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Motile liquid crystal (LC) colloids show peculiar behavior due to the high sensitivity to external stimuli driven by the LC elastic and surface effects. However, few studies focus on harnessing the LC phase transitions to propel colloidal inclusions by the nematic-isotropic (NI) interface. We engineer a quasi-2D active system consisting of solid micron-sized light-absorbent platelets immersed in a thermotropic nematic LC. The platelets self-propel in the presence of light while self-inducing a localized NI phase transition. The sample's temperature, light intensity, and confinement determine three different regimes: a 2D large regime where the platelet-isotropic phase bubble is static and the NI interface remains stable; a compact motile-2D regime where the NI interface lies closer to the platelet's contour; and a motile-3D-confinement regime characterized by the emergence of multipolar configurations of the LC. We perform continuum-theory simulations that predict stationary platelet-LC states when confined in 3D. Our study in an intrinsically far-from-equilibrium landscape is crucial for designing simple synthetic systems that contribute to our understanding of harnessing liquid crystals' phase transitions to propel colloidal inclusions and trigger tunable topological reconfigurations leading to photonic responses., Comment: 29 pages main text. 5 figures main text. Supporting Information included at the end

Accurate interaction potentials between microscopic components such as colloidal particles or cells are crucial to understanding a range of processes, including colloidal crystallization, bacterial colony formation, and cancer metastasis. Even in systems where the precise interaction mechanisms are unknown, effective interactions can be measured to inform simulation and design. However, these measurements are difficult and time-intensive, and often require conditions that are drastically different from in situ conditions of the system of interest. Moreover, existing methods of measuring interparticle potentials rely on constraining a small number of particles at equilibrium, placing limits on which interactions can be measured. We introduce a method for inferring interaction potentials directly from trajectory data of interacting particles. We explicitly solve the equations of motion to find a form of the potential that maximizes the probability of observing a known trajectory. Our method is valid for systems both in and out of equilibrium, is well-suited to large numbers of particles interacting in typical system conditions, and does not assume a functional form of the interaction potential. We apply our method to infer the interactions of colloidal spheres from experimental data, successfully extracting the range and strength of a depletion interaction from the motion of the particles.

Conical surfaces pose an interesting challenge to crystal growth: a crystal growing on a cone can wrap around and meet itself at different radii. We use a disk-packing algorithm to investigate how this closure constraint can geometrically frustrate the growth of single crystals on cones with small opening angles. By varying the crystal seed orientation and cone angle, we find that -- except at special commensurate cone angles -- crystals typically form a seam that runs along the axial direction of the cone, while near the tip, a disordered particle packing forms. We show that the onset of disorder results from a finite-size effect that depends strongly on the circumference and not on the seed orientation or cone angle. This finite-size effect occurs also on cylinders, and we present evidence that on both cylinders and cones, the defect density increases exponentially as circumference decreases. We introduce a simple model for particle attachment at the seam that explains the dependence on the circumference. Our findings suggest that the growth of single crystals can become frustrated even very far from the tip when the cone has a small opening angle. These results may provide insights into the observed geometry of conical crystals in biological and materials applications., Comment: 12 pages, 13 figures

Self-assembly is a vital part of the life cycle of certain icosahedral RNA viruses. Furthermore, the assembly process can be harnessed to make icosahedral virus-like particles (VLPs) from coat protein and RNA in vitro. Although much previous work has explored the effects of RNA-protein interactions on the assembly products, relatively little research has explored the effects of coat-protein concentration. We mix coat protein and RNA from bacteriophage MS2, and we use a combination of gel electrophoresis, dynamic light scattering, and transmission electron microscopy to investigate the assembly products. We show that with increasing coat-protein concentration, the products transition from well-formed MS2 VLPs to ``monster'' particles consisting of multiple partial capsids to RNA-protein condensates consisting of large networks of RNA and partially assembled capsids. We argue that the transition from well-formed to monster particles arises because the assembly follows a nucleation-and-growth pathway in which the nucleation rate depends sensitively on the coat-protein concentration, such that at high protein concentrations, multiple nuclei can form on each RNA strand. To understand the formation of the condensates, which occurs at even higher coat-protein concentrations, we use Monte Carlo simulations with coarse-grained models of capsomers and RNA. These simulations suggest that the the formation of condensates occurs by the adsorption of protein to the RNA followed by the assembly of capsids. Multiple RNA molecules can become trapped when a capsid grows from capsomers attached to two different RNA molecules or when excess protein bridges together growing capsids on different RNA molecules. Our results provide insight into an important biophysical process and could inform design rules for making VLPs for various applications.

Interferometric scattering microscopy (iSCAT) can image the dynamics of nanometer-scale systems. The typical approach to analyzing interferometric images involves intensive processing, which discards data and limits the precision of measurements. We demonstrate an alternative approach: modeling the interferometric point spread function (iPSF) and fitting this model to data within a Bayesian framework. This approach yields best-fit parameters, including the particle's three-dimensional position and polarizability, as well as uncertainties and correlations between these parameters. Building on recent work, we develop a model that is parameterized for rapid fitting. The model is designed to work with Hamiltonian Monte Carlo techniques that leverage automatic differentiation. We validate this approach by fitting the model to interferometric images of colloidal nanoparticles. We apply the method to track a diffusing particle in three dimensions, to directly infer the diffusion coefficient of a nanoparticle without calculating a mean-square displacement, and to quantify the ejection of DNA from an individual lambda phage virus, demonstrating that the approach can be used to infer both static and dynamic properties of nanoscale systems.

Many soft and biological materials display so-called 'soft glassy' dynamics; their constituents undergo anomalous random motions and complex cooperative rearrangements. A recent simulation model of one soft glassy material, a coarsening foam, suggested that the random motions of its bubbles are due to the system configuration moving over a fractal energy landscape in high-dimensional space. Here we show that the salient geometrical features of such high-dimensional fractal landscapes can be explored and reliably quantified, using empirical trajectory data from many degrees of freedom, in a model-free manner. For a mayonnaise-like dense emulsion, analysis of the observed trajectories of oil droplets quantitatively reproduces the high-dimensional fractal geometry of the configuration path and its associated energy minima generated using a computational model. That geometry in turn drives the droplets' complex random motion observed in real space. Our results indicate that experimental studies can elucidate whether the similar dynamics in different soft and biological materials may also be due to fractal landscape dynamics., Comment: 10 pages, 5 figures with Appendices

Photonic balls are spheres tens of micrometers in diameter containing assemblies of nanoparticles or nanopores with a spacing comparable to the wavelength of light. When these nanoscale features are disordered, but still correlated, the photonic balls can show structural color with low angle-dependence. Their colors, combined with the ability to add them to a liquid formulation, make photonic balls a promising new type of pigment particle for paints, coatings, and other applications. However, it is challenging to predict the color of materials made from photonic balls, because the sphere geometry and multiple scattering must be accounted for. To address these challenges, we develop a multiscale modeling approach involving Monte Carlo simulations of multiple scattering at two different scales: we simulate multiple scattering and absorption within a photonic ball and then use the results to simulate multiple scattering and absorption in a film of photonic balls. After validating against experimental spectra, we use the model to show that films of photonic balls scatter light in fundamentally different ways than do homogeneous films of nanopores or nanoparticles, because of their increased surface area and refraction at the interfaces of the balls. Both effects tend to sharply reduce color saturation relative to a homogeneous nanostructured film. We show that saturated colors can be achieved by placing an absorber directly in the photonic balls and mitigating surface roughness. With these design rules, we show that photonic-ball films have an advantage over homogeneous nanostructured films: their colors are even less dependent on the angle.

Disordered nanostructures with correlations on the scale of visible wavelengths can show angle-independent structural colors. These materials could replace dyes in some applications because the color is tunable and resists photobleaching. However, designing nanostructures with a prescribed color is difficult, especially when the application -- cosmetics or displays, for example -- requires specific component materials. A general approach to solving this constrained design problem is modeling and optimization: using a model that predicts the color of a given system, one optimizes the model parameters under constraints to achieve a target color. For this approach to work, the model must make accurate predictions, which is challenging because disordered nanostructures have multiple scattering. To address this challenge, we develop a Monte Carlo model that simulates multiple scattering of light in disordered arrangements of spherical particles or voids. The model produces quantitative agreement with measurements when we account for roughness on the surface of the film, particle polydispersity, and wavelength-dependent absorption in the components. Unlike discrete numerical simulations, our model is parameterized in terms of experimental variables, simplifying the connection between simulation and fabrication. To demonstrate this approach, we reproduce the color of the male mountain bluebird (Sialia currucoides) in an experimental system, using prescribed components and a microstructure that is easy to fabricate. Finally, we use the model to find the limits of angle-independent structural colors for a given system. These results enable an engineering design approach to structural color for many different applications.

Inspired by observations of beads packed on a thin string in such systems as sea-grapes and dental plaque, we study the random sequential adsorption of spheres on a cylinder. We determine the asymptotic fractional coverage of the cylinder as a function of the sole parameter in the problem, the ratio of the sphere radius to the cylinder radius (for a very long cylinder) using a combination of analysis and numerical simulations. Examining the asymptotic structures, we find weak chiral ordering on sufficiently small spatial scales. Experiments involving colloidal microspheres that can attach irreversibly to a silica wire via electrostatic forces or DNA hybridization allow us to verify our predictions for the asymptotic coverage.

Disordered packings of colloidal spheres show angle-independent structural color when the particles are on the scale of the wavelength of visible light. Previous work has shown that the positions of the peaks in the reflectance spectra can be predicted accurately from a single-scattering model that accounts for the effective refractive index of the material. This agreement shows that the main color peak arises from short-range correlations between particles. However, the single-scattering model does not quantitatively reproduce the observed color: the main peak in the reflectance spectrum is much broader and the reflectance at low wavelengths is much larger than predicted by the model. We use a combination of experiment and theory to understand these features. We find that one significant contribution to the breadth of the main peak is light that is scattered, totally internally reflected from the boundary of the sample, and then scattered again. The high reflectance at low wavelengths also results from multiple scattering but can be traced to the increase in the scattering cross-section of individual particles with decreasing wavelength. Both of these effects tend to reduce the saturation of the structural color, which limits the use of these materials in applications. We show that while the single-scattering model cannot reproduce the observed saturations, it can be used to design materials in which multiple scattering is suppressed and the color saturated, even in the absence of absorbing components.

In living cells, molecular motors create activity that enhances the diffusion of particles throughout the cytoplasm, and not just ones attached to the motors. We demonstrate initial steps toward creating artificial cells that mimic this phenomenon. Our system consists of active, Pt-coated Janus particles and passive tracers confined to emulsion droplets. We track the motion of both the active particles and passive tracers in a hydrogen peroxide solution, which serves as the fuel to drive the motion. We first show that correcting for bulk translational and rotational motion of the droplets induced by bubble formation is necessary to accurately track the particles. After drift correction, we find that the active particles show enhanced diffusion in the interior of the droplets and are not captured by the droplet interface. At the particle and hydrogen peroxide concentrations we use, we observe little coupling between the active and passive particles. We discuss the possible reasons for lack of coupling and describe ways to improve the system to more effectively mimic cytoplasmic activity.

Colloidal clusters consist of small numbers of colloidal particles bound by weak, short-range attractions. The equilibrium probability of observing a cluster in a particular geometry is well-described by a statistical mechanical model originally developed for molecules. To explain why this model fits experimental data so well, we derive the partition function classically, with no quantum mechanical considerations. Then, by comparing and contrasting the derivation in particle coordinates with that in center-of-mass coordinates, we physically interpret the terms in the center-of-mass formulation, which is equivalent to the high-temperature partition function for molecules. We discuss, from a purely classical perspective, how and why cluster characteristics such as the symmetry number, moments of inertia, and vibrational frequencies affect the equilibrium probabilities.

A holographic microscope captures interference patterns, or holograms, that encode three-dimensional (3D) information about the object being viewed. Computation is essential to extracting that 3D information. By wrapping low-level scattering codes and taking advantage of Python's data analysis ecosystem, HoloPy makes it easy for experimentalists to use modern, sophisticated inference methods to analyze holograms. The resulting data can be used to understand how small particles or microorganisms move and interact. The project illustrates how computational tools can enable experimental methods and new experiments.

The formation of a viral capsid -- the highly-ordered protein shell that surrounds the genome of a virus -- is the canonical example of self-assembly. The capsids of many positive-sense RNA viruses spontaneously assemble from in vitro mixtures of the coat protein and RNA. The high yield of proper capsids that assemble is remarkable, given their structural complexity: 180 identical proteins must arrange into three distinct local configurations to form an icosahedral capsid with a triangulation number of 3 (T = 3). Despite a wealth of data from structural studies and simulations, even the most fundamental questions about how these structures assemble remain unresolved. Experiments have not determined whether the assembly pathway involves aggregation or nucleation, or how the RNA controls the process. Here we use interferometric scattering microscopy to directly observe the in vitro assembly kinetics of individual, unlabeled capsids of bacteriophage MS2. By measuring how many coat proteins bind to individual MS2 RNA strands over time scales from 1 ms to 900 s, we find that the start of assembly is broadly distributed in time and is followed by a rapid increase in the number of bound proteins. These measurements provide strong evidence for a nucleation-and-growth pathway. We also find that malformed structures assemble when multiple nuclei appear on the same RNA before the first nucleus has finished growing. Our measurements reveal the complex assembly pathways for viral capsids around RNA in quantitative detail, including the nucleation threshold, nucleation time, growth time, and constraints on the critical nucleus size. These results may inform strategies for engineering synthetic capsids or for derailing the assembly of pathogenic viruses., Comment: 41 pages, 20 figures

Hydrophobic PMMA colloidal particles, when dispersed in oil with a relatively high dielectric constant, can become highly charged. In the presence of an interface with a conducting aqueous phase, image charge effects lead to strong binding of colloidal particles to the interface, even though the particles are wetted very little by the aqueous phase. In this paper, we study both the behavior of individual colloidal particles as they approach the interface, and the interactions between particles that are already interfacially bound. We demonstrate that using particles which are minimally wetted by the aqueous phase allows us to isolate and study those interactions which are due solely to charging of the particle surface in oil. Finally, we show that these interactions can be understood by a simple image-charge model in which the particle charge $q$ is the sole fitting parameter.

Previous experiments have shown that spherical colloidal particles relax to equilibrium slowly after they adsorb to a liquid-liquid interface, despite the large interfacial energy gradient driving the adsorption. The slow relaxation has been explained in terms of transient pinning and depinning of the contact line on the surface of the particles. However, the nature of the pinning sites has not been investigated in detail. We use digital holographic microscopy to track a variety of colloidal spheres---inorganic and organic, charge-stabilized and sterically stabilized, aqueous and non-aqueous---as they breach liquid interfaces. We find that nearly all of these particles relax logarithmically in time over timescales much larger than those expected from viscous dissipation alone. By comparing our results to theoretical models of the pinning dynamics, we infer the area per defect to be on the order of a few square nanometers for each of the colloids we examine, whereas the energy per defect can vary from a few $kT$ for non-aqueous and inorganic spheres to tens of $kT$ for aqueous polymer particles. The results suggest that the likely pinning sites are topographical features inherent to colloidal particles---surface roughness in the case of silica particles and grafted polymer "hairs" in the case of polymer particles. We conclude that the slow relaxation must be taken into account in experiments and applications, such as Pickering emulsions, that involve colloids attaching to interfaces. The effect is particularly important for aqueous polymer particles, which pin the contact line strongly., Comment: 26 pages, 7 figures

The effects of contact-line pinning are well-known in macroscopic systems, but are only just beginning to be explored at the microscale in colloidal suspensions. We use digital holography to capture the fast three-dimensional dynamics of micrometer-sized ellipsoids breaching an oil-water interface. We find that the particle angle varies approximately linearly with the height, in contrast to results from simulations based on minimization of the interfacial energy. Using a simple model of the motion of the contact line, we show that the observed coupling between translational and rotational degrees of freedom is likely due to contact-line pinning. We conclude that the dynamics of colloidal particles adsorbing to a liquid interface are not determined by minimization of interfacial energy and viscous dissipation alone; contact-line pinning dictates both the timescale and pathway to equilibrium., Comment: 17 pages, 5 figures, 1 movie

The adsorption of single colloidal microparticles (0.5--1 $\mu$m radius) at a water-oil interface has been recently studied experimentally using digital holographic microscopy [Kaz \textit{et al., Nat. Mater.}, 2012, \textbf{11}, 138--142]. An initially fast adsorption dynamics driven by capillary forces is followed by an unexpectedly slow relaxation to equilibrium that is logarithmic in time and can span hours or days. The slow relaxation kinetics has been attributed to the presence of surface "defects" with nanoscale dimensions (1--5\,nm) that induce multiple metastable configurations of the contact line perimeter. A kinetic model considering thermally activated transitions between such metastable configurations has been proposed [Colosqui \textit{et al., Phys. Rev. Lett.}, 2013, \textbf{111}, 028302] to predict both the relaxation rate and the crossover point to the slow logarithmic regime. However, the adsorption dynamics observed experimentally before the crossover point has remained unstudied. In this work, we propose a Langevin model that is able to describe the entire adsorption process of single colloidal particles by considering metastable states produced by surface defects and thermal motion of the particle and liquid interface. Invoking the fluctuation dissipation theorem, we introduce a drag term that considers significant dissipative forces induced by thermal fluctuations of the liquid interface. Langevin dynamics simulations based on the proposed adsorption model yield close agreement with experimental observations for different microparticles, capturing the crossover from (fast) capillary driven dynamics to (slow) thermally activated kinetics., Comment: 8 pages, 4 figures

Defining the entropy of classical particles raises a number of paradoxes and ambiguities, some of which have been known for over a century. Several, such as Gibbs' paradox, involve the fact that classical particles are distinguishable, and in textbooks these are often `resolved' by appeal to the quantum-mechanical indistinguishability of atoms or molecules of the same type. However, questions then remain of how to correctly define the entropy of large poly-atomic particles such as colloids in suspension, of which no two are exactly alike. By performing experiments on such colloids, one can establish that certain definitions of the classical entropy fit the data, while others in the literature do not. Specifically, the experimental facts point firmly to an `informatic' interpretation that dates back to Gibbs: entropy is determined by the number of microstates that we as observers choose to treat as equivalent when we identify a macrostate. This approach, unlike some others, can account for the existence of colloidal crystals, and for the observed abundances of colloidal clusters of different shapes. We also address some lesser-known paradoxes whereby the physics of colloidal assemblies, which ought to be purely classical, seems to involve quantum mechanics directly. The experimental symptoms of such involvement are predicted to be `isotope effects' in which colloids with different inertial masses, but otherwise identical sizes and properties, show different aggregation statistics. These paradoxes are caused by focussing one's attention on some classical degrees while neglecting others; when all are treated equally, all isotope effects are found to vanish.

High-speed tracking of single particles is a gateway to understanding physical, chemical, and biological processes at the nanoscale. It is also a major experimental challenge, particularly for small, nanometer-scale particles. Although methods such as confocal or fluorescence microscopy offer both high spatial resolution and high signal-to-background ratios, the fluorescence emission lifetime limits the measurement speed, while photobleaching and thermal diffusion limit the duration of measurements. Here we present a tracking method based on elastic light scattering that enables long-duration measurements of nanoparticle dynamics at rates of thousands of frames per second. We contain the particles within a single-mode silica fiber containing a sub-wavelength, nano-fluidic channel and illuminate them using the fiber's strongly confined optical mode. The diffusing particles in this cylinderical geometry are continuously illuminated inside the collection focal plane. We show that the method can track unlabeled dielectric particles as small as 20 nm as well as individual cowpea chlorotic mottle virus (CCMV) virions - 4.6 megadaltons in size - at rates of over 2 kHz for durations of tens of seconds. Our setup is easily incorporated into common optical microscopes and extends their detection range to nanometer-scale particles and macromolecules. The ease-of-use and performance of this technique support its potential for widespread applications in medical diagnostics and micro total analysis systems., Comment: 12 pages, 4 figures, updated references

We study experimentally what is arguably the simplest yet non-trivial colloidal system: two-dimensional clusters of 6 spherical particles bound by depletion interactions. These clusters have multiple, degenerate ground states whose equilibrium distribution is determined by entropic factors, principally the symmetry. We observe the equilibrium rearrangements between ground states as well as all of the low-lying excited states. In contrast to the ground states, the excited states have soft modes and low symmetry, and their occupation probabilities depend on the size of the configuration space reached through internal degrees of freedom, as well as a single "sticky parameter" encapsulating the depth and curvature of the potential. Using a geometrical model that accounts for the entropy of the soft modes and the diffusion rates along them, we accurately reproduce the measured rearrangement rates. The success of this model, which requires no fitting parameters or measurements of the potential, shows that the free-energy landscape of colloidal systems and the dynamics it governs can be understood geometrically., Comment: 13 pages, 6 figures, 3 tables

Colloidal glasses, bird feathers, and beetle scales can all show structural colors arising from short-ranged spatial correlations between scattering centers. Unlike the structural colors arising from Bragg diffraction in ordered materials like opals, the colors of these photonic glasses are independent of orientation, owing to their disordered, isotropic microstructures. However, there are few examples of photonic glasses with angle-independent red colors in nature, and colloidal glasses with particle sizes chosen to yield structural colors in the red show weak color saturation. Using scattering theory, we show that the absence of angle-independent red color can be explained by the tendency of individual particles to backscatter light more strongly in the blue. We discuss how the backscattering resonances of individual particles arise from cavity-like modes, and how they interact with the structural resonances to prevent red. Finally, we use the model to develop design rules for colloidal glasses with red, angle-independent structural colors., Comment: 8 pages, 9 figures

We present a new, high-speed technique to track the three-dimensional translation and rotation of non-spherical colloidal particles. We capture digital holograms of micrometer-scale silica rods and sub-micrometer-scale Janus particles freely diffusing in water, and then fit numerical scattering models based on the discrete dipole approximation to the measured holograms. This inverse-scattering approach allows us to extract the the position and orientation of the particles as a function of time, along with static parameters including the size, shape, and refractive index. The best-fit sizes and refractive indices of both particles agree well with expected values. The technique is able to track the center of mass of the rod to a precision of 35 nm and its orientation to a precision of 1.5$^\circ$, comparable to or better than the precision of other 3D diffusion measurements on non-spherical particles. Furthermore, the measured translational and rotational diffusion coefficients for the silica rods agree with hydrodynamic predictions for a spherocylinder to within 0.3%. We also show that although the Janus particles have only weak optical asymmetry, the technique can track their 2D translation and azimuthal rotation over a depth of field of several micrometers, yielding independent measurements of the effective hydrodynamic radius that agree to within 0.2%. The internal and external consistency of these measurements validate the technique. Because the discrete dipole approximation can model scattering from arbitrarily shaped particles, our technique could be used in a range of applications, including particle tracking, microrheology, and fundamental studies of colloidal self-assembly or microbial motion., Comment: 11 pages, 9 figures, 2 tables

We measure all nonzero elements of the three-dimensional (3D) diffusion tensor D for clusters of colloidal spheres to a precision of 1% or better using digital holographic microscopy. We study both dimers and triangular trimers of spheres, for which no analytical calculations of the diffusion tensor exist. We observe anisotropic rotational and translational diffusion arising from the asymmetries of the clusters. In the case of the three-particle triangular cluster, we also detect a small but statistically significant difference in the rotational diffusion about the two in-plane axes. We attribute this difference to weak breaking of threefold rotational symmetry due to a small amount of particle polydispersity. Our experimental measurements agree well with numerical calculations and show how diffusion constants can be measured under conditions relevant to colloidal self-assembly, where theoretical and even numerical prediction is difficult., Comment: 9 pages, 6 figures, 2 tables. Paper and Supplementary Information revised

Digital holographic microscopy is a fast three-dimensional (3D) imaging tool with many applications in soft matter physics. Recent studies have shown that electromagnetic scattering solutions can be fit to digital holograms to obtain the 3D positions of isolated colloidal spheres with nanometer precision and millisecond temporal resolution. Here we describe the results of new techniques that extend the range of systems that can be studied with fitting. We show that an exact multisphere superposition scattering solution can be used to fit holograms of colloidal clusters containing up to six spheres. We also introduce an approximate and computationally simpler solution, Mie superposition, that is valid for multiple spheres spaced several wavelengths or more from one another. We show that this method can be used to fit holograms of several spheres on an emulsion droplet, and we give a quantitative criterion for assessing its validity., Comment: 20 pages, 6 figures, 1 table

Using experiments and simulations, we investigate the clusters that form when colloidal spheres stick irreversibly to -- or "park" on -- smaller spheres. We use either oppositely charged particles or particles labeled with complementary DNA sequences, and we vary the ratio $\alpha$ of large to small sphere radii. Once bound, the large spheres cannot rearrange, and thus the clusters do not form dense or symmetric packings. Nevertheless, this stochastic aggregation process yields a remarkably narrow distribution of clusters with nearly 90% tetrahedra at $\alpha=2.45$. The high yield of tetrahedra, which reaches 100% in simulations at $\alpha=2.41$, arises not simply because of packing constraints, but also because of the existence of a long-time lower bound that we call the "minimum parking" number. We derive this lower bound from solutions to the classic mathematical problem of spherical covering, and we show that there is a critical size ratio $\alpha_c=(1+\sqrt{2})\approx 2.41$, close to the observed point of maximum yield, where the lower bound equals the upper bound set by packing constraints. The emergence of a critical value in a random aggregation process offers a robust method to assemble uniform clusters for a variety of applications, including metamaterials., Comment: 24 pages, 6 figures

Sphere packing problems have a rich history in both mathematics and physics; yet, relatively few analytical analyses of sphere packings exist, and answers to seemingly simple questions are unknown. Here, we present an analytical method for deriving all packings of n spheres in R^3 satisfying minimal rigidity constraints (>= 3 contacts per sphere and >= 3n-6 total contacts). We derive such packings for n <= 10, and provide a preliminary set of maximal contact packings for 10 < n <= 20. The resultant set of packings has some striking features, among them: (i) all minimally rigid packings for n <= 9 have 3n-6 contacts, (ii) non-rigid packings satisfying minimal rigidity constraints arise for n >= 9, (iii) the number of ground states (i.e. packings with the maximal number of contacts) oscillates with respect to n, (iv) for 10 <= n <= 20 there are only a small number of packings with the maximal number of contacts, and for 10 <= n < 13 these are all commensurate with the HCP lattice. The general method presented here may have applications to other related problems in mathematics, such as the Erdos repeated distance problem and Euclidean distance matrix completion problems., Comment: The updates that have been made to version II are as follows: (i) some updates to the main text have been made, primarily correcting the number of packings reported in PRL, 103, 118303 and in version I of this paper for n = 9,10. (ii) The supplemental information for the paper has been included