# Focal Conic Stacking in Smectic A Liquid Crystals: Smectic Flower and Apollonius Tiling

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## Abstract

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## 1. Introduction

**Figure 1.**(a): three circles of radii ${R}_{1}$, ${R}_{2}$, ${R}_{3}$ are tangent. The figure has been made in the particular case ${R}_{1}={R}_{2}={R}_{3}$. Two circles of Radii ${R}_{4}$ and ${R}_{4}^{\prime}$ are tangent to the three initial circles; (b): coordinates system with the two angles, longitude θ and latitude ϕ.

## 2. Materials and Experimental Setup

#### 2.1. Flower texture

**Figure 2.**(a): experimental observation of the flower texture in TBDA, bar $\sim 100\mu m$; (b): experimental observation of the flower texture in TBDA, bar $\sim 100\mu m$.

#### 2.2. Generation texture

**Figure 3.**(a): experimental observation of some quasi-hexagonal tiling in 8CB, sample thickness $\sim 100\mu m$, bar $\sim 100\mu m$; (b): Apollonius tiling of circles showing 4 generations of sizes using some ${C}^{++}$ numerical simulation.

## 3. The smectic layers

**Figure 4.**(a): three kind of layers for FCD of the first species; (b): representation of a torus: R being the radius of the directing circle and μ the radius of the generating circle; (c): representation of a TFCD (Toric Focal Conic Domain) with some negative Gaussian curvature.

## 4. The Law of Corresponding Cones

**Figure 5.**Illustration of the law of corresponding cones $(l.c.c)$ for two tangent ellipses and their confocal hyperbolae ; (a): 3D representation: the two cones which lie on the two ellipses (blue cones) have one common generatrix; likewise for the two cones, which lie on the two hyperbolae (red cones); adapted from [14]; (b): 2D representation: the line joining the visible focus ${F}_{1}$ of one ellipse to the invisible focus (not physical) of the other ${F}_{2}^{\prime}$ goes exactly through the tangential point M between the two ellipses; adapted from [23].

## 5. Results and Discussion

#### 5.1. Flower texture

#### Model of the experimental texture.

**Figure 6.**Three cones have been drawn; each cone contains some assembly of mutually tangent ellipses. The virtual branches of the hyperbolae have one point of intersection at the apex S of the droplet (a): top view; (b): side view.

#### 5.2. Generation texture

**Figure 7.**(a): torii with $R=2$ for different increasing values of $\mu =1,2,3$ from the left to the right. When $\mu >R$, two sheets exist: layer of type 3 takes place instead of type 2; (b): torii with $\mu =2$ for different increasing values of $R=1,2,3$ from the left to the right. When $R>\mu $, only one sheet exist: layer of type 2 takes place instead of type 3.

#### The variation of μ at fixed R.

#### The variation of R at fixed μ.

**Figure 8.**(a): R=2; when μ increases layers of type 2 transform into layers of type 3; (b): $\mu =2$; when R increases, layers of type 3 transform into layers of type 2.

#### The simulation of the smectic layers.

**Figure 9.**Dupin cyclides simulation: 1rst generation in blue, 2nd in green, 3rd in yellow, 4th in red. (a): Top view ($\theta ={0}^{\circ}$, $\phi ={90}^{\circ}$); (b) some side view ($\theta ={60}^{\circ}$, $\phi ={30}^{\circ}$).

**Figure 10.**(a): simulation of the smectic layers for one single value of μ ($\mu =3$) corresponding to layers of type 3 with two sheets (only one sheet is visible with this point of view); $R=0.9$, $\theta ={10}^{\circ}$, $\phi ={80}^{\circ}$; (b): the same simulation with another point of view ($\theta ={45}^{\circ}$, $\phi ={45}^{\circ}$).

**Figure 11.**simulation of the smectic layers for one single value of μ ($\mu =3$) corresponding to layers of type 3 with two sheets from the second generation of circles; layers are of type 2 for the first generation of circles; $R=0.9$.

**Figure 12.**simulation of the smectic layers for three values of μ ($\mu =1$; $\mu =2$; $\mu =3$) corresponding to layers of type 3 with two sheets; $R=0.9$, $\theta ={80}^{\circ}$, $\phi ={10}^{\circ}$.

## 6. Conclusions

## Acknowledgements

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**MDPI and ACS Style**

Meyer, C.; Le Cunff, L.; Belloul, M.; Foyart, G.
Focal Conic Stacking in Smectic A Liquid Crystals: Smectic Flower and Apollonius Tiling. *Materials* **2009**, *2*, 499-513.
https://doi.org/10.3390/ma2020499

**AMA Style**

Meyer C, Le Cunff L, Belloul M, Foyart G.
Focal Conic Stacking in Smectic A Liquid Crystals: Smectic Flower and Apollonius Tiling. *Materials*. 2009; 2(2):499-513.
https://doi.org/10.3390/ma2020499

**Chicago/Turabian Style**

Meyer, Claire, Loic Le Cunff, Malika Belloul, and Guillaume Foyart.
2009. "Focal Conic Stacking in Smectic A Liquid Crystals: Smectic Flower and Apollonius Tiling" *Materials* 2, no. 2: 499-513.
https://doi.org/10.3390/ma2020499