J. Phys. Chem. B 2001, 105, 12857-12860
12857
Three Distinctive Melting Mechanisms in Isolated Nanoparticles S. J. Zhao,* S. Q. Wang, D. Y. Cheng, and H. Q. Ye Laboratory of Atomic Imaging of Solids, Institute of Metal Research, Chinese Academy of Sciences, Shenyang 110016, China ReceiVed: July 10, 2001; In Final Form: September 28, 2001
The melting properties of size-selected isolated silver nanoparticles of N ) 13-3871 atoms are studied by using molecular dynamics simulations. Three distinctive melting mechanisms are identified. The melting of Ag258-3871 can be explained well by the surface premelting models (SPMs), while the “melting” of Ag13-178 may be described as a transition from a low-energy solidlike structure at low temperatures to a higher-energy liquidlike structure at high temperatures. Acting as a connecting link between such two distinctive melting mechanisms, a new intermediate melting mechanism, in which the melting temperature Tm depresses very slowly while the latent heat of fusion ∆uls has a great enhancement with N decreasing, is identified in Ag120-240.
Introduction There exists a strong need for a microscopic description of the melting process of small particles, since it is not only of scientific interest, but also has some technological implications.1 It has long been known that the particle-size-dependent melting point (Tm) depression occurs when the particle size is on the order of nanometers since it was first reported by Pawlow.2 Recent experiments by Lai et al.3 and Bachels et al.4 revealed that the latent heat of fusion ∆uls also reduced for nanoparticles compared to that of bulk materials. It is crucial to investigate the size dependence of Tm and ∆uls depression in order to facilitate a comprehensive understanding of the melting process of finite material systems.3,4 Several phenomenological models2,5,6 have been proposed to account for the melting point depression. Almost all considered the particle as consisting of “bulk” and “surface” atoms. Nanocalorimetric measurements of the melting properties of both supported tin particles,3 and isolated tin clusters4 demonstrated that the melting point depends nonlinearly on the inverse particle radius r-1, which contrasts with the traditional description of the melting behavior of small particles.2,5,6 The surface premelting models (SPMs) presented by Kofman et al.7 and Sakai,8 which were improved significantly compared to the previous ones, seem to be able to give us a better understanding of the Tm depression for larger particles, whereas for particles smaller than a critical radius rc, the melting cannot be defined in terms of classical thermodynamics. Under these circumstances, phase separation does not occur,9 and the SPMs are not applicable any more. The absence of phase separation leads to certain thermodynamic features that are peculiar to small particles. One is that an ensemble of small particles in the melting region is a mixture of solidlike and liquidlike forms. If one were to observe a single particle, it would be seen to occasionally change form between solidlike and liquidlike. There is a dynamic coexistence of the two states. This has been observed in simulation “experiments”,10,11 and several theories12,13 have been advanced to describe this process. The significant change in the melting process for small neutral particles with r < rc can not only be indicated by the reduced melting temperature but also by the * Corresponding author. E-mail:
[email protected].
enhanced latent heat of fusion.4 However, so far, the particlesize-dependent variations in Tm and ∆uls for small neutral particles with r e rc (especially with r ) rc) is very ambiguous due to few experimental data and the absence of phenomenological model applicable. In this paper, we present extensive molecular-dynamics (MD) simulations of isolated silver particles of N ) 13-3871 atoms. For the first time, the size-dependent variations in Tm and ∆uls for small neutral particles with r e rc are presented, in which three distinctive melting mechanisms are clearly identified. Simulation Method In all of our MD simulations, we used an analytic embeddedatom method (EAM) type potential,14 which was confirmed to be able to describe fractional density change on melting, heat of fusion, linear coefficients of thermal expansion, and heat capacities above room-temperature successfully.14 In constanttemperature simulations by using the damped force method,15 the initial configurations of all isolated particles of N ) 133871 atoms are arranged as the truncated Marks decahedra.16 The simulations were performed by starting at 200 K, at which the system was run for 500 000 integration time steps (∼100 ps), and then the temperature was elevated at a heating rate of 1011 K/s by rescaling the atomic velocities using the Verlet velocity algorithm. After the desired temperature arrived, the atomic configurations were recorded after equilibrating for 500 000 integration time steps. For each of the recorded configurations, another run of 100 000 time steps at the corresponding temperature was performed in order to determine the thermodynamic properties of the system. Results and Discussion Figure 1 shows the caloric curves for the particles with size N ) 13-3871 atoms. Typical signatures of melting obtained from the caloric curves are (i) an upward jump in energy, corresponding to the latent heat of fusion ∆uls, which tends to decrease with reducing particle size for N ) 258-3871 and continues to decrease after an intermittence at N ) 178 where ∆uls has an increase in comparison with that at N ) 258, and
10.1021/jp012638i CCC: $20.00 © 2001 American Chemical Society Published on Web 11/30/2001
12858 J. Phys. Chem. B, Vol. 105, No. 51, 2001
Zhao et al.
Figure 1. Temperature dependence of energy for isolated particles containing 13-3871 silver atoms.
(ii) a continuous decrease of heat capacity C(T) (the slope of the caloric curves) with reducing N until N ) 178, at which C(T) is larger than that at N ) 258. All these results show that two distinctively different melting processes have occurred. For the particles of N ) 258-3871 atoms, the surface layers begin to melt at a critical temperature Tc which is below the melting point Tm of the particle. The thickness of the liquid layer, which depends on local curvature, increases continuously with temperature until a uniform curvature of the solid core is attained at Tm, whereas for the smallest systems of N ) 13-116 atoms, the particles will undergo a transition from a low-energy solidlike structure at low temperatures to a higher-energy liquidlike structure at high temperatures.10-12 In addition, an intermediate melting mechanism, acting as a connecting link between such two distinctive melting mechanisms, exists when the particle size N ≈ 120-240 atoms (radius r ≈ 10.5 ( 1 Å). What happens in those breaks in the caloric curves for the particles with different size? For N ) 258-3871, the particles have a higher proportion of surface atoms, which are more weakly bound and less constrained in their thermal motion17,18 than those in solid core. At high temperatures, some atoms promote out of the outer layers and thus elevate the system’s energy. These promoted atoms are also the ones responsible for liquidlike values of some diagnostic functions (e.g., see the following calculation results of the static structure factor). At Tm, upon a combined effect of thermal fluctuations and an intrinsic elastic instability in the system,19 the crystalline structure of the particle (including the solid core) collapses, and the system properties such as the potential energy and the volume show a stepwise increase, while for N < 258, the melting of the particles cannot be defined in terms of classical thermodynamics and the situations may be very different. This will be addressed in our future research work. In the following, we focus on the investigation of the size dependence of the thermodynamic behaviors of the particles in the three distinctive melting. We first analyze the melting process of the particles of N ) 258-3871 atoms. With respect to the SPM,7,8,20 the melting temperature is taken to be the temperature of equilibrium between the solid sphere core and the concentric liquid shell of a given critical thickness t0, which is an adjustable parameter.3,20 Neglecting the difference between the vapor pressure at the surface of the liquid layer at Tm and the pressure at flat liquid
Figure 2. (a) Melting point of isolated silver particles as a function of the inverse particle radius. The temperature region between the two dotted lines corresponds to the critical temperature region. Inset: r dependence of the melting points of isolated silver particles of N g 258 atoms. The solid line is calculated in terms of eq 2. (b) Latent heat of fusion ∆uls as a function of the inverse particle radius. The experimental value of ∆uls () 0.118 eV/atom) at the bulk melting point of solid is also shown, and the dashed line is only guide for the eye.
surface at To (To is the bulk melting temperature), one obtains:3,20
T0 - T m )
2To ∆µlso
[
σsl
Fs(r - to)
+
(
)]
σlv 1 1 × r Fs F1
(1)
where r is the particle radius, ∆µlso is the latent heat of fusion of bulk solid, and t0 is the critical thickness of the liquid layer at Tm. σsl is the interfacial surface tension between the solid and the liquid, while σlv is that between the liquid and its vapor. Fs and Fl are the densities of the solid and liquid, respectively. Substituting To ) 960.7 °C, ∆µlso ) 1.06 × 109 erg/g, Fs ) 10.49 g/cm3, Fl ) 9.35 g/cm3,21 σsl ) 184 dyn/cm, and σlv ) 910 dyn/cm22 into eq 1, we obtain:
Tm ) 960.7 - 2463 *
(
)
1 1 0.603 * (r - t0) r
(2)
where Tm is in °C and r and t0 are in Å. By adjusting the value of t0, we obtain a best fit to the simulated data in terms of eq 2 when t0 )10.1 Å (see the curves in inset in Figure 2a), which is close to the value (∼10.5 ( 1 Å) directly estimated from the caloric curves. All these results show that the melting process of Ag258-3871 can be described well by the SPM. For particles of r ≈ 10.5 ( 1 Å, the melting
Melting Mechanisms in Isolated Nanoparticles
J. Phys. Chem. B, Vol. 105, No. 51, 2001 12859
Figure 3. Shell-by-shell profiles of static structure factor S for Ag923 at 975, 1000, and 1025 K.
temperature is calculated to be about 368 ( 7 °C, which is shown in Figure 2a by two dotted lines. This sets a lower limit for the particle size at which the melting process can be described by the SPM. In Ag258-3871, melting first occurred at the surface overlayer, and after the outer shell of the particle became into a liquid layer, melting of the whole particle started from the liquid layer to core region very quickly. This can be identified by the calculations of the static structure factor S. In the calculations of S, the particles were divided into a series of concentric shells. The static structure factor S, defined as
S)
1 N
|
2
2 exp(ik B‚bj ∑ r )| j∈L
(3)
is calculated for each of the concentric shells. In the above expressions, L specifies the particular shell, and N is the number of atoms in shell L. k is a prescribed wave vector, and rj is the position of atom j that belongs in shell L. At zero temperature, S equals unity, while in the liquid, it fluctuates close to zero.23 Figure 3 shows the shell-by-shell profiles of S for Ag923 at 975, 1000, and 1025 K. From the core region to the surface, the numbers of the shells are denoted by 1, 2, ..., 12, respectively. It is clear from Figure 3 that melting initiate at the surface. For the two outmost shells, the S values already reaches liquidlike values around a critical temperature Tc ≈ 975 K. When the temperature rises further, the decrease of S is clearly related to the progressive loss of crystalline order from the surface to the core region. When the temperature reaches about 1025 K, the liquid seed already present propagates to the whole particle. While for Ag13-116 in a finite temperature range the S value cannot be determined because it fluctuates quickly between 0 and 1 with the system evolving with time, in such a temperature range, the small system corresponds in statistical thermodynamics to an ensemble that contains two distinct types of states, i.e., solidlike and liquidlike forms, for a small particle. This is a dynamic equilibrium, with individual particles passing between two states, like isomers. Note that this is different from the coexistence of macroscopic phases, which corresponds to having two distinct phases simultaneously present in equilibrium in the same system. Experimental24 and theoretical12 studies show that such a small system may exhibits sharp lower limit of temperature Tf for the thermodynamic stability of the liquidlike form and a higher, sharp upper limit Tm for the thermodynamic stability of the solidlike form. Under these circumstances, the small particle will undergo a transition from a low-energy solidlike structure at Tf to a higher-energy liquidlike structure at Tm.10-12 In the transition temperature range between Tf and
Tm, the two forms are to be observed in equilibrium like two phases or two chemical isomers. Such a transition would be a kind of phase change different from any ordinary phase transition of bulk matter, of any order.10 Here we call such a transition a dynamic coexistence (DCE) “melting”. Such a DCE melting process can also be identified by calculations of the latent heat of fusion ∆uls. The size-dependent depression of ∆uls of small particles, which was first observed by Lai et al.,3 can be reproduced in our MD simulations for the particles of N ) 258-3871 atoms (see the curves in the SPM melting region in Figure 2b). However, when the particle size is reduced further, ∆uls enhancement can be clearly observed (e.g., compare the ∆uls value of Ag55 with that of Ag258 in Figure 2b), which is in qualitative accord with the calorimetric measurement for isolated Sn nanoparticles by Bachels et al. recently.4 It should be noted that in that experiment the conclusion of ∆uls enhancement was drawn from comparing its value of isolated Sn nanoparticles in a molecular beam method4 with that of supported Sn particles formed by evaporation on inert substrate by Lai et al.3 Our simulation results confirm that the latent heat of fusion enhancement is a more general process for neutral particles with r < rc, which is independent of the experimental method as well as the elements investigated. In addition, our results show that an intermediate melting mechanism, acting as a connecting link between the two distinctive melting mechanisms mentioned above, exists at N ≈ 120-240 (or N -1/3 ≈ 0.161-0.203) (see Figure 2a,b). Under these circumstances, Tm depresses very slowly with N decreasing (see the curves in the intermediate melting region in Figure 2a), while ∆uls has a great enhancement (see the peak in the intermediate melting region in Figure 2b), which probably has some important technological implications, for instance, in sintering processes of ultrafine powders. It is interesting to mention that in a caloric measurement of ionized sodium particles of N ) 70-200 atoms in a vacuum, ∆uls was also found to exist a pronounced maximum at the particle size region near 147 atoms.1 Moreover, in our simulations, the value of ∆uls of Ag147 is reduced by 39% compared to that of bulk Ag (see Figure 2b), which can also be compared with the measurement (reduced by about 32% compared to bulk material) of isolated ionized sodium nanoparticles.1 For the particles of N ) 258-3871 atoms, surface melting has been shown to play a key role in the melting. As expected for finite-size systems, the first-order character of the bulk solid-liquid transition is altered by the presence of surface melting. When the particle size decreases further, the interface energy required to form two distinct phases simultaneously may be significant, and it will not be negligible any more.25 The small systems may not be able to tolerate the presence of an interface between two phases because of the relatively large positive contribution of the interface energy to the free energy. Consequently, small particles may not exhibit two distinct phases simultaneously in equilibrium. Below Tf , only the solidlike form is stable; above Tm, only the liquidlike form is stable. In the intermediate temperature range, a collection of such small particles in thermodynamic equilibrium behaves like a statistical ensemble that is a mixture of two kinds of particles: solidlike and liquidlike. In thermodynamic equilibrium, such two states occur in a ratio K ) [solid]/[liquid] ) exp(-∆F/kT) which is fixed by the difference in free energy ∆F between the solid and liquid states.12 Under these circumstances, the small particle will undergo a DCE melting transition, which would be a kind of phase change different from any ordinary phase transition.10
12860 J. Phys. Chem. B, Vol. 105, No. 51, 2001 Acting as a connecting link between such two distinctive melting, the intermediate melting occurs during a finite critical particle-size region. There is probably a complicated interplay and competition between such two distinctive melting in the intermediate melting region, and further investigations, in particular theoretical approaches, for the mechanism of the intermediate melting is of urgent need. Acknowledgment. We acknowledge financial support of this work by the Special Funds for the Major State Basic Research Projects (No. G2000067104) and the National Natural Science Foundation (No. 59831020) of China. References and Notes (1) Schmidt, M.; Kusche, R.; Issendorff, B. V.; Haberland, H. Nature (London) 1998, 393, 238. (2) Pawlow, P. Z. Phys. Chem. 1909, 65, 1. (3) Lai, S. L.; Guo, J. Y.; Petrova, V.; Ramanath, G.; Allen, L. H. Phys. ReV. Lett. 1996, 77, 99. (4) Bachels, T.; Gu¨ntherodt, H.-J.; Scha¨fer, R. Phys. ReV. Lett. 2000, 85, 1250. (5) Reiss, H.; Wilson, I. B. J. Colloid Sci. 1948, 3, 551. (6) Buffat, Ph.; Borel, J.-P. Phys. ReV. A 1976, 13, 2287. (7) Kofman, R.; Cheyssac, P.; Aouaj, A.; Lereah, Y.; Deutscher, G.; David, T. B.; Penisson, J. M.; Bourret, A. Surf. Sci. 1994, 303, 231. (8) Sakai, H. Surf. Sci. 1996, 351, 285. (9) (a) In Large Clusters of Atoms and Molecules; Martin, T. P., Ed.; Kluwer Academic: Dordrecht, 1996. (b) In Microscale Energy Transport;
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