FIG. 6: The orientational correlation function Eq. (12) for different packings with φ$_{j}$ . G$_{6}$ ( r ) vanishes at large r below φ$_{j}$ = 0 . 64, while it approaches a nonzero constant when φ$_{j}$ > 0 . 64. The nonzero asymptote of G$_{6}$ ( r ) is a signature of long-range correlation of orientational order. RCP is a well defined singularity at φ$_{j}$ = 0 . 64 where the orientational symmetry breaking occurs. C. Local orientational disorder We have established that crystalline structures appear in the coexistence region and solid-like branch. The remaining question is what kind of lattice structure dominates in the crystallized packings. Since the differences of the orientational order parameters, especially Q$_{6}$ , are not significant between different perfect lattice clusters (such as the icosahedral, FCC, HCP, BCC, and SC clusters, see [19], Fig. 2) we apply another measure, the local orientational disorder, to identify the crystalline nuclei as defined in M. Bargiel and E. M. Tory, Adv. Powder Technol. 12 , 533 (2001). For a given sphere i , let θ$_{ijk}$ be the angle between the j th and k th neighbors. Furthermore let θ fcc jk , θ hcp jk , θ icos jk be similarly calculated angles for the perfect 13-sphere fragments of FCC, HCP and icosahedral packings, θ bcc jk be the angles for the 8-sphere fragment of a perfect BCC packing, and θ sc jk be the angles for the 6-sphere fragment of a perfect SC packing. The local disorders are defined as following: θ fcc i = √ 1 66 11 ∑ j =1 12 ∑ k = j +1 ( θ$_{ijk}$ − θ fcc jk ) $^{2}$, (13) θ hcp i = √ 1 66 11 ∑ j =1 12 ∑ k = j +1 ( θ$_{ijk}$ − θ hcp jk ) 2 , (14) 23 ① ② ③ ④ ⑤ ⑥ ⑦ ⑧ ⑨ ⑩ ❶ ❷ ❸ ❹ ❺ ❻ ❼ ❽ ❾ ❿ ➀ G ➁ ( r ) r ➂ ➃ ➄ ➅ ➆ ➇ ➈ ➉ ➊ ➋ ➌ ➍ ➎ ➏ ➐ ➑ ➒ ➓ ➔ → ➣ ↔ ↕ ➙ ➛ ➜ ➝ ➞ ➟ ➠ ➡ ➢ ➤ ➥ ➦