figurec [firstpage,color=gray!90,angle=0,scale=0.28, xpos=0in,ypos=-5in]*
Corresponding author<EMAIL_ADDRESS>
${\dagger}$ These authors contributed equally to this work

# LCPOM: Precise Reconstruction of
Polarized Optical Microscopy Images of Liquid Crystals

Chuqiao Chen  Viviana Palacio-Betancur  Sepideh Norouzi  Pablo F. Zubieta Rico
Monirosadat Sadati  Stuart J. Rowan  Pritzker School of Molecular Engineering,
University of Chicago, Chicago, IL 60637  Department of Chemistry, University
of Chicago, Chicago, IL 60637  Juan J. de Pablo  Pritzker School of Molecular
Engineering, University of Chicago, Chicago, IL 60637

###### Abstract

When viewed with a cross-polarized optical microscope (POM), liquid crystals
display interference colors and complex patterns that depend on the material’s
microscopic orientation. That orientation can be manipulated by application of
external fields, which provides the basis for applications in optical display
and sensing technologies. The color patterns themselves have a high
information content. Traditionally, however, calculations of the optical
appearance of liquid crystals have been performed by assuming that a single-
wavelength light source is employed, and reported in a monochromatic scale. In
this work, the original Jones matrix method is extended to calculate the
colored images that arise when a liquid crystal is exposed to a multi-
wavelength source. By accounting for the material properties, the visible
light spectrum and the CIE color matching functions, we demonstrate that the
proposed approach produces colored POM images that are in quantitative
agreement with experimental data. Results are presented for a variety of
systems, including radial, bipolar, and cholesteric droplets, where results of
simulations are compared to experimental microscopy images. The effects of
droplet size, topological defect structure, and droplet orientation are
examined systematically. The technique introduced here generates images that
can be directly compared to experiments, thereby facilitating machine learning
efforts aimed at interpreting LC microscopy images, and paving the way for the
inverse design of materials capable of producing specific internal
microstructures in response to external stimuli.

_K_ eywords Polarized Optical Microscopy $\cdot$ Liquid Crystals $\cdot$
Simulations

##  1 Introduction

When confined between a pair of linear polarizers, liquid crystals (LCs) can
display a wide range of interference colors and complex patterns owing to the
material’s optical birefringence (i.e. the difference between refractive
indices parallel and perpendicular to the molecular axis). [1, 2] The
brightness and color hues are sensitive to the local molecular order, which
can be controlled through external stimuli, including electric fields,
magnetic fields, flows, chemical cues and temperature. [1, 2, 3, 4] On account
of their responsive nature and large optical birefringence, LCs are widely
used in optical devices, ranging from mature technologies such as liquid
crystal displays to state-of-the-art sensors that can detect toxins,
biomolecules, and microplastics. [5, 6, 7, 8, 9, 10]

Liquid crystals are generally characterized using polarized optical microscopy
(POM), which provides a direct measure of the material’s alignment and and is
able to identify any topological defects that might arise in a sample.[8, 11,
12, 13, 14, 15] In confined LC systems, large spatial distortions in the order
field can develop on account of the incompatibility between surface and bulk
orientations, leading to distinct POM images. The realignment of a confined LC
can be triggered by altering the balance between elastic and surface energies;
a minute change in the external environment can completely change the
material’s appearance under POM.[11, 16, 17] Sensing and display devices often
rely solely on the transition between configurations that exhibit different
topological defects (such as bipolar and radial), which are identifiable
through the brightness profile. The substantial color changes that accompany
such transitions are rarely exploited.[15, 18] Understanding how the POM color
patterns of LCs correspond to a particular molecular alignment is of interest
not only from a fundamental perspective, but also for development of next-
generation display and sensing devices.

One method to understand the color texture of POM images is to rely on the
Michel-Levy chart, which tabulates the interference color as a function of
thickness and birefringence. [19, 20, 21] That method, however, is limited to
a uniform orientation of the director field, and is incapable of predicting
the interference color in confined geometries where the alignment exhibits
large spatial variations. In addition, the POM images can change with light
sources and viewing angles, making it difficult to match the color patterns
with the underlying order field and hindering comparisons to experiments with
different setups.

Figure 1: Illustration of the method and plots of key physical properties. (A)
Schematic representation of the method for calculation of a colored POM image
from the order field. In this illustration, a radial droplet with $d=20\mu m$
is computed with $N_{\lambda}=14$ wavelengths. (B) LED light spectrum obtained
from the manufacturer. (C) Refractive indices of 5CB as a function of order
parameter $S$ and wavelength $\lambda$. (D) Transmission ratio as a function
of incident light angle. $Tr_{p}$, $Tr_{s}$ and $\overline{Tr}$ stand for
transmission ratios of p-polarized, s-polarized light and the weighted average
value. See Supplementary materials for details. (E) Color matching functions
from the CIE 1931 standard.

More generally, POM images are calculated without color information using the
Ondris-Crawford method, which produces the brightness profile corresponding to
the LC order field. [22, 23, 24] In this method, the sample is first
discretized into layers whose thickness is much smaller than the wavelength.
Subsequently, the propagation of light is modeled by multiplying the Jones
matrix of each layer, which computes the retardation according to the local LC
alignment. The method is versatile and easy to implement. It has been applied
to many geometries, including droplets and toroids, where it is possible to
reproduce the brightness profile of both nematic and cholesteric LCs. [17, 23,
24, 25, 26] The Ondris-Crawford method can be viewed as the standard approach
for comparisons between experimental POM images and model predictions with
numerical simulations.[11, 13, 17, 27, 28, 29, 30] A major limitation,
however, is that the original formulation assumes a single wavelength, and it
is difficult to compare a monochromatic brightness profile with the color
images that are typically obtained from a white light source with a
distribution of wavelengths in the range between 400-680nm. Note that the
effect of having a broadband light source has been discussed in several
experimental and simulation studies, but reports that include simulated
colored POM images have been limited and the agreement with experiments has
been limited . [26, 27, 28, 29, 31] An exception is provided by the work of
Yoshioka et al., who presented several colored POM images from calculations
that showed good agreement with experiments. [32] Unfortunately, few details
regarding the calculation of the color images were provided in that report. In
this work, we present a systematic methodological study of the computational
generation of POM images (and an accompanying software package), which is
validated through quantitative comparisons to experimental data for a variety
of systems.

##  2 Calculating optical textures

The colored POM images are computed by introducing a color matching function
that combines the information corresponding to multiple wavelengths to produce
RGB values equivalent to the colors perceived by the human eye. [33, 34, 35,
36] In addition, we take into consideration the emission spectrum of the light
source, the dependency of refractive indices on wavelength, and the reflection
at the droplet interface. The accuracy and the applicability of our method are
demonstrated by comparing simulated and experimental POM images of radial,
bipolar, and cholesteric droplets.

When colored POM images are captured in the laboratory, the sample is
typically illuminated with a white light that has a non-uniform spectrum
distribution. The light spectrum differs between laboratories and can alter
the color texture. [37] To produce an accurate color image, the LED spectrum
$I(\lambda)$ for experimental images produced in this work is obtained from
the manufacturer or measured by an optical spectrometer. (Fig. 1A and
Supplementary Fig. S1) In the calculations presented below, the light spectrum
(400 nm - 680 nm) is discretized into $N_{\nu}=20$ intervals and the intensity
profile for each wavelength is computed using the Ondris-Crawford method. [23,
24]

The order field configurations are obtained either from analytical or
numerical solutions, which are then interpolated onto a regular grid with the
desired resolution. Each of the single-wavelength intensity profiles depends
on the local LC alignment and the optical birefringence ($\Delta n$). (Fig.
1A) It is important to note that $\Delta n$ is not a single constant, but a
function of the wavelength ($\lambda$) and the order parameter $S$ (which is a
function of temperature and spatial gradient). Overall, both the ordinary and
extraordinary refractive indices ($n_{o}$ and $n_{e}$ respectively) decrease
with wavelength and saturate in the near-IR region. The quantitative
relationship can be described by a three-band model with constants fitted to
experimental measurements. [4, 38] (Fig. 1C, see Supplementary Materials for
the relevant equations).

In confined droplets, the director field is distorted and the local order
parameter $S$ is smaller near the topological defects, leading to a local drop
in $\Delta n$. Here we assume that the dependence of $n_{e}$ and $n_{o}$ on
$S$ caused by the spatial gradient is equivalent to the variation of $S$
caused by temperature. The following relationship is adapted from the Vuks
equation: [39]

$\Delta n=(n_{e}-n_{o})S/S_{r}$

As an example, for the liquid crystal 4-cyano-4’-phentylbiphenyl (5CB), the
reference state is taken to be $S_{r}=0.76$ at $T=25.1\mathrm{{}^{\circ}}$.
The refractive indices $n_{o}$ and $n_{e}$ are plotted as a function of
$\lambda$ for uniform alignment ($S=0.76$) and near a topological defect
($S=0.1$) (Fig. 1C). It is worth noting that at $S=0.76$, $\Delta n$ decreases
from 0.067 to 0.051 as $\lambda$ increases from $0.40\mu m$ to $0.70\mu m$,
which is significant enough to affect higher order interference colors; this
effect has generally been ignored in previous reports.

In addition to the interference taking place in the bulk of the liquid
crystal, we also consider the light transmission ratio at the water-LC
interface. For simplicity, diffraction and refraction at the interface are
ignored. In this study, the transmission ratio $\overline{Tr}$ is approximated
from the Fresnel equation using the refractive indices of water and 5CB. [38,
40] (Details are available in Supplementary materials) $\overline{Tr}$
decreases with increasing incident angle ($\theta_{i}$), leading to lower
brightness near the periphery of the droplet (Fig. 1F).

To combine these multiple intensity profiles at different wavelengths into an
RGB image, we consider how humans perceive color and how color images are
stored in modern electronic devices. Briefly, the human eye can sense
different wavelengths of light mainly with three types of cone cells in the
retina. [33] These signals are processed by an intricate neural network to
generate a perception of color in the brain. Modern-day electronics represent
color images by assigning tri-stimuli values such as RGB, XYZ, or HSV (Hue
Saturation Value) to each pixel, so that digital displays can allow the human
eye to perceive colors that are relatively independent of the device or the
lighting environment. The matching functions $x_{i}$ ($\lambda$) that
transform wavelength signals to XYZ values were originally determined by the
International Commission on Illumination (in 1931 - CIE 1931 color space), and
they are still widely employed today (Fig. 1E). [34, 35] In this work, the
intensity profiles $P(\mathbf{r},\lambda)$ at $N_{\lambda}$ wavelengths are
weighed by the light intensity $I(\lambda)$ and the matching functions to
obtain an XYZ color image which is then converted to the RGB color image by a
linear transformation. [36]

The image before transformation is calculated by:

$\displaystyle X_{i}(\mathbf{r})$ $\displaystyle=\int
P(\mathbf{r},\lambda)\overline{Tr}^{2}(\mathbf{r},\lambda)I(\lambda)x_{i}(\lambda)d\lambda$
$\displaystyle\approx\sum^{N_{\lambda}}_{j}P(\mathbf{r},\lambda_{j})\overline{Tr}^{2}(\mathbf{r},\lambda_{j})w_{ij}(\lambda_{j})$
(1)

where
$w_{ij}=\int_{\lambda_{j}}^{\lambda_{j+1}}I(\lambda)x_{i}(\lambda)d\lambda$ is
the weight to the $i$th color channel (X, Y, Z) independent of the director
field. $P(\mathbf{r},\lambda_{j})$ represents the single-wavelength intensity
profiles obtained from the Ondris-Crawford method.

In summary, the color image is obtained in four steps: 1) Generate the
director field through analytical or numerical solutions and interpolate it
onto a regular grid with the desired resolution. 2) Compute $N_{\lambda}$
intensity profiles for discretized wavelengths using the Ondris-Crawford
method and multiply by the transmission ratio according to the local
curvature. 3) Weigh by the LED spectrum and color matching functions to get
the XYZ channel images. 4) Transform the image from XYZ color space to RGB
color space and represent the result as a color image.

##  3 Methodology and implementation

Each image ($110\times 110$ pixels) takes less than 20 minutes to compute on a
single Intel Core i5 CPU processor. The sensitivity and speed of this method
can serve for generating a data set that facilitates inverse design or machine
learning. [41, 42]

##  4 Case studies

In this section, we present the diverse applications of the LCPOM Python
package, demonstrating its exceptional capabilities and providing benchmarks
for computational tools seeking to align with experimental observations. Not
only does our software serve as a convincing proof of concept, but it also
offers invaluable insights into crucial considerations such as system size
(sec. 4.1), treatment of topological defects (sec. 4.2), probing the
orientation of nematic morphologies (see sec. 4.3), and moreover, the faithful
reproduction of POM images in cholesteric systems (see sec. 4.4).

###  4.1 Effect of system size

To explore the effect of system size, we have generated POM images of radial
droplets with various diameters ($d$). The radial droplet provides a canonical
example of the interplay between bulk elasticity and surface orientation. In
the presence of a surfactant, the LC molecules orient perpendicular to the
surface of the droplet, resulting in a hedgehog defect at the center of the
droplet. This particular system has been reported to be stable across a wide
range of temperature, different materials, and across different length scales.
The analytical description of the radial director field is
$\mathbf{n}=\mathbf{x}/\left\lVert\mathbf{x}\right\rVert$. The defect is a
divergence of the director field, as pictured in Fig. 1A.

Figure 2: Simulation of the color POM images of radial droplets from the
analytical order field. (A) Simulated images of a radial droplet from
different sizes. (B) Comparison of the image of $d=5.5\pm 0.8\mu$m between
simulation and experiment. (C) Intensity profiles of the RGB color channels
and the radial intensity profile in B. (D) Comparison of the image of
$d=20.3\pm 1.8\pm\mu$m between simulation and experiment. (E) Intensity
profiles of the RGB color channels and the radial intensity profile in D.

The LCPOM results in Fig. 2A show the progression of colors that emerge as the
diameter of the LC droplets is increased from $d=2\mu$m to $20\mu$m. As the
diameter increases from $d=3\mu$m to $6\mu$m, the first-order interference
colors change from yellow to blue (Fig. S2). At $d=6.5\mu$m, a second color
ring emerges from the center of the droplet due to the spatial variation in
the optical path differences. The simulated color textures are highly
sensitive to system size, and a discrepancy in diameter as small as $0.5\mu m$
leads to notable differences in the optical appearance. This level of
sensitivity is not seen in BW intensity profiles, indicating that more
information about the LC order is captured by our proposed method. (Fig. S3)
Moreover, this result highlights the use of a computational tool related to
color to determine the size of an experimental system, or the possibility to
infer the order parameter from microscopy images. More simulated POM images of
this case are provided in Fig. S2.

To reproduce the color texture of experiments in simulations, the
birefringence must be tuned down by $5\%$. This difference is consistent with
the fact that frustrated alignment in curved geometries can lead to an
attenuation of the optical birefringence compared to the uniform bulk samples
in which the refraction indices are measured. Note that the droplet has a
diffuse boundary on the bright-field images, and the determination of size has
an average error of $14\%$ calculated from the FWHM (full width at half
maximum) of the boundary (see Fig. S5). This diffuse appearance is caused by
the diffraction at the interface between materials with mismatching refractive
indices. [40]

Overall, the color dependence on size in simulated images is in good agreement
with experimental results (Fig. 2B, D and S4). The simulated and experimental
images were decomposed into their respective RGB channel contributions (Fig.
2C and E). Given the symmetry of the radial structure, a quantitative
comparison between simulated and experimental images can be achieved by
performing a polar transformation and extracting the radial intensity
profiles. Each of the RGB channels contains contributions from the entire
light spectrum and does not have a simple analytical form. Importantly, the
peak positions of the RGB color channels agree with the ones observed in
experiments for $r/r_{0}<0.8$, which leads to a precise prediction of the
color ring locations, even for large droplets, which exhibit higher-order
interference. On the other hand, peaks close to the boundary ($r/r_{0}>0.8$)
are not observed in experiments, as the intensity decays faster towards the
edge of the droplet than predicted. This is attributed to the diffraction and
fluctuation at the interface which, as discussed above, are not considered in
our algorithm.

Figure 3: Effect of loop orientation in a radial droplet. (A) Simulated radial
droplet with a disclination loop (regions with $S<0.3$) at the center. The
diameter of the loop is approximately $1.9\mu$m. (B-C) Schematic
representation of the orientation of the defect loop with $\alpha=45\degree$
and $\beta=0\degree$ and the corresponding simulated POM image with
$d=17.5\mu$m. (E-F) Schematic representation of the orientation of the defect
loop with $\alpha=45\degree$ and $\beta=145\degree$ and the corresponding
simulated POM image with $d=17.5\mu$m. (D) and (G): Experimental images for
$d=17.5\pm 1.6\mu$m.

###  4.2 Effects of non-point defects

The nature of the defect core is an active field of study, with implications
for the assembly of colloids, active liquid crystals, and photonics. By
definition, the defect core is the divergence of the vector order field, and
can often be treated as a point charge. However, it has been shown
theoretically and in experiments that the topological defects usually do not
take the shape of a point, but appear as a region with diminished order
parameters. [16, 18, 43, 44, 45, 46]. Radial droplets of 5CB typically exhibit
loop disclinations whose diameter is sensitive to the anchoring strength, the
elastic constants (temperature), and the size of the droplet. [46, 47, 48] The
exact topology of the defect and the source of fluctuations have been long-
standing questions that have attracted considerable theoretical interest. [46,
47, 48, 49, 50] In experiments, the defect is usually small and sometimes
appears as a blurry dot due to limitations in the optical resolution. In
contrast to the hedgehog defect from sec. 4.1, a loop disclination is
surrounded by a continuous variation of the director field and has lower
rotational symmetry. This implies that the optical texture should reflect when
a rigid body rotation of the disclination loop occurs, as previously proposed
by de la Cote et al. [51]

To examine this hypothesis, we performed simulations of nematic droplets under
homeotropic anchoring conditions. Calculations of the order field following a
Ginzburg-Landau relaxation yield a scalar and vector order field. The
simulation details can be found in the Supplementary Information. The
orientations in 3D can be described by two angles, $\alpha$ and $\beta$,
because the order fields obey the $D_{\infty h}$ symmetry. Here, $\alpha$ is
the out-of-plane tilt angle between the symmetry axis and the $xy$-plane and
$\beta$ represents the in-plane rotation angle between the $0$-projection of
the symmetry axis and the $y$ axis (Fig. 3A).

As the droplet rotates, the reorientation of the loop creates subtle but clear
changes in the POM image. As expected, the image bears the highest symmetry at
$\alpha=90\degree$ and demonstrates more fuzzy central patterns compared to
the POM image of the analytical form (Fig. S6). A distortion in the optical
texture is observed when $\alpha$ deviates from $90\degree$ (Fig. 3C-F,
Supplementary Movie 1). Importantly, the simulations produce color patterns
that are very similar to those observed in experiments for particular
orientations of the droplet (Fig. 3B-G, Supplementary Movie 2). This agreement
suggests that under appropriate conditions the experiments can be directly
compared to simulations to infer the orientation of the defects, thereby
offering a new way of studying the dynamics and order fluctuations in confined
LC environments.

###  4.3 Elucidation of ambiguous micrographs through perspective sweep

Another ubiquitous configuration in LC microemulsions is the bipolar droplet.
It is characterized by two antipodal surface defects that emerge to satisfy a
parallel molecular orientation tangential to the droplet’s surface. The
transition from a bipolar to a radial configuration can be triggered by adding
a surfactant, which is the principle of operation for many LC-based sensing
devices. [11] Similar to the order field in sec. 4.2, we performed numerical
simulations to generate the vector and scalar order fields of a bipolar
droplet that were then used as input for LCPOM. An advantage of this
computational tool is the control over the viewpoint of an input morphology;
different orientations that yield uncommon micrographs can be probed by this
approach. In this case, simulated images were compared to experimental POM
images of droplets created by dispersing 5CB in a PVA/water solution;
additional experimental details are provided in the SI.

Figure 4: POM images of bipolar droplets with different configurations from
simulations and experiments. (A) Effects of varying the orientation of a
bipolar droplet with $d=20\mu$m. (B-D), from left to right: bright field
image, order field from numerical relaxation, cross-polarized image, simulated
POM image. (B) Droplet with a diameter of $22.9\mu$m. The poles are oriented
close to the $xy$-plane with a small out-of-plane tilt. The simulated image is
produced with $\alpha=30\degree$ and $\beta=55\degree$. (C) Droplet with a
diameter of $13.3\mu$m with the bipolar axis coinciding with the $y$-axis. The
simulated image is produced with $\alpha=50\degree$ and $\beta=0\degree$. (D)
Droplet with a diameter of $12.4\mu$m. The defects are located near the center
of the $xy$-plane aligned with the $z$-axis. The simulated image is produced
with $\alpha=80\degree$ and $\beta=-45\degree$. POM images in C and D are
computed by using the light spectrum in Fig. S1(B).

The bipolar droplets have two defects at opposite poles that obey the
$D_{\infty h}$ symmetry; all the orientations in 3D can be described by the
$\alpha$ and $\beta$ angles from sec. 4.2. Simulated POM images for different
orientations of the bipolar droplet are presented in Fig. 4A. In agreement
with literature reports, simulated images at $\alpha=0\degree$ consist of
concentric rings where the brightness and the outline change as the sample is
rotated in the $xy$-plane. Optical textures similar to these are commonly
reported in experiments. [11, 49, 52] Subtle deviations in the optical texture
are often related to a small tilt of the bipolar axis, i.e. the defects are
tilted out of plane. For instance, it was found that the best agreement
between simulations and experiments correspond to Fig. 4B, where
$\alpha=30\degree$ and $\beta=55\degree$. Note that the spacing between the
pink and green rings and the distortion features near the defects are all
reproduced accurately. Interestingly, screening other orientations yields
unfamiliar optical textures without the concentric rings. As such images have
rarely been associated with bipolar configurations, we performed additional
experiments to confirm the accuracy of LCPOM. Although textures with
concentric rings are observed more often, morphologies corresponding to
orientations with out-of-plane rotation $\alpha>45\degree$ (Fig. 4B-C) are
also confirmed experimentally. It is possible that these textures are reported
less frequently because they are difficult to classify. An alternative
explanation is that bipolar droplets can adopt preferred orientations due to
sedimentation or flow. [21] Nevertheless, the agreement between experiments
and simulations suggests that LCPOM reliably generates POM images of bipolar
droplets at arbitrary orientations, thereby providing a useful tool with which
to interpret POM images and classify droplets that exhibit ambiguous optical
morphologies.

###  4.4 LCPOM with a twist: cholesteric systems

Figure 5: Comparison between simulated and experimental POM images of
cholesteric droplets. The order fields of cholesteric droplets are obtained
from Ginzburg-Landau relaxations with homeotropic boundary conditions. The
number of turns is $N=3.8$ and the diameter of the droplet is set to the
experimental value of $d=17\mu m$. From left to right: isoclinical lines in
red, order field, simulated POM images, and experimental images. The
experimental images are adapted from Krakhalev et al. Scientific Reports
(2017). [27] Order field in (B) is rotated by Euler angles
$(\alpha,\beta)=(70\degree,30\degree)$ relative to (A).

Compared to nematics, cholesterics exhibit additional helical structures along
the orientation of the director. [17, 53] Depending on the droplet diameter,
the helical pitch ($p_{0}$), and the surface interactions, cholesteric
droplets can adopt complex internal morphologies and, as such, provide unique
opportunities for engineering electro-optical and sensing devices. [32, 54,
55, 56] In general, the POM images of cholesteric droplets are highly
sensitive to the droplet orientation, sometimes making it difficult to infer
the exact underlying structure. Here, we computed POM images of a cholesteric
droplet with a number of turns of $N=3.8$, and compared them to experimental
images with $N=3.4$ reported by Krakhalev et al. [27]. The order field was
obtained from a theoretically-informed Monte Carlo simulation with homeotropic
boundary conditions, following the procedure of Palacio-Betancur et al. [56]
The POM calculations are based on the material properties of the E7 mixture.
[27, 57] The images produced in this manner show excellent agreement with
experiments. [27, 28] Note that this match is only obtained when the
birefringence is scaled down by $40\%$ compared to that of bulk E7, implying
that the chiral dopant or the local twist may have led to a decrease in the
birefringence. Another possible reason is that the droplets dispersed in
polymer films may be oblate, and hence the optical path difference could be
overestimated when a spherical geometry is assumed. [27] To examine how the
optical texture changes with droplet orientation, POM images were calculated
at varying angles and compared with experimental results from Krahkhalev et
al. (Supplementary Movie 3). [27] We found that a small change in the
orientation can lead to distinctly different optical textures of the droplet,
yet a good match with experimental results can be obtained when the
orientation is set close to those reported in the original paper (Fig. 5D-F).
In general, controlling the orientation of cholesteric droplets is challenging
in experiments, making it difficult to investigate the optical texture
systematically. By generating high-fidelity color images of complex
structures, LCPOM can help develop a better understanding of structure in
confined cholesterics, which is needed for design of advanced optical devices.

##  5 Discussion and Conclusions

A straightforward method for simulation of color in POM images has been
presented for confined liquid crystals, including droplets. By incorporating
the emission spectrum of the light source, the dependence of refractive
indices on wavelength, the transmission ratio at the droplet interface as well
as the color matching functions, our simulation method is shown to be capable
of generating colored POM images that are in quantitative agreement with
experiments for radial, bipolar and cholesteric droplets. The method provides
a particularly useful tool to validate theoretical models and to interpret
experimental measurements. By comparing computed POM images of the order field
profile obtained from theory or simulations to experimental POM images, one
can gain insights into the governing physics and the balance of various
phenomenological parameters. We envision that the proposed computational tool
will help generate a realistic dataset to aid machine learning efforts aimed
at understanding the structure and dynamics of liquid crystals, and will help
engineer a new generation of LC-based sensing devices where color is used to
extract detailed information about molecular-level sensing events.

## Supplementary Information

Experimental details, and further validation images are provided in the
Supplementary Information.

## Code and Data Availability

The code for LCPOM along with its documentation will be released after beta
testing. Sign up in this form to be notified once it is available.

The files containing the order fields and to reproduce the optical textures in
this work are available upon request from the corresponding author.

## Acknowledgments

C.C. thanks Dr. Neil D. Dolinski for help on spectrum measurement. This work
was primarily supported by the University of Chicago Materials Research
Science and Engineering Center, which is funded by National Science Foundation
under award number DMR-2011854. V.P.B. thanks the Fulbright commission in
Colombia and COLCIENCIAS for support through the PhD student scholarship. M.S.
is supported by National Science Foundation, Division of Materials Research,
Condensed Matter Physics, under the NSF CAREER award 2146428. The authors also
acknowledge the Research Computing Center of the University of Chicago for
computational resources.

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##  Appendix A Materials and experimental methods

To prepare nematic LC droplets, 4-cyano-4’-pentylbiphenyl (5CB, Sigma Aldrich)
was rigorously vortexed in deionized water. Subsequently, 2wt% of Poly (vinyl
alcohol) (PVA, Mw=13k-23k, 87-89% hydrolyzed, Sigma-Aldrich) and sodium
dodecyl sulfate (SDS, Sigma-Aldrich) were added to the 5CB emulsion to
stabilize the droplets and induce planar and homeotropic alignment,
respectively. The droplets were then filled into rectangular borosilicate
capillaries under capillary action ($0.1\times 0.2$ mm ID, VitroCom).

A cross-polarized optical microscope (Zeiss Axioscope 5, Oberkochen, Germany)
equipped with an Axiocam 506 color camera and a 100X objective lens was
employed to image the droplets in the trans- mission mode. The image intensity
distribution for each channel of RGB was obtained using Zencore software.
During the imaging process, the temperature of the droplets was controlled
using a Linkam heating stage (model PE-120), which was connected to a
temperature controller (model T-96) to maintain the temperature at 25°C.

Figures 4c-d were obtained in a different lab. To reduce the mismatch of
refractive indices between 5CB and water, a water/glycerol mixture (50/50
vol%) was used as the continuous phase to disperse the 5CB droplets. 2wt% of
Poly (vinyl alcohol) was added to induce planar anchoring. The droplets were
filled into a circular cell (diameter = 8mm, depth = 0.2mm) to insulate flow.
The images were obtained by Leica DM2700P polarized optical microscope
equipped with MC170 HD 5 camera.

##  Appendix B Computational details

###  B.1 Refractive indices

The extraordinary and ordinary refractive indices are computed from a three-
band model that separates the contributions from $\sigma$ and $\pi$ electrons.
The empirical constants
$g_{0e},g_{1e},g_{2e},g_{0o},g_{1o},g_{2o},\lambda_{0},\lambda_{1},\lambda_{2}$
can be found in the original paper by Wu et al. [4, 57]

$\displaystyle n_{e}(\lambda)$
$\displaystyle=1+g_{0e}\frac{\lambda^{2}\lambda_{0}^{2}}{\lambda^{2}-\lambda_{0}^{2}}+g_{1e}\frac{\lambda^{2}\lambda_{1}^{2}}{\lambda^{2}-\lambda_{1}^{2}}++g_{2e}\frac{\lambda^{2}\lambda_{2}^{2}}{\lambda^{2}-\lambda_{2}^{2}}$
(2) $\displaystyle n_{o}(\lambda)$
$\displaystyle=1+g_{0o}\frac{\lambda^{2}\lambda_{0}^{2}}{\lambda^{2}-\lambda_{0}^{2}}+g_{1o}\frac{\lambda^{2}\lambda_{1}^{2}}{\lambda^{2}-\lambda_{1}^{2}}++g_{2o}\frac{\lambda^{2}\lambda_{2}^{2}}{\lambda^{2}-\lambda_{2}^{2}}$
(3)

###  B.2 Transmission ratio

The transmission ratio depends on the state of polarization, the refractive
indices ($n_{1}$ and $n_{2}$ for the continuous and dispersed phases
respectively) and the incident angle $\theta_{i}$ of the light. In this study,
the transmission ratio at the interface is approximated by the Fresnel
equation.[40]

$\displaystyle Tr_{s}$
$\displaystyle=1-\left|\frac{n_{1}\cos\theta_{i}-n_{2}\cos\theta_{t}}{n_{1}\cos\theta_{i}+n_{2}\cos\theta_{t}}\right|^{2}$
(4) $\displaystyle Tr_{p}$
$\displaystyle=1-\left|\frac{n_{1}\cos\theta_{t}-n_{2}\cos\theta_{i}}{n_{1}\cos\theta_{t}+n_{2}\cos\theta_{i}}\right|^{2}$
(5) $\displaystyle\theta_{t}$
$\displaystyle=\sin^{-1}(n_{1}\sin\theta_{i}/n_{2})$ (6)

Here, the refractive index of water is $n_{1}=1.33$. For simplicity, the
refractive index of liquid crystal $n_{2}$ is taken to be an average value
($n_{2}=(n_{e}+2n_{o})/3$). The Brewster angle is found to be around 59°,
within 1° error from experimental values.[29] The overall transmission
$\overline{Tr}$ is given by:

$\overline{Tr}(\theta_{i},\lambda)=Tr_{p}\cos^{2}\theta_{i}+Tr_{s}\sin^{2}\theta_{i}$
(7)

###  B.3 Continuum simulations of nematic liquid crystals

We adopt a mean-field approach to simulating liquid crystalline order,
described by the tensorial order parameter $\mathbf{Q}(\mathbf{x})$. The free
energy functional includes a short-range Landau polynomial expansion in
$\mathbf{Q}(\mathbf{x})$ to model the isotropic-nematic transition, a long-
range functional that accounts for elastic distortions, and a surface
contribution that imposes a preferred orientation at the boundaries.

$\displaystyle F(\mathbf{Q})=\int d^{3}{\mathbf{x}}$
$\displaystyle\left[\frac{A}{2}\left(1-\frac{U}{3}\right)\mathrm{tr}(\mathbf{Q}^{2})-\frac{AU}{3}\mathrm{tr}(\mathbf{Q}^{3})+\frac{AU}{4}\mathrm{tr}(\mathbf{Q}^{2})^{2}\right.$
$\displaystyle+\left.\frac{L_{1}}{2}\frac{\partial Q_{ij}}{\partial
x_{k}}\frac{\partial Q_{ij}}{\partial
x_{k}}+\frac{L_{5}}{2}\epsilon_{ikl}Q_{ij}\frac{\partial Q_{lj}}{\partial
x_{k}}\right]$ (8) $\displaystyle+\oint$ $\displaystyle
d^{2}{\mathbf{x}}\left[\frac{W_{\parallel}}{2}\left(\mathbf{\bar{Q}-\bar{Q}_{\parallel}}\right)^{2}+\frac{W_{\perp}}{2}\left(\mathbf{Q-Q_{\perp}}\right)^{2}\right].$

The model parameters include the Landau coefficient $A$ that sets the energy
density scale, the dimensionless parameter $U$ that determines the IN
transition (inverse temperature); $L_{1}$ and $L_{5}$ are the elastic
constants related to splay and chiral twist deformations respectively, and
$\epsilon_{ijk}$ is the Levi-Civita operator. The LC orientation at the
surface is imposed to be homeotropic (perpendicular to the surface) or planar
(tangential to the surface), and deviations from that orientation are
penalized with magnitudes $W_{\perp},W_{\parallel}$ respectively. The tensors
$\bar{\mathbf{Q}},\bar{\mathbf{Q}}_{\parallel},\mathbf{Q}_{\perp}$ refer to
projections of $\mathbf{Q}$ and are defined as follows:
$\mathbf{Q}_{\perp}=S(\bm{\nu\nu-\bm{\delta}/3})$,
$\bar{\mathbf{Q}}_{\parallel}=\mathbf{p\cdot\bar{Q}\cdot p}$,
$\bar{\mathbf{Q}}=\mathbf{Q}+S\bm{\delta}/3$, with
$\mathbf{p}=\bm{\delta}-\bm{\nu\nu}$ and $\bm{\nu}$ is the normal unit vector
on the surface. Further details of the free energy densities can be found in
our previous works. [58, 43, 45, 56, 59]

The free energy must satisfy the Euler-Lagrange equations, and the solution is
found by allowing the tensor order parameter to evolve towards equilibrium,
following a Ginzburg-Landau relaxation of $\mathbf{Q}$ of the form,

$\displaystyle\frac{\partial\mathbf{Q}}{\partial
t}=-\frac{1}{\gamma}\left[\frac{\delta
F}{\delta\mathbf{Q}}\right]^{\text{ST}}$ , (9) $\displaystyle\text{with B.C.s
}\left[\frac{\delta F}{\delta\nabla\mathbf{Q}}\cdot\bm{\nu}\right]^{ST}=0$
(10)

where $\gamma$ is the LC rotational diffusion coefficient, and
$\left[\mathbf{A}\right]^{ST}=(\mathbf{A}+\mathbf{A}^{T})/2-\text{tr}(\mathbf{A})\bm{\delta}/3$
is the symmetric traceless operator. The Volterra derivatives are defined by,

$\frac{\delta F}{\delta\mathbf{Q}}=\frac{\partial
F}{\partial\mathbf{Q}}-\frac{\partial}{\partial\mathbf{x}}\cdot\frac{\partial
F}{\partial\mathbf{\nabla Q}}.$ (11)

Numerical solutions to the Ginzburg-Landau relaxation are calculated with in-
house code. The different geometries are discretized into tetrahedral elements
with Cubit [60]. We adopt the Galerkin method of weighted residuals to
transform (9) and (10) into a system of linear algebraic equations. Open-
source libraries like libMesh [61] and SuperLU [62] allow us to implement the
numerical solution method. The scalar $S(\mathbf{x})$ and director
$\mathbf{n}(\mathbf{x})$ order fields are calculated from the eigenvalues and
eigenvectors of $\mathbf{Q}(\mathbf{x})$ once the systems have reached e, and
are used as input parameters in LCPOM.

##  Appendix C Supplementary figures and tables

Figures 3-5 of the main text use numerical solutions of $\mathbf{Q}$ to
calculate the color POM images. The finite element meshes are composed of
quadratic tetrahedral elements. The mesh has $3\times 10^{5}$ elements, and
additional mesh refinement was performed for the cholesteric droplet case, in
which the mesh has $5\times 10^{5}$ elements. The simulation parameters to
obtain the order fields are as follows,

Table S1: Simulation parameters for numerical solutions of the order field System |  $A$ (J/m3) | $U$ |  $L$ (pN) |  $W$ (J/m2) |  Droplet size ($\mu$m)
---|---|---|---|---|---
Radial | $1\times 10^{5}$ | 4 | 6 | $\perp,10^{-3}$ | 17.5
Bipolar | $1\times 10^{5}$ | 4 | 6 | $\parallel,10^{-3}$ | 20.0
Cholesteric | $1\times 10^{5}$ | 3.5 | 16 | $\perp,10^{-4}$ | 17.0
$\hookrightarrow p_{0}=17.0\mu$m |  |  |  |  |
Figure S1: Microscope light sources Figure S2: LCPOM simulations for radial
droplets of various sizes

Figure S3: Comparison of sensitivity between color and BW POM images Figure S4: Comparison between experimental images of radial droplets and LCPOM simulations for the same size system Figure S5: Size determination of radial droplets from bright field images Table S2: Size determination of radial droplets | Diameter $\mathrm{[\mu m]}$ | FWHM left $\mathrm{[\mu m]}$ | FWHM right $\mathrm{[\mu m]}$ | FWHM mean $\mathrm{[\mu m]}$ | % error
---|---|---|---|---|---
1 | 4.4 | 0.6 | 0.9 | 0.8 | 17%
2 | 5.5 | 0.7 | 0.9 | 0.8 | 14%
3 | 8.1 | 1.0 | 1.4 | 1.2 | 15%
4 | 8.6 | 1.0 | 1.7 | 1.3 | 15%
5 | 10.7 | 1.2 | 1.6 | 1.4 | 13%
6 | 16.6 | 1.6 | 1.7 | 1.6 | 10%
7 | 20.3 | 1.6 | 2.0 | 1.8 | 9%
Figure S6: LCPOM simulations of two radial droplets. Order fields obtained
from analytical solution (hedgehog defect) and numerical solution (ring
defect).