Nanosolids,Slushes,and Nanoliquids ARTICLES 1.25 Quantities that show different values in different phase-like forms of the material can be used as order parameters,and 120 ideally one could define suitable restricted thermodynamic 1.15 potentials in terms of such an order parameter that distinguishes the phases.The restricted thermodynamic potential in terms of 110 these parameters would exhibit a double minimum potential over some finite range of temperature or total energy:10 this would 1.05 identify the slush state with the coexistence region.Our system 1.00 does not appear to behave like an ideal system,but that 300600 90012001500 300 600900.12001500 framework does provide a starting point for further analysis of T(K T(K) 23.1 the finite analogs of first-order bulk phase transitions.Our current understanding is less quantitative;however,by analyzing -234 d results for large particles,we developed the following definitions 237 for the solid,liquid,and slush states of clusters and nanopar- M -24.0 ticles:A solid state is the state in which the properties c,B,and -24.3 In k increase almost linearly with temperature.A solid state -24.6 1000K date frted often has small volume and radius,and these properties are smooth functions of temperature.(1)A liquid state is the state 24.9 in which c decreases linearly with temperature,B linearly -25.2 300 increases with temperature or is close to constant,and In 600 980 12001500 300 600 900 12001500 T(的 increases linearly with temperature.A liquid state often has large volume and radius,and these properties are smooth functions Figure 14.Four properties used to characterize the melting transition of of temperature.(2)A slush state is between the solid state and Al3oo:(a)heat capacity,(b)Rg/Rgo)and V/Vo where Rgo)and Vo are the radius of gyration and volume,respectively,at 200 K,(c)coefficient of liquid states,and one or more of the three properties is not a thermal expansion,(d)natural logarithm of isothermal compressibility.The smooth function of temperature. sudden drop after 1600 K in b may be due to the abnormalities in the spline Applying the above definitions of the solid,liquid,and slush fitting functions to obtain dV/dT(see also Figure S-1)since volume is very states to all the particles we studied,Tr and Tm can be roughly smooth after 1600 K.The long dashed and short dashed lines in d are least- squares fits to data points below 600K and above 1000 K.respectively: determined.The results are listed in Table 1.The Tp values for the line at high temperatures has a slightly steeper slope than the one at each property are also listed,where a value in parentheses low temperatures. indicates a deep valley.The Tp values of In k for those particles without sharp peaks but just jumps or drops are determined as reaches a high peak at 700 K and a low peak (a shoulder in the the peak or valley positions of d In k/dT.The Tr and Tm values plot)at 740 K.After 760 K,it again increases almost linearly listed are all determined by drawing straight lines in the c,B, with temperature. and In k plots as we have done for Al3oo since it is hard to Note that an equilibrium mixture of coexisting isomers in a determine these points as unambiguously as one can determine nanoparticle is qualitatively different from the equilibrium the peak temperature To of c.Consequently,the Tf and Tm values between separated phases in a macroscopic system.Neverthe- listed in the table have large uncertainties due to the fact that less,based on the phase definition used for macroscopic phase transitions in finite systems occur in a broad coexistence systems,20.21 it is clear that for Alsoo there are two distinctly region,and there is no freezing or melting point in finite different states,one at low temperatures below 580 K and one systems.Of course,the values also involve uncertainties due at high temperatures above 880 K.At low temperatures the to the inexactness of the potential-energy function and neglect particle has a smaller radius,volume,coefficient of thermal of excited electronic states,but quantitative estimates of those expansion,and compressibility than does the particle at high uncertainties must await further studies. temperatures.Thus,the particle is in the solid state in the low- The Tr and Tm values in Table I can be used to ascertain temperature range and in the liquid state in the high-temperature whether a particle is in a solid,slush,or liquid state at a given range.In these two states,c,B,and In k are almost linear temperature:solid at temperatures below Tr,liquid at temper- functions of temperature after averaging out the weak periodic atures above Tm,and slush at temperatures between Tr and Tm. oscillations.The coexistence region is between 580 and 880 K. Many particles can be viewed as slush even at room temperature; Detailed analysis of these three properties indicates that,from these are marked with asterisks in Table 1.However,the melting one point of view,there is more than one"state"in this region of nanoparticles is different from that of macroscopic systems (for example,see Figure 14d).36.44 but the temperature window which are composed of separated solid and liquid spatial for each"state"can hardly be singled out by analyzing the three domains (like ice and water)in the coexistence region.For physical properties.Following the convention of Beck.Jellinek. nanoparticles,in the coexistence region (i.e.,in the slush state). and Berry,we shall call this situation a slush state. the solid-like and liquid-like structures are in rapid equilibrium In the previous paragraph,we roughly determined the freezing with each other.9.11-13.15.17.1841 From the P(0)values given in temperature Tr and melting temperature Tm for Al300.The same the Supporting Information as a function of temperature,it is analysis was applied to other large particles.For example,for possible to roughly identify how many solid-like structures Al2oo,Tm can be roughly determined as 900 K.We also find contribute to the slush state at a given temperature between Tr that Tr of Al2oo may be as low as 300 K because c,B,and In K and Tm. plots all show a rough region around 400 K,indicating some sort of structural transition in this region.For Al,Tr is about (110)Wales,D.J.;Berry,R.S.Phrys.Rev.Lett.1994.73,2875.Wales, 400 K and Tm is about 900 K.For Als5,Tr is about 480 K and D.J.Energy Landscapes;Cambridge University Press:Cambridge, Tm is about 960 K. 2003;pp440-452 J.AM.CHEM.SOC.VOL 130,NO.38,2008 12709reaches a high peak at 700 K and a low peak (a shoulder in the plot) at 740 K. After 760 K, it again increases almost linearly with temperature. Note that an equilibrium mixture of coexisting isomers in a nanoparticle is qualitatively different from the equilibrium between separated phases in a macroscopic system. Neverthe￾less, based on the phase definition used for macroscopic systems,20,21 it is clear that for Al300 there are two distinctly different states, one at low temperatures below 580 K and one at high temperatures above 880 K. At low temperatures the particle has a smaller radius, volume, coefficient of thermal expansion, and compressibility than does the particle at high temperatures. Thus, the particle is in the solid state in the low￾temperature range and in the liquid state in the high-temperature range. In these two states, c,
, and ln κ are almost linear functions of temperature after averaging out the weak periodic oscillations. The coexistence region is between 580 and 880 K. Detailed analysis of these three properties indicates that, from one point of view, there is more than one “state” in this region (for example, see Figure 14d),36,44 but the temperature window for each “state” can hardly be singled out by analyzing the three physical properties. Following the convention of Beck, Jellinek, and Berry,9 we shall call this situation a slush state. In the previous paragraph, we roughly determined the freezing temperature Tf and melting temperature Tm for Al300. The same analysis was applied to other large particles. For example, for Al200, Tm can be roughly determined as 900 K. We also find that Tf of Al200 may be as low as 300 K because c,
, and ln κ plots all show a rough region around 400 K, indicating some sort of structural transition in this region. For Al177, Tf is about 400 K and Tm is about 900 K. For Al55, Tf is about 480 K and Tm is about 960 K. Quantities that show different values in different phase-like forms of the material can be used as order parameters, and ideally one could define suitable restricted thermodynamic potentials in terms of such an order parameter that distinguishes the phases. The restricted thermodynamic potential in terms of these parameters would exhibit a double minimum potential over some finite range of temperature or total energy;110 this would identify the slush state with the coexistence region. Our system does not appear to behave like an ideal system, but that framework does provide a starting point for further analysis of the finite analogs of first-order bulk phase transitions. Our current understanding is less quantitative; however, by analyzing results for large particles, we developed the following definitions for the solid, liquid, and slush states of clusters and nanopar￾ticles: A solid state is the state in which the properties c,
, and ln κ increase almost linearly with temperature. A solid state often has small volume and radius, and these properties are smooth functions of temperature. (1) A liquid state is the state in which c decreases linearly with temperature,
linearly increases with temperature or is close to constant, and ln φ increases linearly with temperature. A liquid state often has large volume and radius, and these properties are smooth functions of temperature. (2) A slush state is between the solid state and liquid states, and one or more of the three properties is not a smooth function of temperature. Applying the above definitions of the solid, liquid, and slush states to all the particles we studied, Tf and Tm can be roughly determined. The results are listed in Table 1. The Tp values for each property are also listed, where a value in parentheses indicates a deep valley. The Tp values of ln κ for those particles without sharp peaks but just jumps or drops are determined as the peak or valley positions of d ln κ/dT. The Tf and Tm values listed are all determined by drawing straight lines in the c,
, and ln κ plots as we have done for Al300 since it is hard to determine these points as unambiguously as one can determine the peak temperature Tp of c. Consequently, the Tf and Tm values listed in the table have large uncertainties due to the fact that phase transitions in finite systems occur in a broad coexistence region, and there is no freezing or melting point in finite systems.11 Of course, the values also involve uncertainties due to the inexactness of the potential-energy function and neglect of excited electronic states, but quantitative estimates of those uncertainties must await further studies. The Tf and Tm values in Table 1 can be used to ascertain whether a particle is in a solid, slush, or liquid state at a given temperature: solid at temperatures below Tf, liquid at temper￾atures above Tm, and slush at temperatures between Tf and Tm. Many particles can be viewed as slush even at room temperature; these are marked with asterisks in Table 1. However, the melting of nanoparticles is different from that of macroscopic systems which are composed of separated solid and liquid spatial domains (like ice and water) in the coexistence region. For nanoparticles, in the coexistence region (i.e., in the slush state), the solid-like and liquid-like structures are in rapid equilibrium with each other.9,11-13,15,17,18,41 From the P(0) values given in the Supporting Information as a function of temperature, it is possible to roughly identify how many solid-like structures contribute to the slush state at a given temperature between Tf and Tm. (110) Wales, D. J.; Berry, R. S. Phys. ReV. Lett. 1994, 73, 2875. Wales, D. J. Energy Landscapes; Cambridge University Press: Cambridge, 2003; pp 440-452. Figure 14. Four properties used to characterize the melting transition of Al300: (a) heat capacity, (b) Rg/Rg(0) and V/V0 where Rg(0) and V0 are the radius of gyration and volume, respectively, at 200 K, (c) coefficient of thermal expansion, (d) natural logarithm of isothermal compressibility. The sudden drop after 1600 K in b may be due to the abnormalities in the spline fitting functions to obtain dV/dT (see also Figure S-1) since volume is very smooth after 1600 K. The long dashed and short dashed lines in d are least￾squares fits to data points below 600 K and above 1000 K, respectively; the line at high temperatures has a slightly steeper slope than the one at low temperatures. J. AM. CHEM. SOC. 9 VOL. 130, NO. 38, 2008 12709 Nanosolids, Slushes, and Nanoliquids ARTICLES