# From Bend to Splay Dominated Elasticity in Nematics

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Theoretical Framework

#### 2.2. Interaction Potential and Model Details

#### 2.3. Numerical Procedure

#### 2.4. Frames of Reference

#### 2.5. Orientational Order Parameters

## 3. Results and Discussion

#### 3.1. Acute Angle Particles

#### 3.2. Obtuse Angle Particles

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Sketch of the laboratory frame of reference $\left(OXYZ\right)$ and of the particle frame of reference $\left(oxyz\right)$. $\mathit{R}$ is the vector position of the center of mass of the particle in the laboratory frame, and $\mathrm{\Omega}$ is the set of Euler angles that rotate $\left(OXYZ\right)$ into $\left(oxyz\right)$. The red segments represent the director field with a splay distortion of wavenumber $q=0.05\phantom{\rule{0.166667em}{0ex}}{\sigma}^{-1}$.

**Figure 2.**Model particles considered in this work; $\sigma $ is the hard sphere diameter, and $\chi $ is the bend angle, equal to ${45}^{\circ}$ in (

**a**), ${150}^{\circ}$ in (

**b**), ${155}^{\circ}$ in (

**c**) and ${25}^{\circ}$ in (

**d**).

**Figure 3.**(

**a**) Order parameters S and D (inset), and (

**b**) splay, (

**c**) twist and (

**d**) bend elastic constant, as a function of the volume fraction $\varphi $. Values obtained using the non-local form of the ODF, Equation (5) (solid lines), and the local uniaxial form, Equation (6) (dashed lines), for the nematic phase of acute angle V-shaped particles with different values of the bend angle.

**Figure 4.**(

**a**) Polar longitudinal order parameter ${p}_{zZ}$ for V-shaped particles with bend angle $\chi ={45}^{\circ}$ in the presence of a splay deformation, as a function of the deformation wavenumber q, at different values of the packing fraction $\varphi $. (

**b**) Slope of the longitudinal polar order parameter induced by a splay deformation, ${p}_{zZ}\left(q\right)$, calculated at $q=0$, as a function of the volume fraction difference $\varphi -{\varphi}_{N}$, for acute angle V-shaped particles with different values of the bend angle. ${\varphi}_{N}$ is the packing fraction in the nematic phase at the nematic-isotropic boundary. Negative ${p}_{zZ}$ values mean that the apex of a V-shaped particle preferentially points towards the center of the splay deformation (see Figure 1).

**Figure 6.**Elastic constants calculated for acute angle V-shaped with a bend angle $\chi ={45}^{\circ}$ (see inset), as a function of the order parameter S.

**Figure 7.**(

**a**) Order parameters S and D (inset), and (

**b**) splay, (

**c**) twist and (

**d**) bend elastic constant, as a function of the volume fraction $\varphi $. Values obtained using the non-local form of the ODF, Equation (5) (solid lines), and the local uniaxial form, Equation (6) (dashed lines), for the nematic phase of obtuse angle V-shaped particles with different values of the bend angle.

**Figure 8.**Slope of the transversal polar order parameter induced by a bend deformation, ${p}_{xX}\left(q\right)$, calculated at $q=0$, as a function of the packing fraction difference $\varphi -{\varphi}_{N}$, for acute angle V-shaped particles with different values of the bend angle. ${\varphi}_{N}$ is the packing fraction in the nematic phase at the nematic-isotropic boundary. The dashed line shows the results for the curved particle shown in Figure 2c. Positive ${p}_{xX}$ values mean that the apex of a V-shaped particle preferentially points to the opposite direction of the center of a bend deformation.

**Figure 9.**Elastic constants as a function of the volume fraction $\varphi $, for a nematic phase formed by obtuse angle V-shaped particles with a bend angle $\chi ={150}^{\circ}$, red full symbols, and by the curved particles as shown in Figure 2c, blue empty symbols. (For the latter system, the $\varphi $ scale is different from the one in Figure 1a of [37], since in that case, the volume fraction was calculated as $\varphi ={v}_{\mathrm{eff}}\rho $.)

**Table 1.**Geometric parameters of the particles investigated in this work (number of segments M, bend angle $\chi $, geometrical volume ${v}_{0}$, effective ${v}_{\mathrm{eff}}$ volume); packing fractions at the isotropic-nematic coexistence, in the isotropic, ${\varphi}_{\mathrm{I}}$, and in the nematic phase, ${\varphi}_{\mathrm{N}}$.

Particle | M | $\mathit{\chi}{[}^{\circ}]$ | ${\mathit{v}}_{0}$$\left[{\mathit{\sigma}}^{3}\right]$ | ${\mathit{v}}_{\mathbf{eff}}$$\left[{\mathit{\sigma}}^{3}\right]$ | ${\mathit{\varphi}}_{\mathbf{I}}$ | ${\mathit{\varphi}}_{\mathit{n}}$ |
---|---|---|---|---|---|---|

Acute angle V-shaped | 20 | 25 | 10.777 | 13.892 | 0.190 | 0.198 |

20 | 30 | 10.842 | 12.449 | 0.196 | 0.202 | |

20 | 35 | 10.887 | 12.364 | 0.200 | 0.204 | |

20 | 40 | 10.925 | 12.486 | 0.205 | 0.208 | |

20 | 45 | 10.956 | 12.505 | 0.211 | 0.213 | |

Obtuse angle V-shaped | 10 | 140 | 5.760 | 6.485 | 0.247 | 0.255 |

10 | 145 | 5.760 | 6.485 | 0.236 | 0.246 | |

10 | 150 | 5.760 | 6.485 | 0.228 | 0.239 | |

10 | 155 | 5.760 | 6.485 | 0.220 | 0.233 | |

10 | 160 | 5.760 | 6.485 | 0.213 | 0.229 | |

10 | 165 | 5.760 | 6.485 | 0.208 | 0.225 | |

10 | 170 | 5.760 | 6.485 | 0.204 | 0.223 | |

10 | 175 | 5.760 | 6.485 | 0.201 | 0.221 | |

Rod-like | 10 | 180 | 5.760 | 6.485 | 0.200 | 0.221 |

Curved | 10 | 155 | 5.760 | 6.485 | 0.225 | 0.236 |

Tripodal | 30 | 25 | 15.703 | 20.091 | 0.227 | 0.234 |

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**MDPI and ACS Style**

Revignas, D.; Ferrarini, A.
From Bend to Splay Dominated Elasticity in Nematics. *Crystals* **2021**, *11*, 831.
https://doi.org/10.3390/cryst11070831

**AMA Style**

Revignas D, Ferrarini A.
From Bend to Splay Dominated Elasticity in Nematics. *Crystals*. 2021; 11(7):831.
https://doi.org/10.3390/cryst11070831

**Chicago/Turabian Style**

Revignas, Davide, and Alberta Ferrarini.
2021. "From Bend to Splay Dominated Elasticity in Nematics" *Crystals* 11, no. 7: 831.
https://doi.org/10.3390/cryst11070831