# Director Fluctuations in Two-Dimensional Liquid Crystal Disclinations

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Results

## 3. Theoretical Preliminaries

#### 3.1. Splay and Bend Disclinations

#### 3.2. Eigenvalue Problem

#### 3.3. Eigenfunctions and Expansion of the Frank Energy

## 4. Correlation Functions

## 5. Mixed Spiral States

## 6. Two Calculational Procedures

- 1
- 2
- 3
- The change of variables from r to x changed the free energy from a $2\mathrm{D}$ form to a form equivalent to a $1\mathrm{D}$ one.

## 7. Review and Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Sketches of in-plane $\mathbf{c}$-director fields in annular traps with concentric outer and inner bounding circles with (

**a**) perpendicular and (

**b**) tangential boundary conditions.

**Figure 2.**Computer generated images of (

**a**) splay or bend defects and (

**b**) spiral defects under crossed polarizers. The twisting or spiral structure is produced by a combination of splay and bend. The spiral structure was calculated with $\beta =1.5$ and $\mu =800$ corresponding to the orange curve in Figure 9.

**Figure 3.**Phase diagrams for an annulus with (

**a**) tangential and (

**b**) perpendicular BCs. The dividing line between the two defect types is ${\mu}_{c}^{\mathrm{S}}\left(\beta \right)$ in (

**a**) and ${\mu}_{c}^{\mathrm{B}}\left(\beta \right)$ in (

**b**). In (

**a**), the bend (spiral) defect is stable, and in (

**b**), the splay (spiral) defect is stable in the shaded (unshaded) areas.

**Figure 4.**Wavenumbers ${\kappa}_{mn}$ as a function of $\mu $ for our sample configurations with imaginary ${\nu}_{0}$ and ${\mu}_{c}\approx 535$. The dots are obtained by numerical calculation of the zeros ${\kappa}_{01}$ (black), ${\kappa}_{02}$ (gray) and ${\kappa}_{11}$ (blue) of the full function ${Z}_{{\nu}_{m}}$. The black line stems from our approximate analytical solution for ${\kappa}_{01}$ as given in Equation (20). Note the excellent agreement between the black dots and the black line for $\mu $ close to ${\mu}_{c}$. Also note the steep rise of the ${\kappa}_{01}$ near ${\mu}_{c}$.

**Figure 5.**The radial eigenfunctions ${u}_{mn}\left(r\right)$, where m and n are, respectively, the standard integer azimuthal and radial quantum numbers for polar coordinates, for a system with the imaginary value of ${\nu}_{0}$ used in Figure 6c. All plots are for $n=1$, the lowest permitted value for n. On the ordinate, we included a factor $\sqrt{V}$, where $V=\pi {R}_{2}^{2}$ is the total area enclosed by the outer circle of the trap so that the plotted eigenfunctions are dimensionless.

**Figure 6.**The functions ${Z}_{0}\equiv {Z}_{{\nu}_{0}}$ as a function of $\kappa $ for $\mu <{\mu}_{c}$, $\mu ={\mu}_{c}$ and $\mu >{\mu}_{c}$: (

**a**) ${Z}_{0}$ for $\beta =0.8<1$, (

**b**) ${Z}_{1}$ for $\beta =1.25>1$; (

**c**) ${Z}_{0}$ for $\beta =1.25$, and (

**d**) ${Z}_{0}\left(\kappa \right)$ as a function of ${\kappa}^{\prime}=\kappa /i$ for $\mu >{\mu}_{c}$. The rapid oscillations are a consequence of the logarithmic scale. Note that there are no zeros at small $\kappa $ for (

**a**,

**b**) indicating large values of $\epsilon $ both for cases with $\beta <1$ and for modes with $m=1$ (and greater). In addition, the positions of the zeros in (

**a**,

**b**) are fairly insensitive to the value of $\mu $ and to the critical point. In (

**c**), there is a zero in the top curve ($\mu <{\mu}_{c}$) that vanishes at ${\mu}_{c}$ and ceases to exist for $\mu >{\mu}_{c}$. In (

**d**), there is a zero in bottom curve ($\mu >{\mu}_{c}$) that vanishes at ${\mu}_{c}$ and then disappears when $\mu <{\mu}_{c}$. ${Z}_{0}$ has an infinity of larger-value zeros in (

**c**), whereas in (

**d**), it has only one zero when $\mu \ge {\mu}_{c}$.

**Figure 7.**The correlation function $C(r,\varphi )=C(r,r,\varphi ,0)$ defined in Equation (34). The left shows the correlation as a function of dimensionless coordinates in the plane. The green loop indicates an assumed probing radius of $r={R}_{\mathrm{Probe}}$. The right shows a zoom-in on the correlation function along the green loop.

**Figure 8.**Plots of $\Delta F(\beta ,\mu )$ calculated from Equation (37) as a function of $[\mu -{\mu}_{c}\left(\beta \right)]/{\mu}_{c}\left(\beta \right)$ for different values of $\beta $. Note, ${\mu}_{c}\left(x\right)$ grows exponentially as $\beta $ approaches 1, and the regions depicted in this figure are relative to a large value of $\mu $ for small $\beta -1$.

**Figure 9.**Plot of $f(r/{R}_{1})$ for mixed states with $\beta =1.25$ and for different values for $\mu $: $\mu /{\mu}_{c}=1.1$ (red), $\mu /{\mu}_{c}=1.25$ (blue), and $\mu /{\mu}_{c}=1.5$ (orange). Note the peak in amplitude near the core.

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**MDPI and ACS Style**

Stenull, O.; Lubensky, T.C.
Director Fluctuations in Two-Dimensional Liquid Crystal Disclinations. *Crystals* **2022**, *12*, 1.
https://doi.org/10.3390/cryst12010001

**AMA Style**

Stenull O, Lubensky TC.
Director Fluctuations in Two-Dimensional Liquid Crystal Disclinations. *Crystals*. 2022; 12(1):1.
https://doi.org/10.3390/cryst12010001

**Chicago/Turabian Style**

Stenull, Olaf, and Tom C. Lubensky.
2022. "Director Fluctuations in Two-Dimensional Liquid Crystal Disclinations" *Crystals* 12, no. 1: 1.
https://doi.org/10.3390/cryst12010001