# Light-Induced Ring Pattern in a Dye-Doped Nematic Liquid Crystal

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## Abstract

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## 1. Introduction

## 2. Experimental Observations of the Ring Patterns

#### 2.1. Experimental Setup

#### 2.2. Synthesis and Preparation of Dye-Dopant and Liquid Crystal Mixture

#### 2.3. Light-Induced Ring Patterns

## 3. Mathematical Modeling for Photo-Isomerization in Dye-Doped Liquid Crystals

#### 3.1. Adiabatic Elimination and Effective Model

#### 3.2. Homogeneous Illumination and Bifurcation Diagram

#### 3.3. Light-Induced Ring Pattern

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Ring patterns induced by light in a dye-doped liquid crystal cell (DDLCC). (

**a**) Schematic representation of the experimental system. The blue and red bars, respectively, account for the molecules of the liquid crystal and azo-dye. The cell is illuminated by a blue and green beam. The snapshot accounts for the observed ring patterns. A transversal plane in the DDLCC is schematically represented. The areas under higher blue laser irradiation are more disordered, while the zones less illuminated preserve the nematic order. (

**b**) Isomers of the molecule methyl red methyl ester. (

**c**) Two snapshots showing the observed pattern (upper panel) and snapshot with the beam that induces photo-isomerization superimposed (bottom panel).

**Figure 2.**Experimental setup for the dye-doped nematic liquid crystal phototropic transition with a harmless external illumination. (

**a**) A dye-doped liquid crystal cell (DDLCC) is irradiated by a $445$ nm blue laser (excitation light beam) ${L}_{B}$ and illuminated by a $532$ nm green laser (probing light beam) ${L}_{G}$. Two pairs of lenses are placed in a Kepler telescope configuration ${K}_{B}$ and ${K}_{G}$ to expand the laser beam while preserving the collimation. ${\overrightarrow{P}}_{B}$ and ${\overrightarrow{P}}_{G}$ are the polarization of the laser sources. A long-wave pass dichroic mirror $DM$ is used to set both excitation and probing on the same optical line. After the DDLCC, another long-wave pass dichroic mirror is used to filter out the excitation beam. An analyzer in a crossed configuration with respect to $\overrightarrow{{P}_{G}}$. A set of imaging optics $IL$ consisting on a $\times 2$ Kepler telescope and a $\times 7$ zoom lens is used to enhance the image captured by the 1/2″ CCD camera. (

**b**) Absorption spectrum of methyl red methyl ester in dichloromethane 2.0 × 10${}^{-5}$ mol/L. The solid vertical lines account for the wavelength of the exciting and probing light, respectively. Vertical dashed lines account for the absorption maximum.

**Figure 3.**Experimental ring pattern emergence induced by a blue light (with a 445 nm wavelength) applied to a dye-doped liquid crystal cell, E7 NLC with azo-dye methyl red methyl ester at a concentration of 1 wt%. (

**a**) Temporal evolution of transmitted total intensity, measured in the green channel of the CCD camera ($\Delta {I}_{g}$) with respect to the transmitted light without the blue light beam (${I}_{g,0}$). The points were obtained experimentally, and the continuous curve was acquired using the expression $\Delta {I}_{g}\left(t\right)/{I}_{g,0}=A(1-{e}^{t/\tau})$, where $A=2.79$ and $\tau =55.71$ s. Painted areas I and II account for the growth and saturation regions, respectively. (

**b**) Temporal sequence of snapshots in the ring pattern formation process (${t}_{0}=0$ s, ${t}_{1}=1$ s, ${t}_{2}=5$ s, ${t}_{3}=84$ s, ${t}_{4}=360$ s, and ${t}_{5}=570$ s). (

**c**) Spatiotemporal diagram evolution of a diameter cut section.

**Figure 4.**Equilibria ring patterns were experimentally observed for different powers of the blue light beam. After a long period of evolution, snapshots were observed for different powers, as denoted in the lower part of each snapshot.

**Figure 5.**Schematic representation of the bifurcation diagram of the effective model in Equation (15) with constant coefficients. The order parameter S as a function of the bifurcation parameter $\tilde{A}$. ${S}_{0}$, ${S}_{+}$, and ${S}_{-}$ account for the isotropic liquid and nematic phases, respectively. The continuous and dashed lines account for stable and unstable states respectively. ${A}_{sn}$, ${A}_{M}$, and ${A}_{T}$ are the critical points that account for the emergence of the nematic phase; both phases are equally favored, with a transcritical bifurcation of the isotropic liquid phase. ${A}_{sp}$, ${A}_{sp}^{+}$, and ${A}_{sp}^{-}$ account for the spatial instabilities of the homogeneous phases. The painted area shows the region of coexistence between the periodic state and the homogeneous state. The decorated curve explains the amplitude of the patterns.

**Figure 6.**Numerical stationary ring pattern in a dye-doped nematic liquid crystal using the effective model in Equation (9) for $\tilde{A}=-0.5$, $\tilde{B}=0.3$, $\nu =1.05$, $b=0.1$, ${I}_{0}=1.45$, and $w=4$. (

**a**) Contour plot of the squared order parameter S. (

**b**) Profile of the cut of the order parameter S in the diameter of the ring pattern. (

**c**) Surface plot of the squared order parameter S.

**Figure 7.**Numerical light-induced ring pattern in a dye-doped nematic liquid crystal using the model in Equation (9). (

**a**) Temporal evolution of ring pattern using the effective model in Equation (9) for $\tilde{A}=-0.5$, $\tilde{B}=0.3$, $\nu =1.05$, $b=0.1$, ${I}_{0}=1.45$, and $w=4$. (

**b**) Equilibrium of ring patterns numerically obtained for a different forcing strength ${I}_{0}$, and the other parameters are $\tilde{A}=-0.5$, $\tilde{B}=0.3$, $\nu =1.05$, $b=0.1$, and $w=4$.

**Figure 8.**Manipulable ring patterns induced by illumination on a dye-doped liquid crystal sample. (

**a**) Schematic representation of the mechanism for applying the light beam to the dye-doped liquid crystal sample. (

**b**) Schematic representation of the path made by the light beam by adjusting the pitch and yaw of the dichroic mirror. (

**c**) Snapshots of ring patterns observed at different times (${t}_{1}<{t}_{2}<{t}_{3}<{t}_{4}$).

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**MDPI and ACS Style**

Clerc, M.G.; González-Cortés, G.; Hidalgo, P.I.; Letelier, L.A.; Morel, M.J.; Vergara, J.
Light-Induced Ring Pattern in a Dye-Doped Nematic Liquid Crystal. *Appl. Sci.* **2021**, *11*, 5285.
https://doi.org/10.3390/app11115285

**AMA Style**

Clerc MG, González-Cortés G, Hidalgo PI, Letelier LA, Morel MJ, Vergara J.
Light-Induced Ring Pattern in a Dye-Doped Nematic Liquid Crystal. *Applied Sciences*. 2021; 11(11):5285.
https://doi.org/10.3390/app11115285

**Chicago/Turabian Style**

Clerc, Marcel G., Gregorio González-Cortés, Paulina I. Hidalgo, Lucciano A. Letelier, Mauricio J. Morel, and Jorge Vergara.
2021. "Light-Induced Ring Pattern in a Dye-Doped Nematic Liquid Crystal" *Applied Sciences* 11, no. 11: 5285.
https://doi.org/10.3390/app11115285