Frank-Read Mechanism in Nematic Liquid Crystals

Open Access

Frank-Read Mechanism in Nematic Liquid Crystals

Cheng Long, Matthew J. Deutsch, Joseph Angelo, Christopher Culbreath, Hiroshi Yokoyama, Jonathan V. Selinger, and Robin L. B. Selinger

Phys. Rev. X 14, 011044 – Published 11 March 2024

Abstract

In a crystalline solid under mechanical stress, a Frank-Read source is a pinned dislocation segment that repeatedly bows and detaches, generating concentric dislocation loops. We demonstrate that, in nematic liquid crystals, an analogous Frank-Read mechanism can generate concentric disclination loops. Using experiment, simulation, and theory, we study a disclination segment pinned between surface defects on one substrate in a nematic cell. Under applied twist of the nematic director, the pinned segment bows and emits a new disclination loop which expands, leaving the original segment intact; loop emission repeats for each additional 180° of applied twist. We present experimental micrographs showing loop expansion and snap-off, numerical simulations of loop emission under both quasistatic and dynamic loading, and theoretical analysis considering both free energy minimization and the balance of competing forces. We find that the critical stress for disclination loop emission scales as the inverse of segment length and changes as a function of strain rate and temperature, in close analogy to the Frank-Read source mechanism in crystals. Lastly, we discuss how Frank-Read sources could be used to modify microstructural evolution in both passive and active nematics.

Received 14 July 2023
Revised 15 December 2023
Accepted 25 January 2024

DOI:https://doi.org/10.1103/PhysRevX.14.011044

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Disclinations & dislocations Line defects Topological defects

Nematic liquid crystals

Coarse graining Polarized optical microscopy

Polymers & Soft Matter

Authors & Affiliations

Cheng Long ^1,2, Matthew J. Deutsch ¹, Joseph Angelo¹, Christopher Culbreath ¹, Hiroshi Yokoyama ^1,2,*,†, Jonathan V. Selinger ^1,2,*,‡, and Robin L. B. Selinger ^1,2,*,§

¹Advanced Materials and Liquid Crystal Institute, Kent State University, Kent, Ohio 44242, USA
²Physics Department, Kent State University, Kent, Ohio 44242, USA

^*These authors contributed equally to this work.
^†hyokoyam@kent.edu
^‡jselinge@kent.edu
^§rselinge@kent.edu

Popular Summary

When you bend a metal paperclip, it deforms via the motion of defect lines in the underlying crystal. These defects, called dislocations, get stuck—or pinned—at points along their length, that is, by collisions with other defects. Under mechanical stress, dislocations multiply when a pinned segment repeatedly bows outward and snaps off new loops, a mechanism known as a Frank-Read source. This process enables ductile crystalline solids to bend. Here, we show that the Frank-Read mechanism occurs also in nematic liquid crystals, the material used in display screens for phones, computers, and televisions.

Nematics are a structured liquid phase of rod-shaped molecules that align with each other, creating orientational order. Such materials can contain defect lines, called disclinations, where that order is disrupted. We have discovered that in a nematic under a twisting deformation, a pinned disclination can bow outward and snap off new loops. These results demonstrate that the Frank-Read mechanism arises in structured fluids, in close analogy to the mechanism in crystalline solids.

This fascinating connection between two very different types of matter represents a fundamental contribution to the field of materials science, and suggests some interesting potential applications. Frank-Read sources form randomly in metals, but in nematics, we can design and build them in specific locations by inscribing defect pinning points on confining walls. This technique controls precisely where defect loops will be generated under shear, a method potentially useful for new liquid-crystal devices.

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Vol. 14, Iss. 1 — January - March 2024

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Images

Figure 1
Experimental micrographs showing a Frank-Read source in a dynamic liquid-crystal cell, with a disclination segment pinned at two point defects on the lower substrate, before and after an imposed twist strain. (a) Diagram of the dynamic cell. The surface of a convex lens (upper substrate) imposes uniform planar alignment. The anchoring pattern with two point defects is shown on the lower substrate. (b) Viewed from above, the arch appears as a straight line at $t = 0 s$ . (c)–(i) Disclination shape evolution after the angle between the two substrates is increased via mechanical rotation by 144°, showing the Frank-Read mechanism. The defect segment initially bows outward, shown at times (c) $t = 210 s$ , (d) $t = 300 s$ , (e) $t = 400 s$ , (f) $t = 540 s$ , and (g) $t = 740 s$ . (h) Self-intersection occurs at $t = 770 s$ . (i) At $t = 788 s$ , a new loop has snapped off, leaving the initial segment intact.
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Figure 2
Geometry of the simulation cell, for the order-tensor and the lattice simulations. The bottom substrate (purple) has an anchoring pattern with two surface disclinations, while the top substrate (cyan) has uniform planar anchoring, which is initially aligned with the far field of the bottom substrate. A disclination arch (yellow) connects the two surface disclinations. Based on the director field around the disclination arch, the type of the disclination changes continuously from a $+ 1 / 2$ wedge disclination, to a twist disclination and eventually to a $- 1 / 2$ wedge disclination along the line from left to right. The inset graph shows the cross section of the twist disclination in the disclination arch, and the $\hat{Ω}$ vector indicates the rotation axis of the director around the disclination. In the example shown here, the arch is derived from order-tensor simulations.
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Figure 3
Sequence of disclination structures as the top surface is rotated quasistatically, based on simulations of the nematic order tensor. In these views from above, the color indicates the twist angle (in degrees) of the nematic director from the bottom to the top surface. Inside the disclination cores, the twist angle is not uniquely defined because of biaxiality, and, hence, the color is shown as white. (a)–(d) Steady states with $δ ϕ$ fixed at 0°, 90°, 130°, and 150°, respectively. The size of the viewing window is $24 \times 24$ . (e)–(j) Snapshots of the dynamic process when $δ ϕ$ is increased to 160°, which is slightly greater than the threshold angle. The size of the viewing window is $120 \times 120$ . The cyan cylinder located at the top right corner of each image represents the orientation of the top surface.
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Figure 4
Simulations of Frank-Read source behavior at finite temperature, via the Lebwohl-Lasher model, with defect spacing $w = 16$ and cell thickness $d = 10$ . (a)–(c) Disclination loop emission under continuous twist of the planar anchoring orientation at the top surface, with rate $90 ° / 100 k$ time steps and temperature $k_{B} T = 0.1 ε$ . (d) Threshold twist angle $δ ϕ^{*}$ for loop emission increases as a function of applied twist rate. (e)–(g) Disclination loop with fixed subcritical applied twist undergoes stable expansion as temperature increases, due to changes in Frank elasticity and line tension. (h) Threshold twist angle $δ ϕ^{*}$ for loop emission increases as a function of temperature.
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Figure 5
(a) Geometry for the theoretical free energy of the Frank-Read source. The thick red line represents the curved disclination, viewed from above. (b) Stability diagram in terms of angle $δ ϕ$ and dimensionless parameter $α$ , indicating whether the curved disclination is stable, metastable, or unstable (so that it must grow to infinity). Along the gray dashed line, the shape is a semicircle. (c) Comparison of theoretical predictions with nematic-order-tensor simulation results, for different defect spacings $w$ .
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Figure 6
Motion of $y_{top}$ during the unstable expansion process, from nematic-order-tensor simulations. (a) Defect spacing $w = 4$ and angle $δ ϕ = 160 °$ . (b) Defect spacing $w = 8$ and angle $δ ϕ = 108 °$ . The black dashed lines show linear fits in the limit of $y_{top} ≫ w$ .
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