Topological defect coarsening in quenched smectic-$C$ films analyzed using artificial neural networks

Mechanically quenching a thin film of smectic-$C$ liquid crystal results in the formation of a dense array of thousands of topological defects in the director field. The subsequent rapid coarsening of the film texture by the mutual annihilation of defects of opposite sign has been captured using high-speed, polarized light video microscopy. The temporal evolution of the texture has been characterized using an object-detection convolutional neural network to determine the defect locations, and a binary classification network customized to evaluate the brush orientation dynamics around the defects in order to determine their topological signs. At early times following the quench, inherent limits on the spatial resolution result in undercounting of the defects and deviations from expected behavior. At intermediate to late times, the observed annihilation dynamics scale in agreement with theoretical predictions and simulations of the $2\mathrm{D} XY$ model.

Figure 1

Schematic of the film quenching experiment. A thin smectic-$C$ film drawn across a small, circular opening in a sealed chamber is temporarily deformed to a dome by increasing the chamber pressure. A sudden release of the excess pressure then causes the film to return rapidly to its original planar geometry, a mechanical quench that results in the spontaneous formation of topological defects in the director field (inset). The defects are visualized using polarized reflected light microscopy and the coarsening of the defect texture recorded using a high-speed video camera.

Figure 2

Defect coarsening in a mechanically quenched SmC film. The topological defects are located at the points from which the pairs of light and dark brushes (resembling bow ties) emanate. Thousands of $+1$ and $-1$ defects are generated during a typical quench and these mutually annihilate over time. The polarizer and analyzer are decrossed by $60{}^{\circ }$. The intensities of these images, which show only a small part of the film, have been normalized as described in the text.

Figure 3

Normalization of experimental images. (a) Raw experimental image ($t=0.4\mathrm{s}$). (b) Normalized image. The raw images extracted from the quenching videos were normalized to have the same mean image intensity and dynamic range. This procedure ensured that the statistics of all of the experimental images from different quenching events would match those of the training set.

Figure 4

Defect trajectories in a quenched film as determined by the trained neural network (green tracks) compared with defect locations identified manually (white crosses) every 500 frames (at 1 second intervals). The trajectories, which were determined over the course of 3000 frames (an elapsed time of 6 seconds), are superimposed here on the final image in the video sequence, with the surviving $+1$ and $-1$ defects shown respectively in red and blue. The dark speckles are liquid crystal droplets deposited on the chamber window by previously ruptured films. The neural network was trained to ignore these artifacts. The polarizer/analyzer settings are as in Fig. 3.

Figure 5

Bright brush orientation as a function of time for a typical pair of annihilating defects. Quenching occurs at $t=0$ and the pair annihilates at $t=9.6\mathrm{s}$. Over the lifetime of the pair, the brushes around the $+1$ defect (red trace) fluctuate about an azimuth of $135{}^{\circ }$ but do not change their average orientation significantly. The $-1$ defect (blue trace), in contrast, is more orientationally mobile. Initially oriented at $75{}^{\circ }$ in this example, the brushes soon rotate to around $20{}^{\circ }$, where they remain until annihilation. The insets show snapshots of the defects shortly after the quench ($t=0.5\mathrm{s}$) and shortly before annihilation ($t=8\mathrm{s}$). The variations in brush orientation over time are evaluated by a binary classification network in order to determine the topological signs of the defects.

Figure 6

Defect locations predicted by the neural network and classified according to topological sign. The image shows a SmC film six seconds after quenching, with the defects color-coded according to the topological sign predicted by the custom binary classification network. The defect locations determined by the deep learning model proved generally to be highly accurate. In this example, every defect was detected: 18 of strength $+1$ and 21 of strength $-1$. The film diameter is $5\mathrm{mm}$, much larger than the image, the black borders here corresponding to the edges of the microscope field stop. The polarizer/analyzer settings are as in Fig. 3.

Figure 7

Defect number vs time in a quenched smectic-$C$ film. The large number of topological defects generated in a typical quenching experiment decreases over time by the mutual annihilation of $+1, -1$ defect pairs. Images of the film were analyzed starting when the film came back into focus, about 0.2 seconds after the mechanical quench. A fit with the diffusive model (green), which describes algebraic decay of the defect number $N$ with an exponent of 1, predicts coarsening at long times that is faster than observed. The logarithmic correction term of Yurke's model (blue) results in a slower annihilation rate that closely approximates the experimental measurements. Power-law decay with an exponent of 0.9, as suggested by the simulations of Jang et al., is in accord with the Yurke model and matches the data similarly well (red). The experimental data is duplicated here for visual clarity.

Figure 8

Architecture of the $\mathrm{YOLOv}5$ neural network. $\mathrm{YOLOv}5$ comprises three parts, shown in the diagram. The model backbone is a Cross Stage Partial Network (CSPDarknet) for feature extraction. A Path Aggregation Network (PANet) combines these features etc.

Figure 9

Performance of the $\mathrm{YOLOv}5$ model trained to detect topological defects. (a) The precision-recall curve yields a mean Average Precision (mAP) of 0.97, corresponding to a high degree of confidence in defect detection by the trained network. (b) The highest value on the $\mathrm{F}1$ curve (the $\mathrm{F}1$ score) was 0.96. The best $\mathrm{F}1$ score was obtained at a confidence threshold of 0.27.

Figure 10

Sign-agnostic topological defect pair correlation functions computed from experimental and simulation data shortly after a quench ($t=0.2\mathrm{s}$). The correlation function derived from the simulation (blue) grows continuously from zero, while the experimental curve (red) is identically zero below $10\mu \mathrm{m}$, the resolution limit for defect detection by the neural network.