Chapter 9. Argon Clusters

9.1. Introduction

Argon clusters have long been studied to determine the structures of van der Waals aggregates. For small clusters, certain "magic numbers" [1] of atoms have been found which lead to more stable structures than clusters with even one more or one fewer atom. The most stable structure for these clusters is typically the Mackay icosahedron [2].

Bulk solid argon, however, is known [3] to be most stable as a face-centered cubic structure, which is not compatible with the Mackay icosahedral symmetry. To investigate the transition between the small finite cluster behavior and the bulk structure, we modelled a "magic number" cluster of about five million argon atoms using a quenched dynamics process.

9.2. Procedure

We began with a Mackay icosahedral structure of 114 shells, comprising a total of 5,003,879 atoms. The forcefield used was a simple Lennard-Jones 12-6 potential with equilibrium radius (R_e) of 3.82198 Å and well depth (D_e) of 0.23725 kcal/mol, chosen to agree with the bulk lattice spacing and the 0 K heat of vaporization (after correcting for zero-point energy). CMM level 8 was used with a bounding cube 1440 Å on a side; the farfield was updated every 5 timesteps. The initial potential energy of the starting structure using this potential was -9.20x10^6 kcal/mol.

We performed 400 steps (4 ps) of Nosé-Hoover constant-temperature (NVT) dynamics at 80 K with a Nosé time constant \tau_s of 0.1 ps. The kinetic energy of the system rapidly converged to the desired temperature, which was selected to be close to the melting point of bulk argon (at 1 atm), thereby allowing melting of the initial structure while limiting boil-off of surface atoms from the cluster.

After the dynamics run, we quenched the system using dynamics at a temperature of zero K, with all velocities removed after each integration step. Adequate convergence was achieved after 1350 steps. The final energy was -9.29x10^6 kcal/mol, nearly 1% lower than that of the starting structure, indicating that a more stable structure was found. The RMS force on the final structure was 5x10^-4 kcal/mol/Å.

9.3. Results

The final structure was analyzed by comparing simulated diffraction structure factors computed from it to those computed from the initial Mackay icosahedral structure. To look at gross structural aspects, we computed the cluster structure factor (the magnitude of the Fourier transform of the atomic positions) for 0 <= h <= 10 and -10 <= k,l <= 10, using the CMM bounding box as the unit cell. Certain peaks show significant changes, as much as several orders of magnitude, from the initial to the final structure. For example, the peak at (0,7,8) decreases in intensity by a factor of 22, while (2,10,10) increases by a factor of 752.

Sample contour plots of the logarithms of the structure factors for the initial and final structures are shown in Figures 9-1 and 9-2. Both figures show sections through the h=2 plane.

Figure 9-1. h=2 section of log(structure factor) for initial argon cluster.

Figure 9-2. h=2 section of log(structure factor) for final argon cluster

Additional comparisons were made with the energies of smaller, optimized Mackay icosahedra and fcc spheres. A number of small (55 to 3925 atom) clusters were generated. Each cluster was minimized for 1000 steps, which brought all of them essentially to convergence. All energies were computed using exact nonbonds. The potential energy is expected to contain two components: one that varies with the number of atoms (the binding energy) and one that varies with the amount of surface area, or the number of atoms to the 2/3 power (the surface energy). The energy per atom should then be close to linear in the number of atoms to the -1/3 power, and this is in fact observed in Figure 9-3 for both the Mackay icosahedra and fcc spheres.

Figure 9-3. Comparison of potential energies for argon clusters.

The curves appear to cross between 2057 and 2899 atoms, indicating that the fcc structure is more stable after this point. The 5 million atom structure, however, has an energy per atom which, though more negative than all of the smaller clusters, is well above the curves. This would seem to indicate that the structure is in fact not a global minimum for that number of atoms.

9.4. Conclusions

We have shown that molecular dynamics and minimization calculations on systems as large as 5 million atoms are feasible given current software and hardware technology.

The results of these simulations on argon clusters suggest that, even at the achieved local minimum, interesting structural changes are occurring, as evidenced by the changes in peaks in the structure factor. Further analysis of these changes will be necessary.

Although some visualization of these clusters has been performed, bringing to bear the substantial capabilities of the human visual system on the analysis of these structures will require more advanced visualization tools that permit the user to select portions of the cluster to view, rather than overwhelming the limited screen resolution with all 5 million particles.

The primary limitation on improving the search for a global minimum for the 5 million atom cluster was the available minimization technology. Dynamics at 0 K has relatively poor convergence properties, particularly when compared with standard methods such as conjugate gradient minimization. Installation of such improved minimizers in the code is a high priority task.

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Kian-Tat Lim