Narrow Results

Using the Voter and Chen version of an embedded-atom model, derived by fitting simultaneously to experimental data both the diatomic molecule and bulk platinum, we have studied the melting behavior of free, icosahedral, 54-, 55- and 56-atom platinum clusters in the molecular dynamics simulation technique. We present an atom-resolved analysis method that includes physical quantities such as the root-mean-square bond-length fluctuation and coordination number for individual atoms as functions of temperature. The effect of a central atom in the icosahedral structure to the melting process is discussed. The results show that the global minimum structures of the 54-, 55- and 56-atom Pt clusters do not melt at a specific temperature, rather, melting processes take place over a finite temperature range. The heat capacity peaks are not δ-functions, but instead remain finite. An ensemble of clusters in the melting region is a mixture of solid-like and liquid-like clusters.

A theorem is presented concerning the morphogenesis of high-symmetry structures made of three-dimensional morphological units (MU's) free to move in three dimensions or constrained to a surface. All parts of each MU interact non-specifically with the rest of the structure, via an isotropic function of distance. Summing all interactions gives a net figure of merit, ℐ, that depends upon MU positions and orientations. A structure evolves via gradient dynamics, each MU moving down the local gradient of ℐ. The analysis is represented with generality in Fourier space.

A "warping" from a configuration of MU's is a set of MU displacements and/or rotations that slightly perturb nearest neighbor relations; deviations can accrue across the structure, producing large global distortions. A warping behaves qualitatively like a small perturbation, so a warping from a stable equilibrium decays under gradient dynamics. Connection to the Symmetrization Theorem greatly extends the basin of attraction of stable symmetrical configurations. Warped configurations are equivalent as precursors of structure, which helps to understand assembly by accretion.

Numerical illustrations are given in cylindrical geometry, for application to phyllotaxis; and in spherical geometry, for virus capsid structure. For animations of numerical evolutions that find high symmetry via unwarping, see