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 α 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 α = 2.45. The high yield of tetrahedra, which reaches 100% in simulations at α = 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 α c = (1 + √ 2) ≈ 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.

Particles interacting with short-ranged potentials have attracted increasing interest, partly for their ability to model mesoscale systems such as colloids interacting via DNA or depletion. We consider the free-energy landscape of such systems as the range of the potential goes to zero. In this limit, the landscape is entirely defined by geometrical manifolds, plus a single control parameter. These manifolds are fundamental objects that do not depend on the details of the interaction potential and provide the starting point from which any quantity characterizing the system-equilibrium or nonequilibrium-can be computed for arbitrary potentials. To consider dynamical quantities we compute the asymptotic limit of the Fokker-Planck equation and show that it becomes restricted to the low-dimensional manifolds connected by "sticky" boundary conditions. To illustrate our theory, we compute the low-dimensional manifolds for n ≤ 8 identical particles, providing a complete description of the lowest-energy parts of the landscape including floppy modes with up to 2 internal degrees of freedom. The results can be directly tested on colloidal clusters. This limit is a unique approach for understanding energy landscapes, and our hope is that it can also provide insight into finite-range potentials.sticky spheres | self-assembly | transition rates | sticky Brownian motion T he dynamics on free-energy landscapes are a ubiquitous paradigm for characterizing molecular and mesoscopic systems, from atomic clusters, to protein folding, to colloidal clusters (1-4). The predominant strategy for understanding the dynamics on an energy landscape has focused on the stationary points of the energy, the local minima and the transition states, and seeks the dynamical paths that connect these to each other, whereas more recent models generalize to metastable states connected by paths as a Markov state model (5). These techniques have proved to be extremely powerful, giving innumerable insights into the behavior of complex systems (6-13). On the other hand, a major issue has been the difficulty of finding the transition paths, connecting local minima or metastable states to each other, especially given a complex energy landscape in a high-dimensional space. A variety of creative methods have been developed in recent years for efficiently finding transition paths (14-24) but for a given system, there is no guarantee that all such paths have been found.Here we present a different point of view for understanding an energy landscape that occurs when the range over which particles interact is much smaller than their size. Such is the case in certain mesoscale systems, for example, for C 60 molecules (25, 26), or for colloids interacting via depletion (4) or coated with polymers or cDNA strands (27-30). We show that in this limit, the free-energy landscape is described entirely by geometry, plus a single control parameter κ that is a function of the temperature, depth, and curvature of the original potential. This limit is related to the sticky sphe...

An important goal of self-assembly is to achieve a preprogrammed structure with high fidelity.Here, we control the valence of DNA-functionalized emulsions to make linear and branched model polymers, or 'colloidomers'. The distribution of cluster sizes is consistent with a polymerization process in which the droplets achieve their prescribed valence. Conformational dynamics reveals that the chains are freely-jointed, such that the end-to-end length scales with the number of bonds N as N ν , where ν ≈ 3/4, in agreement with the Flory theory in 2D. The chain diffusion coefficient D approximately scales as D ∝ N −ν , as predicted by the Zimm model. Unlike molecular polymers, colloidomers can be repeatedly assembled and disassembled under temperature cycling, allowing for reconfigurable, responsive matter. arXiv:1805.03734v1 [cond-mat.soft] 9 May 2018

We describe and analyze some Monte Carlo methods for manifolds in euclidean space defined by equality and inequality constraints. First, we give an MCMC sampler for probability distributions defined by unnormalized densities on such manifolds. The sampler uses a specific orthogonal projection to the surface that requires only information about the tangent space to the manifold, obtainable from first derivatives of the constraint functions, hence avoiding the need for curvature information or second derivatives. Second, we use the sampler to develop a multistage algorithm to compute integrals over such manifolds. We provide single-run error estimates that avoid the need for multiple independent runs. Computational experiments on various test problems show that the algorithms and error estimates work in practice. The method is applied to compute the entropies of different sticky hard sphere systems. These predict the temperature or interaction energy at which loops of hard sticky spheres become preferable to chains.

We study experimentally what is arguably the simplest yet nontrivial colloidal system: two-dimensional clusters of six 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. DOI: 10.1103/PhysRevLett.114.228301 PACS numbers: 82.70.Dd, 02.40.-k, 05.10.Gg, 05.20.-y Colloidal clusters containing a few particles bound together by weak attractive interactions are among the simplest, nontrivial systems for investigating collective phenomena in condensed matter. Such clusters can equilibrate on experimental time scales and display complex dynamics, yet are small enough that the ground states can be enumerated theoretically, and the positions and motions of all the particles can be measured experimentally. Theoretical and experimental work on isolated threedimensional (3D) colloidal clusters of monodisperse particles has shown how the number of ground states changes with the number of particles N [1-6] and how the free energies of the rigid states are related to entropy-reducing symmetry effects and entropy-enhancing vibrational modes [7][8][9]. The importance of entropy in colloidal clusters stands in stark contrast to the case of atomic clusters, where potential energy effects dominate. The entropically favored clusters are important clues to understanding nucleation barriers in bulk colloidal fluids [4,10] and the local structure of gels [11].However, the excited states and structural rearrangements in such clusters have not yet been studied experimentally. In bulk materials, local structural rearrangements are important to a variety of dynamical phenomena, including the glass transition [12], aging [13,14], epitaxial growth [15], and the jamming transition [16]. A better understanding of the internal dynamics in colloidal clusters could reveal local mechanisms underpinning these bulk phenomena. Only a few experimental studies have explored internal dynamics in colloidal clusters: Perry and coworkers examined transitions between two states of a 3D six-particle cluster of spherical particles [17], Yunker and co-workers studied relations between the vibrational mode structure and the ...

Packing problems, which ask how to arrange a collection of objects in space to meet certain criteria, are important in a great many physical and biological systems, where geometrical arrangements at small scales control behavior at larger ones. In many systems there is no single, optimal packing that dominates, but rather one must understand the entire set of possible packings. As a step in this direction we enumerate rigid clusters of identical hard spheres for n ≤ 14 and clusters with the maximum number of contacts for n ≤ 19. A rigid cluster is one that cannot be continuously deformed while maintaining all contacts. This is a nonlinear notion that arises naturally because such clusters are the metastable states when the spheres interact with a short-range potential, as is the case in many nano-or microscale systems. We believe that our lists are nearly complete, except for a small number of highly singular clusters (linearly floppy but nonlinearly rigid). The data contains some major geometrical surprises, such as the prevalence of hypostatic clusters: those with less than the 3n − 6 contacts generically necessary for rigidity. We discuss these and several other unusual clusters whose geometries may give insight into physical mechanisms, pose mathematical and computational problems, or bring inspiration for designing new materials. Introduction.The study of sphere packings has a long and rich history in mathematics [17,49]. A large body of work has searched for optimal packings, for example, those that maximize the density of an infinite collection of spheres in different dimensions [12,29], or those that minimize an energy or volume functional [26,13,43,53], such as the Thomson problem, which considers electrons on a unit sphere [9]. However, many applications call for knowing the total set of packings-all the possible ways to arrange a given, finite number of spheres to satisfy certain conditions. For example, in condensed-matter physics, spheres are used to model atoms, molecules, colloids, or other units of matter, and one asks how large numbers of units behave collectively. A rich set of phases can emerge, such as crystals, gels, and glasses, and the dynamics of forming these phases or changing between them are often controlled by the geometrical ways of arranging small groups of spheres without overlap [51,42,6,48,41]. Granular materials, such as sand, are also modeled as packings of spheres or other shapes. The total number of mechanically stable packings is argued to give

In this paper, we explore the strong rotation limit of the rotating and stratified Boussinesq equations with periodic boundary conditions when the stratification is order 1 ([Rossby number] Ro = ε, [Froude number] Fr = O(1), as ε → 0). Using the same framework of Embid & Majda (Geophys. Astrophys. Fluid Dyn., vol. 87, 1998, p. 1), we show that the slow dynamics decouples from the fast. Furthermore, we derive equations for the slow dynamics and their conservation laws. The horizontal momentum equations reduce to the two-dimensional Navier–Stokes equations. The equation for the vertically averaged vertical velocity includes a term due to the vertical average of the buoyancy. The buoyancy equation, the only variable to retain its three-dimensionality, is advected by all three two-dimensional slow velocity components. The conservation laws for the slow dynamics include those for the two-dimensional Navier–Stokes equations and a new conserved quantity that describes dynamics between the vertical kinetic energy and the buoyancy. The leading order potential enstrophy is slow while the leading order total energy retains both fast and slow dynamics. We also perform forced numerical simulations of the rotating Boussinesq equations to demonstrate support for three aspects of the theory in the limit Ro → 0: (i) we find the formation and persistence of large-scale columnar Taylor–Proudman flows in the presence of O(1) Froude number; after a spin-up time, (ii) the ratio of the slow total energy to the total energy approaches a constant and that at the smallest Rossby numbers that constant approaches 1 and (iii) the ratio of the slow potential enstrophy to the total potential enstrophy also approaches a constant and that at the lowest Rossby numbers that constant is 1. The results of the numerical simulations indicate that even in the presence of the low wavenumber white noise forcing the dynamics exhibit characteristics of the theory.

We present a theoretical and numerical study of the decay of an internal wave caused by scattering at undulating sea-floor topography, with an eye towards building a simple model in which the decay of internal tides in the ocean can be estimated. As is well known, the interactions of internal waves with irregular boundary shapes lead to a mathematically ill-posed problem, so care needs to be taken to extract meaningful information from this problem. Here, we restrict the problem to two spatial dimensions and build a numerical tool that combines a real-space computation based on the characteristics of the underlying partial differential equation with a spectral computation that satisfies the relevant radiation conditions. Our tool works for finite-amplitude topography but is restricted to subcritical topography slopes. Detailed results are presented for the decay of the gravest vertical internal wave mode as it encounters finite stretches of either sinusoidal topography or random topography defined as a Gaussian random process with a simple power spectrum. A number of scaling laws are identified and a simple expression for the decay rate in terms of the power spectrum is given. Finally, the resulting formulae are applied to an idealized model of sea-floor topography in the ocean, which seems to indicate that this scattering process can provide a rapid decay mechanism for internal tides. However, the present results are restricted to linear fluid dynamics in two spatial dimensions and to uniform stratification, which restricts their direct application to the real ocean.