Spontaneous breaking of chiral symmetry in achiral bent-core liquid crystals: Excluded volume effect

Bent-Core banana-shaped molecules exhibit tilted polar smectic phases with macroscopically chiral layer order even though the constituent molecules are achiral in nature. Here, we show that the excluded volume interactions between the bent-core molecules account for this spontaneous breaking of chiral symmetry in the layer. We have numerically computed excluded volume between two rigid bent-core molecules in a layer using two types of model structures of them and explored the different possible symmetries of the layer that are favored by the excluded volume effect. For both model structures of the molecule, the $C_2$ symmetric layer structure is favored for most values of tilt and bending angle. However, the $C_s$ and $C_1$ point symmetries of the layer are also possible for one of the model structures of the molecules. We have also developed a coupled $XY$-Ising model and performed Monte Carlo simulation to explain the statistical origin of spontaneous chiral symmetry breaking in this system. The coupled $XY$-Ising model accounts for the experimentally observed phase transitions as a function of temperature and electric field.

Figure 1

Schematic representation of the orientation of a BC molecule in a perfectly ordered tilted polar smectic layer where $\theta , \varphi$ and $\psi$ are the Euler angles. The layer normal $\widehat{k}$ is parallel to $z$ axis and the $XY$ plane is the layer plane. The double-headed arrow represents the projection of the long axis on the layer plane. For a perfectly ordered layer, the director $\widehat{n}$ and polar order $\vec{P}$ are parallel to $\widehat{l}$ and $\widehat{p}$ of the molecules, respectively. The unit vector $\widehat{\xi }=(\widehat{k}·\widehat{l})(\widehat{k}\times \widehat{l})/\left|(\widehat{k}·\widehat{l})(\widehat{k}\times \widehat{l})\right|$ represents the tilt direction of a molecule which is perpendicular to the ($\widehat{l},\widehat{k}$) plane.

Figure 2

The model structures (a) bead model (b) hard spherocylinder (HSC) model of a BC molecule used in the computation of excluded volume. The unit vectors $\widehat{l},\widehat{p}$, and $\widehat{m}$ are body fixed axes. The unit vectors $\widehat{u}$ and $\widehat{d}$ represent the orientation of upper and lower arms, respectively.

Figure 3

The variation of the excluded volume $V_{\mathrm{ex}}$ with the relative azimuthal angle $\delta \psi$ between two molecules for an orthogonal smectic layer. Inset shows the difference in $V_{\mathrm{ex}}$ corresponding to $\delta \psi =\pi$ and 0 for different values of $\beta$.

Figure 4

The variation of excluded volume $V_{\mathrm{ex}}$ as a function of roll angle $\psi$ for the HSC model.

Figure 5

The variation of $V_{\mathrm{ex}}$ as a function of roll angle $\psi$ for the bead model. Inset shows the magnified view of the indicated region demonstrating the maximum at $\psi ={180}^{\circ }$ for higher values of $\theta$.

Figure 6

The stability diagram in the $\theta -\beta$ plane representing the regions of stability of the different symmetries of the layers obtained from the excluded volume interactions for bead model of the molecules.

Figure 7

The variation of $V_{\mathrm{ex}}$ with tilt angle $\theta$ at $\psi =0$ or $\pi$ for (a) HSC model and (b) bead model of the molecules.

Figure 8

The favoured tilt angle $\theta$ for the bead model of the BC molecules at different values of bending angle $\beta$.

Figure 9

The representative spin configurations corresponding to the different phases. The blue and red (light gray) arrows denote the orientations of the unit vectors $\widehat{\xi }$ and $\widehat{p}$, respectively. The open circle and filled square symbols represent $+1$ and $-1$ values of the Ising spin, respectively.

Figure 10

The variation of order parameters with temperature for $A=0.5$ at zero electric field. Inset shows the corresponding specific heat variation.

Figure 11

The variation of order parameters with electric field for $T/J=1.8$ and $A=0.5$. Inset shows the magnified view of the region at low field.