Plane radial distribution function

The structure of the system, and in particular the formation of a quasiliquid film can be analyzed by using a plane radial distribution function defined as

$$p_l(r_{\vert\vert})=\left< \frac{1}{n_l} \sum_{i,j \in l } \... ...ij,\vert\vert}-r_{\vert\vert})}{2 \pi r_{\vert\vert}} \right >$$ (6.7)

where $r_{ij,\vert\vert}$ is the component of the $\vec r_{i}-\vec r_{j}$ parallel to the surface plane, $n_l$ is the instantaneous number of atoms in the layer $l$, the sum extends over all particles in the layer $l$, and the angular brackets denote averaging over time (See Fig. 5.19) .

Figure: Radial distribution function $p(r_{\vert\vert})$ of the 6th surface layers of Va(001) system at 1800 K.

In practice, the equilibrium $p_l(r_{\vert\vert})$ function calculated using a histogram method [54,82], based on counting and binning of the atom pair separations. Let $h_k(r,\Delta r)$ to be the number of the atom pairs $(i,j)$ in a $k$th bin (or shell):

$$(k-1)\Delta r \le r_{ij,\vert\vert} \le k\Delta r$$ (6.8)

then the plane radial distribution function (RDF) is obtained from:

$$p_l(r_{\vert\vert})=\left<\frac{A}{n_l} \frac{h_k(r,\Delta r) }{2\pi r_{\vert\vert} \Delta r } \right >$$ (6.9)

where $A$ is the area of a layer, $n_l$ is the instantaneous number of atoms in layer $l$. The two-dimensional radial distribution function $p(r_{\vert\vert})$ for the top surface layers of Va(001) at different temperatures are shown in Figs. 5.20. As seen from the figure the intra-layer structure in these layers changes gradually from crystalline to liquid-like as the temperature increased. Particularly noticeable is disappearance of the crystalline features in $p(r_{\vert\vert})$ corresponding to the second, third and other nearest neighbors. In addition to the heights of the peaks, the area under the $p(r_{\vert\vert})$ curve changes, which reflects the change in the density across the solid-liquid interface.

Figure: Two-dimensional radial distribution function $p(r_{\vert\vert})$ of the five top layers of Va(001) at temperature T=2200K. The layer n=25 corresponds to the adlayer, n=24 corresponds to the (first) surface layer, n=23 corresponds to the second one.

We note that for the adlayer at $T<T_m$ the probability of finding particles with separation beyond the first-neighbor shell is relatively small, indicating a tendency for clustering which persists, though to a smaller extent, even to $T \simeq T_m$.

The plane pair correlation functions of the first surface layer at elevated temperatures for the various low-index faces is shown in Fig. 5.21. It is clear that the crystalline order vanishes gradually and the quasiliquid film thickness increases when the melting point $T_m$ is approached.

Figure: Two-dimensional radial distribution function $p(r_{\vert\vert})$ of the surface layer of $Va(001)$ at various temperatures.

We can conclude relying on the analysis of the $2D$ radial distribution functions of the various faces, that surface premelting begins first on the least packed Va(111) surface at temperature around $2000 \pm 50$ K, e.g. 200 K below the estimated thermodynamical melting point, while at the face Va(001) liquid phase starts nucleate at around $2050 \pm 50$ K, and on the most close packed Va(011) surface noticeable changes in the $p(r_{\vert\vert})$ functions occur at temperature $2150 \pm 50$ K, which is close to the melting point.

Similar results were obtained in studies of surface premelting of low-index faces of fcc metals in computer experiments by H$\ddot a$kinen et al. [83] (Cu), by Chen et al [79] (Ni) and by Carnevali et. al [84] (Au), as well as in some real experiments [23,26,85]. The close packed face (the (111) face of a fcc lattice) preserves its crystalline order up to the melting point. This non-melting behavior of (111) is in striking disagreement with theoretical predictions and results of computer simulations based on a simple type of Lennard-Jones potentials, which are a crude approximation for fcc metals. More sophisticated MD simulations, which use many-body potentials do not confirm the theoretically predicted very pronounced premelting effects for the close packed (111) surface below the triple point [86-88].