J. Phys. Chem. B 2007, 111, 2309-2312
2309
Melting and Glass Transition for Ni Clusters Yuyong Teng, Xianghua Zeng,* and Haiyan Zhang College of Physics Science and Technology, Yangzhou UniVersity, Yangzhou 225002, China
Deyan Sun Department of Physics, East China Normal UniVersity, Shanghai 200062, China ReceiVed: January 4, 2007
The melting of NiN clusters (N ) 29, 50-150) has been investigated by using molecular dynamics (MD) simulations with a quantum corrected Sutton-Chen (Q-SC) many-body potential. Surface melting for Ni147, direct melting for Ni79, and the glass transition for Ni29 have been found, and those melting points are equal to 540, 680, and 940 K, respectively. It shows that the melting temperatures are not only size-dependent but also a symmetrical structure effect; in the neighborhood of the clusters, the cluster with higher symmetry has a higher melting point. From the reciprocal slopes of the caloric curves, the specific heats are obtained as 4.1kB per atom for the liquid and 3.1kB per atom for the solid; these values are not influenced by the cluster size apart in the transition region. The calculated results also show that latent heat of fusion is the dominant effect on the melting temperatures (Tm), and the relationship between S and L is given.
1. Introduction The unusual properties of nanometer-sized clusters have generated a considerable interest as the melting transitions are quite different for the various metals. It is found that the melting temperature decreases dramatically with the decreasing radius of the cluster in the nanometer range.1 Then, some interesting phenomena, such as surface melting,2,3 the glass transition,4,5 solid-solid transitions,3,6 and coexistence between solid and liquid phases7 have been reported. Also, recently, experimental reports showed that small clusters of Sn and Ga in the size range of 17-55 atoms have higher than bulk melting temperature.8,9 Furthermore, it predicts that transition metal clusters have many exciting potential applications in nanoscale electronic devices and catalysis, but experimentally, there are no reports on the melting of the Ni cluster; thus, it attracts much attention. Actually, for the clusters with many atoms, especially for the transition metals, as there are so many valence electrons, it is impossible to calculate all electrons with quantum method; therefore, molecular dynamical simulations, a Monte Carlo (MC) method and an ab initio molecular dynamical simulation, have been used to simulate the thermal properties of different clusters. For nickel clusters, from small clusters (N < 23)10,11 to large clusters (N > 300)12,13, the thermal properties have been discussed by using molecular dynamics (MD) simulations; their works include the studies on the geometry structure of the ground state for small clusters (N < 23) and on the melting and crystallization for the mesoscale regime with N > 750 atoms. Because there are little works on the melting transition for the intermediate number of Ni clusters (N ) 50-150), in order to know more thermal properties of NiN clusters (N ) 29, 50-150) and compare with other kinds of clusters, the melting transitions of NiN clusters (N ) 29, 50-150) have been studied by using MD simulations with the quantum SuttonChen (Q-SC) many-body force field. First, the melting behavior for the Ni clusters will be examined, and then how the melting * Corresponding author. E-mail:
[email protected].
temperature and latent heat of fusion depend on cluster size and lowest energy structure will be carried out. In particular, the influences of latent heat of fusion and entropy of fusion on the melting temperatures and the relationship between the latent heat of fusion and entropy of fusion will be discussed. 2. Formulas and Simulations The structures of Ni clusters have been simulated by using different potentials: the embedded-atom method or related methods,14-17 effective-medium theory calculations,18 tightbinding molecular dynamics calculations,19 the Finnis-Sinclair potential,11 the Sutton-Chen potential,20 ab initio calculations,10,21 the many-body Gupta potential,22 and the quantum Sutton-Chen potential.12 Also, the quantum corrected SuttonChen (Q-SC) many-body force fields (FF)23 with parameters empirically fitted to data on density, cohesive energy, compressibility, and phonon have been applied to study melting, glass transformation, and crystallization for NiCu and CuAg alloys24 and the melting of Ni clusters12 with 336-8007 atoms. Here the quantum Sutton-Chen (Q-SC) force fields have been used to simulate the melting of the NiN clusters (N ) 29, 50-150), as there are fewer parameters, and it is feasible to compare with ready results12 by using the same potential. The potential V is given as follows
V)
[ () 1
a
n
∑i 2∑ j*i r
- cF1/2 i
ij
]
(1)
with
Fi )
() a
∑ j*i r
m
(2)
ij
where rij is the distance between the ith and jth atom, a is the lattice constant, c is a dimensionless parameter, is the parameter with dimension of energy, and m, n are integers. The
10.1021/jp070061k CCC: $37.00 © 2007 American Chemical Society Published on Web 02/10/2007
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Figure 3. Changes of the entropy of fusion with the latent heat of fusion. The solid line is the fit of S to a function of L.
Figure 1. (a) The energy as a function of temperature for the Ni147 cluster as well as snapshots at 0, 900, 910, 920, 930, and 940 K. The dashed line shows the fit of energy to a linear function of temperature for the solid and liquid clusters. (b) The energy as a function of temperature for the Ni79 cluster as well as snapshots at 0, 660, 670, and 680 K. The dashed line shows the fit of energy to a linear function of temperature for the solid and liquid clusters. (c) The energy as a function of temperature for the Ni29 cluster. The dashed line shows the fit of energy to a linear function of temperature for the solid and liquid clusters.
Figure 2. Upper panel: melting temperature as a function of the Ni cluster size. Lower panel: the latent heat of fusion as a function of the Ni cluster size.
square root term in the attractive part of the potential accounts for many-body interactions. For the Ni cluster, the parameters a, c, , m, and n are given as a ) 3.5157 Å, c ) 84.745, ) 7.3767 × 10-3 eV, m ) 5, and n ) 10, respectively.
Figure 4. (a) Energy as a function of temperature for Ni124 and Ni132 clusters. (b) Energy as a function of temperature for Ni102 and Ni129 clusters.
In order to find out the ground state structures of the NiN clusters with the Q-SC potential, we have carried out an extensive search started with various initial structures scaled from the Cambridge Cluster Database.25 The atomic structures are optimized by the steepest descend method for either highly symmetrical structures or low symmetrical structures, and the lowest energy structure of the NiN cluster is chosen as the ground state one. The NiN clusters are simulated using an MD Hamiltonian with constant temperature (T for Hoover),26 constant shape, and constant particle number without periodic boundary conditions. We choose 25 Å as the cutoff distance of the potential, which is much larger than the distances between the atoms for the small clusters. All the atomic trajectories were followed by integrating Newton’s equation of motion for each atom with the Verlet algorithm. The time step in the simulation is 1.0 fs; we have performed the MD run with 1 × 105 steps for equilibration, and the trajectories are recorded in the next 1 × 105 steps with a temperature increment of ∆T ) 100 K and near the melting region with ∆T ) 10 K. To obtain the latent heat of fusion (L) at melting temperature Tm, we fit the potential energy to a linear function of temperature in the solid and liquid phase near the transition region. The entropy of fusion (S) at the transition point is calculated with the relation of S ) L/Tm together with the calculated results for the latent heat of fusion.3,10 3. Results and Discussion A. Melting Transition. To study the melting process, the caloric curve E(T) provides a deep insight into the thermody-
Melting and Glass Transition for Ni Clusters
J. Phys. Chem. B, Vol. 111, No. 9, 2007 2311
TABLE 1: Melting Temperatures and Symmetrical Properties for NiN (N ) 60-77) N
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
group Tm(K)
Cs 540
C2V 560
Cs 520
C1 510
C2V 590
C2V 530
Cs 560
C2V 560
C3V 570
C2V 560
Cs 650
C2V 690
C1 670
Cs 640
C5V 680
D5h 730
Cs 680
C2V 690
namics of the system as shown in Figure 1. Figure 1a gives the caloric curve of Ni147 with T changing from 0 to 1200 K; at the same time, snapshots at 0, 660, 670, and 680 K are presented to explore the changes of the structure. It clearly shows that, except for the transition region of T ) 800-950 K, two linear curves can well fit to the calculated results; the specific heats are obtained from the reciprocal slopes of the curves, which are equal to 3.177kB per atom for the solid phase and 4.166kB per atom for the liquid, very close to the sodium clusters.5 The step in the caloric curves indicates the melting phenomenon, where the coexistence region is from 800 to 950 K and the melting point is around 900 K. At the ground state T ) 0 K, the Ni147 cluster has an icosahedral structure; with the temperature upgrade, the stable structure still keeps until T ) 900 K, and from 900 to 930 K, the formation of the liquid “skin” at the outer layer gradually emerges, which can be seen from the snapshots in Figure 1a; around 940 K, the entire cluster is melted. That means that the melting proceeds from the surface inward and the melting process for the central core is discontinuous: this is a so-called surface melting, which is similar to Ni552 and the mesoscale regime of Ni clusters for fcc structure.12 However, the surface melting is not a general phenomenon for the melting transition of the cluster. Similar to Figure 1a, in Figure 1b, the caloric curves and the snapshot of the Ni79 cluster are presented. At T ) 0 K, the Ni79 cluster is an fcc structure, and the snapshots at T ) 660 and 670 K show that the original structure still keeps until T ) 680 K, where the entire cluster is melted from the step of the caloric curves with the melting point equal to 680 K. Hence, Ni79 clusters melt directly without surface melting, which is different from the reported results12 for the mesoscale fcc Ni clusters. It is hard to say direct melting is related to the initial structure of the clusters, as the mesoscale fcc Ni clusters are melting from the surface inward. The mechanism of the direct melting needs to be studied further and be approved by the experiment. From the reciprocal slope of the curves, the specific heats c for the solid phase and the liquid phase are equal to 3.149kB per atom and 4.143kB per atom, respectively. In addition to the surface melting and direct melting, we have found there is a glass cluster. For a smaller cluster, Ni29 as presented in Figure 1c, at the ground state Ni29, has c3 symmetry structure, from T ) 0 to 400 K; it reflects the solid phase with specific heat c ) 3.118kB per atom; over 540 K, it reflects the liquid phases with specific heat c ) 4.138kB per atom. From low temperature to very high temperature, there is no jump or a sharp change in the caloric curves, so the latent heat is absent for Ni29 and the melting cannot be classified into the first-order phase transition as appeared in other Ni clusters. This phenomenon is similar to Al43 clusters4 and bulk glass, which is called a glass cluster. Also, the glass temperature Tg is 540 K, slightly smaller than 1/3 of the bulk melting point 1728 K and very close to the estimated result of 600 K in ref 12. Compared with the glassy transition of Al43 clusters, the atom number is smaller, and the coexistence region of solid and liquid phases is from 400 to 600 K, and for Al43 clusters, it starts at T ) 400 K and stops at T ) 900 K. Maybe, in the transition region, the heating rate of Ni29 is larger than that of Al43 clusters. For the three clusters Ni29, Ni79, and Ni147, the melting points are equal to 540, 680, and 940 K, respectively; it shows that the melting
temperatures vary with cluster size. Apart in the transition region, for three clusters, the specific heat is not influenced by the cluster size; it is about 4.1kB per atom for the liquid and 3.1kB per atom for the solid. Those results are consistent with some other works.4 B. Structure Effects. The transition for different structures of the cluster was studied to explore the structure of the cluster at the lowest-energy, as it is found that the melting behavior is related to the lowest-energy structures. For sodium clusters, it gave that the size dependence of the melting behavior primarily reflects the geometric structure of the solid clusters.27 For Al clusters, in the size range of 49 < N < 62,28 both polytetrahedral structures for the glue potential and icosahedral structures for the Gupta potential were simulated to study the melting behavior and gave that the melting transition of clusters with polytetrahedral structures is more akin to an ideal glass transition than the finite size equivalent of the bulk first-order melting transition, and icosahedral structures will tend to exhibit well-defined heat capacity peaks akin to bulk melting. To study the melting properties of the Ni cluster, we have systematically calculated NiN clusters with N changing from 50 to 150, and from the caloric curves of all clusters, the melting point temperatures have been extracted, as shown in the upper panel of Figure 2. From the figure, one can clearly find a well-defined peak of Tm at N ) 55, as the addition and reduction of one Ni atom at N ) 55 will greatly change the melting temperature; the melting temperature has an irregular variation with the cluster size, suffering a sharp drop at N ) 56, which was attributed to a structural transition, as found in other references.3,28 It reflects that N ) 55 should be a magic number with melting temperature Tm ) 850 K. Except for the region of N ) 55, on the whole, with the increasing of atoms, the melting temperature increases, and the melting temperatures of the NiN clusters (N ) 50150) range from 500 to 950 K, much lower than the experimental bulk melting temperature 1728 K29 and the theory bulk melting temperature 1700 K; therefore, the melting temperatures are effected by size. For N ) 60-77, there are several small peaks at N ) 61, 64, 68, 71, and 75, which means those clusters have higher melting temperatures and higher thermal stabilities than the neighborhood. As listed in the Table 1, the clusters with a higher melting temperature correspond to much higher symmetries. Compared to the clusters near N ) 61, the group for Ni61 has 4 symmetrical elements; with addition and reduction of one atom, its symmetry reduces with only one element. Ni75 has 20 symmetrical elements; its symmetry is the highest with the maximal melting temperature Tm ) 730 K. Therefore, the symmetrical structure greatly effects the melting, consistent with the results of Al clusters.28 C. Latent Heat of Fusion. To study whether the melting point is related to the energy or entropy, we have studied the relations of the melting point with both the latent heat and the entropy of fusion. The latent heat, which is the energy width of the step at the melting point, is plotted as a function of the atom’s number as shown in lower panel of Figure 2. One can find that the latent heat of fusion correlates with the melting temperatures and that the higher melting temperature corresponds to the larger latent heat of fusion due to the differences in the heat capacity of the solid and the liquid phase. The relations of entropy with latent heat are plotted in Figure 3,
2312 J. Phys. Chem. B, Vol. 111, No. 9, 2007
Teng et al.
and the solid line shows the fit of S to a function of L, which is given as
S ) 1/
(La + b)
(3)
where the fitted parameters a and b are equal to 492.56 K and 6259.31 K/eV, respectively. In Figure 3, the cross lines with arrows show that there are several entropies of fusion corresponding to the same latent heat of fusion and vice versa. To see clearly the influences of the entropy and latent of fusion on the melting, we have calculated the caloric curves of two groups of clusters: one group is Ni124, Ni132 in Figure 4a, and the other is Ni102, Ni129 in Figure 4b. On the one hand, Ni124 and Ni132 clusters with the same latent heat of fusion 0.049 eV correspond to the entropies of fusion 6.15 × 10-5 and 5.96 × 10-5 eV/K, respectively, and their melting temperatures are equal to 800 and 830 K, respectively. It shows that clusters with a smaller entropy of fusion have a higher melting point. On the other hand, Ni102 and Ni129 with the same entropy of S ) 6.15 × 10-5 eV/K are corresponding to the latent heats of 0.044 and 0.051 eV, respectively, the melting temperature for the former is 720 K, and for the latter is 830 K, their difference is 110 K, which indicates that high latent heat of fusion corresponds to a large melting temperature, and the latent heat of fusion is the dominant effect for the melting. 4. Conclusion We have studied the melting of Ni clusters by using an MD method with a Q-SC potential. It turns out that the melting phenomena vary with the different Ni clusters, as described in the paper: surface melting, direct melting, and the glass transition exist; that is, Ni79 clusters melt directly without surface melting, and Ni29 is a glass cluster with the glass temperature Tg ) 540 K. As for the melting of cluster NiN (N ) 50-150), it turns out that the melting temperatures are not only sizedependent but also have a symmetrical structure effect; i.e., the larger the number of atoms, the higher the melting point will be, and in the neighborhood of the clusters, the higher the symmetry of the cluster, the higher the melting point will be. Finally, calculations show that the latent heat of fusion is the major effect on the melting points; the entropy of fusion also influences on the melting, but it is correlated with the latent
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