Stability maps for columnar structures

Figures & data

Figure 2. Computational results for the densest columnar sphere packings. Shown is the packing density Φ as a function of the ratio D/d of tube to sphere diameter, together with the phyllotactic notation of the structures (see [27] for an extended range of D/d values). The shaded region of the diagram is of relevance to the soft sphere packings discussed in this article. Whenever the line-slip packings $(3,\mathbf{2},1)$ and $(3,\mathbf{3},0)$ are not those of highest density their packing fraction is indicated by dotted lines (these are for illustrative purposes only and not the result of computation).

Figure 1. An example of a $(4,4,0)$ uniform (a) columnar structure and its line slip modification (b), generated by packing spheres of diameter d into a tube of diameter D. In the uniform structure, every sphere has an identical neighbourhood with a coordination number z = 6. Its corresponding line slip structure (shown to the right) is characterised by gaps (i.e. loss of contacts), highlighted by the (red) ellipse. For the structure classification we use the phyllotactic notation (see [26, 27] for details).

Figure 3. A schematic version of a part of the full phase diagram presented in [29], showing the structure of minimal enthalpy structure. The horizontal axis (i.e. the pressure p = 0 (hard-sphere) limit) corresponds to the shaded part of Figure 2. The red diamond indicates the value of D/d at which the uniform packing $(4,2,2)$ is found in the hard sphere limit. The regions of the phase diagram indicates which packing – either uniform structures ($U_0,U_1,U_2$) or line-slip ($L_0,L_1,L_2$) has the lowest enthalpy.

Table 1. Critical pressures for reversible transitions.

Structural transition $U_1\iff U_2$	$p_1/{10}^2$	$p_2/{10}^2$	$p_3/{10}^2$
$(2,2,0)\iff (3,2,1)$	0.79	0.40	0.21
$(3,2,1)\iff (4,2,2)$	2.74	2.48	1.47
$(4,3,1)\iff (5,3,2)$	4.70	2.34	0.90
$(3,3,0)\iff (4,3,1)$	3.54	0.87	0.24
$(5,3,2)\iff (6,3,3)$	4.36	2.94	1.18
$(5,4,1)\iff (6,4,2)$	5.36	2.00	0.12
$(4,4,0)\iff (5,4,1)$	5.46	1.02	0.08
$(5,5,0)\iff (6,5,1)$	6.18	0.94	0.08

Note: The values for the pressures $p_1$, $p_2$ and $p_3$ determine the type of transitions encountered upon a change of the size ratio D/d in the stability diagram of Figure 5. This characterises all reversible cases for structures without inner spheres ($D/d<2.713$). The uniform structures $(6,3,3)$ and $(6,4,1)$, which appear above the limit $D/d<2.713$, have also no inner spheres in our cases. All stability diagrams were obtained with a resolution in pressure of $\mathrm{\Delta }p=4\cdot {10}^{-4}$, which determines the uncertainties for all pressures.

Figure 4. Directed network displaying possible structural transitions between uniform structures, in the hard sphere limit. The structures are restricted to packings without internal spheres and are labelled using phyllotactic notation. The point $(0,0,0)$ represents a conceptual origin for the diagram, corresponding to $D/d=0$. Dashed lines are contours of constant D/d – a transition from one structure to another which involves moving along the diagram in a direction perpendicular to the contours implies a change in D/d. Favourable transitions are marked with blue arrows, unfavourable transitions with red arrows and all unstable cases are shown with black dashed arrows.

Figure 5. Schematic displaying the topological features of a stability diagram for reversible transitions. The two uniform arrangements are labelled by $U_1$ and $U_2$ and the intermediate line slip is labelled LS. Solid lines represent discontinuous transitions, continuous transition are marked as dashed lines and the arrows at the lines indicate the direction of the transition. For this type of transition $U_1$ transforms reversibly into $U_2$, showing only hysteresis above a threshold pressure $p_3$. (Taken from previous publication [29])

Figure 6. Schematic stability diagrams for irreversible transitions. The uniform arrangements are labelled by $U_1$, $U_2$ and $U_3$ and the intervening line slip, in each case, is labelled LS. Solid lines represent discontinuous transitions, continuous transition are marked as dashed lines and the arrows at the lines indicate the direction of the transition. (a) Type I transition: Structure $U_1$ is of the type $(l,l/2,l/2)$. (b) Type II transition: Structure $U_2$ is of the type $(l,l,0)$.

Table 2. Critical pressures for irreversible transitions.

Transition $U_1\Rightarrow U_2$	$p_1/{10}^2$	$p_3/{10}^2$	Transition $U_3\Leftarrow U_2$	$p_2/{10}^4$	$p_3/{10}^4$
			$(3,2,1)\Leftarrow (3,3,0)$	0.16	$\cdot \cdot \cdot$
$(4,2,2)\Rightarrow (5,3,2)$	2.89	1.62	$(4,3,1)\Leftarrow (4,4,0)$	0.62	$\cdot \cdot \cdot$
			$(5,4,1)\Leftarrow (5,5,0)$	0.16	$\cdot \cdot \cdot$

Note: The values for the pressures $p_1$, $p_2$ and $p_3$ determine the type of transitions encountered upon a change of the size ratio D/d in the stability diagram of Figure 6. This summarises the characteristics of all irreversible transitions for structures without inner spheres ($D/d<2.713$). The direction of the arrow indicates whether the transition happens at increasing D/d (marked by ⇒) or at decreasing D/d (marked by ⇐). The uncertainties of all pressures are again $\mathrm{\Delta }p=4\cdot {10}^{-5}$ where ‘$\cdot \cdot \cdot$’ indicate that the transition was not directly observed due to the numerical limits of our algorithm.

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