^{1}and Alejandro D. Rey

^{1,a)}

### Abstract

Spider silk fibers have remarkable mechanical properties as a result of an ultraoptimized spinning process. Silk fibers are spun from a lyotropic nematic liquid crystalline anisotropic fluid phase which undergoes significant structural changes throughout the spinning pathway. In the silkextrusion duct, those structural changes are expected to be driven by elastic-mediated interactions between point defects. In this work, the interaction between two point defects of opposite topological charges located on the axis of a cylindrical cavity is studied using a tensor order parameter formalism. Distinct regimes leading to defect annihilation and structural transitions are described in detail. The driving force setting the defects into motion is also examined. The different results suggest that the tensorial approach is primordial in describing the complicated physics of the problem. The phenomenon described is important to the understanding of the process-induced structuring of silk fibers and to defect physics in a more general context.

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. Geometry

B. Tensor order parameter

C. Landau–de Gennes free energy

D. Governing nematodynamic equation

E. Dimensionless quantities

F. Auxiliary conditions

G. Numerical procedure

III. RESULTS AND DISCUSSIONS

IV. CONCLUSIONS

### Key Topics

- Point defects
- 35.0
- Tensor methods
- 25.0
- Crystal defects
- 19.0
- Free energy
- 18.0
- Nematic liquid crystals
- 13.0

## Figures

Schematic of the director fields corresponding to the ERPD (a) and ER (b) structures. In (a), the left and right point defects, respectively, correspond to the hyperbolic and radial hedgehogs. Note that the director is undefined in the core of each point defect.

Schematic of the director fields corresponding to the ERPD (a) and ER (b) structures. In (a), the left and right point defects, respectively, correspond to the hyperbolic and radial hedgehogs. Note that the director is undefined in the core of each point defect.

Evolution of the orientation and alignment fields during the precollision [(a) and (b)], collision [(c) and (d)], and postcollision [(e) and (f)] regimes of two nematic point defects along the axis of cylindrical capillary. Frames (a), (b), (c), (d), (e), and (f) correspond, respectively, to the dimensionless time , 40 585, 40 599, 40 613, and 40 914. The small segments represent the directors and thus the local preferred orientation of the rodlike molecules. The grayscale corresponds to . The white regions are ordered while the black ones, around defects, are disordered. , .

Evolution of the orientation and alignment fields during the precollision [(a) and (b)], collision [(c) and (d)], and postcollision [(e) and (f)] regimes of two nematic point defects along the axis of cylindrical capillary. Frames (a), (b), (c), (d), (e), and (f) correspond, respectively, to the dimensionless time , 40 585, 40 599, 40 613, and 40 914. The small segments represent the directors and thus the local preferred orientation of the rodlike molecules. The grayscale corresponds to . The white regions are ordered while the black ones, around defects, are disordered. , .

Evolution of the scalar order parameter profile, along the axis, during the precollision [(a) and (b)], collision [(c) and (d)], and postcollision [(e) and (f)] regimes. The dimensionless sampling times [(a)–(f)] are , 40 585, 40 599, 40 620, and 40 920. , .

Evolution of the scalar order parameter profile, along the axis, during the precollision [(a) and (b)], collision [(c) and (d)], and postcollision [(e) and (f)] regimes. The dimensionless sampling times [(a)–(f)] are , 40 585, 40 599, 40 620, and 40 920. , .

Reduced position of the point defects along the axis as a function of the reduced time during the annihilation process. (a) , ; (b) , .

Reduced position of the point defects along the axis as a function of the reduced time during the annihilation process. (a) , ; (b) , .

Reduced speed of the point defects as a function of their separating distance during the annihilation process expressed in semilog and linear scales. The speed of the point defects in the precollision and early collision regimes follows an exponential law. The trends for the (a) and (b) cases can be fitted with and , respectively. (a) , ; (b) , .

Reduced speed of the point defects as a function of their separating distance during the annihilation process expressed in semilog and linear scales. The speed of the point defects in the precollision and early collision regimes follows an exponential law. The trends for the (a) and (b) cases can be fitted with and , respectively. (a) , ; (b) , .

Evolution of reduced total free energy (a) and corresponding reduced interaction force (b) as a function of the reduced interdefect distance for , . The circles and squares denote, respectively, the linear and exponential fits.

Evolution of reduced total free energy (a) and corresponding reduced interaction force (b) as a function of the reduced interdefect distance for , . The circles and squares denote, respectively, the linear and exponential fits.

Evolution of reduced total free energy (a) and corresponding reduced interaction force (b) as a function of the reduced interdefect distance for , . The circles and squares denote, respectively, the linear and exponential fits.

Time frame of the reduced total free energy during which the nematic system undergoes its structural transition from the ERPD to the ER configuration through the annihilation of two point defects of opposite topological charges. , . In zone (a), the system presents two defects with distinct cores; in zone (b), the two defects have collapsed into a single defect which gradually disappears; in zone (c) the system is in defect-free ER configuration.

Time frame of the reduced total free energy during which the nematic system undergoes its structural transition from the ERPD to the ER configuration through the annihilation of two point defects of opposite topological charges. , . In zone (a), the system presents two defects with distinct cores; in zone (b), the two defects have collapsed into a single defect which gradually disappears; in zone (c) the system is in defect-free ER configuration.

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