Hard-sphere crystallization gets rarer with increasing dimension

We recently found that crystallization of monodisperse hard spheres from the bulk fluid faces a much higher free-energy barrier in four than in three dimensions at equivalent supersaturation, due to the increased geometrical frustration between the simplex-based fluid order and the crystal [J. A. van Meel, D. Frenkel, and P. Charbonneau, Phys. Rev. E 79, 030201(R) (2009)]. Here, we analyze the microscopic contributions to the fluid-crystal interfacial free energy to understand how the barrier to crystallization changes with dimension. We find the barrier to grow with dimension and we identify the role of polydispersity in preventing crystal formation. The increased fluid stability allows us to study the jamming behavior in four, five, and six dimensions and to compare our observations with two recent theories [C. Song, P. Wang, and H. A. Makse, Nature (London) 453, 629 (2008); G. Parisi and F. Zamponi, Rev. Mod. Phys. (to be published)].

Figure 1

(Color online) Monte Carlo EoS of the fluid and the two densest known ordered phases in 4D [9] (a), 5D (c), and 6D (d) computed at constant V (filled symbols) and constant P (open symbols), with Padé approximants of the virial expansion for the fluid [28], and the Speedy fits to the crystal phase results [39] (solid lines). Insets give $\Delta \mu$ and the common tangent construction for determining coexistence between the fluid and the densest crystal phase. The additional panel for 4D (b) contrasts MC and SPT/CT EoS as well as the resulting coexistence determination (inset).

Figure 2

(Color online) Impact of dimensionality on the decay of orientational order. Top: second to first peak ratio of the radial decay of the orientational order parameter $g_6\left(r\right)$ at fluid coexistence. The line is a guide to the eyes. Bottom: decay of $g_6\left(r\right)$ in the fluid near the hexatic phase in 2D and at coexistence in 3D and 4D. The 5D and 6D plots (not shown) are qualitatively similar to the 4D results.

Figure 3

(Color online) Distributions of second- (a) and third-order (b) $l=6$ invariants for dense crystal phases and the fluid. With increasing dimension the fluid and crystalline local bond-order parameters become increasingly distinct.

Figure 4

(Color online) Top: additional fluid volume per unit area of surface $\Delta v_A={\gamma }_{\text{f}-\text{w}}/P$ from Monte Carlo simulations and the Padé approximant equation of state [28, 58] (points) compared with the virial expansion of Eq. (16) (lines). Bottom: Comparison of ${\gamma }_{\text{x}-\text{w}}/P$ from published simulation data [59, 60] and the Speedy equation of state [39] (points) with the cell-like theory (see text) for different orientations of the 3D fcc crystals (lines).

Figure 5

Additional volume required for placing a wall through a simplex. The disk on the left must be pushed back sufficiently (arrow) to allow for the wall insertion (vertical line).