2.1.

GLYPHS, STREAMLINES AND POLARIZATION MICROSCOPY

In nematic liquid crystals simple director field is usually represented by line segments (Figure 1a). Unfortunately, effectively, this is useful only if visualizing inplane director fields which are twodimensional. The out-of-plane director on a two-dimensional cross section was traditionally described by ‘nails’, where the out-of plane orientation was marked by distinguishing the length of the nail (Figure 1c). With the advances of computer graphics a straightforward method is to draw the director as glyphs of different shapes (Figure1 b&d) where shading is used to represent orientation. In case those deformations are rather weak or more complex in the third dimension, streamlines that follow director field are a better of visualization than glyphs. It should be stressed that its implementation requires careful use of tensorial nature of the director field taking into account the n to –n symmetry of the director [15]. The approach with streamlines is illustrated by a visualizing the twisted boojums that accompany a colloidal particle with planar anchoring in a chromonic liquid crystal [18]. By comparing Figures 1e&f one clearly sees that glyphs hardly indicate the left & right twisting of the nematic director which on contrary is clearly seen in the streamline presentation. The director field glyphs enhanced by color coding are another option to better describe 3D variation of the nematic field [15].

Figure 1:

(a, b) Use of two kinds of glyphs for the presentation of 2D director fields in a cross section of a singular +1/2 disclination and (c,d) a nonsingular +1 twisted disclination. (e,f) Glyph presentation is contrasted to the stream lines for the case of counter-twisted boojums that accompany colloidal sphere enforcing planar anchoring in a chromonic liquid crystal. (g) Simulated polarization microscope picture of a director field induced by such a colloidal particle under crossed polarizers and (h) under crossed polarizers with added lambda plate.

Experimentally, the most common way of observing liquid crystalline structures is by polarized optical transmission microscopy (PM) (e.g. see Ref. 24). This method has a strong potential for recognition of structures primarily in cases when the structures vary mostly in a plane perpendicular to the observation direction. However, even for rather simple 3D nematic fields the method is much less effective in determining details of the structure. The method is also limited to samples that are thinner than the coherence length of the light. To obtain a simulated PM picture usually simple straight ray optics is used to calculate the total phase shift between ordinary and extraordinary component of the ray. The computation is based on slicing the Q-tensor field into long and thin prisms along which effectively, the light-ray is propagated. The local variation of birefringence is described by the Jones matrices [31]. This technique allows for calculations of a transmitted monochromatic and polychromatic light. The simulated white light PM pictures of a two opposite twisted boojums accompanying a planar colloidal particle in a chromonic liquid crystal (Figures 1g&h) demonstrate that beside microscopy with cross polarizers also a picture with inserted lambda wave plate is needed to uncover twisting details of such 3D nematic field [32].

2.2.

DEGREE OF ORDER AND DISCLINATIONS

Defect structures are often visualized by plotting only singular defect lines via drawing isosurfaces of constant nematic degree of order S for some preselected value that is lower as in unperturbed bulk nematic [31, 33]. The Figure 2a illustrates a simple case of a homeotropic colloidal spherical particle encircled by a -1/2 disclination ring. The visualization of the disclination loop via S-isosurface is accompanied by line segment representation of the inplane director field. In contrast, for a trefoil knot-like homeotropic colloidal particle accompanied by two knotted disclination lines (Figure 2b) the director is intentionally omitted as it strongly varies in all three spatial dimensions (for details on the experiment and modeling see Ref. 6).

Figure 2:

Presentation of the -1/2 singular disclination lines that accompany colloidal particles with homeotropic surface anchoring of a nematic media using isosurfaces of S corresponding to the 5% reduction of the bulk order parameter. The Saturn ring disclination loop which encircles the spherical colloidal particle is complemented also by the nematic director field in a cross section (a). In case of knotted colloidal particle, the physical knot is entangled by two nematic field knots (b). To avoid an overcrowded picture the director field is intentionally not presented.

2.3.

SPLAY, BEND AND TWIST

To examine the fine structure of the director field around defects, the elastic nematic deformation modes -splay, bend and twist-can be effectively extracted and visualized by highlighting the degree of specific deformations in the immediate surroundings of the defects without showing complete nematic field. One such parameter that easily visualizes the winding number of disclination lines is the scalar splay-bend parameter S_SB, constructed from second derivatives of the order parameter tensor Q_ij [34]:

where note that summation over repeated indices is assumed. Effectively, the scalar splay-bend parameter S_SB is fully analogous to the divergence of the order- and flexoelectric polarization arising from the deformation of the nematic field in a simplified flexoelectric model [24]. Neglecting the biaxiality and variation of the degree of order S, the tensor form reduces to the well-known vector form:

The first term stems from a strong divergence of splay deformation that corresponds to positive S_SB, while negative valuesof the second term imply strong divergence of the bend deformation. Splay-bend parameter visualization can be applied also for distorted director profiles, as long as the splay and bend deformations reach extremes nearby the disclination lines. The splay-bend parameter proves to be particularly efficient in finding escaped defect lines with integer winding numbers (see Ref. 15). When visualizing twist deformations in chiral and archiral nematic liquid crystals a local twist parameter S_TW, similar to the chiral term in the nematic elastic free-energy is used [8]:

where S is the bulk-order parameter and q₀ is the inverse pitch different from zero only for intrinsically chiral phases, making S_TW zero in the ground state of chiral and achiral nematics. It is convenient for identifying twist variations of singular disclination lines mostly induced by geometrical confinement and for twisted deformations of nonsingular disclinations when the anisotropy of the twist elastic constant (weak) plays the crucial role. For practical purposes both parameters are scaled with the nematic correlation length, thus being dimensionless quantities. Figure 3 illustrates how the splay-bend and twist parameters vary along a -1/2 disclination loop entangling a colloidal dimer in a π-twisted nematic cell [35]. In Figure 4 it is illustrated how the splay-bend parameter S_SB can reveal distortions of a -1/2 disclination loop in a cholesteric phase confined to a spherical droplet with the surface imposing strong homeotropic anchoring. The line cross-sections distort only close to the surface whereas in the bulk they exhibit geometrical distortions known as writhe and twist. The sum of writhe and twist equals the topological invariant – self-linking number - of an arbitrary complex -1/2 disclination loop [12,9].

Figure 3:

Colloidal dimmer confined to π-twisted nematic cell is bound by a (small) circular -1/2 disclination loop and a larger more complex -1/2 disclination loop that locally notably deviates from the -1/2 hyperbolic, as visualized by the splay-bend parameter and twist parameter. Blue color indicates large splay and yellow large bend. Green and violet indicates positive and negative twisting. The two insets show local cross-sections of the larger loop.

Figure 4:

Knotted disclination loop in a cholesteric droplet with homeotropic surface anchoring. Diameter of the droplet is 5 times the value of the intrinsic cholesteric pitch. The splay-bend parameter and nematic order parameter are used to show that profile of the disclination which deviates substantially from the there-fold ribbon when near the surface of the droplet.

2.4.

PONTRYAGIN–THOM SURFACE AND MULTI-PHOTON FLUORESCENCE POLARIZATION MICROSCOPY

The elementary polarization microscopy technique for 2D nematic structures based on counting dark Schlieren brushes was in past years extended to three dimensions by studying three-dimensional regions that share common director orientation, as can be determined by confocal multi-photon flurescence polarization microscopy [27-29]. Chen and collaborators have shown [3,13] that by selecting the light beam direction, the Pontryagin – Thom (PT) surface can be constructed that is formed of points where director is perpendicular to the beam direction. The surface is particularly simple when the system is characterized by a homogenous far-field director and a beam that is parallel to it. The PT surface is closed for nematic fields that have no defects and no confining surfaces. In case of confinement, the PT surface is limited by confining isosurfaces and similarly, if defects are present, they also limit the PT surface. For more details how PT surfaces are linked to topological charges see Ref. 15. In Figure 5, the colloidal particle in a homogenous unwinded cholesteric farfield is decorated by a hopfion, making the PT surface simple yet bounded by the particle. The simulated confocal image of multi-photon PM is shown as well. In Figure 6, a cholesteric droplet with homeotropic surface anchoring contains a disclination loop in the form of a trefoil knot: the PT surface is complex and bounded by the disclination loop and the droplet surface.

Figure 5:

Colloidal particle in the unwinded cholesteric decorated by a hopfion. Director presentations based on (a) glyphs, (b) splay-bend parameter, (c) Pontryagin – Thom surface. (d) Simulated multi-photon fluorescent PM image.

Figure 6:

Trefoil knot in a cholesteric droplet with a diameter 5 times the intrinsic pitch. On the right panel a Pontryagin – Thom surface is plotted together with a color wheel describing the director orientation.