# Nematic and Smectic Phases: Dynamics and Phase Transition

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## Abstract

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## 1. Introduction

## 2. The Smectic Phase

#### 2.1. Model

#### 2.2. Method of Simulation

#### 2.3. Results

## 3. The Nematic Phase

#### 3.1. Model

#### 3.2. Results

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Snapshot of the system at two different temperatures. Molecules are contained in a box with periodic boundary conditions. There is ${N}_{L}=15\times 15\times 30=6750$ sites and 2025 molecules, i.e., a molecule concentration equal to $c=30\%$. Each colored point corresponds to a molecular orientation. (

**Left)**: Initial configuration of the system at a high temperature $T=5$. The system is completely disordered. (

**Right**): Final configuration at low temperature $T=0.1$. The system was cooled down from the disordered phase. Molecules are staged in independent layers and within a layer all molecules have the same orientation (same color). It corresponds to a smectic phase.

**Figure 2.**(

**a**) energy U versus T; (

**b**) diffusion coefficients as functions of T: the upper curve (green) is the perpendicular diffusion coefficient, the lower curve (blue) is the parallel one and the middle curve (red) is the total diffusion coefficient. See text for comments. The molecule concentration c is $c={N}_{s}/{N}_{L}=30\%$, with ${N}_{L}=15\times 15\times 30$. The exchange interactions are ${J}_{\phantom{\rule{-0.166667em}{0ex}}//}=3J$, ${J}_{\perp}=-J$, with $J=1$.

**Figure 3.**Histogram showing the percentage R of sites having $\mathcal{Z}$ nearest neighbors at $T=0.1$, i.e., corresponding to the snapshot at the right of Figure 1. Molecules are distributed such as they have mostly four neighbors, meaning they are staged in layers.

**Figure 4.**Evolution of magnetizations of layer 3 (red) and 14 (blue) and their susceptibilities as functions of temperature. The molecule concentration c is $c={N}_{s}/{N}_{L}=30\%$, with ${N}_{L}=15\times 15\times 30$. The exchange interactions are ${J}_{\phantom{\rule{-0.166667em}{0ex}}//}=3.0$, ${J}_{\perp}=-1.0$.

**Figure 5.**Snapshot of the system at two different temperature. Molecules are contained in a box. There is ${N}_{L}=15\times 15\times 30=6750$ sites and 2025 molecules, i.e., a molecule concentration equal to $c=30\%$. Each colored point corresponds to a molecule orientation. (

**Left**): Initial configuration of the system at high temperature $T=3$ (T is in unit of $J/{k}_{B}$). The system is completely disordered. (

**Right**): Final configuration at low temperature $T=0.1$ obtained by a slow cooling. Note that that there is no positional order like for liquids. It corresponds to a nematic phase. A video showing the cooling of the system is available here.

**Figure 6.**Evolution of energy U and diffusion coefficient D versus T. The molecule concentration c is $c={N}_{s}/{N}_{L}=30\%$, with ${N}_{L}=15\times 15\times 30$. The exchange interactions are ${J}_{1}=-J$, ${J}_{2}=2J$, with $J=1$, and the anisotropy is equal to ${A}_{z}=0.5J$.

**Figure 7.**Evolution of the orientational order parameter and its susceptibility as functions of T. The molecule concentration c is $c={N}_{s}/{N}_{L}=30\%$, with ${N}_{L}=15\times 15\times 30$. The exchange interactions are ${J}_{1}=-J$, ${J}_{2}=2J$, and the anisotropy is equal to ${A}_{z}=0.5J$ ($J=1$).

**Figure 8.**Histogram showing the percentage R of sites having $\mathcal{Z}$ nearest neighbors at $T=0.1$, i.e., corresponding to the snapshot on the right in Figure 5. Molecules have distributed such as they have no neighbors.

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Bailly-Reyre, A.; Diep, H.T.
Nematic and Smectic Phases: Dynamics and Phase Transition. *Symmetry* **2020**, *12*, 1574.
https://doi.org/10.3390/sym12091574

**AMA Style**

Bailly-Reyre A, Diep HT.
Nematic and Smectic Phases: Dynamics and Phase Transition. *Symmetry*. 2020; 12(9):1574.
https://doi.org/10.3390/sym12091574

**Chicago/Turabian Style**

Bailly-Reyre, Aurélien, and Hung T. Diep.
2020. "Nematic and Smectic Phases: Dynamics and Phase Transition" *Symmetry* 12, no. 9: 1574.
https://doi.org/10.3390/sym12091574