Lowering of surface melting temperature in atomic clusters with a nearly closed shell structure

We investigate the interplay of particle number, $N$, and structural properties of selected clusters with $N=12$ up to $N=562$ by employing Gupta potentials parameterized for Aluminum and extensive Monte Carlo simulations. Our analysis focuses on closed shell structures with extra atoms. The latter can put the cluster under a significant stress and we argue that typically such a strained system exhibits a reduced energy barrier for (surface) diffusion of cluster atoms. Consequently, also its surface melting temperature, $T_{\text{S}}$, is reduced, so that $T_{\text{S}}$ separates from and actually falls well below the bulk value. The proposed mechanism may be responsible for the suppression of the surface melting temperature observed in recent experiments.

Figure 1

Characteristic temperatures for clusters of size (a), (d), and (g) $N=12$, (b), (e), and (h) $N=13$, and (c), (f), and (i) $N=14$. Panels (a)–(c) show the specific heat, panels (d)–(f) the Berry parameter, while panels (g)–(i) show the individual rms bond-length fluctuations from Eq. (5). Panels (a)–(f) show data for $\tau =8\times {10}^7$. In panels (g)–(i) dark symbols correspond to $8\times {10}^7$ MC steps while gray points are obtained with $4\times {10}^7$ MC steps per atom.

Figure 2

Ground-state configurations of ${\text{Al}}_{12}$ (left), ${\text{Al}}_{13}$ (center), and ${\text{Al}}_{14}$ (right) as obtained with Gupta potentials. The hole has little impact on the position of the remaining atoms. By contrast, the adatom creates a docking site with a higher coordination number, which is incompatible with the threefold symmetries of the docking sites of the unperturbed ${\text{Al}}_{13}$. Hence, ${\text{Al}}_{14}$ experiences significant strain.

Figure 3

Potential-energy statistics for ${\text{Al}}_{14}$ at $k_{\text{B}}T=0.02\text{eV}$ in the surface molten phase. (a) Convergence of the Berry parameter as a function of MC steps. (b) Sample of the potential energies of accepted configurations during the same MC run as in (a). (c) Histogram of the potential-energy distribution function $w\left(E\right)$ ($2.4\times {10}^6$ energy values, 480 bins).

Figure 4

Temperature dependence of the individual bond fluctuations Eq. (5) for clusters of size (a) $N=15$, (b) $N=16$, and (c) $N=17$. The “solid” bonds with $d_{ij}\le 0.1$ correspond to those between the central atom and the surface atoms. In the surface molten phase all surface atoms are equivalent. Dark symbols correspond to $8\times {10}^7$ MC steps while gray open symbols are obtained with $2\times {10}^7$ MC steps per atom.

Figure 5

Ground-state configurations of (a) ${\text{Al}}_{55}$ and (b) ${\text{Al}}_{56}$ as obtained with Gupta potentials. Large full circles indicate the corner atoms, which remain “solid” in the partially surface molten state of ${\text{Al}}_{56}$. Small open circles [rosette structure (Ref. 49)] show the broken fivefold symmetry caused be the 56th atom absorbed into the surface of the icosahedron.

Figure 6

Temperature dependence of the melting transition for clusters of size (a), (d), and (g) $N=55$, (b), (e), and (h) $N=56$, and (c), (f), and (i) $N=57$. Panels (a)–(c) show the specific heat, panels (d)–(f) the Berry parameter, while panels (g)–(i) show the individual rms bond-length fluctuations from Eq. (5). Panels (a)–(d) show data for $8\times {10}^7$ MC steps per atom; in panels (g)–(i) dark symbols correspond to $8\times {10}^7$ MC steps while gray open symbols are obtained with $4\times {10}^7$ MC steps per atom.

Figure 7

Characteristic temperatures of ${\text{Al}}_N$ associated with self diffusion over the cluster size $N$; for the definitions of $T_{\delta },T_{\text{D}},T_{\text{C}}$ see Table . “a” labels a “confined” central atom, “d” a “confined” central dimer, and “t” a “confined” central trimer. [The onset of the evaporation transition in the investigated systems is found at larger temperatures $T_{\text{evap}}>0.12\text{eV}$ (Ref. 29).]