# Observation of Backflow during the Anihilation of Topologocal Defects in Freely Suspended Smectic Films

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

## 3. Results and Discussion

#### 3.1. Experimental Results

#### 3.2. Discussion and Theoretical Background

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Sample Availability

## Abbreviations

FSF | freely suspended smectic film |

ROI | region of interest |

SmC | smectic C |

## References

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**Figure 1.**(

**a**) Left: Scheme of the film holder mounted on the microscopy table. Right: Schematic of the dye orientation drawn in red in the nematic matrix drawn in blue. $\mathbf{n}$ is the nematic director, $\mathbf{m}$ is the transition moment, and $\mathbf{P}$ is the polarization plane of the excitation beam. (

**b**) Snapshots from the experiment showing the fluorescence intensity recovery of a bleached region (exposed for 8 s to full laser power before $t=0$). The film does not contain defects in the vicinity of the region of interest (ROI). (

**c**) Time dependence of the measured mean intensity of the bleached ROI presented in Figure 1b. The recovery time is about 10 s. The dashed line is an exponential fit.

**Figure 2.**Snapshots of two defect pairs: The $+1$ defect has always tangential orientation of the c-director near the core. The $-1$ defect is initially rotated respective to the partner (in mismatch) so that the director field along a straight line connecting the cores is not uniform. Image (

**a**) was recorded 16 s before annihilation. During the mutual approach, the mismatch angle gradually reduces to zero. This state is almost reached in image (

**b**), 3 s before annihilation. Yellow bars symbolize 50 $\mathsf{\mu}$m. Images were recorded in polarized light with crossed polarizers (black lines) and a diagonal wave plate (slow axis: diagonal yellow line). Bluish regions characterize the c-director in northeast or southwest direction, while orange regions indicate a c-director orientation in southeast or northwest directions. In magenta regions, the c-director is either horizontal or vertical in the images. (

**c**) Defects pair observed in fluorescence mode (top: +1 defect, bottom: −1 defect). The black arrows sketch the c-director. The polarizer orientation is indicated by a black bar. Images (

**a**,

**b**) are courtesy of Péter Salamon and Kirsten Harth.

**Figure 3.**Trajectories of the defects and the flow field in the bleached regions: The snapshots show the pairs at the beginning of the recording, immediately after bleaching stopped. The $+1$ defect, $-1$ defect, and ROI trajectories are shown in red, blue and white respectively. The motion of the ROI near the positive defect (

**a**,

**b**) follows the motion of the defect. The ROI near the negative defect (

**c**,

**d**) does not move noticeably. (

**a**) A single ROI bleach; (

**b**) a double ROI bleach. The trajectories of the $+1$ defect, of the $-1$ defect, and of the ROI are shown in red, blue and white, respectively. The disclinations and ROI are tracked for 8 s. The position of the excitation beam polarization is given by black lines.

**Figure 4.**Distances of two matching defects with respect to the annihilation point. The graphs show the defect separation ${r}_{12}$ and the distances ${r}_{1}$ of the $+1$ defect and ${r}_{2}$ of the $-1$ defect from the point where they finally meet and annihilate. The dashed lines are square-root fits $r=d\sqrt{t}$, and the numbers give d in units of $\mathsf{\mu}$m s${}^{-1/2}$. Image adapted from Reference [52].

**Figure 5.**Flow field in a strongly mismatched pair of defects (mismatch angle approximately 180${}^{\circ}$). Because of the mismatch of the orientations, the defects do not approach each other along the straight interconnection line. The images show the pair with ROI near the $+1$ defect (

**b**) and the $-1$ defect (

**c**). (

**a**) A magnification of the $+1$ defect and ROI from image (

**b**).

**Figure 6.**Advective flow around disclinations in a $\pm 1$ pair. (

**a**) Flow field determined experimentally using the fluorescence bleaching method. The velocity bars are normalized with respect to that of the $+1$ defect (see text). The motion of the $s=+1$ is accompanied by the flow in the same direction of motion. In contrast, the flow is nearly absent in the vicinity of the $s=-1$ defect. (

**b**) Numerical flow velocity field showing the formation of the flow vortices around the $s=+1$ defect. Image (

**b**) reproduced with permission from Svenšek and Žumer [24], copyright American Physical Society.

**Table 1.**Measured flow fields in the images shown in this paper. All velocities are given in $\mathsf{\mu}$m/s. The indices ROI+ and ROI− refer to spots near the disclinations with the respective sign. For Figure 3b, the average of the velocities of both bleached spots is given. The accuracy of the velocity data is of the order of 1 $\mathsf{\mu}$m/s.

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**MDPI and ACS Style**

Missaoui, A.; Lacaze, E.; Eremin, A.; Stannarius, R.
Observation of Backflow during the Anihilation of Topologocal Defects in Freely Suspended Smectic Films. *Crystals* **2021**, *11*, 430.
https://doi.org/10.3390/cryst11040430

**AMA Style**

Missaoui A, Lacaze E, Eremin A, Stannarius R.
Observation of Backflow during the Anihilation of Topologocal Defects in Freely Suspended Smectic Films. *Crystals*. 2021; 11(4):430.
https://doi.org/10.3390/cryst11040430

**Chicago/Turabian Style**

Missaoui, Amine, Emmanuelle Lacaze, Alexey Eremin, and Ralf Stannarius.
2021. "Observation of Backflow during the Anihilation of Topologocal Defects in Freely Suspended Smectic Films" *Crystals* 11, no. 4: 430.
https://doi.org/10.3390/cryst11040430