Geometric frustration in small colloidal clusters

Identification and classification of clusters of dipolar colloids in an external field

Colloids can acquire a dipolar interaction in the presence of an external AC electric field. At high field strength, the particles form strings in the field direction. However, at weaker field strength, competition with isotropic interactions is expected. One means to investigate this interplay between dipolar and isotropic interactions is to consider clusters of such particles. Therefore, we have identified, using the GMIN basin-hopping tool, a rich library of lowest energy clusters of a dipolar colloidal system, where the dipole orientation is fixed to lie along the z axis and the dipole strength is varied for m-membered clusters of 7 ≤ m ≤ 13. In the regime where the isotropic and dipolar interactions are comparable, we find elongated polytetrahedral, octahedral, and spiral clusters as well as a set of non-rigid clusters, which emerge close to the transition to strings. We further implement a search algorithm that identifies these minimum energy clusters in bulk systems using the topological cluster classification [J. Chem. Phys. 139 234506 (2013)]. We demonstrate this methodology with computer simulations, which show instances of these clusters as a function of dipole strength.

Exotic self-assembly of hard spheres in a morphometric solvent

The self-assembly of spheres into geometric structures, under various theoretical conditions, offers valuable insights into complex self-assembly processes in soft systems. Previous studies have utilized pair potentials between spheres to assemble maximum contact clusters in simulations and experiments. The morphometric approach to solvation free energy that we utilize here goes beyond pair potentials; it is a geometry-based theory that incorporates a weighted combination of geometric measures over the solvent accessible surface for solute configurations in a solvent. In this paper, we demonstrate that employing the morphometric model of solvation free energy in simulating the self-assembly of sphere clusters results, under most conditions, in the previously observed maximum contact clusters. Under other conditions, it unveils an assortment of extraordinary sphere configurations, such as double helices and rhombohedra. These exotic structures arise specifically under conditions where the interactions take multibody potentials into account. This investigation establishes a foundation for comprehending the diverse range of geometric forms in self-assembled structures, emphasizing the significance of the morphometric approach in this context.

Real space analysis of colloidal gels: triumphs, challenges and future directions

Colloidal gels constitute an important class of materials found in many contexts and with a wide range of applications. Yet as matter far from equilibrium, gels exhibit a variety of time-dependent behaviours, which can be perplexing, such as an increase in strength prior to catastrophic failure. Remarkably, such complex phenomena are faithfully captured by an extremely simple model-'sticky spheres'. Here we review progress in our understanding of colloidal gels made through the use of real space analysis and particle resolved studies. We consider the challenges of obtaining a suitable experimental system where the refractive index and density of the colloidal particles is matched to that of the solvent. We review work to obtain a particle-level mechanism for rigidity in gels and the evolution of our understanding of time-dependent behaviour, from early-time aggregation to ageing, before considering the response of colloidal gels to deformation and then move on to more complex systems of anisotropic particles and mixtures. Finally we note some more exotic materials with similar properties.

From canyons to valleys: Numerically continuing sticky-hard-sphere clusters to the landscapes of smoother potentials

We study the energy landscapes of particles with short-range attractive interactions as the range of the interactions increases. Starting with the set of local minima for 6≤N≤12 hard spheres that are "sticky," i.e., they interact only when their surfaces are exactly in contact, we use numerical continuation to evolve the local minima (clusters) as the range of the potential increases, using both the Lennard-Jones and Morse families of interaction potentials. As the range increases, clusters merge, until at long ranges only one or two clusters are left. We compare clusters obtained by continuation with different potentials and find that for short and medium ranges, up to about 30% of particle diameter, the continued clusters are nearly identical, both within and across families of potentials. For longer ranges, the clusters vary significantly, with more variation between families of potentials than within a family. We analyze the mechanisms behind the merge events and find that most rearrangements occur when a pair of nonbonded particles comes within the range of the potential. An exception occurs for nonharmonic clusters, i.e., those that have a zero eigenvalue in their Hessian, which undergo a more global rearrangement.

Morphometric Approach to Many-Body Correlations in Hard Spheres

We model the thermodynamics of local structures within the hard sphere liquid at arbitrary volume fractions through the morphometric calculation of n-body correlations. We calculate absolute free energies of local geometric motifs in excellent quantitative agreement with molecular dynamics simulations across the liquid and supercooled liquid regimes. We find a bimodality in the density library of states where fivefold symmetric structures appear lower in free energy than fourfold symmetric structures and from a single reaction path predict a dynamical barrier which scales linearly in the compressibility factor. The method provides a new route to assess changes in the free energy landscape at volume fractions dynamically inaccessible to conventional techniques.

Structural covariance in the hard sphere fluid

We study the joint variability of structural information in a hard sphere fluid biased to avoid crystallisation and form five-fold symmetric geometric motifs. We show that the structural covariance matrix approach, originally proposed for on-lattice liquids [P. Ronceray and P. Harrowell, J. Stat. Mech.: Theory Exp. 2016(8), 084002], can be meaningfully employed to understand structural relationships between different motifs and can predict, within the linear-response regime, structural changes related to motifs distinct from that used to bias the system.

Free energy of singular sticky-sphere clusters

Networks of particles connected by springs model many condensed-matter systems, from colloids interacting with a short-range potential and complex fluids near jamming, to self-assembled lattices and various metamaterials. Under small thermal fluctuations the vibrational entropy of a ground state is given by the harmonic approximation if it has no zero-frequency vibrational modes, yet such singular modes are at the epicenter of many interesting behaviors in the systems above. We consider a system of N spherical particles, and directly account for the singularities that arise in the sticky limit where the pairwise interaction is strong and short ranged. Although the contribution to the partition function from singular clusters diverges in the limit, its asymptotic value can be calculated and depends on only two parameters, characterizing the depth and range of the potential. The result holds for systems that are second-order rigid, a geometric characterization that describes all known ground-state (rigid) sticky clusters. To illustrate the applications of our theory we address the question of emergence: how does crystalline order arise in large systems when it is strongly disfavored in small ones? We calculate the partition functions of all known rigid clusters up to N≤21 and show the cluster landscape is dominated by hyperstatic clusters (those with more than 3N-6 contacts); singular and isostatic clusters are far less frequent, despite their extra vibrational and configurational entropies. Since the most hyperstatic clusters are close to fragments of a close-packed lattice, this underlies the emergence of order in sticky-sphere systems, even those as small as N=10.

Numerical study of cluster formation in binary charged colloids

Cluster formation of oppositely charged colloidal particles is studied numerically. A simple Brownian dynamics method with a screened-Coulomb (Yukawa) potential is employed for numerical simulations. An equilibrium phase which consists of clusters and unassociated particles is obtained. It is shown that the equilibrium association number of clusters and their shapes are determined by charge numbers and charge ratio of the binary particles. The phase diagram of cluster formation for various charge numbers and their ratios is obtained. A simple relation between the association number and the charge ratio is found. It is demonstrated that in the case of high charge ratio the cluster takes a multilayer structure which is highly symmetric. It is also pointed out that the cluster-particle interaction changes dynamically in the cluster formation process, which is involved in the selection of final cluster structure.

A Detailed Study on the Low-Energy Structures of Charged Colloidal Clusters

The target of this investigation is the systematic characterization of the low-energy structures of charged colloidal clusters that may be important to understand the self-assembling process of biomolecules. The aggregation of charged colloidal particles is governed by the attractive short-ranged Morse potential and the Yukawa repulsive tail to describe the long-range charge effect. A global optimization strategy, based on our own evolutionary algorithm, was adopted to discover the low-energy structures of colloidal clusters composed of up to 20 particles. A detailed analysis of the low-energy structures involving charged particles shows that the appearance of the Bernal spiral as the most stable motif occurs, first, at N = 6, but it is favored for larger clusters (N ≥ 13); for 6 ≤ N ≤ 12, there is a competition between the spiral (which is favored for higher charges) and more spherical-like structures. Finally, we study binary clusters composed by two sets of differently charged colloidal particles. Although a great diversity of low-energy structures is observed (especially for aggregates with one of the components in excess), the global minimum is disputed by three structural motifs depending on the composition of the cluster and, in some cases, on the range of the Morse potential.

COLLOIDS. Colloidal matter: Packing, geometry, and entropy

Colloidal particles with well-controlled shapes and interactions are an ideal experimental system for exploring how matter organizes itself. Like atoms and molecules, these particles form bulk phases such as liquids and crystals. But they are more than just crude analogs of atoms; they are a form of matter in their own right, with complex and interesting collective behavior not seen at the atomic scale. Their behavior is affected by geometrical or topological constraints, such as curved surfaces or the shapes of the particles. Because the interactions between the particles are often short-ranged, we can understand the effects of these constraints using geometrical concepts such as packing. The geometrical viewpoint gives us a window into how entropy affects not only the structure of matter, but also the dynamics of how it forms.

Equilibrium fluid-crystal interfacial free energy of bcc-crystallizing aqueous suspensions of polydisperse charged spheres

The interfacial free energy is a central quantity in crystallization from the metastable melt. In suspensions of charged colloidal spheres, nucleation and growth kinetics can be accurately measured from optical experiments. In previous work, from these data effective nonequilibrium values for the interfacial free energy between the emerging bcc nuclei and the adjacent melt in dependence on the chemical potential difference between melt phase and crystal phase were derived using classical nucleation theory (CNT). A strictly linear increase of the interfacial free energy was observed as a function of increased metastability. Here, we further analyze these data for five aqueous suspensions of charged spheres and one binary mixture. We utilize a simple extrapolation scheme and interpret our findings in view of Turnbull's empirical rule. This enables us to present the first systematic experimental estimates for a reduced interfacial free energy, σ(0,bcc), between the bcc-crystal phase and the coexisting equilibrium fluid. Values obtained for σ(0,bcc) are on the order of a few k(B)T. Their values are not correlated to any of the electrostatic interaction parameters but rather show a systematic decrease with increasing size polydispersity and a lower value for the mixture as compared to the pure components. At the same time, σ(0) also shows an approximately linear correlation to the entropy of freezing. The equilibrium interfacial free energy of strictly monodisperse charged spheres may therefore be still greater.

Active dipole clusters: From helical motion to fission

The structure of a finite particle cluster is typically determined by total energy minimization. Here we consider the case where a cluster of soft-sphere dipoles becomes active, i.e., when the individual particles exhibit an additional self-propulsion along their dipole moments. We numerically solve the overdamped equations of motion for soft-sphere dipoles in a solvent. Starting from an initial metastable dipolar cluster, the self-propulsion generates a complex cluster dynamics. The final cluster state has in general a structure widely different to the initial one, the details depend on the model parameters and on the protocol of how the self-propulsion is turned on. The center of mass of the cluster moves on a helical path, the details of which are governed by the initial cluster magnetization. An instantaneous switch to a high self-propulsion leads to fission of the cluster. However, fission does not occur if the self-propulsion is increased slowly to high strengths. Our predictions can be verified through experiments with self-phoretic colloidal Janus particles and for macroscopic self-propelled dipoles in a highly viscous solvent.

Structure and dynamics of model colloidal clusters with short-range attractions

We examine the structure and dynamics of small isolated N-particle clusters interacting via short-ranged Morse potentials. "Ideally prepared ensembles" obtained via exact enumeration studies of sticky hard-sphere packings serve as reference states allowing us to identify key statistical-geometrical properties and to quantitatively characterize how nonequilibrium ensembles prepared by thermal quenches at different rates T[over ̇] differ from their equilibrium counterparts. Studies of equilibrium dynamics show nontrivial temperature dependence: nonexponential relaxation indicates both glassy dynamics and differing stabilities of degenerate clusters with different structures. Our results should be useful for extending recent experimental studies of small colloidal clusters to examine both equilibrium relaxation dynamics at fixed T and a variety of nonequilibrium phenomena.

Kinetics of the dispersion transition and nonergodicity of a system consisting of a disordered porous medium and a nonwetting liquid

An approach has been proposed for the description of the dispersion transition of a nonwetting liquid in confinement. This approach describes intrusion and extrusion processes for the ground state of a disordered porous medium, which is characterized by the formation of a fractal percolation cluster. The observed transition of the system of liquid nanoclusters in confinement to a metastable state in a narrow range of degrees of filling and temperatures has been explained by the appearance of a potential barrier owing to fluctuations of the collective "multiparticle interaction" of liquid nanoclusters in neighboring pores of different sizes on the shell of the fractal percolation cluster of filled pores. The energy of the metastable state forms a potential relief in the space of the porous medium with many maxima and minima. The volume of the dispersed liquid in the metastable state has been calculated within the analytical percolation theory for the ground state with the infinite percolation cluster. The extrusion-time distribution function of pores has been calculated. It has been found that the volume of the nonwetting liquid remaining in the porous medium decreases with time according to a power law. Relaxation in the system under study is a multistep process involving discontinuous equilibrium and overcoming of many local maxima of the potential relief. The formation of the metastable state of the trapped nonwetting liquid has been attributed to the nonergodicity of the disordered porous medium. The model reproduces the observed dependence of the volume of the dispersed liquid both on the degree of filling and on the temperature.

A geometrical approach to computing free-energy landscapes from short-ranged potentials

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.

Novel kinetic trapping in charged colloidal clusters due to self-induced surface charge organization

Colloidal clusters are an unusual state of matter where tunable interactions enable a sufficient reduction in their degrees of freedom that their energy landscapes can become tractable - they form a playground for statistical mechanics and promise unprecedented control of structure on the submicron lengthscale. We study colloidal clusters in a system where a short-ranged polymer-induced attraction drives clustering, while a weak, long-ranged electrostatic repulsion prevents extensive aggregation. We compare experimental yields of cluster structures with theory which assumes simple addition of competing isotropic interactions between the colloids. Here we show that for clusters of size 4 ≤ m ≤ 7, the yield of minimum energy clusters is much less than expected. We attribute this to an anisotropic self-organized surface charge distribution which leads to unexpected kinetic trapping. We introduce a model for the coupling between counterions and binding sites on the colloid surface with which we interpret our findings.

Design principles for self-assembly with short-range interactions

Recent experimental advances have opened up the possibility of equilibrium self-assembly of functionalized nanoblocks with a high degree of controllable specific interactions. Here, we propose design principles for selecting the short-range interactions between self-assembling components to maximize yield. We illustrate the approach with an example from colloidal engineering. We construct an optimal set of local interactions for eight colloidal particles (coated, e.g., with DNA strands) to assemble into a particular polytetrahedral cluster. Maximum yield is attained when the interactions between the colloids follow the design rules: All energetically favorable interactions have the same strength, as do all unfavorable ones, and the number of components and energies fall within the proposed range. In general, it might be necessary to use more component than strictly required for enforcing the ground state configuration. The results motivate design strategies for engineering components that can reliably self-assemble.