Persistent quasiplanar nematic texture: Its properties and topological defects

In the so-called quasiplanar texture of a nematic layer confined between parallel plates with homeotropic anchoring conditions, the director field rotates by $\pi$ between limit surfaces so that field lines have the shape of a dowsing Y-shaped wooden tool. The orientation of the director field at midheight of the layer is arbitrary for symmetry reasons and is thus very sensitive to perturbations. We point out that contrary to accepted ideas the quasiplanar texture can be preserved infinitely in spite of its metastability with respect to the homogeneous homeotropic texture. We propose to call such a long-lived version of the quasiplanar texture the dowser texture. We demonstrate both experimentally and theoretically that in samples of variable thickness, the director field is sensitive to the gradient of the sample thickness through a linear coupling term. As a result, it has a tendency to follow the direction of the thickness gradient. Because of its sensitivity to perturbations we propose to call the midplane director field the dowser field and its tendency to follow the thickness gradient cuneitropism. Under effect of the gradient field, the dowser field obeys the sine-Gordon equation and exhibits domain walls that correspond to the well-known solitonic solutions of the sine-Gordon model.

Figure 1

Scheme of the setup.

Figure 2

Generation of the dowser texture: (a, b) coexistence of the homeotropic and quasiplanar textures separated by the peripheral disclination, (c, d) the dowser texture resulting from the collapse of the peripheral disclination. It contains the residual hyperbolic hedgehog, (e) perspective view of the dowser texture; the shape of the director field is reminiscent of the dowsing Y-shaped tool.

Figure 3

Transition from the homeotropic into the dowser texture observed in a thick enough sample of 5CB: (a–e) shrinking of the peripheral disclination, (f) the dowser texture. It contains the residual hedgehog resulting from the collapse of the disclination.

Figure 4

Diagram of stability of the homeotropic and dowser textures. The dowser texture D is obtained from the homeotropic texture H on the isochoric trajectory HD by an adequate increase of the sample thickness. The dowser texture is preserved along the inverse ${\mathrm{DD}}^{\prime }$ trajectory in spite of its metastability.

Figure 5

Winding of the dowser texture by a rotating magnetic field: (a) initial radial dowser texture with a hedgehog in its center, (b) generation of two $\pi$ walls by application of the magnetic field B, (b–g) winding of the dowser texture by rotation of the magnetic field; the two $\pi$ walls acquire spiral shapes, (h) the dowser texture after suppression of the magnetic field. The insert in picture b is a zoom on a small portion of the $\pi$ wall; arrows represent the dowser field.

Figure 6

Occurrence of $2\pi$ walls during unwinding of the dowser field. (a) Initial radial dowser field with a hedgehog in its center. (b) The dowser field wound in its center by $4\times 2\pi$ by means of a rotating magnetic field. This picture was taken shortly after suppression of the magnetic field at the end of the winding process. (c–f) Continuation of the relaxation at $B=0$. The winding angle in the center is, respectively, $3\times 2\pi$ in (c), $2\times 2\pi$ in (d), $2\pi$ in (e), and 0 in (f).

Figure 7

$2\pi$ walls generated by rotation of the magnetic field by the angle ${\varphi }_{\text{mag}}={125}^{\circ }$ (a), ${180}^{\circ }$ (b), ${225}^{\circ }$ (c), and ${360}^{\circ }$ (d). (e–h) Explanation of the generation of the two $2\pi$ walls by the ${\varphi }_{\text{mag}}={180}^{\circ }$ rotation of the magnetic field shown in (b): (e) initial radial dowser field with a hedgehog in its center, (f) application of the magnetic field, (g) rotation of the magnetic field by ${180}^{\circ }$, (h) formation of the two $2\pi$ walls after suppression of the field.

Figure 8

Collapse of a circular $2\pi$ wall: (a–d) four pictures selected from a video, (e) spatiotemporal cross-section of the video taken along the dotted line defined in (b).

Figure 9

Generation of a straight radial $2\pi$ wall: (a) pattern of isogyres and isochromes of a sample with the residual hedgehog in its center, (b) pattern of isogyres modified by the shift of the hedgehog toward the sample edge, (c) close-up of the $2\pi$ wall connected to the hedgehog.

Figure 10

Relaxation of the dowser field: (a) two $\pi$ walls in the dowser field aligned by the magnetic field B, (b–f) relaxation toward the radial configuration driven by the -$\mathbf{g}·\mathbf{d}$ coupling.

Figure 11

Explanation of the coupling between the dowser field d and the thickness gradient g: (a–c) between parallel plates, the energy of the dowser field does not depend on its orientation, (b–d) in a wedge, the energy depends on the angle $\varphi$ between the dowser field d and the thickness gradient g. For example, the distortion energy of ${\mathbf{d}}_1$ is lower than that of ${\mathbf{d}}_2$.

Figure 12

Analysis of the isogyres pattern in the $2\pi$ wall: (a) enlargement of the rectangle from Fig. 6; (b) dots: data extracted from a, line: fit to the equation (14) with $\varphi$ given by Eq. (12).

Figure 13

Dependence of the $2\pi$ wall's width on its radius.

Figure 14

Effects of anisotropic elasticity in wound dowser field. (a) Identification of the splay, bend, and saddle-splay deformations of the dowser field. (b) interference pattern of a wound dowser field observed in a polarizing microscope. (c) Line labeled “Intensity(x)”: measured intensity; line labeled “a+bsin(cx+d)”: fit to this sin function.