Theory of rotational columnar structures of soft spheres

There is a growing interest in cylindrical structures of hard and soft particles. A promising new method to assemble such structures has recently been introduced by Lee et al. [Lee, Gizynski, and Grzybowski, Adv. Mater. 29, 1704274 (2017)]. They used rapid rotations around a central axis to drive spheres of lower density than the surrounding fluid towards the axis. This resulted in different structures as the number of spheres is varied. Here, we present comprehensive analytic energy calculations for such self-assembled structures, based on a generic soft sphere model, from which we obtain a phase diagram. It displays interesting features, including peritectoid points. These analytic calculations are complemented by preliminary numerical simulations for finite sample sizes with soft spheres. A similar analytic approach could be used to study packings of spheres inside cylinders of fixed dimensions, but with a variation in the number of spheres.

Figure 1

Minimal rotational energy $E_{\text{rot}}$ as a function of dimensionless inverse number density ${\left(\rho d\right)}^{-1}$ for hard sphere packings. The energy of the line-slip packings is given by the solid red line. Vertical black lines indicate the location of homogeneous structures, identified by indices $(l,m,n)$. The straight solid lines (green line) between adjacent homogeneous structures indicates that binary mixtures have a lower energy compared to line-slip packings. The shaded area is the region of all possible hard sphere packings which have a single value of distance $R$ from the central axis; by definition, this precludes binary mixtures. The inset shows examples of a homogeneous and a line-slip structure. See text for an interpretation of the two red arrows.

Figure 2

Minimal energy per sphere for all homogeneous structures as a function of dimensionless inverse number density ${\left(\rho d\right)}^{-1}$ for the case of soft spheres, with harmonic interactions (see text). The main figure shows the result for ${\omega }^{2}M/k=0.55$. The inset shows the energies for ${\omega }^{2}M/k=0.10$, which is close to the hard sphere limit. Common tangents between adjacent homogeneous structures are shown by the black lines and tangent points are shown by black dots. The range over which homogeneous structures are expected are highlighted by shaded strips (colored according to the appropriate homogeneous structure—see key). Outside these ranges the common tangents have a lower energy and binary mixtures are expected.

Figure 3

Calculated phase diagram of homogeneous structures (colored, labeled regions) and their binary mixtures of adjacent homogeneous structures (intervening white space). Most homogeneous regions expand with increasing ${\omega }^{2}M/k$ (i.e., rotation frequency) but the achiral $(3,3,0)$ and $(4,4,0)$ structures vanish in peritectoid points (see the inset). In the case of the (5,5,0) phase only the right-hand boundary can be shown, since it is the last structure without inner spheres. The ${\left(\rho d\right)}^{-1}$ values for the homogeneous structures are in good agreement with simulation results from Ref. [21], indicated by points with dashed horizontal error bars in the hard sphere limit.

Figure 4

The red dots show the energy per sphere from finite-size simulations, for ${\omega }^{2}M/k=0.2$ and $N=24$ spheres. Brown and blue solid lines show the previously analytically calculated energies for the homogeneous $(2,2,0)$ and $(3,2,1)$ structures, and the common tangents (black solid lines) represent the binary mixture of the adjacent structures. At the vertical dashed blue line the homogeneous structure transforms into a binary mixture, whose energy is higher than that of the common tangent (due to finite-size effects). Within the vertical dashed red and brown line line-slip structures are observed.

Figure 5

Numerically computed phase diagram from the finite-size simulation with $N=24$ spheres. In addition to the expected homogeneous structures and binary mixture of $(2,2,0)$ and $(2,1,1)$, a line-slip arrangement, as well as its binary mixture with $(3,2,1)$, are found.