JACS ART I CLES Published on Web 08/26/2008 Nanosolids,Slushes,and Nanoliquids:Characterization of Nanophases in Metal Clusters and Nanoparticles Zhen Hua Li*t and Donald G.Truhlar*+ Department of Chemistry,Fudan University,Shanghai,200433,China and Department of Chemistry and Supercomputing Institute,University of Minnesota, Minneapolis,Minnesota 55455-0431 Received April 3,2008;E-mail:lizhenhua@fudan.edu.cn;truhlar@umn.edu Abstract:One of the keys to understanding the emergent behavior of complex materials and nanoparticles is understanding their phases.Understanding the phases of nanomaterials involves new concepts not present in bulk materials;for example,the phases of nanoparticles are quantum mechanical even when no hydrogen or helium is present.To understand these phases better,molecular dynamics(MD)simulations on size-selected particles employing a realistic analytic many-body potential based on quantum mechanical nanoparticle calculations have been performed to study the temperature-dependent properties and melting transitions of free Aln clusters and nanoparticles with n=10-300 from 200 to 1700 K.By analyzing properties of the particles such as specific heat capacity (c),radius of gyration,volume,coefficient of thermal expansion (B),and isothermal compressibility (K),we developed operational definitions of the solid,slush,and liquid states of metal clusters and nanoparticles.Applying the definitions,which are based on the temperature dependences of c.B,and In k,we determined the temperature domains of the solid,slush,and liquid states of the Al particles.The results show that Al clusters(n<18,diameter of less than 1 nm)are more like molecules,and it is more appropriate to say that they have no melting transition,but Al nanoparticles (n>19,diameter of more than 1 nm)do have a melting transition and are in the liquid state above 900-1000 K.However,all aluminum nanoparticles have a wide temperature interval corresponding to the slush state in which the solid and liquid states coexist in equilibrium,unlike a bulk material where coexistence occurs only at a single temperature (for a given pressure).The commonly accepted operational marker of the melting temperature,namely,the peak position of c,is not unambiguous and not appropriate for characterizing the melting transition for aluminum particles with the exception of a few particle sizes that have a single sharp peak(as a function of temperature)in each of the three properties,c,B,and In K. 1.Introduction we "borrow"concepts from well-studied ones.However,one Metal clusters and nanoparticles,as an intermediate form of must be careful when applying macroconcepts to finite systems matter!-7 between the composing atoms and the corresponding because these concepts may be ill defined for finite systems. bulk materials,have distinct electrical,optical,magnetic. For example,melting is well defined on the macroscale but not chemical,and catalytic properties and have been the subjects on the nanoscale.-13 Molecular dynamics simulations of metal of extensive experimental and theoretical study.Understanding nanoparticles have covered dynamic phase coistenc phe- the evolution of various physical and chemical properties from nomena not present in bulk metals.14- the atomic to the bulk limit is also of great fundamental and Understanding the molecular thermodynamics of nanophases practical interest.Often,when we face a new class of phenomena is a key enabler for the bottom-up approach to nanodesign.For macroscopic systems,a phase is a state with uniform20 or t Fudan University. continuously varying21 physical and chemical properties (in- University of Minnesota. tensive thermodynamic variables)in a well-defined temperature (1)Bonacic-Koutecky,V.;Fantucci,P.;Koutecky,J.Chem.Rev.1991. and pressure range.The change from one phase to another phase 91,1035.de Heer,W.A.Rev.Mod.Phys.1993,65,611. (2)Feldheim,D.L.:Foss.C.A.Metal Nanoparticles:Synthesis, Characterization,and Applications;Marcel Dekker.New York,2002. (8)Berry,R.S.;Jellinek,J.;Natanson,G.Phys.Rev.A 1984,30,919. (3)Buchachenko,A.L.Russ.Chem.Rev.2003,72,375. (9)Beck,T.L.;Jellinek,J.:Berry,R.S.J.Chem.Phrys.1987,87,545. (4)Schmid,G.Nanoparticles:From Theory to Applications;Wiley-VCH: (10)Berry.R.S.:Wales,D.J.Phys.Rev.Lett.1989,63,1156.(a)Wales. Weinheim,2004. D.J.:Berry,R.S.J.Chem.Phys.1990.92.4473.Berry.R.S. (5)Chan,K.-Y.;Ding,J.:Ren,J.:Cheng,S.:Tsang,K.Y.J.Mater. J.Chem.Soc.,Faraday Trans.1990,86,2343.Berry,R.S.Sci.Am. Chem.2004,14,505.Heiz,U.;Bullock,E.L.J.Mater.Chem.2004. 1990,26368. 14.564.O'Hair.R.A.J.:Khairallah.G.N.J.Cluster Sci.2004./5. (11)Berry,R.S.In Clusters of Atoms and Molecules;Haberland,H.. 331. Ed.;Springer Series in Chemical Physics 52;Springer:Berlin,1994: (6)Baletto,F.:Ferrando.R.Rev.Mod.Phys.2005,77.371 p 187.Berry,R.S.Microscale Thermoplrys.Eng.1997,1,1. (7)Astruc,D.:Lu.F.:Aranzaes.T.R.Angew.Chem..Int.Ed.2005. Proykova,A.:Berry,R.S.J.Phys.B:At.Mol.Opt.Phys.2006.39. 44.7852.Watanabe,K.:Menzel,P.;Nilius,N.:Freund,H.-J.Chem. R167. Rev.2006,106,4301.Perepichka,D.F.:Rosei.F.Angew.Chem.. (12)Berry,R.S.C.R.Phys.2002.3,319. Int.Ed.2007.46,6006.Jellinek,J.Faraday Discuss.2008.138.11. (13)Schmidt,M.;Haberland,H.C.R.Phys.2002,3,327. 12698■J.AM.CHEM.S0C.2008,130,12698-12711 10.1021/ja802389d CCC:$40.75 2008 American Chemical Society
Nanosolids, Slushes, and Nanoliquids: Characterization of Nanophases in Metal Clusters and Nanoparticles Zhen Hua Li*,† and Donald G. Truhlar*,‡ Department of Chemistry, Fudan UniVersity, Shanghai, 200433, China and Department of Chemistry and Supercomputing Institute, UniVersity of Minnesota, Minneapolis, Minnesota 55455-0431 Received April 3, 2008; E-mail: lizhenhua@fudan.edu.cn; truhlar@umn.edu Abstract: One of the keys to understanding the emergent behavior of complex materials and nanoparticles is understanding their phases. Understanding the phases of nanomaterials involves new concepts not present in bulk materials; for example, the phases of nanoparticles are quantum mechanical even when no hydrogen or helium is present. To understand these phases better, molecular dynamics (MD) simulations on size-selected particles employing a realistic analytic many-body potential based on quantum mechanical nanoparticle calculations have been performed to study the temperature-dependent properties and melting transitions of free Aln clusters and nanoparticles with n ) 10-300 from 200 to 1700 K. By analyzing properties of the particles such as specific heat capacity (c), radius of gyration, volume, coefficient of thermal expansion (), and isothermal compressibility (κ), we developed operational definitions of the solid, slush, and liquid states of metal clusters and nanoparticles. Applying the definitions, which are based on the temperature dependences of c, , and ln κ, we determined the temperature domains of the solid, slush, and liquid states of the Aln particles. The results show that Aln clusters (n e 18, diameter of less than 1 nm) are more like molecules, and it is more appropriate to say that they have no melting transition, but Aln nanoparticles (n g 19, diameter of more than 1 nm) do have a melting transition and are in the liquid state above 900-1000 K. However, all aluminum nanoparticles have a wide temperature interval corresponding to the slush state in which the solid and liquid states coexist in equilibrium, unlike a bulk material where coexistence occurs only at a single temperature (for a given pressure). The commonly accepted operational marker of the melting temperature, namely, the peak position of c, is not unambiguous and not appropriate for characterizing the melting transition for aluminum particles with the exception of a few particle sizes that have a single sharp peak (as a function of temperature) in each of the three properties, c, , and ln κ. 1. Introduction Metal clusters and nanoparticles, as an intermediate form of matter1-7 between the composing atoms and the corresponding bulk materials, have distinct electrical, optical, magnetic, chemical, and catalytic properties and have been the subjects of extensive experimental and theoretical study. Understanding the evolution of various physical and chemical properties from the atomic to the bulk limit is also of great fundamental and practical interest. Often, when we face a new class of phenomena we “borrow” concepts from well-studied ones. However, one must be careful when applying macroconcepts to finite systems because these concepts may be ill defined for finite systems. For example, melting is well defined on the macroscale but not on the nanoscale.8-13 Molecular dynamics simulations of metal nanoparticles have uncovered dynamic phase coexistence phenomena not present in bulk metals.14-19 Understanding the molecular thermodynamics of nanophases is a key enabler for the bottom-up approach to nanodesign. For macroscopic systems, a phase is a state with uniform20 or continuously varying21 physical and chemical properties (intensive thermodynamic variables) in a well-defined temperature and pressure range. The change from one phase to another phase † Fudan University. ‡ University of Minnesota. (1) Bonacic-Koutecky, V.; Fantucci, P.; Koutecky, J. Chem. ReV. 1991, 91, 1035. de Heer, W. A. ReV. Mod. Phys. 1993, 65, 611. (2) Feldheim, D. L.; Foss, C. A. Metal Nanoparticles: Synthesis, Characterization, and Applications; Marcel Dekker: New York, 2002. (3) Buchachenko, A. L. Russ. Chem. ReV. 2003, 72, 375. (4) Schmid, G. Nanoparticles: From Theory to Applications; Wiley-VCH: Weinheim, 2004. (5) Chan, K.-Y.; Ding, J.; Ren, J.; Cheng, S.; Tsang, K. Y. J. Mater. Chem. 2004, 14, 505. Heiz, U.; Bullock, E. L. J. Mater. Chem. 2004, 14, 564. O’Hair, R. A. J.; Khairallah, G. N. J. Cluster Sci. 2004, 15, 331. (6) Baletto, F.; Ferrando, R. ReV. Mod. Phys. 2005, 77, 371. (7) Astruc, D.; Lu, F.; Aranzaes, T. R. Angew. Chem., Int. Ed. 2005, 44, 7852. Watanabe, K.; Menzel, P.; Nilius, N.; Freund, H.-J. Chem. ReV. 2006, 106, 4301. Perepichka, D. F.; Rosei, F. Angew. Chem., Int. Ed. 2007, 46, 6006. Jellinek, J. Faraday Discuss. 2008, 138, 11. (8) Berry, R. S.; Jellinek, J.; Natanson, G. Phys. ReV. A 1984, 30, 919. (9) Beck, T. L.; Jellinek, J.; Berry, R. S. J. Chem. Phys. 1987, 87, 545. (10) Berry, R. S.; Wales, D. J. Phys. ReV. Lett. 1989, 63, 1156. (a) Wales, D. J.; Berry, R. S. J. Chem. Phys. 1990, 92, 4473. Berry, R. S. J. Chem. Soc., Faraday Trans. 1990, 86, 2343. Berry, R. S. Sci. Am. 1990, 263, 68. (11) Berry, R. S. In Clusters of Atoms and Molecules; Haberland, H., Ed.; Springer Series in Chemical Physics 52; Springer: Berlin, 1994; p 187. Berry, R. S. Microscale Thermophys. Eng. 1997, 1, 1. Proykova, A.; Berry, R. S. J. Phys. B: At. Mol. Opt. Phys. 2006, 39, R167. (12) Berry, R. S. C. R. Phys. 2002, 3, 319. (13) Schmidt, M.; Haberland, H. C. R. Phys. 2002, 3, 327. Published on Web 08/26/2008 12698 9 J. AM. CHEM. SOC. 2008, 130, 12698–12711 10.1021/ja802389d CCC: $40.75 2008 American Chemical Society
Nanosolids,Slushes,and Nanoliquids ARTICLES under equilibrium conditions usually occurs in a very narrow 1.0x10 temperature and pressure range (spontaneously).A small change in temperature or pressure completely changes the phase from one to the other.Thus,the change of phase can be characterized 8.0x10A by a transition temperature Tiran.However,for clusters and nanoparticles ranging in size from several atoms to thousands 今6.0x10 of atoms,the transition from one equilibrium phase to another equilibrium phase occurs gradually in a wider temperature range.11.122 Within this range the two phases are in 4.0x10 —A8=1 keal moll equilibrium with each other.Therefore,the phase change in finite --△s=5 keal moll systems has to be characterized by two temperatures,T1 and T2 with T2>T1.8-10.12 Below T1.the amount of phase 2 is 2.0x10 negligible,while above 72,the amount of phase I is negligible. For a solid-liquid transition.T is the freezing temperature Tr 0.0 and T2 is the melting temperature Tm.Between the two 0 500 1000 1500 2000 2500 3000 temperatures is the solid-liquid coexistence region.419 For TK) macroscopic systems under equilibrium conditions,Tr= Figure 1.Heat capacity of a model system with two nondegenerate states. Tm.8.10,20.2 Experimentally,one convenient way to study a melting transition is to measure the caloric curve (energy as a function others it does not.26.32.35.36.38.2-44 Usually,as in macroscopic of temperature)of the system.13 The heat capacity is the systems,21 the former case has been called a first-order melting derivative of the caloric curve.Caloric curves have been used transition while the latter has often been called a second-order to study the melting of clusters and nanoparticles,.3.24-4 both melting transition,35 although there are also second-order phase experimentally and theoretically,as well as bulk materials.For transitions of other kinds in bulk materials. some cases,the heat capacity curve has a sharp peak,but for The temperature at which the heat capacity curve has a peak or maximum will be called the peak temperature (T)of the (14)Vichare,A.:Kanhere,D.G.J.Plys:Condens.Matter 1998,10, heat capacity.Although the melting temperature is usually taken 3309. as this peak temperature,this is not necessarily a valid procedure (15)Cleveland.C.L.:Luedtke.W.D.:Landman,U.Phys.Rev.B 1999. 60.5065. for finite systems.Consider a model system with only two states. (16)Pochon,S.:MacDonald,K.F.:Knize,R.J.:Zheludev,N.I.Plrys. both nondegenerate,which may be two electronic states or two Reu.Let.2004,92.145702. isomers in equilibrium with each other:the partition function (17)Schebarchov.D.:Hendy.S.C.J.Chem.Phrys.2006.123.104701. (18)Schebarchov,D.:Hendy,S.C.Plrys.Rev.B 2006,73,121402(R) of the system is (19)Alavi,S.:Thompson.D.L.J.Phrys.Chem.A 2006.110,1518. (20)Atkins,P.:Paula,J.D.Atkins'Phrysical Chemistry,7th ed.;Oxford Q=1+e-a7 (1) University Press:New York,2002;p 135. (21)Berry.R.S.:Rice.S.A.:Ross.J.Physical Chemistry.2nd ed.:Topics where As is the energy gap between the two states.Then the in Physical Chemistry Series 12:Oxford University Press:New York. heat capacity of the system is 2000,Pp397,658.See also p347. (22)Sebetci,A.:Guvenc,Z.B.:Kokten.H.Comput.Mater.Sci.2006. △e2/kTr2 35,152. C= (2) (23)Rodumer.E.Chem.Soc.Rev.2006.35.583. (eAukaT+e-AukBT2 (24)Rey,C.:Gallego,L.J.:Garci'a-Rodeja,J.;Alonso.J.A.:Iniguez, M.P.Pys.Reu.B1993.48.8253. Plots of C vs T with two different values of As are depicted in (25)Nayak.S.K.;Khanna,S.N.;Rao.B.K.:Jena,P.J.Phys.:Condens. Figure 1.Since the two states involved can be any two states. Matter1998.10,10853. (26)Schmidt.M.:Kusche.R.:Kronmuiler.W.:von Issendorff.B.: not necessarily a liquid and a solid state,it is clear that observing Haberland,H.Phys.Rev.Lett.1997,79.99.Schmidt,M.;Kusche, a peak in the C curve is not enough to indicate a melting R.:von Issendorff,B.:Haberland,H.Nature 1998.393.238. transition.It may just result from equilibrium between two (27)Sun,D.Y.:Gong,X.G.Phys.Rev.B 1998,57,4730. electronic states or two structural isomers with different energies. (28)Efremov,M.Yu.:Schiettekatte,F.;Zhang,M.;Olson,E.A.:Kwan, A.T.;Berry,R.S.:Allen,L.H.Phys.Rev.Lett.2000,85,3560. Moreover,it is easy to show that at To the population of the (29)Jellinek,J.:Goldberg.A.J.Chem.Phys.2000.113.2570. higher energy state is just 13%(e2/(1+e)).Therefore,even (30)Schmidt,M.:Hippler,Th.:Donges,J.;Kronmuller,W.;von if one can call this a melting transition,it is questionable whether Issendorff,B.:Haberland,H.:Labastie,P.Phys.Rev.Lett.2001. 87.203402.Schmidt,M.;Donges,J.;Hippler,Th.:Haberland,H. the temperature at which C has a peak should be called the Phys.Reu.Let.2003,90.103401. melting temperature since the majority of the system can still (31)Breaux,G.A.:Benirschke,R.C.:Sugai,T.:Kinnear,B.S.:Jarrold, be in the solid state.In a strict sense,since the transition is M.F.Phrys.Rev.Lett.2003.9/,215508. (32)Breaux,G.A.:Hillman.D.A.:Neal,C.M.;Benirschke,R.C.: gradual,there is no melting point at all.Since clusters and Jarrold,M.F.J.Am.Chem.Soc.2004,126,8628. (33)Lai,S.K.:Lin,W.D.:Wu,K.L.;Li,W.H.;Lee,K.C.J.Chem (40)Zhang,W.;Zhang,F.S.;Zhu,Z.Y.Plrys.Rev.B 2006,74,033412 Phys.2004.121,1487. Zhang,W.;Zhang,F.S.;Zhu,Z.Y.Eur.Phys.J.D 2007,43,97. (34)Werner,R.Eur.Phys.J.B 2005.43,47. Zhang.W.:Zhang.F.S.Zhu,Z.Y.Chin.Phys.Len.2007.24. (35)Breaux,G.A.:Cao.B.:Jarrold,M.F.J.Phys.Chem.B 2005,109, 1915. 16575. (41)Duan,H.M.;Ding,F.;Rosen,A.;Harutyunyan,A.R.;Curtarolo, (36)Breaux.G.A.:Neal.C.M.:Cao,B.:Jarrold,M.F.Phrys.Rev.Lett. S.;Bolton,K.Chem.Phys.2007,333,57. 2005,94,173401. (42)Neal,C.M.;Starace,A.K.:Jarrold,M.F.J.Am.Soc.Mass Spectrom. (37)de Bas,B.S.;Ford,M.J.;Cortie,M.B.J.Phys.:Condens.Matter 2007.18.74. 2006.18.55. (43)Neal,C.M.:Starace,A.K.:Jarrold,M.F.;Joshi,K.:Krishnamurty. (38)Joshi,K.;Krishnamurty,S.;Kanhere,D.G.Phys.Rev.Lett.2006, S.:Kanhere,D.G.J.Phys.Chem.C2007,111,17788. 96,135703. (44)Neal,C.M.:Atarace,A.K.;Jarrold,M.F.Phys.Rev.B 2007,76. (39)Noya,E.G.:Doye,J.P.K.:Calvo.F.Plrys.Rev.B 2006.73,125407. 54113. J.AM.CHEM.SOC.VOL.130.NO.38.2008 12699
under equilibrium conditions usually occurs in a very narrow temperature and pressure range (spontaneously). A small change in temperature or pressure completely changes the phase from one to the other. Thus, the change of phase can be characterized by a transition temperature Ttran. However, for clusters and nanoparticles ranging in size from several atoms to thousands of atoms, the transition from one equilibrium phase to another equilibrium phase occurs gradually in a wider temperature range.6,11,12,22,23 Within this range the two phases are in equilibrium with each other. Therefore, the phase change in finite systems has to be characterized by two temperatures, T1 and T2 with T2 > T1. 8-10,12 Below T1, the amount of phase 2 is negligible, while above T2, the amount of phase 1 is negligible. For a solid-liquid transition, T1 is the freezing temperature Tf and T2 is the melting temperature Tm. Between the two temperatures is the solid-liquid coexistence region.14-19 For macroscopic systems under equilibrium conditions, Tf ) Tm. 8,10,20,21 Experimentally, one convenient way to study a melting transition is to measure the caloric curve (energy as a function of temperature) of the system.13 The heat capacity is the derivative of the caloric curve. Caloric curves have been used to study the melting of clusters and nanoparticles,6,13,24-44 both experimentally and theoretically, as well as bulk materials. For some cases, the heat capacity curve has a sharp peak, but for others it does not.26,32,35,36,38,42-44 Usually, as in macroscopic systems,21 the former case has been called a first-order melting transition while the latter has often been called a second-order melting transition,35 although there are also second-order phase transitions of other kinds in bulk materials. The temperature at which the heat capacity curve has a peak or maximum will be called the peak temperature (Tp) of the heat capacity. Although the melting temperature is usually taken as this peak temperature, this is not necessarily a valid procedure for finite systems. Consider a model system with only two states, both nondegenerate, which may be two electronic states or two isomers in equilibrium with each other; the partition function of the system is Q ) 1 + e -∆ε⁄kBT (1) where ∆ε is the energy gap between the two states. Then the heat capacity of the system is C ) ∆ε 2 ⁄ kBT2 (e ∆ε⁄2kBT + e -∆ε⁄2kBT ) 2 (2) Plots of C vs T with two different values of ∆ε are depicted in Figure 1. Since the two states involved can be any two states, not necessarily a liquid and a solid state, it is clear that observing a peak in the C curve is not enough to indicate a melting transition. It may just result from equilibrium between two electronic states or two structural isomers with different energies. Moreover, it is easy to show that at Tp the population of the higher energy state is just 13% (e-2 /(1 + e-2 )). Therefore, even if one can call this a melting transition, it is questionable whether the temperature at which C has a peak should be called the melting temperature since the majority of the system can still be in the solid state. In a strict sense, since the transition is gradual, there is no melting point at all. Since clusters and (14) Vichare, A.; Kanhere, D. G. J. Phys: Condens. Matter 1998, 10, 3309. (15) Cleveland, C. L.; Luedtke, W. D.; Landman, U. Phys. ReV. B 1999, 60, 5065. (16) Pochon, S.; MacDonald, K. F.; Knize, R. J.; Zheludev, N. I. Phys. ReV. Lett. 2004, 92, 145702. (17) Schebarchov, D.; Hendy, S. C. J. Chem. Phys. 2006, 123, 104701. (18) Schebarchov, D.; Hendy, S. C. Phys. ReV. B 2006, 73, 121402(R) (19) Alavi, S.; Thompson, D. L. J. Phys. Chem. A 2006, 110, 1518. (20) Atkins, P.; Paula, J. D. Atkins’ Physical Chemistry, 7th ed.; Oxford University Press: New York, 2002; p 135. (21) Berry, R. S.; Rice, S. A.; Ross, J. Physical Chemistry, 2nd ed.; Topics in Physical Chemistry Series 12; Oxford University Press: New York, 2000; pp 397,658. See also p 347. (22) Sebetci, A.; Gu¨venc, Z. B.; Ko¨kten, H. Comput. Mater. Sci. 2006, 35, 152. (23) Rodumer, E. Chem. Soc. ReV. 2006, 35, 583. (24) Rey, C.; Gallego, L. J.; Garcı´a-Rodeja, J.; Alonso, J. A.; In˜iguez, M. P. Phys. ReV. B 1993, 48, 8253. (25) Nayak, S. K.; Khanna, S. N.; Rao, B. K.; Jena, P. J. Phys.: Condens. Matter 1998, 10, 10853. (26) Schmidt, M.; Kusche, R.; Kronmu¨ler, W.; von Issendorff, B.; Haberland, H. Phys. ReV. Lett. 1997, 79, 99. Schmidt, M.; Kusche, R.; von Issendorff, B.; Haberland, H. Nature 1998, 393, 238. (27) Sun, D. Y.; Gong, X. G. Phys. ReV. B 1998, 57, 4730. (28) Efremov, M. Yu.; Schiettekatte, F.; Zhang, M.; Olson, E. A.; Kwan, A. T.; Berry, R. S.; Allen, L. H. Phys. ReV. Lett. 2000, 85, 3560. (29) Jellinek, J.; Goldberg, A. J. Chem. Phys. 2000, 113, 2570. (30) Schmidt, M.; Hippler, Th.; Donges, J.; Kronmu¨ller, W.; von Issendorff, B.; Haberland, H.; Labastie, P. Phys. ReV. Lett. 2001, 87, 203402. Schmidt, M.; Donges, J.; Hippler, Th.; Haberland, H. Phys. ReV. Lett. 2003, 90, 103401. (31) Breaux, G. A.; Benirschke, R. C.; Sugai, T.; Kinnear, B. S.; Jarrold, M. F. Phys. ReV. Lett. 2003, 91, 215508. (32) Breaux, G. A.; Hillman, D. A.; Neal, C. M.; Benirschke, R. C.; Jarrold, M. F. J. Am. Chem. Soc. 2004, 126, 8628. (33) Lai, S. K.; Lin, W. D.; Wu, K. L.; Li, W. H.; Lee, K. C. J. Chem. Phys. 2004, 121, 1487. (34) Werner, R. Eur. Phys. J. B 2005, 43, 47. (35) Breaux, G. A.; Cao, B.; Jarrold, M. F. J. Phys. Chem. B 2005, 109, 16575. (36) Breaux, G. A.; Neal, C. M.; Cao, B.; Jarrold, M. F. Phys. ReV. Lett. 2005, 94, 173401. (37) de Bas, B. S.; Ford, M. J.; Cortie, M. B. J. Phys.: Condens. Matter 2006, 18, 55. (38) Joshi, K.; Krishnamurty, S.; Kanhere, D. G. Phys. ReV. Lett. 2006, 96, 135703. (39) Noya, E. G.; Doye, J. P. K.; Calvo, F. Phys. ReV. B 2006, 73, 125407. (40) Zhang, W.; Zhang, F. S.; Zhu, Z. Y. Phys. ReV. B 2006, 74, 033412. Zhang, W.; Zhang, F. S.; Zhu, Z. Y. Eur. Phys. J. D 2007, 43, 97. Zhang, W.; Zhang, F. S.; Zhu, Z. Y. Chin. Phys. Lett. 2007, 24, 1915. (41) Duan, H. M.; Ding, F.; Rosen, A.; Harutyunyan, A. R.; Curtarolo, S.; Bolton, K. Chem. Phys. 2007, 333, 57. (42) Neal, C. M.; Starace, A. K.; Jarrold, M. F. J. Am. Soc. Mass Spectrom. 2007, 18, 74. (43) Neal, C. M.; Starace, A. K.; Jarrold, M. F.; Joshi, K.; Krishnamurty, S.; Kanhere, D. G. J. Phys. Chem. C 2007, 111, 17788. (44) Neal, C. M.; Atarace, A. K.; Jarrold, M. F. Phys. ReV. B 2007, 76, 54113. Figure 1. Heat capacity of a model system with two nondegenerate states. J. AM. CHEM. SOC. 9 VOL. 130, NO. 38, 2008 12699 Nanosolids, Slushes, and Nanoliquids ARTICLES
ARTICLES Li and Truhlar nanoparticles are often a mixture of many isomers with similar Their melting has recently been the subject of extensive energies equilibrating with each other.7s -48 their experimental 5.42-44 and theoretical study. 14.19,27.33.34,40.82 86 melting transitions have the same ambiguity Jarrold et al.used multicollision-induced dissociation to measure The distinction between clusters and nanoparticles is not strict, the heat capacities of Al cationic clusters with n=16-48.44 and we use the generic name particles to refer to both of them. 31-38,43 49-63,36 and 63-83.42 They found that for some Aluminum particles have been of great experimenta clusters the heat capacity curve has a well-defined sharp peak, and theoretical4.19.27.29.33.34.39.40.43.57-86 interest for decades. while for others the heat capacity curve is relatively flat and featureless.Taking the temperature To at which C has a (45)Doye,J.P.K.:Calvo.F.Phys.Rev.Lett.2001.86,3570.Doye, maximum as the melting temperature,they found that the J.P.K.:Calvo,F.J.Chem.Phys.2003.119.12680. melting temperature depends greatly on particle size,and even (46)Wang.G.M.:Blaisten-Barojas,E;Roitberg.A.E.J.Chem.Plrys. a change in size by a single atom can make huge differences 2001.115.3640. Monte Carlo439(MC)and molecular dynamics' 4192733.408384 (47)Baletto,F.:Rapallo,A.;Rossi,G.;Ferrando,R.Phys.Rev.B 2004. 69,235421.RossiG.;Rapallo,A.:Mottet.C.:Fortunelli,A.;Baletto. (MD)simulations confirmed the experimental findings.How- F.:Ferrando.R.Phys.Rev.Lett.2004.93.105503. ever,Tp is not enough to characterize a melting transition since (48)Li,Z.H.;Jasper,A.W.:Truhlar,D.G.J.Am.Chem.Soc.2007, the solid and liquid states in finite systems have not been well 129.14899. (49)de Heer,W.A.:Milani,P.:Chatelain,A.Plrys.Rev.Lett.1989,63. defined.Moreover,for those particles with featureless and flat 2834. heat capacity curves,Tp has large uncertainties and should be (50)Lerme.J.:Pellarin.M.:Vialle.J.L.:Baguenard.B.:Broyer.M.Phrys. treated with caution. Rev.Lett.1992,68,2818.Baguenard,B.;Pellarin,M.:Lerme,J.; Vialle,J.L;Broyer,M.J.Chem.Phys.1994,100.754. For the particle sizes studied here,most atoms need to be (51)Martin.T.P.:Naher,U.:Schaber.H.Chem.Phrys.Lett.1992,199. classified as surface atoms rather than as interior atoms with 470. bulk properties characteristic of a macroscopic particle.Except (52)Jarrold.M.F.:Bower.J.E.J.Phys.Chem.1993.97.1746.Jarrold. for a few small clusters,Al13,14 Al3-,83 and Al14.83 available M.F.;Bower,J.E.J.Chem.Phrys.1993,98.2399.Jarrold,M.F.J. Phs.Chem.1995,99,11. simulations of Al cluster melting all use empirical analytical (53)Cha.C.Y.:Gantefor,G.:Eberhardt,W.J.Chem.Phys.1994,100. potential functions,but it is not possible to accurately param- 995.Gantefor,G.;Eberhardt,W.Chem.Phys.Lett.1994,217,600. etrize empirical potentials in the cluster and nanoparticle regime (54)Li,X.:Wu.H.:Wang,X.B.:Wang,L.S.Phys.Rev.Lett.1998,8/. 1909. due to a lack of experimental data for systems with a significant (55)Akola,J.;Manninen,M.;Hakkinen,H.;Landman,U.:Li,X.:Wang. fraction of atoms in nonbulk (e.g.,surface)positions.24.87 L.S.Phys.Rev.B 1999.60.11297.Akola.J.:Manninen.M.: Recently,economical and accurate analytic potentials for Hakkinen,H.;Landman,U.;Li,X.:Wang,L.S.Phys.Rev.B 2000, 62.13216.Kuznetsov,A.E.:Boldyrev,A.I.:Zhai,H.J.;Li,X.: aluminum systems have been developed by fitting to results of Wang,L.S.J.Am.Chem.Soc.2002,124,111791. well-validated electronic-structure calculations'for Al clusters (56)Schnepf.A.:Schnockel,H.Angew.Chem..Int.Ed.2002.41.3532. (57)Jones.R.O.Phrys.Rev.Lett.1991,67.224.Jones.R.O.J.Chem. and nanoparticles as well as to experimental bulk properties. Because pairwise additive potentials are inaccurate for real Phys.1993.99,1194 (58)Cheng,H.-P.;Berry,R.S.;Whetten,R.L.Phrys.Rev.B 1991,43, metals,including metal clusters and metal nanoparticles,these 10647. analytic potentials include many-body effects(i.e.,the potentials (59)Yi,J.Y.:Oh,D.J.U.;Bernholc,J.Phrys.Rev.Lett.1991,67,1248 are not pairwise additive).The many-body (nonpairwise,NP) (60)Rothlisberger.U.:Andreoni,W.:Giannozzi,P./Chem.Phys.1992. 96,1594. potentials named NP-A and NP-B are the most accurate (61)Elbayyariz,Z.;Erkoc,S.Phys.Status Solidi B:Basic Res.1992. potentials available for aluminum systems8 or any metal 170.103. nanoparticles.The potentials have been successfully applied to (62)Peslherbe,G.H.:Hase,W.L.J.Chemn.Phrys.1994,101.8535. aluminum systems to study the vapor-liquid coexistence of (63)Streitz.F.H.:Mintmire.J.W.P/rys.Rev.B 1994.50.11996. (64)Claire,P.d.S.:Peslherbe,G.H.:Hase,W.L.J.Plrys.Chem.1995 Al,energy landscape,48 thermodynamics properties,and 99.8147. reactions of Al particles.Here we use the NP-B potential to (65)Peslherbe.G.H.:Hase.W.L.J.Chemn.Phys.1996.104,9445.Claire. study melting,if we can call it that,of Al particles with n= P.de S.;Hase,W.L.J.Phys.Chem.1996,100,8190. (66)Peslherbe.G.H.:Hase.W.L.J.Chem.Phvs.1996.105.7432. 10-300,which have diameters in the range of about 0.8-2.5 (67)Peslherbe,G.H.:Hase,W.L.J.Phys.Chem.A 2000,104,10566. nm (68)Kumar,V.Phys.Reu.B1998,57.8827. For convenience we call the particles with a diameter less (69)Lloyd,L.D.:Johnston,R.L.Chem.Phrys.1998.236,107.Lloyd. L.D.;Johnston,R.L.J.Chem.Soc.,Dalton Trans.2000,3,307 than ~I nm clusters and those with diameters larger than ~l Lloyd,L.D.:Johnston,R.L.:Roberts,C.:Mortimer-Jones,T.V. nm nanoparticles.For aluminum particles,Al o has a diameter ChemPhysChem 2002,3,408. of about I nm.8 Thus,Al particles with ns 18 will be (70)Ahlrichs,R.;Elliott,S.D.Phys.Chem.Chem.Phrys.1999,1,13 called clusters.and those with n219 will be called nanopar- (71)Rao.B.K.:Jena,P.J.Chem.Phys.1999,111,1890. (72)Dolgounitcheva,O.;Zakrzewski,V.G.;Ortiz,J.V.J.Chem.Plys. ticles.In the present study,by analyzing simulation results,we 1999.111.10762. (73)Turner.G.W.:Johnston.R.L.:Wilson.N.T./Chem.P/rys.2000. (82)Liu,R.S.:Dong.K.J.:Tian,Z.A.;Liu,H.R.:Peng.P.:Yu.A.B. 112.4773. J.Phys.:Condens.Matter 2007,19.19613. (74)Geske,G.D.:Boldyrev,A.I;Li,X.:Wang.L.S.J.Chem.Phys (83)Akola,J.:Manninen.M.Phys.Rev.B 2001,63.193410. 2000.113,5130. (84)Boyiikata,M.:Guivenc,Z.B.Brazilian J.Phys.2006,36,720 (75)Zope,R.R.:Baruah,T.Phrys.Rev.B 2001,64,053202. (85)Puri,P.:Yang,V.J.Phys.Chem.C 2007,111,11776. (76)Deshpande.M.D.:Kanhere.D.G.:Vasiliev,I.:Martin,R.M.Phys (86)Poland,D.J.Chem.Plrys.2007,126,054507. Reu.B2003.035428」 (87)Yang,M.;Jackson,K.A.:Koehler,C.;Frauenheim,T.;Jellinek,J. (77)Joswig,J.-0.:Springborg,M.Phys.Rev.B 2003,68,085408. J.Chem.Pys.2006.124,24308. (78)Schultz,N.E.;Staszewska,G.:Staszewski,P.:Truhlar,D.G.J.Phys. (88)Jasper.A.W.;Staszewski,P.:Staszewski,G.;Schultz,N.E.;Truhlar. Chem.B2004.108.4850. D.G.J.Phys.Chem.B 2004.108.8996.Jasper.A.W.:Schultz. (79)Sebetci,A.;Guivenc,Z.B.Modeling Simul.Mater.Sci.Eng.2005. N.E.;Truhlar,D.G.J.Phys.Chem.B 2005,109,3915. 13,683. (89)Bhatt,D;Jasper,A.W.;Schultz,N.E.;Siepmann,J.I.;Truhlar (80)Peng.P.:Li.G.:Zheng,C.:Han,S.:Liu.R.Sci.China Ser.E 2006. D.G.J.Am.Chem.Soc.2006.128,4224.Bhatt,D.:Schultz.N.E.: 49,385. Jasper,A.W.;Siepmann,J.L;Truhlar,D.G.J.Phys.Chem.B 2006 (81)Li,Z.H.:Bhatt,D.:Schultz,N.E.;Siepmann,J.L:Truhlar,D.G. 110.26135. J.Phys.Chem.C2007.111.16227. (90)Li.Z.H.;Truhlar.D.G.J.Phys.Chem.C 2008,112.11109 12700J.AM.CHEM.S0C.■VOL.130,NO.38.2008
nanoparticles are often a mixture of many isomers with similar energies equilibrating with each other,6,18,22,23,29,37,45-48 their melting transitions have the same ambiguity. The distinction between clusters and nanoparticles is not strict, and we use the generic name particles to refer to both of them. Aluminum particles have been of great experimental36,42-44,49-56 and theoretical14,19,27,29,33,34,39,40,43,57-86 interest for decades. Their melting has recently been the subject of extensive experimental36,42-44 and theoretical study.14,19,27,33,34,40,82-86 Jarrold et al. used multicollision-induced dissociation to measure the heat capacities of Aln cationic clusters with n ) 16-48,44 31-38,43 49-63,36 and 63-83.42 They found that for some clusters the heat capacity curve has a well-defined sharp peak, while for others the heat capacity curve is relatively flat and featureless. Taking the temperature Tp at which C has a maximum as the melting temperature, they found that the melting temperature depends greatly on particle size, and even a change in size by a single atom can make huge differences. Monte Carlo34,39 (MC) and molecular dynamics14,19,27,33,40,83,84 (MD) simulations confirmed the experimental findings. However, Tp is not enough to characterize a melting transition since the solid and liquid states in finite systems have not been well defined. Moreover, for those particles with featureless and flat heat capacity curves, Tp has large uncertainties and should be treated with caution.44 For the particle sizes studied here, most atoms need to be classified as surface atoms rather than as interior atoms with bulk properties characteristic of a macroscopic particle. Except for a few small clusters, Al13, 14 Al13-, 83 and Al14, 83 available simulations of Al cluster melting all use empirical analytical potential functions, but it is not possible to accurately parametrize empirical potentials in the cluster and nanoparticle regime due to a lack of experimental data for systems with a significant fraction of atoms in nonbulk (e.g., surface) positions.24,87 Recently, economical and accurate analytic potentials for aluminum systems have been developed by fitting to results of well-validated electronic-structure calculations78 for Aln clusters and nanoparticles as well as to experimental bulk properties.88 Because pairwise additive potentials are inaccurate for real metals, including metal clusters and metal nanoparticles, these analytic potentials include many-body effects (i.e., the potentials are not pairwise additive). The many-body (nonpairwise, NP) potentials named NP-A and NP-B are the most accurate potentials available for aluminum systems88 or any metal nanoparticles. The potentials have been successfully applied to aluminum systems to study the vapor-liquid coexistence of Al,89 energy landscape,48 thermodynamics properties,81 and reactions90 of Aln particles. Here we use the NP-B potential to study melting, if we can call it that, of Aln particles with n ) 10-300, which have diameters in the range of about 0.8-2.5 nm. For convenience we call the particles with a diameter less than ∼1 nm clusters and those with diameters larger than ∼1 nm nanoparticles. For aluminum particles, Al19 has a diameter of about 1 nm.88,91 Thus, Aln particles with n e 18 will be called clusters, and those with n g 19 will be called nanoparticles. In the present study, by analyzing simulation results, we (45) Doye, J. P. K.; Calvo, F. Phys. ReV. Lett. 2001, 86, 3570. Doye, J. P. K.; Calvo, F. J. Chem. Phys. 2003, 119, 12680. (46) Wang, G. M.; Blaisten-Barojas, E.; Roitberg, A. E. J. Chem. Phys. 2001, 115, 3640. (47) Baletto, F.; Rapallo, A.; Rossi, G.; Ferrando, R. Phys. ReV. B 2004, 69, 235421. Rossi, G.; Rapallo, A.; Mottet, C.; Fortunelli, A.; Baletto, F.; Ferrando, R. Phys. ReV. Lett. 2004, 93, 105503. (48) Li, Z. H.; Jasper, A. W.; Truhlar, D. G. J. Am. Chem. Soc. 2007, 129, 14899. (49) de Heer, W. A.; Milani, P.; Chatelain, A. Phys. ReV. Lett. 1989, 63, 2834. (50) Lerme´, J.; Pellarin, M.; Vialle, J. L.; Baguenard, B.; Broyer, M. Phys. ReV. Lett. 1992, 68, 2818. Baguenard, B.; Pellarin, M.; Lerme´, J.; Vialle, J. L.; Broyer, M. J. Chem. Phys. 1994, 100, 754. (51) Martin, T. P.; Na¨her, U.; Schaber, H. Chem. Phys. Lett. 1992, 199, 470. (52) Jarrold, M. F.; Bower, J. E. J. Phys. Chem. 1993, 97, 1746. Jarrold, M. F.; Bower, J. E. J. Chem. Phys. 1993, 98, 2399. Jarrold, M. F. J. Phys. Chem. 1995, 99, 11. (53) Cha, C. Y.; Gantefo¨r, G.; Eberhardt, W. J. Chem. Phys. 1994, 100, 995. Gantefo¨r, G.; Eberhardt, W. Chem. Phys. Lett. 1994, 217, 600. (54) Li, X.; Wu, H.; Wang, X. B.; Wang, L. S. Phys. ReV. Lett. 1998, 81, 1909. (55) Akola, J.; Manninen, M.; Hakkinen, H.; Landman, U.; Li, X.; Wang, L. S. Phys. ReV. B 1999, 60, 11297. Akola, J.; Manninen, M.; Hakkinen, H.; Landman, U.; Li, X.; Wang, L. S. Phys. ReV. B 2000, 62, 13216. Kuznetsov, A. E.; Boldyrev, A. I.; Zhai, H. J.; Li, X.; Wang, L. S. J. Am. Chem. Soc. 2002, 124, 111791. (56) Schnepf, A.; Schno¨ckel, H. Angew. Chem., Int. Ed. 2002, 41, 3532. (57) Jones, R. O. Phys. ReV. Lett. 1991, 67, 224. Jones, R. O. J. Chem. Phys. 1993, 99, 1194. (58) Cheng, H.-P.; Berry, R. S.; Whetten, R. L. Phys. ReV. B 1991, 43, 10647. (59) Yi, J. Y.; Oh, D. J. U.; Bernholc, J. Phys. ReV. Lett. 1991, 67, 1248. (60) Ro¨thlisberger, U.; Andreoni, W.; Giannozzi, P. J. Chem. Phys. 1992, 96, 1594. (61) Elbayyariz, Z.; Erkoc, S. Phys. Status Solidi B: Basic Res. 1992, 170, 103. (62) Peslherbe, G. H.; Hase, W. L. J. Chem. Phys. 1994, 101, 8535. (63) Streitz, F. H.; Mintmire, J. W. Phys. ReV. B 1994, 50, 11996. (64) Claire, P. d. S.; Peslherbe, G. H.; Hase, W. L. J. Phys. Chem. 1995, 99, 8147. (65) Peslherbe, G. H.; Hase, W. L. J. Chem. Phys. 1996, 104, 9445. Claire, P. de S.; Hase, W. L. J. Phys. Chem. 1996, 100, 8190. (66) Peslherbe, G. H.; Hase, W. L. J. Chem. Phys. 1996, 105, 7432. (67) Peslherbe, G. H.; Hase, W. L. J. Phys. Chem. A 2000, 104, 10566. (68) Kumar, V. Phys. ReV. B 1998, 57, 8827. (69) Lloyd, L. D.; Johnston, R. L. Chem. Phys. 1998, 236, 107. Lloyd, L. D.; Johnston, R. L. J. Chem. Soc., Dalton Trans. 2000, 3, 307. Lloyd, L. D.; Johnston, R. L.; Roberts, C.; Mortimer-Jones, T. V. ChemPhysChem 2002, 3, 408. (70) Ahlrichs, R.; Elliott, S. D. Phys. Chem. Chem. Phys. 1999, 1, 13. (71) Rao, B. K.; Jena, P. J. Chem. Phys. 1999, 111, 1890. (72) Dolgounitcheva, O.; Zakrzewski, V. G.; Ortiz, J. V. J. Chem. Phys. 1999, 111, 10762. (73) Turner, G. W.; Johnston, R. L.; Wilson, N. T. J. Chem. Phys. 2000, 112, 4773. (74) Geske, G. D.; Boldyrev, A. I.; Li, X.; Wang, L. S. J. Chem. Phys. 2000, 113, 5130. (75) Zope, R. R.; Baruah, T. Phys. ReV. B 2001, 64, 053202. (76) Deshpande, M. D.; Kanhere, D. G.; Vasiliev, I.; Martin, R. M. Phys. ReV. B 2003, 035428. (77) Joswig, J.-O.; Springborg, M. Phys. ReV. B 2003, 68, 085408. (78) Schultz, N. E.; Staszewska, G.; Staszewski, P.; Truhlar, D. G. J. Phys. Chem. B 2004, 108, 4850. (79) Sebetci, A.; Gu¨venc¸, Z. B. Modeling Simul. Mater. Sci. Eng. 2005, 13, 683. (80) Peng, P.; Li, G.; Zheng, C.; Han, S.; Liu, R. Sci. China Ser. E 2006, 49, 385. (81) Li, Z. H.; Bhatt, D.; Schultz, N. E.; Siepmann, J. I.; Truhlar, D. G. J. Phys. Chem. C 2007, 111, 16227. (82) Liu, R. S.; Dong, K. J.; Tian, Z. A.; Liu, H. R.; Peng, P.; Yu, A. B. J. Phys.: Condens. Matter 2007, 19, 19613. (83) Akola, J.; Manninen, M. Phys. ReV. B 2001, 63, 193410. (84) Bo¨yu¨kata, M.; Gu¨venc¸, Z. B. Brazilian J. Phys. 2006, 36, 720. (85) Puri, P.; Yang, V. J. Phys. Chem. C 2007, 111, 11776. (86) Poland, D. J. Chem. Phys. 2007, 126, 054507. (87) Yang, M.; Jackson, K. A.; Koehler, C.; Frauenheim, T.; Jellinek, J. J. Chem. Phys. 2006, 124, 24308. (88) Jasper, A. W.; Staszewski, P.; Staszewski, G.; Schultz, N. E.; Truhlar, D. G. J. Phys. Chem. B 2004, 108, 8996. Jasper, A. W.; Schultz, N. E.; Truhlar, D. G. J. Phys. Chem. B 2005, 109, 3915. (89) Bhatt, D; Jasper, A. W.; Schultz, N. E.; Siepmann, J. I.; Truhlar, D. G. J. Am. Chem. Soc. 2006, 128, 4224. Bhatt, D.; Schultz, N. E.; Jasper, A. W.; Siepmann, J. I.; Truhlar, D. G. J. Phys. Chem. B 2006, 110, 26135. (90) Li, Z. H.; Truhlar, D. G. J. Phys. Chem. C 2008, 112, 11109. 12700 J. AM. CHEM. SOC. 9 VOL. 130, NO. 38, 2008 ARTICLES Li and Truhlar
Nanosolids,Slushes,and Nanoliquids ARTICLES will show that it may not be appropriate to call the structural agree well with each other (see Figure S-1 in the Supporting transition or isomerization process of clusters a melting transition Information),and hence,we can use whichever is more convenient while nanoparticles do have a melting transition.(It is only an (we found it is more convenient to use the right-hand side).In accident,but a convenient one,that the border between melting presenting our results we convert the heat capacity to a unitless behavior and no-melting behavior occurs so close to the rather specific heat capacity c defined by arbitrary border we established between nanoparticles and C clusters.)The present investigations indicate that Al nanopar- C= -(3n-3)kg (5) ticles have a wide temperature window of coexistence of solid and liquid states;this coexistence regime is called the slush where the denominator results from the fact that only overall state.12 Definitions of the solid,slush,and liquid states of translational motion was removed in the MD simulations.In the rest of the article we will simply refer to c as the heat capacity,but such particles will be proposed. we should keep in mind that the absolute heat capacity is actually C 2.Simulation Methodology (3)Average Distance to the Center of Mass(RcoM).RcoM is a property to characterize the size of a particle Simulations were run for Al with 10 s ns 300.For n 10-130.all MD simulations were started in the vicinity of the global energy minimum (GM)structures with random initial R.CoM= (6) coordinates and momenta distributed according to the classical phase space distribution of separable harmonic oscillators.64 For n= where r;is the vector position of an atom i and rcoM is the vector 10-65,GM structures obtained previously4s have been used.For position of the center of mass. n=70-130,the same strategy as used in ref 48 was used to locate (4)Radius of Gyration (R).R is another property that can be GM structures.The caloric curve was then studied by heating.For used to characterize the size of a particle larger particles,a search for the GM structure is too expensive: instead,the MD trajectory was started at high temperature with R spherical clusters with atomic coordinates randomly generated in n Ir-rcoMl (7) a sphere with a radius of 16,16,and 19 A for n=177.200,and 300,respectively,and the process simulated is the cooling process. (5)Volume (V).For a spherical object with evenly distributed For each heating simulation the starting temperature is 200 K with mass,it is easy to show that the radius of the sphere has the an increment of 20 K and the ending temperature is 1700 K,while following relationship with the principal moment of inertia ( for cooling simulation the same procedure is reversed. R=V5/2V1/M (8) To determine the local minima that the trajectory visited during the simulation,intermediate configurations were quenched at where M is total mass of the particle.Since a particle need not be random;on average,10%were quenched.Geometries of the spherical,we consider the three principal moments of inertia. quenched structures were optimized. Corresponding to these,there are three radii,R(i=1,2.3).With Details of solving the equations of motion,thermostatting,the these three radii,the volume of a particle can be estimated as27 heating and cooling programs,and optimization are provided in the Supporting Information V -37RR:Rs (9) Several properties have been investigated. (1)Berry Parameter.The Berry parameter.4 is the relative The quantity V was calculated at each step of the molecular root-mean-square fluctuation in the interatomic separation;it is an dynamics simulation and averaged. extension of the original Lindemann parameters used for macro- (6)Coefficient of Thermal Expansion (B) scopic systems.The Berry parameter is calculated by A=器 (10) 4=2∑ki>-<r>2 nm-1) (3) In the current study,the temperature derivatives of V and other <rip properties were obtained by first fitting them with cubic spline functions and then differentiating the fitted spline functions. where r is the distance between two atoms i and j. (7)Isothermal Compressibility (K).99 (2)Heat Capacity.For a macroscopic system the heat capacity at constant volume (C=dEro/dT,where Erot is the total energy of 1(W2)-02 the system)is related to the fluctuation in energy by6 K-kpT (V) (11) where V is calculated by eq 11. C=. Et）-(Ero (4) kg72 3.Results (See page S-3 in the Supporting Information for a discussion of the derivation.)Although the derivation of eq 4 is not directly 3.1.Berry Parameter.Although the Berry parameter (AB) applicable to finite systems,we found that the two sides of eq4 has been widely used to study the melting of clusters and nanoparticlesour simulations show that it is more sensitive to geometrical transitions than other properties (91)Schultz,N.E.:Jasper,A.W.:Bhatt,D.:Siepmann,J.I.;Truhlar. D.G.In Multiscale Simulation Methods for Nanomaterials,Ross R.B..Mohanty,S.,Eds.:Wiley-VCH:Hoboken,NJ,2008;p 169. (96)Hill,T.L.Statistical Mechanics:Principles and Selected Applications: (92)Tanner,G.M.:Bhattacharya,A.;Nayak.S.K;Mahanti,S.D.Plrys. McGraw-Hill:New York,1956;pp 100-101.Rice.O.K.Staristical Reu.E1997.55.322. Mechanics Thermodynamics and Kinetics:W.H.Freeman:San (93)Kaelberer,J.:Etters,R.D J.Chem.Plrys.1977,66,3233.Etters, Francisco,1967:pp 92-93.. R.D.:Kaelberer,J.J.Chem.Phys.1977,66,5112. (97)Ding,F.:Rosen,A.:Bolton,K.Phys.Rev.B 2004.70,75416 (94)Berry,R.S.:Beck.T.L.:Davis.H.L.:Jellinek,J.Adv.Chem.Phrys. (98)Wang,L.:Zhang,Y.:Bian,X.:Chen,Y.Phys.Lett.A 2003.310. 1988.70B.75.Zhou,Y.:Karplus,M.;Ball,K.D.:Berry,R.S. 197. J.Chem.Phs.2002,I16,2323. (99)Pathria,R.K.Staristical Mechanics,2nd ed.:Elsevier:Singapore, (95)Lindemann,F.A.Phys.Z 1910,//609. 1996,p454. J.AM.CHEM.SOC.VOL 130,NO.38,2008 12701
will show that it may not be appropriate to call the structural transition or isomerization process of clusters a melting transition while nanoparticles do have a melting transition. (It is only an accident, but a convenient one, that the border between melting behavior and no-melting behavior occurs so close to the rather arbitrary border we established between nanoparticles and clusters.) The present investigations indicate that Aln nanoparticles have a wide temperature window of coexistence of solid and liquid states; this coexistence regime is called the slush state.9,11,92 Definitions of the solid, slush, and liquid states of such particles will be proposed. 2. Simulation Methodology Simulations were run for Aln with 10 e n e 300. For n ) 10-130, all MD simulations were started in the vicinity of the global energy minimum (GM) structures with random initial coordinates and momenta distributed according to the classical phase space distribution of separable harmonic oscillators.64 For n ) 10-65, GM structures obtained previously48 have been used. For n ) 70-130, the same strategy as used in ref 48 was used to locate GM structures. The caloric curve was then studied by heating. For larger particles, a search for the GM structure is too expensive; instead, the MD trajectory was started at high temperature with spherical clusters with atomic coordinates randomly generated in a sphere with a radius of 16, 16, and 19 Å for n ) 177, 200, and 300, respectively, and the process simulated is the cooling process. For each heating simulation the starting temperature is 200 K with an increment of 20 K and the ending temperature is 1700 K, while for cooling simulation the same procedure is reversed. To determine the local minima that the trajectory visited during the simulation, intermediate configurations were quenched at random; on average, 10% were quenched. Geometries of the quenched structures were optimized. Details of solving the equations of motion, thermostatting, the heating and cooling programs, and optimization are provided in the Supporting Information. Several properties have been investigated. (1) Berry Parameter. The Berry parameter93,94 is the relative root-mean-square fluctuation in the interatomic separation; it is an extension of the original Lindemann parameter95 used for macroscopic systems. The Berry parameter is calculated by ∆B ) 2 n(n - 1)∑i<j √<rij 2 > - < rij>2 <rij> (3) where rij is the distance between two atoms i and j. (2) Heat Capacity. For a macroscopic system the heat capacity at constant volume (C ≡ dETot/dT, where ETot is the total energy of the system) is related to the fluctuation in energy by96 C ) 〈ETot 2 〉 -〈ETot〉2 kBT2 (4) (See page S-3 in the Supporting Information for a discussion of the derivation.) Although the derivation of eq 4 is not directly applicable to finite systems, we found that the two sides of eq 4 agree well with each other (see Figure S-1 in the Supporting Information), and hence, we can use whichever is more convenient (we found it is more convenient to use the right-hand side). In presenting our results we convert the heat capacity to a unitless specific heat capacity c defined by c ) C (3n - 3)kB (5) where the denominator results from the fact that only overall translational motion was removed in the MD simulations. In the rest of the article we will simply refer to c as the heat capacity, but we should keep in mind that the absolute heat capacity is actually C. (3) AVerage Distance to the Center of Mass (RCoM).97 RCoM is a property to characterize the size of a particle RCoM ) 1 n∑i |ri - rCoM| (6) where ri is the vector position of an atom i and rCoM is the vector position of the center of mass. (4) Radius of Gyration (Rg).98 Rg is another property that can be used to characterize the size of a particle Rg )1 n∑i |ri - rCoM| 2 (7) (5) Volume (V). For a spherical object with evenly distributed mass, it is easy to show that the radius of the sphere has the following relationship with the principal moment of inertia (I) R ) √5⁄2√I ⁄ M (8) where M is total mass of the particle. Since a particle need not be spherical, we consider the three principal moments of inertia. Corresponding to these, there are three radii, Ri (i ) 1, 2, 3). With these three radii, the volume of a particle can be estimated as27 V ) 4 3 πR1R2R3 (9) The quantity V was calculated at each step of the molecular dynamics simulation and averaged. (6) Coefficient of Thermal Expansion (). ) 1 V dV dT (10) In the current study, the temperature derivatives of V and other properties were obtained by first fitting them with cubic spline functions and then differentiating the fitted spline functions. (7) Isothermal Compressibility (κ).99 κ ) 1 kBT 〈V2 〉 -〈V〉2 〈V〉 (11) where V is calculated by eq 11. 3. Results 3.1. Berry Parameter. Although the Berry parameter (∆B) has been widely used to study the melting of clusters and nanoparticles,6,9,14,19,27,33,34,84,93,94 our simulations show that it is more sensitive to geometrical transitions than other properties (91) Schultz, N. E.; Jasper, A. W.; Bhatt, D.; Siepmann, J. I.; Truhlar, D. G. In Multiscale Simulation Methods for Nanomaterials; Ross, R. B., Mohanty, S., Eds.; Wiley-VCH: Hoboken, NJ, 2008; p 169. (92) Tanner, G. M.; Bhattacharya, A.; Nayak, S. K.; Mahanti, S. D. Phys. ReV. E 1997, 55, 322. (93) Kaelberer, J.; Etters, R. D J. Chem. Phys. 1977, 66, 3233. Etters, R. D.; Kaelberer, J. J. Chem. Phys. 1977, 66, 5112. (94) Berry, R. S.; Beck, T. L.; Davis, H. L.; Jellinek, J. AdV. Chem. Phys. 1988, 70B, 75. Zhou, Y.; Karplus, M.; Ball, K. D.; Berry, R. S. J. Chem. Phys. 2002, 116, 2323. (95) Lindemann, F. A. Phys. Z 1910, 11, 609. (96) Hill, T. L. Statistical Mechanics: Principles and Selected Applications; McGraw-Hill: New York, 1956; pp 100-101. Rice, O. K. Statistical Mechanics Thermodynamics and Kinetics; W. H. Freeman: San Francisco, 1967; pp 92-93.. (97) Ding, F.; Rosen, A.; Bolton, K. Phys. ReV. B 2004, 70, 75416. (98) Wang, L.; Zhang, Y.; Bian, X.; Chen, Y. Phys. Lett. A 2003, 310, 197. (99) Pathria, R. K. Statistical Mechanics, 2nd ed.; Elsevier: Singapore, 1996; p 454. J. AM. CHEM. SOC. 9 VOL. 130, NO. 38, 2008 12701 Nanosolids, Slushes, and Nanoliquids ARTICLES
ARTICLES Li and Truhlar 130 a we found that consistent results can be obtained with these other 30 128 128 A13840K properties even in runs in which the Berry parameter is not yet converged in individual simulations.and we shall use these other 124 -Al13900K Al66550K 122 properties in the rest of the paper.The general success of this 15 N1340K Al55 600 K N13900K 120 approach may result from high-energy states contributing less 155550K 1.18 to other properties than to the Berry parameter.an interpretation .05 Al56600N 1.18 which is consistent with (but not proved by)our simulation 0.00 1.14 le+7 20+7 1e+7 2e+7 results.The result has general implications for simulations in Time (fs) Time(fs) that often one can achieve a similar understanding of a system 846 442 from one or another observable,but one of these observables 844 842 440 may converge more quickly than the other. 840 3.2.Heat Capacity.Three typical kinds of c curves are plotted .11840K 438 Al13 840 K A13900 -B900K in Figure 4.Curves for all other particles are available in the 172 A185550K -A55550K Supporting Information. Al5s 600K c2.58 Al55600K 170 The first type of c curve has a well-defined peak such that c 168 2.56 increases almost linearly with temperature before the peak and 0 5+6 1e+7 20+7 2+ 50+6 1e+7 20+7 20+7 after the peak c decreases almost linearly with temperature Time (fs) Time(fs) Careful examination of the plots shows that Als1,Al7o,Also. Al90,Al100,Al110,Al120.Al130.Al77,Al200,and Al300 exhibit a Figure 2.Convergence behavior of various properties of Al3 and Alss in quasiplateau in the peak region.For this type of curve,the peak the transition region:(a)Berry parameter,(b)unitless specific heat capacity c,(c)volume,(d)average distance to the center of mass as a function of (plateau)becomes narrower and higher as the particle size simulation time. increases,which demonstrates a trend toward bulk behavior. Plots for Al120.Al130.Al77,Al200.and Al300 (see Supporting that may be used to characterize the system.As just one example Information)are all similar to that for Al30.The trend toward of our findings on this subject,Figure 2a shows that the Berry bulk behavior has been examined before in model systems,102 parameter converges very slowly with simulation time.In fact and the present results are consistent with this previous work. AB obtained at I and 5 ns may differ by more than a factor of All plots for the 11 particles listed above show a bump at about 2.For Alss at 550 K.after 20 ns.AB still shows no sign of 900 K,probably indicating a state change. convergence.However,other properties show much better The second type of c curve,shown in Figure 4b,features a convergence behavior (see Figure 2b-d).In Figure 3 several big bump in the curve rather than a peak.For this type,c properties are plotted as functions of temperature for Alss increases gradually before the bump,where it reaches a obtained with two different simulation times.The plots indicate maximum value and then decreases almost linearly at high that for the Berry parameter different simulation times may give temperatures.The bump does not become narrower as particle very different results(Figure 3a)unless very long simulations size increases.The third type of caloric curve,shown in Figure are run.For the other properties,plots of the property vs T 4c,can be viewed as a superimposition of one or more small obtained with different simulation times almost overlap with peaks before the maximum of the second type of curve. each other.Moreover,the Berry parameter plots indicate that For particles with n<18,the maximum of the peak in c is the most dramatic structural change occurs at about 600 and either so high that the decrease at high temperatures is a part of 550 K for the short time and long time simulation,respectively. the peak tail or so low (for Aljo,Alu,and Alis)that the curve On the other hand,the other plots indicate that the most dramatic goes flat at high temperatures.Putting c plots (Figure 4d)of changes in all the other properties of the nanoparticles occur at Alo-Alis(left)and those of Al9-Al300(right)on two separate about 650 K.Since the diffusion constant is related to the graphs shows that they can be classified into two different average square displacement of an atom groups;for the second group,heat capacities of most particles decrease almost linearly with temperature after 900 K. D-im r()-r0)) (12) The temperature Tp at which c has a maximum is determined as the zero of dc/dT(=dC/dT).where dc/dT is calculated from the Berry parameter is greatly affected by the diffusion of spline fits.For those curves with multiple peaks,we choose individual atoms,which occurs slowly in the transition regime. the one most likely corresponding to a melting transition.For The jumps in the AB plots(Figure 2a)may be due to the jump example,for Al26,Al27,Al38,Al43,and Alss,shown in Figure of an atom to other positions.Beck et al.also noted that Ag 4c,the higher peak temperature is adopted.We find that Tp does can become quite large if any transitions occur between local show a strong dependence on particle size (Table I and Figure potential minima.Indeed,for some clusters where low-energy 5).For many small particles Tp is higher than the bulk melting minima are in equilibrium at low temperatures,48 AB is as large temperature03 of 933 K(dashed line in Figure 5).In agreement as 0.2 at a temperature as low as 200 K (Figure S-14 in the with the experimental findings for Alcations, 36.42-44 we find Supporting Information). that a change of particle size by just one single atom can make Therefore,we focus on other properties and found the heat a very large difference in Tp. capacity,radius,and volume to be particularly useful.For We are cautious about quantitatively comparing our results practical purposes,using the multiple-simulation,multiple- with experiment because the experiments are for Al cations equilibration protocol explained in the Supporting Information, while our analytical potential and simulation are for neutral (100)Einstein.A.Investigations on the Theory of the Brownian Movement: (102)Wales.D.J.;Doye,J.P.K.J.Chem.Phrys.1995.103,3061. Methuen:London,1926:p17.Allen,M.P.:Tildesley,D.J.Computer (103)Chase,M.W..Jr.NIST-JANAF Thermochemical Tables,4th ed.J. Simulation of Liguids;Oxford University Press:Oxford,1987:p 60. Phys.Chem.Ref.Data,Monograph 9:American Institute of Physics: (101)Vollmayr-Lee,K.J.Chem.Phys.2004.121,4781. New York,1998. 12702J.AM.CHEM.S0C.■VOL.130,NO.38.2008
that may be used to characterize the system. As just one example of our findings on this subject, Figure 2a shows that the Berry parameter converges very slowly with simulation time. In fact, ∆B obtained at 1 and 5 ns may differ by more than a factor of 2. For Al55 at 550 K, after 20 ns, ∆B still shows no sign of convergence. However, other properties show much better convergence behavior (see Figure 2b-d). In Figure 3 several properties are plotted as functions of temperature for Al55 obtained with two different simulation times. The plots indicate that for the Berry parameter different simulation times may give very different results (Figure 3a) unless very long simulations are run. For the other properties, plots of the property vs T obtained with different simulation times almost overlap with each other. Moreover, the Berry parameter plots indicate that the most dramatic structural change occurs at about 600 and 550 K for the short time and long time simulation, respectively. On the other hand, the other plots indicate that the most dramatic changes in all the other properties of the nanoparticles occur at about 650 K. Since the diffusion constant is related to the average square displacement of an atom100 D ) lim tf∞ 1 6t 〈|ri (t) - ri (0)|2 〉 (12) the Berry parameter is greatly affected by the diffusion of individual atoms, which occurs slowly in the transition regime. The jumps in the ∆B plots (Figure 2a) may be due to the jump of an atom to other positions.101 Beck et al. also noted that ∆B can become quite large if any transitions occur between local potential minima.9 Indeed, for some clusters where low-energy minima are in equilibrium at low temperatures,48 ∆B is as large as 0.2 at a temperature as low as 200 K (Figure S-14 in the Supporting Information). Therefore, we focus on other properties and found the heat capacity, radius, and volume to be particularly useful. For practical purposes, using the multiple-simulation, multipleequilibration protocol explained in the Supporting Information, we found that consistent results can be obtained with these other properties even in runs in which the Berry parameter is not yet converged in individual simulations, and we shall use these other properties in the rest of the paper. The general success of this approach may result from high-energy states contributing less to other properties than to the Berry parameter, an interpretation which is consistent with (but not proved by) our simulation results. The result has general implications for simulations in that often one can achieve a similar understanding of a system from one or another observable, but one of these observables may converge more quickly than the other. 3.2. Heat Capacity. Three typical kinds of c curves are plotted in Figure 4. Curves for all other particles are available in the Supporting Information. The first type of c curve has a well-defined peak such that c increases almost linearly with temperature before the peak and after the peak c decreases almost linearly with temperature. Careful examination of the plots shows that Al51, Al70, Al80, Al90, Al100, Al110, Al120, Al130, Al177, Al200, and Al300 exhibit a quasiplateau in the peak region. For this type of curve, the peak (plateau) becomes narrower and higher as the particle size increases, which demonstrates a trend toward bulk behavior. Plots for Al120, Al130, Al177, Al200, and Al300 (see Supporting Information) are all similar to that for Al130. The trend toward bulk behavior has been examined before in model systems,102 and the present results are consistent with this previous work. All plots for the 11 particles listed above show a bump at about 900 K, probably indicating a state change. The second type of c curve, shown in Figure 4b, features a big bump in the curve rather than a peak. For this type, c increases gradually before the bump, where it reaches a maximum value and then decreases almost linearly at high temperatures. The bump does not become narrower as particle size increases. The third type of caloric curve, shown in Figure 4c, can be viewed as a superimposition of one or more small peaks before the maximum of the second type of curve. For particles with n e 18, the maximum of the peak in c is either so high that the decrease at high temperatures is a part of the peak tail or so low (for Al10, Al11, and Al18) that the curve goes flat at high temperatures. Putting c plots (Figure 4d) of Al10-Al18 (left) and those of Al19-Al300 (right) on two separate graphs shows that they can be classified into two different groups; for the second group, heat capacities of most particles decrease almost linearly with temperature after 900 K. The temperature Tp at which c has a maximum is determined as the zero of dc/dT () dC/dT), where dc/dT is calculated from spline fits. For those curves with multiple peaks, we choose the one most likely corresponding to a melting transition. For example, for Al26, Al27, Al38, Al43, and Al58, shown in Figure 4c, the higher peak temperature is adopted. We find that Tp does show a strong dependence on particle size (Table 1 and Figure 5). For many small particles Tp is higher than the bulk melting temperature103 of 933 K (dashed line in Figure 5). In agreement with the experimental findings for Aln + cations,36,42-44 we find that a change of particle size by just one single atom can make a very large difference in Tp. We are cautious about quantitatively comparing our results with experiment because the experiments are for Aln + cations while our analytical potential and simulation are for neutral (100) Einstein, A. InVestigations on the Theory of the Brownian MoVement; Methuen: London, 1926; p 17. Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids; Oxford University Press: Oxford, 1987; p 60. (101) Vollmayr-Lee, K. J. Chem. Phys. 2004, 121, 4781. (102) Wales, D. J.; Doye, J. P. K. J. Chem. Phys. 1995, 103, 3061. (103) Chase, M. W., Jr. NIST-JANAF Thermochemical Tables, 4th ed. J. Phys. Chem. Ref. Data, Monograph 9; American Institute of Physics: New York, 1998. Figure 2. Convergence behavior of various properties of Al13 and Al55 in the transition region: (a) Berry parameter, (b) unitless specific heat capacity c, (c) volume, (d) average distance to the center of mass as a function of simulation time. 12702 J. AM. CHEM. SOC. 9 VOL. 130, NO. 38, 2008 ARTICLES Li and Truhlar