Abstract
We report results obtained from Monte Carlo simulations investigating mesophase formation in two model systems of hard pear-shaped particles. The first model considered is a hard variant of the truncated Stone-Expansion model previously shown to form nematic and smectic mesophases when embedded within a 12-6 Gay-Berne-like potential [R. Berardi, M. Ricci and Z. Zannoni. ChemPhysChem, 7:443, 2001]. When stripped of its attractive interactions, however, this system is found to lose its liquid crystalline phases. For particles of length to breadth ratio k = 3, glassy behaviour is seen at high pressures, whereas for k = 5 several bi-layer-like domains are seen, with high intradomain order but little interdomain orientational correlation. For the second model, which uses a parametric shape parameter based on the generalised Gay-Berne formalism, results are presented for particles with elongation k = 3, 4 and 5. Here, the systems with k = 3 and 4 fail to display orientationally ordered phases, but that with k = 5 shows isotropic, nematic and, unusually for a hard-particle model, interdigitated smectic A2 phases.
Computer simulations of hard pear-shaped particles (786.6 KiB)
Introduction
In recent years, flexoelectricity has become an increasingly important feature in the design of materials for use in liquid crystal devices. Flexoelectric behaviour, which leads to field-induced director distortion, results, at a molecular level, from competition between electric and steric dipolar interactions. As well as leading to modified bulk properties, flexoelectricity has been mooted to be a possible driver for switching in devices with bistable anchoring surfaces [1]. Indeed, it has been suggested that the switching mechanism of the zenithally bistable device [2] may rely, in part, on flexoelectric behaviour.
The early studies of Meyer [3] and Prost and Marcerou [4], showed that the mechanisms underlying flexoelectricity can be understood in two ways. In the original explanation from Meyer, flexoelectric behavior was explained in terms of particles with a strong anisotropy in their charge repartition. Thus, it was shown that, upon polarization by an applied field, pear-shaped particles exhibit a splay director distortion, whereas banana-shaped particles exhibit a bend distortion. Subsequently, Prost and Marcerou showed that flexoelectricity could also be obtained using particles with a non-zero quadrupole moment. This did not contradict Meyer’s original work, however, since in reality flexoelectric mesogens are known to possess either one or both of these properties [5].
Although well studied theoretically [6–8], few computer simulations using flexoelectric particles have been performed to date. Whilst particle based simulations showing ferroelectric behaviour are reasonably well established (see, e.g., [9, 10]), models with the dipolar and/or quadrupolar symmetry steric interactions needed for flexoelectric behaviour are relatively scarce.
Neal and co-workers performed one such study using molecules represented by rigid assemblies of three Gay-Berne sites [11]. One of the assemblies considered in Ref. [11] was a triangular arrangement of mutually parallel Gay-Berne sites, leading, overall, to approximately pear-shaped molecules. On compression, a system of such molecules ordered from an isotropic liquid to a smectic arrangement in which the molecular orientations in successive layers were almost perfectly anti-parallel. Subsequently, Stelzer et al. [12] investigated the behaviour of pear-shaped molecules using a model with two interaction sites per particle; each particle comprised a Lennard-Jones site embedded near to one end of a Gay-Berne site. Isotropic, nematic and smectic phases were observed, local antiparallel alignment being seen in the nematic phase. Measurements of the splay and bend flexoelectric coefficients gave a non-zero splay coefficient and, to within error estimates, a zero bend coefficient in accordance with Meyer’s theory. Equivalent simulations by Billeter and Pelcovits [13], using qualitatively the same model but with different energy parametrisations and an alternative method for the calculation of the flexoelectric coefficients, confirmed the results of Ref. [12]. In this case, however, no stable nematic phase was found between the isotropic and (locally antiparallel) smectic A phases.
Whilst the results from these systems proved encouraging, their reliance on multi-site generic potentials remained a relative inefficiency. This was resolved somewhat in recent work by Berardi and some of the current authors [14], in which a single-site model was developed, using Zewdie’s generalisation approach [15, 16], to represent tapered or pear-shaped particles. Here, using the geometrical shape of a Bézier curve as a template for the particle shape, a numerically calculated mesh of contact distance values was fitted using a truncated Stone expansion which, in turn, was employed in the simulations themselves. Results from this study were very encouraging, as both nematic and smectic A phases were found, and, through appropriate manipulation of the well-depth anisotropy terms, equivalent phases with net polar order were generated.
In this paper, we seek to explore the fundamental properties of single-site pear-shaped models such as that used in Ref [14], by investigating mesophase formation in systems of hard, noncentrosymmetric particles. Hard particle simulations have proved to be an effective and efficient testbed for many of the theories of liquid crystal physics [17], and have confirmed that shape anisotropy alone can be sufficient for the onset of nematic and even smectic order. Two distinct systems are described here. The first is a hard version of the truncated Stone expansion potential described in [14]. The second employs a novel approach, based on a parametric variant of the generalized Gay-Berne shape parameter [18], which yields an analytical expression for the contact distance between two pear-shaped objects.
The content of the remainder of this paper is arranged as follows. In subsection A we give a brief description of the truncated Stone expansion potential before presenting and discussing results obtained from Monte Carlo simulations of same. In the following subsection, we introduce the parametric approach for generating shape parameters for non-ellipsoidal particles, and apply it to generate shape parameters for the Bézier pears considered in Ref. [14]. Results obtained from Monte Carlo simulations of such systems are presented in subsection 3. Finally, the two sets of simulations are compared and discussed in Section III.
Discussion and Conclusion
In this paper, we have investigated the mesogenic behaviour of two classes of model hard pear-shaped particles, both based on a target shape built using a Bézier curve. The first model considered used a truncated Stone expansion approach to generate the particle-particle contact distance numerically. Although the Gay-Berne version of this model was well behaved, giving nematic and smectic A mesophases [14], these were not found on removal of the attractive interactions. Rather, the non-convex regions of the contact surfaces induced the particles to interlock, leading to the formation of multi-domain and glassy phases. For this model, therefore, it appears that the nematic-isotropic transition is not driven by particle shape alone: long-ranged orientational order is only seen when the shape is softened somewhat, by the incorporation of attractive interactions.
The second hard-pear model considered here was based on the PHGO approach, a route to non-centrosymmetric shape parameters which we have introduced in this paper. While the PHGO shape parameter is not determined from a full evaluation of the appropriate gaussian integral, the approximation it makes, that locally a non-centrosymmetric particle closely resembles an appropriately chosen ellipsoid, is intuitively reasonable. Furthermore, the computational simplicity and ready transferability of the PHGO model suggest that it may be of considerable utility in the generic modelling of self assembling systems. Here, we have found that the smooth, convex contact surfaces of a PHGO hard pear model yield stable nematic and bilayered smectic A2 phases. Interestingly, these phases are only seen when the particle aspect ratio is increased to k = 5, whereas hard ellipsoid systems are know to form a nematic with k values as low as 2.75 [30]. Future work exploring the behaviour of the PHGO hard pear model will include a more thorough study of its flexoelectric properties, and an investigation into the applicability of the PHGO shape parameter in theoretical approaches commonly used to study liquid crystals.
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