# Multiple Twisted Chiral Nematic Structures in Cylindrical Confinement

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Results

_{1}= q

_{1}R, Q

_{2}= q

_{2}R, Q

_{RT}= q

_{RT}R, k

_{24}= K

_{24}/K, w = RW/K, and the dimensionless free energy is scaled in units of ${F}_{0}=\pi KH.$ Therefore $F\to F/{F}_{0}$, where H is the height of the cylinder. For numerical convenience, we suppose that H is either large in comparison with the period $p=2\pi /{q}_{1}$, or an integer number of p.

#### 2.1. Free Energies of Structures

_{0}and J

_{1}stand for the Bessel functions in the order of zero and one, respectively.

#### 2.2. Landau-Type Analysis

_{RT}of the RT structure if both Q and Q

_{RT}are small. We use Equations (13) and (7) and free energy minimisation yields:

#### 2.3. Numerical Analysis

#### 2.3.1. RZT Structure: Homeotropic Anchoring

_{24}and intrinsic chirality Q for relatively weak anchoring, which we set to w = 1. In Figure 2, we plot Q

_{1}and Q

_{2}equilibrium values (i.e., they determine local minima in F) varying Q between zero and one. For the case Q = 0 (achiral nematic) the RZT structures could be triggered only in the regime ${k}_{24}>{k}_{24}^{\left(e\right)}\equiv 2$. However, for chiral LCs, ${k}_{24}$ efficiently promotes the stability of RZT structures well below ${k}_{24}^{\left(e\right)}$. Furthermore, for ${k}_{24}$= 0, it holds that Q

_{2}= 0 and Q

_{1}= Q. This solution corresponds to the classic cholesteric structure; see Equation (2). Graphs in Figure 2 also reveal that the value of k

_{24}can be extracted experimentally.

#### 2.3.2. RZT Structure: Tangential Anchoring

_{1}(k

_{24}) and Q

_{2}(k

_{24}) on all studied anchoring conditions for two significantly different values of Q, viz., Q = 0.125 and Q = 1. The behaviour is roughly similar for homeotropic and azimuthal anchoring, whereas, for zenithal anchoring, qualitatively different features emerge. In particular, Q

_{1}could even change signs at a critical value of k

_{24,}which we denote by ${k}_{24}^{\left(c\right)}$. Similarly, for a given value of k

_{24}, this crossover could be achieved by varying Q, and we label the corresponding critical value as ${Q}_{c}.$

_{1}= 0 can be observed easily by polarised optical microscopy, this phenomenon may be exploited to measure the splay−bend elastic constant. This is illustrated in Figure 4, where we plot the Q

_{c}(k

_{24}) dependence for different anchoring strengths. Experimentally, one could vary Q by adding a chiral dopant to LC. The reversal of the sign of Q

_{1}exists in the interval 0 < ${k}_{24}$ < 1, well below ${k}_{24}^{\left(e\right)}$. In the strong anchoring limit W → ∞, the graph Q

_{c}(k

_{24}) approaches a straight line.

#### 2.3.3. Relative Stability of RZT and RT Structures

_{24}, Q and w. Due to a broad parameter space, we limit our analysis to a few cases relevant to our study. For example, Figure 4 reveals the parameters for which Q

_{1}= 0 (chirality reversal) is realised for the RZT configuration for zenithal anchoring. It is essential to compare its free energy with the competitive RT structure. Some representative examples are depicted in Figure 5 and Figure 6. In Figure 5, we plot the minimum energies of the competing structures on varying Q for k

_{24}= 0.5 and weak (w = 1) zenithal anchoring for the case exhibiting chirality reversal. In this case, the RZT structure with Q

_{1}< 0 is metastable with respect to RT. However, Figure 5 illustrates the existence of a regime for which the configuration with Q

_{1}< 0 is stable for k

_{24}=0.25. Thus, chirality reversal may be found experimentally in this case. The arrows in Figure 5 approximately indicate the energy of the RZT structure at the reversal of the sign of Q

_{1}, together with the calculated chirality parameters. For lower values of Q, it holds that Q

_{1}< 0, and vice versa. Although the energies for k

_{24}= 0.25 and 0.5 are not very different, the critical value of Q (Q

_{c}= Q, where Q

_{1}changes signs) differs significantly: Q

_{c}= 0.554 for k

_{24}= 0.5, whereas Q

_{c}= 1.097 for k

_{24}= 0.25.

_{1}= Q

_{2}= Q

_{RT}= 1. Simulation details are described in [29]. The polarisations of the polariser and analyser are mutually perpendicular. The angle between the polariser and x-axis (horizontal axis) is 0° or 45°. One sees that the textures are significantly different and that one could easily distinguish these structures by using polarising optical microscopy.

## 3. Conclusions

## 4. Methods

_{11}), twist (K

_{22}), bend (K

_{33}) and saddle-splay (K

_{24}) elastic constant, respectively. The wave vector $q$ reflects the inherent LC chirality.

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

LC: | liquid crystal |

BP: | blue phase |

RZT: | radially z-twisted |

RT: | radially twisted |

NLC: | nematic liquid crystal |

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**Figure 1.**Twisted nematic structures. (

**a**) The radially z-twist deformation. Q

_{1}= 1.0, Q

_{2}= 1.0. The twist is realised both along the ${\overrightarrow{e}}_{\phi}$ and ${\overrightarrow{e}}_{z}$ directions. (

**b**) The radially twisted structure. Here, the twist is realised along ${\overrightarrow{e}}_{r}$. Q

_{RT}= 1.1.

**Figure 2.**Dependence of equilibrium values of Q

_{1}(

**a**) and Q

_{2}(

**b**) on k

_{24}for five different values of the intrinsic chirality Q (denoted by numbers in graphs). Homeotropic anchoring, w = 1.

**Figure 3.**Dependence of Q

_{1}(solid lines) and Q

_{2}(dashed lines) on k

_{24}for Q = 0.125 (

**a**) and Q = 1 (

**b**) and different types of anchoring, labelled by homeotropic (“h”), azimuthal (“ϕ”) and zenithal (“z”). w = 1.

**Figure 4.**The dependence of the critical intrinsic chirality Q

_{c}(where Q

_{1}= 0) on ${k}_{24}$ in the case of zenithal anchoring for different values of anchoring strength. The results were calculated in points labelled with symbols and lines to serve as guides for the eye. From left to right: w = 0.2 (circles), 0.5 (diamonds), 1 (open circles), 2 (stars) and 5 (triangles).

**Figure 5.**Dependence of the minimum energies (thick lines) and chirality parameters (thin lines) of the radially z-twisted (RZT) structure (solid lines) and radially twisted (RT) structure (dashed lines) on the intrinsic chirality Q. k

_{24}= 0.5, zenithal anchoring with w = 1.

**Figure 6.**Dependence of the minimum energies of the RZT and RT structures on the intrinsic chirality Q. Solid lines: k

_{24}= 0.5. Dashed lines: k

_{24}= 0.25. Zenithal anchoring with w = 1. Arrows indicate the sign reversal of Q

_{1}for both values of k

_{24}. For k

_{24}= 0.25, the chirality Q

_{1}reverses its sign in the regime where F

_{RZT}< F

_{RT}.

**Figure 7.**Calculated optical patterns for the RZT structure with Q

_{1}= Q

_{2}= 1. The transmitted polarisation of the polariser is in the x direction (left figure) and at an angle of 45° with respect to the x direction (right figure). Optical data: R = 1 μm, laser light wavelength λ = 445 nm, refraction indices: n

_{o}= 1.544, n

_{e}= 1.821, corresponding to nematic liquid crystal (NLC) E7.

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**MDPI and ACS Style**

Ambrožič, M.; Gudimalla, A.; Rosenblatt, C.; Kralj, S.
Multiple Twisted Chiral Nematic Structures in Cylindrical Confinement. *Crystals* **2020**, *10*, 576.
https://doi.org/10.3390/cryst10070576

**AMA Style**

Ambrožič M, Gudimalla A, Rosenblatt C, Kralj S.
Multiple Twisted Chiral Nematic Structures in Cylindrical Confinement. *Crystals*. 2020; 10(7):576.
https://doi.org/10.3390/cryst10070576

**Chicago/Turabian Style**

Ambrožič, Milan, Apparao Gudimalla, Charles Rosenblatt, and Samo Kralj.
2020. "Multiple Twisted Chiral Nematic Structures in Cylindrical Confinement" *Crystals* 10, no. 7: 576.
https://doi.org/10.3390/cryst10070576