Examples of 3D director imaging

For all FCPM textures:

The stronger the fluorescence signal – the smaller is the angle between the polarizer and local director

Nematic

Cell Undergoing Frederiks Transition

As the voltage is applied, director n in the center of the cell reorients towards the axis Z, which is manifested by a weaker fluorescence.

Figure above also illustrates some limitations of the technique. In an optically anisotropic medium, light generally splits into different modes that propagate with different speed, which makes it difficult to focus the beam. However, in most practical situations, birefringence-induced defocusing is sufficiently small to justify the use of FCPM. It can be roughly estimated as zΔn/n, where z is the depth of scanning, Δn is the difference in the extraordinary and ordinary refractive indices, and n is an average refractive index. With z ~ 15 micron, Δn ~ 0.1, and n ~ 1.6, the characteristic defocusing is only 1 micron. Limited resolution is clearly visible in the vertical FCPM cross-sections near the glass substrates: the decrease in the fluorescence intensity is caused by the finite size of the region where the light is focused rather than by reorientation of n. Finally, light absorption and increased defocusing near the bottom plate make the image somewhat asymmetric along the Z-axis. Low-birefringence LCs and small concentrations of dye help to mitigate these problems.

Below the Frederiks transition is studied in more details. At no applied field the director n is parallel to the axis "y". At small voltages, the structure remains uniform, n_x=0, n_y=1, n_z=0. Above a well-defined threshold of 2.6 V, the electric field reorients from its strictly horizontal orientation towards the z-axis. In the FCPM textures the transition is manifested by the reduced intensity of fluorescent light.

Note that the bands of low intensity are located in the middle of the cell. This is precisely what the theory of Frederiks transition predicts: the maximum deviation of should be in the middle plane. As the field increases, both the amplitude of the director reorientation and the thickness of the reoriented zone should increase. This prediction of the theory is clearly visualized by in the FCPM images of the vertical cross-section of the cell. The experimental FCPM textures are in agreement with the calculated dependencies of director deviation from the plane of cell, angle θ, precented below.

Below we apply FCPM to visualize polar surface instabilities in a nematic LC with positive dielectric anisotropy confined in a cell with homeotropic boundary conditions. The instabilities arise under action of DC electric field due to symmetry braking at cell substrates. The surface instabilities are manifested by appearance of rounded domains due to static director deformations. FCPM vertical cross-sections below clearly show that the director deformations start at electrodes and the symmetry of the deformation depends on the polarity of applied DC-voltage.

Now FCPM is applied to visualize spatially-periodic orientation of director that resulted from periodic distribution of residual charges in a nematic LC layer confined in a cell with planar boundary conditions and subjected to a low-frequency apllied voltage. The used material (MBBA) has negative dielectric anisotropy. For the textures below the rubbing direction is along x. Applied voltage is 8V, frequency f= 70 Hz. The regions with lower intensity in the FCPM texture show where director is reoriented from initial planar orientation (along x) toward the cell normal. Compare the FCPM and PM images below:

• The twist axis is normal to the cell substrates and parallel to the optical axis of the microscope. The specific feature is that for light propagation along the twist axis, the polarization of both ordinary and extraordinary waves can follow the local director when the pitch p of the helicoidal twist is much larger than the wavelength of light (the so-called Mauguin regime).

• We image vertical cross-sections of the TN cell in the regime of ordinary wave excitation (P || n at the irradiated top plate, a), and in the regime of extraordinary wave excitation (P || n at the top plate, b).

• The fluorescence signal is much stronger in the case of extraordinary wave excitation, b, since the light polarization follows the local director. The fluorescence signal is practically uniform in both horizontal and vertical scans of the cell. The macroscopic 90^o- twist is not visualized because of the Mauguin effect.

When the applied voltage exceeds the threshold of Frederiks transition in the TN cell, the director reorients. Both functions θ and Φ change, as the computer simulations demonstrate. In the regions with small θ the effective birefringence is reduced. For example, our estimates show that in the middle plane of a used in our experiment TN cell with ZLI-3412, the effective birefringence is 0.003, well below its original value 0.078 at zero voltage.

Smaller birefringence means that the Mauguin number

M_au = 0.5pΔn/λ

is reduced. The Mauguin condition M_au >> 1 is not satisfied anymore and light polarization does not follow the director twist exactly. The last feature allows one to visualize the orientation of dye in the TN cell, at least partially. At voltages 3V and higher, the fluorescence signal is stronger near the bottom substrate, where At very high voltages >12V the director is vertical, θ = 0 almost everywhere in the cell. Therefore, the fluorescence signal is weak.

The experiment with TN cell shows that the director twist in the direction of observation can be visualized by FCPM if the Mauguin number is not too large. Of course, small birefringence of the LC material helps, as it reduces both the Mauguin number and the defocusing effects.

The planar cholesteric texture is often accompanied by defects of which the most common are oily streaks and dislocations in the systems of layers. FCPM readily visualizes both oily streaks and dislocations. In the FCPM vertical cross-section of cholesteric planar texture below four non-singular λ-disclinations form the so-called Lehmann claster.

These disclinations are in tilted cholesteric lamellae

The FCPM textures below confirm the basic model of an oily streak as a pair of parallel disclinations of strength +1/2 with a wall defect between them. The cholesteric layers either interrupt at this wall, or continuously reorient by 180^o around the cores of the two disclinations. Most importantly, the textures clearly reveal that the two disclinations of strength +1/2 at the base of the oily streak are of the λ - type, according to the Kleman-Friedel terminology.

We stress that the FCPM textures in the figures above show the cross-sections of the cholesteric lamellae with disclination lines; since the lines run parallel to the substrates of the cell, their type and location along the z-axis would be hard to identify by the usual PM.

Recently, researches proposed a new type of colloidal interactions of particles embedded in a liquid crystalline solvent. These interactions arise from orientational elasticity of the anisotropic host and originally were reported for the colloidal inclusions in the bulk of a liquid crystal. At the same time, 2D organization of nanometer- and micrometer-sized colloidal particles at fluid interfaces is a fascinating phenomenon of both fundamental and applied interest. The nature of micron-scale interparticle forces, especially of the attractive nature, remains a subject of ongoing debate, as the regular gravity-capillary mechanism becomes irrelevant when the size of the particles is less than about . In most cases, the interparticle forces are isotropic and cause hexagonal type of ordering. Anisotropic interactions and thus a richer variety of ordered patterns at fluid surfaces can be achieved when the particles are nonspherical. Apparently, the surfaces of anisotropic fluids, i.e., liquid crystals, can also support anisotropic interactions and patterns of colloidal particles. Two-Channel FCPM study reveals that the glycerol droplets are located at the liquid crystal – air interface (see Figure below). The mechanisms responsible for surface behavior of colloidal particles are studied much less than the mechanisms controlling their behavior in the liquid crystal bulk. Recently, we proposed that the surface placement of colloidal particles introduces a new mechanism of the attractive force through the elasto-capillary coupling. Namely, the submerged part of each particle induces elastic distortions in the LC because of the finite surface anchoring at the particle's surface. These elastic distortions result in the vertically-resolved force that acts at the meniscus around the droplets. The increased area of the liquid crystal surface leads to the capillary attraction between the particles and ultimately results in ordered structures, when balanced by elastic repulsion at short separations. The vertical cross-section of the hexagonal structure, reconstructed positions of particles, and director field in the liquid crystal layer are shown below:

One of the homeland security concerns is to develop fast detectors of harmful microorganisms such as Anthrax, thereby eliminating the need to grow the culture for days. Such a technique can be realized using the idea that the growing immune complex (an aggregate formed by the microorganisms to be detected and their specific antibodies) would be accompanied by different types of 3D director fields, depending on how big is this complex as compared to the anchoring extrapolation length. Isolated microbes that are not recognized are not detected while the targeted microbes glued into sufficiently large immune complexes by specific antibodies trigger director changes readily detectable by optical means. To prove the concept, and to refine the detection technique, however, 3D visualization of the director field is absolutely essential, as it allows us to determine light transmittance (which determines a “positive” signal) through the LC sample as the function of director distortions around the immune complex. One can spatially co-localize polarizing microscopy (PM) textures of director distortions in a liquid crystal and the textures of fluorescent signal from fluorescently labeled biological objects (bacteria, viruses, etc.) or beads as their model systems. This allows one for the study of the size and shape of a colloidal inclusion on director distortions in a liquid crystal cell. Notice below that the small particles and their aggregates do not produce visible director distortions, whereas the bigger ones do:

We used fluorescent-labelled (Dragon Green fluorochrome), antigen-coated (streptavidin) latex beads (Bangs Laboratories, Inc.) of diameter 0.56 microns. The used lyotropic liquid crystal is a solution cromolyn (13 wt%) in water. Fluorescent confocal microscopy was used to track the location and size of fluorescently labelled particles. Transmission-mode polarising microscopy was used to detect the director distortions around the complexes. The easy axis of director orientation in the LCLC cassette was aligned either parallel or perpendicular to the polarizer, so that in the absence of distortions, the polarising-microscope texture was dark. The immune complexes observed in the confocal mode were detected by fluorescent markers, Parts A,C,E of the Figure. The polarising-mode textures of the same regions, Parts B,D, F depended on the size of complexes. Complexes (as well as individual non-reacted beads and antibodies) that are smaller than 2 microns in diameter did not cause any noticeable light transmission through the crossed polarizers and the LCLC sample, Part B. In contrast, complexes larger than 2 microns produced noticeable light transmission caused by director distortions in the surrounding LCLC matrix, Part D,F. Note that the director distortions caused by immune complexes of size larger than 2 microns are much larger than then their size (compare Parts E and F) because of the elastic nature of director distortions in LCs.

This material is based upon a multidisciplinary work that is supported by the National Science Foundation and the Intelligence Technology Innovation Center through the joint "Approaches to Combat Terrorism" Program Solicitation NSF 03-569 (DMR-0346348). The work is performed in collaboration with scientists at the Biological Sciences Department of Kent State University (Prof. Chris Woolverton) and at the Northeastern Ohio College of Medicine (Dr. G. Niehaus and Dr. K. Doane).

For more details see Ref. [S.V Shiyanovskii, T. Schneider, I.I. Smalyukh, T. Ishikawa, O. D. Lavrentovich, G. D. Niehaus, K. J. Doane, and C. J. Woolverton, Director distortions and light transmittance around growing immune complexes in lyotropic chromonic liquid crystal and real-time microbe detector, in press (2004)].

In smectic A, the rod-like molecules form layers periodically stacked along n. Director deformations are often in the form of the so-called focal conic domains (FCDs or confocal domains). The term has nothing to do with the confocal technique: it refers to the fact that the frame of the domain is constructed by two confocal defect lines, usually, a pair of mutually perpendicular ellipse and hyperbola. An interesting property (known to mathematicians already in XIX century!) is that the family of curved surfaces (such as smectic layers) can be wrapped around these two lines preserving their equidistance everywhere except at the very defect cores. The 3D configuration of n inside the domain is rather complex: any line that connects a point on the ellipse to a point on the hyperbola, is the local optic axis.

FCPM allows one to clearly reconstruct the basic features of this pattern. Figure 3 shows both "horizontal" xy (parallel to the ellipse) and "vertical" xz (parallel to the hyperbola) cross-sections of the domain. Compare the FCPM and PM textures below.

• FCPM textures in Figs. a and c are (x,y) optical slices taken from the middle-plane of the cell. The intersection of the hyperbola defect with the plane is a spot (marked 'h' in Fig. a) of relatively low intensity.

• The standard PM textures of FCDs, Figs.d and f, show extinction whenever n (=optical axis) is along the polarizer or the analyzer. The maximum intensity is achieved for intermediate orientations. Interchanging the directions of the polarizer and the analyzer leaves the optical texture intact, Figs.d and f. Therefore, the standard PM does not distinguish between two mutually perpendicular director fields, while the FCPM does.

• However, the most striking evidence of the superiority of FCPM over PM is in the ability to resolve the structural features along the direction of view, as discussed below. The maximum intensity is again in the regions where local director is parallel to polarizer. In particular, the surrounding of the FCD is dark, which corresponds to the director oriented along the vertical axis z. In other words, the smectic layers are parallel to the plates far away from the FCDs, as expected.

• The hyperbola defect in Fig.b is seen as a relatively dark line. One would expect that at the core of the defect the liquid crystal and dye molecules are oriented along the hyperbola; since the hyperbola is almost vertical, the intensity of fluorescence should be small. Fig. b also demonstrates that the ellipse plane is in the middle of the cell. The reason is related to the surface anchoring phenomena at the bounding plates. The smectic layers cross the ellipse plane perpendicularly. If the ellipse were located at the boundary, this orientation of layers would strongly contradict the normal surface anchoring of the director. When the elliptic base shifts towards the middle of the cell, the director at the surface becomes closer to vertical orientation.

• Note that there is no perfect symmetry between the top and the bottom halves of the (z,x) slice in Fig. b. The reason is finite absorption of light (that impinges onto the top plate) by the dye-doped material, and defocusing effect that increases with the depth of scanning.

Since mathematically n is well-defined in the vicinity of a FCD, one can calculate the expected FCPM image (taking into account also the finite resolution) and then compare it to the actual FCPM images. Below we perform such computer simulations for the case when the elliptic base of FCD is located at a cell substrate and compare the simulated image to experimental ones for the cell with tangentially degenerate boundary conditions. The resemblance is remarkable.

The FCPM textures above have been simulated for the following layer geometry of Focal Conic domain and director field in the planes of hyperbola and ellipse. The resemblance of the simulated and experimental textures proves that in a cell with tangentially degenerate boundary conditions the elliptic bases of FCDs tend to be located at the LC-substrate interface, as shown below:

We also describe FCPM imaging of structures in the lyotropic counterpart of SmA, the L_α lamellar phase formed by the mixture of cetylpyridinium chloride, hexanol, and brine. The dye Rhodamine 6G in this system is alighed parallel to the lamellae and thus perpendicular to n, in contrast to the cases of thermotropic SmA. Therefore, in the textures below the strong FCPM signal shows where the lamellae are parallel to polarizer P, and week – where the layers are at some angle to polarizer P.

Dynamics of line defects is of great practical importance in metallurgy and lately became a subject of intensive studies also in soft matter systems. Cholesteric liquid crystals with a micron-scale pitch allows one to apply the FCPM technique to the problem of defect dynamics and to literally see features such as kinks and core structure and processes such as glide and climb, with an unsurpassed clarity and resolution. The so-called Peierls-Nabarro friction associated with transformation of a dislocation core hinders glide of the defect across the layers; as a result, the glide is implemented via kinks. FCPM allowed us to reconstruct the 3D structures of different types of the dislocation kinks. We established that the kinks along the dislocations of Burgers vector b=p/2 (here p is the cholesteric pitch) change the level of dislocations by p/4 and p/2; these kinks are confined to the glide plane and are very long, (5-10)p.

In contrast, the kinks along the b=p dislocation are of a typical size p and form cusps in the direction perpendicular to the glide plane.

At the cusp, the nonsingular in the material director field “lambda”- disclinations of opposite signs interchange their ends.