Physicochemical properties of interfaces impose a surface anchoring, and the geometry and topology of confining interfaces determine the director configuration and defects of the LC. In this work, we report the director configuration around a cylindrical object embedded in nematic Sunset Yellow (SSY), a representative lyotropic chromonic LC with the large K24 modulus. The nematic SSY is sandwiched between two flat substrates and a cylinder is placed in the SSY.
The flat boundaries induce a homogeneous directing field orthogonal to the cylinder's axis, while the cylinder aligns neighboring directors parallel to its axis, based on the theory related to K24. These boundary conditions lead to the twisting deformation near the cylinder, and we investigate the deformation experimentally and theoretically, and evaluate K24 of SSY.
Introduction
To address the energy of directors with weak or low void anchorage conditions, the saddle spreading term needs to be intensively investigated. As a successful study (Jeong et al., 2015), our group has proven the importance of the saddle spreading term in director field energy with degenerate anchoring conditions, where boundary directors have no preferred orientation once they touch the boundary. In that study, the resulting double-twist director configuration induced in cylinder, which cannot occur alone with respect to bulk free energies, was taken as strong evidence for the significant effect of the saddle spreading term.
Although this study experimentally confirmed the effect of the saddle term for a concave cylindrical boundary, its result does not complete the proof of the theoretical expression of the saddle term. Unfortunately, it has been difficult to identify and measure the saddle crack free energy effect without an advanced level of director field analysis. Based on our experiments and analysis, this thesis shows that the theoretical effect of the saddle-splay term for the boundary of a convex cylinder is not easily confirmed.
The effect and consequence of hall-game term minimization. a) Surface directors tend to align parallel to the cylinder's axis, to be the configuration of (b), when no other factors affect them. And this board game term's energetic preference for the uniaxial configuration of surface directors is referred to as board game effect.
Background
- Oseen-Frank free energy
- Phenomenological anchoring free energy
- Defects
- Polarimetry for homogeneous field
- Jones matrix calculation
- Lyotropic chromonic liquid crystals
Figures 4(a) and (b) show the expected configuration of surface directors with oblique corners, also called oblique configurations. In particular, when the easy axis is in the plane of a surface, LC molecules are called to have a planar anchoring condition on the surface, and their is written as. 3), are identical, implying that the saddle spreading effect can be interpreted as a planar anchorage with the easy axis parallel to the cylinder axis, and with an anchoring coefficient of.
In addition, when LC molecules prefer to be tangential to a surface, but with no preference for the azimuthal direction, the LC molecules are said to have degenerate planar anchoring on the surface and they are constant, because the easy axis in the case cannot be determined. If the conducting field can avoid its large strain energy at the cost of increasing the free energy of defects or differences, the conducting field forms defects. When a spherical particle is in the homogeneous field of LC with degenerate planar anchoring, a pair of boojum defects, Fig.7 (a), is formed along the direction of the homogeneous field.
When linearly polarized light propagates through a homogeneous guiding field, its polarization can be transformed according to the original direction of polarization and the thickness of the guiding field. However, when the polarization of the light at the entrance to the homogeneous field ( ) is neither parallel nor perpendicular to the routers, the initial linear polarization decomposes into two perpendicular components and they propagate at different speeds due to LC birefringence (Δ. Schematic of the router field that includes defects a) Boojums defects (red) appear on the surface of the spherical particle and along the homogeneous field.
Furthermore, the polarization of light can change depending on the angle of incidence, causing a more complex interaction with directing fields other than the homogeneous field. The change in the polarization of light can be calculated using the Jones matrix when the directing field light propagates through it is known. From that point of view, the director field of known configuration occupying a space can be regarded as a collection of director field configurations occupying only each voxel.
Because each voxel is infinitesimally small, the director field configuration in each voxel can be approximated as a homogeneous field oriented in a known specific direction. Any known directing field can be regarded as a series of infinitesimal homogeneous fields in a known direction. Similarly, director can also be defined as the average orientation of aggregates in a local space.
Ingredients for cells
Sample preparation
Cell production
Coordinate designation for cells
Optical microscopy
Polarimetry setup
Director field and surface directors
Twist deformation in orthogonal cells
The degree of freedom gives the domain an equal chance of rotational distortion from left- or right-handedness. Accordingly, the two turning domains with different hands appeared quite similar, and an interesting defect nestles between the adjacent domains with different handedness, as in Figure 14. These defects also appear with different properties similar to the result reported in (Čopar et al. , 2016).
Now that we have understood the conformation that the director field has in an orthogonal cell, the rest of this thesis deals with questions about its origin and the role of the saddle game effect. The saddle-splay effect is not yet able to identify, due to unquantified bulk free deformation energy. The problem of identifying saddle-splay effect in parallel cells extends to the study of director field in orthogonal cells.
Without the saddle splash term, degenerate planar anchoring and massless energy minimization force the lateral surface (cylinder side) drivers to have significant x components, i.e. obeying only the massless energy minimization, this x -direction side surface director can cause the top surface directors to have ignored x -components associated with unilateral boojum defects as Fig . If the saddle opening effect is also taken into account, only the amount of oblique orientation directors in the bulk and surfaces will be changed.
To this end, to examine the contribution of the saddle-splay effect to surface routers, we first studied the torsional deformation and orientation of the upper surface routers following a previous investigation ( McGinn et al., 2013 ).
Quantitative analysis of director field
This, one simplification and two assumptions, reduces the Euler-Lagrange equation of the total free energy functional to a 2-dimensional Laplace equation (y, K/K z) with a reduced variable z. To solve Laplace's equation analytically, we further simplified the geometry of this equation so that the half cylinder is at the bottom instead of the full cylinder and the cell gap expands to infinity. Explicit relations between the variables as well as the complete y, with solution representation are available in the appendix.
Hall game effect cannot be assumed without a constraint in the boundary condition of the Laplace equation due to the conflict with an early-made requirement for the directors. Keeping that simplification to keep the Laplace equation in 2-dimension, we have explored only a limited possibility in which surface directors can only have oblique angles in approximately flat region of the fiber, referred to as band, but uniaxial in the other area of the fiber's surface, as depicted in Fig. With this new heterogeneous boundary condition, we proceeded to solve the Laplace equation, however numerically due to the analytically challenging boundary conditions.
Unlike the analytical approach, the numerical approach with the new heterogeneous boundary condition assumed the real geometry of the boundary without any simplification. Due to the advantage of numerical analysis, the boundary of the Laplace equation mimics the real system geometry. The surface directors in the band of the heterogeneous boundary are under the boundary condition of the similar form.
As the cell gap increases with other parameters fixed, the middle part of the twisted profile curves as shown in the figure. Analytical solution profile, drawn together. The remaining parameters did not affect the twist profile as much as the gap between the cells, as shown in the figure. If the resulting value is approximately zero, it can be concluded that the saddle stretching effect does not exist outside the fiber or that the simplifications were not appropriate.
Although we confirmed the equilibrium director field in both the parallel cell and the orthogonal cell, we could not qualitatively identify the saddle effect due to the interaction between the bulk elastic deformations. The experiment can be performed with fibers of different radii, which are confirmed to have a degenerate plane anchoring surface, or with a half-cylindrical glass rod of different radii from the manufacturer Hilgenberg, which requires confirmation of the degenerate state. With these boundary changes, the radius dependence of the saddle-crack effect can also be verified.
But even with those horny residual chemicals, the fiber optic was used for parallel cell experiments and the uniaxial configuration was adopted, as it reflected little influence from the horny chemicals. The sawtooth-like fluctuations at the nanometer scale of both profiles originate from the resolution limit of the AFM, and not from the actual fluctuation of the fiber surface.
