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.38.2008that 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, multiple￾equilibration 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