January 03 2004

Comments for referee A's critiques

1. The results obtained on the nearly spherical average shape of the cluster are interesting, but they are not of sufficient broad interest to recommend publication in PRL.

   The spherical average shape is only one feature of our paper. We have several interesting points in our paper:
   1. As described in our paper, there are several possible stable (metastable) structures for gold nanoclusters. Although theoretical calculations have suggested that the icosahedral(Ih) structure is energetically metastable, HREM studies showed that Ih exhibit up to a few thousand atoms. Besides, Cleveland et al. [1] showed that the melting of gold clusters have a precursor of Ih structure, and Chushak and Bartell found the icosahedral structure is stable to mild annealing[4]. Our simulation showed, at least with the glue model, by cooling from liquid, gold clusters with a few thousand atoms form an Ih structure with a missing central atom, which gives more evidence to the Ih preference of gold nanoclusters. We do not regard this consistence of our results with both experimental and some simulation work as trivial.
   2. More than a decade ago, Ercolessi et al. [2] have predicted that Au {111} facets should remain essentially crystalline up to melting, and premelting should not be observed in Ih gold clusters, where only Au {111} facets are present. However, to our knowledge, no serious studies have been taken for Ih gold clusters regarding to the extrordinary stability of Au {111} facets. Our work verified their prediction with detailed quantitative analysis of the surface, yet in addition revealed the influence of the increasing mobility of vertex and edge atoms to Au {111} facets.
   3. Although no faceting transition seperated from the melting transition has been found, our precise surface discrimination and shape averaging methods give out detailed procedure how the sharp facets exhibited at low temperature get rounded and shrink their sizes with increasing temperature. Before this work the surface analysis of small clusters was limited to diffusion.
   4. Our shape averaging method shows at high temperature, the shape of small clusters fluctuate around an equilibrium shape. This allows us in this work, and will allow people in future work to do more quatitative analysis with small clusters at high temperature.
   5. At each temperature, we equilibrated the clusters for 4.3ns, followed with 43ns to collect data. This simulation time is much longer than most previous work (generally collecting data for less than 1ns). This ensures that our systems are fully equilibrated and our data include enough statistics in most cases.

2. there exists previously published work (H.B. Liu et al., Surface Science 491, 88 (2001)), where more cluster sizes and shapes were studied. This published work also address other interesting questions on the melting phenomenon in metal particles that were not discussed in the present paper. Figure 5 of the Surface Science paper already provides theoretical evidence of the nearly spherical shape of icosahedral gold cluster at melting.

   The surface science paper did totally different work, and their conclusions can not cap ours. We give the comparisons and explanations below, which show that the conclusions from the two papers have little in common.
   1. They used the EAM model, and we used the glue model.

   2. They built the clusters with difference structures and heated them up. So our conclusion that Ih structure is favorable has not been presented in their work.

   3. They continuously heated the clusters up, but we equilibrated the clusters at each temperature level. So their simulations gave kinetic properties, and we gave equilibrium, thus more thermodynamic results.
   4. Because of their heating procedure, they drew conclusions according to instantaneous data, but we gave out averaging properties at each temperature level.
   5. They reported a melting zone, but we found the bulk melting is still a first order transition. The melting zone is rather a kinetic phenomenon caused by the heating procedure, not an equilibrium property of ih gold clusters. BTW, our continuous heating simulation of gold nanorod [3] also exhibit such a "melting zone", but our equilibrium study of the same nanorod shows explicitely a first order melting transition.
   6. With our discovery of the shape fluctuaion of small clusters at high temperature, their conclusion of surface disordering, reordering and premelting of ih gold clusters is questionable. They drew this point by visual examination of the instantaneous configurations, yet those configurations are just one sample at each temperature. Our quantitative analysis showed contradictary results, namely no surface melting, as predicted by Ercolessi et al. [2], at least in the sense of equilibrium properties.
   7. As to Fig. 5 in the Surface Science paper, The picture of Ih cluster at 830K shows two large facets, which are larger than any other facets existing in the pictures at lower temperatures. This is also because they showed the instantaneous configurations, which can not represent the actual equilibrium shape at a given temperature. Our paper clarifies this confusing point at provides the method of obtaining equilibrium shapes.

Comments for referee B's critiques

1. Quantum effects are important in this size range and, based on my knowledge of theory, the many-body glue potential will not be appropriate to account for such effects or allow reliable structure calculations in this size regime.

   As Christoph clarified, the glue pontential is an empirical model with parameters fitted to include the quantum effects of electronics. Many studies of small gold clusters have been done with glue potential, most of them are with the sizes smaller than ours. Ercolessi et al. [2] used the glue potential to investigate the structure of small gold particles with N=100-900 atoms. As they pointed out, The reliability of this potential is questionable when N<=200, where several atoms have a small coordination number (<8), while the potential is fitted on bulk and surface where coordination is larger. We studied the clusters with a few thousand atoms, which is far from the questionable size range, and should give reasonable results. Besides, with today's computers, it is still impossible to study such large clusters by ab initio methods, so work can only be done by empirical methods, such as the glue potential, EAM, etc.

2. The use of bond orientational parameters on pages 3 and 4 classified according to symmetry is most likely not commonly understood. Hence these quantities should be defined in sufficient detail, such that non-specialists may understand the reasoning.

   The limited size of PRL paper doesn't allow us to put the details of bond order parameters on. Readers would find the details in the original paper we refered. However, we may add some words to describe the general idea of bond order parameters.

3. The division into surface and bulk layers on page 3 seems rather arbitrary. Considering that surface relaxations die off very fast with deeper layers, it seems to be more reasonable for such small clusters to limit the surface to two or at most three layers, and take the rest as bulk. How does this affect plots 2a and 2b?

   We analyzed all layers and only the surface layer and the interior are shown in Fig. 2 because of the limited size of the paper. Other layers behave trivially between them. The plots are shown below.

   Surface layer

   

   Sub layer 1

   

   Sub layer 2

   

   Sub layer 3

   

   Sub layer 4

   

   Bulk

   

   As referee B suggested, we also treat the outmost first two layers as surface and the rest as bulk to calculate their bond orientational parameters. The plots are shown below.

   Surface (first two layers)

   

   Bulk (all other interior atoms)

   

   The bond order parameters for bulk just act as we expect from the mixing. Those for surface now become odd because now we include the bonds between two layers, whose vibrations severely interfere the values of the bond order parameters.

4. Fig. 2a, what is the meaning of a negative order parameter?

   The negative values can be found in the original paper by Steinhardt et al. (ref. 29). (Should we explain when the negative values present?)

5. From plot 3b it should be possible to extract the activation energy of surface self-diffusion. Does this yield a reasonable value compared to experiments?

   The plot of ln(D) vs. 1/T is shown below.

   

   We fit the lines for 4 temperature ranges according to the emperical Arrhenius law D = D₀ * exp( -E_A/T ):

   • 400K - 600K:
     E_A = 3.42*10^-20 J = 0.21 eV
     D₀ = 5.45*10^-4 Angstrom² / ps
   • 700K - 950K:
     E_A = 1.15*10^-19 J = 0.72 eV
     D₀ = 5.62 Angstrom² / ps
   • 1050K - 1070K:
     E_A = 7.32*10^-19 J = 4.58 eV
     D₀ = 4.36 * 10¹⁹ Angstrom² / ps
   • 1100K - 1200K (liquid):
     E_A = 5.61*10^-20 J = 0.35 eV
     D₀ = 2.14 Angstrom² / ps

   To our knowledge, no experimental activation energy data for Au {111} facets are currently available. We found two papers with the simulation results:

   1. Chushak and Bartell [4] reported their the activation energy for their simulated liquid gold nanoclusters by EAM model (Page 11608, below Eq. 11):

      E_A = 23.88 kJ/mol = 3.97*10^-20 J = 0.25 eV
      D₀ = 27.9*10^-9 m²/s = 2.79 Angstrom² / ps

      Considering the tendency of EAM models to systematically give lower energy values as pointed out by ref. [5] and many other resources, our result for the liquid gold cluster is in very good agreement with theirs.

   2. Boisvert et al. [5] did the first principles calculations for the bulk Au {111} surface at low temperature and the activation energy they found is (Table V): E^A = 0.22 +/- 0.03 eV. Our result of E^A = 0.21 eV for T<=600K is a perfect match with their first principles result.
It is well known that the Arrhenius law is not applicable for high temperature solids (see for instance ref. 6), which is exhibited in our data for 700K-1070K.

6. Fig. 4, is the shape at 600 K the same as at lower temperature? What are the criteria for this shape to be an equilibrium rather than a kinetically limited form? Why is it not highly symmetric if all flats are {111} facets? Also, I do not recognize this cluster to be an icosahedron with 20 (almost) equal {111} facets (claim on page 2).

   The shape at 600K is almost the same as at lower temperature except that the vertices is a little rounded. Those shapes are averaged over very long equilibrating time at each temperature, so it should be an equilibrium form, not a kientically limited one. Because the size of 2624 is not a magic number, the formed ih clusters has 20 {111} facets with different sizes. The clusters with magic numbers we studied are highly symmetric. The pictures in the first row of Fig. 4 are only a profile of cluster surfaces to show the rounding of vertices and edges. We carefully examined the numbers of vertices and edges and the structure of surfaces to ensure what we got are really icoshedra with 20 {111} facets (not necessary to be equal large). Below we show the same average shape of 600K colored with max local curvature from another perspective (The particles are not real surface atoms but the average atom positions in the solid angle cells).

   

REFERENCES