Simulation Methods

The potential used in this study was the Lennard-Jones form[83],

$$E = 4\,\epsilon\sum_{i<j}\left[\left(\sigma\over r_{ij}\right)^{12}-\left(\sigma\over r_{ij}\right)^6\right],$$ (3.1)

where r_ij is the distance between atoms i and j. With $\epsilon$, $\sigma$ and m as the units of energy, distance and mass, respectively, the unit of time is $(m\sigma^2/\epsilon)^\half$. MD simulations of the microcanonical ensemble were conducted using the velocity Verlet algorithm[156]. The initial phase point had zero linear and angular momentum. To prevent evaporation the clusters were placed in a container with walls represented by a LJ type repulsion[153,157]. When an atom hit the wall of the container an equal and opposite force was exerted on the rest of the cluster to prevent linear momentum transfer. As the two forces are collinear, there is also no transfer of angular momentum. The effect of the box is to restrict the phase space considered to bound clusters. The time step used was 0.0465 reduced time units, which is equivalent to $1\times10^{-14}$s for Ar.

The relative root-mean-square interatomic separation $\delta$ was used to assess the degree of melting. It is defined as

$$\delta= {2\over N(N-1)} \sum_{i<j} { \sqrt{\left<R^2_{ij}\right\gt - \left<R_{ij}\right\gt^2 } \over \left<R_{ij}\right\gt },$$ (3.2)

where the angle brackets indicate that an average is taken over the whole trajectory. Lindemann's criterion[158] states that melting is expected when $\delta$ becomes greater than about 0.1.

The kinetic temperature is calculated from the kinetic energy, E_K, via the generalized equipartition theorem:  

$$T_K={2E_K\over k(3N-6)}$$ (3.3)

by taking the mean of T_K over the whole trajectory. This expression is exact in the canonical ensemble, but T_K differs by ${\cal O}(N^{-1})$from the thermodynamic definition of temperature (equation 3.10) in the microcanonical ensemble[159]. STA temperatures were found by averaging T_K over 500 time steps.