ARTICLES Li and Truhlar 100 100 of local-minimum structures (isomers)that the trajectory can (a)Al 200K (b)Al3 200K 90 visit in the simulation temperature range is limited.For example. 400K 400K 600K 600K only64.138.167.217.228.387.907,832.1266.1725.and 80 800 80 800K 2354 nonidentical isomers have been found for Alo-Al2o 1000K 1000K 70 1200K clusters.The number of isomers the trajectory most often visits 200 is even smaller.For example,for Al16 and Al7 more than 90% 1400K 1400K60 1600K 1600K1 0号 of the time the trajectory visits only 8%(74 out of 907)and 30%(247 out of 832)of the low-energy isomers,respectively. 50 6 8四 Therefore,even at high temperatures,these clusters are a mixture of a limited number of isomers. 6 The potential-energy distribution(p(E))of the distinct isomers also indicates that clusters behave more like molecules than 4 nanoparticles do.From the p(E)plots(Figure 9)it is clear that for particles smaller than Al o the energy distribution of isomers 10 is scattered (Alo-Alis)or has well-separated multiple peaks with comparable heights(Al16-Alis).Only for Al particles with 0.0 2 4 .6 81.00.051.01.520 △E(eV) △E(eV) n219 does the distribution for high-energy isomers have a roughly Gaussian distribution,and these isomers can be viewed Figure 12.Percentage (P(E))of quenched structures in the potential-energy as being in one state,the liquid state. range between E to E+oE with OE 0.05 eV:(a)Alu and (b)Al13.The We can conclude that clusters have no melting transition,or abscissa is the relative potential energy to the global minimum structure. in other words,they are too small to melt.3 The measured or simulated caloric curves of these clusters are better understood 100 90 (a)Aln (b)Al3 as resulting from molecule-like isomerization processes than from macroscopic-like melting transitions. P(O) -P(0) 80 4.3.Solid,Slush,and Liguid States in Clusters and +P0.15-0.45) +-1.0-1.15) 70 0.50-0.80j PA1.2-1.5) Nanoparticles.The discussion above raises a basic question: P>1.5) What are the solid and liquid states for clusters and nanopar- ticles?Or even more basically,what is phase for clusters and T50 nanoparticles?Due to the enormous number of atoms in a macroscopic system,the change from one phase to another phase 40 is accompanied by a noncontinuous change in certain physical 30 or chemical properties.The discontinuity is called a first- order phase transition (and characterized by a temperature 20 T.At To,the two phases are in equilibrium and Keg=1.Thus, 10 once these To points have been determined as functions of 0 pressure or volume,the regimes of the phases are known. 300 600 90012001500300 600 90012001500 However,for finite systems,especially clusters and small T(K TK nanoparticles,one has to abandon the concept of a precise T and recognize the fact that different"phases",if there are any, Figure 13.Percentage(P(E))of quenched structures in the three potential- of clusters and nanoparticles are in equilibrium with each other energy ranges corresponding to the three (Al)or four (Al13)different regions as shown in Figure 9:(a)All and (b)Al13.P(0)is the percentage in an appreciable temperature range27The results population of the global minimum structures. for small clusters such as Alu-Al that are discussed above indicate the problematic character of using solely the caloric and the 48 isomers between 0.50 and 0.80 eV(Figure 13a).At curves to characterize the melting transition. 1600 K,25%of the quenched structures are still the GM As particle size increases.the solid and liquid states of structure. nanoparticles can be defined with less ambiguity.We start our For Al13.the peak of c is at 1188 K and the GM structure discussion with the most macroscopic-like particle,Al3o0.For still constitutes about 90%of the quenched structures at this this particle,all properties give consistent results,implying that temperature.At 1600 K,more than 65%the quenched structures there is a state change at about 700 K(Figure 14):(i)c has a are the GM structure.Thus,in the whole simulation temperature very high and sharp peak at about 700 K(Figure 14a).Before range,Al3 is usually in the region of coordinate space around 600 K,c increases almost linearly with temperature.Between the most stable structure,which is a very well-ordered isosa- 760 and 880 K,c reaches a plateau (see also Figure S-3n in the hedral structure.If we call this structure the solid-like structure Supporting Information)and then decreases almost linearly with and the other less stable isomers (about 150 of them between temperature.(ii)Rs and Vjump between 680 and 760 K(Figure 1.0 and 2.0 eV)the liquid-like structures,we can summarize 14b);aside from this region,they are both smooth functions of this situation by saying that Al3 never fully melts in the temperature.(iii)B increases almost linearly with temperature temperature range we studied and neither does Alo,Al,or below 580 K(Figure 14c);after 560 K,the plot shows a small Al2(see also Supporting Information). peak at 600 K.a valley at 640 K.and a high peak at 720 K From another point of view,some of the properties of small After the high peak,B drops and reaches a valley at 780 K. clusters can be understood by analogy to conventional mol- After 820 K,the plot shows an oscillatory behavior;averaging ecules.Usually,when studying molecules,the change from one out these oscillations yields a B that increases almost linearly isomer to another isomer with a distinct structural difference is with temperature.(iv)The logarithm of K(In K)increases almost called an isomerization reaction.For small clusters,the number linearly with temperature below 680 K(Figure 14d).Then it 12708J.AM.CHEM.S0C.■VOL.130,NO.38.2008and the 48 isomers between 0.50 and 0.80 eV (Figure 13a). At 1600 K, 25% of the quenched structures are still the GM structure. For Al13, the peak of c is at 1188 K and the GM structure still constitutes about 90% of the quenched structures at this temperature. At 1600 K, more than 65% the quenched structures are the GM structure. Thus, in the whole simulation temperature range, Al13 is usually in the region of coordinate space around the most stable structure, which is a very well-ordered isosa￾hedral structure. If we call this structure the solid-like structure and the other less stable isomers (about 150 of them between 1.0 and 2.0 eV) the liquid-like structures, we can summarize this situation by saying that Al13 never fully melts in the temperature range we studied and neither does Al10, Al11, or Al12 (see also Supporting Information). From another point of view, some of the properties of small clusters can be understood by analogy to conventional mol￾ecules. Usually, when studying molecules, the change from one isomer to another isomer with a distinct structural difference is called an isomerization reaction. For small clusters, the number of local-minimum structures (isomers) that the trajectory can visit in the simulation temperature range is limited. For example, only 64, 138, 167, 217, 228, 387, 907, 832, 1266, 1725, and 2354 nonidentical isomers have been found for Al10-Al20 clusters. The number of isomers the trajectory most often visits is even smaller. For example, for Al16 and Al17 more than 90% of the time the trajectory visits only 8% (74 out of 907) and 30% (247 out of 832) of the low-energy isomers, respectively. Therefore, even at high temperatures, these clusters are a mixture of a limited number of isomers. The potential-energy distribution (F(E)) of the distinct isomers also indicates that clusters behave more like molecules than nanoparticles do. From the F(E) plots (Figure 9) it is clear that for particles smaller than Al19 the energy distribution of isomers is scattered (Al10-Al15) or has well-separated multiple peaks with comparable heights (Al16-Al18). Only for Aln particles with n g 19 does the distribution for high-energy isomers have a roughly Gaussian distribution, and these isomers can be viewed as being in one state, the liquid state. We can conclude that clusters have no melting transition, or in other words, they are too small to melt.31 The measured or simulated caloric curves of these clusters are better understood as resulting from molecule-like isomerization processes than from macroscopic-like melting transitions. 4.3. Solid, Slush, and Liquid States in Clusters and Nanoparticles. The discussion above raises a basic question: What are the solid and liquid states for clusters and nanopar￾ticles? Or even more basically, what is phase for clusters and nanoparticles?9 Due to the enormous number of atoms in a macroscopic system, the change from one phase to another phase is accompanied by a noncontinuous change in certain physical or chemical properties.9,11,20,21 The discontinuity is called a first￾order phase transition (φ) and characterized by a temperature Tφ. At Tφ, the two phases are in equilibrium and Keq ) 1. Thus, once these Tφ points have been determined as functions of pressure or volume, the regimes of the phases are known. However, for finite systems, especially clusters and small nanoparticles, one has to abandon the concept of a precise Tφ and recognize the fact that different “phases”, if there are any, of clusters and nanoparticles are in equilibrium with each other in an appreciable temperature range.9,11,15-18,25,37,41 The results for small clusters such as Al11-Al13 that are discussed above indicate the problematic character of using solely the caloric curves to characterize the melting transition. As particle size increases, the solid and liquid states of nanoparticles can be defined with less ambiguity. We start our discussion with the most macroscopic-like particle, Al300. For this particle, all properties give consistent results, implying that there is a state change at about 700 K (Figure 14): (i) c has a very high and sharp peak at about 700 K (Figure 14a). Before 600 K, c increases almost linearly with temperature. Between 760 and 880 K, c reaches a plateau (see also Figure S-3n in the Supporting Information) and then decreases almost linearly with temperature. (ii) Rg and V jump between 680 and 760 K (Figure 14b); aside from this region, they are both smooth functions of temperature. (iii)
increases almost linearly with temperature below 580 K (Figure 14c); after 560 K, the plot shows a small peak at 600 K, a valley at 640 K, and a high peak at 720 K. After the high peak,
drops and reaches a valley at 780 K. After 820 K, the plot shows an oscillatory behavior; averaging out these oscillations yields a
that increases almost linearly with temperature. (iv) The logarithm of κ (ln κ) increases almost linearly with temperature below 680 K (Figure 14d). Then it Figure 12. Percentage (P(E)) of quenched structures in the potential-energy range between E to E + δE with δE ) 0.05 eV: (a) Al11 and (b) Al13. The abscissa is the relative potential energy to the global minimum structure. Figure 13. Percentage (P(E)) of quenched structures in the three potential￾energy ranges corresponding to the three (Al11) or four (Al13) different regions as shown in Figure 9: (a) Al11 and (b) Al13. P(0) is the percentage population of the global minimum structures. 12708 J. AM. CHEM. SOC. 9 VOL. 130, NO. 38, 2008 ARTICLES Li and Truhlar