Supplementary Information for

“Standing wave-mediated molecular reorientation and spontaneous

formation of tunable concentric defect arrays in liquid crystal cell”

L. K. Migara, and Jang-Kun Song*

School of Electronic & Electrical Engineering, Sungkyunkwan University, Jangan-Gu, Suwon,

Gyeonggi-do 440-746, Korea

Supplementary Figure S1│Determination of the cell director profile. a, The brightness contrast

between two oppositely oriented domains depends on the viewing direction. When the viewing direction

is in the molecular tilting plane, the two domains appear dark and bright with high contrast (top

illustration). On the other hand, when the viewing direction is normal to the molecular tilting plane, the

two domains appear as having a low contrast (bottom illustration). The brightness variation is caused by

the difference between the domains’ birefringence. By determining the viewing direction at which the

contrast is maximized, the molecular tilting direction can be accurately determined for each domain. b, To

determine the molecular tilting direction in the surrounding area of a defect (□αβγδ), the brightness

profiles were analyzed from the four different viewing directions (A, B, C, and D). On top viewing,

□αβγδ has four bright wings labeled by T(top), B(bottom), L(left), and R(right), and the oppositely

oriented domains are not discernible. c, The oppositely oriented domains are clearly discernible when

viewed from the four oblique viewing directions. When viewed from the A and C directions, the two

domains in the T and B wings have high contrast, but those in the L and R wings have low contrast.

Hence, the molecules in the T and B wings are tilted horizontally in the image (see the red arrows in the

top images). When viewed from the B and D directions, the two domains in the L and R wings have high

contrast, but those in the T and B wings have low contrast. Hence, the molecules in the L and R wings are

tilted vertically in the image (see the red arrows in the bottom images). In this way, one can determine the

director tilting direction in the whole area, and the results are shown in Fig. 2c in the main text.

Supplementary Figure S2│Evidence for the absence of electrohydrodynamic convection (EHC) or

ionic dynamics in the cells used. Low frequency ac voltages were applied to the LC cell, in order to

examine the possibility of the electro-hydrodynamic convection in the cell. a, When 10 V and 100Hz

square wave electric voltage was applied to the cell, random texture with umbilical defects, which is

typical in a vertical LC cell without ionic convective flow, was observed. When decreasing the frequency

down to 0.1 Hz, no change in the umbilical texture was observed, and no instability appeared. Application

of low frequency ac voltage in a LC slab containing ionic compounds usually generates the ionic drift and

the electrohydrodynamic convective (EHC) instability. However, such instability was not observed in the

cell. b, The EHC instability was not observed even at the application of 40 V. Usually, higher voltage

enhances the EHC, when LCs contain sufficient ionic compounds in the cell. We used a commercial

nematic LC mixture (MLC7026) with extremely low ionic concentration, and as a result, ionic convective

flow was not observed. These observations clearly confirm that the electrohydrodynamic convection is

not the origin of the pattern formation in our experiments.

Supplementary Figure S3│Electrohydrodynamic convection (EHC) patterns in an ion-added cell. a,

We prepared LC with high concentration C6 (2 wt%), and exposed UV light for 1 hour to increase the ion

concentration in it. Dye can be an ionic source, and UV irradiation is likely to increase the ion

concentration by decomposing chemicals. When a cell with the ion-added LC was exposed to 40 V 19 Hz

square wave field, net-shaped EHC pattern was observed. b, When the signal was changed to 40 V 19 Hz

pulse train signal, the cell exhibited the periodic standing wave pattern in macroscopic scales. However,

the EHC pattern was significantly reduced, but stripe-shaped EHC pattern was observed in a certain area.

Thus, the ionic motion-induced-EHC pattern can coexist with the periodic standing wave pattern.

Supplementary Figure S4│Simulation of the director profile under pretilt angle and vertical fields.

a, The director profile in the cell was simulated using the Techwiz software (Sanayi Systems, Korea). One

of the main functions in the software is to find the optimum director profile in 2D and 3D space, by

minimizing the total free energy of liquid crystals including the electric free energy and elastic

deformation energy. In this simulation, we aim to find the director profile in the cell with horizontal flow

and vertical electric fields. The software does not include the flow effect. However, we assume that a

weak LC flow may induce a weak bend deformation in the cell having initially vertical alignment, and the

shear effect makes the molecules near the surfaces tilt along the flow direction. The simulation conditions

are dielectric anisotropy (Δε) = -4.2; primary elastic constants, K11 = 16.7, K22 = 7.3, K33 = 18.1; rotational

viscosity (γ1) = 136 mPa sec, the helical power of chiral dopant = 11 μm-1, cell-gap = 3.5 μm, and the

pretilt angle due to the LC flow = 1 ° from vertical axis on both surfaces.

b, The polar and azimuthal angles of director fields before application of electric fields. The polar angle is

set to be 1 ° to consider the flow induced pretilt angle. c, When the electric field is subsequently applied,

the director profile change to that for which the minimal elastic deformation energy is obtained in either

form of the right and left twist deformations. The green and yellow curves correspond to the right and left

twist deformations. In fact, when applying electric fields on the cell with the weakly bend deformation

(shown in b), either of twist deformation with splay deformation (shown in c) or purely splay deformation

(not shown here) can appear depending on the twist and splay elastic constants of LC material. For usual

liquid crystals with lower twist elastic constant than splay elastic constant, the twist deformation has

lower elastic deformation energy, and so, the twist deformation is obtained. d, When a chiral dopant

(R811, 0.2 wt%) was added to the cell, the oppositely oriented domains disappeared. The addition of the

chiral dopant broke the symmetry between the left and right twist deformations, and only the right twist

deformation survived, as suggested by simulations. Compared to the curves of φ in c, that in d has a steep

slope owing to the presence of the chiral dopant.

Supplementary Figure S5│No changes induced after immersing the cell into silicon oil. The periodic

patterns observed: a, before immersing the cell into the glycerin bath, b, after half immersion, and c, after

full immersion. The cell did not exhibit any changes during the immersion process. This confirms that the

periodic pattern is not caused by the substrate vibration. The middle image (b) shows a slight difference

(e.g., higher contrast between two oppositely oriented domains), which is owing to the tilting of the cell

(oblique observation results in the difference).

Meanwhile, the reflected laser beam from the cell surface was projected onto a screen and it was

interfered with another split beam using Michelson interferometer, in order to detect the vibration of the

substrate. However, we could not detect any vibration of surface within the experimental resolution of

Michelson interferometry measurement.

These observations suggest that the standing wave within the cell does not arise from the substrate

vibration.

Supplementary Figure S6│Circular textures observed in a square cell with circular electrodes. We

prepared a cell using two square glass substrates with a circular electrode (~ 5 cm diameter). When a

pulse train signal was applied to the cell, the concentric ring textures were observed, although the

structures were not as periodic as those observed in the cell with square electrodes. This observation

clearly shows that the texture pattern is related to the boundary condition.

Supplementary Figure S7│Determination of the handedness of concentric defects in a cell

containing chiral dopant LCs. We prepared two cells containing R811 and S811 doped LCs,

respectively. The director profile in each cell was analyzed the optical transmittance observation and the

optical compensator method from oblique directions. The red and green dots represent the clockwise and

the anti-clockwise handedness, respectively. The notation follows the same rule explained in Fig. 6 in the

main text. a-b, According to the results of the optical transmittance observation, the handedness of spiral

director orientation was completely opposite in the R811 and S811 LC cells. One can notice that the

defect near the corner in the R811 cell has the clockwise handedness, but that in the S811 cell has the

anti-clockwise handedness. c, The same result was obtained in the optical compensator method. These

observations accord well with the simulation results in Fig. S4.

Supplementary Figure S8│Comparison of standing waves in the LC cell and in other systems, and

boundary conditions. a, The pressure map in the standing waves in the LC cell; it has diagonal nodal

lines (blue arrows), along which the pressure is zero. The p curves represent the pressure profile along the

cell boundary line (line A) and along the diagonal direction (line B). The ∇p curves represent the flow

component along the lines A and B. The p+ and p– stages (top and bottom images) have the opposite

pressures along the boundary. It complies zero normal flux boundary condition with low parallel flux, that

is, it has weak parallel flux only with no normal flux (will be explained in the next paragraph). b, The

pressure map in the standing waves in a rectangular cavity; it has nodal lines parallel to the boundary

lines. The p curve represents the pressure along the boundary. The p+ and p– stages have the same

amplitude. It has zero normal flux boundary condition with strong parallel flux, that is, it has stronger

parallel flux than that in a. c, The vertical displacement map in the standing waves in a rectangular 2D

membrane; it has nodal lines parallel to the boundary and it complies zero displacement boundary

condition, that is, the boundary accords with the node line. The z curve represents the displacement along

the boundary. The p+ and p– stages have the same amplitude.

There are several notable differences between the standing wave in LC cells (a) and those in other

systems (b and c). First, the nodal lines in (a) are rotated by 45° compared with others (b and c). It is due

to different boundary conditions. The wave in 2D membrane (c) follows the zero displacement boundary

condition, that is, the membrane along the boundary line does not vibrate. The wave in the LC cell (a) is

quite similar to that in an air cavity (b) with the exception of the 45° rotation. The difference can be

explained by comparing the two lines A and B in (a); the line A is the boundary line in the wave (a), and

the line B is the diagonal line that is the same as the boundary line in (b). The flux components normal to

the lines are zero in both of the lines A and B (see Fig. 4c in the main text). However, the parallel flux

components are not the same; the line B has larger pressure amplitude than that along line A, and the

corresponding flux intensity (∇p) is also larger along line B than along line A. Hence, the flux along the

boundary in the LC cells (a) is lower than the conventional one in (b); the difference is about 40%. Thus,

a high flux viscosity along the boundary line may be related to the standing wave geometry in the LC

cells.

Second, differently from that in (b), the standing wave in (a) have even modes only; the green

rectangular map in (a) is not allowed, because such a standing wave makes the center of mass vibrate with

time.

Third, the + and – phases are asymmetric in the standing wave in LC cells, owing to different driving

force during the + and – phases.

Supplementary Video S1│The pattern in the cells with and without chiral dopant and its freezing.

The left and right cells contained pure LC and LC added by 0.2 wt% R811, respectively. Initially, a pulse

train signal of 40 V and 26 Hz was applied and the periodic pattern was blinking in accordance with the

pulse signals in both the cells. At ~2 s in the video, the signal was changed into 10V square wave; then

the periodic pattern disappeared in the left cell with pure LC and it was frozen in the right cell with chiral

LC. Thus, we obtained stable periodic patterns with circular director profiles.