Dynamics of hard colloidal cuboids in nematic liquid crystals

We perform dynamic Monte Carlo simulations to investigate the equilibrium dynamics of hard board-like colloidal particles in oblate and prolate nematic liquid crystals. In particular, we characterize the particles' diffusion along the nematic director and perpendicularly to it, and observe a structural relaxation decay that strongly depends on the particle anisotropy. To assess the Gaussianity of their dynamics and eventual occurrence of collective motion, we calculate two- and four-point correlation functions that incorporate the instantaneous values of the diffusion coefficients parallel and perpendicular to the nematic director. Our simulation results highlight the occurrence of Fickian and Gaussian dynamics at short and long times, locate the minimum diffusivity at the self-dual shape, the particle geometry that would preferentially stabilise biaxial nematics, and exclude the existence of dynamically correlated particles.

Figure 1

Model HBPs with thickness $T$, length $L=12T$, and width $W$. The particle width is a simulation parameter with values between $W=T$ (rod-like particles) and $W=L$ (disk-like particles).

Figure 2

Mean square displacement of oblate and prolate HBPs with width-to-thickness ratio indicated in the legend. Frames (a) and (b) refer to the directions parallel and perpendicular to the nematic director, respectively. The vertical dashed lines correspond to $t/\tau =0.1$, 3.3, and 2400. The solid lines highlight the linear dependence of the MSD on time at short and long timescales.

Figure 3

(a) Total ($•$), parallel ($\blacksquare$), and perpendicular ($▴$) diffusion coefficients of oblate and prolate HBPs in nematic LCs. (b) Rotational diffusion coefficients corresponding to the rotation of the axis parallel to $L(◯)$, $W$ ($\square$), and $T$ ($▵$). The vertical dashed line at $W^{*}=\sqrt{L^{*}}\approx 3.46$ indicates the transition from prolate to oblate particle shapes. The solid lines are a guide for the eyes.

Figure 4

Total ($•$), parallel ($\blacksquare$), and perpendicular ($▴$) self-part of the van Hove correlation function of prolate HBPs with $W^{*}=1$. (a), (b), and (c) refer, respectively, to $t/\tau =0.1$, 3.3, and 2400. Dashed lines are Gaussian distributions obtained from Eqs. (6) and (7).

Figure 5

Total ($•$), parallel ($\blacksquare$), and perpendicular ($▴$) self-part of the van Hove correlation function of oblate HBPs with $W^{*}=12$. (a), (b), and (c) refer, respectively, to $t/\tau =0.1$, 3.3, and 2400. Dashed lines are Gaussian distributions obtained from Eqs. (6) and (8).

Figure 6

Self-intermediate scattering functions calculated at $\left|\mathbf{k}\right|=2\pi /T$ for oblate and prolate HBPs, respectively. The particle width-to-thickness ratio is reported in the legend. The solid lines are stretched exponential fits of the form $exp\left[-{(t/t_r)}^{\alpha })\right]$, with $t_r$ the relaxation time and $\alpha$ the stretching exponential (see text for details). The inset shows the relaxation time as a function of the particle anisotropy at $\left|\mathbf{k}\right|=2\pi /T$. The solid lines are guides for the eye and the vertical dashed lines correspond to $t/\tau =0.1$, 3.3, and 2400.

Figure 7

Dynamic susceptibility in ${\mathrm{N}}^{+}$ and ${\mathrm{N}}^{-}$ LCs of HBPs at $\left|\mathbf{k}\right|=2\pi /T$. The particle width-to-thickness ratio is reported in the legend. The vertical dashed lines correspond to $t/\tau =0.1$, 3.3, and 2400.