Melting Behavior of Noble-Metal-Based Bimetallic Clusters∗

The isothermal Brownian-type molecular dynamics simulation was applied to study the melting scenario of noblemetal-based bimetallic clusters. The failure of the simulation results to portend a compatible melting temperature, Tmelt, which is defined in the specific heat CV at its principal peak and in the Lindermann-like parameter at the temperature which it exhibits drastic change, has prompted us to calculate the velocity autocorrelation function VACF or its Fourier-transform, the power spectrum as another useful variable for describing cluster melting. Two bimetallic clusters, namely Ag13Au1 and Ag13Cu1, were selected for illustration. We effected comparative studies of the thermal and dynamic properties of the Ag13Au1, Ag13Cu1 and Ag14 and explored isothermally the VACF and spectral density of individual atoms in each of these clusters as well as the corresponding whole cluster. We examined, in particular, the influence of the impurity atoms Au and Cu. It is observed that the CV of Ag14 displays a prepeak which is absent in Ag13Au1 and Ag13Cu1. The physical origin of this prepeak feature was studied and its presence is ascribed to the migrational relocation of the adatom in the cluster. From the temperature change of VACF and spectral density, we deduced Tmelt. It is found that the calculated Tmelt inferred from VACF and power spectrum agrees quite well with that determined from the main peak position of CV . [DOI: 10.1380/ejssnt.2009.149]


I. INTRODUCTION
A bulk system in a solid phase would transform to a liquid phase if enough heat is transmitted into it until it attains a specific temperature, say T melt . If one were to examine its specific heat C V (T ) as a function of temperature, either experimentally or theoretically, one will find that C V exhibits a sharp peak at T melt , manifesting a thermodynamic first-order phase transition. If, in addition, one proceeds to calculate the relative root-mean-square bond length fluctuation parameter δ(T ) (Lindermann parameter), one will find also that δ increases abruptly at or very close to T melt . Thus the temperature T melt at which both C V and δ undergo drastic changes is commonly defined in the literature as the melting temperature of a bulk system. These congruous characteristic features of C V and δ at T melt are no longer observed for a finite system such as a cluster [1][2][3]. In our recent works [2,3], we have in fact learnt that δ is not a useful parameter for espying T melt since it increases anomalously at a temperature much lower than T melt and may even show rising behavior in multi-steps [2][3][4][5][6][7][8]. The cause for this anomalous behavior is now understood to originate with the permutational isomer transition or, in such case as the 14-atom cluster, the migrational relocation of adatom. In view of these studies, it would appear that there is still a need to seek for another 'phase-transition' variable. In this work, we draw attention to the velocity autocorrelation function (VACF) and its Fourier-transformed function, the power spectrum. These dynamical quantities were recently examined [3] to exhibit thermal variations that may be analyzed to pinpoint a T melt which is compatible with that deduced from C V . As concrete illustrations, we extend our investigation of noble-metal-based bimetal- lic clusters further to Ag 13 M 1 (M=Au and Cu) using the same Brownian-type molecular dynamics (MD) technique as we previously [3] applied to copper-based clusters.

A. Simulation Method
The isothermal Brownian-type MD has been previously introduced and we refer the interested readers to Refs. [1][2][3] for technical details and for formulas used in the simulation. To implement the MD algorithm, the manybody potential which accounts for the interactions between atoms is indispensable. Here we employ as in our previous works [1][2][3] the empirial n-body Gupta potential which is given by  The parameters A ij , p ij , q ij , ξ ij and r (0) ij for the Ag and Ag-Au are taken from Rapallo, et al. [9] and their numerical values are given in Table I. For the Ag-Cu, we refer to our previous article [3] and the work of Mottet, et al. [10] to which we consulted.

B. Thermal and Dynamical quantities
For the thermal and dynamical properties of interest in this work, we calculate the specific heat at constant volume C V and the velocity autocorrelation function C (i) for ith atom in the cluster; these quantities are given by the following equations. (a) Specific heat: The ith atom velocity autocorrelation function and power spectrum: The whole cluster VACF is ) from which we Fourier-transform to obtain the whole cluster Ω(ω). In equations above, M is the total time steps, n is the total number of atoms, v i is the velocity of ith atom in the cluster and ) is ith atom of type a (type b) and E n is the empirical many-body potential defined by Eq. (1). Note that at T = 500 K two atoms, surface or floating, have permuted separately with the center atom (orange−→blue).

III. RESULTS AND DISCUSSION
The ground state structure of both bimetallic clusters have their respective impurity atom resided at the center of the 13-atom icosahedron and their 14th atom "floated" at a site above three triangular shape atoms of which one is the apex atom and two others pertained to a nearby pentagonal ring. Figure 1 displays the C V for the clusters Ag 14 , Ag 13 Au 1 and Ag 13 Cu 1 and the T melt of these clusters which were determined separately from the main maxima of C V (indicated by arrows) are approximately 920, 1096 and 1022 K, respectively. It is observed in the same figure that there is a prepeak in the C V of Ag 14 but, in contrast, no prepeak is found for both bimetallic clusters. The presence of a C V prepeak in Ag 14 can be understood from examining the energy histograms and the cluster structures shown in Figs. 2-4. Generally it can be seen that in the range of temperatures T < 500 K the lowest and first excited states dominate. There are, however, discernible differences. In Ag 14 , the first excited state which describes an atomic structure in which the icosahedron is distorted to permit the adatom residing at a site above four atoms appears first at a lower tempera- ture T = 150 K. In the temperature range 150 ≤ T < 500 K the first excited state increases and is accompanied by the second excited state as well. This thermal behavior is in marked contrast to the bimetallic clusters. For the cluster Ag 13 Au 1 , it is in its lowest energy state up to T = 140 K. Its first excited state can be detected at T = 200 K but on the scale of Fig. 3 (indicated by an arrow) is invisible. Then we observe at T = 300 K the first excited state which persists up to T < 500 K. For this cluster, the second excited state appears at T = 500 K. The cluster Ag 13 Cu 1 behaves similarly except that the second excited state is not observed even up to T = 500 K. Accordingly the migrational relocation of the floating atom that leads to the switching of cluster configurations between the ground, first and possibly second excited states (see Figs. 2-4) is relatively easier and at a higher frequency in Ag 14 (recall the discussion above) than the two Ag-based bimetallic clusters. This structural swap among the lowest, first and second excited states which is in fact a dy- namical process would result in the conspicuous change of the total energy with temperature and hence the emergence of a prepeak at T 300 K. We should remark that at higher temperatures the migrational motion is superimposed by permutational isomer transitions involving the surface and floating atoms. We turn next to the C (i) (t) and its Fourier-transformed spectral density, Ω (i)) (ω). In Fig. 2, we show the structures of lowest energy and lower excited states of Ag 14 , and in the same figure next to it, we describe the probability of occurrence of these states as a function of temperature by the energy histograms. Our simulated results of C (i) and Ω (i) at different temperatures are displayed in Figs. 5-7. A general characteristic of C (i) and Ω (i) for temperatures T < 400 K is that all of fourteen individual atoms exhibit a typical solid-like behavior; the atoms vibrate at a low frequency value ω L 16 rad/ps and this oscillatory behavior is reminiscent of the infra-red molec- ular vibration. The atom at the center of the icosahedron has in addition a high frequency value ω H 41 rad/ps which may be ascribed to the atom being surrounded by an atomic wall. At temperatures T = 500 ∼ 700 K one observes permutations between the central atom and a surface atom. Perhaps more interesting is the change in magnitude of the central atom Ω (c) (ω H ) which declines with increasing temperature. This structural change of Ω (c) (ω H ) was observed previously [3] for Cu 14 and Cu 13 M 1 (M=Au and Ag) and has been used to define T melt . The definition is based on two criteria. The first criterion is related to the magnitude of the central atom Ω (c) (ω H ). Since it declines with increasing temperature, T melt is the temperature when Ω (c) (ω H ) 'dissolves' into the Ω (i) (ω) of the surface (i = s) and floating (i = f ) atoms. The second criterion is associated with the structure of Ω (i) (ω). At T melt , all of fourteen Ω (i) (ω) assume the same structure and share the same low frequency ω L . The simultaneous fulfillment of these two criteria yields ω L 13 rad/ps (second criterion) and T melt 900 K for Ag 14 . The predicted T melt agrees quite well with the T melt 920 K indicated by an arrow in Fig. 1.
In Figs. 8-10 and 11-13, we depict C (i) and Ω (i) of Ag 13 Au 1 and Ag 13 Cu 1 respectively. With respect to those of Ag 14 (see Figs. 5-7), we remark the following similarities and differences: (a) There exist high frequency modes ω H for the Ω (c) of Ag 13 Au 1 and Ag 13 Cu 1 ; the former is located at ω H 34 rad/ps which is lower than the latter ω H 39 rad/ps as well as that of Ag 14 which occurs at ω H 38.5 rad/ps. As in our previous interpretation on Cubased clusters [3], ω H is translated to come from the central atom being caged by the atomic wall of Ag atoms.
(b) Applying the two criteria to the Ω (i) as we did for Ag 14 and also scrutinizing Ω (Fig. 14(a)) of Ag 13 Au 1 , we obtain T melt 1100 K whose value is pretty close to the T melt inferred from C V . In arriving at this solidliquid-like transition temperature we should emphasize that the high frequency mode of Ω (c) decreases weakly with increasing temperature from say ω H 36 rad/ps at T = 100 K to approximately 33 rad/ps at T = 1000 K (temperature just before solid-liquid-like transition), although the magnitude of Ω (c) (ω H ) does not vary much and in fact remains low but undiminished. This characteristic feature of Ω (c) is similar to that of Cu 13 Au 1 [3] but is quite different from the pure Ag 14 cluster. One plausible cause is that the Au atom has a heavier mass which is about twice that of Ag. Since the small-time limit of VACF is inversely proportional to the mass of the atom [3,11], the vehement oscillatory frequency ω H of the Au atom within the cage of Ag atoms is relatively smaller and it goes on even at high temperatures. This scenario is consistent with the observation that the ω H values of the Ω (c) of Ag 14 are relatively higher varying from say ω H 41 rad/ps at T = 100 K to approximately 36 rad/ps at T = 800 K (temperature just before solid-liquid-like transition). The same characteristic feature was previously observed also for Cu 13 Au 1 [3]. There is, however, a slight difference for the latter cluster. Whereas in Cu 13 Au 1 the solid-like behavior is robust even at high temperatures (up to T 1200 K, signaled by the permutations among central and surface atoms) the solid-like behavior is apparently less severe in Ag 13 Au 1 .
Coming to Ag 13 Cu 1 , here the impurity atom Cu is characterized by a lighter mass (about 1.7 times less that of Ag). This cluster displays an entirely different picture in Ω (c) . The low frequency of Ω (c) assumes ω L 21 rad/ps about 7 rad/ps higher com-  pared with the ω L of the surface Ω (s) and floating Ω (f ) , and its high frequency value is now ω H 41 rad/ps at T = 100 K reducing to approximately 37.5 rad/ps at T = 1000 K (temperature just before solidliquid-like transition). Notice that the Ω (c) (ω H ) has a distinctly large magnitude. As a result, the first criterion used for stipulating T melt fails for this cluster. The reason for this anomalously large Ω (c) (ω H ) may be understood qualitatively by referring to Eq. (1). One can estimate the relative importance of the pair (first term) and many-body (second term) contributions by calculating the ratio p ij /q ij [12]. As analyzed generally in the work of Doye [12], one would anticipate a progression from potentials that favour icosahedra to decahedra to close-packed and finally disordered structure as p ij −→ 2q ij or p ij /q ij = 2. Now, in replacing a Ag atom by the impurity Cu atom, the ratio p AgAg /q AgAg = 3.41 increases to p AgCu /q AgCu = 3.81. This implies that there is a tendency for this bimetallic cluster to sustain the icosahedra structure and the central atom Cu whose mass is 1.7 times lighter than Ag atom would have robust contribution coming from the pairwise energy and hence larger Ω (c) (ω H ). This behavior is in plain contrast to Ag 13 Au 1 whose p ij /q ij ratio varies from p AgAg /q AgAg = 3.41 to p AgAu /q AgAu = 2.91. The many-body embedding energy is apparently more crucial.
To predict the T melt , we resort thus to analyzing Fig.  14(b) for the whole cluster Ω(ω) in the range of temperatures 800 ∼ 1200 K. It is readily seen that the high frequency ω H disappears at T 1100 K whose value is reasonably close to the T melt =1096 K determined from the main peak of C V .

IV. CONCLUSIONS
We have performed isothermal Brownian-type molecular dynamics simulation studies for the specific heat and VACF or its Fourier-transformed spectral density of Ag 14 , Ag 13 Au 1 and Ag 13 Cu 1 . We observed a prepeak structure in Ag 14 which is absent in the two Ag-based bimetallic clusters. Our analysis showed that the migrational relocation of the floating atom is the cause for such an interesting feature. For the VACF and power spectrum, we found that they can be used as additional variables for portending the melting temperature of clusters. Upon examining the temperature changes of VACF, Ω (i) and Ω of these three clusters and on the basis of these thermal variations, we predicted their melting temperatures; these predicted values are reasonably close to ones determined at the principal peaks of their respective specific heats. To gain deeper insight into the melting transition, it would be interesting to spur further investigation on such issue as scrutinizing the finite value of Ω (i) (ω) or Ω(ω) at ω = 0, a criterion widely used also in the bulk system to signal a transition to liquid state.