^{1}and Alejandro D. Rey

^{1,a)}

### Abstract

The textures exhibited by nematic liquid crystals confined to cylindrical capillaries under homeotropic anchoring have been studied for nearly thirty years. One of the reasons behind this maintained interest is that the processing of many high-performance fibers including carbon fibers and spider silks involves these textures. Three of these textures, the planar radial with line defect, the planar polar with two line defects (PPLD), and the escape radial (ER), are relatively well understood. A third one, the escape radial with point defects presents, however, some unresolved issues and recent studies have questioned the real nature and dimensionality of the defects involved in this texture. It seems that the defects are not in the form of points but rather in the form of closed lines or rings. This paper presents a detailed study on the connection between point and ring defects in a cylindrical cavity using three-dimensional simulations based on the continuum Landau–de Gennes theory. The results show that true point defects cannot exist in cylindrical cavities and that the merging of two ringlike defects may lead to two qualitatively different stable textures, namely, the ER and PPLD textures. The various results are in qualitative agreement with recent molecular dynamic studies and with theoretical predictions based on experimental observations. The predictions provide new insights on the structural connections between synthetic and biological superfibers.

This work is supported by the Natural Science and Engineering Research Council of Canada (NSERC). One of the authors (G.D.) wishes to acknowledge financial support from NSERC through the CGS program.

I. INTRODUCTION

II. MODELING

A. Computational domain

B. Tensor order parameter

C. Landau de–Gennes free energy

D. Governing nematodynamic equation

E. Dimensionless quantities and auxiliary conditions

F. Computational procedure and postprocessing tools

III. RESULTS AND DISCUSSIONS

A. Stability and existence of ring defects versus point defects in cylindrical cavities

B. Annihilation of a pair of ring defects in a cylindrical cavity

1. ERRD to ER transformation

2. ERRD to PPLD transformation

IV. SUMMARY AND CONCLUSIONS

### Key Topics

- Tensor methods
- 26.0
- Crystal defects
- 25.0
- Point defects
- 20.0
- Eigenvalues
- 16.0
- Nematic liquid crystals
- 10.0

## Figures

Typical textures found in nematic-filled cylindrical capillaries with sidewalls imposing a homeotropic anchoring. (a) transversal view of the planar radial with a line defect texture (PRLD); (b) transversal view of the planar polar with two line defects texture (PPLD); and (c) longitudinal view of the escape radial texture (ER). The (flux) lines are everywhere tangent to the director field while the black dots indicate the presence of line singularities perpendicular to the page.

Typical textures found in nematic-filled cylindrical capillaries with sidewalls imposing a homeotropic anchoring. (a) transversal view of the planar radial with a line defect texture (PRLD); (b) transversal view of the planar polar with two line defects texture (PPLD); and (c) longitudinal view of the escape radial texture (ER). The (flux) lines are everywhere tangent to the director field while the black dots indicate the presence of line singularities perpendicular to the page.

Schematic of the escape radial with point defects texture (ERPD). The flux lines are everywhere tangent to the director field while the black dots indicate the presence of point singularities on the axis of the cavity. (a) The different elementary cross sections given through the ERPD texture; (b) transversal cross section (identical for the radial and hyperbolic point defects); (c) longitudinal cross section. The point defect on the left is said to be radial while bearing the topological charge and the one on the right is hyperbolic and has the charge .

Schematic of the escape radial with point defects texture (ERPD). The flux lines are everywhere tangent to the director field while the black dots indicate the presence of point singularities on the axis of the cavity. (a) The different elementary cross sections given through the ERPD texture; (b) transversal cross section (identical for the radial and hyperbolic point defects); (c) longitudinal cross section. The point defect on the left is said to be radial while bearing the topological charge and the one on the right is hyperbolic and has the charge .

Schematic of the escape radial with ring defects texture (ERRD). The flux lines are everywhere tangent to the director field . (a) The different cross sections taken through the ERRD texture; (b) the transversal cross section (identical for the radial and hyperbolic ring defects); (c) first longitudinal cross section [in (b) and (c) views, the big black dots indicate the passage of the ring singularity]; (d) second longitudinal cross section [normal to the plane given in (c)] in which the dash lines correspond to the ring singularity. In (c) and (d), the defect on the left is the radial ring while the one on the right is the hyperbolic ring .

Schematic of the escape radial with ring defects texture (ERRD). The flux lines are everywhere tangent to the director field . (a) The different cross sections taken through the ERRD texture; (b) the transversal cross section (identical for the radial and hyperbolic ring defects); (c) first longitudinal cross section [in (b) and (c) views, the big black dots indicate the passage of the ring singularity]; (d) second longitudinal cross section [normal to the plane given in (c)] in which the dash lines correspond to the ring singularity. In (c) and (d), the defect on the left is the radial ring while the one on the right is the hyperbolic ring .

(a) Equilibrium -ring defect in a cylindrical capillary subjected to homeotropic anchoring for the parameters and . (b) Enlarged view of the elementary cross section cut through the defect. The surfaces in (a) and (b) correspond to the isolevel of 0.5 of the biaxial parameter .

(a) Equilibrium -ring defect in a cylindrical capillary subjected to homeotropic anchoring for the parameters and . (b) Enlarged view of the elementary cross section cut through the defect. The surfaces in (a) and (b) correspond to the isolevel of 0.5 of the biaxial parameter .

(a) Eigenvalues of the tensor order parameter along the and directions for the same parametric conditions as in Fig. 4. (b) Corresponding variations of the biaxial parameter .

(a) Eigenvalues of the tensor order parameter along the and directions for the same parametric conditions as in Fig. 4. (b) Corresponding variations of the biaxial parameter .

Evolution of the radii of an isolated ring defect confined into a cylindrical capillary with homeotropic anchoring as a function of the cavity radius.

Evolution of the radii of an isolated ring defect confined into a cylindrical capillary with homeotropic anchoring as a function of the cavity radius.

Topological transformations between two interacting ring defects confined into a cylindrical capillary of radius leading to the ER texture. [(a)–(b)] (late) precollision regime, (c) end of collision regime, and (d) postcollision regime. Surfaces are given by the isolevels . Frames (a), (b), (c), and (d) correspond, respectively, to dimensionless time 0, 5500, 5700, and 6350, respectively.

Topological transformations between two interacting ring defects confined into a cylindrical capillary of radius leading to the ER texture. [(a)–(b)] (late) precollision regime, (c) end of collision regime, and (d) postcollision regime. Surfaces are given by the isolevels . Frames (a), (b), (c), and (d) correspond, respectively, to dimensionless time 0, 5500, 5700, and 6350, respectively.

Different cross sections through the tensor order parameter field corresponding to the frame shown in Fig. 7(a). The radial and hyperbolic ring defects are located at and , respectively. Frames (a) and (b) correspond to the planes parallel and normal to the axis of the ring defects. Frame (c) corresponds to the transversal cross section through the ring defect at .

Different cross sections through the tensor order parameter field corresponding to the frame shown in Fig. 7(a). The radial and hyperbolic ring defects are located at and , respectively. Frames (a) and (b) correspond to the planes parallel and normal to the axis of the ring defects. Frame (c) corresponds to the transversal cross section through the ring defect at .

Variations of the tensor order parameter eigenvalues (a) and corresponding biaxial parameter (b) along the axial direction at time zero. The shifted coordinate is centered on the ring defect at .

Variations of the tensor order parameter eigenvalues (a) and corresponding biaxial parameter (b) along the axial direction at time zero. The shifted coordinate is centered on the ring defect at .

Cross section through the tensor order parameter field at dimensionless time 6350 [cf. Fig. 7(d)] showing the ordering in the *chargeless* ring formed after the collision and merging of two oppositely charged ring defects. (a) and (b) correspond to two mutually orthogonal longitudinal cross sections through the defect, while (c) gives the transversal cut at . The field of cuboids shows the mixed escape-planar polarlike nature of the chargeless ring formed at the end of the collision regime.

Cross section through the tensor order parameter field at dimensionless time 6350 [cf. Fig. 7(d)] showing the ordering in the *chargeless* ring formed after the collision and merging of two oppositely charged ring defects. (a) and (b) correspond to two mutually orthogonal longitudinal cross sections through the defect, while (c) gives the transversal cut at . The field of cuboids shows the mixed escape-planar polarlike nature of the chargeless ring formed at the end of the collision regime.

Variations of the tensor order parameter eigenvalues (a) and corresponding biaxial parameter (b) along the axial direction in the chargeless ring during the postcollision regime.

Variations of the tensor order parameter eigenvalues (a) and corresponding biaxial parameter (b) along the axial direction in the chargeless ring during the postcollision regime.

Longitudinal cross section through the tensor order parameter field represented in terms of cuboids and showing the defect-free ER texture that results from the total shrinkage of the chargeless ring defect at the end of the postcollision regime.

Longitudinal cross section through the tensor order parameter field represented in terms of cuboids and showing the defect-free ER texture that results from the total shrinkage of the chargeless ring defect at the end of the postcollision regime.

Topological transformations between two interacting ring defects confined into a cylindrical capillary of radius leading to the PPLD texture. (a) corresponds to time zero and the late precollision regime. (b) shows the configuration in the early collision regime at dimensionless time 2800. The end of the collision regime in which a unique chargeless ring remains is given in (c) for time 3150. Finally, (d) occurs at dimensionless time 11450; the chargeless ring defect has expanded to give the PPLD texture (at least locally). Surfaces are given by the isolevels .

Topological transformations between two interacting ring defects confined into a cylindrical capillary of radius leading to the PPLD texture. (a) corresponds to time zero and the late precollision regime. (b) shows the configuration in the early collision regime at dimensionless time 2800. The end of the collision regime in which a unique chargeless ring remains is given in (c) for time 3150. Finally, (d) occurs at dimensionless time 11450; the chargeless ring defect has expanded to give the PPLD texture (at least locally). Surfaces are given by the isolevels .

(a) Transversal cross section through the tensor order parameter field showing the PPLD texture that occurs when the chargeless ring defect opens at the end of the postcollision regime. (b) and (c) are the corresponding variations of the tensor order parameter eigenvalues and biaxial parameter along the radial direction.

(a) Transversal cross section through the tensor order parameter field showing the PPLD texture that occurs when the chargeless ring defect opens at the end of the postcollision regime. (b) and (c) are the corresponding variations of the tensor order parameter eigenvalues and biaxial parameter along the radial direction.

Schematic phase diagram of the nematic textures found in cylindrical cavities with sidewalls imposing homeotropic anchoring. The textures indicated in parentheses are metastable.

Schematic phase diagram of the nematic textures found in cylindrical cavities with sidewalls imposing homeotropic anchoring. The textures indicated in parentheses are metastable.

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