D. Cluster analysis of crystalline regions and correlation length R 2 g ( s ) = 1 2 s 2 ∑ i,j ( ⃗ r$_{i}$ − ⃗ r$_{j}$ ) 2 , (18) Since the θ$_{i}$ ’s measure the local disorder in the packings compared to a particular lattice structure, a perfect lattice cluster would have a zero value of θ . Any packing with significant amount of certain lattice clusters would indicate a peak centered around the origin in the distribution function of the local disorder θ$_{i}$ corresponding to that particular lattice. The distribution of FCC clusters P ( θ fcc i ) for packings with different φ$_{j}$ is shown in Fig. 7A. We find that FCC and HCP dominate in the crystalline packings for φ$_{j}$ ≥ φ$_{melt}$ . Indeed we observe two prominent peaks in the distribution, one at FCC θ fcc i = 0 and the other at HCP θ fcc i = 0 . 16, while BCC, SC and icosahedral ordering are negligible. The FCC peak dominance indicates that the majority of the clusters are FCC with a small proportion of HCP clusters. Similar to the distributions of local orientational orders shown in Fig. 2C, we find no crystalline clusters in the random packings as evidenced by the Gaussian distributions of P ( θ fcc i ) for φ$_{j}$ ≤ φ$_{rcp}$ as seen in Fig. 7A. In the coexistence region, the distributions are formed by a linear combination of different phases at melting and freezing. The distribution of HCP clusters P ( θ hcp i ) has similar behavior as P ( θ fcc i ), as showed in Fig. 7B. We are able to define crystalline or nearly crystalline clusters in a packing based on the local orientational disorder and mechanical contacts, and visualize them in a 3d plot. The clusters are defined as follows: First, each node in the clusters is a sphere with θ fcc i < θ fcc c or θ hcp i < θ hcp c , where θ fcc c = 0 . 1 and θ hcp c = 0 . 075, as determined in Fig. 7. The definition ensures that the first peak in P ( θ fcc i ) at zero in the distribution function of Fig. 7A is included in this consideration (as well as the analogous analysis for HCP). Next, if any two nodes are in mechanical contact, we build a link between the two nodes. Then the crystalline clusters are those nodes that are linked together. The clusters are visualized in Fig. 1. Based on the definition of the crystalline clusters, the size of the largest cluster in the system and the correlation length of the clusters, ξ , are measured near the melting point. To calculate the correlation length, we first introduce the radius of gyration, R$_{g}$ ( s ), of a cluster consisting of s particles: 25