Self Assembled Clusters of Spheres Related to Spherical Codes

We consider the thermodynamically driven self-assembly of spheres onto the surface of a central sphere. This assembly process forms self-limiting, or terminal, anisotropic clusters (N-clusters) with well defined structures. We use Brownian dynamics to model the assembly of N-clusters varying in size from two to twelve outer spheres, and free energy calculations to predict the expected cluster sizes and shapes as a function of temperature and inner particle diameter. We show that the arrangements of outer spheres at finite temperatures are related to spherical codes, an ideal mathematical sequence of points corresponding to densest possible sphere packings. We demonstrate that temperature and the ratio of the diameters of the inner and outer spheres dictate cluster morphology and dynamics. We find that some N-clusters exhibit collective particle rearrangements, and these collective modes are unique to a given cluster size N. We present a surprising result for the equilibrium structure of a 5-cluster, which prefers an asymmetric square pyramid arrangement over a more symmetric arrangement. Our results suggest a promising way to assemble anisotropic building blocks from constituent colloidal spheres.Comment: 15 pages, 10 figure

Stable Frank-Kasper phases of self-assembled, soft matter spheres

Single molecular species can self-assemble into Frank Kasper (FK) phases, finite approximants of dodecagonal quasicrystals, defying intuitive notions that thermodynamic ground states are maximally symmetric. FK phases are speculated to emerge as the minimal-distortional packings of space-filling spherical domains, but a precise quantitation of this distortion and how it affects assembly thermodynamics remains ambiguous. We use two complementary approaches to demonstrate that the principles driving FK lattice formation in diblock copolymers emerge directly from the strong-stretching theory of spherical domains, in which minimal inter-block area competes with minimal stretching of space-filling chains. The relative stability of FK lattices is studied first using a diblock foam model with unconstrained particle volumes and shapes, which correctly predicts not only the equilibrium {\sigma} lattice, but also the unequal volumes of the equilibrium domains. We then provide a molecular interpretation for these results via self-consistent field theory, illuminating how molecular stiffness regulates the coupling between intra-domain chain configurations and the asymmetry of local packing. These findings shed new light on the role of volume exchange on the formation of distinct FK phases in copolymers, and suggest a paradigm for formation of FK phases in soft matter systems in which unequal domain volumes are selected by the thermodynamic competition between distinct measures of shape asymmetry.Comment: 40 pages, 22 figure

Improved Orientation Sampling for Indexing Diffraction Patterns of Polycrystalline Materials

Orientation mapping is a widely used technique for revealing the microstructure of a polycrystalline sample. The crystalline orientation at each point in the sample is determined by analysis of the diffraction pattern, a process known as pattern indexing. A recent development in pattern indexing is the use of a brute-force approach, whereby diffraction patterns are simulated for a large number of crystalline orientations, and compared against the experimentally observed diffraction pattern in order to determine the most likely orientation. Whilst this method can robust identify orientations in the presence of noise, it has very high computational requirements. In this article, the computational burden is reduced by developing a method for nearly-optimal sampling of orientations. By using the quaternion representation of orientations, it is shown that the optimal sampling problem is equivalent to that of optimally distributing points on a four-dimensional sphere. In doing so, the number of orientation samples needed to achieve a indexing desired accuracy is significantly reduced. Orientation sets at a range of sizes are generated in this way for all Laue groups, and are made available online for easy use.Comment: 11 pages, 7 figure

Simplicial Lattice Study of the 2d Ising CFT

I derive a formulation of the 2-dimensional critical Ising model on non-uniform simplicial lattices. Surprisingly, the derivation leads to a set of geometric constraints that a lattice must satisfy in order for the model to have a well-defined continuum limit. I perform Monte Carlo simulations of the critical Ising model on discretizations of several non-trivial manifolds including a twisted torus and a 2-sphere and I show that the simulations are in agreement with the 2d Ising CFT in the continuum limit. I discuss the inherent benefits of using non-uniform simplicial lattices to study quantum field theory and how the methods developed here can potentially be generalized for use with other theories.Comment: PhD Thesis, Boston University 2023, 61 page

Local Structure in Hard Particle Self-Assembly and Assembly Failure

The relationship between local order and global structure is not often a straightforward one in systems on the nano- and microscale in which interactions are usually weak and thermal fluctuations drive self-assembly. Moreover, structure in systems for which particle symmetry is broken is difficult to describe theoretically on any level higher than a pairwise one, due to the prohibitively high-dimensional nature of the relevant configuration space. However, a thorough understanding of local structure in all phases of soft matter systems is necessary to gain a complete picture of the physics of these systems and to leverage them for technological and materials science applications. In this dissertation, I investigate local structure in systems of anisotropic particles mediated exclusively by entropy maximization. Specifically, I explore the role of local structure in crystallization and its failure by tackling two related lines of inquiry. First, I study the interplay between particle shape and spherical confinement in systems of hard polyhedral particles, to examine locally dense clusters of anisotropic particles and their possible connection to preferred local structures during unconfined self-assembly. I use Monte Carlo simulation methods to find putative densest clusters of the Platonic solids in spherical confinement, for up to N = 60 constituent particles. I find that a spherical boundary suppresses the packing influence of particle shape and produces a robust class of common cluster structures. I also find a range of especially dense clusters at so-called "magic numbers" of constituent particles, and discover that a magic-number cluster of tetrahedra is a prominent motif in the self-assembled structure of tetrahedra, the dodecagonal quasicrystal. Second, I explore the influence of local structure in systems of hard polyhedral particles that fail to crystallize. I use a shape landscape, or a two-dimensional space of particles that are continuously interrelated by a set of shape perturbations, to investigate why slight changes to particle shape sometimes result in the vitrification rather than crystallization of dense monatomic systems of these particles. I show that assembly failure in these systems arises from a multiplicity of competing local structures, each of which is prevalent in ordered phases crystallized by particles that are only slightly different in shape. Thus, systems that fail to assemble do so because they cannot crystallize into any one ordered phase. Third, I demonstrate that fragility in these systems, a technologically relevant measure of glass-forming ability, can be tuned by slight changes to particle shape. I relate this finding to simulations of molecular systems in which fragility is linked to intermolecular bond angle. Finally, I detail the methods and applications of software I developed to detect multi-particle local structure in real space. This software is open-source and in current use, and has already been utilized for local structure detection in several papers by myself and others. I conclude this dissertation by providing an outlook on the implications and future directions of my work.PHDApplied PhysicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/147721/1/erteich_1.pd

Self Assembly Problems of Anisotropic Particles in Soft Matter.

Anisotropic building blocks assembled from colloidal particles are attractive building blocks for self-assembled materials because their complex interactions can be exploited to drive self-assembly. In this dissertation we address the self-assembly of anisotropic particles from multiple novel computational and mathematical angles. First, we accelerate algorithms for modeling systems of anisotropic particles via massively parallel GPUs. We provide a scheme for generating statistically robust pseudo-random numbers that enables GPU acceleration of Brownian and dissipative particle dynamics. We also show how rigid body integration can be accelerated on a GPU. Integrating these two algorithms into a GPU-accelerated molecular dynamics code (HOOMD-blue), make a single GPU the ideal computing environment for modeling the self-assembly of anisotropic nanoparticles. Second, we introduce a new mathematical optimization problem, filling, a hybrid of the familiar shape packing and covering problem, which can be used to model shaped particles. We study the rich mathematical structures of the solution space and provide computational methods for finding optimal solutions for polygons and convex polyhedra. We present a sequence of isosymmetric optimal filling solutions for the Platonic solids. We then consider the filling of a hyper-cone in dimensions two to eight and show the solution remains scale-invariant but dependent on dimension. Third, we study the impact of size variation, polydispersity, on the self-assembly of an anisotropic particle, the polymer-tethered nanosphere, into ordered phases. We show that the local nanoparticle packing motif, icosahedral or crystalline, determines the impact of polydispersity on energy of the system and phase transitions. We show how extensions of the Voronoi tessellation can be calculated and applied to characterize such micro-segregated phases. By applying a Voronoi tessellation, we show that properties of the individual domains can be studied as a function of system properties such as temperature and concentration. Last, we consider the thermodynamically driven self-assembly of terminal clusters of particles. We predict that clusters related to spherical codes, a mathematical sequence of points, can be synthesized via self-assembly. These anisotropic clusters can be tuned to different anisotropies via the ratio of sphere diameters and temperature. The method suggests a rich new way for assembling anisotropic building blocks.Ph.D.Applied Physics and Scientific ComputingUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/91576/1/phillicl_1.pd

A Square Equal-area Map Projection

A novel square equal-area map projection is proposed. The projection combines closed-form forward and inverse solutions with relatively low angular distortion and minimal cusps, a combination of properties not manifested by any previously published square equal-area projection. Thus, the new projection has lower angular distortion than any previously published square equal-area projection with a closed-form solution. Utilizing a quincuncial arrangement, the new projection places the north pole at the center of the square and divides the south pole between its four corners; the projection can be seamlessly tiled. The existence of closed-form solutions makes the projection suitable for real-time visualization applications, both in cartography and in other areas, such as for the display of panoramic images.Comment: 15 pages, 5 figures, 1 tabl

Towards Understanding the Self-assembly of Complicated Particles via Computation.

We develop advanced Monte Carlo sampling schemes and new methods of calculating thermodynamic partition functions that are used to study the self-assembly of complicated ``patchy '' particles. Patchy particles are characterized by their strong anisotropic interactions, which can cause critical slowing down in Monte Carlo simulations of their self-assembly. We prove that detailed balance is maintained for our implementation of Monte Carlo cluster moves that ameliorate critical slowing down and use these simulations to predict the structures self-assembled by patchy tetrominoes. We compare structures predicted from our simulations with those generated by an alternative learning-augmented Monte Carlo approach and show that the learning-augmented approach fails to sample thermodynamic ensembles. We prove one way to maintain detailed balance when parallelizing Monte Carlo using the checkerboard domain decomposition scheme by enumerating the state-to-state transitions for a simple model with general applicability. Our implementation of checkerboard Monte Carlo on graphics processing units enables accelerated sampling of thermodynamic properties and we use it to confirm the fluid-hexatic transition observed at high packing fractions of hard disks. We develop a new method, bottom-up building block assembly, which generates partition functions hierarchically. Bottom-up building block assembly provides a means to answer the question of which structures are favored at a given temperature and allows accelerated prediction of potential energy minimizing structures, which are difficult to determine with Monte Carlo methods. We show how the sequences of clusters generated by bottom-up building block assembly can be used to inform ``assembly pathway engineering'', the design of patchy particles whose assembly propensity is optimized for a target structure. The utility of bottom-up building block assembly is demonstrated for systems of CdTe/CdS tetrahedra, DNA-tethered nanospheres, colloidal analogues of patchy tetrominoes and shape-shifting particles.Ph.D.Chemical EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/91509/1/erjank_1.pd