^{1,a)}and Giorgio J. Moro

^{1,b)}

### Abstract

The feature of local biaxiality of the orientational order in twisted nematics and cholesteric liquid-crystalline phases is faced by modeling the mean field orientational potential on the basis of the surfaceinteraction model [A. Ferrarini, G. J. Moro, P. L. Nordio, and G. R. Luckhurst, Mol. Phys.77, 1 (1992)]. Here we present a tool for the complete parameterization of the potential for general molecular structures and recover the long-pitch approximation usually invoked to model the molecular order in these phases. The method is applied to archetype molecular geometries (an ellipsoidal object, a conical object, a lath-shaped molecule, and the shape’s enantiomers of a propellerlike molecule) in order to evaluate the dependence of the second-rank orientational order parameters on the pitch of the phase. Special emphasis is given to the so-called biaxiality parameter [Z. Yaniv, N. A. P. Vaz, G. Chidichimo, and J. W. Doane, Phys. Rev. Lett.47, 46 (1981)], which can be experimentally determined by the analysis of time-averaged spectra of deuterated probes dissolved in the twisted phase. The model calculations show how probes with different geometries are sensitive to the local biaxiality.

We thank MIUR (PRIN 2005) and INSTM for support. Interesting suggestions from Professor Claudio Zannoni (Department of Chemistry and Industrial Chemistry, University of Bologna, Italy) are acknowledged.

I. INTRODUCTION

II. LOCAL ALIGNMENT IN CHOLESTERICS AND BIAXIALITY PARAMETERS FROM NMR

III. ORIENTATIONAL POTENTIAL FROM SURFACE INTERACTIONS

IV. THE LONG-PITCH LIMIT

V. MODEL CALCULATIONS FOR ARCHETYPE MOLECULES

A. Ellipsoidal molecule

B. Conical molecule

C. Lath-shaped molecule

D. Propellerlike molecule

VI. OUTLINES AND CONCLUSIONS

### Key Topics

- Molecule surface interactions
- 23.0
- Chiral symmetries
- 21.0
- Spatial filtering
- 16.0
- Mean field theory
- 14.0
- Nematic liquid crystals
- 14.0

## Figures

The fixed LF and the position-dependent DF; is the local director in the plane displaced by from the origin of LF along the helical axis . A left-handed twist is sketched.

The fixed LF and the position-dependent DF; is the local director in the plane displaced by from the origin of LF along the helical axis . A left-handed twist is sketched.

(a) Relation between the DF, the MF, and the SMF referred to the point ; the outwards normal to the surface, , is the longitudinal axis of SMF, while the transverse axes are arbitrarily chosen. (b) Relation between the DF, the MF, and the spherical coordinates frame (SCF) referred to the point .

(a) Relation between the DF, the MF, and the SMF referred to the point ; the outwards normal to the surface, , is the longitudinal axis of SMF, while the transverse axes are arbitrarily chosen. (b) Relation between the DF, the MF, and the spherical coordinates frame (SCF) referred to the point .

Archetype molecular geometries considered in the model calculations: (a) revolution ellipsoid, (b) conical object, (c) lath-shaped molecule made of two connected parallelepipeds, and (d) propellerlike molecule made of twisted parallelepipeds.

Archetype molecular geometries considered in the model calculations: (a) revolution ellipsoid, (b) conical object, (c) lath-shaped molecule made of two connected parallelepipeds, and (d) propellerlike molecule made of twisted parallelepipeds.

Pitch dependence of the order parameters and for the ellipsoidal molecule with and . The value for the orientational strength parameter was employed. The profiles are the same for left- or right-handed twists of the cholesteric phase.

Pitch dependence of the order parameters and for the ellipsoidal molecule with and . The value for the orientational strength parameter was employed. The profiles are the same for left- or right-handed twists of the cholesteric phase.

Pitch dependence of the order parameters and for the conical molecule with and . The value for the orientational strength parameter was employed. The profiles are the same for left- or right-handed twists of the cholesteric phase.

Pitch dependence of the order parameters and for the conical molecule with and . The value for the orientational strength parameter was employed. The profiles are the same for left- or right-handed twists of the cholesteric phase.

Pitch dependence of the order parameters , , , and for the lath-shaped molecule with , , , and (circles). The profiles of and of are shown also for the molecule of equal transverse area with , , and (triangles). The value for the orientational strength parameter was employed. The profiles are the same for left- or right-handed twists of the cholesteric phase.

Pitch dependence of the order parameters , , , and for the lath-shaped molecule with , , , and (circles). The profiles of and of are shown also for the molecule of equal transverse area with , , and (triangles). The value for the orientational strength parameter was employed. The profiles are the same for left- or right-handed twists of the cholesteric phase.

Pitch dependence of the order parameters , , , and for the two enantiomers of the propellerlike molecule made of twisted parallelepipeds: (filled circles) and (open circles). The value for the orientational strength parameter was employed. The profiles are referred to a left-handed twist of the phase (for right-handed twist the profiles of the two enantiomers are exchanged).

Pitch dependence of the order parameters , , , and for the two enantiomers of the propellerlike molecule made of twisted parallelepipeds: (filled circles) and (open circles). The value for the orientational strength parameter was employed. The profiles are referred to a left-handed twist of the phase (for right-handed twist the profiles of the two enantiomers are exchanged).

Dependence of the order parameters on the wave vector for the two enantiomers of the propellerlike molecule in the left-handed twisted phase: (filled circles) and (open circles) like in Fig. 7. Dotted lines refer the linear approximation for the order parameters as given in Eq. (42). The arrows in the first panel are placed at a value which corresponds to a pitch of 100 times the major molecular length.

Dependence of the order parameters on the wave vector for the two enantiomers of the propellerlike molecule in the left-handed twisted phase: (filled circles) and (open circles) like in Fig. 7. Dotted lines refer the linear approximation for the order parameters as given in Eq. (42). The arrows in the first panel are placed at a value which corresponds to a pitch of 100 times the major molecular length.

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