Harmonic Order Parameters for Characterizing Complex Particle Morphologies

Order parameters based on spherical harmonics and Fourier coefficients already play a significant role in condensed matter research in the context of systems of spherical or point particles. Here, we extend these types of order parameter to more complex shapes, such as those encountered in nanoscale self-assembly applications. To do so, we build on a powerful set of techniques that originate in the computer science field of "shape matching." We demonstrate how shape matching techniques can be applied to identify unknown structures and create highly-specialized \textit{ad hoc} order parameters. Additionally, we investigate the special symmetry properties of harmonic descriptors, and demonstrate how they can be exploited to provide optimal solutions to certain classes of problems. Our techniques can be applied to particle systems in general, both simulated and experimental, provided the particle positions are known.

[1]  Bing-Yu Chen,et al.  A web-based three-dimensional protein retrieval system by matching visual similarity , 2005, Bioinform..

[2]  D. Grier,et al.  Methods of Digital Video Microscopy for Colloidal Studies , 1996 .

[3]  Michael Engel,et al.  Self-assembly of monatomic complex crystals and quasicrystals with a double-well interaction potential. , 2007, Physical review letters.

[4]  Sharon C Glotzer,et al.  Local ordering of polymer-tethered nanospheres and nanorods and the stabilization of the double gyroid phase. , 2008, The Journal of chemical physics.

[5]  Lora Mak,et al.  An extension of spherical harmonics to region-based rotationally invariant descriptors for molecular shape description and comparison. , 2008, Journal of molecular graphics & modelling.

[6]  Sharon C Glotzer,et al.  Icosahedral packing of polymer-tethered nanospheres and stabilization of the gyroid phase. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[7]  K Schulten,et al.  VMD: visual molecular dynamics. , 1996, Journal of molecular graphics.

[8]  Takeshi Kawasaki,et al.  Correlation between dynamic heterogeneity and medium-range order in two-dimensional glass-forming liquids. , 2007, Physical review letters.

[9]  Shuisheng Jian,et al.  Photonic band-gap engineering of quasiperiodic photonic crystals , 2005 .

[10]  Forward flux sampling-type schemes for simulating rare events: efficiency analysis. , 2006, The Journal of chemical physics.

[11]  U. Gasser,et al.  Local order in a supercooled colloidal fluid observed by confocal microscopy , 2002 .

[12]  Sharon C. Glotzer,et al.  A precise packing sequence for self-assembled convex structures , 2007, Proceedings of the National Academy of Sciences.

[13]  Andrew Schofield,et al.  Real-Space Imaging of Nucleation and Growth in Colloidal Crystallization , 2001, Science.

[14]  Daisuke Kihara,et al.  Potential for Protein Surface Shape Analysis Using Spherical Harmonics and 3D Zernike Descriptors , 2009, Cell Biochemistry and Biophysics.

[15]  Laxmikant V. Kalé,et al.  Scalable molecular dynamics with NAMD , 2005, J. Comput. Chem..

[16]  David R. Nelson,et al.  Dislocation-mediated melting in two dimensions , 1979 .

[17]  Paul J. Besl,et al.  A Method for Registration of 3-D Shapes , 1992, IEEE Trans. Pattern Anal. Mach. Intell..

[18]  David E. Goldberg,et al.  Genetic Algorithms in Search Optimization and Machine Learning , 1988 .

[19]  J Roth,et al.  Solid-phase structures of the dzugutov pair potential. , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[20]  D. Chandler,et al.  Dynamic pathways for viral capsid assembly. , 2005, Biophysical journal.

[21]  G. D. Bergland,et al.  A guided tour of the fast Fourier transform , 1969, IEEE Spectrum.

[22]  Sharon C. Glotzer,et al.  Self-assembly of anisotropic tethered nanoparticle shape amphiphiles , 2005 .

[23]  Alireza Khotanzad,et al.  Invariant Image Recognition by Zernike Moments , 1990, IEEE Trans. Pattern Anal. Mach. Intell..

[24]  Marcus,et al.  Observations of First-Order Liquid-to-Hexatic and Hexatic-to-Solid Phase Transitions in a Confined Colloid Suspension. , 1996, Physical review letters.

[25]  Goran Ungar,et al.  Giant Supramolecular Liquid Crystal Lattice , 2003, Science.

[26]  Ernst,et al.  Search for a correlation length in a simulation of the glass transition. , 1991, Physical review. B, Condensed matter.

[27]  A. Laio,et al.  Escaping free-energy minima , 2002, Proceedings of the National Academy of Sciences of the United States of America.

[28]  E. Kumacheva,et al.  Properties and emerging applications of self-assembled structures made from inorganic nanoparticles. , 2010, Nature nanotechnology.

[29]  Priya Varadan,et al.  Direct Visualization of Long-Range Heterogeneous Structure in Dense Colloidal Gels , 2003 .

[30]  J. A. López del Val,et al.  Principal Components Analysis , 2018, Applied Univariate, Bivariate, and Multivariate Statistics Using Python.

[31]  Martin J. Mohlenkamp.  A fast transform for spherical harmonics , 1997 .

[32]  C. Dellago,et al.  Transition path sampling and the calculation of rate constants , 1998 .

[33]  Stephanie E. A. Gratton,et al.  Imparting size, shape, and composition control of materials for nanomedicine. , 2006, Chemical Society reviews.

[34]  S. Glotzer,et al.  Anisotropy of building blocks and their assembly into complex structures. , 2007, Nature materials.

[35]  Hajime Tanaka,et al.  Frustration on the way to crystallization in glass , 2006 .

[36]  M. Solomon,et al.  Direct visualization of colloidal rod assembly by confocal microscopy. , 2005, Langmuir : the ACS journal of surfaces and colloids.

[37]  Sharon C Glotzer,et al.  Self-assembly of polymer-tethered nanorods. , 2005, Physical review letters.

[38]  Yasuaki Seki,et al.  Biological materials: Structure and mechanical properties , 2008 .

[39]  Sharon C Glotzer,et al.  How do quasicrystals grow? , 2007, Physical review letters.

[40]  Hans-Peter Kriegel,et al.  3D Shape Histograms for Similarity Search and Classification in Spatial Databases , 1999, SSD.

[41]  Zhiyong Tang,et al.  Simulations and analysis of self-assembly of CdTe nanoparticles into wires and sheets. , 2007, Nano letters.

[42]  M. Solomon,et al.  Stacking fault structure in shear-induced colloidal crystallization. , 2006, The Journal of chemical physics.

[43]  G. Fredrickson,et al.  Phase behavior of a blend of polymer-tethered nanoparticles with diblock copolymers. , 2005, The Journal of chemical physics.

[44]  Arthi Jayaraman,et al.  Structure and assembly of dense solutions and melts of single tethered nanoparticles. , 2008, The Journal of chemical physics.

[45]  Richard J. Farris,et al.  Nanostructured Polyethylene-POSS Copolymers: Control of Crystallization and Aggregation , 2002 .

[46]  Marcin Novotni,et al.  3D zernike descriptors for content based shape retrieval , 2003, SM '03.

[47]  Christopher B. Murray,et al.  Structural diversity in binary nanoparticle superlattices , 2006, Nature.

[48]  Aaron S. Keys,et al.  Self-assembly of patchy particles into diamond structures through molecular mimicry. , 2005, Langmuir : the ACS journal of surfaces and colloids.

[49]  Sharon C Glotzer,et al.  Phase diagrams of self-assembled mono-tethered nanospheres from molecular simulation and comparison to surfactants. , 2005, Langmuir : the ACS journal of surfaces and colloids.

[50]  Harold W. Kuhn,et al.  The Hungarian method for the assignment problem , 1955, 50 Years of Integer Programming.

[51]  Edwin L. Thomas,et al.  Anisotropic Micellar Nanoobjects from Reactive Liquid Crystalline Rod−Coil Diblock Copolymers , 2004 .

[52]  Ravi Radhakrishnan,et al.  Effect of the fluid-wall interaction on freezing of confined fluids: Toward the development of a global phase diagram , 2000 .

[53]  Pieter Rein ten Wolde,et al.  Numerical calculation of the rate of crystal nucleation in a Lennard‐Jones system at moderate undercooling , 1996 .

[54]  Sharon C. Glotzer,et al.  Characterizing Structure Through Shape Matching and Applications to Self Assembly , 2010, ArXiv.

[55]  Vincent M. Rotello,et al.  Self-assembly of nanoparticles into structured spherical and network aggregates , 2000, Nature.

[56]  D Frenkel,et al.  Numerical prediction of absolute crystallization rates in hard-sphere colloids. , 2004, The Journal of chemical physics.

[57]  Peter T. Cummings,et al.  Phase transitions in nanoconfined fluids: The evidence from simulation and theory , 2010 .

[58]  Zhiyong Tang,et al.  Self-Assembly of CdTe Nanocrystals into Free-Floating Sheets , 2006, Science.

[59]  Dzugutov.  Formation of a dodecagonal quasicrystalline phase in a simple monatomic liquid. , 1993, Physical review letters.

[60]  E. Thomas,et al.  Self-Assembly of Block Copolymers for Photonic-Bandgap Materials , 2005 .

[61]  S. Glotzer,et al.  Self-Assembly of Patchy Particles. , 2004, Nano letters.

[62]  Szymon Rusinkiewicz,et al.  Rotation Invariant Spherical Harmonic Representation of 3D Shape Descriptors , 2003, Symposium on Geometry Processing.

[63]  Sharon C. Glotzer,et al.  Disordered, quasicrystalline and crystalline phases of densely packed tetrahedra , 2009, Nature.

[64]  Sharon C. Glotzer,et al.  Phase behavior of ditethered nanospheres , 2009, 0907.5015.

[65]  Sharon C Glotzer,et al.  Complex crystal structures formed by the self-assembly of ditethered nanospheres. , 2009, Nano letters.

[66]  Dietmar Saupe,et al.  Tools for 3D-object retrieval: Karhunen-Loeve transform and spherical harmonics , 2001, 2001 IEEE Fourth Workshop on Multimedia Signal Processing (Cat. No.01TH8564).

[67]  Ralph Roskies,et al.  Fourier Descriptors for Plane Closed Curves , 1972, IEEE Transactions on Computers.

[68]  Sharon C. Glotzer,et al.  Tethered Nano Building Blocks: Toward a Conceptual Framework for Nanoparticle Self-Assembly , 2003 .

[69]  R. Larson,et al.  Local stress control of spatiotemporal ordering of colloidal crystals in complex flows. , 2008, Physical review letters.