Director distortions and singularities in inhomogeneous fields

The effect of the electric-field inhomogeneity has been experimentally studied by polarizing microscopy in a homeotropic nematic liquid crystal in confined electrode geometries which may be relevant in display applications. Defects related to tilt inversion have been detected by monitoring the transmitted intensity profile as a function of the applied voltage. The position of the defects could be controlled by an additional magnetic field breaking the symmetry of the original arrangement. The phenomenon has been interpreted via numerical calculation of the director distribution using the continuum theory of nematics. The influence of oblique light incidence and of weak anchoring has also been analyzed. Simulations have provided good qualitative agreement with the observations. The method has turned out to be a sensitive tool to detect small misalignment angles between the magnetic field and the cell plane.

Figure 1

(Color online) Snapshots taken at crossed polarizers demonstrating the electric-field induced pattern in a rectangular pixel of low aspect ratio (type $A$ cell). The applied voltages are (a) $U=0.83U_F$, (b) $U=0.95U_F$, (c) $U=1.03U_F$, and (d) $U=1.17U_F$, respectively.

Figure 2

(Color online) Electroconvection patterns in confined geometry (a) around the threshold voltage $U_c$ and (b) above $U_c$.

Figure 3

Snapshots of the field induced patterns in confined geometry at superimposed electric and magnetic fields in a type $A$ cell at $U=0.8U_F$. The arrows indicate the location of the inversion wall. The magnetic fields are (a) $H=0.71H_F$, (b) $H=0.81H_F$, and (c) $H=0.93H_F$, respectively.

Figure 4

(Color online) Shift $s$ of the defect wall from the center (in units of the sample thickness $d$) versus the magnetic field $H$ (in units of the Freedericksz field $H_F$) at $U=0.8U_F$ in a type $A$ cell. The dotted line marks the location of the electrode edge.

Figure 5

(Color online) Geometry of the cell assumed in the calculations. The thick horizontal lines are the electrodes: infinite at $z=0$, finite $\left(-D/2\le x\le D/2\right)$ at $z=d$. The numerical calculations are done for the area $-L/2\le x\le L/2$, $0\le z\le d$. The thin solid lines are the equipotential lines; some lines of force for the electric field are shown dashed. The thick bars indicate the director orientation; $\theta$ is the director tilt angle. $\alpha$ is the misalignment angle of the magnetic field $\mathbf{H}$. The cell is illuminated by a monochromatic light in the direction $\mathbf{k}$ deviating from the normal incidence by an angle $\delta$.

Figure 6

(Color online) A three-dimensional plot of the calculated a; director tilt angle $\theta \left(x,z\right)$, and b; electric potential $\Phi \left(x,z\right)$ for the applied voltage $U=0.95U_F$ at $H=0$.

Figure 7

(Color online) Calculated $x$-profile of the tilt angle $\theta \left(x,d/2\right)$ in the middle of the cell for various voltages $U$. The vertical dashed lines indicate the electrode edges.

Figure 8

(Color online) Calculated voltage dependence of the maximal tilt angle ${\theta }_{\text{max}}$.

Figure 9

(Color online) Calculated $z$-profile of the tilt angle $\theta \left(0,z\right)$ at the electrode edge, as well as inside and outside the pixel at $U=0.95U_F$.

Figure 10

(Color online) Reconstructed gray scale images at various voltages for the position dependence of the calculated transmitted intensity around the electrode edge at crossed polarizers. The vertical solid lines mark the location of the electrode edges; the dashed line is the center of the electrode.

Figure 11

(Color online) Position dependence of the measured transmitted intensity around the electrode edge at crossed polarizers in a type $A$ cell. The vertical solid lines mark the location of the electrode edge; the dashed line is the center of the electrode.

Figure 12

(Color online) Calculated maximum distortion angle ${\theta }_{\text{max}}$ of the deformation induced by an oblique magnetic field at $U=0$.

Figure 13

(Color online) A three-dimensional plot of the calculated director tilt angle $\theta \left(x,z\right)$ for $U=0.79U_F$, $H=0.7H_F$, and $\alpha =0.5°$.

Figure 14

(Color online) Calculated $x$-profile of the tilt angle $\theta \left(x,d/2\right)$ in the middle of the cell for various $\alpha$ angles at $H=0.7H_F$ and $U=0.79U_F$.

Figure 15

(Color online) Calculated $\alpha$ dependence of the shift of the inversion wall at $H=0.7H_F$ and $U=0.79U_F$.

Figure 16

(Color online) Calculated shift of the inversion wall versus $H$ for various misalignment angles at $U=0.79U_F$. The dashed horizontal line indicates the position of the electrode edge; the vertical line marks the combined Freedericksz threshold $H_{UF}$.

Figure 17

(Color online) Microphotographs of the field induced deformation around an electrode strip in a type $B$ cell at $U=0.79U_F$ for various magnetic fields. The vertical solid lines mark the location of the electrode edges; the dotted line is the center of the electrode. The horizontal bars indicate the shift of the inversion wall.

Figure 18

(Color online) Shift $s$ of the inversion wall from the center (in units of the sample thickness $d$) versus the magnetic field $H$ (in units of the Freedericksz field $H_F$) at various applied voltages in a type $B$ cell. The horizontal line marks the location of the electrode edge; the vertical lines show the threshold magnetic field of the combined electric-magnetic Freedericksz transition. The solid curve is the theoretical shift for $U=0.79U_F$ and ${\alpha }_{fit}=-0.3°$.

Figure 19

(Color online) Shift $s$ of the inversion wall from the center (in units of the sample thickness $d$) versus the magnetic field $H$ (in units of the Freedericksz field $H_F$) at various cell rotation angles $\Delta \alpha =\alpha -{\alpha }_0$ at $U=0.79U_F$ in a type $B$ cell. The horizontal lines mark the location of the electrode edges; the vertical dashed line show the threshold magnetic field of the combined electromagnetic Freedericksz transition.

Figure 20

(Color online) Shift $s$ of the inversion wall from the center (in units of the sample thickness $d$) versus the cell rotation angles $\Delta \alpha =\alpha -{\alpha }_0$ at various magnetic fields $H$ at $U=0.79U_F$ in a type $B$ cell. The lines correspond to a second-order polynomial fit.

Figure 21

(Color online) Shift $s$ of the extinction position from the center (in units of the sample thickness $d$) versus the magnetic field $H$ (in units of the Freedericksz field $H_F$) at $\alpha =-1.5°$ and $U=0.79U_F$ for various light incidence angles $\delta$ in a type $B$ cell. The dashed horizontal line marks the location of the electrode edges; the vertical dotted line shows the threshold magnetic field $H_{UF}$ of the combined electromagnetic Freedericksz transition.

Figure 22

(Color online) Calculated shift $s$ of the inversion wall from the center (in units of the sample thickness $d$) versus the magnetic field $H$ (in units of the Freedericksz field $H_F$) at $\alpha =-0.3°$ and $U=0.79U_F$ for strong as well as for weak anchoring of different strength. The dashed horizontal line marks the location of the electrode edges; the vertical dotted line shows the threshold magnetic field $H_{UF}$ of the combined electric-magnetic Freedericksz transition.