Entropy-Driven Chiral Order in a System of Achiral Bent Particles

Why should achiral particles organize into a helical structure? Here, using theory and molecular dynamics simulations we show that at high concentration crescent-shaped particles interacting through a purely repulsive potential form the twist-bend nematic phase, which features helical order of the twofold symmetry axes of particles, with doubly degenerate handedness. Spontaneous breaking of the chiral symmetry is driven by the entropic gain that derives from the decrease in excluded volume in the helical arrangement. Crucial to this purpose is the concave shape of particles. This study is based on a general formulation of the Onsager theory, which includes biaxiality and polarity of phase and particles, in addition to the space modulation of order. Molecular dynamics simulations corroborate the theoretical predictions and provide further insights into the structure of the helical phase.

Figure 1

Polar orthorombic (a) and helical (b) organization of bent particles. The arrows indicate the orientation of the twofold symmetry axes of particles. (c) Pair configurations where particles have parallel (top) and antiparallel (bottom) $C_2$ axes.

Figure 2

Left: Helical structure of the $N_{\mathrm{TB}}$ phase. A snapshot from a MD trajectory is shown, with color codified according to the projection of the $C_2$ axes of particles on the $X$ axis of the LAB frame (blue—parallel, red—antiparallel). Middle: LAB frame ($X,Z$). The unit vector $\widehat{\mathbf{m}}$, parallel to the local twofold symmetry axis of the phase, performs a helical rotation with pitch $p$ around the unit vector $\widehat{\mathbf{h}}$, parallel to the LAB $Z$ axis. Right: A crescent-shaped particle and the unit vectors $\widehat{\mathbf{a}}$, $\widehat{\mathbf{b}}$, $\widehat{\mathbf{c}}$, attached to it.

Figure 3

Bottom: Order parameters $⟨P_2{⟩}_{\mathbf{a}}$ (open) and $⟨P_1{⟩}_{\mathbf{b}}$ (solid). Top: Helical pitch $p$ (solid) and conical angle $\theta$ (open), as a function of the number density $\overline{\rho }$, calculated for the particles in Fig. 2 using the Onsager-like theory. The images on the left and on the right are snapshots from MD simulations, in the $N$ and in the $N_{\mathrm{TB}}$ phase, respectively; the color code is the same as in Fig. 2.

Figure 4

Longitudinal orientational correlation functions from MD simulations at $\overline{\rho }=0.056/{\sigma }^3$ (see text) as a function of $R_{\parallel }$, the interparticle distance along the helical axis: $S_{bb}^{110}$ (red solid line), $S_{aa}^{221}$ (black short-dashed line) and $S_{aa}^{220}$ (blue dash-dotted line).