Bubble-raft model for a paraboloidal crystal

We investigate crystalline order on a two-dimensional paraboloid of revolution by assembling a single layer of millimeter-sized soap bubbles on the surface of a rotating liquid, thus extending the classic work of Bragg and Nye on planar soap bubble rafts. Topological constraints require crystalline configurations to contain a certain minimum number of topological defects such as disclinations or grain boundary scars whose structure is analyzed as a function of the aspect ratio of the paraboloid. We find the defect structure to agree with theoretical predictions and propose a mechanism for scar nucleation in the presence of large Gaussian curvature.

Figure 1

(Color online) Lateral and top view of a computer reconstruction of two paraboloidal rafts with ${\kappa }_1\approx 0.15{\mathrm{cm}}^{-1}$ [(a), (b)] and ${\kappa }_2\approx 0.32{\mathrm{cm}}^{-1}$ [(c),(d)]. The number of bubbles is $N_1=3813$ and $N_2=3299$, respectively. The color scheme highlights the fivefold (red) and sevenfold (blue) disclinations over sixfold coordinated bubbles (yellow).

Figure 2

(Color online) Translational and orientational correlation functions ($g$ and $g_6$, respectively) for rafts with (a),(b) $\kappa \approx 0.32{\mathrm{cm}}^{-1}$, $a=(0.8410\pm 0.0025)\mathrm{mm}$, and (c),(d) $\kappa \approx 0.15{\mathrm{cm}}^{-1}$, $a=(0.9071\pm 0.0037)\mathrm{mm}$. All the curves are plotted as functions of $r∕a$, where $r$ is the planar distance from the center and $a$ is the bubble radius. The envelope for the crystalline solid decays algebraically (dashed line), while the orientational correlation function approaches the constant value 0.8.

Figure 3

(Color online) An enlarged view of the terminating grain boundary scar shown in Fig. 1d for a system with large Gaussian curvature. The scar starts from the circular perimeter of the vessel and terminates roughly in the center carrying a net $+1$ topological charge. The image of the bubbles (left) shows that they may deform slightly to better fill space, whereas the computer reconstruction of the lattice (right) uses perfect spheres of uniform size.

Figure 4

(Color online) Delaunay triangulation of a portion of the paraboloidal lattice with $\kappa \approx 0.32{\mathrm{cm}}^{-1}$ near the center (top) and the boundary (bottom). Red (lighter) and blue (darker) lines represent the geodesics directed toward first and second neighbors, respectively.