Density functional theory for chiral nematic liquid crystals

Even though chiral nematic phases were the first liquid crystals experimentally observed more than a century ago, the origin of the thermodynamic stability of cholesteric states is still unclear. In this Rapid Communication we address the problem by means of a density functional theory for the equilibrium pitch of chiral particles. When applied to right-handed hard helices, our theory predicts an entropy-driven cholesteric phase, which can be either right or left handed, depending not only on the particle shape but also on the thermodynamic state. We explain the origin of the chiral ordering as an interplay between local nematic alignment and excluded-volume differences between left- and right-handed particle pairs.

Figure 1

(a) Space dependence of the nematic director $\widehat{\mathbf{n}}$ in a cholesteric phase with chiral director $\widehat{\chi }$ and pitch $P$. (b) Hard-helix particle modeled as a collection of 15 partially overlapping hard spheres of diameter $\sigma$, whose centers of mass are equally spaced on a right-handed helix of contour length $L=10\sigma$, inner radius $r=0.4\sigma$, and pitch $p=4\sigma$ [18]. The orientation of the helix can be expressed by means of the unit vector $\widehat{\omega }$ and the internal azimuthal angle $\alpha$.

Figure 2

(a) Free energy per unit volume as a function of the packing fraction $\rho v$ and the chiral wave number $q$ for hard helices with contour length $L/\sigma =10$, helix radius $r/\sigma =0.4$, and pitch $p/\sigma =4$. The white dashed line identifies $q_{\min }$ that minimizes the free energy at fixed packing fraction. (b) Chiral wave number $q_{\min }$ for hard helices with contour length $L/\sigma =10$, helix radius $r/\sigma =0.4$, and helix pitch $p/\sigma =2$, 3, and 4, and (c) the corresponding equilibrium pitch $P=2\pi /q_{\min }$.

Figure 3

Handedness of the cholesteric phase of hard helices with contour length $L/\sigma =10$ and variable helix radius $r$ and pitch $p$. The handedness of the cholesteric phase with respect to that of the constituent hard helices can be the same (solid square), opposite (solid circle), or mixed with a change in handedness from left to right upon increasing the packing fraction (solid triangle). The dashed lines identify the approximate transition between these regimes. Open symbols represent parameters for which such a classification is uncertain within our statistical accuracy.

Figure 4

(a) Difference $\Delta E=E_R-E_L$ between the right- and left-handed excluded volume [cf. Eq. (6)] as a function of the cosine of the angle between the main axes $cos\gamma =\widehat{\omega }·{\widehat{\omega }}^{\prime }$ of the same hard helices as in Figs. 2 and 2. The dashed rectangle identifies the portion of the plot reported in the inset. (b) Estimated difference in free energy $\Delta F_{\chi }$ between right- and left-handed chiral ordering as defined in Eq. (7).