Columnar structures of soft spheres: Metastability and hysteresis

Previously we reported on the stable (i.e., minimal enthalpy) structures of soft monodisperse spheres in a long cylindrical channel. Here we present further simulations, which significantly extend the original phase diagram up to $D/d=2.714$ (ratio of cylinder and sphere diameters), where the nature of densest sphere packing changes. However, macroscopic systems of this kind are not confined to the ideal equilibrium states of this diagram. Consequently, we explore some of the structural transitions to be expected as experimental conditions are varied; these are in general hysteretic. We represent these transitions in a stability diagram for a representative case. Illustrative videos are included in the Supplemental Material.

Figure 1

Two examples of columnar crystals, generated by packing hard spheres of diameter $d$ into a cylinder of diameter $D$. The structure shown on the left is the uniform packing (4, 3, 1) in phyllotactic notation (see Ref. [19]), with every sphere having 6 neighbouring contacts. The structure on the right is the related line-slip structure $(4,\mathbf{3},1)$. The gaps indicated in the second structure correspond to a loss of contacts when compared to the first structure.

Figure 2

Phase diagram for soft sphere packings in cylinders in the range $1.5\le D/d\lesssim 2.7$ and dimensionless pressures $p\le 0.02$. The resolution in the diameter ratio of tube to sphere diameter is $\mathrm{\Delta }D/d=0.0025$ and in the pressure $\mathrm{\Delta }p=0.0004$. Besides the zigzag, the (2, 1, 1) uniform structure and the twisted zigzag, there are 10 uniform (blue shaded) and 14 related line-slip structures (green and brown). Small regions that contain the $(\mathit{2},1,1)$ and the $(\mathit{3},2,1)$ line slips, which were found for hard spheres, are not visible in this phase diagram due to the finite resolution. Discontinuous transitions are indicated by solid black lines, while continuous transitions are indicated by black dashed lines. The diamond symbols at $p=0$ correspond to the hard sphere uniform close-packed structures [19]. The range of the previous results published in Ref. [1] is indicated by the orange arrow above the diagram.

Figure 3

Variation in enthalpy $h$ as a function of $D/d$ for pressure $p=0.007$. The change in $D/d$ allows the packing to continuously (and reversibly) transform between the (3,2,1) uniform structure, the $(3,\mathbf{2},1)$ line slip, and the (4,2,2) uniform structure. The values of $D/d$ at which the nature of the structure changes are indicated by the dashed vertical lines.

Figure 4

Variation in enthalpy $h$ as a function of $D/d$ for pressure $p=0.020$. Changing $D/d$ leads to a reversible and continuous transformation between the (3,2,1) uniform structure and the $(3,\mathbf{2},1)$ line slip for both the forward (increasing $D/d$, blue crosses) and reverse (red circles) trajectories, indicated by the vertical dashed line. In contrast, the transition from the $(3,\mathbf{2},1)$ line-slip structure to (4,2,2) uniform arrangement (thin blue line) is discontinuous and occurs at a lower value of $D/d$ on the reverse trajectory (thick red line).

Figure 5

Variation in enthalpy $h$ as a function of $D/d$ while holding pressure at a constant value of $p=0.026$. The forward trajectory is as before: an increase in $D/d$ leads to a discontinuous transformation from the (3,2,1) uniform structure to the $(3,\mathbf{2},1)$ line slip (thin blue line), a further increase in $D/d$ results in a discontinuous transition to the (4,2,2) uniform structure (thin blue line). The reverse trajectory is remarkable in that the intervening line slip is eliminated. Instead the transition from the (4,2,2) uniform structure to the (3,2,1) uniform structure is via a discontinuous transition (thick red line). The inset shows a zoom on the discontinuous transitions.

Figure 6

(a) The stability diagram for transitions between (3,2,1), (4,2,2), and the associated line slip $(3,\mathbf{2},1)$. The computed transition points are indicated by blue crosses (increasing $D/d$) and red circles (decreasing $D/d$) according to the direction taken. This is indicated by arrows in the accompanying schematic diagram (b). Here the two uniform arrangements are labeled $U_1$ and $U_2$, and the intermediate line-slip arrangement is labeled LS. This diagram represents the topology of (a) but does not preserve geometrical features. Continuous transitions are marked by dashed lines, and discontinuous transitions are represented by solid lines.