Curvature-induced defect unbinding in toroidal geometries

Toroidal templates such as vesicles with hexatic bond orientational order are discussed. The total energy including disclination charges is explicitly computed for hexatic order embedded in a toroidal geometry. Related results apply for tilt or nematic order on the torus in the one Frank constant approximation. Although there is no topological necessity for defects in the ground state, we find that excess disclination defects are nevertheless energetically favored for fat torii or moderate vesicle sizes. Some experimental consequences are discussed.

Figure 1

(Color online) Coordinates $\left(\alpha ,\theta \right)$ defining the torus.

Figure 2

(Color online) Patch of positive Gaussian curvature on the torus (shaded) subtending an azimuthal angle $\Delta \theta$. The surface of the torus is divided into regions of positive and negative Gaussian curvature by the circles labeled $A$ and $B$, on which the Gaussian curvature vanishes [after D. Hilbert and S. Cohn-Vossen, Geometry and the Imagination (Chelsea, New York, 1952)].

Figure 3

(Color online) A plus (filled circle) and a minus (filled square) pair of charges between the plates of a two-dimensional circular capacitor is analogous to a plus-minus pair of disclinations on a torus with the identifications given.

Figure 4

(Color online) Illustration of the calculation discussed in the text: a plus (filled circle) and a minus (filled square) form a disclination dipole on one of the two circles of zero Gaussian curvature. They are then pulled apart until they reach the maximum curvature line (plus) and minimum curvature line (minus).

Figure 5

(Color online) Various contributions to the energy (in units of $K_A∕2$) with aspect ratio $r=\sqrt{2}$ for a disclination dipole separated an angular distance $\alpha$ along the path shown in Fig. 4. The long dashed line is the energy in the absence of defects. The dashed-dotted line is the defect-defect interaction as a function of the separation of the charges. The dashed line is the curvature-defect interaction energy as a function of separation, and the continuous line is the total energy. The core energy contribution, computed from Eq. (40), is very small on this scale. The bending energy is subtracted for clarity.

Figure 6

(Color online) The total energy (in units of $K_A∕2$) for aspect ratio $r=\sqrt{5∕4}$ for varying numbers of defects and $a_P=R_2∕4$. The bending energy at fixed $r$ is subtracted off. The disclination core energy is taken to be $0.1K_A$, which is $0.2$ in the above units (corresponding to $c=0.1$).

Figure 7

(Color online) The total energy, as in Fig. 6, but for the Clifford torus which has aspect ratio $r=\sqrt{2}$.

Figure 8

(Color online) The total energy, as in Fig. 6, but for aspect ratio $r=2.6926$.

Figure 9

(Color online) The total energy (in units of $K_A∕2$) for a torus with aspect ratio $r=\sqrt{5∕4}$ for varying numbers of defects with $R_2∕a_P=100∕\pi$. The disclination core energy is taken to be $K_A$ which is $2$ in the above units (corresponding to $c=1$). The bending energy at fixed $r$ is subtracted.

Figure 10

(Color online) The total energy, as in Fig. 9, but for aspect ratio $r=\sqrt{2}$.

Figure 11

(Color online) The preferred number of defects as a function of the total number of particles for three aspect ratios. The disclination core energy is taken to be $0.1K_A$. The dashed line corresponds to the configuration in Fig. 4.

Figure 12

(Color online) The critical number of particles, above which defects are no longer favored, as a function of the toroidal aspect ratio for vanishing core energy (blue) and $c=0.1$ (green). The analytic estimate for $c=0.1$ is plotted as solid red squares.

Figure 13

(Color online) Illustration of the calculation discussed in the text: a plus disclination (filled circle) and a minus disclination (filled square) form a defect dipole on one of the two zero-curvature circles of the torus. They are then pulled apart in the direction of the maximum curvature line (plus) and the minimum curvature line (minus), respectively.

Figure 14

(Color online) Plot of the energy for the path described by Fig. 13. The disclination core energy is taken to be $0.1K_A$. The aspect ratio is $r=\sqrt{5∕4}$ and $R_2∕a=50$. The optimal number of defect pairs in the final “buckled ring” configuration is $N^{\ast }∕2=7$ for this $M\approx 127000$ particle configuration.

Figure 15

(Color online) The typical ground state configuration for parameters that favor defect proliferation. The arrows indicate the displacement of the equilibrium defect position from the maximal curvature circles.