Nanosolids,Slushes,and Nanoliquids ARTICLES will show that it may not be appropriate to call the structural agree well with each other (see Figure S-1 in the Supporting transition or isomerization process of clusters a melting transition Information),and hence,we can use whichever is more convenient while nanoparticles do have a melting transition.(It is only an (we found it is more convenient to use the right-hand side).In accident,but a convenient one,that the border between melting presenting our results we convert the heat capacity to a unitless behavior and no-melting behavior occurs so close to the rather specific heat capacity c defined by arbitrary border we established between nanoparticles and C clusters.)The present investigations indicate that Al nanopar- C= -(3n-3)kg (5) ticles have a wide temperature window of coexistence of solid and liquid states;this coexistence regime is called the slush where the denominator results from the fact that only overall state.12 Definitions of the solid,slush,and liquid states of translational motion was removed in the MD simulations.In the rest of the article we will simply refer to c as the heat capacity,but such particles will be proposed. we should keep in mind that the absolute heat capacity is actually C 2.Simulation Methodology (3)Average Distance to the Center of Mass(RcoM).RcoM is a property to characterize the size of a particle Simulations were run for Al with 10 s ns 300.For n 10-130.all MD simulations were started in the vicinity of the global energy minimum (GM)structures with random initial R.CoM= (6) coordinates and momenta distributed according to the classical phase space distribution of separable harmonic oscillators.64 For n= where r;is the vector position of an atom i and rcoM is the vector 10-65,GM structures obtained previously4s have been used.For position of the center of mass. n=70-130,the same strategy as used in ref 48 was used to locate (4)Radius of Gyration (R).R is another property that can be GM structures.The caloric curve was then studied by heating.For used to characterize the size of a particle larger particles,a search for the GM structure is too expensive: instead,the MD trajectory was started at high temperature with R spherical clusters with atomic coordinates randomly generated in n Ir-rcoMl (7) a sphere with a radius of 16,16,and 19 A for n=177.200,and 300,respectively,and the process simulated is the cooling process. (5)Volume (V).For a spherical object with evenly distributed For each heating simulation the starting temperature is 200 K with mass,it is easy to show that the radius of the sphere has the an increment of 20 K and the ending temperature is 1700 K,while following relationship with the principal moment of inertia ( for cooling simulation the same procedure is reversed. R=V5/2V1/M (8) To determine the local minima that the trajectory visited during the simulation,intermediate configurations were quenched at where M is total mass of the particle.Since a particle need not be random;on average,10%were quenched.Geometries of the spherical,we consider the three principal moments of inertia. quenched structures were optimized. Corresponding to these,there are three radii,R(i=1,2.3).With Details of solving the equations of motion,thermostatting,the these three radii,the volume of a particle can be estimated as27 heating and cooling programs,and optimization are provided in the Supporting Information V -37RR:Rs (9) Several properties have been investigated. (1)Berry Parameter.The Berry parameter.4 is the relative The quantity V was calculated at each step of the molecular root-mean-square fluctuation in the interatomic separation;it is an dynamics simulation and averaged. extension of the original Lindemann parameters used for macro- (6)Coefficient of Thermal Expansion (B) scopic systems.The Berry parameter is calculated by A=器 (10) 4=2∑ki>-<r>2 nm-1) (3) In the current study,the temperature derivatives of V and other <rip properties were obtained by first fitting them with cubic spline functions and then differentiating the fitted spline functions. where r is the distance between two atoms i and j. (7)Isothermal Compressibility (K).99 (2)Heat Capacity.For a macroscopic system the heat capacity at constant volume (C=dEro/dT,where Erot is the total energy of 1(W2)-02 the system)is related to the fluctuation in energy by6 K-kpT (V) (11) where V is calculated by eq 11. C=. Et）-(Ero (4) kg72 3.Results (See page S-3 in the Supporting Information for a discussion of the derivation.)Although the derivation of eq 4 is not directly 3.1.Berry Parameter.Although the Berry parameter (AB) applicable to finite systems,we found that the two sides of eq4 has been widely used to study the melting of clusters and nanoparticlesour simulations show that it is more sensitive to geometrical transitions than other properties (91)Schultz,N.E.:Jasper,A.W.:Bhatt,D.:Siepmann,J.I.;Truhlar. D.G.In Multiscale Simulation Methods for Nanomaterials,Ross R.B..Mohanty,S.,Eds.:Wiley-VCH:Hoboken,NJ,2008;p 169. (96)Hill,T.L.Statistical Mechanics:Principles and Selected Applications: (92)Tanner,G.M.:Bhattacharya,A.;Nayak.S.K;Mahanti,S.D.Plrys. McGraw-Hill:New York,1956;pp 100-101.Rice.O.K.Staristical Reu.E1997.55.322. Mechanics Thermodynamics and Kinetics:W.H.Freeman:San (93)Kaelberer,J.:Etters,R.D J.Chem.Plrys.1977,66,3233.Etters, Francisco,1967:pp 92-93.. R.D.:Kaelberer,J.J.Chem.Phys.1977,66,5112. (97)Ding,F.:Rosen,A.:Bolton,K.Phys.Rev.B 2004.70,75416 (94)Berry,R.S.:Beck.T.L.:Davis.H.L.:Jellinek,J.Adv.Chem.Phrys. (98)Wang,L.:Zhang,Y.:Bian,X.:Chen,Y.Phys.Lett.A 2003.310. 1988.70B.75.Zhou,Y.:Karplus,M.;Ball,K.D.:Berry,R.S. 197. J.Chem.Phs.2002,I16,2323. (99)Pathria,R.K.Staristical Mechanics,2nd ed.:Elsevier:Singapore, (95)Lindemann,F.A.Phys.Z 1910,//609. 1996,p454. J.AM.CHEM.SOC.VOL 130,NO.38,2008 12701will 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 nanopar￾ticles 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 macro￾scopic systems. The Berry parameter is calculated by ∆B ) 2 n(n - 1)∑i<j √<rij 2 > - < rij>2 <rij> (3) where rij is the distance between two atoms i and j. (2) Heat Capacity. For a macroscopic system the heat capacity at constant volume (C ≡ dETot/dT, where ETot is the total energy of the system) is related to the fluctuation in energy by96 C ) 〈ETot 2 〉 -〈ETot〉2 kBT2 (4) (See page S-3 in the Supporting Information for a discussion of the derivation.) Although the derivation of eq 4 is not directly applicable to finite systems, we found that the two sides of eq 4 agree well with each other (see Figure S-1 in the Supporting Information), and hence, we can use whichever is more convenient (we found it is more convenient to use the right-hand side). In presenting our results we convert the heat capacity to a unitless specific heat capacity c defined by c ) C (3n - 3)kB (5) where the denominator results from the fact that only overall translational motion was removed in the MD simulations. In the rest of the article we will simply refer to c as the heat capacity, but we should keep in mind that the absolute heat capacity is actually C. (3) AVerage Distance to the Center of Mass (RCoM).97 RCoM is a property to characterize the size of a particle RCoM ) 1 n∑i |ri - rCoM| (6) where ri is the vector position of an atom i and rCoM is the vector position of the center of mass. (4) Radius of Gyration (Rg).98 Rg is another property that can be used to characterize the size of a particle Rg )1 n∑i |ri - rCoM| 2 (7) (5) Volume (V). For a spherical object with evenly distributed mass, it is easy to show that the radius of the sphere has the following relationship with the principal moment of inertia (I) R ) √5⁄2√I ⁄ M (8) where M is total mass of the particle. Since a particle need not be spherical, we consider the three principal moments of inertia. Corresponding to these, there are three radii, Ri (i ) 1, 2, 3). With these three radii, the volume of a particle can be estimated as27 V ) 4 3 πR1R2R3 (9) The quantity V was calculated at each step of the molecular dynamics simulation and averaged. (6) Coefficient of Thermal Expansion (
).
) 1 V dV dT (10) In the current study, the temperature derivatives of V and other properties were obtained by first fitting them with cubic spline functions and then differentiating the fitted spline functions. (7) Isothermal Compressibility (κ).99 κ ) 1 kBT 〈V2 〉 -〈V〉2 〈V〉 (11) where V is calculated by eq 11. 3. Results 3.1. Berry Parameter. Although the Berry parameter (∆B) has been widely used to study the melting of clusters and nanoparticles,6,9,14,19,27,33,34,84,93,94 our simulations show that it is more sensitive to geometrical transitions than other properties (91) Schultz, N. E.; Jasper, A. W.; Bhatt, D.; Siepmann, J. I.; Truhlar, D. G. In Multiscale Simulation Methods for Nanomaterials; Ross, R. B., Mohanty, S., Eds.; Wiley-VCH: Hoboken, NJ, 2008; p 169. (92) Tanner, G. M.; Bhattacharya, A.; Nayak, S. K.; Mahanti, S. D. Phys. ReV. E 1997, 55, 322. (93) Kaelberer, J.; Etters, R. D J. Chem. Phys. 1977, 66, 3233. Etters, R. D.; Kaelberer, J. J. Chem. Phys. 1977, 66, 5112. (94) Berry, R. S.; Beck, T. L.; Davis, H. L.; Jellinek, J. AdV. Chem. Phys. 1988, 70B, 75. Zhou, Y.; Karplus, M.; Ball, K. D.; Berry, R. S. J. Chem. Phys. 2002, 116, 2323. (95) Lindemann, F. A. Phys. Z 1910, 11, 609. (96) Hill, T. L. Statistical Mechanics: Principles and Selected Applications; McGraw-Hill: New York, 1956; pp 100-101. Rice, O. K. Statistical Mechanics Thermodynamics and Kinetics; W. H. Freeman: San Francisco, 1967; pp 92-93.. (97) Ding, F.; Rosen, A.; Bolton, K. Phys. ReV. B 2004, 70, 75416. (98) Wang, L.; Zhang, Y.; Bian, X.; Chen, Y. Phys. Lett. A 2003, 310, 197. (99) Pathria, R. K. Statistical Mechanics, 2nd ed.; Elsevier: Singapore, 1996; p 454. J. AM. CHEM. SOC. 9 VOL. 130, NO. 38, 2008 12701 Nanosolids, Slushes, and Nanoliquids ARTICLES