Tilt, polarity, and spontaneous symmetry breaking in liquid crystals

We show through explicit molecular modeling how tilt may be induced in layered mesophases and propagate across the layers, using a concrete representation of flexible tail-core-tail calamitic mesogens in conjunction with the variational cluster expansion. The results demonstrate that spontaneous symmetry breaking observed in smectic liquid crystals—the tilt of the director relative to the layers—can be induced by excluded volume interactions, both in the synclinic and in the anticlinic configurations. @S1063-651X~98!51105-1# PACS number~s!: 61.30.Cz, 64.70.Md The relationship between molecular symmetry and phase symmetry in liquid crystals is very intriguing. The nematic phase—a fluid of orientationally ordered mesogens—is apolar even in cases where the mesogen itself is polar, either in its shape or its electric charge distribution @1#. By contrast, stratified smectic liquid crystals—orientationally ordered fluids wherein the mesogen centers of mass condense into layers—exhibit polarity even when the molecules are apolar; in a tilted smectic-C (SC) phase comprised of apolar mesogens, the phase is polar—the ‘‘ C director’’ specifying the tilt direction, is a unique direction in the SC phase @1#. Additionally, polarity normal to the ‘‘tilt plane’’ is indigenous to the SC phase even when the latter is comprised of apolar mesogens @2#. Thus the relationship between molecular symmetry and phase symmetry appears ambiguous in the sense that the latter cannot always be inferred from the former. In particular, there are instances where the apparent ~local! symmetry of the phase is lower than that of its constituent molecules, a feature that is emblematic of so-called spontaneous symmetry breaking ~SSB!. In this Rapid Communication we focus on attributes of the orientational ordering of flexible molecules in the normal SA and tilted SC phases by starting out with an explicit molecular structure—a generic, three-segment model of a mesogen—and performing calculations using a well-defined statistical-mechanical approximation, the variational cluster expansion @3#. We obtain clear insights into factors that influence the tilt in smectics, the propagation of the tilted ordering across the smectic layers ~synclinic vs anticlinic!, and the sign and magnitude of the spontaneous polarization in the chiral smectic-C (SC) phase. Our findings constitute an explicit derivation of the fundamental propositions of the ‘‘indigenous polarity ~IP! theory’’ of Photinos and Samulski @2,4# and provide a general conceptual framework for understanding polarity and ferroelectricity across the entire spectrum of liquid crystals, in calamitic, discotic, and bent ~‘‘banana-’’ or ‘‘boomerang-shaped’’ ! mesogens. The first efforts to derive the structure of the tilted achiral SC phase from a molecular model date back to the theory of McMillan @5# in which the tilt is a result of end-to-end alignment of permanent dipoles attached obliquely with respect to the mesogen’s ‘‘long-molecular-axis’’ l. The McMillan model, and variants thereof that followed @6#, imply that the appearance of director tilt in the SC phase is inevitably accompanied by strong biasing of the rotational motion about l ~partial rotational freeze-out !@ 7 #. But this picture is in direct conflict with experimental results. In fact, obtaining tilt at the price of rotational biasing undermines rigid-molecule models that have been proposed for the description of spontaneous polarization in tilted smectics @8#. Herein we study the SSB mechanism associated with tilted smectics using idealized, three-segment, model mesogens—a primitive representation of flexible tail-coretail calamitic mesogens ~Fig. 1!. For simplicity we consider two-dimensional phases, in the YZ plane, with perfect smectic layering in the Z direction, i.e., the molecular centers are assumed to be confined on ‘‘smectic lines’’ of length L and constant spacing d. This precludes the description of positional fluctuations in the direction of the layer normal. It is justified, however, in that such fluctuations are not of primary relevance to the SSB mechanism. Molecular flexibility is imparted to the model, while FIG. 1. ~a! Schematic diagram of the three-segment molecular model, showing the segmental axis frames xi , y i , z i , the segmental width D and lengths Li , the total molecular length Lmol, and the core-tail linkage angle b. ~b! Four distinct planar configurations (n5124) of a tail-core-tail mesogen. The closed and open circles indicate the two opposite orientations of the segmental axes xi .