Thin smectic liquid crystalline fibrils of chiral rodlike particles

Inspired by recent experimental work on virus-polymer mixtures, we study the properties of thin smectic fibrils composed of chiral rodlike particles using Monte Carlo simulations. Due to the interplay between surface energy, elastic deformation energy, and entropic effects, the fibril's layers relax into a twisted state. We focus our study on the layers' twist direction and map our results to the antiferromagnetic Ising model. In this view, the chiral interaction mimics an external field that drives the layers to have the same sense of twist. Besides, we determine the free energy difference and barrier height between an alternating and a nonalternating sequence of twisted layers composed of achiral rods and find that an alternating sequence is slightly preferred. We also see that the fibrils contract on increasing the chiral interaction strength and think that further studies on self-assembled functional materials can use our results.

Figure 1

Rendered snapshot of an equilibrated fibril with dimensions $s_M=8$ (number of rods at the side of the hexagon) and $N_M=6$ (number of stacked membranes). Left: side view; right: top view. Different senses of the twists' direction can be seen. The color codes the orientation of rods, and the rendering was done using POV-Ray [10]. Not shown are the depleting spheres that stabilize the structure.

Figure 2

Handedness parameter vs. chiral strength. Fibril dimensions are kept constant at $N_M=4$ and $s_M=6$ and the depletant density is ${\rho }_D=2.31D^{-3}$. Error bars depict the standard deviation.

Figure 3

(Normalized) twist alternation number vs. chiral strength. Fibril dimensions are kept constant at $N_M=4$ and $s_M=6$ and the depletant density is ${\rho }_D=2.31D^{-3}$. Error bars depict the standard deviation. The normalized twist alternation number reaches its maximum already at $\varepsilon =4k_{B}T$, while the plain twist alternation number still decreases and shows large error bars. The reason is that there are two, three, or four left-handed membranes, but in all cases, the number of twist alternations takes its maximum value (1, 0.5, and 0, respectively).

Figure 4

Twist alternation number for the antiferromagnetic Ising model vs. coupling constant for different field strengths.

Figure 5

Fibril length divided by number of stacked membranes vs. chiral strength. Fibril dimensions are kept constant at $N_M=4$ and $s_M=6$ and the depletant density is ${\rho }_D=2.31D^{-3}$. The different symbols (and colors) denote different numbers of left-handed membranes, as indicated. Error bars depict the standard error of the mean.

Figure 6

Reduced potential energy of fibrils vs. chiral strength. Fibril dimensions are kept constant at $N_M=4$ and $s_M=6$ and the depletant density is ${\rho }_D=2.31D^{-3}$. The different symbols (and colors) denote different numbers of left-handed membranes, as indicated. Error bars depict the standard error of the mean.

Figure 7

Edge twist angle vs. chiral strength. Fibril dimensions are kept constant at $N_M=4$ and $s_M=6$ and the depletant density is ${\rho }_D=2.31D^{-3}$. The different symbols denote different numbers of left-handed membranes, as indicated, and the color denotes the handedness [black: right, blue (gray): left]. Error bars depict the standard error of the mean.

Figure 8

Edge twist angle vs. depletant concentration for achiral rods. Fibril dimensions are kept constant at $N_M=4$ and $s_M=6$. The different symbols denote different numbers of left-handed membranes, as indicated, and the color denotes the handedness [black: right; blue (gray): left]. Error bars depict the standard error of the mean.

Figure 9

Hexagonal order parameter vs. depletant concentration for achiral rods. Fibril dimensions are kept constant at $N_M=4$ and $s_M=6$. Error bars are absent for low ${\rho }_D$ and depict the (very small) standard error of the mean from several simulations at high ${\rho }_D$.

Figure 10

Free energy barrier between a fibril with only alternating twist directions (rlrl, $m_s\approx 1.035$) and a fibril with less alternations (rrrl, $m_s\approx 0.964$). Fibril dimensions are $N_M=4$ and $s_M=6$ and the depletant density is ${\rho }_D=2.31D^{-3}$. The $s$ in the definition of $m_s$ is $+1$ if the second membrane is right handed and $-1$ if it is left handed. Umbrella sampling results are analyzed by a weighted histogram analysis method (WHAM) and the umbrella integration technique.

Figure 11

Normalized twist alternation number vs. initial box length for the achiral case ($\varepsilon =0k_{B}T$). Fibril dimensions are $N_M=30$ and $s_M=6$ and the depletant density is ${\rho }_D=2.31D^{-3}$.

Figure 12

Sketch of a fibril of length $L_x$ showing the hypothetical gap between membranes due to a high twist angle ${\phi }_0$. The average distance between membrane center planes is $L_x/N_M$. At the surface, a gap of height $h$ between the membranes appears that depends on the average membrane distance and on the surface twist angle. To avoid the formation of additional surfaces between membranes, the gap needs to be smaller than the depletant diameter. This in turn constrains ${\phi }_0$ to smaller values for larger fibril lengths. Left: ${\phi }_0=0$, right: ${\phi }_0>0$.

Figure 13

Comparison of the handedness parameter for membranes stacked to a fibril (black crosses) and isolated membranes with the same size (blue stars) $s_M=6$ at a depletant density of ${\rho }_D=2.31D^{-3}$.

Figure 14

Comparison of the reduced potential energy for membranes stacked to a fibril and isolated membranes with the same size $s_M=6$ at a depletant density of ${\rho }_D=2.31D^{-3}$. The different symbols (and colors) denote different numbers of left-handed membranes, as indicated.

Figure 15

Comparison of the edge twist angle for membranes stacked to a fibril and isolated membranes with the same size $s_M=6$ at a depletant density of ${\rho }_D=2.31D^{-3}$. We only show results for left-handed membranes and $N^{\ell }=2$ and 4.

Figure 16

Comparison of the area number density for membranes stacked to a fibril and isolated membranes with the same size $s_M=6$ at a depletant density of ${\rho }_D=2.31D^{-3}$. We only show results for left-handed membranes and $N^{\ell }=2$ and 4.

Figure 17

Area number density for isolated membranes vs. chiral strength for different depletant densities (color) and different numbers of rods (symbols: cross: $N=91$; triangle: $N=397$; circle: $N=919$). We only show results for left-handed membranes.

Figure 18

Edge twist angle for isolated membranes vs. chiral strength for different depletant densities (color) and different numbers of rods (symbols: cross: $N=91$; triangle: $N=397$; circle: $N=919$). We only show results for left-handed membranes.

Figure 19

Curvature parameter for isolated membranes vs. number of rods for different depletant densities (color) and a chiral strength of $\varepsilon =6k_{B}T$.