phys.,1994,76:8117
[65]Bhushan B,Koinkar V N.J.Appl.Phys.,1994,75:5741
[66]Bhushan B,Koinkar V N.Wear,1995,180:9
[67]Bhushan B,Koinkar V N.Wear,1995,180:9
[68]路新春,温诗铸,戴K春等.科学通报,1996,41(10):885
[69]路新春,温诗铸,戴K春等.纳米摩擦学进展.北京:清华大学出版社,1996
[70]Wheeler D R.J.AppI.Phys.,1976,47:1123
[71]Pepper S V.J.Appl.Phys.,1979,50:8062
[72]Pepper S V.J.Vac.Soc.Technol.,1982,20:643
[73]Bowdon F P,Tabor D.Friction and Lubrication of Solids.Oxford:Clarendon,1964
[74]山本好夫,水村好次、官川大海.日本机械学会讲演论义集.1974:42
[75]Hirano M,Shinjo K.Phys.Rev.Lett.,1990,B.41:l1837
[76]Shinjo K,Hirano M.Surface Science,1993,283:473
[77]Hirano M,Shinjo K.Wear,1993,168:121
[78]Blatter,Rev.Mod.Phys.66,1125(1994)
[79]J.Higgins and S.Bhattacharya,Phys.Rev.Lett.74,3029(1995).
[80]Grner,Rev.Mod.Phys.60,1129(1988).
[81]Reichhardt and C.J.Olson Reichhardt,Phys.Rev.Lett.92,108301(2004).
[82]Reichardt,C.J.Olson.N.Gronbech-Jensen.and F.Nori,Phys.Rev.Lett.86,4354(2001).
[83]A.Middleton and N.S.Wingreen,Phys.Rev.Lett.71,3198(1993).
[84]Reichhardt and C.J.Olson Reichhardt,Phys.Rev.Lett.90,046802(2003)
[85]Seshadri and R.M.Westervelt,Phys.Rev.B 46,5150(1992)
[86]N.J.Persson.Sliding Friction:Physical Principles and Applications(Springer,Berlin,1998);
[87]N.J.PerssonSurf.Sci.Rep.33,83(1999)
[88]G.Zheng,B.Hu,and G.Hu,Phys.Rev.B 58,5453(1998)
[89]Tekic,O.M.Braun,and B.Hu,Phys.Rev.E 71,026104(2005)
[90]O.M.Braun,A.R.Bishop,and J.Roder,Phys.Rev.Lett.79,3692(1997)
[91]O.M.Braun,H.Zhang,B.Hu,and J.Tekic,Phys.Rev.E 67,066602(2003)
[92]Li,T.Xu,C.Wang,J.Chen,H.Zhou,and H.Liu,Tribol.Int.40,132(2007)
[93]Vanossi,N.Manini,G.Divitini,G.E.Santoro,and E.Tosatt,Phys.Rev.Lett.97,056101(2006)
[94]A.Vanossi,N.Manini,F.Caruso,G.Divitini,G.E.Santoro,and E.Tosatt,Phys.Rev.Bert.99,206101(2007)
[95]A.Vanossi,A.R.Bishop,A.Eranchini,and V.Bortolani,Surf.Sci.566,816(2004)
[96]A.Vanossi,J.Roder,A.R.Bishop,and V.Bortolani,Phys.Rev.E67,016605(2003)
[97]T.S.van Erp et al.,Phys.Rev.B 60,6522(1999).
[98]J.C.Ariyasu and A.R.Bishop,Phys.Rev.B 35,3207(1987).
[99]O.M.Braun et al.,Phys.Rev.Bert.79,3692(1997).
[100]O.Mandel et al.,Nature(London)425,937(2003)
[101]O.M.Braun et al.,Phys.Rev.E 55,3598(1997).
[102]S.Aubry,in:A.R.Bishop,T.Schneider(Eds.),Solitons and CondensedMatter Physics,Springer Series in Solid State Sciences,vol.8,Springer-Verlag,Berlin,1978,pp.264-277
[103]S.Aubry,Physica D 7(1983) 240
[104]S.Aubry,P.Y.Le Daeron,Physica D 8(1983) 381
[105]V.I.Arnold,Mathematical Methods of Classical Mechanics,Springer,NewYork,1997
[106]S.R.Sharma,B.Bergersen,B.Joos,Phys.Rev.B 29(1984)6335
[107]Y.Braiman,J.Baumgarten,J.Jortner,J.Klafter,Phys.Rev.Lett.65(1990) 2398
[108]P.L.Christiansen,A.V.Savin,A.V.Zolotaryuk,Phys.Rev.B 57(1998) 13564
[109]J.E.Dennis,R.B.Schnabel,Numerical Methodfor UnconstrainedOptimizat ion and-Nonlinear Equation,Prentice-Hall,EnglewoodCliffs,N J,1983
[110]D.Vanderbit,S.G.Louie,J.Comp.Phys.56(1984) 259
[111]W.H.Press,S.A.Teukolsky,W.T.Vetterling,B.P.Felannery,Numerical Recipes in Fortran,Cambridge University Press,Cambridge,1992
[112]S.Aubry,in:A.R.Bishop,T.Schneider(Eds.),Solitons andCond ensedMatter Physics,Springer Series in Solid State Sciences,vol.8,Springer-Verlag,Berlin,1978,pp.264-277
[113]S.Aubry,Physica D 7(1983) 240
[114]S.Aubry,P.Y.Le Daeron,Physica D 8(1983) 381
[115]V.I.Arnold,Mathematical Methods of Classical Mechanics,Springer,New York,1997
[116]S.R.Sharma,B.Bergersen,B.Joos,Phys.Rev.B 29(1984) 6335
[117]Y.Braiman,J.Baumgarten,J.Jortner,J.Klafter,Phys.Rev.Lett.65(1990) 2398.
[118]P.L.Christiansen,A.V.Savin,A.V.Zolotaryuk,Phys.Rev.B 57(1998) 13564
[119]J.E.Dennis,R.B.Schnabel,Numerical Method fbrUnconstrainedOptimizat ion and-Nonlinear Equation,Prentice-Hall,EnglewoodCliffs,N J,1983
[120]D.Vanderbit,S.G.Louie,J.Comp.Phys.56(1984) 259
[121]W.H.Press,S.A.Teukolsky,W.T.Vetterling,B.P.Flannery,Numerical Recipes in Fortran,Cambridge University Press,Cambridge,1992.
[122]Borgonovi F,Guarneri I,and Shepelyansky D,Z.Phys.B-Condensedmatter 1990,79:133-142
[123]Hu Bambi and Li Baowen,"Quantum signature of phase transition by breaking of analyticity in Frenkel-Kontorova model",Preprint CNS-9701,Hong Kong Baptist University,March 1997
[124]Hu Bambi,LiBaowen,and Zhang Weimin,"The quantum Frenkel-Kontorova moldel:a squeezed state approach",Preprint CNS-9702,Hong Kong Baptist University,March 1997
[125]Xu Aiguo,Wang Guangrui,Chen Shigang,et al,Phys.Lett.A,1997,233(1):99-104
[126]许爱国,土光瑞,陈式刚,非线性动力学学报,1997,4(4):368-373
[127]Axel F and Aubry S,J.Phys.A,1987,20(14):4873-4889
[2]白春礼.纳米科学与技术.昆明:云南科技出版社,1995
[3]温诗铸,李娜.中国机械过程,1996,7(2):云南科技出版社,1995
[4]温诗铸.纳米摩擦学,清华大学出版社,1998
[5]温诗铸,卜炎,干东英等.纳米机械学研究形状与展望.北京:机械工业出版社,1996
[6]Amontaons G.Memoires de 1' Academie Royal.Amsterdam:Chez Gerald Kuyer,1778:161
[7]Hardy W B,Hardy J K.Phil.Mag.,1919,38:33
[8]Tomlinson G A.Phil Mag.,1929,7:905
[9]温诗铸.科技与出版,1995,4(3):38
[10]Winer W O.Wear,1990,136:19
[11]Arsenault R J,Beeler J B,Esterling D M.Computer Simulation in Material Science.ASM International,1988
[12]Heermann D.W.Computer Simulation Methods in Theoretiacal Physics.Springer-Verlag Press,1990
[13]Allen M D,Tildesley D J,Computer Simualtion of Liquid.Oxfbrd:Clarendon Press,1987
[14]Ciccotti G,Hoover W G,Molecular Dynamics Simulation of Statistical Mechanical System.Elesevier Science Publisher B.V.,1986
[15]Gunstom V,Berendsen W F.Mol.Phy.,1977,34:1311
[16]Allen M D,Tildesley D J.Computer Simulation in Chemical Physics.Netherlands:Kluwer Academic Publishers,1993
[17]Meeyer M,Pontikis V.Computer Simulation in Materials Science,NATO ASI Seris.Netherlands:Kluwer Academic Publishers,1991
[18]Ciccotti G,Ryckaert J P.Comput.Phys.Rep.,1986 4:345
[19]Edberg R,Evans D J,Morriss G P.J.Chem.Phys.,1986,84:6933
[20]Liem S Y,Chen K Y.Surf.Science,1995,328:119
[21]Cummings P T,Evans D J,Ind.Eng.Chem.Res.,1992,31:1237
[22]胡元中,王慧,郭炎等.纳米摩擦学进展.北京:清华大学出版社,1996
[23]胡元中,王慧,郭炎.仪器仪表学报,1995,1:301
[24]张人佶,沈耀.纳米摩擦学进展.北京:清华大学出版社,1996
[25]J.Robertson.Materials Science and Engineering R,37:129-281,(2002)
[26]B.Bhushan.Modern Tribology Handbook,Ⅱ,CRC Press,:787-825,(2001)
[27]H.X.Li,T.Xu,C.B.Wang,J.M.Chen,H.D.Zhou and H.W.Liu,Tribology International 40 132-138(2007)
[28]D.Dowson,History of Tribology,Longman,New York,(1979)
[29]M.Hirano,Atomistic of Friction,Surface Scince Report,60159-201(2006)
[30]M.Dienwieel,G.S.Verhoeven,N.Pradeep,and J.W.M.Frenken Phys.Rev.Lett.92 126101(2004)
[31]Z.G.Zheng,B.Hu and G.Hu,Phys.Rev.B 58 5453-5461(1998)
[32]M.Hirano,Wear 932-940(2003)
[33]H.Wang,Y.Z.Hu,T.Zhang,Tribology International 40 680-686(2007)
[34]P.R.Yang,J.L.Cui,et al.,J.Tribology-transactions of the ASME,130,041502(2008):
[35]T.Strunz and F.J.Elmer,Phys.Rev.E 58,1601,58,1612(1998).
[36]J.B.Sokoloff,Phys.Rev.Lett.71,3450(1993);
J.Phys.Condens.Matter 10,9991(1998).
[37]Y.Braiman,F.Family,and H.G.E.Hentschel,Phys.Rev.B 55,5491(1997).
[38]n.G.E.Hentschel,F.Family,and Y.Braiman,Phys.Rev.Lett.83,104(1999).
[39]J.Tekic,O,M,Braun,and B.Hu,Phys.Rev.E,1 026104(2005)
[40]A.Vanossi,A.R.Bishop,A.Franchini and V.Bortolani,Surface Science,566-568816-820(2004)
[41]C.Reichardt,C.J.Olson,N.Gr?nbech-Jensen,and F.Nori,Phys.Rev.Lett.86,4354(2001)
[42]G.Blatter et al.,Rev.Mod.Phys.66,1125(1994)
[43]M.J.Higgins and S.Bhattacharya,Phys.Rev.Lett.74,3029(1995)
[44]G.Gr(u|¨)ner,Rev.Mod.Phys.60,1129(1988)
[45]C.Reichhardt and C.J.Olson Reichhardt,Phys.Rev.Lett.92,108301(2004)
[46]Y.Guo,Z.H.Qu,Z.Y.Zhang,Phys.Rev.B 73 094118(2006)
[47]B.Hu,L.Yang,ang Y.Zhang,Phys.Rev.Lett.97 124302(2006)
[48]Y.Zolotaryuli,and M.Salerno,Phys.Rev.E 73 066621(2006)
[49]H.C.Hu,S.Y.Lou,and K.W.Chows,Chaos,Solitons Fractals 31 1213-1222(2007)
[50]S.Cuenda,A,Sanchez,and N.R.Quintero,Physica D 223 214-221(2006)
[51]C.L.Wang,W.S.Duan,X.R.Hong,and J.M.Chen,App1.Phys.Lett.93(15)153116(2008)
[52]X.Y.Gao,C.L.Wang,X.R.Hong,and W.S.Duan,Chin.Phys.17 9 3378-09(2008)
[53]C.L.Wang,J.N.Han,X.Y.Gao,W.S.Duan,Commu.Theor.Phys.,51(1) 90-96(2009)
[54]X.Y.Wang,W.S.Duan,M.M.Lin,G.X.Wan,Chaos,Solitons Fractals,30 909-919(2006)
[55]W.S.Duan,J.Parkes,M.M.Lin,Phys.Plasmas,12,022106(2005)
[56]W.S.Duan,G.X.Wan,X.Y.Wang,M.M.Lin,Phys.Plasmas,11 4408-4413(2004)
[57]Homola A M,Israelachvili J N,McGuiggan P M,Gee M L.Proc.Sth International Congress on Tribology,Finland,1989,1
[58]Bowdon F P,Tabor D.Friction and Lubrication,London:Methuen,1967
[59]Johnson K L,Kendall K,Roberts A D.Proc.Roy.Soc.London,1971,A324:301
[60]Ruan J,Bhushan B.J.APPL.Phys.,1993,75:5741
[61]Mate C M,McClelland G M,Erlandsson R,Chiang S.Phys.Rev.Lett.,1987,59:1942
[62]Ruan J,Bhushan B,ASME,J.Tribology.1994,116:378
[63]Erlandsson R,Hadziioannou G,Mate C M et al.J.Chem.Phys.,1988,89:5190
[64]Ruan J,Bhushan B J.Appl.Phys.,1994,76:8117
[65]Bhushan B,Koinkar V N.J.Appl.Phys.,1994,75:5741
[66]Bhushan B,Koinkar V N.Wear,1995,180:9
[67]Bhushan B,Koinkar V N.Wear,1995,180:9
[68]路新春,温诗铸,戴K春等.科学通报,1996,41(10):885
[69]路新春,温诗铸,戴K春等.纳米摩擦学进展.北京:清华大学出版社,1996
[70]Wheeler D R.J.AppI.Phys.,1976,47:1123
[71]Pepper S V.J.Appl.Phys.,1979,50:8062
[72]Pepper S V.J.Vac.Soc.Technol.,1982,20:643
[73]Bowdon F P,Tabor D.Friction and Lubrication of Solids.Oxford:Clarendon,1964
[74]山本好夫,水村好次、官川大海.日本机械学会讲演论义集.1974:42
[75]Hirano M,Shinjo K.Phys.Rev.Lett.,1990,B.41:l1837
[76]Shinjo K,Hirano M.Surface Science,1993,283:473
[77]Hirano M,Shinjo K.Wear,1993,168:121
[78]Blatter,Rev.Mod.Phys.66,1125(1994)
[79]J.Higgins and S.Bhattacharya,Phys.Rev.Lett.74,3029(1995).
[80]Grner,Rev.Mod.Phys.60,1129(1988).
[81]Reichhardt and C.J.Olson Reichhardt,Phys.Rev.Lett.92,108301(2004).
[82]Reichardt,C.J.Olson.N.Gronbech-Jensen.and F.Nori,Phys.Rev.Lett.86,4354(2001).
[83]A.Middleton and N.S.Wingreen,Phys.Rev.Lett.71,3198(1993).
[84]Reichhardt and C.J.Olson Reichhardt,Phys.Rev.Lett.90,046802(2003)
[85]Seshadri and R.M.Westervelt,Phys.Rev.B 46,5150(1992)
[86]N.J.Persson.Sliding Friction:Physical Principles and Applications(Springer,Berlin,1998);
[87]N.J.PerssonSurf.Sci.Rep.33,83(1999)
[88]G.Zheng,B.Hu,and G.Hu,Phys.Rev.B 58,5453(1998)
[89]Tekic,O.M.Braun,and B.Hu,Phys.Rev.E 71,026104(2005)
[90]O.M.Braun,A.R.Bishop,and J.Roder,Phys.Rev.Lett.79,3692(1997)
[91]O.M.Braun,H.Zhang,B.Hu,and J.Tekic,Phys.Rev.E 67,066602(2003)
[92]Li,T.Xu,C.Wang,J.Chen,H.Zhou,and H.Liu,Tribol.Int.40,132(2007)
[93]Vanossi,N.Manini,G.Divitini,G.E.Santoro,and E.Tosatt,Phys.Rev.Lett.97,056101(2006)
[94]A.Vanossi,N.Manini,F.Caruso,G.Divitini,G.E.Santoro,and E.Tosatt,Phys.Rev.Bert.99,206101(2007)
[95]A.Vanossi,A.R.Bishop,A.Eranchini,and V.Bortolani,Surf.Sci.566,816(2004)
[96]A.Vanossi,J.Roder,A.R.Bishop,and V.Bortolani,Phys.Rev.E67,016605(2003)
[97]T.S.van Erp et al.,Phys.Rev.B 60,6522(1999).
[98]J.C.Ariyasu and A.R.Bishop,Phys.Rev.B 35,3207(1987).
[99]O.M.Braun et al.,Phys.Rev.Bert.79,3692(1997).
[100]O.Mandel et al.,Nature(London)425,937(2003)
[101]O.M.Braun et al.,Phys.Rev.E 55,3598(1997).
[102]S.Aubry,in:A.R.Bishop,T.Schneider(Eds.),Solitons and CondensedMatter Physics,Springer Series in Solid State Sciences,vol.8,Springer-Verlag,Berlin,1978,pp.264-277
[103]S.Aubry,Physica D 7(1983) 240
[104]S.Aubry,P.Y.Le Daeron,Physica D 8(1983) 381
[105]V.I.Arnold,Mathematical Methods of Classical Mechanics,Springer,NewYork,1997
[106]S.R.Sharma,B.Bergersen,B.Joos,Phys.Rev.B 29(1984)6335
[107]Y.Braiman,J.Baumgarten,J.Jortner,J.Klafter,Phys.Rev.Lett.65(1990) 2398
[108]P.L.Christiansen,A.V.Savin,A.V.Zolotaryuk,Phys.Rev.B 57(1998) 13564
[109]J.E.Dennis,R.B.Schnabel,Numerical Methodfor UnconstrainedOptimizat ion and-Nonlinear Equation,Prentice-Hall,EnglewoodCliffs,N J,1983
[110]D.Vanderbit,S.G.Louie,J.Comp.Phys.56(1984) 259
[111]W.H.Press,S.A.Teukolsky,W.T.Vetterling,B.P.Felannery,Numerical Recipes in Fortran,Cambridge University Press,Cambridge,1992
[112]S.Aubry,in:A.R.Bishop,T.Schneider(Eds.),Solitons andCond ensedMatter Physics,Springer Series in Solid State Sciences,vol.8,Springer-Verlag,Berlin,1978,pp.264-277
[113]S.Aubry,Physica D 7(1983) 240
[114]S.Aubry,P.Y.Le Daeron,Physica D 8(1983) 381
[115]V.I.Arnold,Mathematical Methods of Classical Mechanics,Springer,New York,1997
[116]S.R.Sharma,B.Bergersen,B.Joos,Phys.Rev.B 29(1984) 6335
[117]Y.Braiman,J.Baumgarten,J.Jortner,J.Klafter,Phys.Rev.Lett.65(1990) 2398.
[118]P.L.Christiansen,A.V.Savin,A.V.Zolotaryuk,Phys.Rev.B 57(1998) 13564
[119]J.E.Dennis,R.B.Schnabel,Numerical Method fbrUnconstrainedOptimizat ion and-Nonlinear Equation,Prentice-Hall,EnglewoodCliffs,N J,1983
[120]D.Vanderbit,S.G.Louie,J.Comp.Phys.56(1984) 259
[121]W.H.Press,S.A.Teukolsky,W.T.Vetterling,B.P.Flannery,Numerical Recipes in Fortran,Cambridge University Press,Cambridge,1992.
[122]Borgonovi F,Guarneri I,and Shepelyansky D,Z.Phys.B-Condensedmatter 1990,79:133-142
[123]Hu Bambi and Li Baowen,"Quantum signature of phase transition by breaking of analyticity in Frenkel-Kontorova model",Preprint CNS-9701,Hong Kong Baptist University,March 1997
[124]Hu Bambi,LiBaowen,and Zhang Weimin,"The quantum Frenkel-Kontorova moldel:a squeezed state approach",Preprint CNS-9702,Hong Kong Baptist University,March 1997
[125]Xu Aiguo,Wang Guangrui,Chen Shigang,et al,Phys.Lett.A,1997,233(1):99-104
[126]许爱国,土光瑞,陈式刚,非线性动力学学报,1997,4(4):368-373
[127]Axel F and Aubry S,J.Phys.A,1987,20(14):4873-4889