Recent breakthroughs in nanotechnology have made it possible to fabricate a rich array of nonspherical particles with arbitrary shape. As demonstrated by numerous studies, anisometry can lead to a variety of translationally- and orientationally-ordered structures. Characterization and quantification of order is therefore crucial for studies of assembled phases of such particles. Although there are well-known measures for characterizing translational order, such as the radial distribution function, structure factor and bond order parameters, no general framework exists for characterizing orientational order especially when formed by geometrically-symmetric objects. (An object is geometrically-symmetric if it is invariant under at least one non-identity orthogonal transformation.) Although group theory provides certain order parameters based on the underlying symmetry group of the mesophases [1, 2], there are issues with this representation: geometrically-distinct instances of the mesophase might correspond to identical values of the tensor order parameter, no statistical estimator is provided for relating the orientations of the particles in the system to bulk orientational order- especially when the symmetry of the particles is different from the symmetry of the phase that they form, and no scheme exists for reproducing the relevant directors from the measured/calculated order parameter except in the case of nematics.

There exist well-known representations for the orientation of a rigid particle, such as polar angles in two dimensions and Euler angles in three dimensions. However, such representations are degenerate for symmetric particles because different values of orientation measures might correspond to identical orientations. We have developed a bijective, non-degenerate description of the orientation of a symmetric particle based on identifying the set of its geometrically-equivalent vectors [3,4] and identifying a proper orientational coordinate from within a finite list of symmetric tensors constructed from these equivalent vectors. We find an upper bound for the rank of such a tensor that depends on the cardinality and simple algebraic properties of the set. A lower bound can be determined by solving an optimization problem and depends on the symmetry. We further propose a general scheme for extracting a scalar order parameter and relevant directors of orientationally-ordered phases formed by systems of symmetric particles. This scheme successfully reproduces the well-known nematic order parameters and we have used it to derive new order parameters for arrangements of regular polyhedra, which we test using computer-generated data. An important merit of our approach is that its derivation is simpler and more intuitive than the classical group-theory derivation of orientational order parameters and instead relies on basic linear algebra and optimization theory.

[1] Fel L. G., Phys. Rev. E, 52, 702 (1995) and references therein

[2] Lubensky T. C., Radzihovsky L., Phys. Rev. E 66, 031704 (2002)

[3] Shelly M., Glotzer S. C., and Palffy-Muhoray P, ‘Generalized Order Parameters for Systems of Orientationally Ordered Anisometric Particles', 2005 APS Meeting, Los Angeles CA

[4] Palffy, P.M. and Zheng, X, unpublished.