Corrected Article: Simulation and observation of line-slip structures in columnar structures of soft spheres [Phys. Rev. E 96, 012610 (2017)]

We present the computed phase diagram of columnar structures of soft spheres under pressure, of which the main feature is the appearance and disappearance of line slips, the shearing of adjacent spirals, as pressure is increased. A comparable experimental observation is made on a column of bubbles under forced drainage, clearly exhibiting the expected line slip.

Figure 1

Examples of columnar hard-sphere structures. For each pair of images, the first shows the arrangement of hard spheres inside a tube, while the second shows a skeleton diagram in which sphere centers are represented by points, and contacts between spheres are indicated by a line joining the sphere centers. The structures shown are (a) a uniform arrangement $(3,2,1)$ and (b) a related line-slip structure $(3,\mathit{2},1)$. The gaps in (b) correspond to a loss of contacts compared to (a); the loss of contacts can also be seen clearly in the skeleton diagrams.

Figure 2

The form of the phase diagram for the range $2.00\le D/d\le 2.22$ and dimensionless pressures $p\le 0.02$. Seven different structures are in the plotted range. With increasing pressure, the widths of the line slips decrease and end in triple points. Discontinuous transitions are displayed as solid black lines, and continuous transitions are represented as dashed lines. The diamond symbols at $p=0$ correspond to the hard-sphere symmetric structures (3,2,1), (3,3,0), and (4,2,2) [16]. Accounting for the possible metastability (see below) of the observed structure, a rough estimate of the experimentally observed line-slip structure is indicated by the ellipse on this phase diagram. Note: there is also a very small $(\mathit{3},2,1)$ line slip region just below $D/d=2.15$ [16], not resolved in the present simulation.

Figure 3

(a) An example of the dimensionless enthalpy $h$ as $D/d$ is varied at a constant pressure $p=0.01$. The plot shows a horizontal cut through the phase diagram of Fig. 2. (b) Compression ratio $C$ of the soft packings as a function of the diameter ratio $D/d$. Only the transition from (3,2,1) to $(3,\mathbf{2},1)$ is continuous and does not involve a change in the slope of $H$ at the phase boundary.

Figure 4

(a) An example of the dimensionless enthalpy $h$ as $p$ is varied at a constant $D/d=2.1$. The plot shows a vertical cut through the phase diagram of Fig. 2. (b) Compression ratio $C$ of the soft packings as a function of the applied pressure. All the transitions shown are discontinuous; however, for the enthalpy the change in the slope at the phase boundaries can only be detected by numerically computing the derivative.

Figure 5

Photograph of columnar structures of bubbles under forced drainage, showing about 40 bubbles. Left: (3,2,1) uniform structure occurring at a relatively low flow rate. Right: bubbles adopt the $(3,\mathit{2},1)$ line-slip structure at a relatively high flow rate. It is similar to the line-slip structure of Fig. 1. The coloring highlights the two interlocked spirals of the (3,2,1) and $(3,\mathit{2},1)$ structures. The ellipse marked on Fig. 2 corresponds roughly to its estimated position on the phase diagram.

Figure 6

Setup for producing columnar foam structures. (a) Gas is introduced at a constant flow rate $q_0$ into a cylindrical column; (b) liquid is added at a constant flow rate $Q$ at the top of the column (forced drainage). $H$ is the height of the entire foam column, $h$ is the height of the submerged foam, and $x$ measures the distance to the liquid pool.

Figure 7

Examples of columnar structures in foams. Part (a) shows the same section of the tube for two different flow rates, corresponding to $\overline{\varphi }=0.03, (3,2,1)$ structure on the right, and $\overline{\varphi }=0.5, (2,\mathit{2},0)$ structure on the left. Part (b) shows a column at two different vertical positions, $x=15\text{cm}$ in the $(2,2,0)$ structure on the left and $x=30\text{cm}$ in the $(3,\mathbf{2},1)$ structure on the right. The bubble size is $d\sim 2.5\text{mm}$. For more details, see also Table 2.