Probing the sensitivity of orientational ordering as a way towards absolute enantiorecognition: Helical-particle solutes in helical-particle nematic solvents

To inquire into whether two enantiomers can be absolutely distinguished on the sole basis of their different degrees of orientational ordering when dissolved in a chiral nematic liquid crystal, several enantiomeric pairs of hard helical particles are dissolved in a cholesteric (nematic) liquid crystal made of slender hard helical particles as well as in a screwlike nematic liquid crystal made of tortuous hard helical particles. While in the former ordinary chiral nematic solvent their nematic order parameters are (almost) coincident, in the latter new chiral nematic solvent the two enantiomeric solutes not only usually have more appreciably different nematic order parameters, but also always have significantly different screwlike order parameters. If also the latter orientational order parameter could be measurable in real experiments, it would constitute an additional decisive piece of information on the way to absolutely distinguishing two enantiomers on the sole basis of their different degrees of orientational order when dissolved in a chiral nematic liquid crystal that is in the apter screwlike nematic phase. Even in that event, however, the general absence of regularity and systematicity in the trend of the orientational order parameters, already manifest in these elementary hard helical-particle binary systems, would make this coveted achievement experimentally arduous without the assistance of very accurate and precise theoretical calculations.

Figure 1

Schematic of the secondary director $\widehat{\mathbf{m}}$ perpendicularly twisting around the primary director $\widehat{n}$ with pitch $p$ coincident in sense and value with that of the helical particles constituting the system in the screwlike nematic phase.

Figure 2

(a) Image of a helical particle composed of 15 partially overlapping spherical beads of diameter $D$ whose centers are arranged along a helical chord of length $L$, radius $r$, and pitch $p$. The mechanical state of such a helical particle can be defined by specifying its frame of reference: the position of the center ${\mathbf{r}}_{\circ }$, and the orientation of the triad of mutually perpendicular unit vectors $\widehat{\mathbf{u}},\widehat{\mathbf{v}}$, and $\widehat{\mathbf{w}}$. The positions of the component spherical beads, when projected on the ($\widehat{\mathbf{v}},\widehat{\mathbf{w}}$) plane, lie on a circumference with center ${\mathbf{r}}_{\circ }$ and radius $r$; the axis along $\widehat{\mathbf{u}}$ is the one along which the helix unrolls with a pitch $p$. (b) Images of a few helical particles considered in the present work: In the top panel there are the two right-handed solvent particles, the one with $r/D=0.2$ and $p/D=9.92$ on the left and the one with $r/D=0.4$ and $p/D=4$ on the right; in the bottom panel there are three left-handed solute particles with $r/D=0.2, p/D=4$, and $n_s=5,9,13$.

Figure 3

The (Saupe) order matrix element ${\mathcal{S}}_{\widehat{\mathbf{u}}\widehat{\mathbf{u}}}^{\sigma }$ of the hard helical-particle solutes both left (red diamond) and right handed (violet square), whose specific helical parameters are indicated within the respective square panel, either in the cholesteric solvent made of right-handed hard helical particles with $r=0.2D;p=9.92D$ (left panels) or in the screwlike nematic solvent made of right-handed hard helical particles with $r=0.4D;p=4D$ (right panels) as a function of the number of the constituent hard spheres $n_s$.

Figure 4

The orientational pair correlation functions $g_{1,\widehat{\mathbf{w}}}^{\sigma ,\mathrm{\Sigma }}\left(r_{\parallel }\right)$ for the left- and right-handed hard helical solute particles, whose helical parameters are indicated within the respective square panel, in the screwlike nematic solvent made of right-handed hard helical particles with $r=0.4D;p=4D$, whose orientational pair correlation function $g_{1,\widehat{\mathbf{w}}}^{\mathrm{\Sigma },\mathrm{\Sigma }}\left(r_{\parallel }\right)$ is also shown in the topmost rectangular panel; in each square panel, results are shown for the several values of $n_s$: 3 (red); 5 (orange); 7 (green); 9 (cyan); 11 (blue); 13 (indigo); 15 (violet).