Structural and Shape Transformation of Gold Nanorods studied by molecular dynamics simulation

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1 Structural and Shape Transformation of Gold Nanorods studied by molecular dynamics simulation Yanting Wang, S. Teitel, and Christoph Dellago Center for Biophysical Modeling and Simulation and Department of Chemistry, University of Utah, 315 South 14 East Room 22, Salt Lake City, Utah , and Department of Physics and Astronomy, University of Rochester, Rochester, New York 14627, and Institute for Experimental Physics, University of Vienna, Boltzmanngasse 5, 19 Vienna, Austria (Dated: May 27, 26) Upon heating elongated gold nanocrystals (nanorods) transform to a shorter and bulkier shape well below melting. Using molecular dynamics simulations we have studied this process for three nanorods consisting of about 3 atoms with similar shape but different exposed surfaces. Our results show that the shape/structure transformation strongly depends on the particular type of surface facets covering the nanorods. While nanorods with {111} surfaces maintain their shape and structure up to the melting temperature, those covered with {11} and {1} facets undergo a transformation during which a change in aspect ratio is accompanied by a reorientation of the underlying face-centered-cubic lattice. This reorientation leads to a crystal mainly covered with highly stable {111} facets. While the transitions of the rods with {11} and {1} surfaces occur through essentially the same sliding plane mechanism, they are initiated at the surface and at an edge, respectively. I. INTRODUCTION The shape and interior structure of nanoscale materials are intimately linked due to their large surface-tovolume ratio. In view of the envisaged technological applications of such materials it is of great importance to understand the effect that shape and surface structure can have on the mechanical and themodynamical properties of nanoparticles. It has been shown recently in experiments 1 3 and simulations 4 7 that in gold nanoparticles the competition between surface and bulk free energetics can drive shape changes and stabilize particular structures. Since surface processes such as roughening and surface melting strongly influence the surface free energetics, they most likely play an important role in the stability of nanosized gold clusters. Bulk metal surfaces can undergo numerous reorganization processes, such as surface relaxation, reconstruction, deconstruction, pre-roughening, roughening, disordering, and melting with increasing temperature 8. Experiments performed by Hoss et al. 9 indicate that the gold {11} surface roughens at 68 K, and melts at 77 K, below the bulk melting temperature of T m = 1337 K 1. Molecular dynamics simulations of this surface carried out by Ercolessi et al. 11 with the glue model 12 found that a liquid- University of Utah University of Rochester University of Vienna Author to whom correspondence should be addressed. Christoph.Dellago@univie.ac.at. like surface disorder appears near T 1 K, below the melting temperature T m = 1357 K of the glue model. In both cases the thickness of the liquid film grows with temperature and diverges at the bulk melting temperature. In contrast, the well-packed gold {111} surface does not melt up to the bulk melting temperature T m as found in molecular dynamics simulations by Carnevali et al. 13. This finding is confirmed by experiments that indicate that the gold {111} surface does not melt up to at least 125 K 14,15. Unlike these two surfaces, the gold {1} surface exhibits a thin, disordered surface flim at T > 117 K as revealed by X-Ray experiments 1,16 The thickness of the liquid film, however, does not grow with temperature. The same incomplete melting of the Au {1} surface was found in simulations by Bilalbegović and Tosatti 17. Recent computer simulations 4,5 indicate that surface instabilities may also play a central role in the transformation of gold nanorods from a long to a shorter and wider shape observed in laser heating experiments 2,3. This shape shape change is accompanied by a bulk reorganization of the nanorod in which the underlying fcclattice reorients in such a way to align with newly formed and particularly stable {111} surface facets. In the simulations, the shape and structural change is initiated at the {11} surface facets possibly related to the roughening transition of these facets. In particular, roughening of the {11} facets may lower or even remove the activation barrier opposing the transformation to the lower free energy state. Similar reconstructions in gold nanorods with different size and geometry were also observed the simulation work of Diao, Gall and Dunn 6,7, who attribute the

2 2 transition to tensile surface stresses. In this paper we report on molecular dynamics simulations that we have carried out to clarify the role played in the transition by the different surface facets. For this purpose, we consider gold nanorods with their sides covered only by one kind of low-index facets, namely, gold {111}, {11}, or {1} facets. These simulations confirm that gold nanorods covered with {111} facets are particularly stable and that nanorods initially covered with other surface facets undergo a shape and structure transformation possibly related to instabilities of the facets or the edges. Some preliminary results of this study have already been included in Ref. 5. The remainder of the paper is organized as follows. In Sec. II we describe the model and the algorithm used to identify surface and bulk atoms with different local structure. The initial structures of the three different nanorods are specified in Sec. III. In Sec. IV we present the results of continuous heating simulations and in Sec. V those of quasi-equilibrium heating simulations, in which the system is heated at a gentler rate. A discussion of the results and some conclusions are provided in Sec. VI. II. MODEL AND METHODS All our simulations were carried out with the so-called many-body glue potential that yields a satisfactory description of the bulk, defect and surface properties of gold 12. In the glue model, the potential energy of a system of N atoms consists of a sum of pair potentials and a many-body glue energy: V = 1 2 i j i φ(r ij ) + i U (n i ). (1) Here the sums run over all particles, r ij = r i r j is the interatomic distance between atoms i and j, and φ(r) is the pair interaction energy. The many-body glue energy U(n i ) depends on the coordination numbers n i defined for all atoms, is a cone region with no other particles inside. A particle is considered to be on the surface if at least one associated hollow cone can be found for this particle. The parameters a = 5 Å and θ = π/3 permit a robust identification of the surface atoms. At high temperatures the shape of nano-sized clusters fluctuates strongly and it is useful to consider the average cluster shape. For spherical clusters, such as icosahedral gold clusters, the 4π solid angle division can be used to calculate the average shape 19. For a gold nanorod, however, due of the large aspect ratio, this method would yield a very nonuniform mesh. Instead of determining the average three-diemensional shape we therefore consider the average cross-section perpendicular to the long axis (the ẑ axis) of the rod. For each instantaneous configuration, the end caps of the rod are discarded, and the remaining surface atoms are projected onto the x y plane. The average cross-sectional shape is then obtained by dividing the x y plane into 1 even polar angles centered at the center of mass of all surface atoms, and averaging the x and y positions for these atoms in each angular division. During the heating process the shape change of the gold nanorods was monitored by calculating the radius of gyration R g (t) = 1 N [r i (t) r c ] 2, (3) where the sum runs over all particles and r c is the center of mass of the rod. The radius of gyration is large for rods with large aspect ratio and reaches its minimum value for a spherical shape. The time evolution of the nanorod structure was followed by calculating bond order parameters 2. The general idea of bond order parameters is to capture the symmetry of bond orientations regardless of the bond lengths. Bonds are defined as the vectors joining a pair of neighboring atoms with an interatomic distance of less than a certain cutoff radius (3.7 Å in our case). The local order parameters associated with a bond r are a set of numbers i n i = j ρ (r ij ), (2) Q lm (r) Y lm (θ(r), φ(r)), (4) where ρ(r) is a short-ranged monotonically decreasing function of the interatomic distance r. Cutoffs of 3.9 Å for ρ(r) and 3.7 Å for φ(r) are used and the equations of motion are integrated with the velocity Verlet algorithm 18 with a time step of 4.3 fs. A cell index method 18 is used to reduce the computational time from order N 2 to order N. To identify surface atoms and distinguish them from atoms in the interior of the nanorod we have developed the so-called cone algorithm 19. For a given particle, an associated cone region is defined as the region inside a cone with a certain side length a and a certain span angle θ, whose vertex resides on the particle. A hollow cone where θ(r) and φ(r) are the polar and azimuthal angles of the bond with respect to an arbitrary reference frame and Y lm (θ(r), φ(r)) are spherical harmonics. Only even-l spherical harmonics are considered so that they are invariant under inversion. Global bond order parameters can then be calculated by averaging Q lm (r) over all bonds: Q lm 1 N b bonds Q lm (r), (5) where N b is the number of bonds. To make the order parameters invariant with respect to rotations of the ref-

3 3 TABLE I: for 3D face-centered-cubic (fcc), hexagonal-close-packed (hcp), simple-cubic (sc), bodycentered-cubic (bcc), liquid, and 2D gold {11} surface, gold {1} surface, and gold {111} surface structures. Geometry Ŵ 4 Ŵ 6 fcc hcp sc bcc liquid 11 surface surface surface erence frame, the second-order invariants are defined as Q l 4π 2l + 1 l m= l and the third-order invariants are defined as W l m 1, m 2, m 3 m 1 + m 2 + m 3 = Qlm 2, (6) ( ) l l l Q m 1 m 2 m lm1 Q lm2 Q lm3, 3 (7) where the coefficients ( ) are the Wigner 3j symbols?. Furthermore, reduced order parameters almost independent on the precise definition of nearest neighbors can be defined: Ŵ l W l ( m Q lm 2) 3/2. (8) We used the four bond order parameters,, Ŵ 4 and Ŵ 6 together to identify structures accurately. The values of these bond order parameters for fcc, hcp, icosahedral, and liquid structures are listed in Table I?. averaged over all bonds can be used to monitor global structural changes. Surface structures of nanomaterials, however, are often quite different from bulk structures. We therefore calculated bond order parameters for the internal atoms and the surface atoms separately. Here, the surface atoms are identified by the cone algorithm mentioned above. The values of the bond order parameters for low-index planar facets are also given in Table I. Based on these bond order parameter values, the following criteria are used to identify the atoms with different local structures. For interior atoms: fcc if >.17 and Ŵ4 <.1; hcp if <.13 and Ŵ4 >.7. For surface atoms: on {111} facet if.7 < <.9 and.8 < Ŵ6 <.2; on {1} facet if <.7 and Ŵ 6 >.2; on {11} facet if >.9 and Ŵ6 <.8. FIG. 1: Initial configuration of the {111} rod relaxed at 5 K. (a) View down the long axis; (b) Side view; (c) Cross-sectional view parallel to the long axis. In (a) and (b), golden atoms represent {111} facets, green atoms represent {1} facets, and gray atoms are on the edges. In (c), golden atoms have a local fcc structure, green atoms have a local hcp structure, and gray atoms have neither. The atoms in the following figures will be colored accordingly. For interior atoms: golden atoms have a local fcc structure, green atoms have a local hcp structure, and gray atoms are neither. For surface atoms: golden atoms are {111} facets, green atoms are {1} facets, red atoms are {11} facets, and gray are others. III. INITIAL CONFIGURATIONS The initial configurations of the gold nanorods where carved out from an fcc-crystal in three different ways to obtain gold rods mostly covered by {111}, {11}, and {1} facets. These manually constructed rods went through a constant temperature MD simulation at T = 5 K for 1 6 steps (4.3 ns) to relax the surface. The potential energy of the final configuration was then minimized using the conjugate gradient method 24 to quench local thermal fluctuations. A. {111} Rod The initial configuration for the gold nanrod mostly covered by gold {111} facets has its long axis along the 111 direction and has an aspect ratio of about 3. This rod has a cross-sectional area of about 2.8 nm 2.8 nm and consists of 3411 atoms. The relaxed and minimized rod at 5 K, colored by its surface bond order parameters, is shown in Fig. 1. The rod is mostly covered by gold {111} facets, with very small {1} facets connecting them along the side and at the cap ends. This rod will be referred to as {111} rod. B. {11} Rod The second rod has its long axis in the 1 direction and is covered with {11} facets on the long sides and with {1} facets on the short sides. This rod has

4 4 27 Rod {111} R g (Å) Rod {1} 18 Rod {11} FIG. 2: Initial configuration of the {11} rod relaxed at 5 K. (a) View down the long axis; (b) Side view; (c) Cross-sectional view parallel to the long axis. In (a) and (b), green atoms represent {1} facets, and most gray atoms are actually on {11} facets. In (c), golden have a local fcc structure, green atoms have a local hcp structure, and gray atoms have neither. FIG. 3: Initial configuration of the {1} rod relaxed at 5 K. (a) View down the long axis; (b) Side view; (c) Cross-sectional view parallel to the long axis. In (a) and (b), green atoms represent {1} facets, and gray atoms are on the edges. In (c), golden atoms have a local fcc structure, green atoms have a local hcp structure, and gray atoms have neither. an aspect ratio of 3, a cross-sectional area of about 2.5 nm 2.5 nm, and consists 2546 atoms. The relaxed and minimized configuration at T = 5 K is shown in Figure 2. In this configuration the {11} facets can not be easily identified automatically, because after surface reconstruction, the atoms immediately underlying {11} atoms move up and become surface atoms. Nevertheless, the reconstructed {11} facets along the rod sides are still easily identified directly by visual inspection. This rod will be referred to as {11} rod. C. {1} Rod The third rod, carved out from the fcc-crystal with the long axis in 1 direction, is covered with six {1} facets. This rod has an aspect ratio of about 3, a crosssectional area of about 2.44 nm 2.44 nm, and consists of 3126 atoms. The relaxed and minimized configuration at T = 5 K is shown in Figure 3. This rod will be referred to as {1} rod FIG. 4: Radius of gyration R g as a function of temperature T for the three different gold nanorods during the continuous heating procedure. IV. RESULTS FOR CONTINUOUS HEATING To model the experimental procedure of laser heating used in the experiments of Link et al. 3, we heat the nanorods continuously from 5 K to complete melting at rate of K/s, roughly in the range comparable to the experimental one 21. The heating is done by rescaling the particle velocities after each molecular dynamics step such that the kinetics energy increases by a fixed amount. The whole heating procedure, in which the total energy of the cluster increases linearly in time, takes less than 1 5 simulation steps corresponding to about 43 ps. The shape changes of the gold nanrods were monitored by following the radius of gyration R g as the cluster is heated (Fig. 4). Here, the instantaneous temperature is defined as T = 2K/3N, where K is the instantaneous total kinetic energy and N is the number of particles. Except for a little decrease below T = 3 K due to a reconstruction of the narrow {1} facets to a {111} structure (green in Fig. 1), the {111} rod does not show a noticeable shape change up to T 12 K, when R g rapidly decrease as the rod melts. In contrast, the shape of {11} rod changes slowly and continuously from about 3 K to 9 K. Even slower change continues from about 9 K to about 115 K, when the rod melts. Compared with the {11} rod, the {1} rod has a faster shape change from about 3 K to about 7 K, then its radius of gyration remains almost constant before the cluster melts at about 125 K. The time evolution of the cluster structure during the heating procedure can be inferred from the average bond order parameters depicted in Fig. 5 as a function of temperature. Consistent with the shape stability of the {111} rod, its bond order parameters change only weakly up to the melting temperature. This remarkable shapestability of the {111} rod is due to the highly stable {111} facets mainly covering the cluster surface. Of course, the elongated {111} rod is far from its equilibrium shape, that, for this particular size, is most likely an almost spherical icosahedron 19,22. High free energy barriers re-

5 Rod {111} Rod {11} Rod {1} (a) (b) FIG. 5: for the {111} rod (a), the {11} rod (b), and the {1} rod (c) during continuous heating. lated to the reorganization of bulk and surface, however, prevent the cluster from relaxing towards this minimum free energy state within the time of the simulation (and of the typical experiment). The behavior of the bond order parameters for the rods mainly covered with the less stable {11} and {1} facets is different. The bond order parameters for the {11} rod soften from about 3 K to about 7 K, corresponding to its continuous shape change. Then, a structural change of the interior from fcc-dominated to hcp-like and back to fcc occurs. This change is marked by a peak of Ŵ 4 between 7 K and 9 K. For the {1} rod, a similar structural change takes place in the slightly lower temperature range from about 4 K to 7 K. During this structural transition the original fcclattice reorients itself in a way to accommodate stable (c) R g (Å) Rod {111} Rod {11} Rod {1} FIG. 6: Radius of gyration R g as a function of temperature T for different gold nanorods during quasi-equilibrium heating. {111} facets on its surface, thus lowering the total free energy of the cluster 5 7. Considering the higher stability of the extended gold {1} surface with respect to the {11} surface it is interesting that the fcc-hcp-fcc transition takes place at a lower temperature for the {1} rod covered with the more stable facets. This unexpected behavior may be a kinetic effect caused by a higher free energy barrier for the transformation of the {1} rod and indicates that different mechanism may be responsible for the transformation of the {11} and {1} rods. Below, we will show that for slower heating the onset for the structural transition of {11} rod the shifts to lower temperatures while that of the {1} remains roughly the same. V. RESULTS FOR QUASI-EQUILIBRIUM HEATING In order to improve our understanding of the phenomena observed in the continuous heating simulations, we have carried out additional calculations with a lower heating rate. In these a quasi-equilibrium heating simulations the configurations relaxed at T = 5 K were first equilibrated at T = 1 K for 4.3 ns. Then, the temperature was incremented from T = 1 K to 12 K in steps of 1 K. At each temperature the rod went through a molecular dynamics simulation of 4.3 ns duration. The shape changes of the gold nanrods were monitored by the following the radius of gyration during the heating procedure (see Fig. 6). In the figure, the vertical dotted lines divide the temperature axis into equal sections. In each section, the radius of gyration is plotted as a function of increasing time for a constant temperature corresponding to the value at the left end point of the section. As in the continuous heating case, the {111} rod does not show any pronounced shape change before melting at T = 11 K, except for a small relaxation at T = 2 K related to the reconstruction of the narrow {1} facets. The {11} rod displays a steady shape change from 1 K to 9 K and then remains stable stable up to the melting

6 6 FIG. 7: Configuration of the {11} rod after quasi-equilibrium heating to 9K. Golden atoms represent {111} facets, green atoms represent {1} facets, and gray atoms are on the edges. FIG. 8: Configuration of the {1} rod after quasi-equilibrium heating to 9K. Golden atoms represent {111} facets, green atoms represent {1} facets, and gray atoms are on the edges. temperature of 11 K. In this intermediate state before melting the {11} rod has an aspect ratio of about 1.8. In contrast to the {11} rod, the {1} rod changes its shape in a more step-like manner from 2 K to 8 K. The resulting intermediate state with an aspect ratio of about 1.9 is then stable up to its melting temperature of T = 12 K. Both the high-temperature intermediate states of the {11} rod and the {1} rod have fcc-lattices reoriented with respect to the starting configuration and are covered with stable {111} facets (see Figs. 7 and 8). Average bond order parameters that are a signature for the local structure of the clusters, are plotted in Figure 9. Essentially, these curves show the same behavior observed in the continuous heating runs and shown in Fig. 4: the structure of the {111} rod changes only weakly up to melting and the fcc-hcp-fcc transition occurs for the {11} rod and the {1} rod. In contrast to the continuous heating procedure, however, the {11} rod undergoes the transition at a considerably lower temperature while the transition temperature for the {1} rod remains unchanged. This re Rod {111} (a) Rod {11} (b) Rod {1} (c) FIG. 9: for (a) rod {111}, (b) rod {11}, and (c) rod {1} during quasi-equilibrium heating. sult is consistent with the lower stability of the {11} facets with respect to the {1} facets and indicates that the higher transition temperature for the {11} rod with faster heating is indeed a kinetic effect. Moreover, for the {1} rod the transition now occurs in a step-like manner over a wider range than in the continuous heating case. All this suggests that the transition may follow different mechanisms for the {11} rod and the {1} rod. To obtain additional insight into the nature of the transition we computed the average cross-sectional shapes of the {11} rod and the {1} rod displayed in Fig. 1. For the {11} rod (panel (a) in Fig. 1) the average cross-sectional shape does not change significantly from 1 K to 2 K. From 3 K to 4 K, however, the four {11} facets are rounded and bulge out. This temperature range coincides with the onset of fcc-hcp-fcc transition. From 5 K to 7 K, and at higher temperatures not shown, the sides of the rod relax and reorganize to form stable {111} facets. Interestingly, the four corners

7 7 2 Rod {11} (a) 2 Rod {1} (b) Y (Å) 1-1 1K 2K 3K 4K 5K 6K 7K Y (Å) 1-1 1K 2K 3K 4K 5K 6K 7K X (Å) X (Å) FIG. 1: Average cross-sectional shapes, viewed down the long axis of the rod, for (a) rod {11} and (b) rod{1}. of the original cross-sectional remain at fixed positions during the whole transformation. This indicates that for the {11} rod the edges do not play an important role during the shape change and that the process driving the transition most likely involves the properties of the {11} facets. In contrast to the {11} rod, the {1} rod has a crosssectional shape that changes mostly at the edges between the flat facets (see Fig. 1(b)). Below T = 5 K, the average shape of the {1} rod only exhibits slight changes with some rounding of the corners. At T = 5 K the average cross-sectional shape rapidly changes from square to rhombic. During this shape change, that yields a rod predominantly covered with {111} facets, the top-left and the bottom-right corners remain at fixed positions while the two other corners bulge out. Throughout, the facets of the rod keep almost perfectly flat and do not disorder. Considering the importance of the free-energy contribution from vertices and edges in the stability of nanomaterials 19,22,23, it is conceivable that the transition mechanism is related to a thermal instability of the edges. The structure of the {11} and {111} rods as they are heated from 1 K is illustrated shown in Fig. 11. For these cross-sectional views thermal fluctuations where quenched by conjugate gradient minimization 24. The atoms are colored according to their local 3D structure. The {11} rod, initially covered by an irregular surface, begin to bend at 2 K when an hcp region starts to form on the bottom side of the rod. This hcp region then propagates through the rod by sliding of {111} planes leaving behind an fcc-region with changed orientation. The structural change in the interior is accompanied by the formation of {111} surface facets. At 7 K the transition is mostly completed an the rod has reached its high-temperature intermediate state (see Figure 7). Unlike the {11} rod, the surface of the {1} rod remains ordered throughout the heating procedure. The transition from fcc to hcp and back to fcc starts at 2 K when layers of atom slide wit respect to each other near the rod center. This configuration remains almost unchanged at 3 K, in contrast to the pronounced change occurring in the {11} rod at this temperature. At the end of 4 K, the middle planes have returned to their FIG. 11: Cross-sectional view for the {11} rod (left) and the {1} rod at various temperatures. Atoms are colored according to local crystal structure: fcc is yellow, hcp is green, and neither is grey. For clarity, teh surface layer and the first sub-surface layer have been removed in this representation. initial fcc structure but with a changed orientation of the underlying lattice. As the hcp layers continue to propagate through the rod, more and more of the surface becomes covered with {111} facets. During the incomplete stages of the transformation the rod is slightly bent, but at 7 K the straight intermediate high-temperature structure is reached (see Figure 8). While the final state is very similar for both for the {11} rod and the {1} rod, the relaxation mechanism towards it is rather different as can be inferred from Figs While for the {11} the fcc-hcp-fcc transition occurs rather quickly at

8 8 low temperature and seems to be favored by the roughened surface, the transition in the {1} rod is spread over a wider temperature range and does not appear to be related to a surface instability. VI. CONCLUSIONS AND DISCUSSION We have performed molecular dynamics simulations of three different gold nanorords heated from a temperature of 5K to melting. The nanorods all consisted of about 3 atoms and had an initial shape with an aspect ratio of 3 but where covered with different surfaces. Our simulations reveal that the thermal stability of the rods strongly depends on the type of exposed surface facets. Rods mostly covered with {111} facets essentially maintain their initial elongated shape up to the melting temperature. Rods covered by {11} and {1} facets, on the other hand, undergo a transition to a shorter and bulkier shape well below melting. During this transition sliding planes cause the interior structure of the rods to change from fcc to hcp and back to fcc. The new fcc lattice is reoriented with respect to the old one leading to a rod mostly covererd with stable {111} facets. While the transitions of the {11} and the {1} rod occur through essentially the same sliding plane mechanism, they re initiated by different events. The result of our simulations indicate that for the transition of the {1} rod the edges seem to play a central role.the transition of the {11} rod, on the other hand, most likely nucleates at the surface facets. In this case, the activation barrier for the structural and shape transformation of the nanorod may be reduced by the roughening transition that occurs at 68K for extended {11} gold surfaces. For finite size surfaces such as cluster facets enhanced fluctuations are expected to lower the roughening temperature? such that it may indeed coincide with the onset of the structure/shape change observed in our {11} rod. The transformation mechanisms discussed above for gold nanorods may well exist in other simple elemental metals. For example, the silver {11} surface has a roughening transition temperature of 6 K and a melting temperature of 1235 K?. The lead {11} surface has a roughening temperature of 415 K and a melting temperature of 61 K?. However, further studied will be required to clarify the role played by surface and edge instabilities in the nucleation of structural and shape transformations in metal nanoclusters. Acknowledgments This work was supported by the Austrian Science Fund (FWF) under Grant No. P17178-N2. 1 Y. Kondo and K. Takayanagi, Phys. Rev. Lett. 79, 3455 (1997). 2 Z. L. Wang, M. B. Mohamed, S. Link, and M. A. El-Sayed, Surf. Sci. 44, L89 (1999). 3 S. Link, C. Burda, B. Nikoobakht, and M. A. El-Sayed, J. Phys. Chem. B 14, 6152 (2). 4 Y. Wang and C. Dellago, J. Phys. Chem. B 17, 9214 (23). 5 Y. Wang, S. Teitel, and C. Dellago, Nano Lett. 5, 2174 (25). 6 J. Diao, K. Gall, and M. L. Dunn, Nat. Mater. 2, 656 (23). 7 J. Diao, K. Gall, and M. L. Dunn, Phys. Rev. B 7, (24). 8 F. D. Di Tolla, E. Tosatti, and F. Ercolessi, Interplay of melting, wetting, overheating and faceting on metal surfaces: theory and simulation, in Monte Carlo and molecular dynamics of condensed matter systems, K. Binder and G. Ciccotti (Eds.), Societá Italiana di Fisica, Bologna, p (1996). 9 A. Hoss, M. Nold, P. von Blanckenhagen, and O. Meyer, Phy. Rev. B 45, 8714 (1992). 1 B. M. Ocko, D. Gibbs, K. G. Huang, D. M. Zehner, and S. G. J. Mochrie, Phys. Rev. B 44, 6429 (1991). 11 F. Ercolessi, S. Iarlori, O. Tomagnini, E. Tosatti, and X. J. Chen, Surf. Sci. 251/252, 645 (1991). 12 F. Ercolessi, M. Parrinello, and E. Tosatti, Philos. Mag. A 58, 213 (1988). 13 P. Carnevali, F. Ercolessi, and E. Tosatti, Phys. Rev. B 36, 671 (1987). 14 K. G. Huang, D. Gibbs, D. M. Zehner, A. R. Sandy, and S. G. J. Mochrie, Phys. Rev. Lett. 65, 3317 (199). 15 G. M. Watson, D. Gibbs, S. Song, A. R. Sandy, S. G. J. Mochrie, and D. M. Zehner, Phys. Rev. B 52, (1995). 16 S. G. J. Mochrie, D. M. Zehner, B. M. Ocko, and D. Gibbs, Phys. Rev. Lett. 64, 2925 (199). 17 G. Bilalbegović and E. Tosatti, Phys. Rev. B 48, 1124 (1993). 18 M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids, Clarendon Press: Oxford, (1987). 19 Y. Wang, S. Teitel, and C. Dellago, J. Chem. Phys. 122, (25). 2 P. J. Steinhardt, D. R. Nelson, and M. Ronchetti, Phys. Rev. B 28, 784 (1983). 21 S. Link, C. Burda, B. Nikoobakht, and M. A. El-Sayed, Chem. Phys. Lett. 315, 12 (1999). 22 Y. Wang, S. Teitel, and C. Dellago, Chem. Phys. Lett. 394, 257 (24). 23 Y. Zhao and B. I. Yakobson, Phys. Rev. Lett. 91, 3551 (23). 24 W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, Cambridge University Press: Cambridge, (1992).