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(n19,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. 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(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. 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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. 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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). 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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>-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 - 2 (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
Nanosolids,Slushes,and Nanoliquids ARTICLES (a)Berry Parameter vs T(K) (b)c vs T (K) 0.40 2.8 0.30 0.20 1.8 0.10 13 0.00 p 300400500600700800 900 300400500600700800900 ◆20ns1traj1ns10traj 20 ns 1 traj-1 ns 10 traj (c)Volume(A)vs T(K) (d)RcoM (A)vs T (K) 925 4.55 905 4.50 885 865 4.45 845 4.40 825 4.35 300400500600700800 900 300400500600700800 900 ◆20ns1traj1ns10tra 20 ns 1 traj -1 ns 10 traj Figure 3.Convergence behavior of various properties of Alss:(a)Berry parameter,(b)unitless specific heat capacity c.(c)volume,(d)average distance to the center of mass as a function of temperature.In parts c and d the results for the two lengths of simulation agree so well that one set of results is hidden behind the other. (a) 2.8 (b) 1.5 2.6 2.4 1.4 2.2 2.0 Alss 1.3 01.8 A 01.2 1.6 1.4 1.1 1.2 1.0 1.0 8 2004006008001000120014001600 Alet T(K) 2004006008001000120014001600 T(K) (c) (d2.o A10 Al-Al30 12 1.8 15 16 A17 14 1 73 7.4 01.2 13 1.1 1.0 1.0 9 2004006008001000120014001600 3006009001200150030060090012001500 T(K) T(0 T(K) Figure 4.Unitless specific heat capacity c curves:(a)one sharp peak.(b)no sharp peak before reaching the linear region at high temperatures,(c)spikes before reaching the linear region at high temperatures,(d)for all particles,left (Alo-Alis)and right(Al19-Al3oo). particles.All simulations with analytical potentials includ- The electronic excitation contribution to the heat capacity can ing the current work failed to predict the more than 100 K drop be estimated using the electronic energy levels for the global of Tp from Alss to Als6.36 If jellium or electronic closed-shell minimum structures calculated in a previous study.81 In the ltaee n eue Supporting Information,we presented results for electronic debatable issue, contributions for several particles;electronic contributions are with the same size can be very different in physical and chemical not negligible and thus may change the position and shape of properties,although such differences should typically become the peak. smaller as particle size increases.ss Another reason for caution 3.3.Average Distance to the Center of Mass,Radius of is that electronic excitation contributions to the heat capacity Gyration,and Volume.Since the solid state is generally more have been omitted in the present simulations since the potential compact than the liquid state,the transition from the solid state is optimized for the ground electronic states of the particles. to the liquid state may lead to significant changes in particle J.AM.CHEM.SOC.VOL 130,NO.38,2008 12703
particles. All simulations34,39,84 with analytical potentials including the current work failed to predict the more than 100 K drop of Tp from Al55 to Al56. 36 If jellium or electronic closed-shell models can be applied to aluminum clusters, which is still a debatable issue,49,50,55,68,71,76 cationic and neutral Aln particles with the same size can be very different in physical and chemical properties, although such differences should typically become smaller as particle size increases.55 Another reason for caution is that electronic excitation contributions to the heat capacity have been omitted in the present simulations since the potential is optimized for the ground electronic states of the particles. The electronic excitation contribution to the heat capacity can be estimated using the electronic energy levels for the global minimum structures calculated in a previous study.81 In the Supporting Information, we presented results for electronic contributions for several particles; electronic contributions are not negligible and thus may change the position and shape of the peak. 3.3. Average Distance to the Center of Mass, Radius of Gyration, and Volume. Since the solid state is generally more compact than the liquid state, the transition from the solid state to the liquid state may lead to significant changes in particle Figure 3. Convergence behavior of various properties of Al55: (a) Berry parameter, (b) unitless specific heat capacity c, (c) volume, (d) average distance to the center of mass as a function of temperature. In parts c and d the results for the two lengths of simulation agree so well that one set of results is hidden behind the other. Figure 4. Unitless specific heat capacity c curves: (a) one sharp peak, (b) no sharp peak before reaching the linear region at high temperatures, (c) spikes before reaching the linear region at high temperatures, (d) for all particles, left (Al10-Al18) and right (Al19-Al300). J. AM. CHEM. SOC. 9 VOL. 130, NO. 38, 2008 12703 Nanosolids, Slushes, and Nanoliquids ARTICLES
ARTICLES Li and Truhlar Table 1.Freezing Temperature (Ti),Melting Temperature (Tm). 1600 Position of Sharp Peaks/Deep Valleys(Tp)in Unitiess Specific Heat T。ofc Capacity(c),Coefficient of Thermal Expansion(B),Natural Logarithm 1400 of Compressibility(In )and Temperature(TPso)at Which P(O)=50 T00 1200 Bulk melting temperature 1000 B 800 10 300717 NA NA 826 1240 320 708 NA 548 907 1300 600 560 1071 1225 840 NA NA 400 1 700 1188 1273 1026 NA NA 14* NA (347) 1353 1187 A 1480 200 NA 1394 A 1186 NA 16* 1446 220 1349 NA 0 17* 1438 NA 0 946 NA 50 100 150 200300 260 624 (311,(384),(456) NA 687 1580 n 600 894 937 26 1062 1420 20* 240 471 1025 542 1360 Figure 5.Peak position(Tp)of heat capacity and temperature at which 740 1127 A NA 1280 1180 P(0)=50(TPo-50)as a function of cluster size.The long dashed line is 0 1119 380 1220 759 66 785 the bulk melting temperature. 1100 520 819 NA 755 905 1100 3.1 936 (427) 543 851 1000 26 613 978 570 960 3.0 (a) 676 72889 617 711,971 (464) 492 544 1060 398 38 9 940 860 0 430 445 940 22 令 (416 (623) 31* (382) 365 1040 2 32* 0, (355) 599 6 800 864 NA 696 2 940 25 28 8 46 38 8 4.9 (b) * (305 36 829 35 38 940 4.8 * 38 6 9* 4 % 460.797 5 8 743 377 368 920 41* W源M M% 8 4.5 824 65 434 840 833 840 (c) 860 5.1 45* 双 品 161030 462 边箱达热00 860 760 5.0 800 4.9 920 9 740 e 820 51 0O1 8 47 820 200 400 600 8001000120014001600 460 蒸 600 582 820 T(K) 2345657 8 0 8 920 920 Figure 6.Three typical radius of gyration(Rg)curves:(a)change gradually 480 667 678 680 645 960 with temperature,(b)obvious jumps,(c)obvious drops 663 1060 1第W1000 608 561 1130 721 300 305. 648 230 1160 radius.We find that Rg is slightly larger than RcoM,and both (382) 686 429 1030 show similar trends with temperature;here we will focus on 60 732 360 345 402 1020 Rg.Figure 6 shows three typical plots of Rg vs T(the rest are in 0 889 (474) 481 1060 the Supporting Information).The majority of the particles 823 (363 388 1060 300 842 (335 360 345 studied in the present work have the first type of Rg curve,shown 1040 64* 240 311 804 (318) 320 NA 1000 in Figure 6a,where Rg increases gradually with temperature. 868 (343) 400 360 100 For some particles,for example,Alj3 and Al19,the curve shows 480 565 460 458 443 940 a semilocalized change of slope,whereas it is almost featureless 80* NA 532 (299).(446 484 NA 900 90* for other particles.The second type of R curve,shown in Figure 100* 8 051 (365).551 (380).524388 940 (327).575 (360).539 d 920 6b,exhibits a clear jump;this is common for large particles, 110 34 572 566 580 501 900 especially for those particles with a well-defined sharp peak in 00 594 8 560 900 heat capacity(AlsI-Als6 and Al7o-Al300).For the third type, 130 46 619 604 900 177 400 652 660 640 d 900 shown in Figure 6c,R drops suddenly at a certain temperature 200 300 657 662 660 d 900 and then increases gradually with temperature(Al27,Al30,Al3s, 300 580 709 717 700 880 Al58,Al59.Al61-Al65). Liquid nanodroplets generally have larger volumes and larger Temperatures are in Kelvin,and particles in slush state at room coefficients of thermal expansion (B)than solid particles with temperature are marked with an asterisk.Tp is the temperature at which the given quantity has a maximum or a minimum (in parentheses).For the same number of atoms.The volume has a similar depen- those In k without sharp peaks,Tp determined is the peak/valley position of dence on temperature as Rg,and all the plots are presented in d In k/dT.Intermediate configurations were not quenched. the Supporting Information.However,it is worth noting that 12704J.AM.CHEM.S0C.■VOL.130,NO.38,2008
radius. We find that Rg is slightly larger than RCoM, and both show similar trends with temperature; here we will focus on Rg. Figure 6 shows three typical plots of Rg vs T (the rest are in the Supporting Information). The majority of the particles studied in the present work have the first type of Rg curve, shown in Figure 6a, where Rg increases gradually with temperature. For some particles, for example, Al13 and Al19, the curve shows a semilocalized change of slope, whereas it is almost featureless for other particles. The second type of Rg curve, shown in Figure 6b, exhibits a clear jump; this is common for large particles, especially for those particles with a well-defined sharp peak in heat capacity (Al51-Al56 and Al70-Al300). For the third type, shown in Figure 6c, Rg drops suddenly at a certain temperature and then increases gradually with temperature (Al27, Al30, Al35, Al58, Al59, Al61-Al65). Liquid nanodroplets generally have larger volumes and larger coefficients of thermal expansion () than solid particles with the same number of atoms. The volume has a similar dependence on temperature as Rg, and all the plots are presented in the Supporting Information. However, it is worth noting that Table 1. Freezing Temperature (Tf), Melting Temperature (Tm), Position of Sharp Peaks/Deep Valleys (Tp) in Unitless Specific Heat Capacity (c), Coefficient of Thermal Expansion (), Natural Logarithm of Compressibility (ln κ), and Temperature (TP50) at Which P(0) ) 50a Tp b n Tf c ln κ c TP50 Tm 10 300 717 NA NA 826 1240 11 320 708 NA 548 907 1300 12 560 1071 1225 840 NA NA 13 700 1188 1273 1026 NA NA 14* NA (347), 1353 NA 1187 NA 1480 15* NA 1394 NA 1186 NA NA 16* NA 1446 NA 220 1349 NA 17* NA 1438 NA 260 946 NA 18* 260 624 (311), (384), (456) NA 687 1580 19 600 894 937 826 1062 1420 20* 240 471, 1025 NA 946 542 1360 21 740 1127 NA NA 1280 1180 22* 280 1119 326 317 380 1220 23 480 759 766 679 785 1100 24 520 819 NA 755 905 1100 25* 260 936 (427), 543 510, 851 543 1000 26 420 613, 978 570 519, 676 569 960 27 420 617, 711, 971 (464) 492 544 1060 28 400 750 NA 452 600 940 29 340 868 432 431 429 860 30 340 943 (416), (623) 430 445 940 31* NA 922 (382) 365 326 1040 32* NA 806 240, (355) 599 429 800 33* NA 864 NA 696 269 940 34* 260 878 346 507 364 960 35* 240 854 (305) 475 368 960 36* NA 829 325 NA 387 940 37* NA 834 248 320 270 940 38* 280 340, 460, 797 415 395 429 920 39* 280 743 377 368 390 920 40* NA 796 NA 405 367 860 41* 280 821 346 335 327 840 42 NA 824 284 434 261 840 43* 240 317, 833 (265) 302 322 840 44 300 835 NA 344 363 860 45* NA 789 NA NA 323 860 46* NA 615 (406) 564 323 760 47* NA 742 NA 486 407 800 48 340 510 541 462 503 920 49 320 580 507 435 389 740 50 380 723 460 447 470 820 51 360 547 517 540 506 820 52 460 597 599 600 582 820 53 440 643 654 660 644 920 54 460 666 674 700 668 920 55 480 667 678 680 645 960 56 480 663 676 680 664 1060 57* 240 608 NA 540 307 1130 58* NA 276, 721 300 305, 648 230 1160 59 300 773 (382) 686 429 1030 60 300 732 360 345 402 1020 61 400 730, 889 (474) 454 481 1060 62 300 823 (363) 310 388 1060 63 300 842 (335) 360 345 1040 64* 240 311, 804 (318) 320 NA 1000 65* 240 352, 868 (343) 400 360 1000 70* NA 420, 480, 565 460 458 443 940 80* NA 532 (299), (446) 484 NA 900 90* 240 340, 561 (365), 551 (380), 524 388 940 100* 240 340, 571 (327), 575 (360), 539 d 920 110 340 572 566 580 501 900 120 500 594 696 568 560 900 130 460 619 632 592 604 900 177 400 652 660 640 d 900 200 300 657 662 660 d 900 300 580 709 717 700 d 880 a Temperatures are in Kelvin, and particles in slush state at room temperature are marked with an asterisk. b Tp is the temperature at which the given quantity has a maximum or a minimum (in parentheses). c For those ln κ without sharp peaks, Tp determined is the peak/valley position of d ln κ/dT. d Intermediate configurations were not quenched. Figure 5. Peak position (Tp) of heat capacity and temperature at which P(0) ) 50 (TP(0))50) as a function of cluster size. The long dashed line is the bulk melting temperature. Figure 6. Three typical radius of gyration (Rg) curves: (a) change gradually with temperature, (b) obvious jumps, (c) obvious drops. 12704 J. AM. CHEM. SOC. 9 VOL. 130, NO. 38, 2008 ARTICLES Li and Truhlar
Nanosolids,Slushes,and Nanoliquids ARTICLES 5e4 -23.0 (a) 6 4e4 23.5 e -24.0 2e-4 ,N18 A14 -24.5 A21 Al36 A45 -25.0 (b) 23.0 (c) -23.5 A61 -24.0 AN120 A62 AN130 Al63 AN177 Al64 -24.5 AN200 A65 N300 (c) -25.0 300 600 900 12001500 300 600 9001200 1500 T(9 T(K) Figure 8.Typical natural logarithm of isothermal compressibility (In K) curves:(a)almost linear in the whole temperature range,(b)with obvious jumps,(c)with sharp peaks.(d)with two almost linear segments joined at about 1000 K. Alet 3.5.Isothermal Compressibility.Isothermal compressibility 200400 600 8001000120014001600 (K)is a function of pressure.The compressibilities calculated T(K) in the present study are the low-pressure limits because it is the free Al particles that are studied.Compressibilities as a Figure 7.Coefficient of thermal expansion (B)curves:(a-c)particles with function of temperature for each particle size are presented in three typical heat capacity curves plotted in Figure 4. the Supporting Information.Here we focus on the natural logarithm of compressibility.For Al1s.Al2,Al36.and Al4s,the In k vs T curve is almost linear in the whole simulation for some particles it seems that the most stable structure at low temperature range (Figure 8a).For some particles,the curve is temperatures,i.e.,the solid-like structure,is not the most smooth but has large curvature (Figure 8b).For most large compact structure,the volume or R.of the particle has a sudden particles the curve has a clear jump or sharp peak at a certain drop,and the coefficient of thermal expansion has a deep valley temperature(Figure 8c).For the particles shown in Figure 8c, (see below)at temperatures where the solid-like structure no before and after the jump or peak,In k is an almost linear longer dominates. function of temperature.The position of the jump or peak almost 3.4.Coefficient of Thermal Expansion.For particles with a overlaps with the peak position of heat capacity.For some well-defined sharp peak in heat capacity,the coefficient of particles,for example,Al61-Al6s as shown in Figure 8d,the thermal expansion B also shows a well-defined sharp peak,as curve has sharp changes at temperatures between 300 and 600 in Figure 7a.The positions of the peaks of B for these particles K.After 600 K,the curve is almost linear until about 1000 K, are almost the same as Tp obtained from c.For those particles then the slope changes,and the curve is again linear.For having almost featureless c curves,however,the B plots give Al62-Al64,the position of the peak between 300 and 600 K is valuable new information,as illustrated in Figure 7b.For in agreement with the small peak in the c curve (see Supporting example,B of Al34 has a sharp peak at about 340 K,and that of Information),while for Al61,there is no clear peak in the c curve Al6 has a deep valley at about 480 K.For most particles,if at this temperature range.Clearly,for the particles shown in one ignores such peaks and valleys,one finds that B generally Figure 8c and 8d,there is a change of state at the point of sharp increases with temperature and then reaches a plateau at high change.Putting all of the plots into one graph,an almost linear temperatures.A clear change of slope is observed for most dependence of In k on T,similar to the functional dependences particles;after the change of slope,B increases more slowly of c and B,can also be seen for most particles above 900-1000 with temperature and the curve shows periodic oscillations. K,but the trend is less clear than for c and B,and the plots are Representative examples are provided by the particles with sizes available in the Supporting Information. between Al61 and Al6s(Figure 7c).Putting all these curves into 3.6.Potential-Energy Distribution of the Quenched one graph shows that for most particles the change of slope is Structures.By quenching intermediate configurations we de- at about 900 K(Figure S-8 in the Supporting Information).This termined what fraction of the time is spent by trajectories in indicates that after 900 K most particles are in a similar state, the configuration space around each local minimum (this time the liquid droplet state. is proportional to the number of quenched structures corre- J.AM.CHEM.SOC.VOL 130,NO.38,2008 12705
for some particles it seems that the most stable structure at low temperatures, i.e., the solid-like structure, is not the most compact structure, the volume or Rg of the particle has a sudden drop, and the coefficient of thermal expansion has a deep valley (see below) at temperatures where the solid-like structure no longer dominates. 3.4. Coefficient of Thermal Expansion. For particles with a well-defined sharp peak in heat capacity, the coefficient of thermal expansion also shows a well-defined sharp peak, as in Figure 7a. The positions of the peaks of for these particles are almost the same as Tp obtained from c. For those particles having almost featureless c curves, however, the plots give valuable new information, as illustrated in Figure 7b. For example, of Al34 has a sharp peak at about 340 K, and that of Al61 has a deep valley at about 480 K. For most particles, if one ignores such peaks and valleys, one finds that generally increases with temperature and then reaches a plateau at high temperatures. A clear change of slope is observed for most particles; after the change of slope, increases more slowly with temperature and the curve shows periodic oscillations. Representative examples are provided by the particles with sizes between Al61 and Al65 (Figure 7c). Putting all these curves into one graph shows that for most particles the change of slope is at about 900 K (Figure S-8 in the Supporting Information). This indicates that after 900 K most particles are in a similar state, the liquid droplet state. 3.5. Isothermal Compressibility. Isothermal compressibility (κ) is a function of pressure. The compressibilities calculated in the present study are the low-pressure limits because it is the free Aln particles that are studied. Compressibilities as a function of temperature for each particle size are presented in the Supporting Information. Here we focus on the natural logarithm of compressibility. For Al18, Al21, Al36, and Al45, the ln κ vs T curve is almost linear in the whole simulation temperature range (Figure 8a). For some particles, the curve is smooth but has large curvature (Figure 8b). For most large particles the curve has a clear jump or sharp peak at a certain temperature (Figure 8c). For the particles shown in Figure 8c, before and after the jump or peak, ln κ is an almost linear function of temperature. The position of the jump or peak almost overlaps with the peak position of heat capacity. For some particles, for example, Al61-Al65 as shown in Figure 8d, the curve has sharp changes at temperatures between 300 and 600 K. After 600 K, the curve is almost linear until about 1000 K, then the slope changes, and the curve is again linear. For Al62-Al64, the position of the peak between 300 and 600 K is in agreement with the small peak in the c curve (see Supporting Information), while for Al61, there is no clear peak in the c curve at this temperature range. Clearly, for the particles shown in Figure 8c and 8d, there is a change of state at the point of sharp change. Putting all of the plots into one graph, an almost linear dependence of ln κ on T, similar to the functional dependences of c and , can also be seen for most particles above 900-1000 K, but the trend is less clear than for c and , and the plots are available in the Supporting Information. 3.6. Potential-Energy Distribution of the Quenched Structures. By quenching intermediate configurations we determined what fraction of the time is spent by trajectories in the configuration space around each local minimum (this time is proportional to the number of quenched structures correFigure 7. Coefficient of thermal expansion () curves: (a-c) particles with three typical heat capacity curves plotted in Figure 4. Figure 8. Typical natural logarithm of isothermal compressibility (ln κ) curves: (a) almost linear in the whole temperature range, (b) with obvious jumps, (c) with sharp peaks, (d) with two almost linear segments joined at about 1000 K. J. AM. CHEM. SOC. 9 VOL. 130, NO. 38, 2008 12705 Nanosolids, Slushes, and Nanoliquids ARTICLES
ARTICLES Li and Truhlar sponding to that minimum).As particle size increases and the 2.0 number of isomers the trajectory can visit grows exponentially, A 1.5 the energy distribution of the isomers becomes quasicontinuous; therefore,it is convenient to discuss these results using an energy A27 (Ae)(3)d scale rather than the ordinal numbers of the isomers.To 5 accomplish this we counted the number of quenched structures 0,0 in each specific energy range.The percentage P(E)is defined as 100 M(E)/Ng.where M(E)is the number of quenched 1.5 structures with potential energy between E and E+OE and Na 1.0 is the total number of quenched configurations at a specific g temperature.The energy resolution oE in the current study is 0.05 ev,and the zero of the energy is the energy of the global 0.6 minimum,GM. 2.0 The P(E)plots of some particles can be found in the 1.5 Supporting Information.We found P(0),the percentage of the 1.0 quenched structures with an energy not higher than that of GM 5 (including GM)by 0.05 eV,is particularly useful(see Support- 0.0 ing Information for the plots and also the next section).Although 0.0 5 1.0 1.5 0 .5 1,0 1.5 2.0 use of a 0.05 eV resolution for E is somewhat arbitrary,the △E(eV) AE(ev) structures with energies not higher than GM by 0.05 eV can be viewed as structures almost degenerate in energy with GM. Figure 9.Potential-energy distribution (p(E))of the nonidentical structures Since these low-energy isomers(including GM)are the domi- quenched in the MD simulations.The abscissa is the relative potential energy to the global minimum structure. nant structures at low temperatures and often have compact and well-ordered structures,48 they probably can be viewed as solid- 100 like structures,while other higher-energy isomers can often be viewed as liquid-like structures.(One must bear in mind that there will hardly be a clear border between solid-like and liquid- Al like structures.)Therefore,the equilibrium constant between Alst the solid state and liquid state is related to P(0)by Als K=100-P0 Alsa (13) P(O) Alss Alsn If we define the temperature at which Keg =1 (P(0)=50)as 20 Alto the population-based melting temperature (Tpso),another quan- Al2 tity characterizing the melting transition of clusters and nano- 0 。444 particles can be defined;such values are given in Table 1. 2003004005006007008009001000 However,for some small particles (for example,Alj3)P(O)is T(9 never under 50,while for some particles (for example,Also) P(0)never exceeds 50 in the simulation temperature range we Figure 10.P(O)(see text for definition)for particles with a single sharp peak in c and B and single sharp peak or obvious jump inIn K. studied.Therefore,for these particles a population-based melting temperature defined this way cannot be obtained.The Tpso values isomer,and ab initio calculations indicate that there is a very are compared to Tp values in Figure 5.It can be seen that for high energy barrier between the two lowest energy structures. those particles with well-defined sharp peaks in heat capacity Therefore,Al3 has extraordinary structural stability.It may Tp of heat capacity and Tpso generally agree well with each other, therefore be called a magic structure or,since it also keeps its while for other particles Tpso generally agrees better with the structure in chemical reactions,a superatom.s Our results Tp of B or In k (Table 1). indicate that the icosahedral structure of Al 3 is dominant as We also analyzed the potential-energy distribution (p(E))of high as 1700 K.As in most studies,we find that the peak of c the distinct isomers in the quenched structures.p(E)is defined occurs at a very high temperature. as dn(E)/dE.where dn(E)is the number of distinct isomers in Among the larger particles,Al19.Al23.Al48,Alsi-Als6,and the energy range E to E+dE.The distribution is obtained by Aluo-Al300 are particles that have a single well-defined peak a Legendre moment method'04 with the of the normalized in c and B,respectively,a clear jump or sharp peak in In k,a Legendre polynomial Px)truncated at /max=30 for Alo-Alis jump in Rg,and,except for Al9,a jump in V:Al9 has a clear and /max=40 for the rest.In Figure 9 we plotted the potential- transition point in Rg and V.Analogously to macroscopic energy distribution(p(E))of distinct isomers for Alo-Al27. systems,the melting transition for these particles can be called a first-order phase transition4 or single-phase transition.3 4.Discussion A common feature of these particles is that the low-energy 4.1.Magic Structures and First-Order Melting isomers between 0 and 0.05 eV are dominant up to about 500 Transitions.Al13 is the most extensively studied of the K.In Figure 10 we plotted P(0)for Al9,Al23,Al48,Als1-Al56, clusters.14.61.72.75.80.83.85 We find that Al3 has a very large energy and Aluo-Al13o as a function of temperature.For these particles, gap(1.01 ev)between the GM and the second lowest energy (105)Bergeron,D.E..Jr.;Morisato,T.:Khanna,S.N.Science 2004,304, 84.Bergeron,D.E..Jr.;Morisato,T.:Khanna,S.N.J.Chem.Phys. (104)Truhlar,D.G.;Blais,N.C.J.Chem.Plrys.1977,67,1532. 2004,121,10456. 12706J.AM.CHEM.S0C.■VOL.130,NO.38,2008
sponding to that minimum). As particle size increases and the number of isomers the trajectory can visit grows exponentially, the energy distribution of the isomers becomes quasicontinuous; therefore, it is convenient to discuss these results using an energy scale rather than the ordinal numbers of the isomers. To accomplish this we counted the number of quenched structures in each specific energy range. The percentage P(E) is defined as 100 M(E)/Nq, where M(E) is the number of quenched structures with potential energy between E and E + δE and Nq is the total number of quenched configurations at a specific temperature. The energy resolution δE in the current study is 0.05 eV, and the zero of the energy is the energy of the global minimum, GM. The P(E) plots of some particles can be found in the Supporting Information. We found P(0), the percentage of the quenched structures with an energy not higher than that of GM (including GM) by 0.05 eV, is particularly useful (see Supporting Information for the plots and also the next section). Although use of a 0.05 eV resolution for δE is somewhat arbitrary, the structures with energies not higher than GM by 0.05 eV can be viewed as structures almost degenerate in energy with GM. Since these low-energy isomers (including GM) are the dominant structures at low temperatures and often have compact and well-ordered structures,48 they probably can be viewed as solidlike structures, while other higher-energy isomers can often be viewed as liquid-like structures. (One must bear in mind that there will hardly be a clear border between solid-like and liquidlike structures.11) Therefore, the equilibrium constant between the solid state and liquid state is related to P(0) by Keq ) 100 - P(0) P(0) (13) If we define the temperature at which Keq ) 1 (P(0) ) 50) as the population-based melting temperature (TP50), another quantity characterizing the melting transition of clusters and nanoparticles can be defined; such values are given in Table 1. However, for some small particles (for example, Al13) P(0) is never under 50, while for some particles (for example, Al80) P(0) never exceeds 50 in the simulation temperature range we studied. Therefore, for these particles a population-based melting temperature defined this way cannot be obtained. The TP50 values are compared to Tp values in Figure 5. It can be seen that for those particles with well-defined sharp peaks in heat capacity Tp of heat capacity and TP50 generally agree well with each other, while for other particles TP50 generally agrees better with the Tp of or ln κ (Table 1). We also analyzed the potential-energy distribution (F(E)) of the distinct isomers in the quenched structures. F(E) is defined as dn(E)/dE, where dn(E) is the number of distinct isomers in the energy range E to E + dE. The distribution is obtained by a Legendre moment method104 with the l of the normalized Legendre polynomial Pl(x) truncated at lmax ) 30 for Al10-Al15 and lmax ) 40 for the rest. In Figure 9 we plotted the potentialenergy distribution (F(E)) of distinct isomers for Al10-Al27. 4. Discussion 4.1. Magic Structures and First-Order Melting Transitions. Al13 is the most extensively studied of the clusters.14,61,72,75,80,83,85 We find that Al13 has a very large energy gap (1.01 eV) between the GM and the second lowest energy isomer, and ab initio calculations indicate that there is a very high energy barrier between the two lowest energy structures.80 Therefore, Al13 has extraordinary structural stability. It may therefore be called a magic structure or, since it also keeps its structure in chemical reactions, a superatom.105 Our results indicate that the icosahedral structure of Al13 is dominant as high as 1700 K. As in most studies, we find that the peak of c occurs at a very high temperature. Among the larger particles, Al19, Al23, Al48, Al51-Al56, and Al110-Al300 are particles that have a single well-defined peak in c and , respectively, a clear jump or sharp peak in ln κ, a jump in Rg, and, except for Al19, a jump in V; Al19 has a clear transition point in Rg and V. Analogously to macroscopic systems, the melting transition for these particles can be called a first-order phase transition42-44 or single-phase transition.33 A common feature of these particles is that the low-energy isomers between 0 and 0.05 eV are dominant up to about 500 K. In Figure 10 we plotted P(0) for Al19, Al23, Al48, Al51-Al56, and Al110-Al130 as a function of temperature. For these particles, (104) Truhlar, D. G.; Blais, N. C. J. Chem. Phys. 1977, 67, 1532. (105) Bergeron, D. E., Jr.; Morisato, T.; Khanna, S. N. Science 2004, 304, 84. Bergeron, D. E., Jr.; Morisato, T.; Khanna, S. N. J. Chem. Phys. 2004, 121, 10456. Figure 9. Potential-energy distribution (F(E)) of the nonidentical structures quenched in the MD simulations. The abscissa is the relative potential energy to the global minimum structure. Figure 10. P(0) (see text for definition) for particles with a single sharp peak in c and and single sharp peak or obvious jump in ln κ. 12706 J. AM. CHEM. SOC. 9 VOL. 130, NO. 38, 2008 ARTICLES Li and Truhlar
Nanosolids,Slushes,and Nanoliquids ARTICLES P(0)is still higher than 50 at 500 K.In the cases of Al9,Al23. 1600 Alis.Als4.Alss,and Al120.there is a large energy gap between the GM and the second lowest energy isomer;we find gaps of 1400 0.66,0.55,0.20,0.70,0.70,and 0.30 eV,respectively.For the 200K other particle sizes,there are several or many low-energy 1500K 1200 isomers that are almost degenerate in energy with the GM and separated from higher-energy structures by a large energy gap: these low-energy isomers also have similar structure and 1000 人。】 probably can be viewed as solid-like structures just like the GM structure.(Recall that a finite-temperature macroscopic solid 800 state is not a single quantum state but rather a large number of quantum states with similar energy and properties;this behavior 600 is seen here at the threshold of emergence.)Als6 is a prototype of such particles.48 For Als6.the GM and second and third 400 lowest-energy structures can be viewed as attaching an alumi- 10 .15 .20 .25 .30 num atom to the icosahedron Alss.There are also 52 other isomers,all with similar geometry,with energy just higher than 1/R(1/Angstrom) the GM by less than 0.07 eV.The rest are higher in energy by Figure 11.Plots of the temperature Tp at which c has a maximum as a 0.41 eV.Our previous study indicates that the properties of the function of 1/R,where R is the particle radius estimated from the radius of Als6 particle can be well represented by the GM in a wide gyration(Rg)by R=V53R Rg values obtained at 200(straight line)and temperature range.Therefore,for those particles with a first- 1500 K (dashed line)are used to make the plots. order melting transition there is a large energy gap between, on the one hand,the GM or the solid-like structures and,on of the deviation from the linear relationship is that for small the other hand,the liquid-like structures. particles Tp is not a very good quantity to characterize the Although an energy gap between isomers is one important melting of a particle (see below).Another possible contributor determinant of the distribution of isomers,the distribution may be that small aluminum particles are not spherical function depends on free energy,which also has an entropic (aluminum nanodroplets are prolate),which is a precondition component,and the entropic contribution determined by the in deriving such relations.The present results confirm that density of vibrational and rotational states in each isomer is macroscopic models such as the Pawlow relation do not explain also important.48 As an apparent consequence,the existence of the interesting properties of finite systems. a large energy gap is not in one-to-one correspondence with 4.2.Clusters (70 does Tp depend linearly on I/R,whereas for temperature,GM structures still have about 70%population(see small particles there is no such relationship.One possible cause Figure 12a).Between 700 and 900 K,isomers between 0.15 and 0.25 eV are the next most abundant and the plot shows a (106)Thomson,W.Philos.Mag.1871,42,448 peak at 0.20 eV.After 900 K,the peak shifts to 0.30 eV.There (107)Pawlow.P.Z.Phys.Chem.1909.65.1. (108)Borel,J.P.Suf.Sci.1981,106,1. are a total of 74 nonidentical isomers in the energy range (109)Wang,L.;Zhang,Y.;Bian,X.:Chen,Y.Phys.Lett.A 2003,310, between 0.15 and 0.45 eV.After 708 K,the Alu particle is a 197. mixture of the GM,the 74 isomers between 0.15 and 0.45 eV, J.AM.CHEM.SOC.VOL 130,NO.38,2008 12707
P(0) is still higher than 50 at 500 K. In the cases of Al19, Al23, Al48, Al54, Al55, and Al120, there is a large energy gap between the GM and the second lowest energy isomer; we find gaps of 0.66, 0.55, 0.20, 0.70, 0.70, and 0.30 eV, respectively. For the other particle sizes, there are several or many low-energy isomers that are almost degenerate in energy with the GM and separated from higher-energy structures by a large energy gap; these low-energy isomers also have similar structure and probably can be viewed as solid-like structures just like the GM structure. (Recall that a finite-temperature macroscopic solid state is not a single quantum state but rather a large number of quantum states with similar energy and properties; this behavior is seen here at the threshold of emergence.) Al56 is a prototype of such particles.48 For Al56, the GM and second and third lowest-energy structures can be viewed as attaching an aluminum atom to the icosahedron Al55. There are also 52 other isomers, all with similar geometry, with energy just higher than the GM by less than 0.07 eV. The rest are higher in energy by 0.41 eV. Our previous study indicates that the properties of the Al56 particle can be well represented by the GM in a wide temperature range.48 Therefore, for those particles with a firstorder melting transition there is a large energy gap between, on the one hand, the GM or the solid-like structures and, on the other hand, the liquid-like structures. Although an energy gap between isomers is one important determinant of the distribution of isomers, the distribution function depends on free energy, which also has an entropic component, and the entropic contribution determined by the density of vibrational and rotational states in each isomer is also important.48 As an apparent consequence, the existence of a large energy gap is not in one-to-one correspondence with the existence of a single sharp peak in the heat capacity. Al38 and Al61 are prototypes for the dominance of the entropic effect. The energy gaps between the two lowest-energy isomers are large, 0.45 and 0.25 eV, respectively. However, Al38 has two small peaks in c before c reaches a maximum, while there is no well-defined peak in c for Al61 (see Table 1 and the Supporting Information). Some studies also found a small peak in c for Al38 and attribute this peak to a premelting process.33 Since the GMs of Al38 and Al61 are closely packed and wellordered structures,48 it seems that the geometrical order in the electronic ground-state GM structure is not an adequate but just a necessary condition for a magic melter, in contrast to the findings in gallium particles.32,38 The entropy effect must be taken into consideration.44-49 Finally, we consider the Gibbs-Thomson relation23,106 and other similar relations such as the Pawlow relation,107,108 which establish a linear relationship between the melting temperature of a particle and the reciprocal of the particle radius (1/R). In Figure 11 the temperature Tp at which the heat capacity has a maximum is plotted as a function of 1/R, where R is estimated as109 R ) √5⁄3Rg . Since Rg is a function of temperature, we use Rg values obtained from two temperatures, 200 and 1500 K, to make the plots. Figure 11 shows that the choice of Rg does not affect the qualitative correlation or lack of correlation between Tp and 1/R much, and it shows that only for particles with size n > 70 does Tp depend linearly on 1/R, whereas for small particles there is no such relationship. One possible cause of the deviation from the linear relationship is that for small particles Tp is not a very good quantity to characterize the melting of a particle (see below). Another possible contributor may be that small aluminum particles are not spherical (aluminum nanodroplets are prolate91), which is a precondition in deriving such relations. The present results confirm that macroscopic models such as the Pawlow relation do not explain the interesting properties of finite systems. 4.2. Clusters (<1 nm, n e 18) Have No Melting Transition. In the Results section, we presented various physical properties of Aln particles that are supposed to change if a particle changes from the solid state to the liquid state. We find startlingly different behavior for different size particles. The simulations provide a much more detailed picture of the phase characteristics than is available experimentally. In particular, as assortment of properties may be monitored. One may use these properties to characterize the melting of clusters and nanoparticles; however, phase transitions in finite systems are more complicated than those in macroscopic systems since phase is not well defined in finite systems. For example, as shown in the Introduction, a peak in the heat capacity curve does not necessarily correspond to a melting transition; it may just result from the equilibrium between isomers, such as in the cases of the clusters Al10, Al11, Al12, and Al13. For each of these four clusters we observe a well-defined peak in the heat capacity curve. The P(E) plots for Al11 and Al13 are presented in Figure 12, and the plots for Al10 and Al12 (given in the Supporting Information) are similar to those for Al11 and Al13, respectively. For these four clusters the energy gap between the GM and the second lowest energy isomer is larger than 0.05 eV, and thus, P(0) is the percentage population of the GM structure. For Al11, the temperature Tp of the peak in c is 708 K (Table 1). At this temperature, GM structures still have about 70% population (see Figure 12a). Between 700 and 900 K, isomers between 0.15 and 0.25 eV are the next most abundant and the plot shows a peak at 0.20 eV. After 900 K, the peak shifts to 0.30 eV. There are a total of 74 nonidentical isomers in the energy range between 0.15 and 0.45 eV. After 708 K, the Al11 particle is a mixture of the GM, the 74 isomers between 0.15 and 0.45 eV, (106) Thomson, W. Philos. Mag. 1871, 42, 448. (107) Pawlow, P. Z. Phys. Chem. 1909, 65, 1. (108) Borel, J. P. Surf. Sci. 1981, 106, 1. (109) Wang, L.; Zhang, Y.; Bian, X.; Chen, Y. Phys. Lett. A 2003, 310, 197. Figure 11. Plots of the temperature Tp at which c has a maximum as a function of 1/R, where R is the particle radius estimated from the radius of gyration (Rg) by R ) √5⁄3Rg. Rg values obtained at 200 (straight line) and 1500 K (dashed line) are used to make the plots. J. AM. CHEM. SOC. 9 VOL. 130, NO. 38, 2008 12707 Nanosolids, Slushes, and Nanoliquids ARTICLES