We do experiments on some of the simplest self-assembling systems, called "colloidal clusters," to understand the basic physics of self-assembly. The clusters consists of a small number (say, fewer than 10 or so) spherical colloidal particles that attract one another. What shapes do they form and why? By answering these questions, we gain insights into how to design particles that can self-assemble into exactly the structures we want.

# Colloidal clusters

## Publications

Clusters of spherical particles are called “colloidal molecules” because they adopt structures that resemble those of true molecules. In this analogy, the particles are the atoms, the attractive interactions between them are bonds, and the different structures that appear in equilibrium are isomers. We take this analogy a step further by doping colloidal molecules with colloidal “isotopes,” particles that have the same size but different bonding energies from the other particles in the system. Our molecules are two-dimensional clusters consisting of polystyrene and silica microspheres held together by depletion interactions. Using a combination of optical microscopy and particle tracking, we examine an ensemble of 4- and 5-particle molecules at different isotope ratios. We find that the isotopes tend to segregate to particular positions in the various isomers. We explain these findings using a statistical mechanical model that accounts for the rotational entropy of the isomers and the different interaction potentials between the different types of particles. The model shows how to optimize the yield of any particular isomer, so as to put the isotopes in desired locations. Our experiments and models show that even in systems of particles with isotropic interactions, the structures of self-assembled molecules can in principle be controlled to a surprisingly high extent.

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.

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.

Colloidal clusters, aggregates of a few micrometer-sized spherical particles, are a model experimental system for understanding the physics of self-assembly and processes such as nucleation. Colloidal clusters are well suited for studies on these topics because they are the simplest colloidal system with internal degrees of freedom. Clusters made from particles that weakly attract one another continually rearrange between different structures. By characterizing these internal dynamics and the structures connected by the rearrangement pathways, we seek to understand the statistical physics underlying self-assembly and equilibration.

In this thesis, we examine the rearrangement dynamics of colloidal clusters and analyze the equilibrium distributions of ground and excited states. We prepare clusters of up to ten microspheres bound by short-range depletion interactions that are tuned to allow equilibration between multiple isostatic arrangements. To study these clusters, we use bright-field and digital holographic microscopy paired with computational post-processing to amass ensemble-averaged and time-averaged probabilities.

We study both two-dimensional (2D) and three-dimensional (3D) clusters composed of either one or two species of particles. To learn about geometrical nucleation barriers, we track rearrangements of particles within freely rotating and translating 3D clusters. We show that rearrangements occur on a timescale of seconds, consistent with diffusion-dominated internal dynamics. To better understand excited states and transition pathways, we track hundreds of rearrangements between degenerate ground states in 2D clusters. We show that the rearrangement rates can be understood using a model with two parameters, which account for the diffusion coefficient along the excited-state rearrangement pathways and the interaction potential. To explore new methods to control self-assembly, we analyze clusters of two species with different masses and different interactions. We find that the interactions allow for control over the intracluster placement of each species, while the masses have no influence. To provide a theoretical framework for understanding these observations, we derive the classical partition function of colloidal clusters in terms of translational, rotational, and vibrational degrees of freedom. We show that the masses of the particles enter the partition function through the kinetic energy but have no effect on the probabilities of states that differ only in where the masses are placed. This result is consistent with our experiments.

Overall, this work shows that the equilibrium distribution of self-assembled colloidal clusters is well-modeled by classical statistical physics, and that the rearrangement dynamics of colloidal clusters can be understood by incorporating diffusion and the effect of the interaction potential. Because both the structures and dynamics can be accurately predicted, these clusters are a promising system for self-assembling novel materials and for studying the emergence of phase transitions.

I discuss experimental progress towards developing a material with an isotropic, negative index of refraction at optical frequencies. The simplest way to make such a material is to create a metafluid, or a disordered collection of subwavelength, isotropic electromagnetic resonators. Small clusters of metal particles, such as tetrahedra, serve as these constituents. What is needed are methods for manufacturing these structures with high precision and in sufficient yield that their resonances are identical.

Jonathan Fan et al. [Science, 328 (5982), 1135-1138, 2010] demonstrated that colloidal self-assembly is a means of preparing electromagnetic resonators from metal nanoparticles. However, the resonances are sensitive to the separation gaps between particles. Standard synthesis routes for metal nanoparticles yield crystals or nanoshells that are inadequate for metafluids due to polydispersity, faceting, and thermal instabilities. To ensure that the separation gaps and resonances are uniform, more monodisperse spherical particles are needed. An additional challenge is the self-assembly of tetrahedral clusters in high yield from these particles. In self-assembly approaches that others have examined previously, the yield of any particular type of cluster is low.

In this dissertation I present solutions to several of these problems, developed in collaboration with my research group and others. We demonstrate that slow chemical etching can transform octahedral gold crystals into ultrasmooth, monodisperse nanospheres. The particles can serve as seeds for the growth of larger octahedra which can in turn be etched. The size of the gold nanospheres can therefore be adjusted as desired. We further show that in colloidal mixtures of two sphere species that strongly bind to one another, the sphere size ratio determines the size distribution of self-assembled clusters. At a critical size ratio, tetrahedral clusters assemble in high yield. We explain the experimentally observed 90% yield with a nonequilibrium “random parking” model based on irreversible binding. Simulations based on this model reveal that 100% yield of tetrahedra is possible in principle. Finally, we combine these results and present methods for the self-assembly and purification of tetrahedral plasmonic nanoclusters, the simplest building blocks for isotropic metafluids.

We establish size limitations for assembling structures of controlled size and shape out of colloidal particles with short-ranged interactions. Through simulations we show that structures with highly variable shapes made out of dozens of particles can form with high yield, as long as each particle in the structure binds only to the particles in their local environment. To understand this, we identify the excited states that compete with the ground-state structure and demonstrate that these excited states have a completely topological characterization, valid when the interparticle interactions are short-ranged. This allows complete enumeration of the energy landscape and gives bounds on how large a colloidal structure can assemble with high yield. For large structures the yield can be significant, even with hundreds of particles.

In this thesis, I discuss engineering colloidal particles to have specific, isotropic interactions and studying their cluster geometries in equilibrium. I discuss light scattering experiments showing that a highly specific protein, Dscam, is unstable against thermal aggregation. This result lead me to use DNA instead to control interparticle specificity. I coated 1-micron diameter polystyrene particles uniformly with DNA. I used fluorescence microscopy with oxygen-scavenging enzymes to observe these particles self-assembling in clusters. These experiments show that a packing of 6 spheres that is rarely seen in a single-component system is observed very often in an optimized 3-species system. Then I show experiments using the same 3 species but 9 total particles, finding that the equilibrium yields of the most likely cluster relative to other stable clusters are lower than at 6 particles. I conclude from these experiments that optimizing the assembly of an otherwise unlikely configuration may require nearly as many species as particles. Finally, I investigate the scalability of self-assembly of particles with isotropic and specific interactions theoretically. I use both exact and approximate partition functions to show that spheres with specific interactions can have energy landscapes with thermodynamically large numbers of strictly local minima relative to the number of their ground states. Compared to single-component systems, these systems of many different species may spend much more time in kinetic traps and never reach their ground states. Finally, I discuss briefly some directions for further study, including questions of how the results in this thesis may be related to protein folding and complex formation.

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 αc=(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.

We measure all nonzero elements of the three-dimensional 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.

We discuss digital holographic microscopy (DHM), a 3D imaging technique capable of measuring the positions of micron-sized colloidal particles with nanometer precision and sub-millisecond temporal resolution. We use exact electromagnetic scattering solutions to model holograms of multiple colloidal spheres. While the Lorenz-Mie solution for scattering by isolated spheres has previously been used to model digital holograms, we apply for the first time an exact multisphere superposition scattering model that is capable of modeling holograms from spheres that are sufficiently close together to exhibit optical coupling.

The ability to design and assemble three-dimensional structures from colloidal particles is limited by the absence of specific directional bonds. As a result, complex or low-coordination structures, common in atomic and molecular systems, are rare in the colloidal domain. Here we demonstrate a general method for creating the colloidal analogues of atoms with valence: colloidal particles with chemically distinct surface patches that imitate hybridized atomic orbitals, including sp, sp2, sp3, sp3d, sp3d2 and sp3d3. Functionalized with DNA with single-stranded sticky ends, patches on different particles can form highly directional bonds through programmable, specific and reversible DNA hybridization. These features allow the particles to self-assemble into |[lsquo]|colloidal molecules|[rsquo]| with triangular, tetrahedral and other bonding symmetries, and should also give access to a rich variety of new microstructured colloidal materials.

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 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 analyze holograms of several spheres on an emulsion droplet, and we give a quantitative criterion for assessing its validity.

The energetics and assembly pathways of small clusters may yield insights into processes occurring at the earliest stages of nucleation. We use a model system consisting of micrometer-sized, spherical colloidal particles to study the structure and dynamics of small clusters, where the number of particles is small (N <= 10). The particles interact through a short-range depletion attraction with a depth of a few k_B T. We describe two methods to form colloidal clusters, one based on isolating the particles in microwells and another based on directly assembling clusters in the gas phase using optical tweezers. We use the first technique to obtain ensemble-averaged probabilities of cluster structures as a function of N. These experiments show that clusters with symmetries compatible with crystalline order are rarely formed under equilibrium conditions. We use the second technique to study the dynamics of the clusters, and in particular how they transition between free-energy minima. To monitor the clusters we use a fast three-dimensional imaging technique, digital holographic microscopy, that can resolve the positions of each particle in the cluster with 30-45 nm precision on millisecond timescales. The real-space measurements allow us to obtain estimates for the lifetimes of the energy minima and the transition states. It is not yet clear whether the observed dynamics are relevant for small nuclei, which may not have sufficient time to transition between states before other particles or clusters attach to them. However, the measurements do provide some glimpses into how systems containing a small number of particles traverse their free-energy landscape.

Plasmonic nanoparticle assemblies are a materials platform in which optical modes, resonant frequencies, and near-field intensities can be specified by the number and position of nanoparticles in a cluster. A current challenge is to achieve clusters with higher yields and new types of shapes. In this Letter, we show that a broad range of plasmonic nanoshell nanoclusters can be assembled onto a lithographically defined elastomeric substrate with relatively high yields using templated assembly. We assemble and measure the optical properties of three cluster types: Fano-resonant heptamers, linear chains, and rings of nanoparticles. The yield of heptamer clusters is measured to be over 30%. The assembly of plasmonic nanoclusters on an elastomer paves the way for new classes of plasmonic nanocircuits and colloidal metamaterials that can be transfer-printed onto various substrate media.

We discuss a new method for simultaneously probing translational, rotational, and vibrational dynamics in dilute colloidal suspensions using digital holographic microscopy (DHM). We record digital holograms of clusters of 1-μm-diameter colloidal spheres interacting through short-range attractions, and we fit the holograms to an exact model of the scattering from multiple spheres. The model, based on the T-matrix formulation, accounts for multiple scattering and near-field coupling. We also explicitly account for the non-asymptotic radial decay of the scattered fields, allowing us to accurately fit holograms recorded with the focal plane located as little as 15 μm from the particle. Applying the fitting technique to a time-series of holograms of Brownian dimers allows simultaneous measurement of six dynamical modes — three translational, two rotational, and one vibrational — on timescales ranging from 10−3 to 1 s. We measure the translational and rotational diffusion constants to a precision of 0.6%, and we use the vibrational data to measure the interaction potential between the spheres to a precision of ∼50 nm in separation distance. Finally, we show that the fitting technique can be used to measure dynamics of clusters containing three or more spheres.

DNA nanotechnology provides a versatile foundation for the chemical assembly of nanostructures. Plasmonic nanoparticle assemblies are of particular interest because they can be tailored to exhibit a broad range of electromagnetic phenomena. In this Letter, we report the assembly of DNA-functionalized nanoparticles into heteropentamer clusters, which consist of a smaller gold sphere surrounded by a ring of four larger spheres. Magnetic and Fano-like resonances are observed in individual clusters. The DNA plays a dual role: it selectively assembles the clusters in solution and functions as an insulating spacer between the conductive nanoparticles. These particle assemblies can be generalized to a new class of DNA-enabled plasmonic heterostructures that comprise various active and passive materials and other forms of DNA scaffolding.

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 $\mathbb{R}^3$ satisfying minimal rigidity constraints ($\geq 3$ contacts per sphere and $\geq 3n-6$ total contacts). We derive such packings for $n \leq 10$ and provide a preliminary set of maximum contact packings for $10 < n \leq 20$. The resultant set of packings has some striking features; among them are the following: (i) all minimally rigid packings for $n \leq 9$ have exactly $3n-6$ contacts; (ii) nonrigid packings satisfying minimal rigidity constraints arise for $n \geq 9$; (iii) the number of ground states (i.e., packings with the maximum number of contacts) oscillates with respect to $n$; (iv) for $10 \leq n \leq 20$ there are only a small number of packings with the maximum number of contacts, and for $10 \leq n < 13$ these are all commensurate with the hexagonal close-packed lattice. The general method presented here may have applications to other related problems in mathematics, such as the Erdös repeated distance problem and Euclidean distance matrix completion problems.

The self-assembly of colloids is an alternative to top-down processing that enables the fabrication of nanostructures. We show that self-assembled clusters of metal-dielectric spheres are the basis for nanophotonic structures. By tailoring the number and position of spheres in close-packed clusters, plasmon modes exhibiting strong magnetic and Fano-like resonances emerge. The use of identical spheres simplifies cluster assembly and facilitates the fabrication of highly symmetric structures. Dielectric spacers are used to tailor the interparticle spacing in these clusters to be approximately 2 nanometers. These types of chemically synthesized nanoparticle clusters can be generalized to other two- and three-dimensional structures and can serve as building blocks for new metamaterials.

Assemblies of strongly interacting metallic nanoparticles are the basis for plasmonic nanostructure engineering. We demonstrate that clusters of four identical spherical particles self-assembled into a close-packed asymmetric quadrumer support strong Fano-like interference. This feature is highly sensitive to the polarization of the incident electric field due to orientation-dependent coupling between particles in the cluster. This structure demonstrates how careful design of self-assembled colloidal systems can lead to the creation of new plasmonic modes and the enabling of interference effects in plasmonic systems.

We describe a procedure to synthesize colloidal clusters with polyhedral morphologies in high yield (liter quantities at up to 70% purity) using a combination of emulsion polymerization and inorganic surface chemistry. We show that the synthesis initially used for silica-polystyrene hybrid clusters can be generalized to create clusters from other inorganic and polymer particles. We also show that high yields of particular morphologies can be obtained by precise control of the inorganic seed particle size, a finding that can be explained using a hard-sphere packing model. These clusters can be further chemically modified for a variety of applications. Introducing a cross-linker leads to colloidal clusters that can be index matched in an appropriate solvent, allowing them to be used for particle tracking or optical studies of colloidal self-assembly. Also, depositing a thin silica layer on these colloids allows the surface properties to be controlled using silane chemistry.

We have developed a new method to produce hybrid particles with polyhedral shapes in very high yield (liter quantities at up to 70% purity) using a combination of emulsion polymerization and inorganic surface chemistry. The procedure has been generalized to create complex geometries, including hybrid line segments, triangles, tetrahedra, octahedra, and more. The optical properties of these particles are tailored for studying their dynamics and self-assembly. For example, we produce systems that consist of index-matched spheres allowing us to define the position of each elementary particle in three-dimensional space. We present some preliminary studies on the self-assembly of these complex shaped systems based on electron and optical microscopy.

We calculate the ground states of hard-sphere clusters, in which n identical hard spherical particles bind by isotropic short-ranged attraction. Combining graph theoretic enumeration with basic geometry, we analytically solve for clusters of n<=10 particles satisfying minimal rigidity constraints. For n<=9 the ground state degeneracy increases exponentially with n, but for n>9 the degeneracy decreases due to the formation of structures with >3n-6 contacts. Interestingly, for n=10 and possibly at n=11 and n=12, the ground states of this system are subsets of hexagonal close-packed crystals. The ground states are not icosahedra at n=12 and n=13. We relate our results to the structure and thermodynamics of suspensions of colloidal particles with short-ranged attractions.

I discuss how colloidal particles organize when they are confined by emulsion droplets. In these systems, the interplay between surface tension and interparticle repulsion drives the formation of complex, non-crystalline 3D arrangements. These can be classified into three groups: colloidosomes, or Pickering emulsions, structures that form when particles are bound to the interface of a spherical droplet; colloidal clusters, small polyhedral configurations of colloids formed by capillary forces generated in an evaporating emulsion droplet; and supraparticles, hall-shaped crystallites formed in the interior of emulsion droplets. I discuss the preparation, properties, and structure of each of these systems, using relevant results from geometry to describe how the particles organize.

Most of the colloidal clusters have been produced from oil-in-water emulsions with identical microspheres dispersed in oil droplets. Here, we present new types of binary colloidal clusters from phase-inverted water-in-oil emulsions using various combinations of two different colloids with several size ratios: monodisperse silica or polystyrene microspheres for larger particles and silica or titania nanoparticles for smaller particles. Obviously, a better understanding of how finite groups of different colloids self-organize in a confined geometry may help us control the structure of matter at multiple length scales. In addition, since aqueous dispersions have much better phase stability, we could produce much more diverse colloidal materials from water-in-oil emulsions rather than from oil-in-water emulsions. Interestingly, the configurations of the large microspheres were not changed by the presence of the small particles. However, the arrangement of the smaller particles was strongly dependent on the nature of the interparticle interactions. The experimentally observed structural evolutions were consistent with the numerical simulations calculated using Surface Evolver. These clusters with nonisotropic structures can be used as building blocks for novel colloidal structures with unusual properties or by themselves as light scatterers, diffusers, and complex adaptive matter exhibiting emergent behavior.