Polyhedral Symmetries Research Articles

Symmetry-group-protected microfluidics

Modern micromanipulation techniques typically involve trapping using electromagnetic, acoustic, or flow fields that produce stresses on the trapped particles thereby precluding stress-free manipulations. Here, we show that by employing polyhedral symmetries in a multichannel microfluidic design, we can separate the tasks of displacing and trapping a particle into two distinct sets of flow operations, each characterized and protected by their unique groups of symmetries. By combining only the displacing uniform flow modes to entrain and move targeted particles in arbitrary directions, we were able to realize symmetry-protected, stress-free micromanipulation in 3D. Furthermore, we engineered complex, microscale paths by programming and controlling the flow within each channel in real time, resulting in multiple particles simultaneously following desired paths in the absence of any supervision or feedback. Our work therefore provides a general symmetry-group-based framework for understanding and engineering microfluidics and a novel platform for 3D stress-free manipulations. Published by the American Physical Society 2024

Symmetry Measure of Truss Structures with Disturbed Higher-order Symmetries

This paper covers aspects of quantifying the symmetry of two- and three-dimensional elastic bar-and-joint structures. The concept of symmetry as a quantitative property instead of a binary question of 'yes' or 'no' is widely accepted and thoroughly investigated, for example, in molecular physics but also in engineering sciences, mainly in chemical engineering. Similarly to most of the articles written on this topic, our method is also based on the comparison of specific metrics of the analyzed structure and a reference one, i.e., which possesses the desired (perfect) symmetry. The deviation of the analyzed (imperfect) structure from the reference structure is quantified by one scalar. The novelty in our approach is that we consider not just the relative position of the nodes but also the normal stiffness of the truss members, even for structures with higher-order, i.e., polyhedral symmetries. For both geometric and material properties to be accounted for, the eigenvalues of the stiffness matrix were chosen as metrics. The difficulty lies in finding the reference structure which will be carried out based on energy principles.

Quantifying microstructural evolution via time-dependent reduced-dimension metrics based on hierarchical n-point polytope functions.

We devise reduced-dimension metrics for effectively measuring the distance between two points (i.e., microstructures) in the microstructure space and quantifying the pathway associated with microstructural evolution, based on a recently introduced set of hierarchical n-point polytope functions P_{n}. The P_{n} functions provide the probability of finding particular n-point configurations associated with regular n polytopes in the material system, and are a special subset of the standard n-point correlation functions S_{n} that effectively decompose the structural features in the system into regular polyhedral basis with different symmetries. The nth order metric Ω_{n} is defined as the L_{1} norm associated with the P_{n} functions of two distinct microstructures. By choosing a reference initial state (i.e., a microstructure associated with t_{0}=0), the Ω_{n}(t) metrics quantify the evolution of distinct polyhedral symmetries and can in principle capture emerging polyhedral symmetries that are not apparent in the initial state. To demonstrate their utility, we apply the Ω_{n} metrics to a two-dimensional binary system undergoing spinodal decomposition to extract the phase separation dynamics via the temporal scaling behavior of the corresponding Ω_{n}(t), which reveals mechanisms governing the evolution. Moreover, we employ Ω_{n}(t) to analyze pattern evolution during vapor deposition of phase-separating alloy films with different surface contact angles, which exhibit rich evolution dynamics including both unstable and oscillating patterns. The Ω_{n} metrics have potential applications in establishing quantitative processing-structure-property relationships, as well as real-time processing control and optimization of complex heterogeneous material systems.

General liquid-crystal theory for anisotropically shaped molecules: Symmetry, orientational order parameters, and system free energy.

A general theory of liquid crystals is presented, starting from the group-theory symmetry analysis of the constituting molecules. A particular attention is paid to the type of elastic free-energies and their relationships with the molecular symmetries. The orientational order-parameter tensors are identified for each molecular symmetry, in a consideration of consistently keeping the leading, characteristic elastic free energies in a model. The order parameters are expressed in terms of symmetric traceless tensors, some of high orders, for all major molecular symmetries, including seven groups of axial symmetries and seven groups of polyhedral symmetries. For spatially inhomogeneous liquid crystals, the couplings of these tensors in the elastic energies are derived by expanding the interaction energies between these molecules. The aim is to provide a general view of the molecular symmetries of individual molecules, orientational order parameters characterizing the orientational distribution functions, and the elastic free energies, all under one single group-theory approach.

Classifying Local Anisotropy Formed by Rigid Molecules: Symmetries and Tensors

We consider an infinitesimal volume where there are many rigid molecules of the same kind and discuss the description and classification of the local anisotropy in this volume by tensors. First, we examine the symmetry of a rigid molecule, which is described by a point group in $SO(3)$. For each point group in $SO(3)$, we find the tensors invariant under the rotations in the group. These tensors shall be symmetric and traceless. We write down the explicit expressions. The order parameters to describe the local anisotropy are then chosen as some of the invariant tensors averaged about the density function. Next, we discuss the classification of local anisotropy by the symmetry of the whole infinitesimal volume. This mesoscopic symmetry can be recognized by the value of the order parameter tensors in the sense of maximum entropy state. For some sets of order parameter tensors involving different molecular symmetries, we give the classification of mesoscopic symmetries, in which the threefold, fourfold, and polyhedral symmetries are examined.

Generic first-order phase transitions between isotropic and orientational phases with polyhedral symmetries.

Polyhedral nematics are examples of exotic orientational phases that possess a complex internal symmetry, representing highly nontrivial ways of rotational symmetry breaking, and are subject to current experimental pursuits in colloidal and molecular systems. The classification of these phases has been known for a long time; however, their transitions to the disordered isotropic liquid phase remain largely unexplored, except for a few symmetries. In this work, we utilize a recently introduced non-Abelian gauge theory to explore the nature of the underlying nematic-isotropic transition for all three-dimensional polyhedral nematics. The gauge theory can readily be applied to nematic phases with an arbitrary point-group symmetry, including those where traditional Landau methods and the associated lattice models may become too involved to implement owing to a prohibitive order-parameter tensor of high rank or (the absence of) mirror symmetries. By means of exhaustive Monte Carlo simulations, we find that the nematic-isotropic transition is generically first-order for all polyhedral symmetries. Moreover, we show that this universal result is fully consistent with our expectation from a renormalization group approach, as well as with other lattice models for symmetries already studied in the literature. We argue that extreme fine tuning is required to promote those transitions to second-order ones. We also comment on the nature of phase transitions breaking the O(3) symmetry in general cases.

Dimpled Polyhedral Colloids Formed by Colloidal Crystal Templating.

Many approaches to the self-assembly of interesting microstructures rely on particles with engineered shapes. We create dimpled solid particles by molding oil droplets in the interstices of a close-packed colloidal crystal and polymerizing them in situ, resulting in particles containing multiple spherical dimples arranged with various polyhedral symmetries. Monodisperse micrometer-sized droplets of 3-methacryloxypropyltrimethoxysilane (TPM) are mixed with an excess of polystyrene (PS) microspheres (2.58 μm) and allowed to sediment, forming colloidal crystals with TPM droplets inside their interstitial sites. When these crystals are compressed by partial drying, the trapped droplets wet the multiple microspheres surrounding them, forming a three-dimensional capillary bridge with the symmetry of the interstitial spaces, and then mild heating triggers polymerization in situ. Depending on the initial particle volume fraction and extent of drying, a high yield of dimpled particles having different symmetries including tetrahedra and cubes is obtained. The fractional yield of different shapes varies with the size ratio of the TPM droplets and the PS microspheres forming the template lattice. Sedimentation velocity fractionation methods are effective for enrichment of specific symmetries but not complete purification. Our approach for forming polyhedral particles should be readily scalable to larger samples and smaller sized particles if desired.

Platonic solids generate their four-dimensional analogues

This paper shows how regular convex 4-polytopes - the analogues of the Platonic solids in four dimensions - can be constructed from three-dimensional considerations concerning the Platonic solids alone. Via the Cartan-Dieudonné theorem, the reflective symmetries of the Platonic solids generate rotations. In a Clifford algebra framework, the space of spinors generating such three-dimensional rotations has a natural four-dimensional Euclidean structure. The spinors arising from the Platonic solids can thus in turn be interpreted as vertices in four-dimensional space, giving a simple construction of the four-dimensional polytopes 16-cell, 24-cell, the F4 root system and the 600-cell. In particular, these polytopes have `mysterious' symmetries, that are almost trivial when seen from the three-dimensional spinorial point of view. In fact, all these induced polytopes are also known to be root systems and thus generate rank-4 Coxeter groups, which can be shown to be a general property of the spinor construction. These considerations thus also apply to other root systems such as A(1)\oplus I(2)(n) which induces I(2)(n)\oplus I(2)(n), explaining the existence of the grand antiprism and the snub 24-cell, as well as their symmetries. These results are discussed in the wider mathematical context of Arnold's trinities and the McKay correspondence. These results are thus a novel link between the geometries of three and four dimensions, with interesting potential applications on both sides of the correspondence, to real three-dimensional systems with polyhedral symmetries such as (quasi)crystals and viruses, as well as four-dimensional geometries arising for instance in Grand Unified Theories and string and M-theory.

Continuous symmetry analyses: Cnv and Dn measures of molecules, complexes, and proteins

The continuous symmetry methodology has been developed to provide a quantitative estimation for the degree of a selected symmetry point-group in a molecular structure. Previous developed measures included the C(S), C(n), S(n) (including C(i)) symmetries, several polyhedral symmetries, and the related chirality measure. Motivated by the abundance and importance of C(nv) symmetry in many key molecules and by the abundance of D(n) symmetry in many organometallic complexes and in many protein clusters, measures for these symmetries, based on the general definition of the continuous symmetry measure, were developed. Detailed description of the derivation of the measures, along with plenty of examples for their application, is provided.

Exploiting polyhedral symmetries in social choice

A large amount of literature in social choice theory deals with quantifying the probability of certain election outcomes. One way of computing the probability of a specific voting situation under the Impartial Anonymous Culture assumption is via counting integral points in polyhedra. Here, Ehrhart theory can help, but unfortunately the dimension and complexity of the involved polyhedra grows rapidly with the number of candidates. However, if we exploit available polyhedral symmetries, some computations become possible that previously were infeasible. We show this in three well known examples: Condorcet’s paradox, Condorcet efficiency of plurality voting and in Plurality voting vs Plurality Runoff.

Continuous Symmetry Measures. VI. The Relations Between Polyhedral Point-Group/Subgroup Symmetries

We extend our analysis of the symmetry content of the classical polyhedra [1] to the analysis of the degree of polyhedral subgroup symmetries. The quantitative levels of the hierarchical polyhedral symmetries series of Oh, D4h and D2h of hexacoordinated structures, as well as the relations between them, serve as an example. A distinction is made between two types of measures: quantitative evaluation of the degree of symmetry, and quantitative evaluation of the degree of content of a reference shape.

Thermal expansion of alumnus perovskite ScAlO<sub>3</sub>

Thermal structure deformation of ScAlO3 perovskite was studied by in situ X-ray single crystal structure analysis at high temperatures. As temperature increases, all structure parameters of the orthorhombic perovskite shift very slightly toward the ideal polyhedral symmetries. The cell volume shows a quadratic thermal expansion expressed by V(T)/V293=1+1.151×10−5T+9.218×10−9T2. The expansions of ScO8 and void space are significant for the cell volume expansion because they occupy large volume in the unit cell. The tilting and rotation of AlO6 octahedral linkage are a little changed by the anisotropic thermal motion of O1 and O2. Higher order anisotropic anharmonic thermal parameters of γpqr and δpqrs of O1 and O2 atoms are increased noticeably with temperature. Anharmonic motions of oxygen atoms, inducing the tilting and rotation of AlO6 octahedra mainly control the thermal changes of perovskite structure.

Ensembles of Singularities Generated by Surfaces with Polyhedral Symmetry

The role of the symmetries in the topology of sets of Lagrangian singularities is studied in a simple physical model: the envelope of the rays emanating from a convex wave front invariant under the action of polyhedral groups. New point singularities are found of integer index and located at the vertices of the polyhedron or of its reciprocal. This remarkable layout results from the interplay between the symmetries of the singularities, the polyhedral symmetries, and the topology of the wave front. An application to fine-particle magnetic systems is given.

Symbolic symmetry verification for harmonic functions invariant under polyhedral symmetries

Methods for using symbolic computation systems to verify polyhedral symmetries of harmonic functions on the sphere are presented. The particular case of icosahedral symmetry, which is important in the structure of small spherical viruses, fullerenes, and quasi‐crystals, is examined in detail. Previous work by the authors, in which general expressions for polyhedral harmonics in terms of spherical harmonics were derived, is verified by checking the symmetry of all icosahedral harmonics up to order 44. © 1995 American Institute of Physics.

An embedded atom method study of the equilibrium shapes of small platinum and palladium clusters

The embedded atom method (EAM) was used to determine the equilibrium shapes of small platinum and palladium clusters (N=5–60 atoms). The stability of various polyhedral symmetries was performed at 0K. A search for the presence of any other lower energy structures was also carried out. Pt clusters reconstructed to lower energy clusters even at the magic numbers. Pd also reconstructed to lower energy structures except at the magic number sizes where the icosahedron was found to be the most stable.

An investigation of the shape of small platinum clusters using the embedded atom method

The equilibrium shape and structure of small platinum clusters with 5–60 atoms was investigated using the embedded atom method (EAM). An analysis of the stability of the clusters with polyhedral symmetries indicated that, except at the magic number sizes, the clusters spontaneously reconstructed to more stable shapes at 0 K. Even the magic number icosahedra and cubo-octahedra were unstable upon annealing at 500 K. The clusters which reconstruct to lower energy structures appear to be highly asymmetric and irregular. Many different structures were found to have similar energies. The annealed clusters also display surface features not found in bulk platinum.

Topological aspects of chemically significant polyhedra

This paper considers both static and dynamic properties of chemically significant polyhedra. Static properties of polyhedra consider relationships between the numbers and degrees/sizes of polyhedral vertices, edges, and faces; polyhedral symmetries; and numbers of topologically distinct polyhedra of various types. Dynamic properties of polyhedra involve studies of polyhedral isomerizations from both macroscopic and microscopic points of view. Macroscopic aspects of polyhedral isomerization can be described by graphs called topological representations in which the vertices correspond to different permutational isomers and the edges to single degenerate polyhedral isomerization steps. Such topological representations are presented for isomerizations of polyhedra having five, six, and eight vertices. Microscopic aspects of polyhedral isomerizations arise from consideration of the details of polyhedral topology, such as the topological aspects of diamond-square-diamond processes. In this connection, Gale diagrams are useful for describing isomerizations of five- and six-vertex polyhedra, including the Berry pseudorotation of a trigonal bipyramid through a square pyramid intermediate and the Bailar or Ray and Dutt twists of an octahedron through a trigonal prism intermediate.