Umbilical defect dynamics in an inhomogeneous nematic liquid crystal layer

Electrically driven nematic liquid crystals layers are ideal contexts for studying the interactions of local topological defects, umbilical defects. In homogeneous samples the number of defects is expected to decrease inversely proportional to time as a result of defect-pair interaction law, so-called coarsening process. Experimentally, we characterize the coarsening dynamics in samples containing glass beads as spacers and show that the inclusion of such imperfections changes the exponent of the coarsening law. Moreover, we demonstrate that beads that are slightly deformed alter the surrounding molecular distribution and attract vortices of both topological charges, thus, presenting a mainly quadrupolar behavior. Theoretically, based on a model of vortices diluted in a dipolar medium, a $\frac{2}{3}$ exponent is inferred, which is consistent with the experimental observations.

Figure 1

Snapshot of umbilical defects of opposite charges observed in a nematic liquid crystal layer within circular crossed polarizers (CCP). Umbilical defect of positive (negative) charge has circular (square) shape.

Figure 2

Schematic representation of the experimental setup of a nematic liquid crystal layer with negative dielectric constant and homeotropic anchoring under the influence of a vertical voltage. The essential parts of the setup are emphasized. Crossed polarizers, either linear or circular, are used in order to analyze the liquid crystal texture. Two types of liquid crystal cells have been studied: (i) homogeneous and (ii) inhomogeneous (with glass beads) sample.

Figure 3

Creation and annihilation dynamics of umbilical defects in a homogeneous nematic liquid crystal layer under two crossed linear polarizers. The temporal sequence of snapshots from left to right and top to bottom (a)–(e) corresponds to driving the cell from zero voltage to a voltage $V_0$ beyond the Fréedericksz transition threshold. The bottom numbers in each panel account for the respective frame. The temporal increment of each frame corresponds to $400\mu \mathrm{s}$. The interception of four black brushes gives the position of umbilical defects with topological charge $\pm 1$. (a) Liquid crystal cell without applied voltage, showing the orientation, respectively, of the polarizer (P) and analyzer (A). (b) Emergence of orientational domains after $800\mu \mathrm{s}$ the voltage is switched to $V_0=15V_{\text{rms}}$. (c) Creation of vortices through reorganization of domains; circumferences and respective numbers account for the different topological charge of the defects. The interception of two black brushes gives the position of the defects with topological charge $\pm \frac{1}{2}$. (d) Diluted gas of vortices. (e) Vortex pair and (f) temporal evolution of the number of vortices $N\left(t\right)$.

Figure 4

Coarsening dynamics in a homogeneous cell. Number of umbilical defects as a function of time. The solid black and dashed curves are, respectively, the experimental evolution of $N\left(t\right)$ and the fitting curve $N_f\left(t\right)=\beta t^{-\alpha }+N_{\infty }$ with $\alpha =0.981\pm 0.001, \beta =5.7\times {10}^3\pm 2\times {10}^2$, and $N_{\infty }=70\pm 2$.

Figure 5

Number of umbilical defects as a function of time. Coarsening process of umbilical defects in a homogeneous cell with different driven voltages (a) $V_0=60$ V, (b) $V_0=80$ V, (d) $V_0=100$ V, using the same frequency 100 Hz. The black and red curves account for the experimental data and fitting curve formula (9), respectively. Critical fitting exponent $\alpha$ (d), asymptotic number of vortices $N_{\infty }$ (d), and $\beta$ (e) as function of applied voltage.

Figure 6

Umbilical defect annihilation dynamics in an inhomogeneous liquid crystal cell with glass beads and in-between two linear crossed polarizers. Temporal sequence of snapshots from left to right and top to bottom depicts the vortex evolution starting from the switch-on of the driving voltage. The temporal increment of each frame corresponds to 0.33 s. The position of the umbilical defects is given by the interception of four black brushes. The white dashed circumference accounts for the position of a glass bead.

Figure 7

Coarsening process of umbilical defects in an inhomogeneous sample analyzed over different observation zones, I and III. (a), (d) Voronoi diagram of glass beads in the observed zone, I and III, respectively. (b), (e) Histogram of the mutual distance of the glass beads. The solid curve is a fitting curve using a Rayleigh distribution. (c), (f) Corresponding scaling curves of the number of defects vs normalized time. Black points stand for the experimental data, dashed lines correspond to fitting curves of the form $N\left(t\right)=\beta /t^{\alpha }+N_{\infty }$ with $\alpha =0.60$ and 0.25, respectively.

Figure 8

Interaction between a glass bead and umbilical defects observed under linear crossed polarizers. (a) Sequences of temporal snapshots. The bottom numbers in each panel account for the respective frame. The temporal increment of each frame corresponds to $0.07\mathrm{s}$. Umbilical defects of a positive (circular shape) and negative (square shape) charge under circular crossed polarizers are recognized and monitored. The dashed circles account for the umbilical defects. (b) Trajectory of the vortices: the dashed points (red) indicate the trajectory of defects, the dashed straight line accounts for the initial distance between defects.

Figure 9

Quadrupolar structure generated by a glass bead in the liquid crystal medium. Experimental characterization of the collision points of the umbilical defects for different experimental realizations. In most of the realizations only one defect collides with the bead. The dark (blue) and light (red) points account for the collision points of positive, respectively, negative vortices. The color areas highlight the different collision regions of the bead.

Figure 10

Schematic representation of the director field lines induced by a perfect spherical bead (left panel) and a slightly deformed bead (right panel). Upper and lower panels show a side and top view of the correspondingly induced defects.

Figure 11

Optical micrograph of active and passive glass beads. Snapshots of a liquid crystal sample within linear crossed polarizers without (top panel) and with (bottom panel) voltage. The lateral beads are passive, while the central bead is active.

Figure 12

Schematic representation of the vortex-pair interaction in the presence of glass beads. The index $k$ accounts for the $k\mathrm{th}$ glass bead. ${\mathbf{r}}_{\mathbf{i}}$ is the vector position of $i\mathrm{th}$ vortex. ${\mathbf{l}}_{\mathbf{k}}$ is the vector position of the $k\mathrm{th}$ glass bead. The inset is an experimental snapshot obtained with the sample in-between crossed linear polarizers.